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 http://ufdc.ufl.edu/AA00026382/00001
Material Information
 Title:
 Solute transport in heterogeneous porous formations
 Creator:
 Demmy, George Gary, 1966
 Publication Date:
 1999
 Language:
 English
 Physical Description:
 xv, 138 leaves : ; 29 cm.
Subjects
 Subjects / Keywords:
 Conductivity ( jstor )
Covariance ( jstor ) Lagrangian function ( jstor ) Simulations ( jstor ) Solutes ( jstor ) Statistical discrepancies ( jstor ) Statistics ( jstor ) Trajectories ( jstor ) Travel time ( jstor ) Velocity ( jstor ) Agricultural and Biological Engineering thesis, Ph. D ( lcsh ) Dissertations, Academic  Agricultural and Biological Engineering  UF ( lcsh )
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 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1999.
 Bibliography:
 Includes bibliographical references (leaves 134137).
 General Note:
 Printout.
 General Note:
 Vita.
 Statement of Responsibility:
 by George Gary Demmy, Jr.
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SOLUTE TRANSPORT IN HETEROGENEOUS POROUS FORMATIONS
By
GEORGE GARY DEMMY, JR.
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1999
For my parents.
ACKNOWLEDGMENTS
Dissertations have but one name in the author slot, but anyone who has written one will readily testify that it represents the work and support of myriad people. Mary Hall freed my mind from many of the administrative hassles students encounter. Dawn Mendoza appointed, unappointed, and reappointed me numerous times throughout my tenure, and made the business end of things run as painlessly as possible. Ralph Hoffmann kept me in keys and desks and office space, and helped me get things wired when they needed wiring. Wally Hunter, system administrator and Unix hacker, deserves as much praise as I can heap upon him for keeping the network up and running and the data contained with in it safe and available.
The quality of an institution and its faculty is reflected in the students they attract. I have enjoyed interacting with the students of the Hydrologic Sciences Academic Cluster, and this interaction has enhanced my efforts. Liyong Li, Yan Zhang, and Xavier Foussereau shared the same computer lab trenches as I, and my work cannot be separated from their influence.
Many rewarding research experiences have been associated with collaborations made with Jim Jawitz. Jim is a colleague whose outlook and expertise is somewhat different than my own, but it is precisely these differences that have made the collaborations so rewarding. I shall seek to continue working with Jim through out my career.
Dr. Andrew I. James provided access to computer codes he developed in the course of his own research and technical assistance.
ill
Dr. Kenneth Campbell deserves special recognition not only for his exemplary service on my committee, but also for the example he has set as an engineer, a scientist, and a professor.
I have appreciated Dr. Mike Annable's wit and example through the years and have enjoyed interacting with him in a wide variety of settings, whether in the classroom, on the football field, or elsewhere.
Dr. Rao has always challenged me with questions that require looking beyond a simple formula and that evade a fillintheblank answer. Should I become scientist of any repute, Dr. Rao would be the man to blame.
Dr. Kirk Hatfield is my source. He has been a source of wisdom, sympathy, advice, and funding. I have greatly enjoyed working with Dr. Hatfield as both teaching and research assistant.
Dr. Gia Destouni taught a few stochastic subsurface hydrology lectures in the spring of 1994 that would forever change my thinking. She introduced the Lagrangian approach to transport with which I have been obsessed ever since. Dr. Destouni pursues fruitful collaboration vigorously, and this aggressive, yet collaborative, style is one that I now seek to emulate.
Dr. Vladimir Cvetkovic has generously hosted three visits to the Kungl Tekniska H6gskolan in Stockholm, Sweden. His considerable generosity flows from his intense interest in stochastic Lagrangian transport.
Dr. Sten Berglund has been a friend and mentor through this arduous journey. It is no exaggeration to say that without his help and encouragement this document would not exist.
Dr. Wendy Graham deserves special recognition for her tireless faith and support. She has allowed me a free reign to pursue whatever interests I chose. In hindsight, this method coupled, with my own personality quirks and spectrum of interests, may not
iv
be the most efficient for producing degrees and papers, but it has produced several interesting adventures.
George and Ellen Demmy, my parents, have provided love and all manner of support throughout my life, academic and otherwise. They both instilled in me a love of learning and an insatiable curiosity about all things that serve me well, and will remain among my best traits.
Finally, I'd like to extend my thanks and love to Celine Bufkin, for her love, encouragement, support, and vision.
v
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ........................... .... iii
LIST OF TABLES ......................... ..... ... .... ix
LIST OF FIGURES ............... .... .......... x
ABSTRACT ................. ..................... xv
CHAPTERS
1 INTRODUCTION ........................... 1
1.1 Primary Contribution ............ ..... ..... 1
1.2 Historical Context ........................ 1
1.3 Research Goals ......................... 3
1.3.1 Lagrangian Velocity Mean and Covariance ....... 3 1.3.2 Injection Mode Analysis .................. 4
1.3.3 Heterogeneous Reaction Parameter .......... 4
1.3.4 Application to "Reality" ................. 4
2 LAGRANGIANEULERIAN FLOW FIELD RELATIONSHIP .. 5
2.1 Introduction .................. ....... 5
2.2 Flow Field Partitioning Strategies .............. 7
2.2.1 Simple Subdivision .................. 8
2.2.2 General Subdivision .................. 10
2.3 Descriptions of Fields ................... ... 12
2.4 Numerical Experiments ................... .. 13
2.4.1 The Log Conductivity Field .............. 14
2.4.2 Velocity Field Generation ............... 18
2.4.3 Trajectory Generation ................. 19
2.4.4 Monte Carlo Simulation ................ 19
2.4.5 Sampling Strategy ................... 20
2.4.6 Simulation Results ................... 22
2.4.7 Log Conductivity Statistics .............. 23
2.4.8 Velocity Statistics ................... 29
2.4.9 Head and Head Gradient ............... 40
2.5 Theory .............................. 40
2.5.1 Effective Conductivity ................. 40
vi
2.5.2 Eulerian Velocity Field ................ 42
2.5.3 Streamtube Weights .................. 48
2.5.4 Control Plane Oriented Velocity Moments .... ... 51 2.5.5 Control Time Oriented Velocity Moments ....... 55 2.5.6 Log Conductivity Moments .............. 59
2.6 Evaluation of Simulation and Theoretical Results ....... 60
2.6.1 Mean Velocities in Space ............... 60
2.6.2 Velocity Variances in Space .............. 60
2.6.3 Mean Velocities in Time ................ 61
2.6.4 Velocity Variances in Time .............. 62
2.6.5 Log Conductivity Statistics .............. 62
2.7 General Conclusions .................. 63
3 NONREACTIVE SOLUTE TRANSPORT ............. 65
3.1 Introduction .................. ....... 65
3.2 Injection Mode and Streamtube Selection .......... 65
3.2.1 Uniform Resident Injection .............. 65
3.2.2 Injection in Flux ................... .. 67
3.3 Integrated Streamtube Properties .............. 67
3.3.1 Displacements ................... ... 68
3.3.2 Travel Times ................... .... 69
3.4 Solute Transport ................... ...... 70
3.5 Evaluation of Expressions ................... 71
3.5.1 Displacements ................... ... 71
3.5.2 Travel Times ................... .... 72
3.6 General Conclusions ................... .... 75
4 REACTIVE SOLUTE TRANSPORT ................ 79
4.1 Introduction ...... ................... .. 79
4.2 Temporal Moments ................... .... 79
4.3 Conceptual Overview of Solute Transport .......... 80
4.4 Simulation of Reactive Solute Transport ........... 83
4.4.1 Reaction Parameter .................. 84
4.4.2 Reaction Flow Path .................. 85
4.4.3 Reaction Flow PathTravel Time Cross Moment 93
4.5 Theory of Reactive Solute Transport ............. 93
4.5.1 Mean Reaction Flow Path ............... 96
4.5.2 Reaction Flow Path Variance ............. 97
4.6 Discussion ................... ....... 98
4.7 Spatial Moments. ................... ..... 99
4.7.1 Injection Mode ................... ... 105
4.7.2 Discussion ................... ..... 106
vii
5 APPLICATION OF CONCEPTS ............ ........ 108
5.1 Introduction .......... ................. 108
5.2 Theory .............. .... ............ 108
5.2.1 Spatial Moments ................ . 109
5.2.2 Relative Degradation ............ ... 110
5.3 Application: Simulation .............. . 111
5.3.1 Results ....... ........ .. .. 113
5.3.2 Discussion ........ . . . .... 113
5.4 Application: Field Data .................. 115
5.4.1 Results and Discussion: Center of Mass ....... 116 5.4.2 Results and Discussion: Degradation ......... 117
5.5 General Conclusions ............... . 117
6 GENERAL CONCLUSIONS ................... ... 119
APPENDICES
A EULERIAN VELOCITY DISTRIBUTION PROPERTIES .... 125
B VELOCITY LOG CONDUCTIVITY CORRELATION ...... 127
C CONSISTENT FIRST ORDER APPROXIMATION ....... 131
C.1 Travel Time ................... ........ 131
C.2 Displacements .......................... 133
REFERENCES ................................... 134
BIOGRAPHICAL SKETCH ................... ........ 138
V111iii
LIST OF TABLES
Table page
2.1 Comparison of target log conductivity values and observed values along
areaweighted Eulerian trajectories. ................ 26
2.2 Comparison of target log conductivity values for areaweighted Eulerian
trajectories in space and observed areaweighted Lagrangian trajectories in time. ......... ... .... ........ .. 29
2.3 Comparison of a priori Eulerian velocity variance to observed values.
The decimal representation of the exact fractional values of the predicted variance are provided for convenience. ........... 35
2.4 Comparison of observed log hydraulic conductivity variances to estimators. Observed value is zeroslope line regressed through data. The decimal representation of the fractional value is included for convenience ................ .. ..... .......... 63
5.1 The center of mass in meters of steady plumes estimated by traditional
trapezoidal integration of spatial concentration data and by the mean
v/k estimator Equation (5.7). .................. .. 116
5.2 Degradation rates relative to naphthalene and absolute degradation
rates in reciprocal days as estimated by Equation (5.9) and as reported by MacIntyre et al. [1993]. NR denotes results not reported in MacIntyre et al. [1993]. Naphthalene degradation reported to be
0.0064 d1 .................. ...... ....... 117
ix
LIST OF FIGURES
Figure page
2.1 Schematic representation of an aquifer. ................. 6
2.2 Simple subdivision of the flow field by one of four strategies: A) equalarea straight line, B) equalarea streamline, C) equalflow streamline,
D) equalflow straight line. .................. .... 9
2.3 Four different partitioning strategies for the same flow field: A) area
weight Eulerian trajectory, B) area weight Lagrangian trajectory, C) flow weight Lagrangian trajectory, D) flow weight Eulerian
trajectory .. .. . ... .. .. .. 11
2.4 Log conductivity distribution from a 41400 node realization generated
by the turningbands method compared to the standard normal distribution. Target distribution parameters are ln k = 0 and U~k = 1. 16
2.5 Estimated log conductivity longitudinal covariance estimated from a
realization generated by the turningbands method compared to an exponential covariance function. Variance of log conductivity an k = 1 and correlation length Aln k = 1. Spacing for log conductivity values
was Alnk/5 ....... ...... ... .... .. ...... 17
2.6 Mean log conductivity as function of displacement in mean flow direction for the four different trajectory collections and Ulnk = 1/2 (a)
and a k = 1 (b). ........................... 24
2.7 Log conductivity variance as function of displacement in mean flow
direction for the four different trajectory collections and Ulnk = 1/2 (a) and Ink = 1 (b). Variances are normalized by target variance of
Eulerian log conductivity field an.k. .................. 25
2.8 Mean log conductivity as function of travel time to control planes along
mean flow direction for the four different trajectory collections and
nk = 1/2 (a) and l nk= 1 (b) ..................... .... 27
2.9 Log conductivity variance as function of travel time to control planes
along mean flow direction for the four different trajectory collections and a1 k = 1/2 (a) and Ink = 1 (b). Variances are normalized by
target variance of Eulerian log conductivity field n k. . ... 28
U~YbVYIYI~IIVV VI UIUI~IL ~b VI~l~~UIILI 1~;1l 'Ck
2.10 Comparison of positiondependent mean velocity estimators to data for
the four collections of trajectories for ank = 1/2 (a) and r2nk = 1
(b). Velocities are normalized by U .. ............... 30
2.11 Comparison of the positiondependent velocity variance estimators to
data for the four collections of trajectories for aInk = 1/2 (a) and
aOnk = 1 (b). The variances are normalized by U2bunk. ........ 31
2.12 Comparison of spatial Eulerian (AW/ET) and Lagrangian trajectory
(FW/LT) velocity covariance data to covariance model. Data are taken from the a nk = 1 set of realizations. Covariances presented
as dimensionless lag correlation functions. ......... ..34
2.13 Comparison of timedependent mean velocity estimators to data for
the four collections of trajectories for ank = 1/2 (a) and Ink = 1
(b). Velocities are normalized by U........... . . 36
2.14 Comparison of timedependent velocity variance estimators to data for
the four collections of trajectories for aonk = 1/2 (a) and ank = 1
(b). The variances are normalized by U2bok. ............. 37
2.15 Comparison of temporal Lagrangian trajectory (AW/LT) velocity covariance data to covariance model. Data are taken from the aUnk 1 set of realizations. Covariances presented as dimensionless lag correlation functions. ....... ............ ....... 38
2.16 Normalized log initial velocities for twodimensional model aquifer compared to the standard normal distribution. The random variable was normalized by the sample mean and standard deviation to illustrate
the lognormality. ................ . . 39
2.17 Comparison of velocity covariance expression Equation (2.25) to the
approximation Equation (2.13) for aInk = 1. Covariances are presented as dimensionless correlation functions. ............ 45
2.18 Components of the velocity covariance function. ............ 46
3.1 Comparison of mean displacements for the two injection modes or
streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT) statistics correspond to a uniform resident injection. The top figure contains the aink = 1/2
data and the bottom figure contains the an k = 1 data. ....... 73
xi
3.2 Comparison of displacement variances for the two injection modes or
streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT) statistics correspond to a uniform resident injection. The top figure contains the oan4k = 1/2 data and the bottom figure contains the Unk = 1 data. Variances
are normalized by AnkUnk. ....... ... .......... 74
3.3 Comparison of mean arrival times for the two injection modes or streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT) statistics correspond to a uniform resident injection. The top figure contains the a k = 1/2 data and the
bottom figure contains the a k = 1 data. .............. 76
3.4 Comparison of arrival time variances for the two injection modes or
streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT) statistics correspond to a uniform resident injection. The top figure contains the an k = 1/2 data and the bottom figure contains the ank = 1 data. Variances
are normalized by (U/(Al\kaClnk))2 .. .................... .....77
3.5 Travel time estimate error as function of displacement associated with
assuming a uniform resident injection when injection is in flux for
aUnk = {1/2,1} ................ ...... .. ...... 78
4.1 Mean reaction parameter values for different values of correlation parameter / and different trajectory collections for the ank = 1 set of simulations. Top figure is P = {1, +1}. Middle figure is / = {1/2,+1/2}. Bottom figure is / = 0 (no correlation). Trajectory collections are areaweighted Lagrangian (AL), flowweighted
Lagrangian (AF) and areaweighted Eulerian (AE). ......... 86
4.2 Mean reaction parameter values for different values of correlation parameter / and different trajectory collections for the ank = 1/2 set of simulations. Top figure is / = {1,+1}. Middle figure is = { 1/2, +1/2}. Bottom figure is / = 0 (no correlation). Trajectory collections are areaweighted Lagrangian (AL), flowweighted
Lagrangian (AF) and areaweighted Eulerian (AE). ......... 87
4.3 Mean reaction path observed values and estimators for different values
of correlation parameter / and different trajectory collections for the nk = 1 set of simulations. Top figure is / = {1, +1). Middle figure is / = {1/2, +1/2}. Bottom figure is 3 = 0 (no correlation).
Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF). .................. ... 89
xii
4.4 Mean reaction path observed values and estimators for different values
of correlation parameter p and different trajectory collections for the
k = 1/2 set of simulations. Top figure is / = {1, +1}. Middle
figure is P = {1/2, +1/2}. Bottom figure is 3 = 0 (no correlation).
Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF). ..... ... ................. 90
4.5 Reaction path variance observed values and estimators for different
values of correlation parameter / and different trajectory collections for the CUn k = 1 set of simulations. Top figure is 3 = {1, +1}.
Bottom figure is 3 = {1/2, +1/2}. Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF). .. 91
4.6 Reaction path variance values and estimators for different values of
correlation parameter p and different trajectory collections for the crnk = 1/2 set of simulations. Top figure is 3= {1, +1}. Bottom figure is / = {1/2, +1/2}. Trajectory collections are areaweighted
Lagrangian (AL) and flowweighted Lagrangian (AF) ......... 92
4.7 Correlation between the reaction flow path and travel time for different
values of correlation parameter / and different trajectory collections for the crank = 1 set of simulations. Top figure is / = {1, +1}.
Bottom figure is /3 = {1/2, +1/2}. Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF). 94
4.8 Correlation between the reaction flow path and travel time for different
values of correlation parameter /3 and different trajectory collections for the cank = 1/2 set of simulations. Top figure is / = {1, +1}.
Bottom figure is / = {1/2, +1/2}. Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF). .. 95
4.9 Assumed timedependent log conductivity covariance function along
Lagrangian trajectories (Equation (4.29)) compared to simulation data for variances of log conductivity aink = {1/2, 1}. The Lagrangian velocity covariance function is Equation (2.61). ...... 103
4.10 Hypothesized timedependent Eulerian log conductivity covariance functions for Un2k = {1/2, 1} compared to simulation data and Equation (4.29). ................. ............. 104
4.11 Mean reaction flow path as a function of nonreactive travel time compared to estimator Equation (4.25) for correlation parameter values
/3= {1,0,1}. .................. .... ...... 104
4.12 Reaction flow path variance a function of nonreactive travel time for correlation parameter values 3 = {1, 0, 1} .............. 105
xiii
4.13 Mean reaction flow path as a function of nonreactive travel time compared to estimator Equation (4.34) for correlation parameter values
= {1/2, 0, 1/2} (top figure) and 0 = {1, 0, 1} (bottom figure).
2nk =1................. .............. 107
5.1 Resident concentration profile averaged along a transect orthogonal to
the mean flow direction for plumes for ornk = {0.0, 0.1,0.5, 1.0}. .. 114
B.1 Comparison of velocity and log conductivity correlation functions from
Graham and McLaughlin [1989b] and Rubin and Dagan [1992] and a simplified exponential model. All functions are normalized by KJaIk/O. The apparent "roughness" of the Rubin and Dagan [1992] function is due to estimation errors associated with inferring
their function from graphical data. .................. 128
B.2 Comparison of the simplified velocitylog conductivity correlation function to simulation data for or = {1/2, 1} ................ 129
xiv
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SOLUTE TRANSPORT IN HETEROGENEOUS POROUS FORMATIONS
By
George Gary Demmy, Jr.
December 1999
Chair: Wendy Graham
Major Department: Agricultural and Biological Engineering
This work quantifies relationships between the spatial, or Eulerian, distribution of the properties of a chemically and physically heterogeneous porous medium and those as observed along the natural, or Lagrangian, trajectories that a fluid particle traces in a steady and irrotational flow. From these relationships, expressions that relate the transport of solutes through the porous medium along the natural trajectories to the aforementioned Eulerian distributions are developed. The effects of injection mode upon global measures of transport as reflected by the temporal moments of breakthrough curves and spatial moments of a solute plume are developed. The coupled effects of correlation of a linear equilibrium sorption to the underlying log hydraulic conductivity field and injection mode on the evolving temporal moments of mass breakthrough curve and the coupled effects of correlation of a firstorder decay coefficient and injection mode upon the spatial moments of a solute plume are examined.
xv
CHAPTER 1
INTRODUCTION
1.1 Primary Contribution
The primary contribution of this work is documenting a consistent approach to the development of estimators of solute transport in the Lagrangian framework. This consistent approach is to identify the collection of Lagrangian trajectories associated with a stationary Eulerian velocity field for which the Lagrangian velocities are also stationary. This is a modest extension of work pioneered by myriad hydrologists. The benefit derived from this contribution is enhanced understanding of transport of solutes in heterogeneous velocity fields.
1.2 Historical Context
A grant to study the impact of heterogeneous source morphology upon subsequent solute transport initiated this research. In seeking a suitable framework and "toolset" with which to attack this problem, there appeared what seemed to be fundamental inconsistencies in the literature. Given the peculiar tendencies of the author, the "larger issues" could not be addressed until these apparent inconsistencies were resolved.
Gedeon Dagan developed a robust theory of solute transport using a stochasticLagrangian framework [Dagan, 1982a, b]. In fact, an entire "school" of stochastic subsurface hydrology sprung from his work.
Allen Shapiro and Vladimir Cvetkovic wrote Stochastic analysis of solute arrival time in heterogeneous porous media in 1988 [Shapiro and Cvetkovic, 1988]. The authors made an explicit assumption often implicitly made in the Lagrangian transport literature, namely that a fluid parcel deviates little from its mean trajectory in weakly
1
2
heterogeneous conductivity fields, and that the Lagrangian and Eulerian velocities are approximately equal. They consistently applied this concept, developing expressions for arrival time means and variances for nonreactive solutes in the "Standard Model Aquifer," a hypothetical aquifer characterized by a heterogeneous conductivity field with a prescribed correlation structure and subject to certain prescribed boundary conditions (see Section 2.1). These equations predicted that the mean arrival time for solute subject to a uniform resident injection is given by = x/Vh (1.1)
where angle brackets <> denote an ensemble average operator and Vh is the harmonic mean Lagrangian velocity. From the small deviation assumption, the Lagrangian and Eulerian velocities are approximately the same, thus the estimated harmonic mean Lagrangian velocity is equal to the estimated harmonic mean Eulerian velocity. However, the mean arrival time for uniform resident injection of solute after travelling several integral scales is given by
<7>= x/U (1.2)
where U is the arithmetic mean Eulerian velocity. This expression is also the "consistent firstorder approximation" (CFOA) of the travel time. Widespread adoption of the CFOA through out the stochasticLagrangian transport literature probably stems Feynman's requirement: we must reproduce what we already know. The harmonic mean of a positive definite process is always less than the arithmetic mean, so Equation (1.1) systematically overpredicts travel times for large displacements. Disturbing was that the estimators derived by Shapiro and Cvetkovic [1988] using clear, consistent, intuitive and logical methods seemed to violate Feynman's rule, especially in light of the success enjoyed by the Dagan school.
3
This harmonic/arithmetic mean discrepancy was explicitly noted by Dagan et al. [1992], who relegated the Shapiro and Cvetkovic [1988] result to applicability in areas "close enough to the input zone" [Dagan et al., 1992 p. 1374].
Cvetkovic et al. [1996] developed a semianalytical expression that described the transition from the "near field" harmonic mean velocity to the "far field" expression based upon the these known "endpoints" and an estimate of the Lagrangian velocity integral scale. Their detailed analysis of the Lagrangian velocity field revealed that the Lagrangian velocity is nonstationary in displacements in space, and resulted in a nonlinear propagation of the mean arrival time with distance.
This was strange. Why was the Lagrangian velocity field nonstationary when the Eulerian was stationary? It was commonly assumed that the stationarity of one implied the stationarity of the other. What had started as a simple preliminary literature review resulted in a quandry. The paths followed in pursuit of this quandry led to the "research goals" of the dissertation.
1.3 Research Goals
There were four broad objectives specified for this work. All centered around the flow of water and the transport of solutes in aquifers characterized by heterogeneous hydraulic conductivity fields.
1.3.1 Lagrangian Velocity Mean and Covariance
A primary goal of this research was to develop a "Lagrangian covariance function" for the "Standard Model Aquifer". This was extended slightly to the mean and covariance of the spacestationary velocities along equal flowweight streamlines and timestationary velocities along equal areaweight streamlines. Quantification of this covariance function greatly simplifies the development of equations, or estimators, that describe the movement of water and solutes in the heterogeneous velocity fields associated with the "Standard Model Aquifer."
4
1.3.2 Injection Mode Analysis
The stochasticLagrangian framework is convenient for the analysis of the effects of the method by which solute is introduced into the flow field, or injection mode, upon the subeqent transport. These effects of injection mode upon nonreactive solute transport are characterized by the mean and variance of a breakthrough curve, or mass arrival distribution, and by the mean and variance of mass displacement. A goal of this research was to develop estimators for these quantities.
1.3.3 Heterogeneous Reaction Parameter
Set as a goal were estimators for the mean and variance of a heterogeneous "reaction parameter" integrated along trajectories of stationary velocty. These estimators were used to evaluate the coupled effects of injection mode and a heterogeneous linear equilbirium sorption process in a stochasticLagrangian framework.
1.3.4 Application to "Reality"
A further goal was to develop a "traditional" estimator for the center of mass of a continuously injected solute plume subject to firstorder decay and to evaluate data from a natural attenuation experiment and numerical experiments. Estimator performance was evaluated using the results of the prevous developments.
CHAPTER 2
LAGRANGIANEULERIAN FLOW FIELD RELATIONSHIP
Develop an intuitive judgment and understanding for everything.
Miyamoto Musashi1
2.1 Introduction
Consider the steady, welldeveloped flow of water in an aquifer meeting Bear's [1972] definition of a porous medium. In most natural settings, gentle gradients induce a flow wellquantified by Darcy's law. Knowledge of the water fluid properties, relative permeability of porous medium, and potential gradients in space and time would assure a reasonable estimate corresponding water fluxes.2
One way to record this knowledge would be to create maps of these different properties in space at different times. This framework is commonly called the Eulerian framework, and is the most common approach to hydrologic problems. Another way to record this knowledge is to map properties along moving coordinates in time. This approach is commonly called the Lagrangian framework, and its application may without controversy be described as less widespread than Eulerian approaches. Dagan [1989] and Gelhar [1993] review development of flow of fluids and transport of contaminants in porous media in both of these frameworks.
1 Musashi [1982 p. 49]
2 The concepts associated with the following discussion are quite general, and are applicable to a much larger variety of environmental transport problems. The emphasis upon groundwater follows from an interest in simplicity.
5
6
Consider now flow of water in a more specific model aquifer: the steady, irrotational, and Darcian flow of water of homogeneous fluid properties in a finite rectangular twodimensional aquifer of heterogeneous hydraulic conductivity (see Figure 2.1). Mobile water 0 completely saturates the homogeneous pore space. An orthogonal
y
no flow boundary
\rando conductivity field
Cartesian coordinate system lies parallel to the aquifer boundaries, and x and y de)  ,. 0.
.raConstant potential boundaries located at x = 0 and =Lmaintainaconstantaverage potential gradient J across the domain, such that water flows into the aquifer at x = 0 and out of the aquifer at i = L at a volumetric discharge rate Q (dimensions [L3T1]). For an intuitive dimensional consistency, allow the aquifer some small and
7
uniform thickness in the vertical direction, say h. The total area of the vertical plane at x = 0 through which water enters the aquifer is A. Define the spatiallyaveraged groundwater seepage velocity at this inlet plane (IP) as U fA q[a]da
fA da (2.1) OA
For a large class of hydrologic applications, water quantity is the sole concern. For these problems, the "route" taken by the water is of little interest, as is, say the distribution of ages or residence times. Spatial and temporal distributions of hydraulic properties hold some interest, but only in how they relate to effective bulk properties.
However, a large class of practical hydrologic problems exist for which the trajectories of water parcels across the flow domain is of great importance. Many of these problems involve the fate and transport of contaminants through the aquifer system. The fundamental problem is this: to relate an Eulerian flow field to its Lagrangian, or trajectorybased, equivalent.
2.2 Flow Field Partitioning Strategies A streamline is a hydrodynamic entity that is at all points tangent to the velocity vector. For twodimensional flow, a streamtube is an object defined by two streamlines. Stream surfaces, the intersection of which form streamlines, define threedimensional streamtubes. Infinite collections of streamlines exist that subdivide the flow field into collections of streamtubes. Again, the interest of clarity suggests a focus upon twodimensional flow. For a lucid discussion of streamlines, stream functions, and stream surfaces in two and threedimensional flows, see Bear [1972].
Impervious boundaries are streamlines, thus the entire model domain is a streamtube by definition. Viewed as a "black box," Eulerian and Lagrangian distinctions have little meaning. This black box approach is the purview of reactor engineering,
an important and welldeveloped discipline, from which a great deal of insight and useful mathematics may be drawn. For an excellent review, especially with respect to concepts related to transfer functions, see Jury and Roth [1990].
For a large domain, the scale at which chemical reactions takes place is generally much smaller than that of the domain. That is, processes such as sorption, microbial and chemical degradation, and the like, are dependent upon chemical potentials across distances on the order of the pore scale to perhaps that of a "microDarcy" scale, defined as the smallest volume for which Darcy's law is generally applicable. However, the scale of interest tends to be much larger, say that of a well, or its capture zone or radius of influence, or perhaps the interface across which groundwater discharges into a surface water body, or, perhaps the Floridan aquifer. In order to consider these reactions in detail, it is necessary to resolve the larger domain into smaller subdomains. For modeling real systems, the scale and degree to which the domain is subdivided should be based upon the available data.
2.2.1 Simple Subdivision
Consider the water flowing into the model aquifer across the inlet, or injection, plane (IP). Assume it is of interest to divide the domain into two subdomains. There are many ways to do this. One would be to divide the domain "down the middle," that is, at a line parallel to the x axis located y = Ly/2 (curve A Figure 2.2). This division creates two subdomains of equal volume and gives an equal "volume weight" to each subdomain. The division also creates two subIPs of equal area and gives an equal "area weight" to each subdomain. For a perfectly homogeneous porous medium, the groundwater velocity vectors will be parallel to this line, and the discharges through each subdomain will be the same as well. Moreover, there will be no advective transfer of water across the boundary. However, the introduction of conductivity heterogeneity in the transverse direction will almost certainly lead to deviations of the streamlines, and to streamlines crossing the boundary. This implies that while
Ly
L /2 A
B
yo C
Yo D
0
Figure 2.2: Simple subdivision of the flow field by one of four strategies: A) equalarea straight line, B) equalarea streamline, C) equalflow streamline, D) equalflow straight line.
the volume of water contained in each subdomain is constant, the discharge through a plane perpendicular to the partition is a function of plane position in x.
Consider now a streamline originating at (0, Ly/2) (curve B on Figure 2.2). For a homogeneous medium, this corresponds to an equal volume partitioning. That is, the domain centerline is a streamline. However, transverse heterogeneity would likely induce changes in the velocity field such that the streamline would not remain on the centerline. For general transverse heterogeneity, the flow rate entering the two domains will not be equal, in general. Thus, the flow weight and the volume weight of the two subdomains are unequal in this case. Again, the magnitude of the areas through which water enters the subdomains are equal.
Consider now a streamline originating at some coordinate (0, yo) such that the volumetric flow rate is equal for both domains (curve C on Figure 2.2). This streamline corresponds to the stream function
F[ Yo T [0, Ly] T [O, 0]
[0, ya] = (2.2)
10
This is an equal partitioning with respect to the flow field, or the stream function. This sort of partitioning arises quite naturally in fluid mechanics and in numerical modeling of heterogeneous flow systems using streamtube approaches (see, e.g., Theile et al. [1996]). The flow weight of each subdomain is equal, although the volume weights are different.
A fourth scheme would be to extend a line from (0, yo) to (Li, yo) (curve D Figure 2.2). In this case, the subdomains share only equal flow rates at x = 0. In subsequent discussions, the phrase Eulerian trajectory will refer to such a straight line trajectory, and the phrase Lagrangian trajectory will refer to trajectory of a streamline. The phrase area weight will refer to an equal area partitioning scheme at a definition plane, and the phrase flow weight will refer to an equal discharge partitioning scheme. At this point, it is important to note that no mention has been made of boundary conditions as they relate to transport equations. These partitioning schemes are simply ways of subdividing a flow field. An area weight Eulerian trajectory scheme corresponds to traditional spectral approaches that assume small deviations about a mean trajectory. The use of trajectory is quite general, and does not imply a "natural" path or streamline.
2.2.2 General Subdivision
Consider dividing the flow field into N partitions according to the strategies described (see Figure 2.3). Theile et al. [1996] demonstrated that streamlines may represent streamtubes. Therefore, the remainder of this discussion will focus upon streamline properties, with the implication that these properties may be extended to streamtubes as well. A primary purpose of subdividing the domain is to represent the continuum of domain properties with a discrete set of data. A regular finitedifference grid may be thought of as a set of points at equal spacing in space along a collection of areaweighted Eulerian trajectories. An alternative method would be to create spacings based upon travel times along the trajectory, where travel time is defined as
.........~ ~ ~ ~ ~ ~ ~~. ......   ..... .A D
B C
Figure 2.3: Four different partitioning strategies for the same flow field: A) area weight Eulerian trajectory, B) area weight Lagrangian trajectory, C) flow weight
 Lagrangian trajectory, D) flow weight Eulerian trajectory.
12
the path integral of the inverse velocity over the trajectory r ol] = ds (2.3)
where is a unit vector that is at all points the parallel to the trajectory. It is clear that this sort of Lagrangian description has little meaning in a static system, and is in this sense less general than Eulerian descriptions. Thus, Lagrangian description is a tool reserved for dynamic systems and is a supplement to Eulerian descriptions. It is worthy to note the distinction between a description, a general approach, and a trajectory, a specific entity used in a description.
To summarize, a primary objective is to approximate a steady and irrotational flow field continuum with a discrete set of trajectories. The trajectories follow either a straight line (Eulerian trajectory) or a streamline (Lagrangian trajectory). The trajectories carry either an equal flowweight or an equal injection areaweight.
2.3 Descriptions of Fields
Consider once again the model aquifer depicted in Figure 2.1. As mentioned in the introduction of this chapter, the aquifer has certain features in which a hydrologist might be interested, e.g., the intrinsic permeability or the porosity. These features may be thought of as a field in which parameters vary continuously. This is a fundamental concept of continuum mechanics and is discussed in a hydrologic context by Bear [1972], Dagan [1989], and Gelhar [1993]. The ultimate description of any such field would be a map of parameter values valid at any point, time, or scale. In lieu of this "ultimate description," stochastic hydrologists characterize spatial heterogeneity with conditional probability density functions that reproduce the dominant characteristics of the field and observed values at given locations. The entire subject of geostatistics is devoted to this end (see Journel and Huijbregts [1978]). An Eulerian field refers to areaweighted Euleriantrajectory oriented field. A Lagrangian field refers to a flowweighted Lagrangiantrajectory oriented field.
13
It is instructive to illustrate these concepts with an example and to interpret the results of this illustration.
2.4 Numerical Experiments
The illustration of these concepts requires a heterogeneous flow field and collections of trajectories, both Eulerian and Lagrangian, that correspond to this flow field. The requirement that the flow field be based upon "reference fields" of known properties enhances the possibility of theoretical interpretation drawn from the large body of literature concerning flow and solute transport in idealized porous media. A further requirement that our reference fields bear some resemblance to what might be expected in nature enhances the possibility of useful applicability.
A numerical simulation code modeled steady groundwater flow in a hypothetical heterogeneous twodimensional aquifer. A particle tracking code traced Eulerian and Lagrangian trajectories across this aquifer, recording data that included the local groundwater velocity and local hydraulic conductivity at intervals in space and time along the different trajectories. Statistics, specifically mean, variance, and covariance, summarize the results of these "measurements."
The conceptual view of the aquifer is that given by Figure 2.1. To generalize the system, correlation length, or integral scale, of the log conductivity field Aln k serves as a characteristic length with which to normalize length scales. The characteristic time Alnk/U serves to normalize time scales, where U is the arithmetic mean velocity defined in Equation (2.1). The definition of an integral scale is the integral from zero to infinity of the correlation function R, or,
A = R[s]ds (2.4) 0O
14
The design criteria for the model aquifer are as follows In k correlation length Aln k = 1
effective conductivity Ke = 1
average gradient J = 1/50
mobile water content 0 = 1/5
From these criteria and Darcy's law, the areaaveraged velocity is U = KeJ/O = 1/10. Clearly, the parameter values are artificial and selected for convenience, but values for Ke in md1 and 0 fall in ranges typical for sand [Domenico and Schwartz, 1990].
Two 5000 replicate Monte Carlo simulations of aquifers with log conductivity variances of 1/2 and 1, respectively, comprised the suite of aquifer simulations. The following sections describe the experimental design and the results.
2.4.1 The Log Conductivity Field
The foundation upon which to build the illustration is a heterogenous hydraulic conductivity field. A commonlyused stationary lognormally distributed and exponentially correlated isotropic model serves as the Eulerian hydraulic conductivity reference field (see e.g., Freeze [1975] or Gelhar and Axness [1983]). Working from a hypothesized conductivity field is preferable to generating a hypothetical velocity field with specified characteristics (e.g., Bellin et al. [1994]) because an objective of this work is to describe the underlying conductivity field along Lagrangian trajectories and relate its Lagrangian description to its Eulerian properties.
For completeness, it is pertinent to briefly review description of a heterogeneous field by treating the process as being "random" Measured values of hydraulic conductivity of virtually any natural porous formation will vary from place to place. In the absence of measurement error, this variability in the measured values reflects the heterogeneity of myriad physical factors and processes that control the hydraulic conductivity. As mentioned, measurements of hydraulic conductivity in the field often follow a lognormal distribution. In these cases, it is useful to conceive of the
15
conductivity field as a realization of a lognormal random process. The process is said to be stationary if the ensemble mean and variance do not vary in space. The process is nonstationary if some trend exists in the mean or variance. In practice, only single realization is available, and the "ensemble" properties are ultimately unknowable. However, if the field is large compared to the scale of variability, one may assume ergodicity, or, that the ensemble parameters may be estimated from a sample drawn from the single realization since there exists a the possibility of a large number of statisticallyindependent observation points. For a more detailed development of these concepts from the hydrologist's perspective, see Gelhar [1993]. For a more formal development of these concepts, see Papoulis [1991].
The model of hydraulic conductivity variability is a stationary, exponentiallycorrelated isotropic lognormal distribution. A lognormal random variable is one for which its natural logarithm is normally distributed. As such, three parameters summarize the statistical properties of the adopted conductivity model, namely the mean of log k Iln k, the variance of log k U2n k, and the correlation length of log k Aln k. The covariance model is given by
Clnk[r] = nkexp[r/Alnk] (2.5) where r is the separation between two points in the log conductivity field. Use of this simple covariance model requires stationarity of variance, and usually implies a stationarity of the mean as well. The moments of the "real space" conductivity field are given by
= exp[nplnk + n22nk/2] (2.6) where n is the order of the moment.
In the absence of a perfectly described physical aquifer, the method of turningbands generated equallylikely realizations of a hypothetical conductivity field with
16
prescribed ensemble statistics [Tompson et al., 1989]. Turningbands was capable of producing two and threedimensional fields with the desired statistical properties and was available as a portable and easily modified code. Figure 2.4 illustrates the lognormality of the a typical twodimensional log conductivity field realization. Figure 2.5 compares an estimate of the longitudinal covariance to the hypothesized model. In order to capture the features of the log conductivity spatial variability, the
log conductivity
standard normal 0.8
S0.6
7.
5 0.4
E
0.2
3 2 1 0 1 2 3 log conductivity
Figure 2.4: Log conductivity distribution from a 41400 node realization generated by the turningbands method compared to the standard normal distribution. Target distribution parameters are 1lnk = 0 and r2nk = 1.
turningbands algorithm generated five log conductivity observations per correlation length. This level of discretization falls in the range of other similar sets of simulations. Cvetkovic et al. [1998] used a discretization of 4 nodes per Alnk for simulations of fields with oank as large as 1. Bellin et al. [1992] used a discretization of 8 nodes per Alnk for simulations of fields with a1,nk as large as 1.6.
17
estimated In k covariance + exponential covariance 0.8
a 0.6
C
0
o 0.4
0.2 '
4
  ta
0
0 1 2 3
r/AInk
Figure 2.5: Estimated log conductivity longitudinal covariance estimated from a realization generated by the turningbands method compared to an exponential covari2 I
ance function. Variance of log conductivity Calnk = 1 and correlation length AInk = 1. Spacing for log conductivity values was Alnk/5.
Trajectories covered several characteristic lengths Al, k and several characteristic times Aln k/U to capture both nearfield and farfield behavior in space and Lagrangian time. The first requirement was easy to enforce by making the domain size equal to or greater than the required number of correlation lengths. The second requirement was somewhat more problematic, since the distance traveled in some number of Alnk/U varies from trajectory to trajectory. Preliminary simulations indicated that a domain 40Alnk long was adequate to capture travel up to 10 characteristic time scales Alnk/U in 2000 trajectory realizations generated in the most heterogeneous field considered (a2nk = 1). Thus, the selected domain extent was 40Ank
The conceptual model aquifer is a large, bounded, primarily twodimensional, steady flow field. As such, it should exhibit a variety of head, head gradient, and velocity nonstationarities due to the boundary conditions (see Figure 2.1). While
18
researchers such as Rubin and Dagan [1988],[1989], and Osnes [1998] have investigated such nonstationarities, the additional complication of a rigorous consideration of these sorts of nonstationarities would unduly obfuscate the very simple concepts illustrated here. To minimize the effect of these nonstationarities, "solute transport" is constrained to take place some distance from the boundaries of the numerical domain. Following the example of Bellin et al. [1992], a 3Alnk "buffer" was established on both sides of the required 40 for a total longitudinal domain length of 46Alnk. Preliminary simulations indicated that the maximum transverse displacement of a streamline in a an k = 1 field was around one third the longitudinal displacement. For a streamline originating on the aquifer centerline parallel to the x axis, an aquifer width of 36Aln k was deemed adequate to contain large transverse displacements over a 40Alnk longitudinal displacement and provide a similar "buffer" to mitigate boundary effects.
Combining the "physical" domain requirements with the discretization required to capture the log conductivity variability resulted in a numerical grid 230 by 180 for a twodimensional aquifer and a numerical domain with 41400 nodes. A threedimensional analysis under the same assumptions would have required a numerical domain of more than 7 million nodes. Simulations of this magnitude were untenable in light of the available computational resources.
2.4.2 Velocity Field Generation
A mixed finiteelement scheme solved coupled velocity and pressure equations derived from Darcy's law and the continuity equation. The code took input parameters that matched the design criteria and the log conductivity field generated by turningbands. Andrew I. James wrote and tested the core of the code. James et al. [1997] used a derivative version of this code to generate flow fields for synthetic tracer tests. A mixed finiteelement solution to the flow field was specified, following the results and arguments of Mos6 et al. [1994]. The mixed finiteelement scheme is known
19
to provide an accurate velocity field solution for groundwater flow in heterogeneous conductivity fields, one that is more accurate than other, more traditional, schemes [Mos6 et al., 1994].
2.4.3 Trajectory Generation
An objective of this research was to record data, such as local velocity and and conductivity, along different collections of trajectories, both Eulerian and Lagrangian. These trajectories were traced using a "particletracking" scheme based upon a semianalytical method presented by Pollock [1988]. This method assumes that velocity varies linearly over an element, and that the velocity in some direction, say i, is independent of displacements in orthogonal directions. From these assumptions, it is a relatively simple exercise to determine the displacement of a "particle" in a linear velocity field for a given time, or the time required to make a certain displacement. A computer code recorded particle position, time, total path length along trajectory, velocity, local conductivity, and a value from another random field with the same statistical properties as, but uncorrelated to, the log conductivity field at several control planes equallyspaced along the mean flow direction and at several "control times" after injection along both trajectories. This uncorrelated random variable was recorded for reactive transport simulations, to be discussed in subseqent chapters.
2.4.4 Monte Carlo Simulation
The possibility of theoretical interpretation of the results is enhanced by requiring nearergodic samples from the population. That is, the samples must reproduce the population statistics. To do this, the number of statistically independent replicates drawn from the velocity and conductivity fields must be quite large. Rather than attempt a massive singlerealization, the following method generates a large number of statistically independent realizations
20
generate a large, twodimensional conductivity field with the desired statistical
properties
generate a flow field by imposing a constant mean gradient across the saturated
conductivity field
trace the trajectories of two particles injected at a common "control location",
one trajectory is a straight line or Eulerian trajectory and the other a streamline or Lagrangian trajectory, and record properties at equal lags in space and in time along trajectories that pass through "inner domains" that are relatively
undisturbed by nonstationarities induced by the boundaries
At this juncture, it is perhaps worth mentioning a minor "philosophical" point. These simulations do not represent a real aquifer, per se, but rather a hypothetical aquifer with known properties. The goal is understand the convolution of the physics of the groundwater flow and a known heterogeneity that has analogs in nature, and it is hoped that the reader finds some merit to this modest goal. The 41400 conductivity values needed for any single realization do not represent 41400 measurements taken from some real aquifer. The Monte Carlo aspect of this experiment reenforces this point.
2.4.5 Sampling Strategy
Consider "dropping particles" into a flow field, with no particular regard to location. These particles move along streamlines without interaction with the porous medium. Any one location into which one is dropped is as equallylikely as any other. The trajectories associated with these particles carry an equal area weight, analogous to equalarea subdivision of an injection plane. Consider now, the same disposal of particles, but this time an area is assigned to each particle at the injection point such that an equal discharge is associated with each particle, and its associated trajectory. For a homogeneous mobile water content, this area is inversely proportional to the local velocity normal to the area. Thus, each particle may be thought to carry an equal
21
flow weight. In this way, it is possible to "map" from one collection of trajectories to another.
At a given control plane or time, the areaweighted mean and variance of some property over N realizations, say v was calculated using the familiar estimators3 S N
i=1
N
var[va] = N N ) (2.8) i= 1
While these particular estimators seem obvious, it is important to realize that "dropping the particle" into the flow field assigns an equal area weight to each trajectory. For a single realization, this is equivalent to starting each particle with equidistant spacing. These estimators are based upon an assumption of equallylikely statisticallyindependent samples (see, e.g., Mood et al. [1974]). In order to draw even a few hundred independent samples from a single realization, the required flow domain would be enormous. This was a primary consideration for adopting Monte Carlo simulation using a minimal domain.
The flowweighted mean property v was calculated via
v N= p=u[O] (2.9) Ei=1 Ui[O]
3 Press et al. [1988] have criticized the variance estimator in Equation (2.8) as being prone to accumulation of roundoff error for large data sets. The rather substantial size of the data sets are naturally a concern. Comparision of the variance calculated by a corrected twopass method presented by Press et al. [1988] and the "traditional" estimator yielded no difference between the two methods within the first five significant digits.
22
where ui[0] is the initial velocity recorded for trajectory i. The variance by
var[vf = i=1 ui[O](v Vf)2 (2.10) z,1 U[0]
An alternative method for generating a collection of streamlines would be to identify the streamline corresponding to the "center" stream function expressed in Equation (2.2). In this case, the estimators of the equal flow statistics would be Equations (2.7) and (2.8). The equal area statistics would be given by
a = ] (2.11) 1i= 1/ui[]
and
var[va] Ei=(Vi a)2/ (2.12) =~= 1/ui[O]
The "reference" collection of the trajectories selected uniformly in space was specified in the simulations for convenience.
2.4.6 Simulation Results
The following sections present the results of two 5000 replicate Monte Carlo simulations. Detailed discussion of these results is deferred until after the development of some theoretical models with which to analyze these results. However, some heuristics are provided to familiarize the reader with some of the concepts developed in subsequent sections and chapters.
A summary of the notation used with the following results provides clarity. AW denotes an areaweighted trajectory and FW denotes a flowweighted trajectory ET denotes an Eulerian trajectory and LT denotes a Lagrangian. A particular weighttrajectory combination is concatenated the with a /. Thus, an areaweighted Eulerian trajectory is denoted as AW/LT.
For a given simulation, that is, for one set of 5000 replicates for a log conductivity field of some given arlk,, the statistics of the log conductivity and velocity along the
23
four different trajectories are calculated at intervals in time and in space. Thus, for any one property, say log conductivity, and any one statistic, say the mean, there will be two sets of results, namely that for a2nk = 1/2 and that for o2nk = 1. Associated with each set will be two subsets, namely statistics along the four trajectories recorded at space or time intervals.
For convenience, the notation and the trajectories to which each notation corresponds as depicted in Figures 2.2 and 2.3 is summarized here: AW/ET areaweighted Eulerian trajectory (Figures 2.2 and 2.3 A) AW/LT areaweighted Lagrangian trajectory (Figures 2.2 and 2.3 B) FW/LT flowweighted Lagrangian trajectory (Figures 2.2 and 2.3 C) FW/ET flowweighted Eulerian trajectory (Figures 2.2 and 2.3 D)
2.4.7 Log Conductivity Statistics
The log conductivity field serves as the "foundation" upon which the simulations are built, as well as the theory. The data for the areaweighted Eulerian trajectories correspond to those design criteria specified. The log conductivity mean and variance for AW/ET appear stationary in space (see Figures 2.6 and 2.7). That is, they appear to "bounce around" their "target" values with no apparent trend. Moreover, the statistics of FW/LT appear likewise stationary, but the mean is somewhat larger and the variance slightly smaller than AW/ET. The magnitude of the differences appear proportional to aOnk. A comparison of the "target" values, e.g., the specified inputs into the turningbands algorithm, and the observed ensemble values along the trajectories is presented in Table 2.1. The "observed" entry in Table 2.1 represents the statistic average along the trajectory, as estimated by regressing a zeroslope line through the data using a generalized leastsquares method. There is close correspondence between the target and observed values, supporting an ergodic hypothesis for the log conductivity statistics. That is, sample statistics are an accurately estimation
24
0.6
0.5 FW/LT + AW/LT x FW/ET
0.4 AW/ET n Eq. 2.72 reference 
0.3 *
 5 r~ ~  fx
XE
0.1
0.1
a
0.2
0 10 20 30 40
x/Al k
0.6
0.5     FW/LT +
+ + AW/LT x + + N < FW/ET
0.4 + + + AW/ET o xx x Eq. 2.72 xx reference 0.3 x
S0.2
x 0.1
00
0.1 x
b
0.2
0 10 20 30 40 X/Aln k
Figure 2.6: Mean log conductivity as function of displacement in mean flow direction for the four different trajectory collections and a1k = 1/2 (a) and oa = 1 (b).
25
1.1
1.05 ae iFW/LT + o 5 m AW/LT x
0.95 +0 + +
o x FW/ET
+ + +x + X 4 X X +0
C4 0.9 + + x+
!  ........................................4..........
0.85
a
0.8
0.75
0.7
0 10 20 30 40
X/AIn k
1.1
1.05 FW/LT + AW/LT x S ii IN WM[ 0 N FW/ET 1 .......  AW/ET
Sx9 0 reference ... 95 xx + + x Eq. 2.73
0.95 : xx x x x + + + X++
x + + xCN 0.9 +
b +
0.85
b
0.8
0.750.7
0 10 20 30 40
X/AXlnk
Figure 2.7: Log conductivity variance as function of displacement in mean flow direction for the four different trajectory collections and onk = 1/2 (a) and a2 = 1 (b). Variances are normalized by target variance of Eulerian log conductivity field an k"
26
Table 2.1: Comparison of target log conductivity values and observed values along areaweighted Eulerian trajectories.
parameter target observed
IAlnk('12nk = 1/2) 0 0.003
nk( 1nfk = 1) 0 0.005
12nk 1/2 0.498
an k 1 1.002
of the population statistics. Again, this is important for a quantitative theoretical analysis of these results.
The AW/LT and the FW/ET exhibit distinctly nonstationary behavior in the mean and less distinct behavior in variance as a function of displacement along the mean flow direction (see Figures 2.6 and 2.7). At the injection point, the AW/ET and AW/LT statistics are identical and the FW/ET and FW/LT statistics are identical, because these trajectories carry the same weight and the points in consideration are identical (see e.g., Figure 2.2). At very large displacements, one would expect the statistics of like trajectories to assume similar characteristics, regardless of weight. As the trajectory traverses several integral scales that characterize the heterogeneity in question, "information" about its starting location, is diminished. In the case of the FW/ET, the statistics "start" at the same place as FW/LT, and eventually "end" at the same place as the AW/ET. Similarly, the AW/LT statistics start where do the AW/ET and end where do the FW/LT. In fact, the data exhibit this behavior (see Figures 2.6 and 2.7).
The log conductivity statistics in time show somewhat different behavior. Only the AW/LT appear stationary in time (see Figures 2.8 and 2.9). There is an important observation to be gleaned from these results. The AW/LT statistics, in addition to appearing stationary, also appear to have a similar value to the input AW/ET statistics (see Table 2.2). Recall, the AW/LT statistics are clearly nonstationary in space, and are clearly distinct from those of the AW/ET. This observation greatly
27
0.4 FW/LT + AW/LT x FW/ET x AW/ET o est AW/LT 0.2 w++
A x
et A +o 0 xx
0.2
0 DD
E 0.2
0.4 a
0 2 4 6 8 10 Ut/I
0.4 + FW/LT + + AW/LT x ++ FW/ET w + AW/ET o + + est AW/LT an 0.2 +
'5 ++++
E+
0 Cx
0 m E 0.2
0.4 b 0 00
0 2 4 6 8 10
Ut/I
Figure 2.8: Mean log conductivity as function of travel time to control planes along mean flow direction for the four different trajectory collections and ank = 1/2 (a) and ~nk : 1 (b).
28
1.1 I
0
1.05 FW/LT + + x + x O AW/LT x
x Oxx+ x ox FW/ET K xx xa+a + X~ AW/ET o .......... +.. .... .s ........ reference  X X )K A TX X XOn
a + a x xx a K X X + ++X +00 0 + X X X + S4E 0.95 + 0 ++ xx N++ x xx+ N0 0 + + +
N NN NN N ++
0.9 N +
0.85
a
0.8
0 2 4 6 8 10
tU/Aln k
1.1
1.05 x O FW/LT + x n AW/LT x x x 0 x o o FW/ET
X X X 0 XX X+ XN xxx x x x Nxf~o' AW/ET o
1 x o xx    xx reference 0 xx x + N + x~ +xX X Ox ++NX X x + N ++ + +
xx95 ++ x+ + x+ xxx +
0.9 + + +' 00+ 0 +
0 + N
+0 + ++ N NN
0.85 K
NI b
0.8
0 2 4 6 8 10
tU/Alnk
Figure 2.9: Log conductivity variance as function of travel time to control planes along mean flow direction for the four different trajectory collections and arnk = 1/2
(a) and an k = 1 (b). Variances are normalized by target variance of Eulerian log conductivity field cr .lnk
29
Table 2.2: Comparison of target log conductivity values for areaweighted Eulerian trajectories in space and observed areaweighted Lagrangian trajectories in time.
parameter target observed
I.nk( 1nk = 1/2) 0 0.000
Inlnk(12nk = 1) 0 0.017
a1nk 1/2 0.497
o2 k 1 1.000
simplifies relating the Eulerian and Lagrangian fields, as shall be shown. The "longtime" asymptotic mean log conductivity values for both Eulerian and Lagrangian trajectories are somewhat less than their "largedisplacement" counterparts, and the magnitude of the difference is proportional to oalk (compare Figures 2.6 and 2.8). This is consistent with the harmonic averaging implicit with a timeaveraged timedependent velocity and the arithmetic averaging implicit with a spaceaveraged spacedependent velocity. Recall, we map the quantities onto the trajectories by following velocities, whether along "natural" streamlines (Lagrangian trajectories) or not (Eulerian trajectories). A simple heuristic is as follows. Low conductivities correspond to low velocities. For an equaltime spacing (or equallylikely observations in time) more "measurements" will come from low velocities, since the particle tracing the trajectory spends more time going slow. For an equaldistance spacing more measurements will come from high velocities, since the particle "covers more ground" while going fast. This is perhaps more intuitive when discussed in terms of the velocity field itself.
2.4.8 Velocity Statistics
The mean velocities along the different trajectories exhibit a similar behavior as do the mean log conductivities along the same trajectories (see Figure 2.10 and compare Figure 2.6). The AW/ET exhibits apparently stationary behavior in the mean and variance (see Figure 2.11) in accordance to established theory (e.g., Dagan [1989]). The AW/LT exhibits nonstationary behavior in the mean as observed by Cvetkovic et al. [1996]. Again, the AW/ET and FW/LT trajectories appear to be stationary
30
1.45
1.4
1.35 a FW/LT + AW/LT x FW/ET
1.3 A
AW/ET o Eq. 2.35 .1.25 Eq. 2.43 Eq. 2.44 1.2++ + 1 + reference . 1. + x + + xx
1.15 xxX x x
1.1
1.05
0.95
0 10 20 30 40 /Aln k
1.45
+ 1
1.4
1.35 xx FW/LT + i x,, AW/LT x 1.3 FW/ET x AW/ET o x, Eq. 2.35  1.25 Eq. 2.43  Eq. 2.44
1.2 b reference 1.15 :
1.1
1.05
0
0.95
0 10 20 30 40
z/AI k
Figure 2.10: Comparison of positiondependent mean velocity estimators to data for the four collections of trajectories for Uank = 1/2 (a) and Uank = 1 (b). Velocities are normalized by U.
31
2
1.8 a FW/LT + AW/LT x
FW/ET x
AW/ET a
1.6 Eq. 2.37 + + Eq. 2.55 + + Eq. 2.57 + xx+x + Eq. 2.13 L 1.4 ++ .....x..x + x x + 2 S+ + + + x S ." x x x +* S1.2 xx+ x S" + *
... X. x X x TL
1.8 + x* x + FW/LT + + ++ x x Cx +x x AW/LT x x xx + x + x+ FW/ET x x x +x AW/ET S1.6 x xx + Eq. 2.37 Eq. 2.55 S Eq. 2.57 L 1 b Eq. 2.13 ...........
1.4 b
S1.24 X
1o du
0.8
0 10 20 30 40
/A, lnk
Figure 2.11: Comparison of the positiondependent velocity variance estimators to data for the four collections of trajectories for onk = 1/2 (a) and U nk = 1 (b). The variances are normalized by U2bn k Ink' +
32
in both the mean and variance. The velocity variances exhibit behavior to similar to that of the means for their corresponding trajectories and quite different than the behavior exhibited by the log conductivity variances (compare Figures 2.11 and 2.10 and compare Figures 2.11 and 2.7). There is a qualitative similarity between the nonstationary velocity means and the nonstationary velocity variances. A heuristic is that the velocity statistic "starts" at one value "in the nearfield" and transitions to an asymptotic value "in the farfield." The following sections demonstrate that these endpoints and the transition are predictable.
The qualitative dissimilarity between the log conductivity variance and velocity variance is an artifact of examining the log conductivity values on the one hand and the "real space" velocity values on the other. Consider the FW/LT log conductivity mean and variance for the ank = 1 simulation. The FW/LT mean log conductivity is larger than that of the AW/ET, but the variances are roughly the same value. In fact, regressing a zeroslope line through the data results in a "mean" FW/LT variance that is slightly less than that of the AW/ET. However, the "real space" conductivity mean and variance are both functions of both the mean and variance of the log conductivity process (recall Equation (2.6)). A cursory examination of the data in conjunction with Equation (2.6) reveal that the mean and variance of the "real space" conductivity along FW/LT are both higher than their AW/ET counterparts, and the magnitude of this difference is proportional to Unk The mean velocities in Figure 2.10 are normalized by the arithmetic mean velocity U predicted from the input parameters for the flow simulator. That the mean velocity along AW/ET closely follows 1 is a good indication that the effective conductivity expression is accurate. Notice that the FW/LT and farfield AW/LT values are higher than AW/ET, and the magnitude is proportional to the variance of the log conductivity (see Figure 2.10). This behavior is intuitively correct. Consider a
33
particle injected into a flow field. Constraint of the trajectory to an Eulerian trajectory forces the trajectory through lowvelocity areas that a natural streamline, or Lagrangian trajectory, would have otherwise bypassed. Of course, some streamlines pass through even the lowest velocity areas, but these are few in an equalflow sense. Thus, equalarea weighting preferentially weights low velocity zones. That there are implications for sampling schemes in heterogeneous media, especially in context of multilevel samplers, should be intuitively obvious to even the most casual observer. A discussion of some of these implications is made in a subsequent section, after these concepts are developed further in a more quantitative and theoretical framework.
It is appropriate to evaluate the adopted expression for the AW/ET longitudinal covariance function suggested by Dagan and Cvetkovic [1993], namely
Cua[r] = O nkb[e]U2 exp [b[e]r/Alnk] (2.13) where r is the separation between points along the ET, and b[e] is a function of the anisotropy ratio e. The anisotropy ratio itself is a function of the correlation lengths of the log conductivity in the different directions. The primarily consideration is that b may be derived from theoretical considerations and is not some ad hoc fitting parameter. The value of this function for a twodimensional isotropic aquifer is 3/8. Again, for further background on this function, see Dagan and Cvetkovic [1993]. In the limit r = 0, the covariance reduces to the point variance, which for the model case is o~ = 3U2In k/8. For the remainder of this discussion, a reference to b contains its dependence upon e implicitly. The subscript ua may be decoded as follows: for a subscript tw, the t is the trajectory and property (u for Eulerian velocity, and v for Lagrangian velocity) and the w is the weight (a for area and f for flow). Thus, C,, is the covariance function for velocities along Eulerian trajectories. We present this covariance function and that covariance function estimated from the assumedstationary collection of AW/ETs in Figure 2.12 The perceptive observer will note
34
AW/ET +
FW/LT x
Eq. 2.13 .
0.8
c 0.6
0 0
8 0.4
X
0.2 +
X '
+ .
0 5 10 15
r/A lnk
Figure 2.12: Comparison of spatial Eulerian (AW/ET) and Lagrangian trajectory (FW/LT) velocity covariance data to covariance model. Data are taken from the onk = 1 set of realizations. Covariances presented as dimensionless lag correlation functions.
that the analytical expression overestimates the covariance at short lags and underestimates the covariance at longer lags. This discrepancy and its implications are discussed in some detail in a subsequent section.
The velocity variances in Figure 2.11 are normalized by the zerolag covariance predicted by placing the input simulation parameters into Equation 2.13. The AW/ET velocity variance appears wellpredicted by this expression, as evidenced by a relatively good fit to 1 for both values of onk in Figure 2.11. A regression of a zeroslope line through these data indicates that the a priori estimate under predicts the observed values by about five percent for the larger a2nk (see Table 2.3) The apparent stationarity of the velocities along the FW/LTs suggest estimation of the covariance function for these velocities as for the AW/ET (see Figure 2.12). The correlation length of the FW/LT is very close to that of the AW/ET, although the actual value is probably slightly higher. Recall the definition of the correlation length as given by
35
Table 2.3: Comparison of a priori Eulerian velocity variance to observed values. The decimal representation of the exact fractional values of the predicted variance are provided for convenience.
parameter predicted observed (G nk = 1/2) 3/1600=0.001875 0.001871 O'u(O nk = 1) 3/800=0.00375 0.00394
Equation (2.4). The correlation function at each considered lag along the estimated functions is systematically larger for FW/LT, even though the magnitude of the difference is small. Integrating these functions would result in a slightly larger value for FW/LT, and thus a larger correlation length. In the case there is no variability in the log conductivity, Eulerian and Lagrangian trajectories coincide. Moreover, an equalarea weight corresponds to an equal flow weight. Thus, the estimated correlation functions from "either set" would be identical. From this conclude that there is some dependence of the correlation length of the FW/LT trajectories upon Unk' Cvetkovic et al. [1996] observed similar behavior in log velocity correlation functions estimated from AW/LTs with a stationarity assumption for the nonstationary AW/LT velocities. Thus, this is an slight extention of this work by identifying the stationary streamlinebased velocity field from which to work, and work from that field. Moreover, this supports their findings of an increasing correlation length with aInk Understanding the relationship between the variability of the log conductivity field and the correlation length of the associated velocity field is of fundamental importance to completely understanding the nature of flow in heterogeneous media. Sadly, these "secrets" remain uncovered and must be left this to future research efforts.
The velocity means as a function of time exhibit qualitatively similar behavior as do the log conductivity means in time (see Figure 2.13 and compare Figure 2.8). Again, only the AW/LT exhibits stationary behavior in time for both the mean and variance (see Figure 2.14) as did the AW/LT log conductivity values. The mean and variance of the AW/LT in time are approximately those of the AW/ET in space
36
1.4 'FW/LT + AW/LT x FW/ET = 1.3 AW/ET o Eq. 2.63
Eq. 2.59 1.2 Eq. 2.67 Eq. 2.66
tU/Alnk
1.1 ++FW/LT ++,
AW/ET x Eq. 2.63
Eq. 2.590.9 Aa: ... ...
0.9 a
0.8
0.7 anmamam
0 2 4 6 8 10 tU/Aln k
1.4 FW/LT + AW/LT x
1.3 FW/ET AW/ET o S,,Eq. 2.63 'K + Eq. 2.59 1.2 Eq. 2.67 ..........
Eq. 2.66 ) + ++
1.1 X +++;+XX X xxxxxxxX XxxXx
0.9 o "
0 8o ... .... "
0.7
0 2 4 6 8 10 tU/Al k
Figure 2.13: Comparison of timedependent mean velocity estimators to data for the four collections of trajectories for u~nk= 1/2 (a) and a"nk = 1 (b). Velocities are normalized by U.
37
FW/LT + AW/LT x 1.8 FW/ET AW/ET o Eq. 2.65 .1.6 Eq. 2.61 Eq. 2.70 N Eq. 2.69 Eq 269 4 1.4
1.2 + ++ ) k+ +. + + x X Xxx+ +.x + x C 4 P 1 ..   ..  ,
0.8 .. ..
0.6
a
0.4
0 2 4 6 8 10 tU/An k
FW/LT + AW/LT x 1.8 1 FW/ET w AW/ET o Eq. 2.65 1.6 Eq. 2.61 .
Eq. 2.70 "14 1 .4 Eq. 2.69 .
a 1 + ++
1.2 + ++ + x x x + xx N 1  0.8
0.8 a
b
0.4
0 2 4 6 8 10 tU/Aln k
Figure 2.14: Comparison of timedependent velocity variance estimators to data for the four collections of trajectories for onk = 1/2 (a) and oink = 1 (b). The variances are normalized by U2b n k
38
for both simulations. This result is important, because it implies the possibility of mapping from AW/ET to AW/LT. We estimated the covariance function for the stationary AW/LT velocities (see Figure 2.15). As an estimate of the covariance
AW/LT
Eq. 2.61 0.8
r 0.6
0D+
++
8 0.4
0.2+ ++++
0I
0 2 4 6 8 10 tU/Aln k
Figure 2.15: Comparison of temporal Lagrangian trajectory (AW/LT) velocity covariance data to covariance model. Data are taken from the an k = 1 set of realizations. Covariances presented as dimensionless lag correlation functions.
function, the substitution r = Ut was made in Equation 2.13 and the result compared to the estimated covariance function (see Figure 2.15). The closedform predictor behaves similarly to its spatial counterpart: earlytime correlation is underpredicted and longtime correlation is overpredicted (compare Figures 2.15 and 2.12).
For completeness, one other aspect of the velocity field was examined, namely the distribution of initial velocities (see Figure 2.16). The logarithm of the initial velocity was rankordered and subject to the transformation In v In v
In v' lnv (2.14) In v
39
0.8 In V
standard normal . 0.6
0.4
0.2
0 "
3 2 1 0 1 2 3
log velocity
Figure 2.16: Normalized log initial velocities for twodimensional model aquifer compared to the standard normal distribution. The random variable was normalized by the sample mean and standard deviation to illustrate the lognormality. where Inv is the sample mean and in,, is the sample standard deviation. Working here with the logarithm, rather than the velocity itself, better demonstrates the apparent lognormality of the velocity fields. Recall, the logarithm of a lognormal random variable is itself normally distributed. Figure 2.16 indicates that the transformed log initial velocities are welldescribed by a standard normal distribution. The lognormality of the velocities seems logical since the conductivity field is itself lognormally distributed. However, the velocity is in general a product of random variables: the conductivity, the head gradient, and the mobile water content. The logarithm of the velocity is the sum of random variables. Sums of random variables tend toward normality by the central limit theorem (see Mood et al. [1974]), and thus, products of random variables tend towards lognormality. It would be interesting to test this
40
hypothesis by examining the velocity characteristics of flow fields arising from different log conductivity distributions (e.g., fractal, uniform, white noise, etc.), and this is left to future work.
2.4.9 Head and Head Gradient
Head and head gradient information along the trajectories was not recorded. This is a regrettable oversight, since much of the behavior of the flow water and the transport of solutes in heterogeneous media can be summarized as the interplay of head, head gradient and conductivity covariance functions. The literature concerning Eulerian head and head gradient fields is extensive and is summarized in large part by Gelhar [1993]. In fact, much is drawn from this body of literature for some of the theoretical developments in the following sections. That this information was not recorded detracts mainly from the completeness of this work, and further investigations are left to future works.
2.5 Theory
2.5.1 Effective Conductivity
As with the simulations, the theoretical developments are founded on the bedrock of the log conductivity field. Section 2.4.7 demonstrates that the simulation results well match the "target" ensemble quantities. From this field is had directly a derived parameter, namely the a priori effective conductivity Ke. An expression for the effective conductivity, similar to that given by Gelhar and Axzness [1983], is derived here to illustrate the mechanics of different techniques employed through out these derivations. Consider a onedimensional Darcy's law of the form q = k d(2.15)
dx
where q is the specific discharge, k is the hydraulic conductivity, and is the hydraulic head. Consider a large stationary conductivity field such that the log conductivity
41
may be resolved into a mean, and a zeromean perturbation, e.g., In k = l.ink + f. Let the head gradient likewise be resolved into a mean and zeromean perturbation, e.g., d /dx = J(1 + j). Inserting these expressions into Equation (2.15) yields q = exp[Allnk + f]J(1 + j)
(2.16)
= KgJ exp[f](1 + j)
Expand the exponential term in this equation as a Taylor series about zero, truncating the series at secondorder, and expand the product of the perturbation terms q KgJ(1 + f + f2/2)(1 + j)
(2.17)
= KgJ(1 + f + f2/2 +j + fj + f2j/2)
Take the expected value of this expression, dropping terms higher than secondorder, resulting in an expression
= KJ(1+ ank/2+ < fj >) (2.18) Consider this equation and the three terms in the parenthesis. The first is a "mean" term, the second is a "contribution" due to the variability of the conductivity field and the third is the log conductivity and head gradient crosscovariance. Gelhar [1993] evaluates this expression for isotropic log conductivity fields using spectral methods. The result, which is independent of the shape of the correlation function, is surprisingly simple [Gelhar, 1993]
= 2n k/d (2.19) where d is the dimension of the domain. This negative correlation between head gradient and log conductivity has a profound effect on the flow. The effective conductivity may be defined as that value when multiplied by the spatial mean gradient results in the observed specific discharge (flow rate divided by discharge area). The a priori effective conductivity is the effective conductivity estimated from Equations 2.18 and 2.19. With d = 2, we have for twodimensional conductivity fields, the a priori
42
effective conductivity is simply Ke = Kg. Notice that the effective conductivity can be significantly less than the arithmetic mean conductivity Ka = Kgexp[aonk/2].
The definition of the effective conductivity is quite general, and a general effective conductivity expression is denoted Ke = Kgc where c is the "correction" due to the aforementioned processes. Under this definition, there is always some effective conductivity value, regardless of nonstationarities in boundaries or flow domains, but its magnitude may depend upon several factors. There is no general a priori effective conductivity estimator. The estimator given here is limited to stationary head and conductivity fields.
2.5.2 Eulerian Velocity Field
The relationship between the Eulerian conductivity field and its associated Eulerian velocity field has been one of the more thoroughly examined aspects of subsurface hydrology. For the example problem, closedform expressions are sought for the Eulerian, or spatial, mean velocity U, the Eulerian velocity covariance C,a[r], and its point distribution, given that the underlying conductivity field is large, stationary, lognormallydistributed, exponentially correlated, and characterized by parameters lnk, a12nk, and A2, and subject to the conditions prescribed for the model aquifer (see Section 2.1 and Figure 2.1).
From the development of the effective conductivity relationship in the preceding section, an expression for the mean Eulerian velocity is
U cJ= (2.20)
0
where c = 1 for a twodimensional isotropic aquifer. The c is retained to emphasize that this expression is not the "consistent firstorder approximation" of the velocity that would result from dropping the a2nk and < fj > terms in Equation (2.18). For the twodimensional isotropic aquifer, which is the model case, these two terms happen to be equal in magnitude and opposite in sign, and the consistent firstorder
43
expression and effective expression are the same. This damping effect of the negative head gradient and log hydraulic conductivity is largely responsible for the robustness of these perturbation expansions. The truncated terms are not negligible in the sense their magnitude is small. For instance, for ank = 1, their magnitude is onehalf that of the mean. This brings us to an important philosophical point. A primary usefulness of these expansions is to identify the dominant "components" of the process. A purely "consistent" approach could lead to neglect of some important process. For instance, consider a strict, firstorder expansion of Equation (2.16). Equation (2.17) would contain no f2 term, and Equation (2.18) would contain no aUnk term. The resulting effective expression would seriously underestimate the specific discharge, since the contribution of the conductivity variability would "carry no weight" with respect to the other contributing processes. Analysis of the neglected higher order terms is necessary for a more complete understanding of the process. The general goal of the approach, however, is to identify the dominant subprocesses that contribute to the larger process, and the relative importance, or weight, of each subprocess.
Expressions for the Eulerian velocity covariance for the model system have been given by various researchers, including Graham and McLaughlin [1989a] and Rubin [1990]. Shapiro and Cvetkovic [1988] develop an inverse velocity covariance function using a perturbation expansion of the inverse flow velocity and spectral methods. Consider once again the perturbed Darcy's law expression given by Equation (2.16). Dividing this by the homogeneous mobile water content, neglecting secondorder terms and higher, the meanremoved expression, or "velocity perturbation," is S= J (f + j) (2.21)
0
44
If these processes are stationary stochastic processes, they can be represented represented by a FourierStieltjes integral (see e.g., Bakr et al. [1978]) u'=J exp[iM x]dZ,[M] (2.22)
where i = v I, M is the wave number vector, and dZ,, is the complex random amplitude of the velocity perturbation. The FourierStieltjes representation of Equation (2.21) is
KJ
dZ,, = (dZf + dZj) (2.23) Compare this expression to that derived by Shapiro and Cvetkovic [1988] (their Equation (A4))
0
dZ/= K (dZf + dZ) (2.24) The spectra that result from these complex random amplitudes are structurally the same, differing by the square of the mean values by which they are multiplied (e.g., by (KgJ/O)2 on the one hand and (O/(KJ))2 on the other). In general, the correlation structure of the inverse of a random function is not the same as that of the random function itself. The firstorder series approximation of the inverse velocity is linear in the velocity itself (e.g., 1/v 1 v), and thus the velocity correlation characteristics are recovered as an approximation of the inverse velocity correlation characteristics. For the case of the exponentially correlated isotropic conductivity field, the longitudinal velocity covariance of a twodimensional field is given by (Shapiro and Cvetkovic [1988])
C,[s] = 3ank(KJ/O)2 (exp[s](s2 + 3 + 3s4) + s2/2 3s4) (2.25)
45
where s = x/Alnk. The corresponding expression for a threedimensional field is
C,[s] = 80ank(KgJ/o)2 (exp[s](s2 + 5s3 + 12s4 + 12s5) + s3 12s5)
(2.26)
Equation (2.25) provides a superior match to the data in comparison to Equation (2.13) (see Figure 2.17). However, the expression for the Eulerian covariance
approximation full expression estimated covariance +
0.8
C:
4
0.6
"t 0.4
0.2 ',
0
0 2 4 6 8 10 r/A lnk
Figure 2.17: Comparison of velocity covariance expression Equation (2.25) to the approximation Equation (2.13) for aln k = 1. Covariances are presented as dimensionless correlation functions. given in Equation (2.13) captures the dominant features of the velocity covariance structure, including the correlation length and the point velocity variance of Equation (2.25) in a simple and familiar form.
As noted, however, Equation (2.13) exhibits some systematic differences from the observed behavior. The data show a rapid initial decay, followed by a longrange persistence (see Figure 2.17). This behavior is examined with an expression adapted
46
from Dagan [1984]
Rua[r] = Onk (RIn k k [rdr d[rr2 (2.27) where R1nk is the log conductivity correlation function, RlnkO is the log conductivity and head crosscorrelation function, and 'y, is the head variogram. From this expression, glean that the velocity covariance is a process comprised of three dominant subprocesses, namely Rlnk, RInk, and yo. At small displacements, all three of these subcomponents exhibit rapid trend towards zero (see Figure 2.18). The RInk, and 7y
log conductivity
In kgradient head gradient 0.5
C
O
.4j
0
0.5
0 2 4 6 8 10 r/Aln k
Figure 2.18: Components of the velocity covariance function.
expressions exhibit effects that persist well beyond that of the log conductivity field. The longrange correlation effects of the head field in the mean direction of flow are cannot be precisely predicted with the simple exponential function Equation (2.13). In order to recover the proper correlation length, the approximation must systematically overpredict the velocity correlation at small separations. The "cost" of the
47
approximation, however, will realize significant returns in the form of simplicity, as shown in subsequent sections.
Previous works have focused upon the moments of the Eulerian velocity field that result from mean gradients imposed upon heterogeneous conductivity fields. These works shall be extended slightly with a hypothesis about a specific case that allows a variety of insights to be drawn.
Hypothesis: The point velocity distribution of an Eulerian velocity field induced by an stationary mean gradient J across a lognormally distributed, exponentially correlated log hydraulic conductivity field characterized by parameters Pink, a2nk, and in k and saturated with mobile water at a constant and homogeneous content 0 is lognormal with approximate parameters
MIn u = In K9Jc (2.28)
0 V1 + b2n k
and
2nu = In[1 + boa k] (2.29) (see Appendix A for details of development). A fundamental assumption in addition to the lognormality hypothesis is that the "real space" moments of u are Pu = U and Ua U2ba,2k. Equation (2.13) is adopted as an approximation of the velocity covarince function.
In lieu of a rigorous proof, the results of a numerical experiment in which statistically independent velocities are drawn uniformly in space from simulated flow fields are presented (see Figure 2.16). The normalization of the velocities in Figure 2.16 than that of previous figures in that the log velocities are normalized by the sample mean and standard deviations to emphasize the lognormality of the velocity distribution. Normalization by the moments predicted by Equations (2.28) and (2.29) would have resulted in a lognormal distribution function that deviates from the standard
48
normal. This indicates that the moments I. = U and a2 = U2bou k may require some further investigation. The adequacy of these estimates are discussed in subsequent sections.
2.5.3 Streamtube Weights
The reference velocity field is now to be subdivided into streamtubes. Each streamtube may be considered to carry some "weight" in relation to other streamtubes for a given collection. Consider the partitioning of the flow field into N streamlines separated by an equal distance along the IP. This is an equalarea weighted partitioning scheme, and results in N subIPs of equal area, but dissimilar discharge. The discharge through subIP i is di = OAaui (2.30)
where Aa = A/N. The relative flowweight of this streamtube i is given by wfi = ui/U (2.31) An equalflow partitioning scheme results in in N subIPs of equal discharge, but of dissimilar area. The discharge through subIP i is d = OAaivi
(2.32)
=Q/N
Solving for the area yields
d
Aai = (2.33) Ovi
The relative areaweight of this streamtube i is wai = U/vi (2.34)
49
The perceptive reader will notice a slight notational change between the two partitioning schemes. A lowercase u represents velocities drawn uniformly in space and a lowercase v represents velocities drawn uniformly in flow. The distinction is important. It is through these subIPs that water enters the streamtubes and a description of the subIP properties is essential to streamtube, and thus flow field, descriptions. If from a collection of N subIPs the random selection, or drawing, of one subIP is equally likely as any other, the weight assigned to any one observation from M samples would be 1/M. Assume that all of the subIPs are sampled. The average subIP discharge is the same for the equalarea and equalflow partitioning schemes, and equal to d = OAU/N. However, the discharge variance is zero for equalflow and nonzero for equalarea, when the velocity field is heterogeneous. The weights expressed in Equations (2.34) and (2.31) allow the properties of one collection of streamtubes to be mapped to the other collection of streamtubes. These weights combined with the assumed values of the Eulerian velocity field permit derivation of properties of the initial velocities of the equalflow partitioning schemes. The expected value of v must equal the expected value flowweighted u. Thus, = V
U
=< U>
1
U (+ u) (2.35) = U(1 + ba2n k)
K, Jc 2
S Jc (1 + bl Ik)
0
50
The variance of v is given by var[v] =< v2 > < v >2. The expected value of v2 must equal the expected value of flowweighted u2. That is =< UU2
U
1 9
= 1 exp[3Pini + 1,2nu
U 2 (2.36)
1 exp[3(pinu + c2nu/2) + 312nu] = U2(1 + bak)
Thus, the variance of v is given by var[v] = V2b2, k (2.37)
It is worthy to note that these results do not follow from an assumption that v = u2/U. That is, the velocity along the Lagrangian trajectory is not simply the flowweighted velocity along an Eulerian trajectory.
Hypothesis: The point velocity distribution along a flowweighted Lagrangian trajectory in the model stationary Eulerian velocity field is lognormal with approximate parameters
Pin v = In [KJ 1 + b2 nk (2.38) and
O nv= n[1 + b Ink] (2.39)
Moreover, the process is assumed secondorder stationary in displacements along the mean direction of flow, that is, at control planes, and is characterized by the covariance function
CV,[r] = V2ba k exp[br/Alk] (2.40)
51
where r is the lag along the mean flow direction. Again this follows from the requirement that the distribution match the assumed "real space" v moments p, = V = U(1 + bulnk) and a = V2bcr' k. The correlation structure of the FW/LT velocities is similar to that of AW/ET (see Figure 2.12).
Employing these velocity distribution hypotheses, it is now possible to derive some relationships between the different flow field partitioning schemes. From observation, it is inferred that the velocities along areaweighted Euleriantrajectories and flowweighted Lagrangiantrajectories are secondorder stationary in displacements along the mean flow direction (recall Figures 2.10 and 2.11). Having adopted or postulated expressions for these first two moments, it is now possible to derive expressions for the nonstationary velocity means and variances of the areaweighted Lagrangian and the flowweighted Eulerian streamtubes.
2.5.4 Control Plane Oriented Velocity Moments
Cvetkovic et al. [1996] postulated a semianalytical estimator for a nonstationary mean velocity along areaweighted Lagrangiantrajectory observed at control planes. We would like to work directly from the assumed stationary flowweighted Lagrangiantrajectory velocity field. The general expression for the expected value of the areaweighted Lagrangiantrajectory velocity is
U
'a[x] =< vv[x]> (2.41) v [0]
Resolve the stationary mean velocities into the mean of the flowweighted Lagrangiantrajectory field and zeromean perturbation.
0a[x] =< V(1 + v'[zx) > (2.42) V(l + v'[01
52
Expand the denominator in a series to second order va[X] =
2 Cv[r]
= U(1 + o vf ) (2.43) V2 V2
= U(1 + ba2nk(1 exp[br/Alnkl)) In a similar fashion, an expression is derived for the nonstationary mean velocity along a flowweighted Euleriantrajectory. In this case, it is the stationary Euleriantrajectory velocity field that serves as a basis. The mean of the perturbation expansion is the Eulerian mean velocity. It is interesting to note that this result requires no series expansion.
u[0]
zil[z] =< Uu[z] >
1< U(1 + u'[O])U(1 + u'[x]) > U C[r] (2.44) = U(1 + )
U2
= U(1 + bulnk exp[br/Al k])
The nonstationary velocity variances along the respective crossweighted trajectories are derived in a similar fashion. The general expression for the areaweighted Lagrangiantrajectory velocity variance is
U
var[va,[x]] =< (v[x] 1 a[x])2 > (2.45) v[0]
This equation may be rewritten
var[va[x]] =< U V[x]2> U[X]2 (2.46) v[O]
Resolve the first term of the preceding equation into perturbation expressions U V2(1 + v'[x])2
< v[x]2 >= U < V(+[ (2.47) v [0] V(1 + V[0])
53
Expanding the denominator in a series, taking the expected value, and neglecting thirdorder and higher moments yields
U
< v[]2>= UV(1 + 2bo nk(1 + exp] bx/Alnk])) (2.48) In order to improve the quality of the estimate, the righthand side (RHS) of the preceding equation is required to match the known extreme values of the lefthand (LHS) side. The LHS at X = 0 is
U
< vov[0]2 > = U v [0] (2.49) = UV
The RHS at X = 0 is
RHS[0] = UV(1 + 2bo,2 k(1 exp[b0/Al k]))
(2.50)
= UV
The statistical independence of the initial velocity and the velocities as x goes to infinity are exploited to calculate the LHS as x 4+ c LHS[oo] =< > v [0]
= o2 V2 (2.51) = UV(1 + bunk)2 The RHS as x + oo is
RHS[oo] = UV(1 + 2ba nk(1 exp[boo/Alnk]))
(2.52)
= UV(1 + 2ban k) The RHS is deficient a term (ba nk)2. Rewrite the RHS thus
RHS = UV(1 + (2balnk + (blk)2)(1 exp[br/Al}k])) (2.53)
54
Substituting Equations (2.53) and (2.43) into Equation (2.45) yields
var[va[x]] = UV(1 + (2ba2nk + (bu2nk)2)(1 exp[br/Alnk]))
(2.54)
(U(1 + bonk(1 exp[br/Alk ])))2
Further expansion of this expression offers no intuitively obvious simplifications or insights in its entirety. However, this expression may be approximated by dropping a few terms thus
var[va[x]] = V2bnk U2bnk (2b n k + (bank)2) exp[br/Alnlk] (2.55)
Two unconventional techniques were employed in this derivation that shall now be defended. The first is requiring the series expansion to match the known endpoints. Simple expressions that condense the system behavior into an compact form that allows the relative contribution of the component processes to be easily grasped are sought. The series expansion and subsequent truncation is itself an approximation. The systematic error introduced by "missing" a known endpoint is distracting. In the case of the velocity covariance, the shape of the correlation function was "sacrificed" for simplicity, but the correlation length was retained. Had the simplified expression neither matched the shape of the data nor reproduced the correlation length, the expression would have been somewhat more questionable. In this case, the expression is required meet the observed endpoint values at the "cost" of the rigor of the series expansion and truncation.
The second simplification is the ad hoc dropping of terms to simplify the expression. Again, simple expressions that allow the relative contribution of the different processes be easily grasped are sought. It would be difficult to hypothesize Equation (2.55) a priori as an empirical expression. A primary function of the series expansion is to indicate the dominant processes and give an estimate of their relative contributions. This hybrid approach is useful for the purposes here. These techniques
55
are not a panacea, however, and the approximations have only been tested for the log conductivity variances tested in this work.
The nonstationary flowweighted Euleriantrajectory velocity variance may be derived in a similar fashion. These results are presented directly. The complete expression is
var [uf [x]] = U2 (1 + b n + (2b2nk +3 (b2n2 (bbo ( nk)) (1 exp [br/Alk))
(U (1 + boInk exp [br/Alnk]))2
(2.56)
and an approximation of the above obtained by neglecting a few terms is
var [u1 [x]] = U2bo2nk (1 + (2b2nk + (bunk)2) exp [br/Alnk]) (2.57)
2.5.5 Control Time Oriented Velocity Moments
The displacementinvariant velocity covariances for u and v permitted derivation of expressions for the first two moments of the respective velocities for the four different trajectory collections. However, only the areaweighted Lagrangiantrajectory collection appears to be stationary in time. This is perhaps not intuitively obvious, as the velocity is clearly not stationary in displacements along the mean direction of flow (e.g., Equations (2.43) and (2.54)) because there is an implicit distribution of times associated with the establishment of a reference injection plane. After long travel distances and times, the areaweighted Lagrangiantrajectory point velocity statistics are equal to the flowweighted Lagrangiantrajectory point statistics. At the injection plane, the areaweighted Lagrangiantrajectory point velocity statistics are identically equal to the areaweighted Euleriantrajectory point velocity statistics. These relationships can be exploited to derive equations that provide a few insights about how properties change in Lagrangian time, and to derive an expression for a Lagrangiantime dependent velocity covariance function.
56
The areaweighted velocity at the IP is U by definition. Since the areaweighted Lagrangiantrajectory velocity is stationary in time, its mean in time is U regardless of position. At long travel times, the point velocity is stationary in displacements as well. For constant means in space and time, the following relationship holds
T [t]dt= X d)1 (2.58) This expression states that the harmonic mean velocity in space equals the arithmetic mean velocity in time. Returning now to the model aquifer, invoke the hypothesis of lognormality of the velocities along either trajectory. The harmonic mean of v[x] is given by
Vh = exp[plnv alnv/2]
= exp[,ilnv] .exp[ainv]1/2
1 (2.59)
= U 1 + ba 1
I1 + ba.2n k
=U
Similarly, the harmonic mean of u[x] is given by
U
Uh = + 2 (2.60)
1 + b ln k
Although derived by different means, this result is equal to the result given by Shapiro and Cvetkovic [1988]. These harmonic means are the asymptotic mean velocities of the two trajectories in time. The initial timedependent velocities are identical to the initial spacedependent velocities, an artifact of starting all of the trajectories at the injection plane at an implied time t = 0. This indicates that the velocity covariance in time along the areaweighted Lagrangian trajectory is C,,[t = 0] U2ba2nk. As mentioned, at large times, the statistical properties of the Lagrangiantrajectories converge. Making a change of variables x = Vht = Ut in Equation (2.40) and substituting the known Ca[t = 0] for the value of Crf[r = 0] in yields an
57
expression for the stationary areaweighted Lagrangiantrajectory velocity covariance
Cva[t] = U2bUank exp[bUt/Alnk] (2.61) Similar expressions have been derived before by other researchers (see e.g., Dagan [1989]), but the conceptual development is quite different. Rather than seek to approximate the covariance function, an approach to quantify it directly from properties of the system is demonstrated. Moreover, the quantity U is perceived to be the harmonic mean of the Lagrangiantrajectory velocity, and that it is quantitatively equal to the areaaverage velocity of the flow field. This is an important point.
The mean velocity and velocity variance along the flowweighted Lagrangiantrajectory is found in a manner similar to those in space, but there are some conceptual issues that should be addressed. It is the collection of areaweighted Lagrangiantrajectories that exhibits a stationary velocity. The mean velocity is v[0]
0pr[t]=< U v[t]> (2.62) Expand the velocities into a mean and a zeromean perturbation. The mean here is the time average mean, or the harmonic mean of the space average.
Vft [t] = U < U (1 + v' [0]) U (1 + v' [t]) >
= U t1 + (2.63) = U (1 + bInk exp [bUt/Aln k])
This expression requires no series expansion. For the control plane oriented velocity statistics, the mapping was from flowweighted to areaweighted trajectories. Equation 2.63 is identical in form to Equation 2.44 with the substitution r = Ut. Recognize
58
the quantitative similarity and write velocity variance equations directly The complete expression is
var [v []]= U2 (1 + bank + (2bonk + 3 (blk)2 + (ba k)) 3 (1 exp [bUt/Alnk]))
(U (1 + bo, k exp [br/Alnk]))2
(2.64)
and the approximation is
var [vf [t]] U2bIk (i 1+ (2blnk + (bok )2) exp [bUt/Alnk]) (2.65)
Calculation of the mean velocities along the Eulerian trajectories is problematic in the absence of a stationary velocity covariance function. However, note that the areaweighted velocities start at U and that the flowweighted velocities start at V and both progress to U/(1 + ban k). Recognizing the similarity shared by the different mean time estimators, write approximate expressions directly
at t] 1 + b (1 + bor k exp [bUt/Alnk]) (2.66) and
at [t]= 1 +blk (1 + (2boInk + b 1nk) (exp [bUt/Alnk])) (2.67) Again, devoid of a stationary covariance function for the Eulerian velocity field in time, intuitive arguments lead us to anticipate an asymptotic u variance in time to be
at = U,2bank
U= 2 (2.68)
1 + ban k k
The initial velocity variances are known from the covariance functions in space. It is possible to anticipate forms of the nonstationary variance equations similar to those found previously, and include them for the sake of symmetry and completeness. The
59
areaweighted Euleriantrajectory timedependent velocity variance approximation is
var[ua[t]]= ( U l )b bnk (1 + (2bo~k + (bo2nk)) exp [bUt/AXIk]) (2.69) The fluxweighted Euleriantrajectory timedependent velocity variance approximation is
var[uf[t]] =
1 Ub2 k) b,2nk (1 + (3bu,2k + 3 (b2nk 2 + (bu2 k )3 exp [bUt/Alnk]) (2.70)
2.5.6 Log Conductivity Moments
The first two moments of the log conductivity along those trajectories for which the moments are stationary at control planes located along the mean direction of flow or at control times are sought. Working from a stationary Eulerian reference field, the mean and variance along the AW/ET are llnk and a2,k, respectively. Infer that the FW/LT is stationary as well, analogous to the velocity field results. The flowweighted mean log conductivity perturbation is given by u[0]f Kexp[f]J(1 + j)O
< > = < f>
U OKJ (2.71) =<(1 + f + f2/2 +...)(1 + j)f> /c Expanding, retaining secondorder terms, and taking the expected value yields
< If>= nk(1 1/d)/c (2.72) which, for twodimensional fields, is lnk/2. Similarly, the variance is
60
Finally, note that the AW/LT log conductivity appears stationary in time as does the velocity. Moreover, it appears to have values equal in magnitude to its AW/ET counterpart, as does the velocity.
Unlike the velocity, expressions for the trajectorybased crosscovariance of the velocity and log hydraulic conductivity that are necessary for developing the nonstationary log hydraulic conductivity moments were not developed. This is left for future work.
2.6 Evaluation of Simulation and Theoretical Results
2.6.1 Mean Velocities in Space
Figure 2.10 presents the control plane oriented mean velocities for the four different collections of streamtubes. The mean velocities are normalized by U = K9J/1 = 1/10 and this value was used in the mean velocity expressions as well. That is, the parameter values are the input parameters to the simulations. Other input parameter values are b = 3/8 and a2nk = {1/2, 1} depending upon the set of simulations. The deviation of the mean of areaweighted Euleriantrajectory data from 1 are an indication of the quality of the effective conductivity expression.
The areaweighted Euleriantrajectory mean velocity appears stationary, in accordance with established theory, and its value appears to be well predicted by the effective conductivity relationship. The fluxweighted Lagrangiantrajectory mean velocity appears stationary, supporting the stationarity hypothesis. Moreover, the value of the mean velocity appears to be well predicted for both variances of log conductivity. Additionally, the nonstationary mean velocity estimators appear to accurately reproduce the transition between the asymptotic velocities.
2.6.2 Velocity Variances in Space
Figure 2.11 presents the control plane oriented velocity variances for the four different collections of streamtubes. The variances are normalized by the areaweighted
61
Euleriantrajectory velocity variance estimated from the input parameters to the simulations. That is, by
U2b,2nk = (KgJ/0)2ba0nk (2.74)
As with the mean velocity estimators, the variance estimators employ only the simulation input parameters.
Once again, the areaweighted Euleriantrajectory variance exhibited its presumed behavior, and supports the well established fact that the Eulerian velocity field is fairly well understood in space. The hypothesized flowweighted Lagrangiantrajectory velocity variance is somewhat higher that the data for both an k. A possible reason is the negative correlation between log conductivity and head gradient is not weighted properly in the collection of trajectories skewed towards higher log conductivity values.
The other two estimators also suffer from this bias in the asymptotic Lagrangiantrajectory velocity variance, although the transition between initial and final values for both seems to be well characterized.
2.6.3 Mean Velocities in Time
Figure 2.13 presents the timedependent mean velocities for the four different collections of streamtubes. Again, the mean velocities are normalized by U as estimated from the simulation input parameters. Only the areaweighted Lagrangiantrajectory mean velocity exhibited a constant mean behavior in time. Its estimated harmonic mean is U, and the estimate seems good, although one might argue that there is a slight overpredictive bias.
The estimated harmonic mean of Eulerian trajectories appear to have a stronger under predictive bias. The transition of the flowweighted Lagrangiantrajectory appears to be captured better than those of the Eulerian trajectories. It should be noted that the Lagrangian trajectory transition was based upon an expression for
62
a stationary velocity field, whereas the Eulerian trajectories are ad hoc estimators, based upon a transition between known end points.
2.6.4 Velocity Variances in Time
Figure 2.14 presents the timedependent velocity variances for the four different collections of streamtubes. The variances are normalized by the areaweighted Lagrangiantrajectory velocity variance estimated from the input parameters to the simulations. The areaweighted Lagrangianvelocity appears to be stationary in the variance. The data seem to lie slightly above the estimated value.
The overprediction of the initial fluxweighted velocity variance has far less impact upon the timedependent velocity variances. This combined with what appears to be a good estimate of the longtime Euleriantrajectory velocity variance yields a better match to the simulations than one might expect from the means. However, there is a strong initial decline in the variance not captured by the simple exponential expressions. The areaweighted Lagrangiantrajectory covariance function exhibits a similar rapid decay. The early time behavior is likely dominated by the conductivity covariance. As the trajectory experiences a wide range of conductivity variability, this effect diminishes, and the longer range correlations related induced by the head field begin to dominate.
2.6.5 Log Conductivity Statistics
The predictors of the stationary log conductivity statistics worked well for both the spacedependent and timedependent means although there seemed to be a slight overpredictive bias for the flowweighted trajectory (see Figures 2.6 and2.8). The AW/ET log conductivity variance was "well predicted" in that the turningbands method performed well, as discussed in Sections 2.4.1 and 2.4.7. The log conductivity variance of the FW/LT appeared to be lower than that of the AW/ET, as predicted by
63
Equation (2.73), but the magnitude of under prediction was larger than that observed (see Table 2.4).
Table 2.4: Comparison of observed log hydraulic conductivity variances to estimators. Observed value is zeroslope line regressed through data. The decimal representation of the fractional value is included for convenience.
a2nk = 1/2
parameter predicted observed AW/ETal, k[z] 1/2 0.497 FW/LTU k[x] 7/16=0.4375 0.484 AW/LTank[t] 1/2 0.498 UInk = 1
parameter predicted observed AW/ET Ink[x] 1 1.002 FW/LTn k[x] 3/4 0.944 AW/LTuk [t] 1 1.000
2.7 General Conclusions
In an steady irrotational flow field, a fluid parcel trajectory follows a hydrodynamic streamline. The flow field properties as observed along streamlines can differ radically than those observed "uniformly in space." For the model system, a stationary Eulerian log conductivity field, this work identified the trajectory collections for which certain properties appear statistically stationary to second order. For displacements along the mean flow direction, point velocity and log conductivity appear stationary along streamlines separated by equal discharges. For displacements in travel time, these same properties appear stationary along streamlines that are selected uniformly in space. The stationarity of these properties facilitates the development of simple expressions for the statistics of displacement and travel times, as we shall show in following chapters.
An immediate result of the preceding analysis is that the bulk of the flow field in the sense of total discharge travels at a velocity that is locally greater than the spatial
64
average, and this disparity increases with heterogeneity of the underlying conductivity field. This might be described as "preferential flow." A practical implication for the transport of kinetically sorbing or interacting solutes is that the local velocity for the majority of the flow may occur in a regime significantly higher than that estimated as the flow field average. An understanding of the medium heterogeneity is essential if laboratory experiments are to be "scaled up."
It is intuitively obvious, and has certainly been noted before, that the bulk of the flow passes through an increasingly small portion of the total "swept volume" as the medium heterogeneity increases. Again, the practical implications are likewise obvious. Consider some aquifer remediation effort that involves flushing a "contaminated zone" with some fluid. This might be as simple as the upgradient water in the case of pumpandtreat or as extravagant as an surfactantenhanced microemulsion. In heterogeneous flow systems, the bulk of the flushing fluid contacts a disproportionately small volume of the total swept volume. For contaminants that tend to reside in areas of high conductivity, the serves to enhance the efficacy of remediation efforts. Contaminants that tend to reside in areas of low permeability, conversely, would be more difficult to remove. While these conclusions may be drawn from "common sense," application of the concepts developed here may help to better understand these issues quantitatively.
CHAPTER 3
NONREACTIVE SOLUTE TRANSPORT
3.1 Introduction
The previous chapter saw the derivation of the relationship between the velocities that lie along Lagrangian trajectories and the parameters that describe a simple model Eulerian flow field. These Lagrangian relationships are employed to derive a few functions that characterize global measures of solute transport: namely the statistics of mass displacement in time and the statistics of mass arrival times at control planes. The frameworks are are that of Dagan [1982b] and that of Shapiro and Cvetkovic [1988], respectively. The recovery of a few wellknown results are anticipated, in addition to some that are new. However, the approach is novel in that the work is based upon trajectorybased statistical properties of the flow field.
3.2 Injection Mode and Streamtube Selection
Demmy et al. [1999] noted the relationship between boundary conditions and sampling of streamlines. A clarification of this relationship and a generalization the results of that work are sought. Unlike the previous chapter, the Eulerian trajectories are not considered in the context of an actual transport trajectory, as they are of little practical importance. However, the results of Shapiro and Cvetkovic [1988] correspond to areaweighted Euleriantrajectories, as will be shown. Therefore, trajectory, Lagrangian trajectory, streamtube, and streamline are used interchangeably.
3.2.1 Uniform Resident Injection
Consider the application of mass at a constant density co across the entire face of the injection plane (IP) of the model aquifer (see Figure 2.1). Let the mass occupy
65
66
some small Ax that is invariant with displacements along the IP. Following the terminology of Kreft and Zuber [1978], this injection mode is defined to be a uniform resident injection. Uniform refers to the homogeneous mass density in space and resident refers to the volume of resident fluid into which the solute is introduced. Subsequent references to a resident injection will refer to a uniform resident injection unless otherwise noted. Now, this injection mode has nothing to do with the flow field per se. That is, the mass "magically appears" and occupies a fixed volume at a fixed concentration without being influenced by local water fluxes. Consider the flow field to be divided into some number of streamtubes, say N. Moreover, let these streamtubes carry an equal areaweight at the IP. For a small depth of injection Ax, this injection mode has assigned an equal mass weight to each streamtube with respect to the total solute mass. The total mass of solute is Mr = coOhLAx (3.1)
The mass weight for any one streamtube i is m 1 (3.2) M N
This value is identically equal to the areaweight of the streamtube, and constant for all in the collection.
Consider now dividing the streamtube into N streamtubes that all carry the same flow weight. The mass in some streamtube i is given by mi ai
M L
U (3.3)
vi
67
3.2.2 Injection in Flux
Consider now maintaining the IP at co for some brief interval At. A mass
Mf = coOhLUAt (3.4) enters the domain and the different streamtube collections. This injection mode is a uniform injection in flux. Uniform, again, refers to the uniform mass density and "in flux" refers to the influent water that carries the solute into the flow domain. A mass
L
mi = coOhviAt (3.5)
N
enters equalarea streamtube i. This corresponds to a mass weight of mi ui
(3.6)
M U
A mass
mi = co0haiviAt
= co QAt (3.7)
N
enters equalflow streamtube i. This corresponds to a mass weight of 1/N.
3.3 Integrated Streamtube Properties
Let us return to the notion of the streamtubes as pure hydrodynamic entities. It is possible to discuss properties of these streamtubes with no mention of injection modes. Two properties hold some hydrodynamic interest. The first is the velocity integrated over time. For for some streamtube i this is Xi [t] = vi[t]dt (3.8) A velocity integrated over time is a "displacement" with a dimension of length. The second the inverse velocity integrated over some distance. For some streamtube i this
68
is
t[ = (3.9) oiX vi[]
This integral has a dimension of time. These integrals are given in terms of the x velocity component as a function of longitudinal displacement or time. It is suggested that future works examine the details of a more general approach, e.g., displacements along the trajectory.
3.3.1 Displacements
Consider a model flow field such as the one given in the previous chapter. Quantification of the displacement and travel time statistics for areaweighted and flowweighted trajectories is sought. Previous results indicate that the velocity for a collection of areaweighted streamtubes is stationary in time, characterized by mean U and approximate covariance U2ba 2nk exp[bUt/An k]. The expected value of Equation (3.8) is
=< v[t]dt>
=Uf (3.10) = Ut
Although the velocity of the flowweighted streamtube collection is not stationary in time, the results of the previous chapter are employed to map the properties of the areaweighted collection to the flowweighted collection.
=< U fv[t]dt> (3.11) Recognize that this equation may be rearranged to be the integral of Equation (2.63) in time. Substitute the RHS of Equation (2.63) into the preceding equation and
69
integrate to yield < xf[i] >= Ut + AlInkI2nk(1 exp[bUt/Alnk]) (3.12) The displacement variance for the areaweighted streamtube collection is given by
var[xa[t]] =< Xz[t]2> 2
=< Ci[t'],t"]dt'dt"> (Ut)2 = e t C rt at[t t]dt'dt" (U t)2
2 / j Cat [s]dt'ds (Ut)2 (3.13) = 2 (t s)CIat[s]ds (Ut)2 = 2AInk k2nk Ut + lk (exp[bUt/Aink] 1)
3.3.2 Travel Times
Expressions for the travel time statistics are sought, in analogy to those sought for displacement. The reference collection of velocities from which these expressions will be derived is the stationary velocities of the equal flow streamtubes, or the equalflow weight Lagrangian trajectories. The mean travel time for these streamtubes is given by
1 dx
1 dx (3.14) Vh
U
The mean travel time for the areaweighted streamtubes is / U dx
a v[] (3.15)
(x + Alnk12 nk(1 exp[bx/Alnk]))
U
70
The arrival time variance for the flowweighted collection of streamtubes is
var[tf [x]] =< X> (3.16) ] v [x'] o v [x"](3.16) Approximate the inverse velocity covariance contained in the first term of the RHS of Equation 3.16 with the a series expansion of the perturbation equation
1 1
"] V2(1 + v'[x'])(1 + v'[x"11])
=< (1 v'[x'] + v' [']2 .. .)(1 '[x"] + v'[x"]2 >
V21 (3.17)
= V(1 + 2ao2 + Cvf[x', x"] + HOT)
1 ( Cvif [X', X"])
U2 V2
where HOT is the collection of higher order terms. Inserting this expression into Equation 3.16 yields an expression for the travel time variance for fluxweighted streamlines
var[tf[zx]] = + A (exp[bx/A2k] )) (318)
3.4 Solute Transport
Expressions for the statistics of the integrated trajectory properties of displacement and travel time have been derived in the absence of an explicit reference to solute transport to demonstrate that these properties exist simultaneously in the same flow field. However, they are intimately related to solute transport. Consider the transport of an infinitesimal particle whose movement is constrained to follow a streamline. Moreover, at any time or position on the streamline, the particle velocity is identically equal to that of the underlying velocity field. Its displacement from x = 0 in t = T is given by Equation (3.8) and the time required to travel from x = 0 to x = X is given by Equation (3.9). For the same flow field, two collections of streamtubes have been discussed, namely areaweighted and flowweighted. Placing
71
one particle into each areaweighted streamtube at x = 0 and t = 0 is analogous to the a uniform resident mass injection. Equations (3.10) and (3.13) express the mean particle displacement and particle displacement variance as a function of time. Equation (3.15) expresses the mean particle travel time at different control planes.
Similarly, placing one particle into each flowweighted streamtube at x = 0 and t = 0 is analogous to the a uniform injection in flux. Equation (3.12) expresses the mean particle displacement as a function of time. Equations (3.14) and (3.18) express the mean particle travel time and particle travel time variance at different control planes.
In addition to these two injection modes, there are hybrid injection modes. For example, consider if the introduction were somewhat random, or particle mass were some deterministic function of the injection velocity, or some combination. In fact, it is left to the imaginative reader to conceive of different injection mode possibilities and the physical systems to which these correspond.
3.5 Evaluation of Expressions
In the previous chapter, a numerical experiment used to test certain hypothesis about the relationship between Lagrangian and Eulerian descriptions of flow fields was described. Those data are employed here. In the comparison of the derived expressions to the data, the functions are based upon the simulation input parameters, and all normalization is carried out using these same parameters. Therefore, the expressions represent a priori estimates of the simulation results, based, of course, on known input parameters.
3.5.1 Displacements
Figure 3.1 presents the model estimates of mean mass displacements for the two injection modes. The flux injection preferentially selects, or mass weights, high velocity areas of the flow domain, and the propagation of the center of mass is characterized
72
by a higher velocity. The timevarying mean velocity expression for the flowweighted Lagrangiantrajectories (Equation (2.63)) is, in fact, an expression for the mean velocity of a plume resulting from a uniform flux injection. The longtime asymptotic slope of the fluxweighted mean displacement is U, as is easily verified by taking limit of Equation (2.63) as t + oo. The mean displacement functions predict the observed behavior, and also provide an excellent fit to the data.
The displacement variance data are interesting (see Figure 3.2). Injection mode seems to have little effect upon the displacement variance in time. This is not to say that injection mode has little effect upon the spatial distribution of the plume in time. At a given time t, a plume injected in flux will have traveled a greater distance, on average. After traveling the same distance, the displacement variance of the fluxinjected plume will be less than that of the resident injection. Thus, the "spreading as function of mean displacement" is less in the case of the fluxinjected solute. The areaweighted trajectory displacement variance overpredicts the observed displacement variances, and the longtime slopes appear to have different values. After 10 characteristic times, the exponential term in Equation (3.13) retains a little more that two percent of its original value. This difference does not seem large enough to account for the observed discrepancy. The variance of the stationary areaweighted Lagrangiantrajectory velocity field in time appears to be well predicted. Thus, the problem may lie with the rather poor description of the shorttime and longtime correlation exhibited by the observed velocity covariance given by the assumed model.
3.5.2 Travel Times
Figure 3.3 shows good agreement between the estimated and observed mean travel times. The injection influx shows earlier arrival than that of the uniform resident injection. Uniform resident injection weights all areas of the IP equally. Large areas that contribute little to the overall flow receive and equal mass weighting as those
73
10
x4
8 AW/LT + FW/LT x "" Eq. 3.12 Eq 3.10 .................X .44
4 .4 .4
x
2
0 I I
0 2 4 6 8 10
tU/ An k
10 .4
8 AW/LT + FW/LT x Eq. 3.12 . .X +
E q 3.10 ...................
k .4
XX
( 4
.4
x"
0
0 2 4 6 8 10
tU/ An k
Figure 3.1: Comparison of mean displacements for the two injection modes or streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT) statistics correspond to a uniform resident injection. The top figure contains the rnk = 1/2 data and the bottom figure contains the cink = 1 data.
74
15
AW/LT + ,* FW/LT x Eq. 3.13 0
0 2 4 6 8 10
tU/Aln k
5
AW/LT + +x FW/LT x L +xt Eq. 3.13   +x
+X
10 +Xx normalized by +
0 2 4 6 8 10 tU/An k
correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT)
2 2
75
areas that contribute more significantly to the overall flow. However, as the solute moves through the domain and experiences the full range of velocity variability along its trajectory, the "slow start" is damped out and the instantaneous average point velocity for all for the particles is V and the harmonic mean Vh = U. The travel time variances appear to be better matched than the displacement variances (see 3.4). This is reasonable in light of the better apparent match of the assumed velocity covariance function in space than that in time. The uniform injection appears to exhibit less nonlinear behavior than the injection in flux. The uniform injection variance is reasonably expected to be larger than that of the flux injection, due to the skewed massweighting of lowvelocity areas by the uniform resident injection.
3.6 General Conclusions
It is clear that the injection mode effects are most prevalent in the "near field" where the initial velocities and travel times or displacements are well correlated. After the plume travels several correlation scales, these injection mode effects diminish. The near field and the neartofar transitional scales are quite interesting, as these intermediate scales are on the order of sites associated with certain agricultural settings such as some feed lots and waste holding facilities, urban land uses such as gas stations and dry cleaners, and their associated remediation efforts, and a plethora of "field scale" experiments. These scales are examined in the perspective of the "error" in travel time estimate associated with assuming a resident injection when the "real" injection is in flux
e[x] = AlnkUln2k(1 exp[bx/Alnk])/x (3.19) given by subtracting Equation (3.15) by Equation (3.14) and dividing the difference by Equation (3.14) (see Figure 3.5) The "near field" might be characterized as the first five to ten correlation lengths, where the injection mode effects are quite prevalent, even for modest log conductivity variability. These effects diminish less rapidly over
76
40
30 AW/LT + X FW/LT x Eq. 3.15  +
Eq. 3.14
0
0 10 20 30 40 X/Alnk
40 .
30 AW/LT + FW/LT x x Eq. 3.15. Eq. 3.14 ...............
20
10
0 10 20 30 40 x/Alk
Figure 3.3: Comparison of mean arrival times for the two injection modes or streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT) statistics correspond to a uniform resident injection. The top figure contains the aUink  1/2 data and the bottom figure contains the 02k =1 data.
In k =I data
77
80
+
tx
60 AW/LT + .x FW/LT x x C14Eq. 3.18  x
S40.*x
20 40
'+Xx.
.4X
S0 0x
+ +
x.
+ X + x..'
20 .*+
++
0 AW/L +
0 10 20 30 40
In k
4.
+x
60 A"
0 10 20 30 40
78
0.4 1
:n.k = 1/2 0.3
t 0.2
0.1
0 I
0 10 20 30 40
/Al nk
Figure 3.5: Travel time estimate error as function of displacement associated with assuming a uniform resident injection when injection is in flux for Cink = {1/2, 1}. a "transition" region between 10 and 30 correlation lengths. Beyond 30 correlation lengths, the difference between the two modes is less than a few percent.
Much of the Lagrangian transport literature is predicated upon a uniform resident injection, but the socalled consistent firstorder travel time estimate that is regularly employed is quantitatively that for a flux injection. For large displacements, this error is small, and the simplification afforded by this approximation is considerable (compare Equation (3.14) and Equation (3.15)). For many practical applications, an error of ten, or even more, percent might be considered negligible. Therefore, it is not suggested that this approximation be abandoned, but rather phrasing such as "inconsistent" firstorder approximation be applied in future works, since this Lagrangian expression is inconsistently based upon the Eulerian mean velocity.
CHAPTER 4
REACTIVE SOLUTE TRANSPORT
4.1 Introduction
The concepts developed in the previous chapters are employed to analyze reactive solute transport. The focus is upon global measures of transport, rather than predictions of local quantities. Particular processes of interest are injection mode effects, the effects of correlation of the local "reaction parameter" to the log conductivity field, and the interplay of these injection mode and correlation effects. A suite of numerical experiments illustrates these effects. Additionally, expressions are developed for temporal moments of mass breakthrough curves for a solute subject to linear equilibrium sorption for the different injection modes that reflect the observed behavior in relatively simple closedforms.
Spatial moments of sorbing solutes are not directly treated in this work. The experimental design was targeted directly towards a temporal moment analysis, and this design decision has some implications for spatial moment analysis. These decision and implications are discussed, in hopes that future works might be guided by this analysis. Some of the properties of the travel time dependent reaction parameter are discussed, as are some of the implications for solute transport.
4.2 Temporal Moments
The concepts developed in the preceding chapters are applied to examine the effect of injection mode upon the temporal moments of the breakthrough curve of a solute subject to linear equilibrium sorption. A uniform mean gradient induces a steady irrotational flow in a uniformly saturated porous medium characterized by an exponentially correlated lognormal conductivity distribution. This flow field is divided
79
80
into collections of noninteracting streamtubes. Dispersion at the streamtube scale is considered to be negligible in its effect on the aggregatescale temporal moments (e.g., Dagan [1989]). The sorption process is related to a generic reaction parameter that is heterogeneous and may be correlated to the log conductivity field.
The previous chapters demonstrated the relationship between streamtube selection strategies (equalarea or equalflow) and injection mode (uniform resident and influx). The reference collection is equalflow streamtubes, since the associated velocity and log conductivity statistics are stationary in displacements along the mean flow direction.
The primary objective of this chapter is to examine the effect of injection mode upon the temporal moments of the mass breakthrough curve of a solute subject to linear equilibrium sorption in a heterogeneous flow and sorption field. A brief review the conceptual model of solute transport is given. General expressions are presented for the first two temporal moments of the breakthrough curves associated with the uniform injection of a solute both in the influent fluid flux and in the resident fluid. The results of the reactive solute transport simulations are presented, followed by the development of some analytical expressions that help explain the observed behavior.
4.3 Conceptual Overview of Solute Transport
Consider a general multidimensional flow field that is resolved into a collection of noninteracting streamtubes. Transport along an individual streamtube is characterized with a transfer function y[t; x] (see Jury and Roth [1990] and Cvetkovic et al. [1998]). For a conservative and nonreactive solute, the transfer function is simply
y[t; x] = 6 [t T[x]] (4.1) where 6 is the Dirac delta function and r is the travel time. The physical interpretation of this transfer function is this: a particle located at x = 0 "arrives" at x = X at time t = T[X]. We conceive of this travel time as not only the time required to
81
travel to some point x, but as an intrinsic integrated streamtube property, namely the inverse velocity integrated along the trajectory (e.g., Equation (3.9)).
Recall reaction flow path concept of Cvetkovic et al. [1998]. The reaction flow path, parameterized in space, is defined as x P[x]dx (4.2) o V [x]
where P[x] is some reaction parameter of interest. For linear equilibrium sorption, P is a linear partitioning coefficient. The reaction flow path is also an integrated streamtube property. As such, any reference to p properties in general are broadly applicable to any distributed reaction parameter, perhaps a firstorder decay coefficient or a Langmuir sorption parameter. However, the focus here is upon linear equilibrium sorption to promote a ready and intuitive grasp of the pertinent concepts. For this case, p may be thought of as the time that the solute is sorbed. Thus the arrival time of a particle undergoing a linear equilibrium sorption process is the sum of the nonreactive travel time 7 and the time sorbed M. The transfer function characterizing this transport and reaction process is (see Cvetkovic et al. [1998])
y[t; x] = 6 [t 7[x] P[x]] (4.3)
Following the work of Cvetkovic et al. [1998], the adopted reaction parameter model is
P = Pg exp[w] exp[P In k] (4.4)
where Pg is the geometric mean of P, w is a zeromean, exponentiallycorrelated normal random variable with variance of and correlation length A, = Alnk that is uncorrelated to the log conductivity field, and 3 is a strength of correlation parameter relating P to the underlying log conductivity field. The equality A,, = Alnk was specified for convenience.
82
Two specific injection modes are considered: uniform injection influx and uniform resident injection. For the uniform injection influx, each streamtube receives an equal amount of solute mass (Equation (3.5)), and thus mass weight (Equation (3.6)). Uniform resident injection distributes the mass uniformly in space, but each streamtube receives a different mass and mass weight (Equation (3.3)). The mass breakthrough for some streamtube i is given by
mi[t; x] = mi[0; 0]y[t; x] (4.5)
An ergodic condition is assumed such that aggregate properties may be thought of as ensemble properties in the statistical sense. The aggregate mass breakthrough is given by taking the expected value of Equation (4.5). For injection influx, the expected solute breakthrough curve is given by = M <[t; x] > (4.6) For uniform resident injection, the expected solute breakthrough curve is given by = MU (4.7) where U is the arithmetic mean, or spatially averaged, velocity and v[O] is the velocity at the injection point. The temporal moment of this breakthrough curve is sought. The n temporal moment is given by 8n
tn = lim(1)n (4.8) s+O aSn
where S is the Laplace transform of S and s the corresponding Laplace space variable. The Laplace transform of Equation (4.3) is given by '[s; x] = exp [s (T[x] + p[x])] (4.9)
83
Two temporal moments of interest for the influx breakthrough curve are the mean tlf =<7> + (4.10) and the variance
t2f = T + 2 + 2a,, (4.11) For uniform resident injection, the moments are tlf = + (4.12) and the variance
t2f =< U(T +)2/v[0] > < U/v[0] >2 < UI/v[0] >2 (4.13) From these equations, it is seen that the mean and variance of the expected reactive travel time distribution is comprised of the first two moments of the nonreactive travel time and the reaction flow path and the cross moment of the travel time and reaction flow path.
4.4 Simulation of Reactive Solute Transport
The results of the simulations described in Chapter 2 are employed to simulate the transport of a reactive solute. As mentioned in that chapter, information was recorded along two particle trajectories: the first was a natural trajectory corresponding to a hydrodynamic streamline, and the second was a straight line parallel to the mean flow direction. Data were recorded at equal increments in time after injection, and at "control planes" perpendicular to the mean flow at equal displacements from the injection plane. Among these data were the velocity, the hydraulic conductivity, and an observation of a lognormal random field with the same statistical and correlation properties as the log hydraulic conductivity field.
84
4.4.1 Reaction Parameter
From these data, trajectorybased P fields were constructed using Equation (4.4). In this manner, any variety of correlation relationships may be analyzed using the same set of simulation data. There is a dearth of information about the structure of sorption fields in natural systems (see, e.g. Tompson [1993] or Jawitz [1999]). Thus, numerically convenient parameter values were selected. This suite of experiments examined correlation parameter values of = {1, 1/2, 0,1/2, 1} in log conductivity fields of variances ank = {1/2, 1}. The extreme correlation parameters / = {1, 1} are arguably physically implausible, yet they are included to demonstrate the effect of correlation without the "noise" associated with an uncorrelated component. The remaining three values, 0 = {1/2, 0, 1/2} might be considered a range of more plausible values. For consistency, the variance of point log reaction parameter was held constant and equal to that of the log conductivity field by requiring a, = (1k,2)a12nk. The expected value of P was 1. This is a slightly different approach than that taken with the log hydraulic conductivity, where the geometric mean was fixed at K, = 1. In the case of the twodimensional isotropic log conductivity field, the effective conductivity is equal to the geometric mean. The expected value of the reaction value was fixed < P >= 1, because in some practical applications P is assumed to be linearly related to some property of interest, and the "amount" of that property is to be estimated based upon the temporal moments of linearly sorbing solutes, or tracers (see e.g., Jin et al. [1995], Annable et al. [1998], and Jawitz [1999]).
The point average of reaction parameter P as observed along a Lagrangian trajectory is a function of injection mode, correlation to the log conductivity and the variance of the log conductivity. As with the log conductivity, the equalflux streamtubes exhibit an apparently stationary mean reaction parameter in displacement along the mean flow direction, whereas the equalarea streamtubes exhibit distinctly nonstationary behavior that is analogous to that of the log conductivity (see Figures 4.1 and
85
4.2). These observations, of course, are not surprising, since the reaction parameter is a function of the log conductivity. Positive correlation between the log conductivity and the reaction parameter (0 > 0) results in a reaction parameter trajectory mean that is larger than the uniform spatial mean. Negative correlation (3 < 0) results in a reaction parameter trajectorymean that is smaller than the uniform spatial mean. No correlation (p = 0) results in equal trajectorybased and uniform spatial mean reaction parameter estimates. A decrease in the system variability (e.g., the variance of log conductivity ank) results in a smaller difference between the volume average and the trajectory averages.
4.4.2 Reaction Flow Path
A simple trapezoidal rule was used to calculate the reaction flow path p along a solute particle trajectory. For a collection of n trajectories, there were n p "observations" at each control plane. For an intuitive grasp of what the statistics of the reaction path "means", recall that for linear equilibrium sorption, the reaction path corresponds to the time that the solute is sorbed. The sum of the the nonreactive travel time and the sorbed time equals the total reactive travel time of the solute from the point of injection to the point of observation. See Cvetkovic et al. [1998] for further application of the statistics of the reaction flow path.
The mean and the variance of the reaction flow path are presented for Lagrangian trajectories that carry either an equal flow weight or an equal area weight at a reference plane. Recall that the equal flow collection can be thought to correspond to a uniform injection influx and that the equal area collection can be thought to correspond to a uniform injection into the resident fluid. The coupled effect of injection mode and reaction parameter correlation to the log conductivity are illustrated by examining reactive flow fields with five different correlation characteristics. The first case is that in which the reaction parameter is completely determined by the

Full Text 
36
Figure 2.13: Comparison of timedependent mean velocity estimators to data for the
four collections of trajectories for afnk = 1/2 (a) and
normalized by U.
18
researchers such as Rubin and Dagan [1988],[1989], and Osnes [1998] have investi
gated such nonstationarities, the additional complication of a rigorous consideration
of these sorts of nonstationarities would unduly obfuscate the very simple concepts
illustrated here. To minimize the effect of these nonstationarities, solute transport
is constrained to take place some distance from the boundaries of the numerical do
main. Following the example of Beilin et al. [1992], a 3Ainjt buffer was established
on both sides of the required 40 for a total longitudinal domain length of 46Ain fc.
Preliminary simulations indicated that the maximum transverse displacement of a
streamline in a <4 k = 1 field was around one third the longitudinal displacement.
For a streamline originating on the aquifer centerline parallel to the x axis, an aquifer
width of 36Alnjt was deemed adequate to contain large transverse displacements over a
40Alnfc longitudinal displacement and provide a similar buffer to mitigate boundary
effects.
Combining the physical domain requirements with the discretization required
to capture the log conductivity variability resulted in a numerical grid 230 by 180
for a twodimensional aquifer and a numerical domain with 41400 nodes. A three
dimensional analysis under the same assumptions would have required a numerical
domain of more than 7 million nodes. Simulations of this magnitude were untenable
in light of the available computational resources.
2.4.2 Velocity Field Generation
A mixed finiteelement scheme solved coupled velocity and pressure equations de
rived from Darcys law and the continuity equation. The code took input parameters
that matched the design criteria and the log conductivity field generated by turning
bands. Andrew I. James wrote and tested the core of the code. James et al. [1997]
used a derivative version of this code to generate flow fields for synthetic tracer tests.
A mixed finiteelement solution to the flow field was specified, following the results
and arguments of Mos et al. [1994]. The mixed finiteelement scheme is known
15
conductivity field as a realization of a lognormal random process. The process is
said to be stationary if the ensemble mean and variance do not vary in space. The
process is nonstationary if some trend exists in the mean or variance. In practice,
only single realization is available, and the ensemble properties are ultimately un
knowable. However, if the field is large compared to the scale of variability, one may
assume ergodicity, or, that the ensemble parameters may be estimated from a sample
drawn from the single realization since there exists a the possibility of a large num
ber of statisticallyindependent observation points. For a more detailed development
of these concepts from the hydrologists perspective, see Gelhar [1993]. For a more
formal development of these concepts, see Papoulis [1991].
The model of hydraulic conductivity variability is a stationary, exponentially
correlated isotropic lognormal distribution. A lognormal random variable is one for
which its natural logarithm is normally distributed. As such, three parameters sum
marize the statistical properties of the adopted conductivity model, namely the mean
of log k /iinjt, the variance of log k ofnfc, and the correlation length of log k \\nk The
covariance model is given by
c,*[r] = ffL. tl
(2.5)
where r is the separation between two points in the log conductivity field. Use of
this simple covariance model requires stationarity of variance, and usually implies a
stationarity of the mean as well. The moments of the real space conductivity field
are given by
= exp[ii(iit + n2ffft/2]
(2.6)
where n is the order of the moment.
In the absence of a perfectly described physical aquifer, the method of turning
bands generated equallylikely realizations of a hypothetical conductivity field with
20
generate a large, twodimensional conductivity field with the desired statistical
properties
generate a flow field by imposing a constant mean gradient across the saturated
conductivity field
trace the trajectories of two particles injected at a common control location,
one trajectory is a straight line or Eulerian trajectory and the other a streamline
or Lagrangian trajectory, and record properties at equal lags in space and in
time along trajectories that pass through inner domains that are relatively
undisturbed by nonstationarities induced by the boundaries
At this juncture, it is perhaps worth mentioning a minor philosophical point.
These simulations do not represent a real aquifer, per se, but rather a hypothetical
aquifer with known properties. The goal is understand the convolution of the physics
of the groundwater flow and a known heterogeneity that has analogs in nature, and it
is hoped that the reader finds some merit to this modest goal. The 41400 conductivity
values needed for any single realization do not represent 41400 measurements taken
from some real aquifer. The Monte Carlo aspect of this experiment reenforces this
point.
2.4.5 Sampling Strategy
Consider dropping particles into a flow field, with no particular regard to lo
cation. These particles move along streamlines without interaction with the porous
medium. Any one location into which one is dropped is as equallylikely as any other.
The trajectories associated with these particles carry an equal area weight, analogous
to equalarea subdivision of an injection plane. Consider now, the same disposal of
particles, but this time an area is assigned to each particle at the injection point such
that an equal discharge is associated with each particle, and its associated trajectory.
For a homogeneous mobile water content, this area is inversely proportional to the lo
cal velocity normal to the area. Thus, each particle may be thought to carry an equal
85
4.2). These observations, of course, are not surprising, since the reaction parameter is
a function of the log conductivity. Positive correlation between the log conductivity
and the reaction parameter (Â¡3 > 0) results in a reaction parameter trajectory mean
that is larger than the uniform spatial mean. Negative correlation (Â¡3 < 0) results in
a reaction parameter trajectorymean that is smaller than the uniform spatial mean.
No correlation (Â¡3 = 0) results in equal trajectorybased and uniform spatial mean
reaction parameter estimates. A decrease in the system variability (e.g., the variance
of log conductivity cr,2nfc) results in a smaller difference between the volume average
and the trajectory averages.
4.4.2 Reaction Flow Path
A simple trapezoidal rule was used to calculate the reaction flow path // along a
solute particle trajectory. For a collection of n trajectories, there were n // obser
vations at each control plane. For an intuitive grasp of what the statistics of the
reaction path means, recall that for linear equilibrium sorption, the reaction path
corresponds to the time that the solute is sorbed. The sum of the the nonreactive
travel time and the sorbed time equals the total reactive travel time of the solute
from the point of injection to the point of observation. See Cvetkovic et al. [1998] for
further application of the statistics of the reaction flow path.
The mean and the variance of the reaction flow path are presented for Lagrangian
trajectories that carry either an equal flow weight or an equal area weight at a ref
erence plane. Recall that the equal flow collection can be thought to correspond
to a uniform injection influx and that the equal area collection can be thought to
correspond to a uniform injection into the resident fluid. The coupled effect of injec
tion mode and reaction parameter correlation to the log conductivity are illustrated
by examining reactive flow fields with five different correlation characteristics. The
first case is that in which the reaction parameter is completely determined by the
34
1
0.8
.I '6
*4
8 0.4
0.2
0
0 5 10 15
f /^ln k
Figure 2.12: Comparison of spatial Eulerian (AW/ET) and Lagrangian trajectory
(FW/LT) velocity covariance data to covariance model. Data are taken from the
cr,2nfc = 1 set of realizations. Covariances presented as dimensionless lag correlation
functions.
that the analytical expression overestimates the covariance at short lags and under
estimates the covariance at longer lags. This discrepancy and its implications are
discussed in some detail in a subsequent section.
The velocity variances in Figure 2.11 are normalized by the zerolag covariance pre
dicted by placing the input simulation parameters into Equation 2.13. The AW/ET
velocity variance appears wellpredicted by this expression, as evidenced by a rela
tively good fit to 1 for both values of a^nk in Figure 2.11. A regression of a zeroslope
line through these data indicates that the a priori estimate under predicts the ob
served values by about five percent for the larger ofnfc (see Table 2.3) The apparent
stationarity of the velocities along the FW/LTs suggest estimation of the covariance
function for these velocities as for the AW/ET (see Figure 2.12). The correlation
length of the FW/LT is very close to that of the AW/ET, although the actual value
is probably slightly higher. Recall the definition of the correlation length as given by
SOLUTE TRANSPORT IN HETEROGENEOUS POROUS FORMATIONS
By
GEORGE GARY DEMMY, JR.
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1999
82
Two specific injection modes are considered: uniform injection influx and uniform
resident injection. For the uniform injection influx, each streamtube receives an equal
amount of solute mass (Equation (3.5)), and thus mass weight (Equation (3.6)). Uni
form resident injection distributes the mass uniformly in space, but each streamtube
receives a different mass and mass weight (Equation (3.3)). The mass breakthrough
for some streamtube i is given by
m[t; x] = rrii[0; 0]y[i; x] (4.5)
An ergodic condition is assumed such that aggregate properties may be thought of
as ensemble properties in the statistical sense. The aggregate mass breakthrough
is given by taking the expected value of Equation (4.5). For injection influx, the
expected solute breakthrough curve is given by
= M (4.6)
For uniform resident injection, the expected solute breakthrough curve is given by
= MU <'y[t]x)/v[Q\> (4.7)
where U is the arithmetic mean, or spatially averaged, velocity and v[0] is the velocity
at the injection point. The temporal moment of this breakthrough curve is sought.
The n temporal moment is given by
tn = lim(1)Tl^<5[s;x]> (4.8)
where S is the Laplace transform of S and s the corresponding Laplace space variable.
The Laplace transform of Equation (4.3) is given by
7[s; x\ = exp [s (r[x] + //[x])]
(4.9)
SOLUTE TRANSPORT IN HETEROGENEOUS POROUS FORMATIONS
By
GEORGE GARY DEMMY, JR.
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1999
For my parents.
ACKNOWLEDGMENTS
Dissertations have but one name in the author slot, but anyone who has writ
ten one will readily testify that it represents the work and support of myriad peo
ple. Mary Hall freed my mind from many of the administrative hassles students
encounter. Dawn Mendoza appointed, unappointed, and reappointed me numerous
times throughout my tenure, and made the business end of things run as painlessly
as possible. Ralph Hoffmann kept me in keys and desks and office space, and helped
me get things wired when they needed wiring. Wally Hunter, system administrator
and Unix hacker, deserves as much praise as 1 can heap upon him for keeping the
network up and running and the data contained with in it safe and available.
The quality of an institution and its faculty is reflected in the students they
attract. 1 have enjoyed interacting with the students of the Hydrologic Sciences
Academic Cluster, and this interaction has enhanced my efforts. Liyong Li, Yan
Zhang, and Xavier Foussereau shared the same computer lab trenches as I, and my
work cannot be separated from their influence.
Many rewarding research experiences have been associated with collaborations
made with Jim Jawitz. Jim is a colleague whose outlook and expertise is somewhat
different than my own, but it is precisely these differences that have made the col
laborations so rewarding. 1 shall seek to continue working with Jim through out my
career.
Dr. Andrew I. James provided access to computer codes he developed in the course
of his own research and technical assistance.
in
Dr. Kenneth Campbell deserves special recognition not only for his exemplary
service on my committee, but also for the example he has set as an engineer, a
scientist, and a professor.
I have appreciated Dr. Mike Annables wit and example through the years and
have enjoyed interacting with him in a wide variety of settings, whether in the class
room, on the football field, or elsewhere.
Dr. Rao has always challenged me with questions that require looking beyond a
simple formula and that evade a fillintheblank answer. Should I become scientist
of any repute, Dr. Rao would be the man to blame.
Dr. Kirk Hatfield is my source. He has been a source of wisdom, sympathy, advice,
and funding. I have greatly enjoyed working with Dr. Hatfield as both teaching and
research assistant.
Dr. Gia Destouni taught a few stochastic subsurface hydrology lectures in the
spring of 1994 that would forever change my thinking. She introduced the Lagrangian
approach to transport with which I have been obsessed ever since. Dr. Destouni
pursues fruitful collaboration vigorously, and this aggressive, yet collaborative, style
is one that I now seek to emulate.
Dr. Vladimir Cvetkovic has generously hosted three visits to the Kungl Tekniska
Hogskolan in Stockholm, Sweden. His considerable generosity flows from his intense
interest in stochastic Lagrangian transport.
Dr. Sten Berglund has been a friend and mentor through this arduous journey.
It is no exaggeration to say that without his help and encouragement this document
would not exist.
Dr. Wendy Graham deserves special recognition for her tireless faith and support.
She has allowed me a free reign to pursue whatever interests I chose. In hindsight, this
method coupled, with my own personality quirks and spectrum of interests, may not
IV
be the most efficient for producing degrees and papers, but it has produced several
interesting adventures.
George and Ellen Demmy, my parents, have provided love and all manner of
support throughout my life, academic and otherwise. They both instilled in me a
love of learning and an insatiable curiosity about all things that serve me well, and
will remain among my best traits.
Finally, Id like to extend my thanks and love to Celine Bufkin, for her love,
encouragement, support, and vision.
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS in
LIST OF TABLES ix
LIST OF FIGURES x
ABSTRACT xv
CHAPTERS
1 INTRODUCTION 1
1.1 Primary Contribution 1
1.2 Historical Context 1
1.3 Research Goals 3
1.3.1 Lagrangian Velocity Mean and Covariance 3
1.3.2 Injection Mode Analysis 4
1.3.3 Heterogeneous Reaction Parameter 4
1.3.4 Application to Reality 4
2 LAGRANGIANEULERIAN FLOW FIELD RELATIONSHIP ... 5
2.1 Introduction 5
2.2 Flow Field Partitioning Strategies 7
2.2.1 Simple Subdivision 8
2.2.2 General Subdivision 10
2.3 Descriptions of Fields 12
2.4 Numerical Experiments 13
2.4.1 The Log Conductivity Field 14
2.4.2 Velocity Field Generation 18
2.4.3 Trajectory Generation 19
2.4.4 Monte Carlo Simulation 19
2.4.5 Sampling Strategy 20
2.4.6 Simulation Results 22
2.4.7 Log Conductivity Statistics 23
2.4.8 Velocity Statistics 29
2.4.9 Head and Head Gradient 40
2.5 Theory 40
2.5.1 Effective Conductivity 40
vi
2.5.2 Eulerian Velocity Field 42
2.5.3 Streamtube Weights 48
2.5.4 Control Plane Oriented Velocity Moments 51
2.5.5 Control Time Oriented Velocity Moments 55
2.5.6 Log Conductivity Moments 59
2.6 Evaluation of Simulation and Theoretical Results 60
2.6.1 Mean Velocities in Space 60
2.6.2 Velocity Variances in Space 60
2.6.3 Mean Velocities in Time 61
2.6.4 Velocity Variances in Time 62
2.6.5 Log Conductivity Statistics 62
2.7 General Conclusions 63
3 NONREACTIVE SOLUTE TRANSPORT 65
3.1 Introduction 65
3.2 Injection Mode and Streamtube Selection 65
3.2.1 Uniform Resident Injection 65
3.2.2 Injection in Flux 67
3.3 Integrated Streamtube Properties 67
3.3.1 Displacements 68
3.3.2 Travel Times 69
3.4 Solute Transport 70
3.5 Evaluation of Expressions 71
3.5.1 Displacements 71
3.5.2 Travel Times 72
3.6 General Conclusions 75
4 REACTIVE SOLUTE TRANSPORT 79
4.1 Introduction 79
4.2 Temporal Moments 79
4.3 Conceptual Overview of Solute Transport 80
4.4 Simulation of Reactive Solute Transport 83
4.4.1 Reaction Parameter 84
4.4.2 Reaction Flow Path 85
4.4.3 Reaction Flow PathTravel Time Cross Moment ... 93
4.5 Theory of Reactive Solute Transport 93
4.5.1 Mean Reaction Flow Path 96
4.5.2 Reaction Flow Path Variance 97
4.6 Discussion 98
4.7 Spatial Moments 99
4.7.1 Injection Mode 105
4.7.2 Discussion 106
vii
5 APPLICATION OF CONCEPTS 108
5.1 Introduction 108
5.2 Theory 108
5.2.1 Spatial Moments 109
5.2.2 Relative Degradation 110
5.3 Application: Simulation Ill
5.3.1 Results 113
5.3.2 Discussion 113
5.4 Application: Field Data 115
5.4.1 Results and Discussion: Center of Mass 116
5.4.2 Results and Discussion: Degradation 117
5.5 General Conclusions 117
6 GENERAL CONCLUSIONS 119
APPENDICES
A EULERIAN VELOCITY DISTRIBUTION PROPERTIES 125
B VELOCITY LOG CONDUCTIVITY CORRELATION 127
C CONSISTENT FIRST ORDER APPROXIMATION 131
C.l Travel Time 131
C.2 Displacements 133
REFERENCES 134
BIOGRAPHICAL SKETCH 138
viii
LIST OF TABLES
Table page
2.1 Comparison of target log conductivity values and observed values along
areaweighted Eulerian trajectories 26
2.2 Comparison of target log conductivity values for areaweighted Eulerian
trajectories in space and observed areaweighted Lagrangian trajec
tories in time 29
2.3 Comparison of a priori Eulerian velocity variance to observed values.
The decimal representation of the exact fractional values of the pre
dicted variance are provided for convenience 35
2.4 Comparison of observed log hydraulic conductivity variances to estima
tors. Observed value is zeroslope line regressed through data. The
decimal representation of the fractional value is included for conve
nience 63
5.1 The center of mass in meters of steady plumes estimated by traditional
trapezoidal integration of spatial concentration data and by the mean
v/k estimator Equation (5.7) 116
5.2 Degradation rates relative to naphthalene and absolute degradation
rates in reciprocal days as estimated by Equation (5.9) and as re
ported by MacIntyre et al. [1993]. NR denotes results not reported
in MacIntyre et al. [1993]. Naphthalene degradation reported to be
0.0064 d1 117
IX
LIST OF FIGURES
Figure page
2.1 Schematic representation of an aquifer 6
2.2 Simple subdivision of the flow field by one of four strategies: A) equal
area straight line, B) equalarea streamline, C) equalflow streamline,
D) equalflow straight line 9
2.3 Four different partitioning strategies for the same flow field: A) area
weight Eulerian trajectory, B) area weight Lagrangian trajectory,
C) flow weight Lagrangian trajectory, D) flow weight Eulerian
trajectory 11
2.4 Log conductivity distribution from a 41400 node realization generated
by the turningbands method compared to the standard normal dis
tribution. Target distribution parameters are fi\nk = 0 and ofnk = 1. 16
2.5 Estimated log conductivity longitudinal covariance estimated from a
realization generated by the turningbands method compared to an
exponential covariance function. Variance of log conductivity =
1 and correlation length X\nk = 1. Spacing for log conductivity values
was Ainfc/5 17
2.6 Mean log conductivity as function of displacement in mean flow direc
tion for the four different trajectory collections and ofnfc = 1/2 (a)
and lk = 1 (b) 24
2.7 Log conductivity variance as function of displacement in mean flow
direction for the four different trajectory collections and = 1/2
(a) and
Eulerian log conductivity field ofnfc 25
2.8 Mean log conductivity as function of travel time to control planes along
mean flow direction for the four different trajectory collections and
\nk = V2 (a) and afnk = 1 (b) 27
2.9 Log conductivity variance as function of travel time to control planes
along mean flow direction for the four different trajectory collections
and ofnA. = 1/2 (a) and a?nk = 1 (b). Variances are normalized by
target variance of Eulerian log conductivity field cr?nk 28
x
2.10 Comparison of positiondependent mean velocity estimators to data for
the four collections of trajectories for a2nk = 1/2 (a) and a2nk = 1
(b). Velocities are normalized by U 30
2.11 Comparison of the positiondependent velocity variance estimators to
data for the four collections of trajectories for elk =1/2 (a) and
a2nk = 1 (b). The variances are normalized by U2ba2nk 31
2.12 Comparison of spatial Eulerian (AW/ET) and Lagrangian trajectory
(FW/LT) velocity covariance data to covariance model. Data are
taken from the a2nk = 1 set of realizations. Covariances presented
as dimensionless lag correlation functions 34
2.13 Comparison of timedependent mean velocity estimators to data for
the four collections of trajectories for a2nk = 1/2 (a) and a2nk = 1
(b). Velocities are normalized by U 36
2.14 Comparison of timedependent velocity variance estimators to data for
the four collections of trajectories for a2nk = 1/2 (a) and a2nk = 1
(b). The variances are normalized by U2ba2nk 37
2.15 Comparison of temporal Lagrangian trajectory (AW/LT) velocity co
variance data to covariance model. Data are taken from the a2nk = 1
set of realizations. Covariances presented as dimensionless lag cor
relation functions 38
2.16 Normalized log initial velocities for twodimensional model aquifer com
pared to the standard normal distribution. The random variable was
normalized by the sample mean and standard deviation to illustrate
the lognormality 39
2.17 Comparison of velocity covariance expression Equation (2.25) to the
approximation Equation (2.13) for a2nk = 1. Covariances are pre
sented as dimensionless correlation functions 45
2.18 Components of the velocity covariance function 46
3.1 Comparison of mean displacements for the two injection modes or
streamtube collections. The flowweighted Lagrangian trajectory
(FW/LT) statistics correspond to a flux injection, and the area
weighed Lagrangian trajectory (AW/LT) statistics correspond to a
uniform resident injection. The top figure contains the a2nk = 1/2
data and the bottom figure contains the <72nfc = 1 data 73
xi
3.2 Comparison of displacement variances for the two injection modes or
streamtube collections. The flowweighted Lagrangian trajectory
(FW/LT) statistics correspond to a flux injection, and the area
weighed Lagrangian trajectory (AW/LT) statistics correspond to a
uniform resident injection. The top figure contains the a?nk = 1/2
data and the bottom figure contains the ofnfc = 1 data. Variances
are normalized by A2n kk 74
3.3 Comparison of mean arrival times for the two injection modes or stream
tube collections. The flowweighted Lagrangian trajectory (FW/LT)
statistics correspond to a flux injection, and the areaweighed La
grangian trajectory (AW/LT) statistics correspond to a uniform res
ident injection. The top figure contains the ofnA. = 1/2 data and the
bottom figure contains the ofnfc = 1 data 76
3.4 Comparison of arrival time variances for the two injection modes or
streamtube collections. The flowweighted Lagrangian trajectory
(FW/LT) statistics correspond to a flux injection, and the area
weighed Lagrangian trajectory (AW/LT) statistics correspond to a
uniform resident injection. The top figure contains the a?nk = 1/2
data and the bottom figure contains the ofnfc = 1 data. Variances
are normalized by (U/(\ink&ink))2 77
3.5 Travel time estimate error as function of displacement associated with
assuming a uniform resident injection when injection is in flux for
4.1 Mean reaction parameter values for different values of correlation pa
rameter (3 and different trajectory collections for the ofnfc = 1 set
of simulations. Top figure is (3 {1,+1}. Middle figure is Â¡3 =
{1/2,+1/2}. Bottom figure is (3 0 (no correlation). Trajec
tory collections are areaweighted Lagrangian (AL), flowweighted
Lagrangian (AF) and areaweighted Eulerian (AE) 86
4.2 Mean reaction parameter values for different values of correlation pa
rameter Â¡3 and different trajectory collections for the ofnfc = 1/2
set of simulations. Top figure is Â¡3 = {1,+1}. Middle figure is
f3 = { 1/2,+1/2}. Bottom figure is Â¡3 0 (no correlation). Trajec
tory collections are areaweighted Lagrangian (AL), flowweighted
Lagrangian (AF) and areaweighted Eulerian (AE) 87
4.3 Mean reaction path observed values and estimators for different values
of correlation parameter Â¡3 and different trajectory collections for
the cr2n k 1 set of simulations. Top figure is Â¡3 = {1, +1}. Middle
figure is (3 = {1/2, +1/2}. Bottom figure is Â¡3 = 0 (no correlation).
Trajectory collections are areaweighted Lagrangian (AL) and flow
weighted Lagrangian (AF)
xii
89
4.4 Mean reaction path observed values and estimators for different values
of correlation parameter /3 and different trajectory collections for the
a?nk = 1/2 set of simulations. Top figure is f3 = {1,+1}. Middle
figure is Â£ {1/2, +1/2}. Bottom figure is Â¡3 = 0 (no correlation).
Trajectory collections are areaweighted Lagrangian (AL) and flow
weighted Lagrangian (AF) 90
4.5 Reaction path variance observed values and estimators for different
values of correlation parameter Â¡3 and different trajectory collections
for the ofale = 1 set of simulations. Top figure is /3 = {1,+1}.
Bottom figure is /3 (1/2, +1/2}. Trajectory collections are area
weighted Lagrangian (AL) and flowweighted Lagrangian (AF). . 91
4.6 Reaction path variance values and estimators for different values of
correlation parameter /3 and different trajectory collections for the
ofnk = 1/2 set of simulations. Top figure is Â¡3 = { 1,+1}. Bottom
figure is 13 = {1/2, +1/2}. Trajectory collections are areaweighted
Lagrangian (AL) and flowweighted Lagrangian (AF) 92
4.7 Correlation between the reaction flow path and travel time for different
values of correlation parameter Â£ and different trajectory collections
for the cr,2nfe = 1 set of simulations. Top figure is /3 = { 1,+1}.
Bottom figure is (3 = {1/2,+1/2}. Trajectory collections are area
weighted Lagrangian (AL) and flowweighted Lagrangian (AF). . 94
4.8 Correlation between the reaction flow path and travel time for different
values of correlation parameter (3 and different trajectory collections
for the a?nk = 1/2 set of simulations. Top figure is Â¡3 = { 1, +1}
Bottom figure is Â¡3 = { 1/2, +1/2}. Trajectory collections are area
weighted Lagrangian (AL) and flowweighted Lagrangian (AF). . 95
4.9 Assumed timedependent log conductivity covariance function along
Lagrangian trajectories (Equation (4.29)) compared to simulation
data for variances of log conductivity afnfc = {1/2,1}. The La
grangian velocity covariance function is Equation (2.61) 103
4.10 Hypothesized timedependent Eulerian log conductivity covariance func
tions for \nk = {1/2,1} compared to simulation data and Equa
tion (4.29) 104
4.11 Mean reaction flow path as a function of nonreactive travel time com
pared to estimator Equation (4.25) for correlation parameter values
Â£ = {1,0,1} 104
4.12 Reaction flow path variance a function of nonreactive travel time for
correlation parameter values Â£ = { 1,0,1}
xiii
105
4.13 Mean reaction flow path as a function of nonreactive travel time com
pared to estimator Equation (4.34) for correlation parameter values
(3 {1/2,0,1/2} (top figure) and /3 = {1,0,1} (bottom figure).
5.1 Resident concentration profile averaged along a transect orthogonal to
the mean flow direction for plumes for a^nk = {0.0,0.1,0.5,1.0}. . 114
B.l Comparison of velocity and log conductivity correlation functions from
Graham and McLaughlin [1989b] and Rubin and Dagan [1992] and
a simplified exponential model. All functions are normalized by
KgJafnk/9. The apparent roughness of the Rubin and Dagan
[1992] function is due to estimation errors associated with inferring
their function from graphical data 128
B.2 Comparison of the simplified velocitylog conductivity correlation func
tion to simulation data for aj {1/2,1} 129
xiv
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SOLUTE TRANSPORT IN HETEROGENEOUS POROUS FORMATIONS
By
George Gary Demmy, Jr.
December 1999
Chair: Wendy Graham
Major Department: Agricultural and Biological Engineering
This work quantifies relationships between the spatial, or Eulerian, distribution of
the properties of a chemically and physically heterogeneous porous medium and those
as observed along the natural, or Lagrangian, trajectories that a fluid particle traces
in a steady and irrotational flow. From these relationships, expressions that relate
the transport of solutes through the porous medium along the natural trajectories
to the aforementioned Eulerian distributions are developed. The effects of injection
mode upon global measures of transport as reflected by the temporal moments of
breakthrough curves and spatial moments of a solute plume are developed. The
coupled effects of correlation of a linear equilibrium sorption to the underlying log
hydraulic conductivity field and injection mode on the evolving temporal moments
of mass breakthrough curve and the coupled effects of correlation of a firstorder
decay coefficient and injection mode upon the spatial moments of a solute plume are
examined.
xv
CHAPTER 1
INTRODUCTION
1.1 Primary Contribution
The primary contribution of this work is documenting a consistent approach to
the development of estimators of solute transport in the Lagrangian framework. This
consistent approach is to identify the collection of Lagrangian trajectories associated
with a stationary Eulerian velocity field for which the Lagrangian velocities are also
stationary. This is a modest extension of work pioneered by myriad hydrologists.
The benefit derived from this contribution is enhanced understanding of transport of
solutes in heterogeneous velocity fields.
1.2 Historical Context
A grant to study the impact of heterogeneous source morphology upon subse
quent solute transport initiated this research. In seeking a suitable framework and
toolset with which to attack this problem, there appeared what seemed to be funda
mental inconsistencies in the literature. Given the peculiar tendencies of the author,
the larger issues could not be addressed until these apparent inconsistencies were
resolved.
Gedeon Dagan developed a robust theory of solute transport using a stochastic
Lagrangian framework [Dagan, 1982a, b]. In fact, an entire school of stochastic
subsurface hydrology sprung from his work.
Allen Shapiro and Vladimir Cvetkovic wrote Stochastic analysis of solute arrival
time in heterogeneous porous media in 1988 [Shapiro and Cvetkovic, 1988]. The au
thors made an explicit assumption often implicitly made in the Lagrangian transport
literature, namely that a fluid parcel deviates little from its mean trajectory in weakly
1
2
heterogeneous conductivity fields, and that the Lagrangian and Eulerian velocities are
approximately equal. They consistently applied this concept, developing expressions
for arrival time means and variances for nonreactive solutes in the Standard Model
Aquifer, a hypothetical aquifer characterized by a heterogeneous conductivity field
with a prescribed correlation structure and subject to certain prescribed boundary
conditions (see Section 2.1). These equations predicted that the mean arrival time
for solute subject to a uniform resident injection is given by
= x/Vh (1.1)
where angle brackets <> denote an ensemble average operator and Vj, is the harmonic
mean Lagrangian velocity. From the small deviation assumption, the Lagrangian
and Eulerian velocities are approximately the same, thus the estimated harmonic
mean Lagrangian velocity is equal to the estimated harmonic mean Eulerian velocity.
However, the mean arrival time for uniform resident injection of solute after travelling
several integral scales is given by
x/U (1.2)
where U is the arithmetic mean Eulerian velocity. This expression is also the con
sistent firstorder approximation (CFOA) of the travel time. Widespread adoption
of the CFOA through out the stochasticLagrangian transport literature probably
stems Feynmans requirement: we must reproduce what we already know. The har
monic mean of a positive definite process is always less than the arithmetic mean, so
Equation (1.1) systematically overpredicts travel times for large displacements. Dis
turbing was that the estimators derived by Shapiro and Cvetkovic [1988] using clear,
consistent, intuitive and logical methods seemed to violate Feynmans rule, especially
in light of the success enjoyed by the Dagan school.
3
This harmonic/arithmetic mean discrepancy was explicitly noted by Dagan et al.
[1992], who relegated the Shapiro and Cvetkovic [1988] result to applicability in areas
close enough to the input zone [Dagan et al, 1992 p. 1374].
Cvetkovic et al. [1996] developed a semianalytical expression that described the
transition from the near field harmonic mean velocity to the far field expression
based upon the these known endpoints and an estimate of the Lagrangian velocity
integral scale. Their detailed analysis of the Lagrangian velocity field revealed that
the Lagrangian velocity is nonstationary in displacements in space, and resulted in a
nonlinear propagation of the mean arrival time with distance.
This was strange. Why was the Lagrangian velocity field nonstationary when
the Eulerian was stationary? It was commonly assumed that the stationarity of one
implied the stationarity of the other. What had started as a simple preliminary
literature review resulted in a quandry. The paths followed in pursuit of this quandry
led to the research goals of the dissertation.
1.3 Research Goals
There were four broad objectives specified for this work. All centered around the
flow of water and the transport of solutes in aquifers characterized by heterogeneous
hydraulic conductivity fields.
1.3.1 Lagrangian Velocity Mean and Covariance
A primary goal of this research was to develop a Lagrangian covariance function
for the Standard Model Aquifer. This was extended slightly to the mean and
covariance of the spacestationary velocities along equal flowweight streamlines and
timestationary velocities along equal areaweight streamlines. Quantification of this
covariance function greatly simplifies the development of equations, or estimators,
that describe the movement of water and solutes in the heterogeneous velocity fields
associated with the Standard Model Aquifer.
4
1.3.2 Injection Mode Analysis
The stochasticLagrangian framework is convenient for the analysis of the effects
of the method by which solute is introduced into the flow field, or injection mode,
upon the subeqent transport. These effects of injection mode upon nonreactive solute
transport are characterized by the mean and variance of a breakthrough curve, or
mass arrival distribution, and by the mean and variance of mass displacement. A
goal of this research was to develop estimators for these quantities.
1.3.3 Heterogeneous Reaction Parameter
Set as a goal were estimators for the mean and variance of a heterogeneous reac
tion parameter integrated along trajectories of stationary velocty. These estimators
were used to evaluate the coupled effects of injection mode and a heterogeneous linear
equilbirium sorption process in a stochasticLagrangian framework.
1.3.4 Application to Reality
A further goal was to develop a traditional estimator for the center of mass
of a continuously injected solute plume subject to firstorder decay and to evaluate
data from a natural attenuation experiment and numerical experiments. Estimator
performance was evaluated using the results of the prevous developments.
CHAPTER 2
LAGRANGIANEULERIAN FLOW FIELD RELATIONSHIP
Develop an intuitive judgment and understanding for everything.
Miyamoto Musashi1
2.1 Introduction
Consider the steady, welldeveloped flow of water in an aquifer meeting Bears
[1972] definition of a porous medium. In most natural settings, gentle gradients
induce a flow wellquantified by Darcys law. Knowledge of the water fluid properties,
relative permeability of porous medium, and potential gradients in space and time
would assure a reasonable estimate corresponding water fluxes.2
One way to record this knowledge would be to create maps of these different
properties in space at different times. This framework is commonly called the Eulerian
framework, and is the most common approach to hydrologic problems. Another way
to record this knowledge is to map properties along moving coordinates in time.
This approach is commonly called the Lagrangian framework, and its application
may without controversy be described as less widespread than Eulerian approaches.
Dagan [1989] and Gelhar [1993] review development of flow of fluids and transport of
contaminants in porous media in both of these frameworks.
1 Musashi [1982 p. 49]
2 The concepts associated with the following discussion are quite general, and are
applicable to a much larger variety of environmental transport problems. The em
phasis upon groundwater follows from an interest in simplicity.
5
6
Consider now flow of water in a more specific model aquifer: the steady, irrota
tional, and Darcian flow of water of homogeneous fluid properties in a finite rectangu
lar twodimensional aquifer of heterogeneous hydraulic conductivity (see Figure 2.1).
Mobile water 9 completely saturates the homogeneous pore space. An orthogonal
Figure 2.1: Schematic representation of an aquifer.
Cartesian coordinate system lies parallel to the aquifer boundaries, and x and y de
note its directions. Parallel and impervious boundaries lie at y = 0 and y = Ly.
Constant potential boundaries located at x = 0 and x = Lx maintain a constant av
erage potential gradient J across the domain, such that water flows into the aquifer at
x = 0 and out of the aquifer at x = Lx at a volumetric discharge rate Q (dimensions
[.L3T~1]). For an intuitive dimensional consistency, allow the aquifer some small and
7
uniform thickness in the vertical direction, say h. The total area of the vertical plane
at x = 0 through which water enters the aquifer is A. Define the spatiallyaveraged
groundwater seepage velocity at this inlet plane (IP) as
jj = 14 (Â¡[a\da
Lda (2.1)
Q_
6A
For a large class of hydrologic applications, water quantity is the sole concern.
For these problems, the route taken by the water is of little interest, as is, say
the distribution of ages or residence times. Spatial and temporal distributions of
hydraulic properties hold some interest, but only in how they relate to effective bulk
properties.
However, a large class of practical hydrologic problems exist for which the trajec
tories of water parcels across the flow domain is of great importance. Many of these
problems involve the fate and transport of contaminants through the aquifer system.
The fundamental problem is this: to relate an Eulerian flow field to its Lagrangian,
or trajectorybased, equivalent.
2.2 Flow Field Partitioning Strategies
A streamline is a hydrodynamic entity that is at all points tangent to the ve
locity vector. For twodimensional flow, a streamtube is an object defined by two
streamlines. Stream surfaces, the intersection of which form streamlines, define three
dimensional streamtubes. Infinite collections of streamlines exist that subdivide the
flow field into collections of streamtubes. Again, the interest of clarity suggests a focus
upon twodimensional flow. For a lucid discussion of streamlines, stream functions,
and stream surfaces in two and threedimensional flows, see Bear [1972],
Impervious boundaries are streamlines, thus the entire model domain is a stream
tube by definition. Viewed as a black box, Eulerian and Lagrangian distinctions
have little meaning. This black box approach is the purview of reactor engineering,
8
an important and welldeveloped discipline, from which a great deal of insight and
useful mathematics may be drawn. For an excellent review, especially with respect
to concepts related to transfer functions, see Jury and Roth [1990].
For a large domain, the scale at which chemical reactions takes place is generally
much smaller than that of the domain. That is, processes such as sorption, microbial
and chemical degradation, and the like, are dependent upon chemical potentials across
distances on the order of the pore scale to perhaps that of a microDarcy scale,
defined as the smallest volume for which Darcys law is generally applicable. However,
the scale of interest tends to be much larger, say that of a well, or its capture zone
or radius of influence, or perhaps the interface across which groundwater discharges
into a surface water body, or, perhaps the Floridan aquifer. In order to consider
these reactions in detail, it is necessary to resolve the larger domain into smaller
subdomains. For modeling real systems, the scale and degree to which the domain
is subdivided should be based upon the available data.
2.2.1 Simple Subdivision
Consider the water flowing into the model aquifer across the inlet, or injection,
plane (IP). Assume it is of interest to divide the domain into two subdomains. There
are many ways to do this. One would be to divide the domain down the middle, that
is, at a line parallel to the x axis located y = Ly/2 (curve A Figure 2.2). This division
creates two subdomains of equal volume and gives an equal volume weight to each
subdomain. The division also creates two subIPs of equal area and gives an equal
area weight to each subdomain. For a perfectly homogeneous porous medium, the
groundwater velocity vectors will be parallel to this line, and the discharges through
each subdomain will be the same as well. Moreover, there will be no advective
transfer of water across the boundary. However, the introduction of conductivity
heterogeneity in the transverse direction will almost certainly lead to deviations of
the streamlines, and to streamlines crossing the boundary. This implies that while
9
Figure 2.2: Simple subdivision of the flow field by one of four strategies: A) equal
area straight line, B) equalarea streamline, C) equalflow streamline, D) equalflow
straight line.
the volume of water contained in each subdomain is constant, the discharge through
a plane perpendicular to the partition is a function of plane position in x.
Consider now a streamline originating at (0,Ly/2) (curve B on Figure 2.2). For
a homogeneous medium, this corresponds to an equal volume partitioning. That is,
the domain centerline is a streamline. However, transverse heterogeneity would likely
induce changes in the velocity field such that the streamline would not remain on
the centerline. For general transverse heterogeneity, the flow rate entering the two
domains will not be equal, in general. Thus, the flow weight and the volume weight
of the two subdomains are unequal in this case. Again, the magnitude of the areas
through which water enters the subdomains are equal.
Consider now a streamline originating at some coordinate (0, j/0) such that the vol
umetric flow rate is equal for both domains (curve C on Figure 2.2). This streamline
corresponds to the stream function
^[0,2/0]
m Ly] *[0,0]
2
(2.2)
10
This is an equal partitioning with respect to the flow field, or the stream function.
This sort of partitioning arises quite naturally in fluid mechanics and in numerical
modeling of heterogeneous flow systems using streamtube approaches (see, e.g., Theile
et al. [1996]). The flow weight of each subdomain is equal, although the volume
weights are different.
A fourth scheme would be to extend a line from (0,yo) to (Lx,y0) (curve D Fig
ure 2.2). In this case, the subdomains share only equal flow rates at x 0. In
subsequent discussions, the phrase Eulerian trajectory will refer to such a straight
line trajectory, and the phrase Lagrangian trajectory will refer to trajectory of a
streamline. The phrase area weight will refer to an equal area partitioning scheme
at a definition plane, and the phrase flow weight will refer to an equal discharge par
titioning scheme. At this point, it is important to note that no mention has been
made of boundary conditions as they relate to transport equations. These partition
ing schemes are simply ways of subdividing a flow field. An area weight Eulerian
trajectory scheme corresponds to traditional spectral approaches that assume small
deviations about a mean trajectory. The use of trajectory is quite general, and does
not imply a natural path or streamline.
2.2.2 General Subdivision
Consider dividing the flow field into N partitions according to the strategies de
scribed (see Figure 2.3). Theile et al. [1996] demonstrated that streamlines may
represent streamtubes. Therefore, the remainder of this discussion will focus upon
streamline properties, with the implication that these properties may be extended to
streamtubes as well. A primary purpose of subdividing the domain is to represent the
continuum of domain properties with a discrete set of data. A regular finitedifference
grid may be thought of as a set of points at equal spacing in space along a collection
of areaweighted Eulerian trajectories. An alternative method would be to create
spacings based upon travel times along the trajectory, where travel time is defined as
11
Figure 2.3: Four different partitioning strategies for the same flow field: A) area
weight Eulerian trajectory, B) area weight Lagrangian trajectory, C) flow weight
 Lagrangian trajectory, D) flow weight Eulerian trajectory.
12
the path integral of the inverse velocity over the trajectory
rV\=f ^
Jo
(2.3)
o V[Cl S
where s is a unit vector that is at all points the parallel to the trajectory. It is clear
that this sort of Lagrangian description has little meaning in a static system, and is
in this sense less general than Eulerian descriptions. Thus, Lagrangian description
is a tool reserved for dynamic systems and is a supplement to Eulerian descriptions.
It is worthy to note the distinction between a description, a general approach, and a
trajectory, a specific entity used in a description.
To summarize, a primary objective is to approximate a steady and irrotational
flow field continuum with a discrete set of trajectories. The trajectories follow either
a straight line (Eulerian trajectory) or a streamline (Lagrangian trajectory). The
trajectories carry either an equal flowweight or an equal injection areaweight.
2.3 Descriptions of Fields
Consider once again the model aquifer depicted in Figure 2.1. As mentioned in the
introduction of this chapter, the aquifer has certain features in which a hydrologist
might be interested, e.g., the intrinsic permeability or the porosity. These features
may be thought of as a field in which parameters vary continuously. This is a funda
mental concept of continuum mechanics and is discussed in a hydrologic context by
Bear [1972], Dagan [1989], and Gelhar [1993]. The ultimate description of any such
field would be a map of parameter values valid at any point, time, or scale. In lieu of
this ultimate description, stochastic hydrologists characterize spatial heterogeneity
with conditional probability density functions that reproduce the dominant charac
teristics of the field and observed values at given locations. The entire subject of
geostatistics is devoted to this end (see Journel and Huijbregts [1978]). An Eulerian
field refers to areaweighted Euleriantrajectory oriented field. A Lagrangian field
refers to a flowweighted Lagrangiantrajectory oriented field.
13
It is instructive to illustrate these concepts with an example and to interpret the
results of this illustration.
2.4 Numerical Experiments
The illustration of these concepts requires a heterogeneous flow field and col
lections of trajectories, both Eulerian and Lagrangian, that correspond to this flow
field. The requirement that the flow field be based upon reference fields of known
properties enhances the possibility of theoretical interpretation drawn from the large
body of literature concerning flow and solute transport in idealized porous media. A
further requirement that our reference fields bear some resemblance to what might
be expected in nature enhances the possibility of useful applicability.
A numerical simulation code modeled steady groundwater flow in a hypothetical
heterogeneous twodimensional aquifer. A particle tracking code traced Eulerian and
Lagrangian trajectories across this aquifer, recording data that included the local
groundwater velocity and local hydraulic conductivity at intervals in space and time
along the different trajectories. Statistics, specifically mean, variance, and covariance,
summarize the results of these measurements.
The conceptual view of the aquifer is that given by Figure 2.1. To generalize the
system, correlation length, or integral scale, of the log conductivity field serves
as a characteristic length with which to normalize length scales. The characteristic
time Aink/U serves to normalize time scales, where U is the arithmetic mean velocity
defined in Equation (2.1). The definition of an integral scale is the integral from zero
to infinity of the correlation function R, or,
roo
A = / f?[s]ds
Jo
(2.4)
14
The design criteria for the model aquifer are as follows
In A: correlation length Aifc = 1
effective conductivity Ke = 1
average gradient J = 1/50
mobile water content 6 = 1/5
From these criteria and Darcys law, the areaaveraged velocity is U = KeJÂ¡9 = 1/10.
Clearly, the parameter values are artificial and selected for convenience, but values
for Ke in md~l and 6 fall in ranges typical for sand [Domenico and Schwartz, 1990].
Two 5000 replicate Monte Carlo simulations of aquifers with log conductivity
variances of 1/2 and 1, respectively, comprised the suite of aquifer simulations. The
following sections describe the experimental design and the results.
2.4.1 The Log Conductivity Field
The foundation upon which to build the illustration is a heterogenous hydraulic
conductivity field. A commonlyused stationary lognormally distributed and expo
nentially correlated isotropic model serves as the Eulerian hydraulic conductivity
reference field (see e.g., Freeze [1975] or Gelhar and Axness [1983]). Working from a
hypothesized conductivity field is preferable to generating a hypothetical velocity field
with specified characteristics (e.g., Beilin et al. [1994]) because an objective of this
work is to describe the underlying conductivity field along Lagrangian trajectories
and relate its Lagrangian description to its Eulerian properties.
For completeness, it is pertinent to briefly review description of a heterogeneous
field by treating the process as being random Measured values of hydraulic con
ductivity of virtually any natural porous formation will vary from place to place.
In the absence of measurement error, this variability in the measured values reflects
the heterogeneity of myriad physical factors and processes that control the hydraulic
conductivity. As mentioned, measurements of hydraulic conductivity in the field of
ten follow a lognormal distribution. In these cases, it is useful to conceive of the
15
conductivity field as a realization of a lognormal random process. The process is
said to be stationary if the ensemble mean and variance do not vary in space. The
process is nonstationary if some trend exists in the mean or variance. In practice,
only single realization is available, and the ensemble properties are ultimately un
knowable. However, if the field is large compared to the scale of variability, one may
assume ergodicity, or, that the ensemble parameters may be estimated from a sample
drawn from the single realization since there exists a the possibility of a large num
ber of statisticallyindependent observation points. For a more detailed development
of these concepts from the hydrologists perspective, see Gelhar [1993]. For a more
formal development of these concepts, see Papoulis [1991].
The model of hydraulic conductivity variability is a stationary, exponentially
correlated isotropic lognormal distribution. A lognormal random variable is one for
which its natural logarithm is normally distributed. As such, three parameters sum
marize the statistical properties of the adopted conductivity model, namely the mean
of log k /iinjt, the variance of log k ofnfc, and the correlation length of log k \\nk The
covariance model is given by
c,*[r] = ffL. tl
(2.5)
where r is the separation between two points in the log conductivity field. Use of
this simple covariance model requires stationarity of variance, and usually implies a
stationarity of the mean as well. The moments of the real space conductivity field
are given by
= exp[ii(iit + n2ffft/2]
(2.6)
where n is the order of the moment.
In the absence of a perfectly described physical aquifer, the method of turning
bands generated equallylikely realizations of a hypothetical conductivity field with
16
prescribed ensemble statistics [Tompson et al., 1989]. Turningbands was capable
of producing two and threedimensional fields with the desired statistical proper
ties and was available as a portable and easily modified code. Figure 2.4 illustrates
the lognormality of the a typical twodimensional log conductivity field realization.
Figure 2.5 compares an estimate of the longitudinal covariance to the hypothesized
model. In order to capture the features of the log conductivity spatial variability, the
log conductivity
Figure 2.4: Log conductivity distribution from a 41400 node realization generated
by the turningbands method compared to the standard normal distribution. Target
distribution parameters are /in* = 0 and ofnfc = 1.
turningbands algorithm generated five log conductivity observations per correlation
length. This level of discretization falls in the range of other similar sets of simula
tions. Cvetkovic et al. [1998] used a discretization of 4 nodes per Ain for simulations
of fields with
per Aink for simulations of fields with ofnJfc as large as 1.6.
17
1
0.8
o
c
to
(0
>
o
0.4
0.2
0
0 12 3
T / ^ln k
Figure 2.5: Estimated log conductivity longitudinal covariance estimated from a re
alization generated by the turningbands method compared to an exponential covari
ance function. Variance of log conductivity ofnfc = 1 and correlation length X\nk = 1.
Spacing for log conductivity values was Ainfc/5.
Trajectories covered several characteristic lengths A]nfc and several characteristic
times Ai k/U to capture both nearfield and farfield behavior in space and Lagrangian
time. The first requirement was easy to enforce by making the domain size equal to or
greater than the required number of correlation lengths. The second requirement was
somewhat more problematic, since the distance traveled in some number of X\nk/U
varies from trajectory to trajectory. Preliminary simulations indicated that a domain
40 Ain A: long was adequate to capture travel up to 10 characteristic time scales Ainjt/U
in 2000 trajectory realizations generated in the most heterogeneous field considered
(ofnfc = !) Thus, the selected domain extent was 40A[nfc.
The conceptual model aquifer is a large, bounded, primarily twodimensional,
steady flow field. As such, it should exhibit a variety of head, head gradient, and
velocity nonstationarities due to the boundary conditions (see Figure 2.1). While
estimated In k covariance
exponential covariance
18
researchers such as Rubin and Dagan [1988],[1989], and Osnes [1998] have investi
gated such nonstationarities, the additional complication of a rigorous consideration
of these sorts of nonstationarities would unduly obfuscate the very simple concepts
illustrated here. To minimize the effect of these nonstationarities, solute transport
is constrained to take place some distance from the boundaries of the numerical do
main. Following the example of Beilin et al. [1992], a 3Ainjt buffer was established
on both sides of the required 40 for a total longitudinal domain length of 46Ain fc.
Preliminary simulations indicated that the maximum transverse displacement of a
streamline in a <4 k = 1 field was around one third the longitudinal displacement.
For a streamline originating on the aquifer centerline parallel to the x axis, an aquifer
width of 36Alnjt was deemed adequate to contain large transverse displacements over a
40Alnfc longitudinal displacement and provide a similar buffer to mitigate boundary
effects.
Combining the physical domain requirements with the discretization required
to capture the log conductivity variability resulted in a numerical grid 230 by 180
for a twodimensional aquifer and a numerical domain with 41400 nodes. A three
dimensional analysis under the same assumptions would have required a numerical
domain of more than 7 million nodes. Simulations of this magnitude were untenable
in light of the available computational resources.
2.4.2 Velocity Field Generation
A mixed finiteelement scheme solved coupled velocity and pressure equations de
rived from Darcys law and the continuity equation. The code took input parameters
that matched the design criteria and the log conductivity field generated by turning
bands. Andrew I. James wrote and tested the core of the code. James et al. [1997]
used a derivative version of this code to generate flow fields for synthetic tracer tests.
A mixed finiteelement solution to the flow field was specified, following the results
and arguments of Mos et al. [1994]. The mixed finiteelement scheme is known
19
to provide an accurate velocity field solution for groundwater flow in heterogeneous
conductivity fields, one that is more accurate than other, more traditional, schemes
[Mos et al., 1994].
2.4.3 Trajectory Generation
An objective of this research was to record data, such as local velocity and and
conductivity, along different collections of trajectories, both Eulerian and Lagrangian.
These trajectories were traced using a particletracking scheme based upon a semi
analytical method presented by Pollock [1988]. This method assumes that velocity
varies linearly over an element, and that the velocity in some direction, say i, is
independent of displacements in orthogonal directions. From these assumptions, it is
a relatively simple exercise to determine the displacement of a particle in a linear
velocity field for a given time, or the time required to make a certain displacement.
A computer code recorded particle position, time, total path length along trajectory,
velocity, local conductivity, and a value from another random field with the same
statistical properties as, but uncorrelated to, the log conductivity field at several
control planes equallyspaced along the mean flow direction and at several control
times after injection along both trajectories. This uncorrelated random variable was
recorded for reactive transport simulations, to be discussed in subseqent chapters.
2.4.4 Monte Carlo Simulation
The possibility of theoretical interpretation of the results is enhanced by requiring
nearergodic samples from the population. That is, the samples must reproduce the
population statistics. To do this, the number of statistically independent replicates
drawn from the velocity and conductivity fields must be quite large. Rather than
attempt a massive singlerealization, the following method generates a large number
of statistically independent realizations
20
generate a large, twodimensional conductivity field with the desired statistical
properties
generate a flow field by imposing a constant mean gradient across the saturated
conductivity field
trace the trajectories of two particles injected at a common control location,
one trajectory is a straight line or Eulerian trajectory and the other a streamline
or Lagrangian trajectory, and record properties at equal lags in space and in
time along trajectories that pass through inner domains that are relatively
undisturbed by nonstationarities induced by the boundaries
At this juncture, it is perhaps worth mentioning a minor philosophical point.
These simulations do not represent a real aquifer, per se, but rather a hypothetical
aquifer with known properties. The goal is understand the convolution of the physics
of the groundwater flow and a known heterogeneity that has analogs in nature, and it
is hoped that the reader finds some merit to this modest goal. The 41400 conductivity
values needed for any single realization do not represent 41400 measurements taken
from some real aquifer. The Monte Carlo aspect of this experiment reenforces this
point.
2.4.5 Sampling Strategy
Consider dropping particles into a flow field, with no particular regard to lo
cation. These particles move along streamlines without interaction with the porous
medium. Any one location into which one is dropped is as equallylikely as any other.
The trajectories associated with these particles carry an equal area weight, analogous
to equalarea subdivision of an injection plane. Consider now, the same disposal of
particles, but this time an area is assigned to each particle at the injection point such
that an equal discharge is associated with each particle, and its associated trajectory.
For a homogeneous mobile water content, this area is inversely proportional to the lo
cal velocity normal to the area. Thus, each particle may be thought to carry an equal
21
flow weight. In this way, it is possible to map from one collection of trajectories to
another.
At a given control plane or time, the areaweighted mean and variance of some
property over N realizations, say v was calculated using the familiar estimators3
= (2'7)
varM = jyiry(vi Nvl) (28)
21
While these particular estimators seem obvious, it is important to realize that drop
ping the particle into the flow field assigns an equal area weight to each trajectory.
For a single realization, this is equivalent to starting each particle with equidis
tant spacing. These estimators are based upon an assumption of equallylikely
statisticallyindependent samples (see, e.g., Mood et al. [1974]). In order to draw
even a few hundred independent samples from a single realization, the required flow
domain would be enormous. This was a primary consideration for adopting Monte
Carlo simulation using a minimal domain.
The flowweighted mean property v was calculated via
vf
EZ=ii[0fo
EÂ£it*[o]
(2.9)
3 Press et al. [1988] have criticized the variance estimator in Equation (2.8) as
being prone to accumulation of roundoff error for large data sets. The rather sub
stantial size of the data sets are naturally a concern. Comparision of the variance
calculated by a corrected twopass method presented by Press et al. [1988] and the
traditional estimator yielded no difference between the two methods within the
first five significant digits.
22
where u[0] is the initial velocity recorded for trajectory i. The variance by
E^MO*/)2
var[u/] =
Efci Â¡[o]
(2.10)
An alternative method for generating a collection of streamlines would be to iden
tify the streamline corresponding to the center stream function expressed in Equa
tion (2.2). In this case, the estimators of the equal flow statistics would be Equa
tions (2.7) and (2.8). The equal area statistics would be given by
Ei=i i/i[o]
(2.11)
and
ElViVa^/um
var[ua] =
e2=i i/i[o]
(2.12)
The reference collection of the trajectories selected uniformly in space was specified
in the simulations for convenience.
2.4.6 Simulation Results
The following sections present the results of two 5000 replicate Monte Carlo simu
lations. Detailed discussion of these results is deferred until after the development of
some theoretical models with which to analyze these results. However, some heuris
tics are provided to familiarize the reader with some of the concepts developed in
subsequent sections and chapters.
A summary of the notation used with the following results provides clarity. AW
denotes an areaweighted trajectory and FW denotes a flowweighted trajectory ET
denotes an Eulerian trajectory and LT denotes a Lagrangian. A particular weight
trajectory combination is concatenated the with a /. Thus, an areaweighted Eulerian
trajectory is denoted as AW/LT.
For a given simulation, that is, for one set of 5000 replicates for a log conductivity
field of some given a?nk, the statistics of the log conductivity and velocity along the
23
four different trajectories are calculated at intervals in time and in space. Thus, for
any one property, say log conductivity, and any one statistic, say the mean, there will
be two sets of results, namely that for ofnfe = 1/2 and that for afnfc = 1. Associated
with each set will be two subsets, namely statistics along the four trajectories recorded
at space or time intervals.
For convenience, the notation and the trajectories to which each notation corre
sponds as depicted in Figures 2.2 and 2.3 is summarized here:
AW/ET areaweighted Eulerian trajectory (Figures 2.2 and 2.3 A)
AW/LT areaweighted Lagrangian trajectory (Figures 2.2 and 2.3 B)
FW/LT flowweighted Lagrangian trajectory (Figures 2.2 and 2.3 C)
FW/ET flowweighted Eulerian trajectory (Figures 2.2 and 2.3 D)
2.4.7 Log Conductivity Statistics
The log conductivity field serves as the foundation upon which the simulations
are built, as well as the theory. The data for the areaweighted Eulerian trajectories
correspond to those design criteria specified. The log conductivity mean and vari
ance for AW/ET appear stationary in space (see Figures 2.6 and 2.7). That is, they
appear to bounce around their target values with no apparent trend. Moreover,
the statistics of FW/LT appear likewise stationary, but the mean is somewhat larger
and the variance slightly smaller than AW/ET. The magnitude of the differences
appear proportional to ofnA:. A comparison of the target values, e.g., the specified
inputs into the turningbands algorithm, and the observed ensemble values along the
trajectories is presented in Table 2.1. The observed entry in Table 2.1 represents
the statistic average along the trajectory, as estimated by regressing a zeroslope line
through the data using a generalized leastsquares method. There is close correspon
dence between the target and observed values, supporting an ergodic hypothesis for
the log conductivity statistics. That is, sample statistics are an accurately estimation
24
FW/LT
AW/LT
FW/ET
AW/ET
Eq. 2.72
reference
FW/LT
AW/LT
FW/ET
AW/ET
Eq. 2.72
reference
+
X
X
X
X
Figure 2.6: Mean log conductivity as function of displacement in mean flow direction
for the four different trajectory collections and a?nk = 1/2 (a) and rfnk = 1 (b).
25
10
20
T'/'^ln k
30
FW/LT
AW/LT
FW/ET
AW/ET
reference
Eq. 2.73
40
FW/LT
AW/LT
FW/ET
AW/ET
reference
Eq. 2.73
Figure 2.7: Log conductivity variance as function of displacement in mean flow direc
tion for the four different trajectory collections and aÂ¡nk = 1/2 (a) and a^nk = 1 (b).
Variances are normalized by target variance of Eulerian log conductivity field
26
Table 2.1: Comparison of target log conductivity values and observed values along
areaweighted Eulerian trajectories.
target observed
parameter
Min k{Kk = 1/2)
Mlnfc^lnfc = 1)
k
0 0.003
0 0.005
1/2 0.498
1 1.002
of the population statistics. Again, this is important for a quantitative theoretical
analysis of these results.
The AW/LT and the FW/ET exhibit distinctly nonstationary behavior in the
mean and less distinct behavior in variance as a function of displacement along the
mean flow direction (see Figures 2.6 and 2.7). At the injection point, the AW/ET and
AW/LT statistics are identical and the FW/ET and FW/LT statistics are identical,
because these trajectories carry the same weight and the points in consideration are
identical (see e.g., Figure 2.2). At very large displacements, one would expect the
statistics of like trajectories to assume similar characteristics, regardless of weight.
As the trajectory traverses several integral scales that characterize the heterogeneity
in question, information about its starting location, is diminished. In the case of
the FW/ET, the statistics start at the same place as FW/LT, and eventually end
at the same place as the AW/ET. Similarly, the AW/LT statistics start where do the
AW/ET and end where do the FW/LT. In fact, the data exhibit this behavior (see
Figures 2.6 and 2.7).
The log conductivity statistics in time show somewhat different behavior. Only
the AW/LT appear stationary in time (see Figures 2.8 and 2.9). There is an impor
tant observation to be gleaned from these results. The AW/LT statistics, in addition
to appearing stationary, also appear to have a similar value to the input AW/ET
statistics (see Table 2.2). Recall, the AW/LT statistics are clearly nonstationary in
space, and are clearly distinct from those of the AW/ET. This observation greatly
27
FW/LT
AW/LT
FW/ET
AW/ET
est AW/LT
X
X
FW/LT +
AW/LT x
FW/ET *
AW/ET
est AW/LT 
0 2 4 6 8 10
Ut/1
>
o
13
T3
C
o
o
O)
o
c
CO
CD
E
0.4 
0.2 
* +
+
X
X
X
X
t&tSZ&SM ++.+.+..:..T+t^x+*++.H;..
fTr *XXxx xxxxxXxXxxxxxx**?
xxxxxxxXxx x
0.2 
w
X
a_
0.4 b
**
qd
Q.
dQqb *******x
xPa?DOorf^^95fi^gg<
Figure 2.8: Mean log conductivity as function of travel time to control planes along
mean flow direction for the four different trajectory collections and afnk = 1/2 (a)
and a\nk = 1 (b)
28
O 2 4 6 8 10
tU/X infc
FW/LT +
AW/LT x
FW/ET *
AW/ET
reference
tU/Xink
FW/LT +
AW/LT x
FW/ET *
AW/ET
reference
Figure 2.9: Log conductivity variance as function of travel time to control planes
along mean flow direction for the four different trajectory collections and afnk = 1/2
(a) and = 1 (b). Variances are normalized by target variance of Eulerian log
conductivity field a,2nfc.
29
Table 2.2: Comparison of target log conductivity values for areaweighted Eulerian
trajectories in space and observed areaweighted Lagrangian trajectories in time.
parameter target observed
WnkKk = 1/2) 0.000
/hnfc(0fnfc = 1) 0 017
simplifies relating the Eulerian and Lagrangian fields, as shall be shown. The long
time asymptotic mean log conductivity values for both Eulerian and Lagrangian
trajectories are somewhat less than their largedisplacement counterparts, and the
magnitude of the difference is proportional to ofnfc (compare Figures 2.6 and 2.8).
This is consistent with the harmonic averaging implicit with a timeaveraged time
dependent velocity and the arithmetic averaging implicit with a spaceaveraged space
dependent velocity. Recall, we map the quantities onto the trajectories by following
velocities, whether along natural streamlines (Lagrangian trajectories) or not (Eu
lerian trajectories). A simple heuristic is as follows. Low conductivities correspond to
low velocities. For an equaltime spacing (or equallylikely observations in time) more
measurements will come from low velocities, since the particle tracing the trajec
tory spends more time going slow. For an equaldistance spacing more measurements
will come from high velocities, since the particle covers more ground while going
fast. This is perhaps more intuitive when discussed in terms of the velocity field itself.
2.4.8 Velocity Statistics
The mean velocities along the different trajectories exhibit a similar behavior as do
the mean log conductivities along the same trajectories (see Figure 2.10 and compare
Figure 2.6). The AW/ET exhibits apparently stationary behavior in the mean and
variance (see Figure 2.11) in accordance to established theory (e.g., Dagan [1989]).
The AW/LT exhibits nonstationary behavior in the mean as observed by Cvetkovic
et al. [1996]. Again, the AW/ET and FW/LT trajectories appear to be stationary
30
FW/LT
AW/LT
FW/ET
AW/ET
Eq. 2.35
Eq. 2.43
Eq. 2.44
reference
+
FW/LT
AW/LT
FW/ET
AW/ET
Eq. 2.35
Eq. 2.43
Eq. 2.44
reference
D
Figure 2.10: Comparison of positiondependent mean velocity estimators to data for
the four collections of trajectories for afnJfc = 1/2 (a) and ofnJfc = 1 (b). Velocities are
normalized by U.
31
FW/LT
AW/LT
FW/ET
AW/ET
Eq. 2.37
Eq. 2.55
Eq. 2.57
Eq. 2.13
FW/LT
AW/LT
FW/ET
AW/ET
Eq. 2.37
Eq. 2.55
Eq. 2.57
Eq. 2.13
^/^ln k
x
Figure 2.11: Comparison of the positiondependent velocity variance estimators to
data for the four collections of trajectories for afnk = 1/2 (a) and c>fnA. = 1 (b). The
variances are normalized by U2bafnk.
32
in both the mean and variance. The velocity variances exhibit behavior to similar
to that of the means for their corresponding trajectories and quite different than the
behavior exhibited by the log conductivity variances (compare Figures 2.11 and 2.10
and compare Figures 2.11 and 2.7). There is a qualitative similarity between the
nonstationary velocity means and the nonstationary velocity variances. A heuristic
is that the velocity statistic starts at one value in the nearfield and transitions
to an asymptotic value in the farfield. The following sections demonstrate that
these endpoints and the transition are predictable.
The qualitative dissimilarity between the log conductivity variance and velocity
variance is an artifact of examining the log conductivity values on the one hand and
the real space velocity values on the other. Consider the FW/LT log conductivity
mean and variance for the ofnfc = 1 simulation. The FW/LT mean log conductivity
is larger than that of the AW/ET, but the variances are roughly the same value.
In fact, regressing a zeroslope line through the data results in a mean FW/LT
variance that is slightly less than that of the AW/ET. However, the real space
conductivity mean and variance are both functions of both the mean and variance of
the log conductivity process (recall Equation (2.6)). A cursory examination of the
data in conjunction with Equation (2.6) reveal that the mean and variance of the real
space conductivity along FW/LT are both higher than their AW/ET counterparts,
and the magnitude of this difference is proportional to ofnfc.
The mean velocities in Figure 2.10 are normalized by the arithmetic mean veloc
ity U predicted from the input parameters for the flow simulator. That the mean
velocity along AW/ET closely follows 1 is a good indication that the effective con
ductivity expression is accurate. Notice that the FW/LT and farfield AW/LT values
are higher than AW/ET, and the magnitude is proportional to the variance of the
log conductivity (see Figure 2.10). This behavior is intuitively correct. Consider a
33
particle injected into a flow field. Constraint of the trajectory to an Eulerian tra
jectory forces the trajectory through lowvelocity areas that a natural streamline, or
Lagrangian trajectory, would have otherwise bypassed. Of course, some streamlines
pass through even the lowest velocity areas, but these are few in an equalflow sense.
Thus, equalarea weighting preferentially weights low velocity zones. That there are
implications for sampling schemes in heterogeneous media, especially in context of
multilevel samplers, should be intuitively obvious to even the most casual observer.
A discussion of some of these implications is made in a subsequent section, after these
concepts are developed further in a more quantitative and theoretical framework.
It is appropriate to evaluate the adopted expression for the AW/ET longitudinal
covariance function suggested by Dagan and Cvetkovic [1993], namely
Cua[r] = afnlcb[e]U2 exp [&[e]r/Ainfc] (2.13)
where r is the separation between points along the ET, and b[e] is a function of the
anisotropy ratio e. The anisotropy ratio itself is a function of the correlation lengths
of the log conductivity in the different directions. The primarily consideration is
that b may be derived from theoretical considerations and is not some ad hoc fitting
parameter. The value of this function for a twodimensional isotropic aquifer is 3/8.
Again, for further background on this function, see Dagan and Cvetkovic [1993]. In
the limit r = 0, the covariance reduces to the point variance, which for the model
case is al = 3U2afnk/8. For the remainder of this discussion, a reference to b contains
its dependence upon e implicitly. The subscript ua may be decoded as follows: for
a subscript tw, the t is the trajectory and property (u for Eulerian velocity, and v
for Lagrangian velocity) and the w is the weight (a for area and / for flow). Thus,
Cua is the covariance function for velocities along Eulerian trajectories. We present
this covariance function and that covariance function estimated from the assumed
stationary collection of AW/ETs in Figure 2.12 The perceptive observer will note
34
1
0.8
.I '6
*4
8 0.4
0.2
0
0 5 10 15
f /^ln k
Figure 2.12: Comparison of spatial Eulerian (AW/ET) and Lagrangian trajectory
(FW/LT) velocity covariance data to covariance model. Data are taken from the
cr,2nfc = 1 set of realizations. Covariances presented as dimensionless lag correlation
functions.
that the analytical expression overestimates the covariance at short lags and under
estimates the covariance at longer lags. This discrepancy and its implications are
discussed in some detail in a subsequent section.
The velocity variances in Figure 2.11 are normalized by the zerolag covariance pre
dicted by placing the input simulation parameters into Equation 2.13. The AW/ET
velocity variance appears wellpredicted by this expression, as evidenced by a rela
tively good fit to 1 for both values of a^nk in Figure 2.11. A regression of a zeroslope
line through these data indicates that the a priori estimate under predicts the ob
served values by about five percent for the larger ofnfc (see Table 2.3) The apparent
stationarity of the velocities along the FW/LTs suggest estimation of the covariance
function for these velocities as for the AW/ET (see Figure 2.12). The correlation
length of the FW/LT is very close to that of the AW/ET, although the actual value
is probably slightly higher. Recall the definition of the correlation length as given by
35
Table 2.3: Comparison of a priori Eulerian velocity variance to observed values. The
decimal representation of the exact fractional values of the predicted variance are
provided for convenience.
parameter predicted observed
^Kfc = l/2) 3/1600=0.001875 0.001871
crl((7\nk = 1) 3/800=0.00375 0.00394
Equation (2.4). The correlation function at each considered lag along the estimated
functions is systematically larger for FW/LT, even though the magnitude of the dif
ference is small. Integrating these functions would result in a slightly larger value
for FW/LT, and thus a larger correlation length. In the case there is no variability
in the log conductivity, Eulerian and Lagrangian trajectories coincide. Moreover, an
equalarea weight corresponds to an equal flow weight. Thus, the estimated correla
tion functions from either set would be identical. From this conclude that there
is some dependence of the correlation length of the FW/LT trajectories upon afnk.
Cvetkovic et al. [1996] observed similar behavior in log velocity correlation functions
estimated from AW/LTs with a stationarity assumption for the nonstationary AW/LT
velocities. Thus, this is an slight extention of this work by identifying the station
ary streamlinebased velocity field from which to work, and work from that field.
Moreover, this supports their findings of an increasing correlation length with ofnfc.
Understanding the relationship between the variability of the log conductivity field
and the correlation length of the associated velocity field is of fundamental impor
tance to completely understanding the nature of flow in heterogeneous media. Sadly,
these secrets remain uncovered and must be left this to future research efforts.
The velocity means as a function of time exhibit qualitatively similar behavior
as do the log conductivity means in time (see Figure 2.13 and compare Figure 2.8).
Again, only the AW/LT exhibits stationary behavior in time for both the mean and
variance (see Figure 2.14) as did the AW/LT log conductivity values. The mean
and variance of the AW/LT in time are approximately those of the AW/ET in space
36
Figure 2.13: Comparison of timedependent mean velocity estimators to data for the
four collections of trajectories for afnk = 1/2 (a) and
normalized by U.
37

a
s
b
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
FW/LT +
AW/LT x
FW/ET *
AW/ET
Eq. 2.65
Eq. 2.61
Eq. 2.70
Eq. 2.69
+4',
+. +
*+ +7+ + +
XX _X. "xx"x
.5
a _ x** **& ****
b "SSgg!. ** *
t^QSS. B mm
sftv.:##Baeae??
10
tU/X\nk
a
Ji
c
o S
b
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0 2 4 6 8 10
++* V
FW/LT +
AW/LT x
FW/ET *
AW/ET
Eq. 2.65
Eq. 2.61
Eq. 2.70
Eq. 2.69
V M ^
'+ + +
+ \ +
\ xx xxvv++ +
* x\ X xV.V+ +* *
+ + *
A* *
x x.\. x.
\ \ *
9 \x
g w** \ x
. V. *'*
XX )K ^ X
'"'DCIlbt3QQQSoF5DD*8Da"88a H8BB85fiSSS*8HB83
tU /\\n k
Figure 2.14: Comparison of timedependent velocity variance estimators to data for
the four collections of trajectories for a2nk = 1/2 (a) and ofnfc = 1 (b). The variances
are normalized by U2ba2nk.
38
for both simulations. This result is important, because it implies the possibility of
mapping from AW/ET to AW/LT. We estimated the covariance function for the
stationary AW/LT velocities (see Figure 2.15). As an estimate of the covariance
1
0.8
 0.6
4
_ro
k_
8 0.4
0.2
0
0 2 4 6 8 10
tU/X\nk
Figure 2.15: Comparison of temporal Lagrangian trajectory (AW/LT) velocity covari
ance data to covariance model. Data are taken from the ofnfc = 1 set of realizations.
Covariances presented as dimensionless lag correlation functions.
function, the substitution r = Ut was made in Equation 2.13 and the result compared
to the estimated covariance function (see Figure 2.15). The closedform predictor
behaves similarly to its spatial counterpart: earlytime correlation is underpredicted
and longtime correlation is overpredicted (compare Figures 2.15 and 2.12).
For completeness, one other aspect of the velocity field was examined, namely the
distribution of initial velocities (see Figure 2.16). The logarithm of the initial velocity
was rankordered and subject to the transformation
. lnu \nv
lnu =
^ln v
(2.14)
39
log velocity
Figure 2.16: Normalized log initial velocities for twodimensional model aquifer com
pared to the standard normal distribution. The random variable was normalized by
the sample mean and standard deviation to illustrate the lognormality.
where lnu is the sample mean and Ein is the sample standard deviation. Work
ing here with the logarithm, rather than the velocity itself, better demonstrates the
apparent lognormality of the velocity fields. Recall, the logarithm of a lognormal
random variable is itself normally distributed. Figure 2.16 indicates that the trans
formed log initial velocities are welldescribed by a standard normal distribution. The
lognormality of the velocities seems logical since the conductivity field is itself lognor
mally distributed. However, the velocity is in general a product of random variables:
the conductivity, the head gradient, and the mobile water content. The logarithm of
the velocity is the sum of random variables. Sums of random variables tend toward
normality by the central limit theorem (see Mood et al. [1974]), and thus, products
of random variables tend towards lognormality. It would be interesting to test this
40
hypothesis by examining the velocity characteristics of flow fields arising from differ
ent log conductivity distributions (e.g., fractal, uniform, white noise, etc.), and this
is left to future work.
2.4.9 Head and Head Gradient
Head and head gradient information along the trajectories was not recorded. This
is a regrettable oversight, since much of the behavior of the flow water and the trans
port of solutes in heterogeneous media can be summarized as the interplay of head,
head gradient and conductivity covariance functions. The literature concerning Eu
lerian head and head gradient fields is extensive and is summarized in large part by
Gelhar [1993]. In fact, much is drawn from this body of literature for some of the
theoretical developments in the following sections. That this information was not
recorded detracts mainly from the completeness of this work, and further investiga
tions are left to future works.
2.5 Theory
2.5.1 Effective Conductivity
As with the simulations, the theoretical developments are founded on the bedrock
of the log conductivity field. Section 2.4.7 demonstrates that the simulation results
well match the target ensemble quantities. From this field is had directly a derived
parameter, namely the a priori effective conductivity Ke. An expression for the
effective conductivity, similar to that given by Gelhar and Axness [1983], is derived
here to illustrate the mechanics of different techniques employed through out these
derivations. Consider a onedimensional Darcys law of the form
Q =
dx
(2.15)
where q is the specific discharge, k is the hydraulic conductivity, and is the hydraulic
head. Consider a large stationary conductivity field such that the log conductivity
41
may be resolved into a mean, and a zeromean perturbation, e.g., In A: = p\nk + /.
Let the head gradient likewise be resolved into a mean and zeromean perturbation,
e.g., d(j)/dx = J( 1 + j). Inserting these expressions into Equation (2.15) yields
q = exp^in* + f]J{l+j)
(2.16)
= KgJ exp[/](l + j)
Expand the exponential term in this equation as a Taylor series about zero, truncating
the series at secondorder, and expand the product of the perturbation terms
q~K,J(l+f + P/2)(l+j)
(2.17)
= A',./(! + / + p/2 + j + fj + Pi/2)
Take the expected value of this expression, dropping terms higher than secondorder,
resulting in an expression
= KgJ(l + ofnA,/2f ) (2.18)
Consider this equation and the three terms in the parenthesis. The first is a mean
term, the second is a contribution due to the variability of the conductivity field
and the third is the log conductivity and head gradient crosscovariance. Gelhar
[1993] evaluates this expression for isotropic log conductivity fields using spectral
methods. The result, which is independent of the shape of the correlation function,
is surprisingly simple [Gelhar, 1993]
=o&Jd (2.19)
where d is the dimension of the domain. This negative correlation between head gradi
ent and log conductivity has a profound effect on the flow. The effective conductivity
may be defined as that value when multiplied by the spatial mean gradient results
in the observed specific discharge (flow rate divided by discharge area). The a pri
ori effective conductivity is the effective conductivity estimated from Equations 2.18
and 2.19. With d = 2, we have for twodimensional conductivity fields, the a priori
42
effective conductivity is simply Ke Kg. Notice that the effective conductivity can
be significantly less than the arithmetic mean conductivity Ka = Kg exp[cr12nfc/2].
The definition of the effective conductivity is quite general, and a general effective
conductivity expression is denoted Ke = Kgc where c is the correction due to
the aforementioned processes. Under this definition, there is always some effective
conductivity value, regardless of nonstationarities in boundaries or flow domains, but
its magnitude may depend upon several factors. There is no general a priori effective
conductivity estimator. The estimator given here is limited to stationary head and
conductivity fields.
2.5.2 Eulerian Velocity Field
The relationship between the Eulerian conductivity field and its associated Eule
rian velocity field has been one of the more thoroughly examined aspects of subsurface
hydrology. For the example problem, closedform expressions are sought for the Eu
lerian, or spatial, mean velocity U, the Eulerian velocity covariance Cua[r], and its
point distribution, given that the underlying conductivity field is large, stationary,
lognormallydistributed, exponentially correlated, and characterized by parameters
P\nk, ofnk, and A^nJt, and subject to the conditions prescribed for the model aquifer
(see Section 2.1 and Figure 2.1).
From the development of the effective conductivity relationship in the preceding
section, an expression for the mean Eulerian velocity is
U = fr (2.20)
where c = 1 for a twodimensional isotropic aquifer. The c is retained to emphasize
that this expression is not the consistent firstorder approximation of the velocity
that would result from dropping the afnk and < fj > terms in Equation (2.18).
For the twodimensional isotropic aquifer, which is the model case, these two terms
happen to be equal in magnitude and opposite in sign, and the consistent firstorder
43
expression and effective expression are the same. This damping effect of the negative
head gradient and log hydraulic conductivity is largely responsible for the robustness
of these perturbation expansions. The truncated terms are not negligible in the sense
their magnitude is small. For instance, for a2nk = 1, their magnitude is onehalf that
of the mean. This brings us to an important philosophical point. A primary usefulness
of these expansions is to identify the dominant components of the process. A purely
consistent approach could lead to neglect of some important process. For instance,
consider a strict, firstorder expansion of Equation (2.16). Equation (2.17) would
contain no f2 term, and Equation (2.18) would contain no o2nk term. The resulting
effective expression would seriously underestimate the specific discharge, since the
contribution of the conductivity variability would carry no weight with respect to
the other contributing processes. Analysis of the neglected higher order terms is
necessary for a more complete understanding of the process. The general goal of the
approach, however, is to identify the dominant subprocesses that contribute to the
larger process, and the relative importance, or weight, of each subprocess.
Expressions for the Eulerian velocity covariance for the model system have been
given by various researchers, including Graham, and McLaughlin [1989a] and Rubin
[1990]. Shapiro and Cvetkovic [1988] develop an inverse velocity covariance function
using a perturbation expansion of the inverse flow velocity and spectral methods.
Consider once again the perturbed Darcys law expression given by Equation (2.16).
Dividing this by the homogeneous mobile water content, neglecting secondorder
terms and higher, the meanremoved expression, or velocity perturbation, is
(2.21)
44
If these processes are stationary stochastic processes, they can be represented repre
sented by a FourierStieltjes integral (see e.g., Bakr et al. [1978])
poo
u'= exp[fM x]dZu/[M]
J OO
(2.22)
where i \fl, M is the wave number vector, and dZu' is the complex random
amplitude of the velocity perturbation. The FourierStieltjes representation of Equa
tion (2.21) is
dZu, = ^(dZf + dZj)
6
(2.23)
Compare this expression to that derived by Shapiro and Cvetkovic [1988] (their Equa
tion (A4))
dZi/u^JjidZf + dZj) (2.24)
The spectra that result from these complex random amplitudes are structurally the
same, differing by the square of the mean values by which they are multiplied (e.g., by
(KgJ/9)2 on the one hand and (9/(KgJ))2 on the other). In general, the correlation
structure of the inverse of a random function is not the same as that of the random
function itself. The firstorder series approximation of the inverse velocity is linear in
the velocity itself (e.g., 1/v ~ 1 v), and thus the velocity correlation characteristics
are recovered as an approximation of the inverse velocity correlation characteristics.
For the case of the exponentially correlated isotropic conductivity field, the longitudi
nal velocity covariance of a twodimensional field is given by (Shapiro and Cvetkovic
[1988])
Cu[s] = 3afnk(KgJ/6)2 (exp[s](s 2 + 3s 3 + 3s 4) + s 2/2 3s 4) (2.25)
45
where s = x/\\nk The corresponding expression for a threedimensional field is
Cu[s] = 8alk(KgJ/d)2 (exp[s](s2 + 5s~3 + 12s'4 + 12s~5) + s"3 12S5)
(2.26)
Equation (2.25) provides a superior match to the data in comparison to Equa
tion (2.13) (see Figure 2.17). However, the expression for the Eulerian covariance
Figure 2.17: Comparison of velocity covariance expression Equation (2.25) to the ap
proximation Equation (2.13) for ofnfc = 1. Covariances are presented as dimensionless
correlation functions.
given in Equation (2.13) captures the dominant features of the velocity covariance
structure, including the correlation length and the point velocity variance of Equa
tion (2.25) in a simple and familiar form.
As noted, however, Equation (2.13) exhibits some systematic differences from the
observed behavior. The data show a rapid initial decay, followed by a longrange
persistence (see Figure 2.17). This behavior is examined with an expression adapted
46
from Dagan [1984]
7> r i 2 r> ri dR\nktt>[r] d 7tf>[r]N\ ^ o7\
Rua[r] = ffinfc ^lnfcM j (227)
where RÂ¡nk is the log conductivity correlation function, Rink is the log conductivity
and head crosscorrelation function, and 7^ is the head variogram. From this ex
pression, glean that the velocity covariance is a process comprised of three dominant
subprocesses, namely R\nk, R\nk, and 7^. At small displacements, all three of these
subcomponents exhibit rapid trend towards zero (see Figure 2.18). The R\nk
Figure 2.18: Components of the velocity covariance function.
expressions exhibit effects that persist well beyond that of the log conductivity field.
The longrange correlation effects of the head field in the mean direction of flow are
cannot be precisely predicted with the simple exponential function Equation (2.13).
In order to recover the proper correlation length, the approximation must system
atically overpredict the velocity correlation at small separations. The cost of the
47
approximation, however, will realize significant returns in the form of simplicity, as
shown in subsequent sections.
Previous works have focused upon the moments of the Eulerian velocity field that
result from mean gradients imposed upon heterogeneous conductivity fields. These
works shall be extended slightly with a hypothesis about a specific case that allows
a variety of insights to be drawn.
Hypothesis: The point velocity distribution of an Eulerian velocity field induced
by an stationary mean gradient J across a lognormally distributed, exponentially
correlated log hydraulic conductivity field characterized by parameters fi\nk, crfnk,
and A^n k and saturated with mobile water at a constant and homogeneous content 9
is lognormal with approximate parameters
(2.
and
/hnu In
KgJC
dV 1 + bl
In k
Ohxu = ln[! + blk\
(2.29)
(see Appendix A for details of development). A fundamental assumption in addition
to the lognormality hypothesis is that the real space moments of u are \iu = U
and o\ = U2ba2nk. Equation (2.13) is adopted as an approximation of the velocity
covarince function.
In lieu of a rigorous proof, the results of a numerical experiment in which statisti
cally independent velocities are drawn uniformly in space from simulated flow fields
are presented (see Figure 2.16). The normalization of the velocities in Figure 2.16
than that of previous figures in that the log velocities are normalized by the sample
mean and standard deviations to emphasize the lognormality of the velocity distribu
tion. Normalization by the moments predicted by Equations (2.28) and (2.29) would
have resulted in a lognormal distribution function that deviates from the standard
48
normal. This indicates that the moments /zu = U and o\ = U2ba2nk may require some
further investigation. The adequacy of these estimates are discussed in subsequent
sections.
2.5.3 Streamtube Weights
The reference velocity field is now to be subdivided into streamtubes. Each
streamtube may be considered to carry some weight in relation to other streamtubes
for a given collection. Consider the partitioning of the flow field into N streamlines
separated by an equal distance along the IP. This is an equalarea weighted parti
tioning scheme, and results in N subIPs of equal area, but dissimilar discharge. The
discharge through subIP i is
di 9/\aUi (2.30)
where A a = A/N. The relative flowweight of this streamtube i is given by
wfi = Ui/U
(2.31)
An equalflow partitioning scheme results in in N subIPs of equal discharge, but
of dissimilar area. The discharge through subIP i is
d = 9/\a,iVi
Q/N
Solving for the area yields
(2.32)
Aa' = fe
The relative areaweight of this streamtube i is
(2.33)
Wai ~ U/Vi
(2.34)
49
The perceptive reader will notice a slight notational change between the two parti
tioning schemes. A lowercase u represents velocities drawn uniformly in space and a
lowercase v represents velocities drawn uniformly in flow. The distinction is impor
tant. It is through these subIPs that water enters the streamtubes and a description
of the subIP properties is essential to streamtube, and thus flow field, descriptions.
If from a collection of N subIPs the random selection, or drawing, of one subIP is
equally likely as any other, the weight assigned to any one observation from M samples
would be 1/M. Assume that all of the subIPs are sampled. The average subIP dis
charge is the same for the equalarea and equalflow partitioning schemes, and equal
to d = 6AU/N. However, the discharge variance is zero for equalflow and nonzero
for equalarea, when the velocity field is heterogeneous. The weights expressed in
Equations (2.34) and (2.31) allow the properties of one collection of streamtubes to
be mapped to the other collection of streamtubes. These weights combined with the
assumed values of the Eulerian velocity field permit derivation of properties of the
initial velocities of the equalflow partitioning schemes. The expected value of v must
equal the expected value flowweighted u. Thus,
= V
u
= < U>
U
= ij(al+^)
= U(l + baÂ¡nk)
= ^( i + K*)
(2.35)
50
The variance of v is given by var[i>] = 2. The expected value of v2 must
equal the expected value of flowweighted u2. That is
=
1 ro 9 2 l
= exp[3/iinu + alnu]
= y exp[3(x,nu + ofnu/2) + 3ofnJ
= U*(l + bo&k)
U 2
(2.36)
Thus, the variance of v is given by
var[u] = V2ba]
lnfc
(2.37)
It is worthy to note that these results do not follow from an assumption that v u2/U.
That is, the velocity along the Lagrangian trajectory is not simply the flowweighted
velocity along an Eulerian trajectory.
Hypothesis: The point velocity distribution along a flowweighted Lagrangian
trajectory in the model stationary Eulerian velocity field is lognormal with approxi
mate parameters
(2.38)
and
= M1 + ba\n k\
(2.39)
Moreover, the process is assumed secondorder stationary in displacements along
the mean direction of flow, that is, at control planes, and is characterized by the
covariance function
CVf[r\ = V2ba2nkexp[br/Xlnh\
(2.40)
51
where r is the lag along the mean flow direction. Again this follows from the require
ment that the distribution match the assumed real space v moments /i = V =
U(1 + bo\nk) and a\ = V2bafnk. The correlation structure of the FW/LT velocities is
similar to that of AW/ET (see Figure 2.12).
Employing these velocity distribution hypotheses, it is now possible to derive some
relationships between the different flow field partitioning schemes. From observation,
it is inferred that the velocities along areaweighted Euleriantrajectories and flow
weighted Lagrangiantrajectories are secondorder stationary in displacements along
the mean flow direction (recall Figures 2.10 and 2.11). Having adopted or postulated
expressions for these first two moments, it is now possible to derive expressions for
the nonstationary velocity means and variances of the areaweighted Lagrangian and
the flowweighted Eulerian streamtubes.
2.5.4 Control Plane Oriented Velocity Moments
Cvetkovic et al. [1996] postulated a semianalytical estimator for a nonstation
ary mean velocity along areaweighted Lagrangiantrajectory observed at control
planes. We would like to work directly from the assumed stationary flowweighted
Lagrangiantrajectory velocity field. The general expression for the expected value
of the areaweighted Lagrangiantrajectory velocity is
Va[x] =< v[z] > (241)
Resolve the stationary mean velocities into the mean of the flowweighted Lagrangian
trajectory field and zeromean perturbation.
U
V{l+v'[Q])
va[x] =<
V(1 T i/[x]) >
(2.42)
52
Expand the denominator in a series to second order
va[x] =
(2.43)
= U( 1 + exp[6r/AiJ))
In a similar fashion, an expression is derived for the nonstationary mean velocity
along a flowweighted Euleriantrajectory. In this case, it is the stationary Eulerian
trajectory velocity field that serves as a basis. The mean of the perturbation expan
sion is the Eulerian mean velocity. It is interesting to note that this result requires
no series expansion.
(2.44)
= [/(! + ba?nkexp[br/\ink})
The nonstationary velocity variances along the respective crossweighted trajec
tories are derived in a similar fashion. The general expression for the areaweighted
Lagrangiantrajectory velocity variance is
(2.45)
This equation may be rewritten
(2.46)
Resolve the first term of the preceding equation into perturbation expressions
(2.47)
53
Expanding the denominator in a series, taking the expected value, and neglecting
thirdorder and higher moments yields
<^]2>= UV(l + 2ba2nk(l + exp] bx/\\nk])) (2.48)
In order to improve the quality of the estimate, the righthand side (RHS) of the
preceding equation is required to match the known extreme values of the lefthand
(LHS) side. The LHS at X = 0 is
<^rn[0]2> = U
n[0] 1 J 1 J
= UV
(2.49)
(2.50)
The RHS at X = 0 is
RHS[0] = UV( 1 + 2&a2nfc(l exp[60/Aln*]))
= uv
The statistical independence of the initial velocity and the velocities as x goes to
infinity are exploited to calculate the LHS as x > oo
U
LHS[oo] =<  >< n[oo]2 >
= I + v2
= UV(l + bo2nk)2
The RHS as x > oo is
RHS[oo] = UV( 1 + 25cT]2nfc(l exp[5oo/Alnfc]))
= UV{ 1 + 2 ba2nk)
The RHS is deficient a term (5cr2nfc)2. Rewrite the RHS thus
(2.51)
(2.52)
RHS UV(1 + (2ba2nk + (6a2nfc)2)(l exp[br/\lnk}))
(2.53)
54
Substituting Equations (2.53) and (2.43) into Equation (2.45) yields
var[ua[x]] = UV{ 1 + (2ba?nk + {ba?ak)2)(l exp[5r/Ainjfc]))
(2.54)
~{U{ 1 + ba?nk(l exp[5r/Alnfc])))2
Further expansion of this expression offers no intuitively obvious simplifications or
insights in its entirety. However, this expression may be approximated by dropping
a few terms thus
var[ua[:r]] = V2ba?nk U2ba?nk(2ba?nk + {bafnk)2) exp[6r/Ainfc] (2.55)
Two unconventional techniques were employed in this derivation that shall now be
defended. The first is requiring the series expansion to match the known endpoints.
Simple expressions that condense the system behavior into an compact form that
allows the relative contribution of the component processes to be easily grasped are
sought. The series expansion and subsequent truncation is itself an approximation.
The systematic error introduced by missing a known endpoint is distracting. In the
case of the velocity covariance, the shape of the correlation function was sacrificed
for simplicity, but the correlation length was retained. Had the simplified expression
neither matched the shape of the data nor reproduced the correlation length, the
expression would have been somewhat more questionable. In this case, the expression
is required meet the observed endpoint values at the cost of the rigor of the series
expansion and truncation.
The second simplification is the ad hoc dropping of terms to simplify the expres
sion. Again, simple expressions that allow the relative contribution of the different
processes be easily grasped are sought. It would be difficult to hypothesize Equa
tion (2.55) a priori as an empirical expression. A primary function of the series
expansion is to indicate the dominant processes and give an estimate of their relative
contributions. This hybrid approach is useful for the purposes here. These techniques
55
are not a panacea, however, and the approximations have only been tested for the
log conductivity variances tested in this work.
The nonstationary flowweighted Euleriantrajectory velocity variance may be de
rived in a similar fashion. These results are presented directly. The complete expres
sion is
var [uf [x]] = U2 (l + ba?nk + (2bafnk + 3 (bafnk)2 + (krjj,*)3) (1 exp [br/Alnfc]))
 (U (1 + 6a,2nfcexp[6r/AinJt]))2
(2.56)
and an approximation of the above obtained by neglecting a few terms is
var [uf [*]] = U2ba?nk (l + (2bafnk + (6a,2nJt)2) exp [6r/Alnfc]) (2.57)
2.5.5 Control Time Oriented Velocity Moments
The displacementinvariant velocity covariances for u and v permitted derivation
of expressions for the first two moments of the respective velocities for the four dif
ferent trajectory collections. However, only the areaweighted Lagrangiantrajectory
collection appears to be stationary in time. This is perhaps not intuitively obvious,
as the velocity is clearly not stationary in displacements along the mean direction
of flow (e.g., Equations (2.43) and (2.54)) because there is an implicit distribution
of times associated with the establishment of a reference injection plane. After long
travel distances and times, the areaweighted Lagrangiantrajectory point velocity
statistics are equal to the flowweighted Lagrangiantrajectory point statistics. At
the injection plane, the areaweighted Lagrangiantrajectory point velocity statistics
are identically equal to the areaweighted Euleriantrajectory point velocity statistics.
These relationships can be exploited to derive equations that provide a few insights
about how properties change in Lagrangian time, and to derive an expression for a
Lagrangiantime dependent velocity covariance function.
56
The areaweighted velocity at the IP is U by definition. Since the areaweighted
Lagrangiantrajectory velocity is stationary in time, its mean in time is U regardless
of position. At long travel times, the point velocity is stationary in displacements as
well. For constant means in space and time, the following relationship holds
1
T
(2.58)
This expression states that the harmonic mean velocity in space equals the arithmetic
mean velocity in time. Returning now to the model aquifer, invoke the hypothesis of
lognormality of the velocities along either trajectory. The harmonic mean of u[x] is
given by
Vh = exp[/qn crinu/2]
= exp[pin] exp[crin]1/2
= u^l + bol
= U
;ln k
V1 + ba\i
In k
(2.59)
Similarly, the harmonic mean of u[x] is given by
Uh
u
1 + b(Jlk
(2.60)
Although derived by different means, this result is equal to the result given by Shapiro
and Cvetkovic [1988]. These harmonic means are the asymptotic mean velocities of
the two trajectories in time. The initial timedependent velocities are identical to
the initial spacedependent velocities, an artifact of starting all of the trajectories
at the injection plane at an implied time t = 0. This indicates that the velocity
covariance in time along the areaweighted Lagrangian trajectory is Cva[t = 0] =
U2bafnk As mentioned, at large times, the statistical properties of the Lagrangian
trajectories converge. Making a change of variables x = V^t Ut in Equation (2.40)
and substituting the known Cva[t = 0] for the value of Cvj[r 0] in yields an
57
expression for the stationary areaweighted Lagrangiantrajectory velocity covariance
CJi] = U2balÂ¡k C'>:p[M/i/Ai]
(2.61)
Similar expressions have been derived before by other researchers (see e.g., Dagan
[1989]), but the conceptual development is quite different. Rather than seek to ap
proximate the covariance function, an approach to quantify it directly from properties
of the system is demonstrated. Moreover, the quantity U is perceived to be the har
monic mean of the Lagrangiantrajectory velocity, and that it is quantitatively equal
to the areaaverage velocity of the flow field. This is an important point.
The mean velocity and velocity variance along the flowweighted Lagrangian
trajectory is found in a manner similar to those in space, but there are some concep
tual issues that should be addressed. It is the collection of areaweighted Lagrangian
trajectories that exhibits a stationary velocity. The mean velocity is
. u[0] ,
Vft[t] =<jfv[t]>
(2.62)
Expand the velocities into a mean and a zeromean perturbation. The mean here is
the time average mean, or the harmonic mean of the space average.
W = ^ <^(1 +'[0])r/ (! + '[(])>
(2.63)
U (l + bainkexp[bUt/\lnk])
This expression requires no series expansion. For the control plane oriented velocity
statistics, the mapping was from flowweighted to areaweighted trajectories. Equa
tion 2.63 is identical in form to Equation 2.44 with the substitution r = Ut. Recognize
58
the quantitative similarity and write velocity variance equations directly The com
plete expression is
var [vf [t]] = U2 (l + bafnk + (2bafnk + 3 (ba?Dk)2 + (btfU)3) (1 exp [bUt/\lnk]f)
 {U (1 + bafnk exp [6r/Alnfc]))2
(2.64)
and the approximation is
var [vf [t]] = U2bafnk (l + (2bafnk + {ba?nk)2^) exp [bUt/Alnfc]) (2.65)
Calculation of the mean velocities along the Eulerian trajectories is problematic
in the absence of a stationary velocity covariance function. However, note that the
areaweighted velocities start at U and that the flowweighted velocities start at V
and both progress to U/(l + bo\nk). Recognizing the similarity shared by the different
mean time estimators, write approximate expressions directly
at [i] = 1 (1 + ba?n k exp [bUt/Aln*]) (2.66)
and
Uat [^]
x + ba2 (! + (2blk + Wnfc) (eXP [bUt/XInfc]))
(2.67)
Again, devoid of a stationary covariance function for the Eulerian velocity field in
time, intuitive arguments lead us to anticipate an asymptotic u variance in time to
be
a:
ut
uzMnk
u
! + Kn k
h\n k
(2.68)
The initial velocity variances are known from the covariance functions in space. It is
possible to anticipate forms of the nonstationary variance equations similar to those
found previously, and include them for the sake of symmetry and completeness. The
59
areaweighted Euleriantrajectory timedependent velocity variance approximation is
varK[t)] = (l+U ) bol, (1 + {2bolt+ (bait))exP[bUt/Kt]) (2.69)
The fluxweighted Euleriantrajectory timedependent velocity variance approxima
tion is
var[u/[t]] =
(t+TkT5) b(J'nk 1 + (36CTl2nfc + 3 + (b0^kY) exP [bUt/Xlfc])
(2.70)
2.5.6 Log Conductivity Moments
The first two moments of the log conductivity along those trajectories for which
the moments are stationary at control planes located along the mean direction of
flow or at control times are sought. Working from a stationary Eulerian reference
field, the mean and variance along the AW/ET are fi\nk and afnk, respectively. Infer
that the FW/LT is stationary as well, analogous to the velocity field results. The
flowweighted mean log conductivity perturbation is given by
[0] f ^ ^gexp[/]J(1+ j)9
u 1 9KeJ 1 (2.71)
=<(l + / + /2/2 + ..)(l+j)/>/c
Expanding, retaining secondorder terms, and taking the expected value yields
<^f>=olk(\l/d)/c (2.72)
which, for twodimensional fields, is ofnfc/2. Similarly, the variance is
<^f> <^/>2= Kt(l 1 (2.73)
which, for twodimensional fields, is afnfc(l
60
Finally, note that the AW/LT log conductivity appears stationary in time as does
the velocity. Moreover, it appears to have values equal in magnitude to its AW/ET
counterpart, as does the velocity.
Unlike the velocity, expressions for the trajectorybased crosscovariance of the
velocity and log hydraulic conductivity that are necessary for developing the non
stationary log hydraulic conductivity moments were not developed. This is left for
future work.
2.6 Evaluation of Simulation and Theoretical Results
2.6.1 Mean Velocities in Space
Figure 2.10 presents the control plane oriented mean velocities for the four different
collections of streamtubes. The mean velocities are normalized by U = KgJ/6 1/10
and this value was used in the mean velocity expressions as well. That is, the parame
ter values are the input parameters to the simulations. Other input parameter values
are b = 3/8 and o^nk = {1/2,1} depending upon the set of simulations. The deviation
of the mean of areaweighted Euleriantrajectory data from 1 are an indication of the
quality of the effective conductivity expression.
The areaweighted Euleriantrajectory mean velocity appears stationary, in ac
cordance with established theory, and its value appears to be well predicted by the
effective conductivity relationship. The fluxweighted Lagrangiantrajectory mean
velocity appears stationary, supporting the stationarity hypothesis. Moreover, the
value of the mean velocity appears to be well predicted for both variances of log
conductivity. Additionally, the nonstationary mean velocity estimators appear to
accurately reproduce the transition between the asymptotic velocities.
2.6.2 Velocity Variances in Space
Figure 2.11 presents the control plane oriented velocity variances for the four dif
ferent collections of streamtubes. The variances are normalized by the areaweighted
61
Euleriantrajectory velocity variance estimated from the input parameters to the sim
ulations. That is, by
u2 f< = (KgJ/e)2bolh (2.74)
As with the mean velocity estimators, the variance estimators employ only the sim
ulation input parameters.
Once again, the areaweighted Euleriantrajectory variance exhibited its pre
sumed behavior, and supports the well established fact that the Eulerian velocity
field is fairly well understood in space. The hypothesized flowweighted Lagrangian
trajectory velocity variance is somewhat higher that the data for both
reason is the negative correlation between log conductivity and head gradient is not
weighted properly in the collection of trajectories skewed towards higher log conduc
tivity values.
The other two estimators also suffer from this bias in the asymptotic Lagrangian
trajectory velocity variance, although the transition between initial and final values
for both seems to be well characterized.
2.6.3 Mean Velocities in Time
Figure 2.13 presents the timedependent mean velocities for the four different col
lections of streamtubes. Again, the mean velocities are normalized by U as estimated
from the simulation input parameters. Only the areaweighted Lagrangiantrajectory
mean velocity exhibited a constant mean behavior in time. Its estimated harmonic
mean is U, and the estimate seems good, although one might argue that there is a
slight overpredictive bias.
The estimated harmonic mean of Eulerian trajectories appear to have a stronger
under predictive bias. The transition of the flowweighted Lagrangiantrajectory
appears to be captured better than those of the Eulerian trajectories. It should be
noted that the Lagrangian trajectory transition was based upon an expression for
62
a stationary velocity field, whereas the Eulerian trajectories are ad hoc estimators,
based upon a transition between known end points.
2.6.4 Velocity Variances in Time
Figure 2.14 presents the timedependent velocity variances for the four differ
ent collections of streamtubes. The variances are normalized by the areaweighted
Lagrangiantrajectory velocity variance estimated from the input parameters to the
simulations. The areaweighted Lagrangianvelocity appears to be stationary in the
variance. The data seem to lie slightly above the estimated value.
The overprediction of the initial fluxweighted velocity variance has far less impact
upon the timedependent velocity variances. This combined with what appears to
be a good estimate of the longtime Euleriantrajectory velocity variance yields a
better match to the simulations than one might expect from the means. However,
there is a strong initial decline in the variance not captured by the simple exponential
expressions. The areaweighted Lagrangiantrajectory covariance function exhibits a
similar rapid decay. The early time behavior is likely dominated by the conductivity
covariance. As the trajectory experiences a wide range of conductivity variability,
this effect diminishes, and the longer range correlations related induced by the head
field begin to dominate.
2.6.5 Log Conductivity Statistics
The predictors of the stationary log conductivity statistics worked well for both
the spacedependent and timedependent means although there seemed to be a slight
overpredictive bias for the flowweighted trajectory (see Figures 2.6 and2.8). The
AW/ET log conductivity variance was well predicted in that the turningbands
method performed well, as discussed in Sections 2.4.1 and 2.4.7. The log conductivity
variance of the FW/LT appeared to be lower than that of the AW/ET, as predicted by
63
Equation (2.73), but the magnitude of under prediction was larger than that observed
(see Table 2.4).
Table 2.4: Comparison of observed log hydraulic conductivity variances to estimators.
Observed value is zeroslope line regressed through data. The decimal representation
of the fractional value is included for convenience.
parameter
predicted
observed
AW/ETffjU*]
1/2
0.497
FW/LTfffni[i]
7/16=0.4375
0.484
AW/LTafnt[tl
1/2
0.498
\nk = 1
parameter
predicted
observed
AW/ETffjUM
1
1.002
FW/LToU*]
3/4
0.944
AW/LTVfnt[i]
1
1.000
2.7 General Conclusions
In an steady irrotational flow field, a fluid parcel trajectory follows a hydrodynamic
streamline. The flow field properties as observed along streamlines can differ radically
than those observed uniformly in space. For the model system, a stationary Eu
lerian log conductivity field, this work identified the trajectory collections for which
certain properties appear statistically stationary to second order. For displacements
along the mean flow direction, point velocity and log conductivity appear stationary
along streamlines separated by equal discharges. For displacements in travel time,
these same properties appear stationary along streamlines that are selected uniformly
in space. The stationarity of these properties facilitates the development of simple
expressions for the statistics of displacement and travel times, as we shall show in
following chapters.
An immediate result of the preceding analysis is that the bulk of the flow field in
the sense of total discharge travels at a velocity that is locally greater than the spatial
64
average, and this disparity increases with heterogeneity of the underlying conductivity
field. This might be described as preferential flow. A practical implication for the
transport of kinetically sorbing or interacting solutes is that the local velocity for the
majority of the flow may occur in a regime significantly higher than that estimated
as the flow field average. An understanding of the medium heterogeneity is essential
if laboratory experiments are to be scaled up.
It is intuitively obvious, and has certainly been noted before, that the bulk of the
flow passes through an increasingly small portion of the total swept volume as the
medium heterogeneity increases. Again, the practical implications are likewise obvi
ous. Consider some aquifer remediation effort that involves flushing a contaminated
zone with some fluid. This might be as simple as the upgradient water in the case of
pumpandtreat or as extravagant as an surfactantenhanced microemulsion. In het
erogeneous flow systems, the bulk of the flushing fluid contacts a disproportionately
small volume of the total swept volume. For contaminants that tend to reside in areas
of high conductivity, the serves to enhance the efficacy of remediation efforts. Con
taminants that tend to reside in areas of low permeability, conversely, would be more
difficult to remove. While these conclusions may be drawn from common sense,
application of the concepts developed here may help to better understand these issues
quantitatively.
CHAPTER 3
NONREACTIVE SOLUTE TRANSPORT
3.1 Introduction
The previous chapter saw the derivation of the relationship between the velocities
that lie along Lagrangian trajectories and the parameters that describe a simple
model Eulerian flow field. These Lagrangian relationships are employed to derive
a few functions that characterize global measures of solute transport: namely the
statistics of mass displacement in time and the statistics of mass arrival times at
control planes. The frameworks are are that of Dagan [1982b] and that of Shapiro
and Cvetkovic [1988], respectively. The recovery of a few wellknown results are
anticipated, in addition to some that are new. However, the approach is novel in that
the work is based upon trajectorybased statistical properties of the flow field.
3.2 Injection Mode and Streamtube Selection
Demmy et al. [1999] noted the relationship between boundary conditions and
sampling of streamlines. A clarification of this relationship and a generalization the
results of that work are sought. Unlike the previous chapter, the Eulerian trajecto
ries are not considered in the context of an actual transport trajectory, as they are
of little practical importance. However, the results of Shapiro and Cvetkovic [1988]
correspond to areaweighted Euleriantrajectories, as will be shown. Therefore, tra
jectory, Lagrangian trajectory, streamtube, and streamline are used interchangeably.
3.2.1 Uniform Resident Injection
Consider the application of mass at a constant density c0 across the entire face of
the injection plane (IP) of the model aquifer (see Figure 2.1). Let the mass occupy
65
66
some small Ax that is invariant with displacements along the IP. Following the ter
minology of Kreft and Zuber [1978], this injection mode is defined to be a uniform
resident injection. Uniform refers to the homogeneous mass density in space and
resident refers to the volume of resident fluid into which the solute is introduced.
Subsequent references to a resident injection will refer to a uniform resident injection
unless otherwise noted. Now, this injection mode has nothing to do with the flow field
per se. That is, the mass magically appears and occupies a fixed volume at a fixed
concentration without being influenced by local water fluxes. Consider the flow field
to be divided into some number of streamtubes, say N. Moreover, let these stream
tubes carry an equal areaweight at the IP. For a small depth of injection Ax, this
injection mode has assigned an equal mass weight to each streamtube with respect
to the total solute mass. The total mass of solute is
Mr coOhLAx
(3,1)
The mass weight for any one streamtube i is
mi 1
M =
(3.2)
This value is identically equal to the areaweight of the streamtube, and constant for
all in the collection.
Consider now dividing the streamtube into N streamtubes that all carry the same
flow weight. The mass in some streamtube i is given by
m a
~M = ~L
_ U
Vi
(3,3)
67
3.2.2 Injection in Flux
Consider now maintaining the IP at Co for some brief interval At. A mass
Mf = c09hLUAt
(3.4)
enters the domain and the different streamtube collections. This injection mode is a
uniform injection in flux. Uniform, again, refers to the uniform mass density and in
flux refers to the influent water that carries the solute into the flow domain. A mass
rrii = c09hViAt
enters equalarea streamtube i. This corresponds to a mass weight of
rrii Ut
~M = U
A mass
rrii codhciiViAt
= cod At
enters equalflow streamtube i. This corresponds to a mass weight of l/N.
(3.5)
(3.6)
(3.7)
3.3 Integrated Streamtube Properties
Let us return to the notion of the streamtubes as pure hydrodynamic entities. It
is possible to discuss properties of these streamtubes with no mention of injection
modes. Two properties hold some hydrodynamic interest. The first is the velocity
integrated over time. For for some streamtube i this is
x
:[t] = f Vi[t]dt
Jo
(3.8)
A velocity integrated over time is a displacement with a dimension of length. The
second the inverse velocity integrated over some distance. For some streamtube i this
68
is
(39)
This integral has a dimension of time. These integrals are given in terms of the x
velocity component as a function of longitudinal displacement or time. It is suggested
that future works examine the details of a more general approach, e.g., displacements
along the trajectory.
3.3.1 Displacements
Consider a model flow field such as the one given in the previous chapter. Quan
tification of the displacement and travel time statistics for areaweighted and flow
weighted trajectories is sought. Previous results indicate that the velocity for a
collection of areaweighted streamtubes is stationary in time, characterized by mean
U and approximate covariance U2bo\nk exp[bUt/X\nk] The expected value of Equa
tion (3.8) is
(3.10)
= Ut
Although the velocity of the flowweighted streamtube collection is not stationary in
time, the results of the previous chapter are employed to map the properties of the
areaweighted collection to the flowweighted collection.
(3.11)
Recognize that this equation may be rearranged to be the integral of Equation (2.63)
in time. Substitute the RHS of Equation (2.63) into the preceding equation and
69
integrate to yield
= Ut + Aifc(T2nA(l exp[M/i/Alnl])
(3.12)
The displacement variance for the areaweighted streamtube collection is given by
var[xa[t]]  2
v[t"]dt"> {Ut)2
(3.13)
3.3.2 Travel Times
Expressions for the travel time statistics are sought, in analogy to those sought for
displacement. The reference collection of velocities from which these expressions will
be derived is the stationary velocities of the equal flow streamtubes, or the equalflow
weight Lagrangian trajectories. The mean travel time for these streamtubes is given
by
(3.14)
X
U
The mean travel time for the areaweighted streamtubes is
(3.15)
70
The arrival time variance for the flowweighted collection of streamtubes is
(316)
var
Approximate the inverse velocity covariance contained in the first term of the RHS
of Equation 3.16 with the a series expansion of the perturbation equation
1 1
<
v[x']v[x"}
> =<
>
V2(l + u'[a:'])(l 1 v'[x"])
=< T(i + vyf AA + V? )>
= lj(l + 2 al + C,[*',A + HOT)
(3.17)
_1_ / Cvf[x',x"}\
u2 V v2 )
where HOT is the collection of higher order terms. Inserting this expression into
Equation 3.16 yields an expression for the travel time variance for fluxweighted
streamlines
var[t/[x]] + ^p(exp[kr/A?nJt] 1)^J (3.18)
3.4 Solute Transport
Expressions for the statistics of the integrated trajectory properties of displace
ment and travel time have been derived in the absence of an explicit reference to
solute transport to demonstrate that these properties exist simultaneously in the
same flow field. However, they are intimately related to solute transport. Consider
the transport of an infinitesimal particle whose movement is constrained to follow a
streamline. Moreover, at any time or position on the streamline, the particle velocity
is identically equal to that of the underlying velocity field. Its displacement from
x = 0 in t = T is given by Equation (3.8) and the time required to travel from x = 0
to x = X is given by Equation (3.9). For the same flow field, two collections of
streamtubes have been discussed, namely areaweighted and flowweighted. Placing
71
one particle into each areaweighted streamtube at x = 0 and t 0 is analogous
to the a uniform resident mass injection. Equations (3.10) and (3.13) express the
mean particle displacement and particle displacement variance as a function of time.
Equation (3.15) expresses the mean particle travel time at different control planes.
Similarly, placing one particle into each flowweighted streamtube at x = 0 and
t = 0 is analogous to the a uniform injection in flux. Equation (3.12) expresses
the mean particle displacement as a function of time. Equations (3.14) and (3.18)
express the mean particle travel time and particle travel time variance at different
control planes.
In addition to these two injection modes, there are hybrid injection modes. For
example, consider if the introduction were somewhat random, or particle mass were
some deterministic function of the injection velocity, or some combination. In fact,
it is left to the imaginative reader to conceive of different injection mode possibilities
and the physical systems to which these correspond.
3.5 Evaluation of Expressions
In the previous chapter, a numerical experiment used to test certain hypothesis
about the relationship between Lagrangian and Eulerian descriptions of flow fields
was described. Those data are employed here. In the comparison of the derived
expressions to the data, the functions are based upon the simulation input parameters,
and all normalization is carried out using these same parameters. Therefore, the
expressions represent a priori estimates of the simulation results, based, of course,
on known input parameters.
3.5.1 Displacements
Figure 3.1 presents the model estimates of mean mass displacements for the two
injection modes. The flux injection preferentially selects, or mass weights, high veloc
ity areas of the flow domain, and the propagation of the center of mass is characterized
72
by a higher velocity. The timevarying mean velocity expression for the flowweighted
Lagrangiantrajectories (Equation (2.63)) is, in fact, an expression for the mean ve
locity of a plume resulting from a uniform flux injection. The longtime asymptotic
slope of the fluxweighted mean displacement is U, as is easily verified by taking limit
of Equation (2.63) as t oo. The mean displacement functions predict the observed
behavior, and also provide an excellent fit to the data.
The displacement variance data are interesting (see Figure 3.2). Injection mode
seems to have little effect upon the displacement variance in time. This is not to
say that injection mode has little effect upon the spatial distribution of the plume
in time. At a given time t, a plume injected in flux will have traveled a greater
distance, on average. After traveling the same distance, the displacement variance
of the fluxinjected plume will be less than that of the resident injection. Thus,
the spreading as function of mean displacement is less in the case of the flux
injected solute. The areaweighted trajectory displacement variance overpredicts the
observed displacement variances, and the longtime slopes appear to have different
values. After 10 characteristic times, the exponential term in Equation (3.13) retains
a little more that two percent of its original value. This difference does not seem large
enough to account for the observed discrepancy. The variance of the stationary area
weighted Lagrangiantrajectory velocity field in time appears to be well predicted.
Thus, the problem may lie with the rather poor description of the shorttime and long
time correlation exhibited by the observed velocity covariance given by the assumed
model.
3.5.2 Travel Times
Figure 3.3 shows good agreement between the estimated and observed mean travel
times. The injection influx shows earlier arrival than that of the uniform resident
injection. Uniform resident injection weights all areas of the IP equally. Large areas
that contribute little to the overall flow receive and equal mass weighting as those
73
G
10
8 
7*
.X A
AW/LT +
FW/LT x
Eq. 3.12
Eq. 3.10
.*,+
x A
,*v
,xV
.X 4
,v
.V
.*X .4
.**/
.X4+
X .4
.X ,4
.X ,4
X ,4
X ,+
jf
.x4
X?+
X+
x+
.X
:*.+
.x
X/
.X X
10
tU/X\nk
x.
c
10
8 
.X
,*x
V
AW/LT
FW/LT
Eq. 3.12
Eq. 3.10
X ..+
X ..+
.x .+
.'X A
,xv /
,x* /
,x* ++
'X .4
.X
* *
4
x 4
X 4
X 4
/X
** *
X
.'X 4
X .4
/X .4
.X ,4
* y*
4
X.4
**
x+
o *
10
tU/X\nk
Figure 3.1: Comparison of mean displacements for the two injection modes or stream
tube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics cor
respond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT)
statistics correspond to a uniform resident injection. The top figure contains the
74
15
C)
a
c
b
JÂ£
e
?
IM H
b
10
AW/LT +
FW/LT x
Eq. 3.13
&
//
/Â¥
.* Â¥
'Â¥
Â¥
/ *
V*
*
4%
y
4 6
tU/X infc
10
15
10
C
b
S
c
CN H
b
AW/LT +
FW/LT x
Eq. 3.13
+x
+x
/ A
/ A
/ **
V
,v
/A
vÂ¥
***Â£
:***
4 6
tU/X\nk
10
Figure 3.2: Comparison of displacement variances for the two injection modes or
streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics
correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT)
statistics correspond to a uniform resident injection. The top figure contains the
a\nk 1/2 data and the bottom figure contains the ofnjt = 1 data. Variances are
normalized by \fnkofnk.
75
areas that contribute more significantly to the overall flow. However, as the solute
moves through the domain and experiences the full range of velocity variability along
its trajectory, the slow start is damped out and the instantaneous average point
velocity for all for the particles is V and the harmonic mean V/, = U. The travel time
variances appear to be better matched than the displacement variances (see 3.4). This
is reasonable in light of the better apparent match of the assumed velocity covariance
function in space than that in time. The uniform injection appears to exhibit less
nonlinear behavior than the injection in flux. The uniform injection variance is
reasonably expected to be larger than that of the flux injection, due to the skewed
massweighting of lowvelocity areas by the uniform resident injection.
3.6 General Conclusions
It is clear that the injection mode effects are most prevalent in the near field
where the initial velocities and travel times or displacements are well correlated. Af
ter the plume travels several correlation scales, these injection mode effects diminish.
The near field and the neartofar transitional scales are quite interesting, as these
intermediate scales are on the order of sites associated with certain agricultural set
tings such as some feed lots and waste holding facilities, urban land uses such as gas
stations and dry cleaners, and their associated remediation efforts, and a plethora of
field scale experiments. These scales are examined in the perspective of the error
in travel time estimate associated with assuming a resident injection when the real
injection is in flux
e[x] = \\nktfnk{l exp[6x/Ainfc])/a; (3.19)
given by subtracting Equation (3.15) by Equation (3.14) and dividing the difference
by Equation (3.14) (see Figure 3.5) The near field might be characterized as the first
five to ten correlation lengths, where the injection mode effects are quite prevalent,
even for modest log conductivity variability. These effects diminish less rapidly over
76
40
30
Vi
c
20
t
10
0
0 10 20 30 40
^/^ln k
40
30
Vi
G
^ 20
t
10
0
0 10 20 30 40
E / ^ln k
Figure 3.3: Comparison of mean arrival times for the two injection modes or stream
tube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics cor
respond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT)
statistics correspond to a uniform resident injection. The top figure contains the
a\nk = 1/2 data and the bottom figure contains the afnk = 1 data.
AW/LT
FW/LT
Eq. 3.15
Eq. 3.14
+*
jyr
d?
&
+v*
A'
X
d?
&
&
+y
AW/LT
FW/LT
Eq. 3.15
Eq. 3.14
+x
J
J?
+X
Jr
J
Jc
**
Jr
ft
jr
Jk
J
77
60
C
b
Â£ 40
S
b
20
AW/LT +
FW/LT x
Eq. 3.18
+ X'
Ay
+ x .
+ X
+ X *'
+ X.'
. X/
+ vxy
V
+ X..*
+ X.*
+ X/
+ X.'
+ *
+ X'
+ x
+ X'
, + X
. + X
++ X'
+ x*
+ x*
X
+ X'
+
+y
10
20
% / X\n fc
30
40
Figure 3.4: Comparison of arrival time variances for the two injection modes or
streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics
correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT)
statistics correspond to a uniform resident injection. The top figure contains the
inJt = 1/2 data and the bottom figure contains the ofnfc = 1 data. Variances are
normalized by {U/ (\\nkoXnk))2.
78
Figure 3.5: Travel time estimate error as function of displacement associated with
assuming a uniform resident injection when injection is in flux for = {1/2,1}
a transition region between 10 and 30 correlation lengths. Beyond 30 correlation
lengths, the difference between the two modes is less than a few percent.
Much of the Lagrangian transport literature is predicated upon a uniform resident
injection, but the socalled consistent firstorder travel time estimate that is regularly
employed is quantitatively that for a flux injection. For large displacements, this
error is small, and the simplification afforded by this approximation is considerable
(compare Equation (3.14) and Equation (3.15)). For many practical applications,
an error of ten, or even more, percent might be considered negligible. Therefore,
it is not suggested that this approximation be abandoned, but rather phrasing such
as inconsistent firstorder approximation be applied in future works, since this
Lagrangian expression is inconsistently based upon the Eulerian mean velocity.
CHAPTER 4
REACTIVE SOLUTE TRANSPORT
4.1 Introduction
The concepts developed in the previous chapters are employed to analyze reac
tive solute transport. The focus is upon global measures of transport, rather than
predictions of local quantities. Particular processes of interest are injection mode
effects, the effects of correlation of the local reaction parameter to the log con
ductivity field, and the interplay of these injection mode and correlation effects. A
suite of numerical experiments illustrates these effects. Additionally, expressions are
developed for temporal moments of mass breakthrough curves for a solute subject to
linear equilibrium sorption for the different injection modes that reflect the observed
behavior in relatively simple closedforms.
Spatial moments of sorbing solutes are not directly treated in this work. The
experimental design was targeted directly towards a temporal moment analysis, and
this design decision has some implications for spatial moment analysis. These decision
and implications are discussed, in hopes that future works might be guided by this
analysis. Some of the properties of the travel time dependent reaction parameter are
discussed, as are some of the implications for solute transport.
4.2 Temporal Moments
The concepts developed in the preceding chapters are applied to examine the effect
of injection mode upon the temporal moments of the breakthrough curve of a solute
subject to linear equilibrium sorption. A uniform mean gradient induces a steady
irrotational flow in a uniformly saturated porous medium characterized by an expo
nentially correlated lognormal conductivity distribution. This flow field is divided
79
80
into collections of noninteracting streamtubes. Dispersion at the streamtube scale
is considered to be negligible in its effect on the aggregatescale temporal moments
(e.g., Dagan [1989]). The sorption process is related to a generic reaction parameter
that is heterogeneous and may be correlated to the log conductivity field.
The previous chapters demonstrated the relationship between streamtube selec
tion strategies (equalarea or equalflow) and injection mode (uniform resident and
influx). The reference collection is equalflow streamtubes, since the associated ve
locity and log conductivity statistics are stationary in displacements along the mean
flow direction.
The primary objective of this chapter is to examine the effect of injection mode
upon the temporal moments of the mass breakthrough curve of a solute subject to
linear equilibrium sorption in a heterogeneous flow and sorption field. A brief review
the conceptual model of solute transport is given. General expressions are presented
for the first two temporal moments of the breakthrough curves associated with the
uniform injection of a solute both in the influent fluid flux and in the resident fluid.
The results of the reactive solute transport simulations are presented, followed by the
development of some analytical expressions that help explain the observed behavior.
4.3 Conceptual Overview of Solute Transport
Consider a general multidimensional flow field that is resolved into a collection
of noninteracting streamtubes. Transport along an individual streamtube is charac
terized with a transfer function y[t; x\ (see Jury and Roth [1990] and Cvetkovic et al.
[1998]). For a conservative and nonreactive solute, the transfer function is simply
7[t, x\ = 5 [t t[x]} (4.1)
where 6 is the Dirac delta function and r is the travel time. The physical interpreta
tion of this transfer function is this: a particle located at x = 0 arrives at x X
at time t = r[X]. We conceive of this travel time as not only the time required to
81
travel to some point x, but as an intrinsic integrated streamtube property, namely
the inverse velocity integrated along the trajectory (e.g., Equation (3.9)).
Recall reaction flow path concept of Cvetkovic et al. [1998]. The reaction flow
path, parameterized in space, is defined as
(4.2)
where P[x] is some reaction parameter of interest. For linear equilibrium sorption,
P is a linear partitioning coefficient. The reaction flow path is also an integrated
streamtube property. As such, any reference to n properties in general are broadly
applicable to any distributed reaction parameter, perhaps a firstorder decay coef
ficient or a Langmuir sorption parameter. However, the focus here is upon linear
equilibrium sorption to promote a ready and intuitive grasp of the pertinent con
cepts. For this case, Â¡i may be thought of as the time that the solute is sorbed. Thus
the arrival time of a particle undergoing a linear equilibrium sorption process is the
sum of the nonreactive travel time r and the time sorbed fi. The transfer function
characterizing this transport and reaction process is (see Cvetkovic et al. [1998])
T[*i *] = l< T[x] lAA]
(4.3)
Following the work of Cvetkovic et al. [1998], the adopted reaction parameter
model is
P = Pg exp[io] exp[/3 In k]
(4.4)
where Pg is the geometric mean of P, w is a zeromean, exponentiallycorrelated
normal random variable with variance and correlation length A, = Ain* that is
uncorrelated to the log conductivity field, and /? is a strength of correlation parameter
relating P to the underlying log conductivity field. The equality A^ = X\nk was
specified for convenience.
82
Two specific injection modes are considered: uniform injection influx and uniform
resident injection. For the uniform injection influx, each streamtube receives an equal
amount of solute mass (Equation (3.5)), and thus mass weight (Equation (3.6)). Uni
form resident injection distributes the mass uniformly in space, but each streamtube
receives a different mass and mass weight (Equation (3.3)). The mass breakthrough
for some streamtube i is given by
m[t; x] = rrii[0; 0]y[i; x] (4.5)
An ergodic condition is assumed such that aggregate properties may be thought of
as ensemble properties in the statistical sense. The aggregate mass breakthrough
is given by taking the expected value of Equation (4.5). For injection influx, the
expected solute breakthrough curve is given by
= M (4.6)
For uniform resident injection, the expected solute breakthrough curve is given by
= MU <'y[t]x)/v[Q\> (4.7)
where U is the arithmetic mean, or spatially averaged, velocity and v[0] is the velocity
at the injection point. The temporal moment of this breakthrough curve is sought.
The n temporal moment is given by
tn = lim(1)Tl^<5[s;x]> (4.8)
where S is the Laplace transform of S and s the corresponding Laplace space variable.
The Laplace transform of Equation (4.3) is given by
7[s; x\ = exp [s (r[x] + //[x])]
(4.9)
83
Two temporal moments of interest for the influx breakthrough curve are the mean
tif = + < fx> (410)
and the variance
h f + 2 oTil (411)
For uniform resident injection, the moments are
t\f =< Ut/v[ 0] > + < Ufi/v[ 0] > (4.12)
and the variance
hf 2 [0]>2 (413)
From these equations, it is seen that the mean and variance of the expected reactive
travel time distribution is comprised of the first two moments of the nonreactive travel
time and the reaction flow path and the cross moment of the travel time and reaction
flow path.
4.4 Simulation of Reactive Solute Transport
The results of the simulations described in Chapter 2 are employed to simulate the
transport of a reactive solute. As mentioned in that chapter, information was recorded
along two particle trajectories: the first was a natural trajectory corresponding to a
hydrodynamic streamline, and the second was a straight line parallel to the mean
flow direction. Data were recorded at equal increments in time after injection, and
at control planes perpendicular to the mean flow at equal displacements from the
injection plane. Among these data were the velocity, the hydraulic conductivity, and
an observation of a lognormal random field with the same statistical and correlation
properties as the log hydraulic conductivity field.
84
4.4.1 Reaction Parameter
From these data, trajectorybased P fields were constructed using Equation (4.4).
In this manner, any variety of correlation relationships may be analyzed using the
same set of simulation data. There is a dearth of information about the structure of
sorption fields in natural systems (see, e.g. Tompson [1993] or Jawitz [1999]). Thus,
numerically convenient parameter values were selected. This suite of experiments ex
amined correlation parameter values of /3 = {1, 1/2,0,1/2,1} in log conductivity
fields of variances a^nk = {1/2,1}. The extreme correlation parameters /3 = {1,1}
are arguably physically implausible, yet they are included to demonstrate the effect
of correlation without the noise associated with an uncorrelated component. The
remaining three values, Â¡3 = {1/2,0,1/2} might be considered a range of more plau
sible values. For consistency, the variance of point log reaction parameter was held
constant and equal to that of the log conductivity field by requiring cr[} = (1 /32)afnk.
The expected value of P was 1. This is a slightly different approach than that taken
with the log hydraulic conductivity, where the geometric mean was fixed at Kg = 1.
In the case of the twodimensional isotropic log conductivity field, the effective con
ductivity is equal to the geometric mean. The expected value of the reaction value
was fixed < P >= 1, because in some practical applications P is assumed to be lin
early related to some property of interest, and the amount of that property is to
be estimated based upon the temporal moments of linearly sorbing solutes, or tracers
(see e.g., Jin et al. [1995], Annable et al. [1998], and Jawitz [1999]).
The point average of reaction parameter P as observed along a Lagrangian trajec
tory is a function of injection mode, correlation to the log conductivity and the vari
ance of the log conductivity. As with the log conductivity, the equalflux streamtubes
exhibit an apparently stationary mean reaction parameter in displacement along the
mean flow direction, whereas the equalarea streamtubes exhibit distinctly nonsta
tionary behavior that is analogous to that of the log conductivity (see Figures 4.1 and
85
4.2). These observations, of course, are not surprising, since the reaction parameter is
a function of the log conductivity. Positive correlation between the log conductivity
and the reaction parameter (Â¡3 > 0) results in a reaction parameter trajectory mean
that is larger than the uniform spatial mean. Negative correlation (Â¡3 < 0) results in
a reaction parameter trajectorymean that is smaller than the uniform spatial mean.
No correlation (Â¡3 = 0) results in equal trajectorybased and uniform spatial mean
reaction parameter estimates. A decrease in the system variability (e.g., the variance
of log conductivity cr,2nfc) results in a smaller difference between the volume average
and the trajectory averages.
4.4.2 Reaction Flow Path
A simple trapezoidal rule was used to calculate the reaction flow path // along a
solute particle trajectory. For a collection of n trajectories, there were n // obser
vations at each control plane. For an intuitive grasp of what the statistics of the
reaction path means, recall that for linear equilibrium sorption, the reaction path
corresponds to the time that the solute is sorbed. The sum of the the nonreactive
travel time and the sorbed time equals the total reactive travel time of the solute
from the point of injection to the point of observation. See Cvetkovic et al. [1998] for
further application of the statistics of the reaction flow path.
The mean and the variance of the reaction flow path are presented for Lagrangian
trajectories that carry either an equal flow weight or an equal area weight at a ref
erence plane. Recall that the equal flow collection can be thought to correspond
to a uniform injection influx and that the equal area collection can be thought to
correspond to a uniform injection into the resident fluid. The coupled effect of injec
tion mode and reaction parameter correlation to the log conductivity are illustrated
by examining reactive flow fields with five different correlation characteristics. The
first case is that in which the reaction parameter is completely determined by the
86
Figure 4.1: Mean reaction parameter values for different values of correlation pa
rameter /? and different trajectory collections for the a^nk = 1 set of simulations.
Top figure is Â¡3 = {1,+1}. Middle figure is /3 = { 1/2,+1/2}. Bottom figure is
Â¡3 = 0 (no correlation). Trajectory collections are areaweighted Lagrangian (AL),
flowweighted Lagrangian (AF) and areaweighted Eulerian (AE).
87
O 10 20 30 40
%/^In k
Figure 4.2: Mean reaction parameter values for different values of correlation pa
rameter /3 and different trajectory collections for the crfnk 1/2 set of simulations.
Top figure is Â¡3 {1,+1}. Middle figure is Â¡3 = {1/2,+1/2}. Bottom figure is
/3 = 0 (no correlation). Trajectory collections are areaweighted Lagrangian (AL),
flowweighted Lagrangian (AF) and areaweighted Eulerian (AE).
88
conductivity field (0 = {1,1}). Considered are the two subcases, namely that of
perfect negative correlation (/3 = 1) and perfect positive correlation (0 = 1).
For Â¡3 = 1, the equal area collection exhibits nonlinear behavior analogous to
that of travel time (see Figure 4.3 and 4.4). However, this nonlinearity is enhanced by
the correlation of high reaction parameter values in the low conductivity areas. For
the equal flow collection, however, the nonlinearity is mitigated, and // appears to be
a linear function of displacement. For 0 = 1, the both trajectory collections exhibit
the same linear behavior. Thus positive correlation appears to mitigate the effect of
injection mode on the mean behavior. The reaction path variance is strongly affected
by the sign of the correlation (see Figures 4.5 and 4.6). For negative correlation, there
is a distinct difference between the injection mode. However, positive correlation
again mitigates the difference in modes. Moreover, the positive correlation greatly
reduces the reaction path variance. For linear equilibrium sorption, this implies a
reduction of the reactive travel time variance.
The next case considered is that of no correlation between the reaction param
eter and the log conductivity (/3 = 0). The reaction path mean exhibits the same
behavior as does nonreactive travel time. The nonlinearity in the mean for the equal
area streamtubes is attributable to the nonstationarity in the associated velocity field.
However, the combined effects of heterogeneous velocity and sorption manifest them
selves in the reaction flow path variance. The reaction flow path variance is higher
than that of the nonreactive travel time, even though the scaled means are the same.
The 0 = 0 reaction flow path variance lies in between that of the positive and
negative correlation. For the lower total system variability case o oÂ¡nk = 1/2, the
magnitude of the effect of correlation appears nearly symmetric (see Figure 4.6). For
CTinjt = 1, however, there the effect of negative correlation seems to have a greater
impact upon both total variability and the effect of injection mode (see Figure 4.5).
89
O 10 20 30 40
Â£/^ln k
Figure 4.3: Mean reaction path observed values and estimators for different values
of correlation parameter Â¡3 and different trajectory collections for the afnk = 1 set of
simulations. Top figure is Â¡3 = {1,+1}. Middle figure is [3 = { 1/2, +1/2}. Bottom
figure is (3 = 0 (no correlation). Trajectory collections are areaweighted Lagrangian
(AL) and flowweighted Lagrangian (AF).
90
% / ^ln k
Figure 4.4: Mean reaction path observed values and estimators for different values of
correlation parameter /3 and different trajectory collections for the Kk = 1/2 set of
simulations. Top figure is /3 { 1, +1}. Middle figure is /3 {1/2, +1/2}. Bottom
figure is /? = 0 (no correlation). Trajectory collections are areaweighted Lagrangian
(AL) and flowweighted Lagrangian (AF).
91
O 10 20 30 40
*^/^ln k
%/'bn k
Figure 4.5: Reaction path variance observed values and estimators for different values
of correlation parameter f3 and different trajectory collections for the
simulations. Top figure is Â¡3 = { 1,+1}. Bottom figure is ^ {1/2,+1/2}. Tra
jectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian
(AF).
92
Figure 4.6: Reaction path variance values and estimators for different values of cor
relation parameter /3 and different trajectory collections for the cr^ = 1/2 set of
simulations. Top figure is Â¡3 = {1, 41} Bottom figure is = { 1/2,+1/2}. Tra
jectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian
(AF).
93
4.4.3 Reaction Flow PathTravel Time Cross Moment
The reaction flow pathnonreactive travel time cross moment cr^ contributes to
the variance of the expected solute flux (see Equations (4.11) and (4.13)). For linear
equilibrium sorption, this reflects the correlation between the time sorbed and the
time traveling. The correlation of the reaction flow path and the travel time is little
affected by injection mode (see Figures 4.7 and 4.8). It is, however, sensitive to the
correlation parameter /3. As with the reaction flow path statistics, the deviation of
from the Â¡3 = 0 is greater in the case of positive correlation for the same value
of /?. The total system variability seems to diminish the effect of correlation upon
o^T. This is most readily apparent from a comparison of the two (3 {1,1} cases
for the different values of ofnA. (compare the top figures in Figures 4.7 and 4.8). The
correlations for the correlated case shift towards that of the (3 0 in the higher log
conductivity variance field, and the Â¡3 0 case is slightly lower for the greater
all cases except {3 = 1, Lh = ! the correlations appear to plateau at nonzero values.
This is not surprising, since both r and /i are integrals of the inverse Lagrangian
velocity.
4.5 Theory of Reactive Solute Transport
The concepts of the previous chapters are applied to extend the work of researchers
including Cvetkovic et al. [1998] and Demmy et al. [1999]. These results are used
to interpret the results of the numerical experiments. Expressions are sought for
the mean and variance of the reaction flow path for the two different collections
of streamtubes. Cvetkovic et al. [1998] have developed expressions for equal area
streamtubes in unbounded multidimensional flow fields. Their mean // expression
neglects the correlation between the reaction parameter and the log conductivity,
and is a linear function of displacement. Their variance expression is based upon
a series expansion of the inverse Eulerian velocity perturbation. This work extends
this work with consideration the correlation between the reaction parameter and log
94
*r/^ln k
Figure 4.7: Correlation between the reaction flow path and travel time for different
values of correlation parameter Â¡3 and different trajectory collections for the ofnA. = 1
set of simulations. Top figure is ft = {1,41}. Bottom figure is /3 = { 1/2, 41/2}.
Trajectory collections are areaweighted Lagrangian (AL) and flowweighted La
grangian (AF).
95
% / *bn k
Figure 4.8: Correlation between the reaction flow path and travel time for differ
ent values of correlation parameter (3 and different trajectory collections for the
a\nk = 1/2 set of simulations. Top figure is f3 = { 1,+1}. Bottom figure is
Â¡3 { 1/2,+1/2}. Trajectory collections are areaweighted Lagrangian (AL) and
flowweighted Lagrangian (AF).
96
conductivity in the calculation of the mean, working from Lagrangian properties that
are stationary in displacements along the mean flow direction, and examination of
the effects of injection mode.
4.5.1 Mean Reaction Flow Path
Substitution of Equation (4.4) into Equation (4.2) and taking the expected value
yields
= P* exp[cr
/2] f
Jo
exp[/3 In k[x]]
v[x]
dx
(4.14)
The stationary Lagrangian serves as a reference field. The log conductivity field is
decomposed into its mean and mean removed perturbation / and the velocity into
its mean and mean removed perturbation thus
<
. p* r 2 /91 f /exp[/3(/qnfc + /[g])]\ ,
n> Pg expK/2] ^ ^ V(l + v'[x]) )d ^
Define Pg = P* exp[/xinjt], and expand the nonlinear terms of the previous equation
into series, take the expected value retaining second order terms thus
Pg exp[cr^/2]
< fi >=
V
1(1 fioi, + Pcj/2 + ai)x
(4.16)
Recall that aj = afnk. The / subscript is used for clarity. This expression captures
the linearity of the observed mean fi (see Figure 4.3), but shows a heavy dependence
upon the sign of the correlation parameter /3 that is not apparent in the data. The
mean reaction flow path for the equal flow streamtubes appears to be dependent only
upon the mean value of the reaction parameter, e.g.
Pg exP[<4/2 + 020//2]
=
{l + av)x
V
(4.17)
97
For the hypothesized Lagrangian distribution V = U(1 + ba2) and a\ = ba2, thus the
Equation reduces to
= Pgexp[al/2 + /32a2/2]x/U (4.18)
dx
(4.19)
The equal area streamtube mean /x is
Manipulations similar to that which yielded Equation (4.16) yield
< Uy.lv[0] >= j\ 1 + 2al + /?VJ/2 + C,[l] 2/3C/[x])di (4.20)
Substituting the simplified relationship CvÂ¡[x] = a2 exp[x/(2X)]/2 yields
< Vp/v[0] >=^Â¡^1(1 + 2ba) + /}V,/2)x ^
+ 2(3\crf(exp[x/(2\)] 1) + Acr^(l exp[6x/A])
Notice that the first collection of terms, or the linear portion, of the preceding equa
tion is the same as that in Equation (4.16). Similar arguments are employed to yield
the equal area streamtube mean /x estimator
< Un/v[0\ >
Pgexp[al/2 + p2aj/2]
U
x
(4.22)
+ 2/?Aa^(exp[:r/(2A)] 1) + Ac^(l exp[bx/\})
4.5.2 Reaction Flow Path Variance
Adapting an expression for the reaction flow path variance given by Cvetkovic
et al. [1998] (their Equation (B4)) to the simplified velocity and velocitylog conduc
tivity functions central to this work, write
[x] = 2P2 exp[al\/U2 [ {x s)(Ctu[s] + P2Cf[s] 4pCfv[s\ + C^ds (4.23)
Jo
98
For the special case of = (1 (32)cr2f, the expression integrates to
ofa] =2A2ct2P2 exp[(l p2)a}]/U2{exp[x/\f] 1
+ ^(exp[6s/A/] 1) + 8/3(1 exp[/(2A/)]) (4.24)
+ (2 4/3)x/A/}
A simple and intuitive reaction flow path variance estimator for the equalarea
streamtube collection is untenable in this framework. This is due in part to the
nonlinear reaction flow path mean.
4.6 Discussion
Figures 4.3 and 4.4 demonstrate a close match between Equations (4.18) and
(4.22) and simulation data. The parameters placed into the estimators are the target
values selected for the simulation, and thus represent an a priori estimate of the
mean reaction flow path. Equation (4.22) captures not only the nonlinearity of the
nonreactive travel time, but the nonlinearities induced by reaction parameter and log
conductivity correlation. While the absolute difference between Equations (4.18)
and (4.22) appear small, consider that the scale of the simulations is fairly large. The
solute has traveled over forty correlation scales of the log conductivity. In settings
where the the scale of conductivity heterogeneity are large compared to the overall
scale, the relative impact of injection mode and reaction parameter correlation to the
log conductivity will have a relatively greater effect.
The equalflux streamtube reaction flow path variance estimator (Equation (4.24))
shows relatively poor performance in the more heterogeneous system (Figure 4.5),
compared to that in the less heterogeneous system (Figure 4.6). Moreover, this ex
pression does not work well at all for values of positive correlation, yielding physically
implausible reaction flow path variances (cP < 0) for correlation parameter values
greater that 1/2. The lack of robustness is perhaps attributable to an inadequate
99
representation of the myriad interacting processes by the firstorder perturbation ex
pansions.
Injection of solute in the influent fluid mitigates the effect of correlation on the
mean and the nonlinearity of mean nonreactive travel time. For tracer experiments
that determine the volume or area of some property based upon mean arrival times
of reactive and nonreactive tracers, correlation will have little effect upon the first
temporal moment of the solute breakthrough curve if and only if the solute is uni
formly injected in flux. However, injections that preferentially favor lowflux areas
may exhibit strong nonlinearities, and be particularly sensitive to negative correla
tions between the local conductivity and reaction parameter.
The contribution of the /x variance to the second moment of the reactive solute
travel time is extremely sensitive to correlation between the reaction parameter and
the conductivity. Experiments that rely upon higher order moments of tracer ex
periments are, therefore, questionable if there is no plnfc correlation information.
Whereas injection mode dominates the correlation behavior in the mean, correlation
dominates injection mode in the reactive travel time variance.
4.7 Spatial Moments
Symmetry would dictate proceeding with a corresponding spatial moment analysis
for transport of solute subject to a linear equilibrium sorption in a heterogeneous
reaction parameter field. However, this process presents a particular complication
that thwarts the development of relatively simple expressions following the same
methods those previously employed. In the case of the travel time statistics, the
control structures are two control planes. A fluid and solute parcel travels between
these two control planes, sweeping out a fixed and constant volume, without regard to
correlation properties, reaction parameters, or processes. The time required to sweep
this volume, of course, is highly dependent upon these factors. For a clock time
100
control structure, a solute parcel will trace different volumes for different processes
and correlation structures.
The reaction parameter model is comprised of two random processes. The first
is that of the log conductivity field and the second is a random component that
has the same distributional properties as, but is otherwise uncorrelated to, the log
conductivity field. In the case of the temporal moments, the weighted covariance
functions of the log conductivity and this random component were employed directly.
These functions can be parameterized, of course, in terms of the nonreactive travel
time. However, these travel time parameterized covariance functions time are not
immediately applicable to sorption analysis in the way that covariance parameterized
in clock time would be, because the displacement for a given clock time is highly
dependent upon the correlation of the reaction parameter to the log conductivity
field. This is not to say that sorptiondependent spatial moment analysis is not viable
in this framework. In fact, Cvetkovic et al. [1998] analyzed the spatial moments
of a solute subject to nonlinear equilibrium sorption in a heterogeneous sorption
field in this framework. However, the sorption parameter and the underlying log
conductivity field were uncorrelated. Further clock time oriented are developments
left for future work. Expressions involving the travel time dependent reaction flow
path are developed and applications to and implications for first order decay are
discussed.
Continuing the theme of this dissertation, expressions based upon the trajectory
based covariance functions are sought. The timedependent reaction flow path for
the selected reaction parameter model is related to the log conductivity covariance
function, as shall be demonstrated in the following.
As in the previous chapter, transport along Lagrangian trajectories is considered.
Assumed are secondorder stationary velocities along areaweighted trajectories in
time and secondorder stationary velocities along flowweighted trajectories in space.
101
The log hydraulic conductivity is assumed to be similarly stationary. Moreover, the
point random process Pg exp[ic] is assumed likewise stationary, since it is uncorrelated
to the random log conductivity. From these assumptions, the expected value of the
timedependent reaction flow path is
< > =
= Pg exp[<4/2 + /?2
The variance is given by
var[/i[r]] =< Pg exp [w[t]\ exp [(3 In k[t}} dt
Jo (4.26)
Pg exp [w[t]\ exp [P In k[t]] dt' < /z[t] >2
Consider the integrals in the previous equation. The exact expression is denoted by
I and is approximated by the truncated series expansions of the exponential terms
thus
I Pg f f < (1 + tu[]) (1 + w[t']) (1 + p\nfc[t]) (1 + p\nk[t']) > dtdt' (4.27)
y Jo Jo
Note that a possibly more elegant approach would be to explore the Lagrangian
features of the conductivity field, rather than the log conductivity field, and work
from a stationary conductivity field covariance function and a similar uncorrelated
field. This would obviate the need for the expansions. This suggested inquiry is left
to future works. Expanding the above expression, taking the expected value, and
dropping products of covariances yields
I = Pg [T [{l + Cw[t,t']+ p2CXnk[t,t'))dtdt' (4.28)
Jo Jo
A few observations allow the adoption of an approximate expression for the time
dependent Lagrangian covariance function C\Dk[t, t']. Relatively simple expressions
that capture the behavior of the dominant processes are sought here. These expres
sions might facilitate future works that deemphasize simplicity for greater rigor and
f
< Pg exp[it;[i]] exp[/3 In &[Â£]] > dt
(4.25)
102
perhaps a better fit to the data. The Eulerian log conductivity covariance is the
foundation upon which the rest of this work has been built. The model field is ex
ponentially correlated in space, with correlation length \\nk The Eulerian velocity
covariance exhibits a much longer correlation length due to the dissipation of the
head across the field (see previous chapters). It is reasonable, therefore, to expect
that the Lagrangian log conductivity covariance correlation length will lie somewhere
in between that of the Eulerian log conductivity and the Lagrangian velocity.
At zero lag, the areaweighted Lagrangiantrajectory log conductivity covariance
is equal to the variance of the Eulerian field. This value is ofnfc, by definition. Thus,
an exponential approximation of the Lagrangian log conductivity covariance function
is
C\nkiat[t} = cr?nkexp[b'Ut/\lnk] (4.29)
where b' is a function similar to b that relates the correlation length to a correla
tion time. The approximation b' 1 is made, which is equivalent to assuming that
the Lagrangian and Eulerian correlation lengths are equal is space and substituting
x = Vht Ut into the spacedependent Eulerian log conductivity covariance func
tion. Comparison of Equation (4.29) with b' = 1 to covariance functions estimated
from log conductivity observations taken along areaweighted Lagrangiantrajectories
generated in the numerical experiments described in Chapter 2 shows good qualita
tive agreement (see Figure 4.9). The data exhibit an apparent dependence of the
Lagrangian correlation length upon the variance of the log conductivity field that is
not captured by Equation(4.29). One might expect similar behavior in the Eulerian
field a priori, since the appropriate substitution into the Eulerian covariance function
would be x Uht = Ut/( 1 + ba^nk). The higher variance of log conductivity results
in a lower harmonic mean average velocity, implying it takes longer to to travel a
unit length in more heterogeneous fields and a larger Lagrangian correlation time. In
103
1
0.8
0.6
I 0.4
g
0.2
0
0 2 1 1 1
0 2 4 6 8 10
Ut/X\nk
Figure 4.9: Assumed timedependent log conductivity covariance function along La
grangian trajectories (Equation (4.29)) compared to simulation data for variances
of log conductivity afnk {1/2,1}. The Lagrangian velocity covariance function is
Equation (2.61).
x \
\ x
V x
1/2
1
Lagrangian In k
Lagrangian V
\++xx
\ ++Â¥xx.
Xx
.T++X+xxXxxxxxXv
*4,..^ ^/xxxxx* Jv+^iv
+++++++*+;t+++ *+*
fact, this is the behavior observed in the simulations (see Figure 4.10) The data are
compelling enough to suggest further work. However, we shall accept Equation (4.29)
with b' = 1 as a suitable approximation.
Substituting Equation (4.29) with t = \t t'\ into Equation (4.28), integrating
and inserting the result into Equation (4.26) yields
r r 11 ^^"9 ^lnfc T cru,)A[n j. r tt /\ r\
var[/i[rj] = *1 (Ut/\lnk ~ (1 exp[t/r/AinfcJ) 30^
pK + ^Jr2
Neither this equation nor Equation (4.25) show a dependence upon the sign of Â¡3.
For the mean reaction flow path, the simulations show a weak dependence upon
correlation parameter (see Figure 4.11). The linear propagation of the mean reaction
flow path in time is due to the stationarity of the local reaction parameter along
the areaweighted Lagrangian trajectories. Again, it should be noted that time
104
c
o
o
o
0.8
0.6
2 0.4
0.2
0.2
%
\\\
+\\
W\
V?*
1/2
1
Lagrangian In k
Eulerian In k 1/2
Eulerian In k 1
0 
i'4x_*xxxxxxXxxx>
*4:+^=E i
.xx xxxxxxxvxx ~ v,.
4 6
Ut/X\nk
10
Figure 4.10: Hypothesized timedependent Eulerian log conductivity covariance func
tions for ofnk {1/2,1} compared to simulation data and Equation (4.29).
A
a.
V
Ut/X\nk
Figure 4.11: Mean reaction flow path as a function of nonreactive travel time com
pared to estimator Equation (4.25) for correlation parameter values /3 = { 1,0,1}.
105
refers to nonreactive travel time. Reaction flow path variance, however, is strongly
affected by correlation, and Equation (4.30) is completely inappropriate (see Fig
ure 4.12). Moreover, this equation can give physically implausible values at large
Figure 4.12: Reaction flow path variance a function of nonreactive travel time for
correlation parameter values Â¡3 = {1,0,1}.
values of time. This is a result of the series approximations required for the raw
second moment term in the variance, and the lack of such approximations in the
first moment squared term.
4.7.1 Injection Mode
The coupled effects of injection mode and correlation on the mean travel time
dependent reaction flow path are examined. Using the relationships developed in
Chapter 3, the hypothesis that the statistics reaction flow path corresponding to the
equalflow streamtubes are given by flowweighting the equalarea properties is made.
106
Thus, the mean reaction flow path for a flux injection is given by
<^Â§a>=<^Â§[ nm> (i)
In taking the expected value, the crosscorrelation of the initial velocity and the that
portion of the reaction parameter correlated to the log conductivity field must be
considered. Using nowfamiliar expansions, the expected value of the flowweighted
reaction flow path is
< >= 'W\1+pClu[t)/v)dt
U Jo
(4.32)
where Cfu is the covariance of the initial velocity perturbation and the log con
ductivity perturbation. Once again, this covariance shall be approximated with an
exponential function of the form (see Appendix B)
Ur
Cfu[r) = ^
exp
2A
In k
(4.33)
Substituting this term into Equation (4.32) and integrating yields
< ^H^[r] >=
U
U
(ut + /3a12nfcAinfcexp[
]
2A]nfc
(4.34)
4.7.2 Discussion
Similar to its space dependent counterpart, the mean time dependent reaction flow
path is sensitive to injection mode. However, it is the uniform resident injection that
appears to mitigate correlation effects upon the mean reaction flow path. Correlation
has a strong effect of variance, and it is quite likely that these effects dominate any
injection mode effects, as was the case for the space dependent //.
107
Ut/X infc
Figure 4.13: Mean reaction flow path as a function of nonreactive travel time
compared to estimator Equation (4.34) for correlation parameter values Â¡3 =
{ 1/2,0,1/2} (top figure) and /3 = { 1,0,1} (bottom figure). afDk = 1.
CHAPTER 5
APPLICATION OF CONCEPTS
5.1 Introduction
An illustration of a practical application of trajectory based flow and transport
modeling is sought. Useful and fairly robust relationships, namely predictors of spatial
moments of a continuously injected plume subject to a first order decay, using simpler
streamtube concepts (e.g. Jury and Roth [1990]). However, the observed bias in
prediction shall be explained using the more complete trajectory based approach.
In light of the robustness of the simple relationships developed here, they are used
to estimate relative degradation rates for different solute species and applied to field
data collected at a fieldscale macrodispersion and natural attenuation experiment
conducted in Columbus, Mississippi (see Stauffer et al. [1994] and Stauffer et al.
[1997]).
5.2 Theory
Consider a subsurface source of groundwater contamination that releases solute
into the aqueous phase at a constant concentration. An example of such a source is
a poorly mobile nonaqueous phase liquid, comprised of relatively insoluble organic
compounds, distributed at some nearresidual saturation through out a portion of
some aquifer. The constant assumption applies when the rate of change of a prop
erty of interest, such as the aqueous phase concentration at the source, is small in
comparison to the temporal scales associated with transport and decay.
The actual groundwater flow field is modeld as a steady irrotational flow field that
may be resolved into an aggregation of streamtubes. Neglecting the effect of local
108
109
dispersive processes, an advectiondecay equation with streamtube specific parame
ters is assumed to describe the fate and transport of the solute along the streamtube
trajectory. This advectiondecay equation is
dc dc
di = vTxkc
(5.1)
subject to c[x,t < 0] = 0 and c[x = 0,t => 0] = Co, where v is an effective advective
velocity [LT1], and k is a firstorder decay constant [T1]. The assumption of
effective streamtube parameters is the simplification that separates this method from
the more general consideration of properties varying along the trajectory.
Although much of this language is similar to that used in earlier chapters, the
fundamental difference is that the streamtube has constant effective parameters. This
is analgous to an assuption of stationarity of properties, but is different in significant
ways, as shall be shown.
For advective transport in a single streamtube, the resident concentration is equal
to the fluxaveraged concentration. However, the difference between these detection
modes manifests itself when concentrations are averaged over several streamtubes.
For the remainder of this discussion, any reference to a concentration will implicitly
imply a reference to a resident concentration. The solution to this equation for the
given initial and boundary conditions is
Q
= exp [kx/v] h[vt x] (5.2)
Co
where h[x] is the Heaviside step function, with the properties h[x] 0 for x < 0 and
h[x] 1 for x > 0.
5.2.1 Spatial Moments
The defintion of the raw spatial moment of order n of curve c is
/
n
L
oo xnc[x]dx
(5.3)
110
A normalized moment of order n is a raw moment of order n divided by the zeroth
moment, and is denoted by dropping the prime (i.e., (j,n). The zeroth spatial moment
of this concentration curve is given by
lilt] = j[l exp[kt\]
(5.4)
The zeroth spatial moment of a contaminant plume related to the total mass in
solution. The first normalized spatial moment, or center of mass, is given by
Hi[t] =
'nil exp[kt](kt + 1)
.hi 1 exp[/ci]
(5.5)
In the limit t oo fix = v/k.
The concentration given in Equation (5.2) is a function of this asymptotic center
of mass v/k. Solving Equation (5.2) with h[] = 1 for v/k yields
v x
k ln[c/c0]
(5.6)
Thus, an observation of the concentration at any point of a steady streamtube con
centration curve is an observation of its center of mass, given that c0 and x are
known.
Assume that independent samples can be drawn uniformly from the collection of
streamtubes that originate from a source c0 located at x = 0. From the linearity of
first moments, the mean center of mass may be estimated from N samples by the
simple estimator
x
1 N
= ~y
/V
Xi
N jy/ In [cj/co]
(5.7)
5.2.2 Relative Degradation
Consider two solutes a and b subject to a firstorder decay process, characterized
by rate constants ka and kb, respectively. The relative degradation rate of solute a to
Ill
b is defined as
kr
ka/kb
(5.8)
Let these solutes be released into a flow field at a steady rate, and allow the solutes
to reach a steady state in which the concentration profiles for each do not vary in
time. If within a streamtube i, the streamtube effective velocity Vi is a approximately
equal for solutes a and b, the relative degradation rate of a to b in streamtube i is
given by
kri (u/kb) / (n/ka)
(5.9)
Mlbt/Ml ai
That is, the relative degradation rate is given by the ratio of the first normalized
spatial moment of the steadystate concentration curve of solute b to that of a. The
average relative degradation rate is estimated as the arithmetic mean of kri. Notice
that this is not the ratio of the mean first spatial moments.
5.3 Application: Simulation
The estimator was tested by simulating the release of a solute into a heterogeneous
twodimensional aquifer using the United States Geological Survey flow and transport
code MOC3D [Konikow et ai, 1996]. The length units used herein are a dimensionless
quantity resulting from the normalization of the fundamental length by that of the
correlation length of the exponentially correlated log hydraulic conductivity field. In
the interest of clarity, however, this dimensionless quantity will be referred to as a
correlation length A.
Let the reference Cartesian coordinate system be oriented such that the xdirection
is parallel with the mean direction of flow, and the ydirection is orthogonal to that
and lies within the plane formed by the aquifer. Let the origin lie at the lower left
corner of the aquifer. The rectangular aquifer extended 15 A x and 25 A in y, and was
112
subdivided into 10 nodes per A in both direction. Constant heads maintained at the
x = 0A and x = 15A boundaries imposed a steady mean gradient of J = 0.01. Noflow
conditions were maintained at the y = 0A and y = 25A boundaries. To simulate the
release of a contaminant at constant concentration, the influent water concentration
was assumed to be a constant concentration Co = 1. This source was centered on
the x = 0 boundary, and was 10A wide.
A turningbands algorithm generated a synthetic heterogeneous log conductivity
field used in the simulations [Tompson et al., 1989]. The output of the turningbands
generator was a realization of an exponentiallycorrelated standard normal process
(/x = 0,
0.5, and 1.0 via the transformation
/ =
where o\nk is the target standard deviation of the log conductivity field / and 2 is the
standard normal process. The aquifer porosity was assumed to be equal to the mobile
water content and equal to a constant value n = 0.2. The effective conductivity
of a twodimensional exponentially correlated log conductivity field is, to a good
approximation, the geometric mean conductivity Kg. For p = 0, Kg = 1, in units of
(dimensionless) correlation lengths per characteristic time. The independence of the
effective conductivity from its variance convieniently allows the variance to be changed
without substantially changing the bulk flow properties. Thus, the anticipated Darcy
flux for the aquifer, regardless of crin*, was q= 0.01 A per characteristic time, and the
characteristic filtration velocity was uÂ¡, = 0.05A per characteristic time. The subscript
b refers to a bulk property, to be distinguished from streamtube properties discussed
earlier.
The solute was assumed to undergo a constant decay process characterized by a
firstorder rate coefficient k = 0.05 in units of an inverse characteristic time defined
113
by Vf/X. This value yields what might be termed a bulk first spatial moment
Vb/k = 1. For a homogeneous conductivity field, this is numerically equivalent to the
first longitunial spatial moment of the steadystate plume.
The steadystate concentration field was sampled by taking the reported nu
merical concentration values as the local resident concentration. The plume center
ofmass was estimated by the estimator presented in Equation (5.7). The actual
first longitudinal spatial moment was calculated using a trapezoidal rule integration.
A fate and transport simulations were carried out in 6 sets of hydraulic conductivity
fields corresponding to 6 standard normal process realizations.
5.3.1 Results
For the case of a homogeneous conductivity field, any one nonzero concentration
measurement used with the estimator given in Equation (5.7) returned a value for the
first longitudinal spatial moment that for all practical purposes was equal to Vb/k, and
five percent less than that predicted by the numerical integration. As the variance of
log conductivity increased, however, both the numerical integration and the estimator
tended to systematically predict values greater than Vb/k. Additionally, the estimator
systematically underpredicted the value given by the numerical integration, and by
an amount which generally increased with variance of the log conductivity.
5.3.2 Discussion
This systematic increase in the the first longitudinal spatial moment of the plume
with increasing log hydraulic condutivity variance, despite a fairly constant vÂ¡, is
explained by considering the nature of the underlying Lagrianian velocity field. The
steadystate concentration plume is, in essence, a map of travel times from the source.
As previously stated, Cvetkovic et al. [1996] demonstrated that the statisics of such
a field are nonstationary. For small displacements from the source, the mean travel
114
o 1
O 1 2 3 4 5
x/X
O 1 2,3 4 5
X/X
Figure 5.1: Resident concentration profile averaged along a transect orthogonal to
the mean flow direction for plumes for lk = {0.0,0.1,0.5,1.0}.
time propagates
dr/dx = 1 /vh
(5.11)
where vb is the harmonic mean velocity. For regions distant from the source region,
the mean travel time propagates
dr/dx = l/va (5.12)
where va is the arithmetic mean velocity, and equal in value to vb, the bulk filtration
velocity. The harmonic mean velocity is necessarily less than or equal to the arith
metic mean velocity, which in turn indicates that the mean travel time propagates as
a slower rate near the source.
Additionally, the correlation of the Lagrangian velocity persists over greater dis
tances than that of the underlying conductivity field. Solute which is introduced in
115
areas of high local flux tend to move away from the source at relatively high veloci
ties. The net effect of what might be termed a preferential flow of solute in these
higher velocity streamtubes, is to induce a tail on a transverselyaveraged resident
concentration profile, which increases with variance of log conductivity.
These effects are illustrated by comparing concentration profile created by averag
ing concentrations along planes normal to the mean direction of flow to concentration
profiles given by c/cq = exp[kx/v] and c = /c0 exp[/cr[x]], where r is given by
t\x] = (x 4 Kk^ink (1 exp[kr/AInfc])) (5.13)
va
Here Ah,*, is the correlation length of the log conductivity distribution and b is a shape
factor for the conductivity anisotropy.
5.4 Application: Field Data
Data from an elaborate experiment conducted at Columbus Air Force Base, Mis
sissippi were analyzed using the results dervied in the previous section. The objective
the experiment was to characterize the natural attenuation of certain hydrocarbons
in groundwater emanating from subsurface nonaqueous phase liquid (NAPL) sources
Stauffer et al. [1997]. A NAPL hydrocarbon source comprised of decane, benzene,
toluene, ethyl benzene, para xylene (pxylene), and naphthalene was emplaced at the
site of the MADE [Boggs et al., 1992] and MADE2 [MacIntyre et al, 1993] experi
ments. This source released these poorlysoluble organic solutes into the groundwater,
and the resultant plumes were monitored at downgradient multilevel sampler loca
tions. The center of mass of the steadystate plume was estimated using a trapezoidal
rule integration of concentration data in space and by a simple averaging of pointwise
observations of concentration via Equation (5.7). Relative degradation and absolute
degradation rates of the different constituents are estimated using Equation (5.9).
116
5.4.1 Results and Discussion: Center of Mass
The center of mass of the steady state plumes emanating from the emplaced
contaminant source was estimated by two methods. The first might be considered
a traditional approach, namely simple trapezoidal rule integrations of the data. A
known initial concentration for each species was assumed known. This concentration
was calculated from initial source mass fractions and aqueous solubilities given in
Stauffer et al. [1997] and assuming Raoults law (see Table 5.1). The second approach
was to use Equation (5.7) (see Table 5.1). The same initial concentration used in the
spatial integration was used with the estimator approach. Displacements were taken
as the distance from the multilevel sampler to the source zone.
benzene toluene ethyl benzene pxylene naphthalene
integration 4.0 2.0 3.1 2.3 3.8
v/k 5.3 1.7 2.5 1.9 3.1
Table 5.1: The center of mass in meters of steady plumes estimated by traditional
trapezoidal integration of spatial concentration data and by the mean v/k estimator
Equation (5.7).
Equation (5.7) gives results similar to those of the trapezoidal rule integration,
while offering some compelling computational advantages. While the spatial integra
tion is conceptually simple, it is computationally demanding for unequally spaced
data in that the different trapezoids must be calculated and summed for each
integration. New data require recalculation of these trapezoids. Of course, these
calculations are trivial for computer codes, but tedious for back of the envelope
calculations. Equation (5.7) is a fairly simple calculation for any scientific calculator,
and provides an excellent tool for a rapid estimate of the center of mass from sparse
and scattered local concentration. New data are easily incorporated into existing sets
without the need of a global recalculation.
117
5.4.2 Results and Discussion: Degradation
The relative degradation and absolute degradation of the different contaminants
were estimated using Equation (5.9) and degradation rates reported by MacIntyre
et al. [1993]. These reported benzene, pxylene, and naphthalene degradation rates
were estimated from another experiment conducted at the site in which a cocktail of
hydrocarbons was injected into the groundwater and the fate of the resultant plume
was monitored (see Table 5.2). This experiment was conducted before the NAPL
benzene
toluene
ethyl benzene
pxylene
relative
0.87
3.4
1.5
2.0
estimated
0.0056
0.022
0.0094
0.013
reported
0.0070
NR
NR
0.011
Table 5.2: Degradation rates relative to naphthalene and absolute degradation rates
in reciprocal days as estimated by Equation (5.9) and as reported by MacIntyre et al.
[1993]. NR denotes results not reported in MacIntyre et al. [1993]. Naphthalene
degradation reported to be 0.0064 d~x.
source was emplaced, and is in that sense independent of the concentration data as
sociated with the natural attenuation experiment. Nevertheless, the rates estimated
from the natural attenuation data were similar to those reported by MacIntyre et al.
[1993]. The technique used here is quite simple, and does not require regressions or
nonlinear fits to transport models.
5.5 General Conclusions
These estimators, while fairly simple and apparently robust, are limited in appli
cability to quasisteady plumes, and it is not clear the extent to which such plumes
exist. Sites where poorly mobile NAPLs have been known to exist for long times are
likely candidates, especially if decay rates are known to be large in comparsion to
transport time scales.
118
Local dispersion was neglected, and local dispersion certainly has a significant
effect on the local concentration variability. However, its neglect does not appear to
diminish the results here to the point of uselessness.
The nonstationarity of equalarea Lagrangian trajectories in a steady flow field
was neglected. For highly heterogeneous conductivity fields, low velocity areas are
disproportionately sampled with respect to the flow average velocity. For a spatially
uniform decay process, this would tend to underestimate the plume extent, since the
observed values will tend to be associated with long travel times. It should be
noted that this sampling bias is intrinsic to multilevel samplers.
The correlation of velocity and decay coefficient was neglected. A strong negative
correlation could result in a significant underprediction of the center of mass in a
highly heterogeneous field. Again, it should be noted that this is partially due to the
sampling bias assumed by the equalweight multilevel sampler sampling strategy that
will preferentially sample low velocity areas with respect to the flow weighted mean.
CHAPTER 6
GENERAL CONCLUSIONS
Any discipline in which you can publish a paper on boundary conditions
is an immature discipline. Sten Berglund1
Many of the results of this study are quantitatively and qualitatively the same
as those of previous studies. For instance, chemical engineers have long known that
the mean residence time of the swept volume of a reactor for a flux injection is
L/U, where L is the reactor length and U is the mean filtration velocity. However,
hydrologists seek to understand the effects of the heterogeneity of the reactor itself
and the interplay of the processes as they manifest themselves between x = 0 and
x = L.
An old premise served as the conceptual foundation of this work, namely that
a system, in this case a heterogeneous flow field, may be observed in two ways:
along straightline, or Eulerian, trajectories and along natural or Lagrangian
trajectories along hydrodynamic streamlines. The flow field was decomposed into an
aggregation of elements, either as area elements that lie between Eulerian trajectories
or hydrodynamic streamtubes that lie in between streamlines. Two criteria defined
four different collections of the flow field elements, namely that at some definition
plane, each element has either an equal area through which water enters the element,
or an equal volume of water enters the element per unit time. The imaginative
1 From a conversation with the author about the acceptance of Demmy et al. [1999]
for publication.
119
120
reader might contrive a plethora of trajectory and trajectory weighting schemes. For
compactness, the following notation summarizes the four different trajectories
AW/ET equalarea weighted Eulerian, or straight line, trajectory
AW/LT equalarea weighted Lagrangian, or streamline, trajectory
FW/ET equalflow weighted Eulerian, or straight line, trajectory
FW/LT equalflow weighted Lagrangian, or streamline, trajectory
Myriad methods exist for mapping properties observed along the different trajec
tories. That is, the value of some property may be observed along the trajectory and
recorded at different intervals in space or time. A traditional method is a regular Eule
rian gridding that divides the flow field into equallyspaced trajectories and records
information at regular intervals along the trajectory. Two specific parameterizations
are considered: equal intervals in displacement along the mean flow direction and
equal intervals in advective travel time.
Employing these concepts, the behavior of different flow field properties was exam
ined. These properties, such as log conductivity and local velocity, were observed
along the different trajectories at control planes, or planes equally spaced in the
mean flow direction, or control times, or at the displacement that an indivisible
fluid parcel would be after traveling for some reference time. For a steady and irrota
tional flow field resulting from a constant mean gradient applied across a stationary
Eulerian log conductivity field of uniform and constant mobile water content, the
trajectories for which the statistics certain properties appeared to be secondorder
stationary were identified. That is, the mean and variance of the property appeared
to have a constant value at all observation points along the trajectory. The log con
ductivity and velocity observed at control planes appeared stationary for the equally
spaced Eulerian trajectories (AW/ET) and Lagrangian trajectories separated by an
121
equal discharge (FW/LT). For equallyspaced control times, only Lagrangian trajec
tories equallyspaced at a reference plane (AW/LT) appeared to exhibit stationary
log conductivity and velocity statistics.
The stationarity of the properties greatly simplifies the development of predictors
of solute transport. When taking the expected value of the integral of some property
along a trajectory, say the integral of velocity in time, the timevarying velocity
function may be replaced with its time average.
As implied by the stationarity of the statistics along some trajectory collections,
but not others, the numerical statistics of properties varied between the different
collections. Adopting simple and accurate models for the statistical properties of
some property in a reference collection, the statistics of the properties along other
collections, including the observed nonstationarities, were well predicted. This had
significant application in the evaluation of the effect of injection mode upon solute
transport.
Two methods by which solute might be uniformly introduced into a system were
considered. The first was a resident injection were a fixed volume at the system inlet
was uniformly filled with solute at some reference concentration. The second was to
allow influent water at some reference concentration to carry the mass into the system.
The displacement and travel time statistics of AW/LT correspond to solute transport
associated with a uniform resident injection. The transport of solute injected in flux
is described by the displacement and travel time statistics of the FW/LT.
The socalled consistent firstorder approximations of the inverse velocity were
shown to be built upon a flawed premise, but are nonetheless quantitatively cor
rect and robust. The premise is that the Lagrangian flow field may be accurately
approximated by the Eulerian flow field. While this may be true for the weakest
of heterogeneities, this approximation breaks down rapidly with increasing hetero
geneity if it is consistently applied. As inconsistently formulated, the approximation
122
< 1/ti >~ l/U is quantitatively equal to the inverse of the harmonic mean of the
Lagrangian velocity < 1/v >= 1/14. Moreover, the timeaverage Lagrangian veloc
ity is quantitatively equal to the space average Eulerian velocity, and these equalities
contribute to the robustness of other approximations. Were the Lagrangian field to
be consistently approximated by the Eulerian, travel times would be greatly over
predicted and displacements greatly underpredicted. It is, in fact, the consistent
inconsistent approximation of the Lagrangian by the Eulerian that contributes to
much of the observed robustness of the Lagrangian transport theory. While not
many new equations were proposed or so many new phenomena were unearthed,
it is hoped that a modicum of understanding was fostered. This is the primary
contribution of this work.
These concepts were illustrated with an analysis of the transport of an solute ex
periencing a linear equilibrium sorption in a field characterized by a variable sorption
coefficient. While the process has been studied before, it is important to illustrate
the new concepts and approaches here with familiar examples. The sorption study
was restricted to transport as characterized by mass breakthrough at control planes.
An interesting new result is that injection mode strongly affects the propagation of
the mean breakthrough time. A uniform resident injection exhibits a nonlinearity
in the mean, and this nonlinearity is predictably enhanced by negative correlation
between the sorption coefficient and the underlying log conductivity field. Injection
in flux, however, results in a linear propagation of mean arrival time in space, and
mitigates the effect upon correlation. Arrival time variance, however, is dominated by
the correlation between sorption coefficient and the log conductivity field. Injection
mode has comparatively little effect. This has profound implications for tracer test
analyses based upon the arrival time variance of breakthrough curves.
The concepts developed in this work are further illustrated with the develop
ment and analysis of an estimator of the center of mass of a plume resulting from
123
continuously injected solute subject to a firstorder decay. The assumption of non
interacting streamtubes with constant effective properties allows solution of the
advectiondecay equation for a single streamtube. An ergodiclike hypothesis al
lows this solution to a distribution of streamtubes. This is what might be termed
a classic streamtube approach. While surprisingly robust and perhaps worthy of
application in field settings, the estimator shows a bias explained using the trajectory
based analysis developed in this work.
One measure of quality of a work is the number of questions that it answers. Per
haps another is the number of interesting questions that it begs. Several issues are
left for further study. A few specific questions are addressed first, then a few more
general. The relationship between an Eulerian reference system and its Lagrangian
counterpart remains incompletely understood. A primary area that requires attention
is the increasing Lagrangian velocity correlation length with increasing log conduc
tivity variance. A great deal of work remains for predicting the spatial moments of a
reactive solute.
Numerical constraints limited this study to relatively mild heterogeneity. How
ever, a highly heterogeneous medium, might be replaceable by two or more media
characterized by lesser heterogeneity, and a systematic replacement system would
greatly enhance computational efforts. The turning bands method employed in the
generation of the random fields was computationally expensive, in comparison to,
say, the particle tracking algorithms or data analysis programs, and to the flow solver
for weak heterogeneity. The method requires the generation of several independent
normal line processes. This problem is wellsuited to parallel and distributed compu
tational techniques, and an easytouse implementation would certainly find wide use.
Several research groups are working on parallel and distributed flow codes, and the
research community would benefit greatly from a wider dissemination of or greater
access to these codes.
124
Thinking more broadly, some of the concepts employed here might find application
in other fields. The flow of water through a wetland system might be reasonably
approximated as a twodimensional aquiferlike object. Transport of contaminants
might be modeled using similar techniques, especially if the transport time scales
are relatively short compared to system variability. In general, heterogeneity will
imply some sort of preferential flow, and the identification of preferential flows in
heterogeneous systems should be paramount consideration in the design of monitoring
systems.
APPENDIX A
EULERIAN VELOCITY DISTRIBUTION PROPERTIES
Consider lognormally distributed random variable V with moments < V > U
and var(V) = S. These moments are related to the parameters fx and a2 via
U = exp(/r + c2/2) (A.l)
S exp(2/x + 2cr2) exp(2/r + a2) (A.2)
Let M = exp(/r) and Z exp(cr2). Rewrite the previous equations as
U = MZ1/2 (A.3)
S = M2(Z2 Z) (A.4)
Solving for M and Z yields
M =
U2
VsTTp
(A.5)
z =
s + u2
u2
(A.6)
Expanding the definitions of M and Z and taking the natural log of both sides of the
previous two equations yields
H In
t2 = In
u2 \
Vs + u2)
(A7)
fS + U2\
(A.8)
( u2 )
125
126
The effective conductivity of a porous medium may be defined as the numerical
conductivity value that satisfies the Darcy relationship
Q = KeJ
(A.9)
where Q is the specific discharge and J is the difference in hydraulic head across the
formation. There are several expressions for the effective conductivity of a stationary
heterogenous conductivity fields (see Gelhar [1993]). The general form of the effective
conductivity is
Kc = KJ{alt,g)
(A.10)
where Kg and crfnk are the geometric mean and the variance of the log conductivity
field, respectively, g is a function of aquifer anisotropy, and / is a function that scales
the geometric mean to the proper effective value. For a twodimensional isotropic
aquifer, /(ofnJt) = 1. Thus, for an aquifer of homogeneous mobile water content 6,
an estimate of the spatial mean seepage velocity U is given by
(All)
The point velocity variance for such a field is S bU2afnk, where b 3/8.
Substitution of Equation (A. 11) into this expression yields
02
(A.12)
Substitution of Equations (A.11) and (A.12) into Equations (A.7) and (A.8)
fi = In
KgJ
0\/l + b(jfni
(A.13)
a2 = ln(l + ba2nk)
(A.14)
APPENDIX B
VELOCITY LOG CONDUCTIVITY CORRELATION
A simplified correlation relationship between the velocity field and the log con
ductivity is sought. Graham and McLaughlin [1989b] and Rubin and Dagan [1992]
derived expressions based upon perturbation expansions of Darcys law and spectral
methods. Graham and McLaughlin [1989b] assume a holetype covariance function
for the log conductivity field, in order to assure a complementary stationary head gra
dient field. Rubin and Dagan [1992] assume an exponentially correlated anisotropic
log conductivity field.
The velocitylog conductivity covariance functions for both analytical cases shows
qualitatively similar behavior (see Figure B.l), and an simplified estimator would be
applicable for both log conductivity correlation models. Although these rigorously
developed estimators are attractive theoretically, the mundane and oftneglected task
of mapping theoretical results into application often requires simplifying assumptions.
Presented for your consideration is the covariance function of Graham and McLaughlin
[1989b] (their Equation (B5)
P./K] = KrfJ/e^aOMaQ l(aC)2ifoK] ,
z (B.l)
(a/2) ((KM] <%K]) }
where a = 7r/(4A/) and is the modified Bessel function of order i. The integral
scale of this covariance function is Ay. An exponential approximation to this function
that reproduces the integral scale, general trend, and point covariance of that given
by Graham and McLaughlin [1989b] is sought. A function of the form
pvf\A = w 7 exp[x/(2A)] (B.2)
127
128
r/Xf
Figure B.l: Comparison of velocity and log conductivity correlation functions from
Graham and McLaughlin [1989b] and Rubin and Dagan [1992] and a simplified expo
nential model. All functions are normalized by KgJa^nk/6. The apparent roughness
of the Rubin and Dagan [1992] function is due to estimation errors associated with
inferring their function from graphical data.
129
meets these criteria (see Figure B.l).
The covariance function presented by Rubin and Dagan [1992] is the sum of
three twofunction products that involve integrations of Bessel functions. For two
dimensional porous systems, the simplified function condenses that monumental work
into an elegant and concise form.
The simplified expression is tested against correlation functions estimated from
the velocity and log conductivity data taken from the simulations (see Figure B.2).
The simplified estimator gives a reasonable a priori estimate of the velocity and
Figure B.2: Comparison of the simplified velocitylog conductivity correlation func
tion to simulation data for aj {1/2,1}.
log conductivity correlation observed along the equalflux Lagrangian trajectories.
The reasonable match to analytical expressions and the observed data enhance our
confidence in this simplified estimator.
The results of Rubin and Dagan [1992] (their Figure 7) indicate that anisotropy
decreases the point covariance and increases the long range correlation effects. Yet
130
the functional form appears to remain exponential. Therefore, it is suggested that
future work focus upon generalizing the simple results found here to threedimensions
and general anisotropy.
APPENDIX C
CONSISTENT FIRST ORDER APPROXIMATION
C.l Travel Time
Usage of a consistent firstorder approximation for travel time permeates the
stochastic Lagrangian literature. This approximation is quantitatively correct for
collections of equalflux streamtubes. However, the premise upon which it is built is
incorrect. Consider a stationary lognormal process V. The lognormal distribution is
selected for convenience, as a random variable with values guaranteed to be greater
than zero is required. Since V is stationary by definition, then so is 1/V.
Consider the expected value of the integral
roo
]>=< / dx/V[x]>
Jo
(C.l)
The order of evaluation may be interchanged, taking the expected value of the argu
ment of the integral thus:
roo
[x] > = / < dx/V[x] >
Jo
= x/Vh
(C.2)
where 14 is the point harmonic mean of the random process V. Consider now a
perturbation expression for the random process V = (1 + v) where V is the mean of
V, and v is a zeromean random process representing the perturbation of the process
about its mean. Inserting this expression into Equation (C.l) yields
y{l + v[x})>
(C.3)
131
132
Expand (1 + v) in a series,
1 r
=^r / < (1 u[x] + v[x]2 )> dx
VJ o
(C.4)
Should this expression be truncated at first order, the leading term < r x/V is left.
However, from Equation (C.2), the exact expression is < r >= x/Vh. < r >= x/V
is a wellestablished result in the literature for the first temporal moment of a solute
plume injected in flux Kreft and Zuber [1978].
The answer to this riddle lies in the examination of the properties of the flow
field. Even if the deviations are relatively small from the straightline, or Eulerian,
trajectory, the statistics of the properties recorded along the actual, or Lagrangian,
trajectory can be quite different. For instance, the mean velocity recorded along
equalflow weight streamlines in an quasiinfinite constant mean gradient flow in
a exponentially correlated lognormal log conductivity field is approximately V
(1 + bafnk)U where U is the spatially averaged Eulerian velocity. The variance of
the the velocity perturbation along these trajectories is approximately a% bafnk.
Evaluating the expected value in Equation (C.4) and retaining the second order term
yields
=1(1 + a2)
(C.5)
Substituting our approximate values into the preceding equation yields
(C.6)
= x/U
The socalled smalldeviation assumption is a poor one. The approximation of the
Lagrangian field by the Eulerian field works because the correlation properties are
quite similar and the coincidence that the harmonic mean Lagrangian velocity is
approximately equal to the arithmetic mean Eulerian velocity.
133
C.2 Displacements
The center of mass of a solute plume uniformly injected in space is known to
move = Ut, where U, again, is the spatially averaged Eulerian velocity. If the
solute parcels move along Lagrangian trajectories, and the statistics of Lagrangian
and Eulerian fields are known to be quite different, why does this work?
The answer is that the temporally averaged velocity of equalarea Lagrangian
streamtubes is stationary in time and equal to the spatially averaged Eulerian ve
locity. In fact, the temporally averaged Eulerian velocity is lower than its spatially
averaged counterpart. So once again, the smalldisplacement assumption, if consis
tently applied, would result in a very poor result. What is remarkable is that the
correlation properties of the two fields are so similar, and herein lies the strength of
the approximations of Lagrangian fields with Eulerian fields.
REFERENCES
Annable, M. D., J. W. Jawitz, P. S. C. Rao, D. P. Dai, H. K. Kim, and A. L.
Wood. Field evaluation of interfacial and partitioning tracers for characterization
of effective naplwater contact areas. Ground Water, 36(3):495502, 1998.
Bakr, A. A., L. W. Gelhar, A. L. Gutjahr, and J. R. MacMillan. Stochastic analysis of
spatial variability in subsurface flows 1. Comparison of one and threedimensional
flows. Water Resour. Res., 14(2):263271, 1978.
Bear, J. Dynamics of Fluids in Porous Media. Dover Publications, Inc., New York,
1972.
Beilin, A., Y. Rubin, and A. Rinaldo. EulerianLagrangian approach for modeling of
flow and transport in heterogeneous geological formations. Water Resour. Res., 30
(11) :29132924, 1994.
Beilin, A., P. Salandin, and A. Rinaldo. Simulation of dispersion in heterogeneous
porous formations: Statistics, firstorder theories, convergence of computations.
Water Resour. Res., 28(9):22112227, 1992.
Boggs, J. M., S. C. Young, L. M. Beard, L. W. Gelhar, K. R. Rehfeldt, and E. E.
Adams. Field study of dispersion in a heterogeneous aquifer 1. Overview and site
description. Water Resour. Res., 28(12):32813291, 1992.
Cvetkovic, V. D., H. Cheng, and X.H. Wen. Analysis of nonlinear effects on tracer
migration in heterogeneous aquifers using Lagrangian travel time statistics. Water
Resour. Res., 32(6):16711680, 1996.
Cvetkovic, V., G. Dagan, and H. Cheng. Contaminant transport in aquifers with
spatially variable hydraulic and sorption properties. Proc. R. Soc. London Ser. A,
454:21732207, 1998.
Dagan, G. Stochastic modeling of groundwater flow by unconditional and conditional
probabilites 1. Conditional simulation and the direct problem. Water Resour. Res.,
18:835848, 1982a.
134
135
Dagan, G. Stochastic modeling of groundwater flow by unconditional and conditional
probabilites 2. The solute transport. Water Resour. Res., 18:813833, 1982b.
Dagan, G. Solute transport in heterogeneous formations. J. Fluid Mech., 145:151
177, 1984.
Dagan, G. Flow and Transport in Porous Formations. SpringerVerlag, New York,
1989.
Dagan, G. and V. Cvetkovic. Spatial moments of a kinetically sorbing solute plume
in a heterogeneous aquifer. Water Resour. Res., 29(12):40534061, 1993.
Dagan, G., V. Cvetkovic, and A. Shapiro. A solute flux approach to transport in
heterogeneous formations 1. The general framework. Water Resour. Res., 28(5):
13691376, 1992.
Demmy, G., S. Berglund, and W. Graham. Injection mode implications for non
reactive solute transport in porous media: Analysis in a stochastic Lagrangian
framework. Water Resour. Res., 35:19651974, 1999.
Domenico, P. A. and F. W. Schwartz. Physical and Chemical Hydrogeology. John
Wiley and Sons, New York, 1990.
Freeze, R. A. A stochasticconceptual analysis of onedimensional groundwater flow
in nonuniform homogeneous media. Water Resour. Res., 11 (5):725741, 1975.
Gelhar, L. W. Stochastic Subsurface Hydrology. PrenticeHall, Inc., Englewood Cliffs,
New Jersey, 1993.
Gelhar, L. W. and C. L. Axness. Threedimensional stochastic analysis of macrodis
persion in aquifers. Water Resour. Res., 19(1):161180, 1983.
Graham, W. and D. McLaughlin. Stochastic analysis of nonstationary subsurface
solute transport 1. Unconditional moments. Water Resour. Res., 25(2):215232,
1989a.
Graham, W. and D. McLaughlin. Stochastic analysis of nonstationary subsurface
solute transport 2. Conditional moments. Water Resour. Res., 25(11):23312355,
1989b.
James, A. L, W. Graham, K. Hatfield, P. S. C. Rao, and M. D. Annable. Opti
mal estimation of residual nonaqueous phase liquid saturations using partitioning
tracer concentration data. Water Resour. Res., 33(12):26212636, 1997.
136
Jawitz, J. W. Aquifer contaminant source zone characterization with partitioning
tracers and remediation with singlephase microemulsion flushing. PhD thesis, Uni
versity of Florida, Gainesville, Florida, 1999.
Jin, M., M. Delshad, V. Dwarakanath, D. C. McKinney, G. A. Pope, K. Sepehrnoori,
and C. E. Tilburg. Partitioning tracer test for detection, estimation, and reme
diation performance assessment of subsurface nonaqueous phase liquids. Water
Resour. Res., 31(5):12011211, 1995.
Journel, A. and C. Huijbregts. Mining Geostatistics. Academic Press, New York,
1978.
Jury, W. A. and K. Roth. Transfer Functions and Solute Movement Through Soil.
Birkhuser Verlag, Basel, 1990.
Konikow, L. F., D. J. Goode, and G. Z. Hornberger. A threedimensional methodof
characteristics solutetransport model (MOC3D). Waterresources investigations
report 964267, United States Geological Survey, Restion, Virginia, 1996.
Kreft, A. and A. Zuber. On the physical meaning of the dispersion equation and
its solutions for different initial and boundary conditions. Chem. Eng. Sci., 33:
14711480, 1978.
MacIntyre, W. G., M. Boggs, C. Antworth, and T. B. Stauffer. Degradation kinetics
of aromatic organic solutes introduced into a heterogeneous aquifer. Water Resour.
Res., 29(12):40454051, 1993.
Mood, A. M., F. A. Graybill, and D. C. Boes. Introduction to the Theory of Statistics.
McGrawHill Publishing Company, New York, 1974.
Mos, R., P. Siegel, P. Ackerer, and G. Chavent. Application of the mixed hybrid
finite element approximation in a groundwater flow model: Luxury or necessity?
Water Resour. Res., 30(11):30013012, 1994.
Musashi, M. A Book of Five Rings. The Overlook Press, Woodstock, New York,
1982.
Osnes, H. Stochastic analysis of velocity variability in bounded rectangular hetero
geneous aquifers. Adv. Water Resour., 21:203215, 1998.
Papoulis, A. Probability, Random Variables, and Stochastic Processes. McGrawHill
Publishing Company, New York, 1991.
Pollock, D. W. Semianalytical computation of path lines for finitedifference models.
Ground Water, 26(6):743750, 1988.
137
Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical
Recipes in C. Cambridge University Press, Cambridge, England, 1988.
Rubin, Y. Stochastic modeling of macrodispersion in heterogeneous porous media.
Water Resour. Res., 26(7):16891697, 1990.
Rubin, Y. and G. Dagan. Stochastic analysis of boundaries effects on head spatial
variability in heterogeneous aquifers 1. Constant head bounary. Water Resour.
Res., 24(10):16891697, 1988.
Rubin, Y. and G. Dagan. Stochastic analysis of boundaries effects on head spatial
variability in heterogeneous aquifers 2. Impervious boundary. Water Resour. Res.,
25(4) :707712, 1989.
Rubin, Y. and G. Dagan. A note on the head and velocity convariances in three
dimensional flow through heterogeneous anisotropic porous media. Water Resour.
Res., 28(5):14631470, 1992.
Shapiro, A. M. and V. D. Cvetkovic. Stochastic analysis of solute arrival time in
heterogeneous porous media. Water Resour. Res., 24(10): 17111718, 1988.
Stauffer, T. B., C. P. Antworth, R. G. Young, W. G. MacIntyre, J. M. Boggs, and
L. M. Beard. Degradation of aromatic hydrocarbons in an aquifer during a field
experiment demonstrating the feasibility of remediation by natural attenuation.
Technical report, Armstrong Laboratory, United States Air Force, Tyndall Air
Force Base, Florida, 1994.
Stauffer, T. B., J. M. Boggs, and W. G. MacIntyre. Ten years of research in ground
water transport studies at Columbus Air Force Base, Mississippi. In Sayler, G.,
J. Sanseverino, and K. L. Davis, editors, Biotechnology in the Sustainable Environ
ment, Plenum Press, New York, 1997.
Theile, M. R., R. P. Batycky, M. J. Blunt, and F. M. Orr, Jr. Simulating flow in
heterogeneous systems using streamtubes and streamlines. SPERE, 11:512, 1996.
Tompson, A. F. B. Numerical simulation of chemical migration in physically and
chemically heterogeneous porous media. Water Resour. Res., 29(ll):37093726,
1993.
Tompson, A. F. B., R. Ababou, and L. W. Gelhar. Implementation of the three
dimensional turning bands random field generator. Water Resour. Res., 25(10):
22272243, 1989.
BIOGRAPHICAL SKETCH
George Demmy was born to George and Ellen Demmy 23 November 1966 in
Lakeland, Florida. He received a Bachelor of Arts in Physics and German Literature
from the Florida State University in 1990, a Master of Engineering in Agricultural
Engineering from the University of Florida in 1993, and Doctor of Philosophy in
Agricultural and Biological Engineering from the University of Florida in 1999.
138
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Wendy JJrailam,
Professor of Agrie
gineermg
and Biological En
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
P.S.C Rao/ CoChair
Lee A. Rieth Distinguished Professor of
Environmental Engineering
Purdue University
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Kenneth L. Campbell
Professor of Agricultural and Biological En
gineering
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Kirk Hatfield
Associate Professor of Civil Engineering
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Michael D. Annable
Associate Professor of Environmental En
gineering Sciences
This dissertation was submitted to the Graduate Faculty of the College of En
gineering and to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
December 1999
.
M. J. Ohanian
Dean, College of Engineering
Winfred M. Phillips
Dean, Graduate School
116
5.4.1 Results and Discussion: Center of Mass
The center of mass of the steady state plumes emanating from the emplaced
contaminant source was estimated by two methods. The first might be considered
a traditional approach, namely simple trapezoidal rule integrations of the data. A
known initial concentration for each species was assumed known. This concentration
was calculated from initial source mass fractions and aqueous solubilities given in
Stauffer et al. [1997] and assuming Raoults law (see Table 5.1). The second approach
was to use Equation (5.7) (see Table 5.1). The same initial concentration used in the
spatial integration was used with the estimator approach. Displacements were taken
as the distance from the multilevel sampler to the source zone.
benzene toluene ethyl benzene pxylene naphthalene
integration 4.0 2.0 3.1 2.3 3.8
v/k 5.3 1.7 2.5 1.9 3.1
Table 5.1: The center of mass in meters of steady plumes estimated by traditional
trapezoidal integration of spatial concentration data and by the mean v/k estimator
Equation (5.7).
Equation (5.7) gives results similar to those of the trapezoidal rule integration,
while offering some compelling computational advantages. While the spatial integra
tion is conceptually simple, it is computationally demanding for unequally spaced
data in that the different trapezoids must be calculated and summed for each
integration. New data require recalculation of these trapezoids. Of course, these
calculations are trivial for computer codes, but tedious for back of the envelope
calculations. Equation (5.7) is a fairly simple calculation for any scientific calculator,
and provides an excellent tool for a rapid estimate of the center of mass from sparse
and scattered local concentration. New data are easily incorporated into existing sets
without the need of a global recalculation.
60
Finally, note that the AW/LT log conductivity appears stationary in time as does
the velocity. Moreover, it appears to have values equal in magnitude to its AW/ET
counterpart, as does the velocity.
Unlike the velocity, expressions for the trajectorybased crosscovariance of the
velocity and log hydraulic conductivity that are necessary for developing the non
stationary log hydraulic conductivity moments were not developed. This is left for
future work.
2.6 Evaluation of Simulation and Theoretical Results
2.6.1 Mean Velocities in Space
Figure 2.10 presents the control plane oriented mean velocities for the four different
collections of streamtubes. The mean velocities are normalized by U = KgJ/6 1/10
and this value was used in the mean velocity expressions as well. That is, the parame
ter values are the input parameters to the simulations. Other input parameter values
are b = 3/8 and o^nk = {1/2,1} depending upon the set of simulations. The deviation
of the mean of areaweighted Euleriantrajectory data from 1 are an indication of the
quality of the effective conductivity expression.
The areaweighted Euleriantrajectory mean velocity appears stationary, in ac
cordance with established theory, and its value appears to be well predicted by the
effective conductivity relationship. The fluxweighted Lagrangiantrajectory mean
velocity appears stationary, supporting the stationarity hypothesis. Moreover, the
value of the mean velocity appears to be well predicted for both variances of log
conductivity. Additionally, the nonstationary mean velocity estimators appear to
accurately reproduce the transition between the asymptotic velocities.
2.6.2 Velocity Variances in Space
Figure 2.11 presents the control plane oriented velocity variances for the four dif
ferent collections of streamtubes. The variances are normalized by the areaweighted
122
< 1/ti >~ l/U is quantitatively equal to the inverse of the harmonic mean of the
Lagrangian velocity < 1/v >= 1/14. Moreover, the timeaverage Lagrangian veloc
ity is quantitatively equal to the space average Eulerian velocity, and these equalities
contribute to the robustness of other approximations. Were the Lagrangian field to
be consistently approximated by the Eulerian, travel times would be greatly over
predicted and displacements greatly underpredicted. It is, in fact, the consistent
inconsistent approximation of the Lagrangian by the Eulerian that contributes to
much of the observed robustness of the Lagrangian transport theory. While not
many new equations were proposed or so many new phenomena were unearthed,
it is hoped that a modicum of understanding was fostered. This is the primary
contribution of this work.
These concepts were illustrated with an analysis of the transport of an solute ex
periencing a linear equilibrium sorption in a field characterized by a variable sorption
coefficient. While the process has been studied before, it is important to illustrate
the new concepts and approaches here with familiar examples. The sorption study
was restricted to transport as characterized by mass breakthrough at control planes.
An interesting new result is that injection mode strongly affects the propagation of
the mean breakthrough time. A uniform resident injection exhibits a nonlinearity
in the mean, and this nonlinearity is predictably enhanced by negative correlation
between the sorption coefficient and the underlying log conductivity field. Injection
in flux, however, results in a linear propagation of mean arrival time in space, and
mitigates the effect upon correlation. Arrival time variance, however, is dominated by
the correlation between sorption coefficient and the log conductivity field. Injection
mode has comparatively little effect. This has profound implications for tracer test
analyses based upon the arrival time variance of breakthrough curves.
The concepts developed in this work are further illustrated with the develop
ment and analysis of an estimator of the center of mass of a plume resulting from
4
1.3.2 Injection Mode Analysis
The stochasticLagrangian framework is convenient for the analysis of the effects
of the method by which solute is introduced into the flow field, or injection mode,
upon the subeqent transport. These effects of injection mode upon nonreactive solute
transport are characterized by the mean and variance of a breakthrough curve, or
mass arrival distribution, and by the mean and variance of mass displacement. A
goal of this research was to develop estimators for these quantities.
1.3.3 Heterogeneous Reaction Parameter
Set as a goal were estimators for the mean and variance of a heterogeneous reac
tion parameter integrated along trajectories of stationary velocty. These estimators
were used to evaluate the coupled effects of injection mode and a heterogeneous linear
equilbirium sorption process in a stochasticLagrangian framework.
1.3.4 Application to Reality
A further goal was to develop a traditional estimator for the center of mass
of a continuously injected solute plume subject to firstorder decay and to evaluate
data from a natural attenuation experiment and numerical experiments. Estimator
performance was evaluated using the results of the prevous developments.
54
Substituting Equations (2.53) and (2.43) into Equation (2.45) yields
var[ua[x]] = UV{ 1 + (2ba?nk + {ba?ak)2)(l exp[5r/Ainjfc]))
(2.54)
~{U{ 1 + ba?nk(l exp[5r/Alnfc])))2
Further expansion of this expression offers no intuitively obvious simplifications or
insights in its entirety. However, this expression may be approximated by dropping
a few terms thus
var[ua[:r]] = V2ba?nk U2ba?nk(2ba?nk + {bafnk)2) exp[6r/Ainfc] (2.55)
Two unconventional techniques were employed in this derivation that shall now be
defended. The first is requiring the series expansion to match the known endpoints.
Simple expressions that condense the system behavior into an compact form that
allows the relative contribution of the component processes to be easily grasped are
sought. The series expansion and subsequent truncation is itself an approximation.
The systematic error introduced by missing a known endpoint is distracting. In the
case of the velocity covariance, the shape of the correlation function was sacrificed
for simplicity, but the correlation length was retained. Had the simplified expression
neither matched the shape of the data nor reproduced the correlation length, the
expression would have been somewhat more questionable. In this case, the expression
is required meet the observed endpoint values at the cost of the rigor of the series
expansion and truncation.
The second simplification is the ad hoc dropping of terms to simplify the expres
sion. Again, simple expressions that allow the relative contribution of the different
processes be easily grasped are sought. It would be difficult to hypothesize Equa
tion (2.55) a priori as an empirical expression. A primary function of the series
expansion is to indicate the dominant processes and give an estimate of their relative
contributions. This hybrid approach is useful for the purposes here. These techniques
92
Figure 4.6: Reaction path variance values and estimators for different values of cor
relation parameter /3 and different trajectory collections for the cr^ = 1/2 set of
simulations. Top figure is Â¡3 = {1, 41} Bottom figure is = { 1/2,+1/2}. Tra
jectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian
(AF).
113
by Vf/X. This value yields what might be termed a bulk first spatial moment
Vb/k = 1. For a homogeneous conductivity field, this is numerically equivalent to the
first longitunial spatial moment of the steadystate plume.
The steadystate concentration field was sampled by taking the reported nu
merical concentration values as the local resident concentration. The plume center
ofmass was estimated by the estimator presented in Equation (5.7). The actual
first longitudinal spatial moment was calculated using a trapezoidal rule integration.
A fate and transport simulations were carried out in 6 sets of hydraulic conductivity
fields corresponding to 6 standard normal process realizations.
5.3.1 Results
For the case of a homogeneous conductivity field, any one nonzero concentration
measurement used with the estimator given in Equation (5.7) returned a value for the
first longitudinal spatial moment that for all practical purposes was equal to Vb/k, and
five percent less than that predicted by the numerical integration. As the variance of
log conductivity increased, however, both the numerical integration and the estimator
tended to systematically predict values greater than Vb/k. Additionally, the estimator
systematically underpredicted the value given by the numerical integration, and by
an amount which generally increased with variance of the log conductivity.
5.3.2 Discussion
This systematic increase in the the first longitudinal spatial moment of the plume
with increasing log hydraulic condutivity variance, despite a fairly constant vÂ¡, is
explained by considering the nature of the underlying Lagrianian velocity field. The
steadystate concentration plume is, in essence, a map of travel times from the source.
As previously stated, Cvetkovic et al. [1996] demonstrated that the statisics of such
a field are nonstationary. For small displacements from the source, the mean travel
117
5.4.2 Results and Discussion: Degradation
The relative degradation and absolute degradation of the different contaminants
were estimated using Equation (5.9) and degradation rates reported by MacIntyre
et al. [1993]. These reported benzene, pxylene, and naphthalene degradation rates
were estimated from another experiment conducted at the site in which a cocktail of
hydrocarbons was injected into the groundwater and the fate of the resultant plume
was monitored (see Table 5.2). This experiment was conducted before the NAPL
benzene
toluene
ethyl benzene
pxylene
relative
0.87
3.4
1.5
2.0
estimated
0.0056
0.022
0.0094
0.013
reported
0.0070
NR
NR
0.011
Table 5.2: Degradation rates relative to naphthalene and absolute degradation rates
in reciprocal days as estimated by Equation (5.9) and as reported by MacIntyre et al.
[1993]. NR denotes results not reported in MacIntyre et al. [1993]. Naphthalene
degradation reported to be 0.0064 d~x.
source was emplaced, and is in that sense independent of the concentration data as
sociated with the natural attenuation experiment. Nevertheless, the rates estimated
from the natural attenuation data were similar to those reported by MacIntyre et al.
[1993]. The technique used here is quite simple, and does not require regressions or
nonlinear fits to transport models.
5.5 General Conclusions
These estimators, while fairly simple and apparently robust, are limited in appli
cability to quasisteady plumes, and it is not clear the extent to which such plumes
exist. Sites where poorly mobile NAPLs have been known to exist for long times are
likely candidates, especially if decay rates are known to be large in comparsion to
transport time scales.
109
dispersive processes, an advectiondecay equation with streamtube specific parame
ters is assumed to describe the fate and transport of the solute along the streamtube
trajectory. This advectiondecay equation is
dc dc
di = vTxkc
(5.1)
subject to c[x,t < 0] = 0 and c[x = 0,t => 0] = Co, where v is an effective advective
velocity [LT1], and k is a firstorder decay constant [T1]. The assumption of
effective streamtube parameters is the simplification that separates this method from
the more general consideration of properties varying along the trajectory.
Although much of this language is similar to that used in earlier chapters, the
fundamental difference is that the streamtube has constant effective parameters. This
is analgous to an assuption of stationarity of properties, but is different in significant
ways, as shall be shown.
For advective transport in a single streamtube, the resident concentration is equal
to the fluxaveraged concentration. However, the difference between these detection
modes manifests itself when concentrations are averaged over several streamtubes.
For the remainder of this discussion, any reference to a concentration will implicitly
imply a reference to a resident concentration. The solution to this equation for the
given initial and boundary conditions is
Q
= exp [kx/v] h[vt x] (5.2)
Co
where h[x] is the Heaviside step function, with the properties h[x] 0 for x < 0 and
h[x] 1 for x > 0.
5.2.1 Spatial Moments
The defintion of the raw spatial moment of order n of curve c is
/
n
L
oo xnc[x]dx
(5.3)
29
Table 2.2: Comparison of target log conductivity values for areaweighted Eulerian
trajectories in space and observed areaweighted Lagrangian trajectories in time.
parameter target observed
WnkKk = 1/2) 0.000
/hnfc(0fnfc = 1) 0 017
simplifies relating the Eulerian and Lagrangian fields, as shall be shown. The long
time asymptotic mean log conductivity values for both Eulerian and Lagrangian
trajectories are somewhat less than their largedisplacement counterparts, and the
magnitude of the difference is proportional to ofnfc (compare Figures 2.6 and 2.8).
This is consistent with the harmonic averaging implicit with a timeaveraged time
dependent velocity and the arithmetic averaging implicit with a spaceaveraged space
dependent velocity. Recall, we map the quantities onto the trajectories by following
velocities, whether along natural streamlines (Lagrangian trajectories) or not (Eu
lerian trajectories). A simple heuristic is as follows. Low conductivities correspond to
low velocities. For an equaltime spacing (or equallylikely observations in time) more
measurements will come from low velocities, since the particle tracing the trajec
tory spends more time going slow. For an equaldistance spacing more measurements
will come from high velocities, since the particle covers more ground while going
fast. This is perhaps more intuitive when discussed in terms of the velocity field itself.
2.4.8 Velocity Statistics
The mean velocities along the different trajectories exhibit a similar behavior as do
the mean log conductivities along the same trajectories (see Figure 2.10 and compare
Figure 2.6). The AW/ET exhibits apparently stationary behavior in the mean and
variance (see Figure 2.11) in accordance to established theory (e.g., Dagan [1989]).
The AW/LT exhibits nonstationary behavior in the mean as observed by Cvetkovic
et al. [1996]. Again, the AW/ET and FW/LT trajectories appear to be stationary
22
where u[0] is the initial velocity recorded for trajectory i. The variance by
E^MO*/)2
var[u/] =
Efci Â¡[o]
(2.10)
An alternative method for generating a collection of streamlines would be to iden
tify the streamline corresponding to the center stream function expressed in Equa
tion (2.2). In this case, the estimators of the equal flow statistics would be Equa
tions (2.7) and (2.8). The equal area statistics would be given by
Ei=i i/i[o]
(2.11)
and
ElViVa^/um
var[ua] =
e2=i i/i[o]
(2.12)
The reference collection of the trajectories selected uniformly in space was specified
in the simulations for convenience.
2.4.6 Simulation Results
The following sections present the results of two 5000 replicate Monte Carlo simu
lations. Detailed discussion of these results is deferred until after the development of
some theoretical models with which to analyze these results. However, some heuris
tics are provided to familiarize the reader with some of the concepts developed in
subsequent sections and chapters.
A summary of the notation used with the following results provides clarity. AW
denotes an areaweighted trajectory and FW denotes a flowweighted trajectory ET
denotes an Eulerian trajectory and LT denotes a Lagrangian. A particular weight
trajectory combination is concatenated the with a /. Thus, an areaweighted Eulerian
trajectory is denoted as AW/LT.
For a given simulation, that is, for one set of 5000 replicates for a log conductivity
field of some given a?nk, the statistics of the log conductivity and velocity along the
Ill
b is defined as
kr
ka/kb
(5.8)
Let these solutes be released into a flow field at a steady rate, and allow the solutes
to reach a steady state in which the concentration profiles for each do not vary in
time. If within a streamtube i, the streamtube effective velocity Vi is a approximately
equal for solutes a and b, the relative degradation rate of a to b in streamtube i is
given by
kri (u/kb) / (n/ka)
(5.9)
Mlbt/Ml ai
That is, the relative degradation rate is given by the ratio of the first normalized
spatial moment of the steadystate concentration curve of solute b to that of a. The
average relative degradation rate is estimated as the arithmetic mean of kri. Notice
that this is not the ratio of the mean first spatial moments.
5.3 Application: Simulation
The estimator was tested by simulating the release of a solute into a heterogeneous
twodimensional aquifer using the United States Geological Survey flow and transport
code MOC3D [Konikow et ai, 1996]. The length units used herein are a dimensionless
quantity resulting from the normalization of the fundamental length by that of the
correlation length of the exponentially correlated log hydraulic conductivity field. In
the interest of clarity, however, this dimensionless quantity will be referred to as a
correlation length A.
Let the reference Cartesian coordinate system be oriented such that the xdirection
is parallel with the mean direction of flow, and the ydirection is orthogonal to that
and lies within the plane formed by the aquifer. Let the origin lie at the lower left
corner of the aquifer. The rectangular aquifer extended 15 A x and 25 A in y, and was
133
C.2 Displacements
The center of mass of a solute plume uniformly injected in space is known to
move = Ut, where U, again, is the spatially averaged Eulerian velocity. If the
solute parcels move along Lagrangian trajectories, and the statistics of Lagrangian
and Eulerian fields are known to be quite different, why does this work?
The answer is that the temporally averaged velocity of equalarea Lagrangian
streamtubes is stationary in time and equal to the spatially averaged Eulerian ve
locity. In fact, the temporally averaged Eulerian velocity is lower than its spatially
averaged counterpart. So once again, the smalldisplacement assumption, if consis
tently applied, would result in a very poor result. What is remarkable is that the
correlation properties of the two fields are so similar, and herein lies the strength of
the approximations of Lagrangian fields with Eulerian fields.
103
1
0.8
0.6
I 0.4
g
0.2
0
0 2 1 1 1
0 2 4 6 8 10
Ut/X\nk
Figure 4.9: Assumed timedependent log conductivity covariance function along La
grangian trajectories (Equation (4.29)) compared to simulation data for variances
of log conductivity afnk {1/2,1}. The Lagrangian velocity covariance function is
Equation (2.61).
x \
\ x
V x
1/2
1
Lagrangian In k
Lagrangian V
\++xx
\ ++Â¥xx.
Xx
.T++X+xxXxxxxxXv
*4,..^ ^/xxxxx* Jv+^iv
+++++++*+;t+++ *+*
fact, this is the behavior observed in the simulations (see Figure 4.10) The data are
compelling enough to suggest further work. However, we shall accept Equation (4.29)
with b' = 1 as a suitable approximation.
Substituting Equation (4.29) with t = \t t'\ into Equation (4.28), integrating
and inserting the result into Equation (4.26) yields
r r 11 ^^"9 ^lnfc T cru,)A[n j. r tt /\ r\
var[/i[rj] = *1 (Ut/\lnk ~ (1 exp[t/r/AinfcJ) 30^
pK + ^Jr2
Neither this equation nor Equation (4.25) show a dependence upon the sign of Â¡3.
For the mean reaction flow path, the simulations show a weak dependence upon
correlation parameter (see Figure 4.11). The linear propagation of the mean reaction
flow path in time is due to the stationarity of the local reaction parameter along
the areaweighted Lagrangian trajectories. Again, it should be noted that time
26
Table 2.1: Comparison of target log conductivity values and observed values along
areaweighted Eulerian trajectories.
target observed
parameter
Min k{Kk = 1/2)
Mlnfc^lnfc = 1)
k
0 0.003
0 0.005
1/2 0.498
1 1.002
of the population statistics. Again, this is important for a quantitative theoretical
analysis of these results.
The AW/LT and the FW/ET exhibit distinctly nonstationary behavior in the
mean and less distinct behavior in variance as a function of displacement along the
mean flow direction (see Figures 2.6 and 2.7). At the injection point, the AW/ET and
AW/LT statistics are identical and the FW/ET and FW/LT statistics are identical,
because these trajectories carry the same weight and the points in consideration are
identical (see e.g., Figure 2.2). At very large displacements, one would expect the
statistics of like trajectories to assume similar characteristics, regardless of weight.
As the trajectory traverses several integral scales that characterize the heterogeneity
in question, information about its starting location, is diminished. In the case of
the FW/ET, the statistics start at the same place as FW/LT, and eventually end
at the same place as the AW/ET. Similarly, the AW/LT statistics start where do the
AW/ET and end where do the FW/LT. In fact, the data exhibit this behavior (see
Figures 2.6 and 2.7).
The log conductivity statistics in time show somewhat different behavior. Only
the AW/LT appear stationary in time (see Figures 2.8 and 2.9). There is an impor
tant observation to be gleaned from these results. The AW/LT statistics, in addition
to appearing stationary, also appear to have a similar value to the input AW/ET
statistics (see Table 2.2). Recall, the AW/LT statistics are clearly nonstationary in
space, and are clearly distinct from those of the AW/ET. This observation greatly
96
conductivity in the calculation of the mean, working from Lagrangian properties that
are stationary in displacements along the mean flow direction, and examination of
the effects of injection mode.
4.5.1 Mean Reaction Flow Path
Substitution of Equation (4.4) into Equation (4.2) and taking the expected value
yields
= P* exp[cr
/2] f
Jo
exp[/3 In k[x]]
v[x]
dx
(4.14)
The stationary Lagrangian serves as a reference field. The log conductivity field is
decomposed into its mean and mean removed perturbation / and the velocity into
its mean and mean removed perturbation thus
<
. p* r 2 /91 f /exp[/3(/qnfc + /[g])]\ ,
n> Pg expK/2] ^ ^ V(l + v'[x]) )d ^
Define Pg = P* exp[/xinjt], and expand the nonlinear terms of the previous equation
into series, take the expected value retaining second order terms thus
Pg exp[cr^/2]
< fi >=
V
1(1 fioi, + Pcj/2 + ai)x
(4.16)
Recall that aj = afnk. The / subscript is used for clarity. This expression captures
the linearity of the observed mean fi (see Figure 4.3), but shows a heavy dependence
upon the sign of the correlation parameter /3 that is not apparent in the data. The
mean reaction flow path for the equal flow streamtubes appears to be dependent only
upon the mean value of the reaction parameter, e.g.
Pg exP[<4/2 + 020//2]
=
{l + av)x
V
(4.17)
5 APPLICATION OF CONCEPTS 108
5.1 Introduction 108
5.2 Theory 108
5.2.1 Spatial Moments 109
5.2.2 Relative Degradation 110
5.3 Application: Simulation Ill
5.3.1 Results 113
5.3.2 Discussion 113
5.4 Application: Field Data 115
5.4.1 Results and Discussion: Center of Mass 116
5.4.2 Results and Discussion: Degradation 117
5.5 General Conclusions 117
6 GENERAL CONCLUSIONS 119
APPENDICES
A EULERIAN VELOCITY DISTRIBUTION PROPERTIES 125
B VELOCITY LOG CONDUCTIVITY CORRELATION 127
C CONSISTENT FIRST ORDER APPROXIMATION 131
C.l Travel Time 131
C.2 Displacements 133
REFERENCES 134
BIOGRAPHICAL SKETCH 138
viii
128
r/Xf
Figure B.l: Comparison of velocity and log conductivity correlation functions from
Graham and McLaughlin [1989b] and Rubin and Dagan [1992] and a simplified expo
nential model. All functions are normalized by KgJa^nk/6. The apparent roughness
of the Rubin and Dagan [1992] function is due to estimation errors associated with
inferring their function from graphical data.
28
O 2 4 6 8 10
tU/X infc
FW/LT +
AW/LT x
FW/ET *
AW/ET
reference
tU/Xink
FW/LT +
AW/LT x
FW/ET *
AW/ET
reference
Figure 2.9: Log conductivity variance as function of travel time to control planes
along mean flow direction for the four different trajectory collections and afnk = 1/2
(a) and = 1 (b). Variances are normalized by target variance of Eulerian log
conductivity field a,2nfc.
67
3.2.2 Injection in Flux
Consider now maintaining the IP at Co for some brief interval At. A mass
Mf = c09hLUAt
(3.4)
enters the domain and the different streamtube collections. This injection mode is a
uniform injection in flux. Uniform, again, refers to the uniform mass density and in
flux refers to the influent water that carries the solute into the flow domain. A mass
rrii = c09hViAt
enters equalarea streamtube i. This corresponds to a mass weight of
rrii Ut
~M = U
A mass
rrii codhciiViAt
= cod At
enters equalflow streamtube i. This corresponds to a mass weight of l/N.
(3.5)
(3.6)
(3.7)
3.3 Integrated Streamtube Properties
Let us return to the notion of the streamtubes as pure hydrodynamic entities. It
is possible to discuss properties of these streamtubes with no mention of injection
modes. Two properties hold some hydrodynamic interest. The first is the velocity
integrated over time. For for some streamtube i this is
x
:[t] = f Vi[t]dt
Jo
(3.8)
A velocity integrated over time is a displacement with a dimension of length. The
second the inverse velocity integrated over some distance. For some streamtube i this
72
by a higher velocity. The timevarying mean velocity expression for the flowweighted
Lagrangiantrajectories (Equation (2.63)) is, in fact, an expression for the mean ve
locity of a plume resulting from a uniform flux injection. The longtime asymptotic
slope of the fluxweighted mean displacement is U, as is easily verified by taking limit
of Equation (2.63) as t oo. The mean displacement functions predict the observed
behavior, and also provide an excellent fit to the data.
The displacement variance data are interesting (see Figure 3.2). Injection mode
seems to have little effect upon the displacement variance in time. This is not to
say that injection mode has little effect upon the spatial distribution of the plume
in time. At a given time t, a plume injected in flux will have traveled a greater
distance, on average. After traveling the same distance, the displacement variance
of the fluxinjected plume will be less than that of the resident injection. Thus,
the spreading as function of mean displacement is less in the case of the flux
injected solute. The areaweighted trajectory displacement variance overpredicts the
observed displacement variances, and the longtime slopes appear to have different
values. After 10 characteristic times, the exponential term in Equation (3.13) retains
a little more that two percent of its original value. This difference does not seem large
enough to account for the observed discrepancy. The variance of the stationary area
weighted Lagrangiantrajectory velocity field in time appears to be well predicted.
Thus, the problem may lie with the rather poor description of the shorttime and long
time correlation exhibited by the observed velocity covariance given by the assumed
model.
3.5.2 Travel Times
Figure 3.3 shows good agreement between the estimated and observed mean travel
times. The injection influx shows earlier arrival than that of the uniform resident
injection. Uniform resident injection weights all areas of the IP equally. Large areas
that contribute little to the overall flow receive and equal mass weighting as those
101
The log hydraulic conductivity is assumed to be similarly stationary. Moreover, the
point random process Pg exp[ic] is assumed likewise stationary, since it is uncorrelated
to the random log conductivity. From these assumptions, the expected value of the
timedependent reaction flow path is
< > =
= Pg exp[<4/2 + /?2
The variance is given by
var[/i[r]] =< Pg exp [w[t]\ exp [(3 In k[t}} dt
Jo (4.26)
Pg exp [w[t]\ exp [P In k[t]] dt' < /z[t] >2
Consider the integrals in the previous equation. The exact expression is denoted by
I and is approximated by the truncated series expansions of the exponential terms
thus
I Pg f f < (1 + tu[]) (1 + w[t']) (1 + p\nfc[t]) (1 + p\nk[t']) > dtdt' (4.27)
y Jo Jo
Note that a possibly more elegant approach would be to explore the Lagrangian
features of the conductivity field, rather than the log conductivity field, and work
from a stationary conductivity field covariance function and a similar uncorrelated
field. This would obviate the need for the expansions. This suggested inquiry is left
to future works. Expanding the above expression, taking the expected value, and
dropping products of covariances yields
I = Pg [T [{l + Cw[t,t']+ p2CXnk[t,t'))dtdt' (4.28)
Jo Jo
A few observations allow the adoption of an approximate expression for the time
dependent Lagrangian covariance function C\Dk[t, t']. Relatively simple expressions
that capture the behavior of the dominant processes are sought here. These expres
sions might facilitate future works that deemphasize simplicity for greater rigor and
f
< Pg exp[it;[i]] exp[/3 In &[Â£]] > dt
(4.25)
45
where s = x/\\nk The corresponding expression for a threedimensional field is
Cu[s] = 8alk(KgJ/d)2 (exp[s](s2 + 5s~3 + 12s'4 + 12s~5) + s"3 12S5)
(2.26)
Equation (2.25) provides a superior match to the data in comparison to Equa
tion (2.13) (see Figure 2.17). However, the expression for the Eulerian covariance
Figure 2.17: Comparison of velocity covariance expression Equation (2.25) to the ap
proximation Equation (2.13) for ofnfc = 1. Covariances are presented as dimensionless
correlation functions.
given in Equation (2.13) captures the dominant features of the velocity covariance
structure, including the correlation length and the point velocity variance of Equa
tion (2.25) in a simple and familiar form.
As noted, however, Equation (2.13) exhibits some systematic differences from the
observed behavior. The data show a rapid initial decay, followed by a longrange
persistence (see Figure 2.17). This behavior is examined with an expression adapted
123
continuously injected solute subject to a firstorder decay. The assumption of non
interacting streamtubes with constant effective properties allows solution of the
advectiondecay equation for a single streamtube. An ergodiclike hypothesis al
lows this solution to a distribution of streamtubes. This is what might be termed
a classic streamtube approach. While surprisingly robust and perhaps worthy of
application in field settings, the estimator shows a bias explained using the trajectory
based analysis developed in this work.
One measure of quality of a work is the number of questions that it answers. Per
haps another is the number of interesting questions that it begs. Several issues are
left for further study. A few specific questions are addressed first, then a few more
general. The relationship between an Eulerian reference system and its Lagrangian
counterpart remains incompletely understood. A primary area that requires attention
is the increasing Lagrangian velocity correlation length with increasing log conduc
tivity variance. A great deal of work remains for predicting the spatial moments of a
reactive solute.
Numerical constraints limited this study to relatively mild heterogeneity. How
ever, a highly heterogeneous medium, might be replaceable by two or more media
characterized by lesser heterogeneity, and a systematic replacement system would
greatly enhance computational efforts. The turning bands method employed in the
generation of the random fields was computationally expensive, in comparison to,
say, the particle tracking algorithms or data analysis programs, and to the flow solver
for weak heterogeneity. The method requires the generation of several independent
normal line processes. This problem is wellsuited to parallel and distributed compu
tational techniques, and an easytouse implementation would certainly find wide use.
Several research groups are working on parallel and distributed flow codes, and the
research community would benefit greatly from a wider dissemination of or greater
access to these codes.
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS in
LIST OF TABLES ix
LIST OF FIGURES x
ABSTRACT xv
CHAPTERS
1 INTRODUCTION 1
1.1 Primary Contribution 1
1.2 Historical Context 1
1.3 Research Goals 3
1.3.1 Lagrangian Velocity Mean and Covariance 3
1.3.2 Injection Mode Analysis 4
1.3.3 Heterogeneous Reaction Parameter 4
1.3.4 Application to Reality 4
2 LAGRANGIANEULERIAN FLOW FIELD RELATIONSHIP ... 5
2.1 Introduction 5
2.2 Flow Field Partitioning Strategies 7
2.2.1 Simple Subdivision 8
2.2.2 General Subdivision 10
2.3 Descriptions of Fields 12
2.4 Numerical Experiments 13
2.4.1 The Log Conductivity Field 14
2.4.2 Velocity Field Generation 18
2.4.3 Trajectory Generation 19
2.4.4 Monte Carlo Simulation 19
2.4.5 Sampling Strategy 20
2.4.6 Simulation Results 22
2.4.7 Log Conductivity Statistics 23
2.4.8 Velocity Statistics 29
2.4.9 Head and Head Gradient 40
2.5 Theory 40
2.5.1 Effective Conductivity 40
vi
69
integrate to yield
= Ut + Aifc(T2nA(l exp[M/i/Alnl])
(3.12)
The displacement variance for the areaweighted streamtube collection is given by
var[xa[t]]  2
v[t"]dt"> {Ut)2
(3.13)
3.3.2 Travel Times
Expressions for the travel time statistics are sought, in analogy to those sought for
displacement. The reference collection of velocities from which these expressions will
be derived is the stationary velocities of the equal flow streamtubes, or the equalflow
weight Lagrangian trajectories. The mean travel time for these streamtubes is given
by
(3.14)
X
U
The mean travel time for the areaweighted streamtubes is
(3.15)
137
Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical
Recipes in C. Cambridge University Press, Cambridge, England, 1988.
Rubin, Y. Stochastic modeling of macrodispersion in heterogeneous porous media.
Water Resour. Res., 26(7):16891697, 1990.
Rubin, Y. and G. Dagan. Stochastic analysis of boundaries effects on head spatial
variability in heterogeneous aquifers 1. Constant head bounary. Water Resour.
Res., 24(10):16891697, 1988.
Rubin, Y. and G. Dagan. Stochastic analysis of boundaries effects on head spatial
variability in heterogeneous aquifers 2. Impervious boundary. Water Resour. Res.,
25(4) :707712, 1989.
Rubin, Y. and G. Dagan. A note on the head and velocity convariances in three
dimensional flow through heterogeneous anisotropic porous media. Water Resour.
Res., 28(5):14631470, 1992.
Shapiro, A. M. and V. D. Cvetkovic. Stochastic analysis of solute arrival time in
heterogeneous porous media. Water Resour. Res., 24(10): 17111718, 1988.
Stauffer, T. B., C. P. Antworth, R. G. Young, W. G. MacIntyre, J. M. Boggs, and
L. M. Beard. Degradation of aromatic hydrocarbons in an aquifer during a field
experiment demonstrating the feasibility of remediation by natural attenuation.
Technical report, Armstrong Laboratory, United States Air Force, Tyndall Air
Force Base, Florida, 1994.
Stauffer, T. B., J. M. Boggs, and W. G. MacIntyre. Ten years of research in ground
water transport studies at Columbus Air Force Base, Mississippi. In Sayler, G.,
J. Sanseverino, and K. L. Davis, editors, Biotechnology in the Sustainable Environ
ment, Plenum Press, New York, 1997.
Theile, M. R., R. P. Batycky, M. J. Blunt, and F. M. Orr, Jr. Simulating flow in
heterogeneous systems using streamtubes and streamlines. SPERE, 11:512, 1996.
Tompson, A. F. B. Numerical simulation of chemical migration in physically and
chemically heterogeneous porous media. Water Resour. Res., 29(ll):37093726,
1993.
Tompson, A. F. B., R. Ababou, and L. W. Gelhar. Implementation of the three
dimensional turning bands random field generator. Water Resour. Res., 25(10):
22272243, 1989.
CHAPTER 3
NONREACTIVE SOLUTE TRANSPORT
3.1 Introduction
The previous chapter saw the derivation of the relationship between the velocities
that lie along Lagrangian trajectories and the parameters that describe a simple
model Eulerian flow field. These Lagrangian relationships are employed to derive
a few functions that characterize global measures of solute transport: namely the
statistics of mass displacement in time and the statistics of mass arrival times at
control planes. The frameworks are are that of Dagan [1982b] and that of Shapiro
and Cvetkovic [1988], respectively. The recovery of a few wellknown results are
anticipated, in addition to some that are new. However, the approach is novel in that
the work is based upon trajectorybased statistical properties of the flow field.
3.2 Injection Mode and Streamtube Selection
Demmy et al. [1999] noted the relationship between boundary conditions and
sampling of streamlines. A clarification of this relationship and a generalization the
results of that work are sought. Unlike the previous chapter, the Eulerian trajecto
ries are not considered in the context of an actual transport trajectory, as they are
of little practical importance. However, the results of Shapiro and Cvetkovic [1988]
correspond to areaweighted Euleriantrajectories, as will be shown. Therefore, tra
jectory, Lagrangian trajectory, streamtube, and streamline are used interchangeably.
3.2.1 Uniform Resident Injection
Consider the application of mass at a constant density c0 across the entire face of
the injection plane (IP) of the model aquifer (see Figure 2.1). Let the mass occupy
65
be the most efficient for producing degrees and papers, but it has produced several
interesting adventures.
George and Ellen Demmy, my parents, have provided love and all manner of
support throughout my life, academic and otherwise. They both instilled in me a
love of learning and an insatiable curiosity about all things that serve me well, and
will remain among my best traits.
Finally, Id like to extend my thanks and love to Celine Bufkin, for her love,
encouragement, support, and vision.
2
heterogeneous conductivity fields, and that the Lagrangian and Eulerian velocities are
approximately equal. They consistently applied this concept, developing expressions
for arrival time means and variances for nonreactive solutes in the Standard Model
Aquifer, a hypothetical aquifer characterized by a heterogeneous conductivity field
with a prescribed correlation structure and subject to certain prescribed boundary
conditions (see Section 2.1). These equations predicted that the mean arrival time
for solute subject to a uniform resident injection is given by
= x/Vh (1.1)
where angle brackets <> denote an ensemble average operator and Vj, is the harmonic
mean Lagrangian velocity. From the small deviation assumption, the Lagrangian
and Eulerian velocities are approximately the same, thus the estimated harmonic
mean Lagrangian velocity is equal to the estimated harmonic mean Eulerian velocity.
However, the mean arrival time for uniform resident injection of solute after travelling
several integral scales is given by
x/U (1.2)
where U is the arithmetic mean Eulerian velocity. This expression is also the con
sistent firstorder approximation (CFOA) of the travel time. Widespread adoption
of the CFOA through out the stochasticLagrangian transport literature probably
stems Feynmans requirement: we must reproduce what we already know. The har
monic mean of a positive definite process is always less than the arithmetic mean, so
Equation (1.1) systematically overpredicts travel times for large displacements. Dis
turbing was that the estimators derived by Shapiro and Cvetkovic [1988] using clear,
consistent, intuitive and logical methods seemed to violate Feynmans rule, especially
in light of the success enjoyed by the Dagan school.
74
15
C)
a
c
b
JÂ£
e
?
IM H
b
10
AW/LT +
FW/LT x
Eq. 3.13
&
//
/Â¥
.* Â¥
'Â¥
Â¥
/ *
V*
*
4%
y
4 6
tU/X infc
10
15
10
C
b
S
c
CN H
b
AW/LT +
FW/LT x
Eq. 3.13
+x
+x
/ A
/ A
/ **
V
,v
/A
vÂ¥
***Â£
:***
4 6
tU/X\nk
10
Figure 3.2: Comparison of displacement variances for the two injection modes or
streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics
correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT)
statistics correspond to a uniform resident injection. The top figure contains the
a\nk 1/2 data and the bottom figure contains the ofnjt = 1 data. Variances are
normalized by \fnkofnk.
40
hypothesis by examining the velocity characteristics of flow fields arising from differ
ent log conductivity distributions (e.g., fractal, uniform, white noise, etc.), and this
is left to future work.
2.4.9 Head and Head Gradient
Head and head gradient information along the trajectories was not recorded. This
is a regrettable oversight, since much of the behavior of the flow water and the trans
port of solutes in heterogeneous media can be summarized as the interplay of head,
head gradient and conductivity covariance functions. The literature concerning Eu
lerian head and head gradient fields is extensive and is summarized in large part by
Gelhar [1993]. In fact, much is drawn from this body of literature for some of the
theoretical developments in the following sections. That this information was not
recorded detracts mainly from the completeness of this work, and further investiga
tions are left to future works.
2.5 Theory
2.5.1 Effective Conductivity
As with the simulations, the theoretical developments are founded on the bedrock
of the log conductivity field. Section 2.4.7 demonstrates that the simulation results
well match the target ensemble quantities. From this field is had directly a derived
parameter, namely the a priori effective conductivity Ke. An expression for the
effective conductivity, similar to that given by Gelhar and Axness [1983], is derived
here to illustrate the mechanics of different techniques employed through out these
derivations. Consider a onedimensional Darcys law of the form
Q =
dx
(2.15)
where q is the specific discharge, k is the hydraulic conductivity, and is the hydraulic
head. Consider a large stationary conductivity field such that the log conductivity
64
average, and this disparity increases with heterogeneity of the underlying conductivity
field. This might be described as preferential flow. A practical implication for the
transport of kinetically sorbing or interacting solutes is that the local velocity for the
majority of the flow may occur in a regime significantly higher than that estimated
as the flow field average. An understanding of the medium heterogeneity is essential
if laboratory experiments are to be scaled up.
It is intuitively obvious, and has certainly been noted before, that the bulk of the
flow passes through an increasingly small portion of the total swept volume as the
medium heterogeneity increases. Again, the practical implications are likewise obvi
ous. Consider some aquifer remediation effort that involves flushing a contaminated
zone with some fluid. This might be as simple as the upgradient water in the case of
pumpandtreat or as extravagant as an surfactantenhanced microemulsion. In het
erogeneous flow systems, the bulk of the flushing fluid contacts a disproportionately
small volume of the total swept volume. For contaminants that tend to reside in areas
of high conductivity, the serves to enhance the efficacy of remediation efforts. Con
taminants that tend to reside in areas of low permeability, conversely, would be more
difficult to remove. While these conclusions may be drawn from common sense,
application of the concepts developed here may help to better understand these issues
quantitatively.
120
reader might contrive a plethora of trajectory and trajectory weighting schemes. For
compactness, the following notation summarizes the four different trajectories
AW/ET equalarea weighted Eulerian, or straight line, trajectory
AW/LT equalarea weighted Lagrangian, or streamline, trajectory
FW/ET equalflow weighted Eulerian, or straight line, trajectory
FW/LT equalflow weighted Lagrangian, or streamline, trajectory
Myriad methods exist for mapping properties observed along the different trajec
tories. That is, the value of some property may be observed along the trajectory and
recorded at different intervals in space or time. A traditional method is a regular Eule
rian gridding that divides the flow field into equallyspaced trajectories and records
information at regular intervals along the trajectory. Two specific parameterizations
are considered: equal intervals in displacement along the mean flow direction and
equal intervals in advective travel time.
Employing these concepts, the behavior of different flow field properties was exam
ined. These properties, such as log conductivity and local velocity, were observed
along the different trajectories at control planes, or planes equally spaced in the
mean flow direction, or control times, or at the displacement that an indivisible
fluid parcel would be after traveling for some reference time. For a steady and irrota
tional flow field resulting from a constant mean gradient applied across a stationary
Eulerian log conductivity field of uniform and constant mobile water content, the
trajectories for which the statistics certain properties appeared to be secondorder
stationary were identified. That is, the mean and variance of the property appeared
to have a constant value at all observation points along the trajectory. The log con
ductivity and velocity observed at control planes appeared stationary for the equally
spaced Eulerian trajectories (AW/ET) and Lagrangian trajectories separated by an
114
o 1
O 1 2 3 4 5
x/X
O 1 2,3 4 5
X/X
Figure 5.1: Resident concentration profile averaged along a transect orthogonal to
the mean flow direction for plumes for lk = {0.0,0.1,0.5,1.0}.
time propagates
dr/dx = 1 /vh
(5.11)
where vb is the harmonic mean velocity. For regions distant from the source region,
the mean travel time propagates
dr/dx = l/va (5.12)
where va is the arithmetic mean velocity, and equal in value to vb, the bulk filtration
velocity. The harmonic mean velocity is necessarily less than or equal to the arith
metic mean velocity, which in turn indicates that the mean travel time propagates as
a slower rate near the source.
Additionally, the correlation of the Lagrangian velocity persists over greater dis
tances than that of the underlying conductivity field. Solute which is introduced in
130
the functional form appears to remain exponential. Therefore, it is suggested that
future work focus upon generalizing the simple results found here to threedimensions
and general anisotropy.
APPENDIX B
VELOCITY LOG CONDUCTIVITY CORRELATION
A simplified correlation relationship between the velocity field and the log con
ductivity is sought. Graham and McLaughlin [1989b] and Rubin and Dagan [1992]
derived expressions based upon perturbation expansions of Darcys law and spectral
methods. Graham and McLaughlin [1989b] assume a holetype covariance function
for the log conductivity field, in order to assure a complementary stationary head gra
dient field. Rubin and Dagan [1992] assume an exponentially correlated anisotropic
log conductivity field.
The velocitylog conductivity covariance functions for both analytical cases shows
qualitatively similar behavior (see Figure B.l), and an simplified estimator would be
applicable for both log conductivity correlation models. Although these rigorously
developed estimators are attractive theoretically, the mundane and oftneglected task
of mapping theoretical results into application often requires simplifying assumptions.
Presented for your consideration is the covariance function of Graham and McLaughlin
[1989b] (their Equation (B5)
P./K] = KrfJ/e^aOMaQ l(aC)2ifoK] ,
z (B.l)
(a/2) ((KM] <%K]) }
where a = 7r/(4A/) and is the modified Bessel function of order i. The integral
scale of this covariance function is Ay. An exponential approximation to this function
that reproduces the integral scale, general trend, and point covariance of that given
by Graham and McLaughlin [1989b] is sought. A function of the form
pvf\A = w 7 exp[x/(2A)] (B.2)
127
75
areas that contribute more significantly to the overall flow. However, as the solute
moves through the domain and experiences the full range of velocity variability along
its trajectory, the slow start is damped out and the instantaneous average point
velocity for all for the particles is V and the harmonic mean V/, = U. The travel time
variances appear to be better matched than the displacement variances (see 3.4). This
is reasonable in light of the better apparent match of the assumed velocity covariance
function in space than that in time. The uniform injection appears to exhibit less
nonlinear behavior than the injection in flux. The uniform injection variance is
reasonably expected to be larger than that of the flux injection, due to the skewed
massweighting of lowvelocity areas by the uniform resident injection.
3.6 General Conclusions
It is clear that the injection mode effects are most prevalent in the near field
where the initial velocities and travel times or displacements are well correlated. Af
ter the plume travels several correlation scales, these injection mode effects diminish.
The near field and the neartofar transitional scales are quite interesting, as these
intermediate scales are on the order of sites associated with certain agricultural set
tings such as some feed lots and waste holding facilities, urban land uses such as gas
stations and dry cleaners, and their associated remediation efforts, and a plethora of
field scale experiments. These scales are examined in the perspective of the error
in travel time estimate associated with assuming a resident injection when the real
injection is in flux
e[x] = \\nktfnk{l exp[6x/Ainfc])/a; (3.19)
given by subtracting Equation (3.15) by Equation (3.14) and dividing the difference
by Equation (3.14) (see Figure 3.5) The near field might be characterized as the first
five to ten correlation lengths, where the injection mode effects are quite prevalent,
even for modest log conductivity variability. These effects diminish less rapidly over
BIOGRAPHICAL SKETCH
George Demmy was born to George and Ellen Demmy 23 November 1966 in
Lakeland, Florida. He received a Bachelor of Arts in Physics and German Literature
from the Florida State University in 1990, a Master of Engineering in Agricultural
Engineering from the University of Florida in 1993, and Doctor of Philosophy in
Agricultural and Biological Engineering from the University of Florida in 1999.
138
3
This harmonic/arithmetic mean discrepancy was explicitly noted by Dagan et al.
[1992], who relegated the Shapiro and Cvetkovic [1988] result to applicability in areas
close enough to the input zone [Dagan et al, 1992 p. 1374].
Cvetkovic et al. [1996] developed a semianalytical expression that described the
transition from the near field harmonic mean velocity to the far field expression
based upon the these known endpoints and an estimate of the Lagrangian velocity
integral scale. Their detailed analysis of the Lagrangian velocity field revealed that
the Lagrangian velocity is nonstationary in displacements in space, and resulted in a
nonlinear propagation of the mean arrival time with distance.
This was strange. Why was the Lagrangian velocity field nonstationary when
the Eulerian was stationary? It was commonly assumed that the stationarity of one
implied the stationarity of the other. What had started as a simple preliminary
literature review resulted in a quandry. The paths followed in pursuit of this quandry
led to the research goals of the dissertation.
1.3 Research Goals
There were four broad objectives specified for this work. All centered around the
flow of water and the transport of solutes in aquifers characterized by heterogeneous
hydraulic conductivity fields.
1.3.1 Lagrangian Velocity Mean and Covariance
A primary goal of this research was to develop a Lagrangian covariance function
for the Standard Model Aquifer. This was extended slightly to the mean and
covariance of the spacestationary velocities along equal flowweight streamlines and
timestationary velocities along equal areaweight streamlines. Quantification of this
covariance function greatly simplifies the development of equations, or estimators,
that describe the movement of water and solutes in the heterogeneous velocity fields
associated with the Standard Model Aquifer.
CHAPTER 5
APPLICATION OF CONCEPTS
5.1 Introduction
An illustration of a practical application of trajectory based flow and transport
modeling is sought. Useful and fairly robust relationships, namely predictors of spatial
moments of a continuously injected plume subject to a first order decay, using simpler
streamtube concepts (e.g. Jury and Roth [1990]). However, the observed bias in
prediction shall be explained using the more complete trajectory based approach.
In light of the robustness of the simple relationships developed here, they are used
to estimate relative degradation rates for different solute species and applied to field
data collected at a fieldscale macrodispersion and natural attenuation experiment
conducted in Columbus, Mississippi (see Stauffer et al. [1994] and Stauffer et al.
[1997]).
5.2 Theory
Consider a subsurface source of groundwater contamination that releases solute
into the aqueous phase at a constant concentration. An example of such a source is
a poorly mobile nonaqueous phase liquid, comprised of relatively insoluble organic
compounds, distributed at some nearresidual saturation through out a portion of
some aquifer. The constant assumption applies when the rate of change of a prop
erty of interest, such as the aqueous phase concentration at the source, is small in
comparison to the temporal scales associated with transport and decay.
The actual groundwater flow field is modeld as a steady irrotational flow field that
may be resolved into an aggregation of streamtubes. Neglecting the effect of local
108
41
may be resolved into a mean, and a zeromean perturbation, e.g., In A: = p\nk + /.
Let the head gradient likewise be resolved into a mean and zeromean perturbation,
e.g., d(j)/dx = J( 1 + j). Inserting these expressions into Equation (2.15) yields
q = exp^in* + f]J{l+j)
(2.16)
= KgJ exp[/](l + j)
Expand the exponential term in this equation as a Taylor series about zero, truncating
the series at secondorder, and expand the product of the perturbation terms
q~K,J(l+f + P/2)(l+j)
(2.17)
= A',./(! + / + p/2 + j + fj + Pi/2)
Take the expected value of this expression, dropping terms higher than secondorder,
resulting in an expression
= KgJ(l + ofnA,/2f ) (2.18)
Consider this equation and the three terms in the parenthesis. The first is a mean
term, the second is a contribution due to the variability of the conductivity field
and the third is the log conductivity and head gradient crosscovariance. Gelhar
[1993] evaluates this expression for isotropic log conductivity fields using spectral
methods. The result, which is independent of the shape of the correlation function,
is surprisingly simple [Gelhar, 1993]
=o&Jd (2.19)
where d is the dimension of the domain. This negative correlation between head gradi
ent and log conductivity has a profound effect on the flow. The effective conductivity
may be defined as that value when multiplied by the spatial mean gradient results
in the observed specific discharge (flow rate divided by discharge area). The a pri
ori effective conductivity is the effective conductivity estimated from Equations 2.18
and 2.19. With d = 2, we have for twodimensional conductivity fields, the a priori
121
equal discharge (FW/LT). For equallyspaced control times, only Lagrangian trajec
tories equallyspaced at a reference plane (AW/LT) appeared to exhibit stationary
log conductivity and velocity statistics.
The stationarity of the properties greatly simplifies the development of predictors
of solute transport. When taking the expected value of the integral of some property
along a trajectory, say the integral of velocity in time, the timevarying velocity
function may be replaced with its time average.
As implied by the stationarity of the statistics along some trajectory collections,
but not others, the numerical statistics of properties varied between the different
collections. Adopting simple and accurate models for the statistical properties of
some property in a reference collection, the statistics of the properties along other
collections, including the observed nonstationarities, were well predicted. This had
significant application in the evaluation of the effect of injection mode upon solute
transport.
Two methods by which solute might be uniformly introduced into a system were
considered. The first was a resident injection were a fixed volume at the system inlet
was uniformly filled with solute at some reference concentration. The second was to
allow influent water at some reference concentration to carry the mass into the system.
The displacement and travel time statistics of AW/LT correspond to solute transport
associated with a uniform resident injection. The transport of solute injected in flux
is described by the displacement and travel time statistics of the FW/LT.
The socalled consistent firstorder approximations of the inverse velocity were
shown to be built upon a flawed premise, but are nonetheless quantitatively cor
rect and robust. The premise is that the Lagrangian flow field may be accurately
approximated by the Eulerian flow field. While this may be true for the weakest
of heterogeneities, this approximation breaks down rapidly with increasing hetero
geneity if it is consistently applied. As inconsistently formulated, the approximation
33
particle injected into a flow field. Constraint of the trajectory to an Eulerian tra
jectory forces the trajectory through lowvelocity areas that a natural streamline, or
Lagrangian trajectory, would have otherwise bypassed. Of course, some streamlines
pass through even the lowest velocity areas, but these are few in an equalflow sense.
Thus, equalarea weighting preferentially weights low velocity zones. That there are
implications for sampling schemes in heterogeneous media, especially in context of
multilevel samplers, should be intuitively obvious to even the most casual observer.
A discussion of some of these implications is made in a subsequent section, after these
concepts are developed further in a more quantitative and theoretical framework.
It is appropriate to evaluate the adopted expression for the AW/ET longitudinal
covariance function suggested by Dagan and Cvetkovic [1993], namely
Cua[r] = afnlcb[e]U2 exp [&[e]r/Ainfc] (2.13)
where r is the separation between points along the ET, and b[e] is a function of the
anisotropy ratio e. The anisotropy ratio itself is a function of the correlation lengths
of the log conductivity in the different directions. The primarily consideration is
that b may be derived from theoretical considerations and is not some ad hoc fitting
parameter. The value of this function for a twodimensional isotropic aquifer is 3/8.
Again, for further background on this function, see Dagan and Cvetkovic [1993]. In
the limit r = 0, the covariance reduces to the point variance, which for the model
case is al = 3U2afnk/8. For the remainder of this discussion, a reference to b contains
its dependence upon e implicitly. The subscript ua may be decoded as follows: for
a subscript tw, the t is the trajectory and property (u for Eulerian velocity, and v
for Lagrangian velocity) and the w is the weight (a for area and / for flow). Thus,
Cua is the covariance function for velocities along Eulerian trajectories. We present
this covariance function and that covariance function estimated from the assumed
stationary collection of AW/ETs in Figure 2.12 The perceptive observer will note
2.10 Comparison of positiondependent mean velocity estimators to data for
the four collections of trajectories for a2nk = 1/2 (a) and a2nk = 1
(b). Velocities are normalized by U 30
2.11 Comparison of the positiondependent velocity variance estimators to
data for the four collections of trajectories for elk =1/2 (a) and
a2nk = 1 (b). The variances are normalized by U2ba2nk 31
2.12 Comparison of spatial Eulerian (AW/ET) and Lagrangian trajectory
(FW/LT) velocity covariance data to covariance model. Data are
taken from the a2nk = 1 set of realizations. Covariances presented
as dimensionless lag correlation functions 34
2.13 Comparison of timedependent mean velocity estimators to data for
the four collections of trajectories for a2nk = 1/2 (a) and a2nk = 1
(b). Velocities are normalized by U 36
2.14 Comparison of timedependent velocity variance estimators to data for
the four collections of trajectories for a2nk = 1/2 (a) and a2nk = 1
(b). The variances are normalized by U2ba2nk 37
2.15 Comparison of temporal Lagrangian trajectory (AW/LT) velocity co
variance data to covariance model. Data are taken from the a2nk = 1
set of realizations. Covariances presented as dimensionless lag cor
relation functions 38
2.16 Normalized log initial velocities for twodimensional model aquifer com
pared to the standard normal distribution. The random variable was
normalized by the sample mean and standard deviation to illustrate
the lognormality 39
2.17 Comparison of velocity covariance expression Equation (2.25) to the
approximation Equation (2.13) for a2nk = 1. Covariances are pre
sented as dimensionless correlation functions 45
2.18 Components of the velocity covariance function 46
3.1 Comparison of mean displacements for the two injection modes or
streamtube collections. The flowweighted Lagrangian trajectory
(FW/LT) statistics correspond to a flux injection, and the area
weighed Lagrangian trajectory (AW/LT) statistics correspond to a
uniform resident injection. The top figure contains the a2nk = 1/2
data and the bottom figure contains the <72nfc = 1 data 73
xi
95
% / *bn k
Figure 4.8: Correlation between the reaction flow path and travel time for differ
ent values of correlation parameter (3 and different trajectory collections for the
a\nk = 1/2 set of simulations. Top figure is f3 = { 1,+1}. Bottom figure is
Â¡3 { 1/2,+1/2}. Trajectory collections are areaweighted Lagrangian (AL) and
flowweighted Lagrangian (AF).
66
some small Ax that is invariant with displacements along the IP. Following the ter
minology of Kreft and Zuber [1978], this injection mode is defined to be a uniform
resident injection. Uniform refers to the homogeneous mass density in space and
resident refers to the volume of resident fluid into which the solute is introduced.
Subsequent references to a resident injection will refer to a uniform resident injection
unless otherwise noted. Now, this injection mode has nothing to do with the flow field
per se. That is, the mass magically appears and occupies a fixed volume at a fixed
concentration without being influenced by local water fluxes. Consider the flow field
to be divided into some number of streamtubes, say N. Moreover, let these stream
tubes carry an equal areaweight at the IP. For a small depth of injection Ax, this
injection mode has assigned an equal mass weight to each streamtube with respect
to the total solute mass. The total mass of solute is
Mr coOhLAx
(3,1)
The mass weight for any one streamtube i is
mi 1
M =
(3.2)
This value is identically equal to the areaweight of the streamtube, and constant for
all in the collection.
Consider now dividing the streamtube into N streamtubes that all carry the same
flow weight. The mass in some streamtube i is given by
m a
~M = ~L
_ U
Vi
(3,3)
8
an important and welldeveloped discipline, from which a great deal of insight and
useful mathematics may be drawn. For an excellent review, especially with respect
to concepts related to transfer functions, see Jury and Roth [1990].
For a large domain, the scale at which chemical reactions takes place is generally
much smaller than that of the domain. That is, processes such as sorption, microbial
and chemical degradation, and the like, are dependent upon chemical potentials across
distances on the order of the pore scale to perhaps that of a microDarcy scale,
defined as the smallest volume for which Darcys law is generally applicable. However,
the scale of interest tends to be much larger, say that of a well, or its capture zone
or radius of influence, or perhaps the interface across which groundwater discharges
into a surface water body, or, perhaps the Floridan aquifer. In order to consider
these reactions in detail, it is necessary to resolve the larger domain into smaller
subdomains. For modeling real systems, the scale and degree to which the domain
is subdivided should be based upon the available data.
2.2.1 Simple Subdivision
Consider the water flowing into the model aquifer across the inlet, or injection,
plane (IP). Assume it is of interest to divide the domain into two subdomains. There
are many ways to do this. One would be to divide the domain down the middle, that
is, at a line parallel to the x axis located y = Ly/2 (curve A Figure 2.2). This division
creates two subdomains of equal volume and gives an equal volume weight to each
subdomain. The division also creates two subIPs of equal area and gives an equal
area weight to each subdomain. For a perfectly homogeneous porous medium, the
groundwater velocity vectors will be parallel to this line, and the discharges through
each subdomain will be the same as well. Moreover, there will be no advective
transfer of water across the boundary. However, the introduction of conductivity
heterogeneity in the transverse direction will almost certainly lead to deviations of
the streamlines, and to streamlines crossing the boundary. This implies that while
102
perhaps a better fit to the data. The Eulerian log conductivity covariance is the
foundation upon which the rest of this work has been built. The model field is ex
ponentially correlated in space, with correlation length \\nk The Eulerian velocity
covariance exhibits a much longer correlation length due to the dissipation of the
head across the field (see previous chapters). It is reasonable, therefore, to expect
that the Lagrangian log conductivity covariance correlation length will lie somewhere
in between that of the Eulerian log conductivity and the Lagrangian velocity.
At zero lag, the areaweighted Lagrangiantrajectory log conductivity covariance
is equal to the variance of the Eulerian field. This value is ofnfc, by definition. Thus,
an exponential approximation of the Lagrangian log conductivity covariance function
is
C\nkiat[t} = cr?nkexp[b'Ut/\lnk] (4.29)
where b' is a function similar to b that relates the correlation length to a correla
tion time. The approximation b' 1 is made, which is equivalent to assuming that
the Lagrangian and Eulerian correlation lengths are equal is space and substituting
x = Vht Ut into the spacedependent Eulerian log conductivity covariance func
tion. Comparison of Equation (4.29) with b' = 1 to covariance functions estimated
from log conductivity observations taken along areaweighted Lagrangiantrajectories
generated in the numerical experiments described in Chapter 2 shows good qualita
tive agreement (see Figure 4.9). The data exhibit an apparent dependence of the
Lagrangian correlation length upon the variance of the log conductivity field that is
not captured by Equation(4.29). One might expect similar behavior in the Eulerian
field a priori, since the appropriate substitution into the Eulerian covariance function
would be x Uht = Ut/( 1 + ba^nk). The higher variance of log conductivity results
in a lower harmonic mean average velocity, implying it takes longer to to travel a
unit length in more heterogeneous fields and a larger Lagrangian correlation time. In
58
the quantitative similarity and write velocity variance equations directly The com
plete expression is
var [vf [t]] = U2 (l + bafnk + (2bafnk + 3 (ba?Dk)2 + (btfU)3) (1 exp [bUt/\lnk]f)
 {U (1 + bafnk exp [6r/Alnfc]))2
(2.64)
and the approximation is
var [vf [t]] = U2bafnk (l + (2bafnk + {ba?nk)2^) exp [bUt/Alnfc]) (2.65)
Calculation of the mean velocities along the Eulerian trajectories is problematic
in the absence of a stationary velocity covariance function. However, note that the
areaweighted velocities start at U and that the flowweighted velocities start at V
and both progress to U/(l + bo\nk). Recognizing the similarity shared by the different
mean time estimators, write approximate expressions directly
at [i] = 1 (1 + ba?n k exp [bUt/Aln*]) (2.66)
and
Uat [^]
x + ba2 (! + (2blk + Wnfc) (eXP [bUt/XInfc]))
(2.67)
Again, devoid of a stationary covariance function for the Eulerian velocity field in
time, intuitive arguments lead us to anticipate an asymptotic u variance in time to
be
a:
ut
uzMnk
u
! + Kn k
h\n k
(2.68)
The initial velocity variances are known from the covariance functions in space. It is
possible to anticipate forms of the nonstationary variance equations similar to those
found previously, and include them for the sake of symmetry and completeness. The
11
Figure 2.3: Four different partitioning strategies for the same flow field: A) area
weight Eulerian trajectory, B) area weight Lagrangian trajectory, C) flow weight
 Lagrangian trajectory, D) flow weight Eulerian trajectory.
7
uniform thickness in the vertical direction, say h. The total area of the vertical plane
at x = 0 through which water enters the aquifer is A. Define the spatiallyaveraged
groundwater seepage velocity at this inlet plane (IP) as
jj = 14 (Â¡[a\da
Lda (2.1)
Q_
6A
For a large class of hydrologic applications, water quantity is the sole concern.
For these problems, the route taken by the water is of little interest, as is, say
the distribution of ages or residence times. Spatial and temporal distributions of
hydraulic properties hold some interest, but only in how they relate to effective bulk
properties.
However, a large class of practical hydrologic problems exist for which the trajec
tories of water parcels across the flow domain is of great importance. Many of these
problems involve the fate and transport of contaminants through the aquifer system.
The fundamental problem is this: to relate an Eulerian flow field to its Lagrangian,
or trajectorybased, equivalent.
2.2 Flow Field Partitioning Strategies
A streamline is a hydrodynamic entity that is at all points tangent to the ve
locity vector. For twodimensional flow, a streamtube is an object defined by two
streamlines. Stream surfaces, the intersection of which form streamlines, define three
dimensional streamtubes. Infinite collections of streamlines exist that subdivide the
flow field into collections of streamtubes. Again, the interest of clarity suggests a focus
upon twodimensional flow. For a lucid discussion of streamlines, stream functions,
and stream surfaces in two and threedimensional flows, see Bear [1972],
Impervious boundaries are streamlines, thus the entire model domain is a stream
tube by definition. Viewed as a black box, Eulerian and Lagrangian distinctions
have little meaning. This black box approach is the purview of reactor engineering,
23
four different trajectories are calculated at intervals in time and in space. Thus, for
any one property, say log conductivity, and any one statistic, say the mean, there will
be two sets of results, namely that for ofnfe = 1/2 and that for afnfc = 1. Associated
with each set will be two subsets, namely statistics along the four trajectories recorded
at space or time intervals.
For convenience, the notation and the trajectories to which each notation corre
sponds as depicted in Figures 2.2 and 2.3 is summarized here:
AW/ET areaweighted Eulerian trajectory (Figures 2.2 and 2.3 A)
AW/LT areaweighted Lagrangian trajectory (Figures 2.2 and 2.3 B)
FW/LT flowweighted Lagrangian trajectory (Figures 2.2 and 2.3 C)
FW/ET flowweighted Eulerian trajectory (Figures 2.2 and 2.3 D)
2.4.7 Log Conductivity Statistics
The log conductivity field serves as the foundation upon which the simulations
are built, as well as the theory. The data for the areaweighted Eulerian trajectories
correspond to those design criteria specified. The log conductivity mean and vari
ance for AW/ET appear stationary in space (see Figures 2.6 and 2.7). That is, they
appear to bounce around their target values with no apparent trend. Moreover,
the statistics of FW/LT appear likewise stationary, but the mean is somewhat larger
and the variance slightly smaller than AW/ET. The magnitude of the differences
appear proportional to ofnA:. A comparison of the target values, e.g., the specified
inputs into the turningbands algorithm, and the observed ensemble values along the
trajectories is presented in Table 2.1. The observed entry in Table 2.1 represents
the statistic average along the trajectory, as estimated by regressing a zeroslope line
through the data using a generalized leastsquares method. There is close correspon
dence between the target and observed values, supporting an ergodic hypothesis for
the log conductivity statistics. That is, sample statistics are an accurately estimation
38
for both simulations. This result is important, because it implies the possibility of
mapping from AW/ET to AW/LT. We estimated the covariance function for the
stationary AW/LT velocities (see Figure 2.15). As an estimate of the covariance
1
0.8
 0.6
4
_ro
k_
8 0.4
0.2
0
0 2 4 6 8 10
tU/X\nk
Figure 2.15: Comparison of temporal Lagrangian trajectory (AW/LT) velocity covari
ance data to covariance model. Data are taken from the ofnfc = 1 set of realizations.
Covariances presented as dimensionless lag correlation functions.
function, the substitution r = Ut was made in Equation 2.13 and the result compared
to the estimated covariance function (see Figure 2.15). The closedform predictor
behaves similarly to its spatial counterpart: earlytime correlation is underpredicted
and longtime correlation is overpredicted (compare Figures 2.15 and 2.12).
For completeness, one other aspect of the velocity field was examined, namely the
distribution of initial velocities (see Figure 2.16). The logarithm of the initial velocity
was rankordered and subject to the transformation
. lnu \nv
lnu =
^ln v
(2.14)
105
refers to nonreactive travel time. Reaction flow path variance, however, is strongly
affected by correlation, and Equation (4.30) is completely inappropriate (see Fig
ure 4.12). Moreover, this equation can give physically implausible values at large
Figure 4.12: Reaction flow path variance a function of nonreactive travel time for
correlation parameter values Â¡3 = {1,0,1}.
values of time. This is a result of the series approximations required for the raw
second moment term in the variance, and the lack of such approximations in the
first moment squared term.
4.7.1 Injection Mode
The coupled effects of injection mode and correlation on the mean travel time
dependent reaction flow path are examined. Using the relationships developed in
Chapter 3, the hypothesis that the statistics reaction flow path corresponding to the
equalflow streamtubes are given by flowweighting the equalarea properties is made.
CHAPTER 2
LAGRANGIANEULERIAN FLOW FIELD RELATIONSHIP
Develop an intuitive judgment and understanding for everything.
Miyamoto Musashi1
2.1 Introduction
Consider the steady, welldeveloped flow of water in an aquifer meeting Bears
[1972] definition of a porous medium. In most natural settings, gentle gradients
induce a flow wellquantified by Darcys law. Knowledge of the water fluid properties,
relative permeability of porous medium, and potential gradients in space and time
would assure a reasonable estimate corresponding water fluxes.2
One way to record this knowledge would be to create maps of these different
properties in space at different times. This framework is commonly called the Eulerian
framework, and is the most common approach to hydrologic problems. Another way
to record this knowledge is to map properties along moving coordinates in time.
This approach is commonly called the Lagrangian framework, and its application
may without controversy be described as less widespread than Eulerian approaches.
Dagan [1989] and Gelhar [1993] review development of flow of fluids and transport of
contaminants in porous media in both of these frameworks.
1 Musashi [1982 p. 49]
2 The concepts associated with the following discussion are quite general, and are
applicable to a much larger variety of environmental transport problems. The em
phasis upon groundwater follows from an interest in simplicity.
5
136
Jawitz, J. W. Aquifer contaminant source zone characterization with partitioning
tracers and remediation with singlephase microemulsion flushing. PhD thesis, Uni
versity of Florida, Gainesville, Florida, 1999.
Jin, M., M. Delshad, V. Dwarakanath, D. C. McKinney, G. A. Pope, K. Sepehrnoori,
and C. E. Tilburg. Partitioning tracer test for detection, estimation, and reme
diation performance assessment of subsurface nonaqueous phase liquids. Water
Resour. Res., 31(5):12011211, 1995.
Journel, A. and C. Huijbregts. Mining Geostatistics. Academic Press, New York,
1978.
Jury, W. A. and K. Roth. Transfer Functions and Solute Movement Through Soil.
Birkhuser Verlag, Basel, 1990.
Konikow, L. F., D. J. Goode, and G. Z. Hornberger. A threedimensional methodof
characteristics solutetransport model (MOC3D). Waterresources investigations
report 964267, United States Geological Survey, Restion, Virginia, 1996.
Kreft, A. and A. Zuber. On the physical meaning of the dispersion equation and
its solutions for different initial and boundary conditions. Chem. Eng. Sci., 33:
14711480, 1978.
MacIntyre, W. G., M. Boggs, C. Antworth, and T. B. Stauffer. Degradation kinetics
of aromatic organic solutes introduced into a heterogeneous aquifer. Water Resour.
Res., 29(12):40454051, 1993.
Mood, A. M., F. A. Graybill, and D. C. Boes. Introduction to the Theory of Statistics.
McGrawHill Publishing Company, New York, 1974.
Mos, R., P. Siegel, P. Ackerer, and G. Chavent. Application of the mixed hybrid
finite element approximation in a groundwater flow model: Luxury or necessity?
Water Resour. Res., 30(11):30013012, 1994.
Musashi, M. A Book of Five Rings. The Overlook Press, Woodstock, New York,
1982.
Osnes, H. Stochastic analysis of velocity variability in bounded rectangular hetero
geneous aquifers. Adv. Water Resour., 21:203215, 1998.
Papoulis, A. Probability, Random Variables, and Stochastic Processes. McGrawHill
Publishing Company, New York, 1991.
Pollock, D. W. Semianalytical computation of path lines for finitedifference models.
Ground Water, 26(6):743750, 1988.
57
expression for the stationary areaweighted Lagrangiantrajectory velocity covariance
CJi] = U2balÂ¡k C'>:p[M/i/Ai]
(2.61)
Similar expressions have been derived before by other researchers (see e.g., Dagan
[1989]), but the conceptual development is quite different. Rather than seek to ap
proximate the covariance function, an approach to quantify it directly from properties
of the system is demonstrated. Moreover, the quantity U is perceived to be the har
monic mean of the Lagrangiantrajectory velocity, and that it is quantitatively equal
to the areaaverage velocity of the flow field. This is an important point.
The mean velocity and velocity variance along the flowweighted Lagrangian
trajectory is found in a manner similar to those in space, but there are some concep
tual issues that should be addressed. It is the collection of areaweighted Lagrangian
trajectories that exhibits a stationary velocity. The mean velocity is
. u[0] ,
Vft[t] =<jfv[t]>
(2.62)
Expand the velocities into a mean and a zeromean perturbation. The mean here is
the time average mean, or the harmonic mean of the space average.
W = ^ <^(1 +'[0])r/ (! + '[(])>
(2.63)
U (l + bainkexp[bUt/\lnk])
This expression requires no series expansion. For the control plane oriented velocity
statistics, the mapping was from flowweighted to areaweighted trajectories. Equa
tion 2.63 is identical in form to Equation 2.44 with the substitution r = Ut. Recognize
39
log velocity
Figure 2.16: Normalized log initial velocities for twodimensional model aquifer com
pared to the standard normal distribution. The random variable was normalized by
the sample mean and standard deviation to illustrate the lognormality.
where lnu is the sample mean and Ein is the sample standard deviation. Work
ing here with the logarithm, rather than the velocity itself, better demonstrates the
apparent lognormality of the velocity fields. Recall, the logarithm of a lognormal
random variable is itself normally distributed. Figure 2.16 indicates that the trans
formed log initial velocities are welldescribed by a standard normal distribution. The
lognormality of the velocities seems logical since the conductivity field is itself lognor
mally distributed. However, the velocity is in general a product of random variables:
the conductivity, the head gradient, and the mobile water content. The logarithm of
the velocity is the sum of random variables. Sums of random variables tend toward
normality by the central limit theorem (see Mood et al. [1974]), and thus, products
of random variables tend towards lognormality. It would be interesting to test this
47
approximation, however, will realize significant returns in the form of simplicity, as
shown in subsequent sections.
Previous works have focused upon the moments of the Eulerian velocity field that
result from mean gradients imposed upon heterogeneous conductivity fields. These
works shall be extended slightly with a hypothesis about a specific case that allows
a variety of insights to be drawn.
Hypothesis: The point velocity distribution of an Eulerian velocity field induced
by an stationary mean gradient J across a lognormally distributed, exponentially
correlated log hydraulic conductivity field characterized by parameters fi\nk, crfnk,
and A^n k and saturated with mobile water at a constant and homogeneous content 9
is lognormal with approximate parameters
(2.
and
/hnu In
KgJC
dV 1 + bl
In k
Ohxu = ln[! + blk\
(2.29)
(see Appendix A for details of development). A fundamental assumption in addition
to the lognormality hypothesis is that the real space moments of u are \iu = U
and o\ = U2ba2nk. Equation (2.13) is adopted as an approximation of the velocity
covarince function.
In lieu of a rigorous proof, the results of a numerical experiment in which statisti
cally independent velocities are drawn uniformly in space from simulated flow fields
are presented (see Figure 2.16). The normalization of the velocities in Figure 2.16
than that of previous figures in that the log velocities are normalized by the sample
mean and standard deviations to emphasize the lognormality of the velocity distribu
tion. Normalization by the moments predicted by Equations (2.28) and (2.29) would
have resulted in a lognormal distribution function that deviates from the standard
17
1
0.8
o
c
to
(0
>
o
0.4
0.2
0
0 12 3
T / ^ln k
Figure 2.5: Estimated log conductivity longitudinal covariance estimated from a re
alization generated by the turningbands method compared to an exponential covari
ance function. Variance of log conductivity ofnfc = 1 and correlation length X\nk = 1.
Spacing for log conductivity values was Ainfc/5.
Trajectories covered several characteristic lengths A]nfc and several characteristic
times Ai k/U to capture both nearfield and farfield behavior in space and Lagrangian
time. The first requirement was easy to enforce by making the domain size equal to or
greater than the required number of correlation lengths. The second requirement was
somewhat more problematic, since the distance traveled in some number of X\nk/U
varies from trajectory to trajectory. Preliminary simulations indicated that a domain
40 Ain A: long was adequate to capture travel up to 10 characteristic time scales Ainjt/U
in 2000 trajectory realizations generated in the most heterogeneous field considered
(ofnfc = !) Thus, the selected domain extent was 40A[nfc.
The conceptual model aquifer is a large, bounded, primarily twodimensional,
steady flow field. As such, it should exhibit a variety of head, head gradient, and
velocity nonstationarities due to the boundary conditions (see Figure 2.1). While
estimated In k covariance
exponential covariance
13
It is instructive to illustrate these concepts with an example and to interpret the
results of this illustration.
2.4 Numerical Experiments
The illustration of these concepts requires a heterogeneous flow field and col
lections of trajectories, both Eulerian and Lagrangian, that correspond to this flow
field. The requirement that the flow field be based upon reference fields of known
properties enhances the possibility of theoretical interpretation drawn from the large
body of literature concerning flow and solute transport in idealized porous media. A
further requirement that our reference fields bear some resemblance to what might
be expected in nature enhances the possibility of useful applicability.
A numerical simulation code modeled steady groundwater flow in a hypothetical
heterogeneous twodimensional aquifer. A particle tracking code traced Eulerian and
Lagrangian trajectories across this aquifer, recording data that included the local
groundwater velocity and local hydraulic conductivity at intervals in space and time
along the different trajectories. Statistics, specifically mean, variance, and covariance,
summarize the results of these measurements.
The conceptual view of the aquifer is that given by Figure 2.1. To generalize the
system, correlation length, or integral scale, of the log conductivity field serves
as a characteristic length with which to normalize length scales. The characteristic
time Aink/U serves to normalize time scales, where U is the arithmetic mean velocity
defined in Equation (2.1). The definition of an integral scale is the integral from zero
to infinity of the correlation function R, or,
roo
A = / f?[s]ds
Jo
(2.4)
81
travel to some point x, but as an intrinsic integrated streamtube property, namely
the inverse velocity integrated along the trajectory (e.g., Equation (3.9)).
Recall reaction flow path concept of Cvetkovic et al. [1998]. The reaction flow
path, parameterized in space, is defined as
(4.2)
where P[x] is some reaction parameter of interest. For linear equilibrium sorption,
P is a linear partitioning coefficient. The reaction flow path is also an integrated
streamtube property. As such, any reference to n properties in general are broadly
applicable to any distributed reaction parameter, perhaps a firstorder decay coef
ficient or a Langmuir sorption parameter. However, the focus here is upon linear
equilibrium sorption to promote a ready and intuitive grasp of the pertinent con
cepts. For this case, Â¡i may be thought of as the time that the solute is sorbed. Thus
the arrival time of a particle undergoing a linear equilibrium sorption process is the
sum of the nonreactive travel time r and the time sorbed fi. The transfer function
characterizing this transport and reaction process is (see Cvetkovic et al. [1998])
T[*i *] = l< T[x] lAA]
(4.3)
Following the work of Cvetkovic et al. [1998], the adopted reaction parameter
model is
P = Pg exp[io] exp[/3 In k]
(4.4)
where Pg is the geometric mean of P, w is a zeromean, exponentiallycorrelated
normal random variable with variance and correlation length A, = Ain* that is
uncorrelated to the log conductivity field, and /? is a strength of correlation parameter
relating P to the underlying log conductivity field. The equality A^ = X\nk was
specified for convenience.
55
are not a panacea, however, and the approximations have only been tested for the
log conductivity variances tested in this work.
The nonstationary flowweighted Euleriantrajectory velocity variance may be de
rived in a similar fashion. These results are presented directly. The complete expres
sion is
var [uf [x]] = U2 (l + ba?nk + (2bafnk + 3 (bafnk)2 + (krjj,*)3) (1 exp [br/Alnfc]))
 (U (1 + 6a,2nfcexp[6r/AinJt]))2
(2.56)
and an approximation of the above obtained by neglecting a few terms is
var [uf [*]] = U2ba?nk (l + (2bafnk + (6a,2nJt)2) exp [6r/Alnfc]) (2.57)
2.5.5 Control Time Oriented Velocity Moments
The displacementinvariant velocity covariances for u and v permitted derivation
of expressions for the first two moments of the respective velocities for the four dif
ferent trajectory collections. However, only the areaweighted Lagrangiantrajectory
collection appears to be stationary in time. This is perhaps not intuitively obvious,
as the velocity is clearly not stationary in displacements along the mean direction
of flow (e.g., Equations (2.43) and (2.54)) because there is an implicit distribution
of times associated with the establishment of a reference injection plane. After long
travel distances and times, the areaweighted Lagrangiantrajectory point velocity
statistics are equal to the flowweighted Lagrangiantrajectory point statistics. At
the injection plane, the areaweighted Lagrangiantrajectory point velocity statistics
are identically equal to the areaweighted Euleriantrajectory point velocity statistics.
These relationships can be exploited to derive equations that provide a few insights
about how properties change in Lagrangian time, and to derive an expression for a
Lagrangiantime dependent velocity covariance function.
49
The perceptive reader will notice a slight notational change between the two parti
tioning schemes. A lowercase u represents velocities drawn uniformly in space and a
lowercase v represents velocities drawn uniformly in flow. The distinction is impor
tant. It is through these subIPs that water enters the streamtubes and a description
of the subIP properties is essential to streamtube, and thus flow field, descriptions.
If from a collection of N subIPs the random selection, or drawing, of one subIP is
equally likely as any other, the weight assigned to any one observation from M samples
would be 1/M. Assume that all of the subIPs are sampled. The average subIP dis
charge is the same for the equalarea and equalflow partitioning schemes, and equal
to d = 6AU/N. However, the discharge variance is zero for equalflow and nonzero
for equalarea, when the velocity field is heterogeneous. The weights expressed in
Equations (2.34) and (2.31) allow the properties of one collection of streamtubes to
be mapped to the other collection of streamtubes. These weights combined with the
assumed values of the Eulerian velocity field permit derivation of properties of the
initial velocities of the equalflow partitioning schemes. The expected value of v must
equal the expected value flowweighted u. Thus,
= V
u
= < U>
U
= ij(al+^)
= U(l + baÂ¡nk)
= ^( i + K*)
(2.35)
6
Consider now flow of water in a more specific model aquifer: the steady, irrota
tional, and Darcian flow of water of homogeneous fluid properties in a finite rectangu
lar twodimensional aquifer of heterogeneous hydraulic conductivity (see Figure 2.1).
Mobile water 9 completely saturates the homogeneous pore space. An orthogonal
Figure 2.1: Schematic representation of an aquifer.
Cartesian coordinate system lies parallel to the aquifer boundaries, and x and y de
note its directions. Parallel and impervious boundaries lie at y = 0 and y = Ly.
Constant potential boundaries located at x = 0 and x = Lx maintain a constant av
erage potential gradient J across the domain, such that water flows into the aquifer at
x = 0 and out of the aquifer at x = Lx at a volumetric discharge rate Q (dimensions
[.L3T~1]). For an intuitive dimensional consistency, allow the aquifer some small and
CHAPTER 1
INTRODUCTION
1.1 Primary Contribution
The primary contribution of this work is documenting a consistent approach to
the development of estimators of solute transport in the Lagrangian framework. This
consistent approach is to identify the collection of Lagrangian trajectories associated
with a stationary Eulerian velocity field for which the Lagrangian velocities are also
stationary. This is a modest extension of work pioneered by myriad hydrologists.
The benefit derived from this contribution is enhanced understanding of transport of
solutes in heterogeneous velocity fields.
1.2 Historical Context
A grant to study the impact of heterogeneous source morphology upon subse
quent solute transport initiated this research. In seeking a suitable framework and
toolset with which to attack this problem, there appeared what seemed to be funda
mental inconsistencies in the literature. Given the peculiar tendencies of the author,
the larger issues could not be addressed until these apparent inconsistencies were
resolved.
Gedeon Dagan developed a robust theory of solute transport using a stochastic
Lagrangian framework [Dagan, 1982a, b]. In fact, an entire school of stochastic
subsurface hydrology sprung from his work.
Allen Shapiro and Vladimir Cvetkovic wrote Stochastic analysis of solute arrival
time in heterogeneous porous media in 1988 [Shapiro and Cvetkovic, 1988]. The au
thors made an explicit assumption often implicitly made in the Lagrangian transport
literature, namely that a fluid parcel deviates little from its mean trajectory in weakly
1
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SOLUTE TRANSPORT IN HETEROGENEOUS POROUS FORMATIONS
By
George Gary Demmy, Jr.
December 1999
Chair: Wendy Graham
Major Department: Agricultural and Biological Engineering
This work quantifies relationships between the spatial, or Eulerian, distribution of
the properties of a chemically and physically heterogeneous porous medium and those
as observed along the natural, or Lagrangian, trajectories that a fluid particle traces
in a steady and irrotational flow. From these relationships, expressions that relate
the transport of solutes through the porous medium along the natural trajectories
to the aforementioned Eulerian distributions are developed. The effects of injection
mode upon global measures of transport as reflected by the temporal moments of
breakthrough curves and spatial moments of a solute plume are developed. The
coupled effects of correlation of a linear equilibrium sorption to the underlying log
hydraulic conductivity field and injection mode on the evolving temporal moments
of mass breakthrough curve and the coupled effects of correlation of a firstorder
decay coefficient and injection mode upon the spatial moments of a solute plume are
examined.
xv
9
Figure 2.2: Simple subdivision of the flow field by one of four strategies: A) equal
area straight line, B) equalarea streamline, C) equalflow streamline, D) equalflow
straight line.
the volume of water contained in each subdomain is constant, the discharge through
a plane perpendicular to the partition is a function of plane position in x.
Consider now a streamline originating at (0,Ly/2) (curve B on Figure 2.2). For
a homogeneous medium, this corresponds to an equal volume partitioning. That is,
the domain centerline is a streamline. However, transverse heterogeneity would likely
induce changes in the velocity field such that the streamline would not remain on
the centerline. For general transverse heterogeneity, the flow rate entering the two
domains will not be equal, in general. Thus, the flow weight and the volume weight
of the two subdomains are unequal in this case. Again, the magnitude of the areas
through which water enters the subdomains are equal.
Consider now a streamline originating at some coordinate (0, j/0) such that the vol
umetric flow rate is equal for both domains (curve C on Figure 2.2). This streamline
corresponds to the stream function
^[0,2/0]
m Ly] *[0,0]
2
(2.2)
42
effective conductivity is simply Ke Kg. Notice that the effective conductivity can
be significantly less than the arithmetic mean conductivity Ka = Kg exp[cr12nfc/2].
The definition of the effective conductivity is quite general, and a general effective
conductivity expression is denoted Ke = Kgc where c is the correction due to
the aforementioned processes. Under this definition, there is always some effective
conductivity value, regardless of nonstationarities in boundaries or flow domains, but
its magnitude may depend upon several factors. There is no general a priori effective
conductivity estimator. The estimator given here is limited to stationary head and
conductivity fields.
2.5.2 Eulerian Velocity Field
The relationship between the Eulerian conductivity field and its associated Eule
rian velocity field has been one of the more thoroughly examined aspects of subsurface
hydrology. For the example problem, closedform expressions are sought for the Eu
lerian, or spatial, mean velocity U, the Eulerian velocity covariance Cua[r], and its
point distribution, given that the underlying conductivity field is large, stationary,
lognormallydistributed, exponentially correlated, and characterized by parameters
P\nk, ofnk, and A^nJt, and subject to the conditions prescribed for the model aquifer
(see Section 2.1 and Figure 2.1).
From the development of the effective conductivity relationship in the preceding
section, an expression for the mean Eulerian velocity is
U = fr (2.20)
where c = 1 for a twodimensional isotropic aquifer. The c is retained to emphasize
that this expression is not the consistent firstorder approximation of the velocity
that would result from dropping the afnk and < fj > terms in Equation (2.18).
For the twodimensional isotropic aquifer, which is the model case, these two terms
happen to be equal in magnitude and opposite in sign, and the consistent firstorder
44
If these processes are stationary stochastic processes, they can be represented repre
sented by a FourierStieltjes integral (see e.g., Bakr et al. [1978])
poo
u'= exp[fM x]dZu/[M]
J OO
(2.22)
where i \fl, M is the wave number vector, and dZu' is the complex random
amplitude of the velocity perturbation. The FourierStieltjes representation of Equa
tion (2.21) is
dZu, = ^(dZf + dZj)
6
(2.23)
Compare this expression to that derived by Shapiro and Cvetkovic [1988] (their Equa
tion (A4))
dZi/u^JjidZf + dZj) (2.24)
The spectra that result from these complex random amplitudes are structurally the
same, differing by the square of the mean values by which they are multiplied (e.g., by
(KgJ/9)2 on the one hand and (9/(KgJ))2 on the other). In general, the correlation
structure of the inverse of a random function is not the same as that of the random
function itself. The firstorder series approximation of the inverse velocity is linear in
the velocity itself (e.g., 1/v ~ 1 v), and thus the velocity correlation characteristics
are recovered as an approximation of the inverse velocity correlation characteristics.
For the case of the exponentially correlated isotropic conductivity field, the longitudi
nal velocity covariance of a twodimensional field is given by (Shapiro and Cvetkovic
[1988])
Cu[s] = 3afnk(KgJ/6)2 (exp[s](s 2 + 3s 3 + 3s 4) + s 2/2 3s 4) (2.25)
71
one particle into each areaweighted streamtube at x = 0 and t 0 is analogous
to the a uniform resident mass injection. Equations (3.10) and (3.13) express the
mean particle displacement and particle displacement variance as a function of time.
Equation (3.15) expresses the mean particle travel time at different control planes.
Similarly, placing one particle into each flowweighted streamtube at x = 0 and
t = 0 is analogous to the a uniform injection in flux. Equation (3.12) expresses
the mean particle displacement as a function of time. Equations (3.14) and (3.18)
express the mean particle travel time and particle travel time variance at different
control planes.
In addition to these two injection modes, there are hybrid injection modes. For
example, consider if the introduction were somewhat random, or particle mass were
some deterministic function of the injection velocity, or some combination. In fact,
it is left to the imaginative reader to conceive of different injection mode possibilities
and the physical systems to which these correspond.
3.5 Evaluation of Expressions
In the previous chapter, a numerical experiment used to test certain hypothesis
about the relationship between Lagrangian and Eulerian descriptions of flow fields
was described. Those data are employed here. In the comparison of the derived
expressions to the data, the functions are based upon the simulation input parameters,
and all normalization is carried out using these same parameters. Therefore, the
expressions represent a priori estimates of the simulation results, based, of course,
on known input parameters.
3.5.1 Displacements
Figure 3.1 presents the model estimates of mean mass displacements for the two
injection modes. The flux injection preferentially selects, or mass weights, high veloc
ity areas of the flow domain, and the propagation of the center of mass is characterized
90
% / ^ln k
Figure 4.4: Mean reaction path observed values and estimators for different values of
correlation parameter /3 and different trajectory collections for the Kk = 1/2 set of
simulations. Top figure is /3 { 1, +1}. Middle figure is /3 {1/2, +1/2}. Bottom
figure is /? = 0 (no correlation). Trajectory collections are areaweighted Lagrangian
(AL) and flowweighted Lagrangian (AF).
73
G
10
8 
7*
.X A
AW/LT +
FW/LT x
Eq. 3.12
Eq. 3.10
.*,+
x A
,*v
,xV
.X 4
,v
.V
.*X .4
.**/
.X4+
X .4
.X ,4
.X ,4
X ,4
X ,+
jf
.x4
X?+
X+
x+
.X
:*.+
.x
X/
.X X
10
tU/X\nk
x.
c
10
8 
.X
,*x
V
AW/LT
FW/LT
Eq. 3.12
Eq. 3.10
X ..+
X ..+
.x .+
.'X A
,xv /
,x* /
,x* ++
'X .4
.X
* *
4
x 4
X 4
X 4
/X
** *
X
.'X 4
X .4
/X .4
.X ,4
* y*
4
X.4
**
x+
o *
10
tU/X\nk
Figure 3.1: Comparison of mean displacements for the two injection modes or stream
tube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics cor
respond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT)
statistics correspond to a uniform resident injection. The top figure contains the
For my parents.
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FILES
Dr. Kenneth Campbell deserves special recognition not only for his exemplary
service on my committee, but also for the example he has set as an engineer, a
scientist, and a professor.
I have appreciated Dr. Mike Annables wit and example through the years and
have enjoyed interacting with him in a wide variety of settings, whether in the class
room, on the football field, or elsewhere.
Dr. Rao has always challenged me with questions that require looking beyond a
simple formula and that evade a fillintheblank answer. Should I become scientist
of any repute, Dr. Rao would be the man to blame.
Dr. Kirk Hatfield is my source. He has been a source of wisdom, sympathy, advice,
and funding. I have greatly enjoyed working with Dr. Hatfield as both teaching and
research assistant.
Dr. Gia Destouni taught a few stochastic subsurface hydrology lectures in the
spring of 1994 that would forever change my thinking. She introduced the Lagrangian
approach to transport with which I have been obsessed ever since. Dr. Destouni
pursues fruitful collaboration vigorously, and this aggressive, yet collaborative, style
is one that I now seek to emulate.
Dr. Vladimir Cvetkovic has generously hosted three visits to the Kungl Tekniska
Hogskolan in Stockholm, Sweden. His considerable generosity flows from his intense
interest in stochastic Lagrangian transport.
Dr. Sten Berglund has been a friend and mentor through this arduous journey.
It is no exaggeration to say that without his help and encouragement this document
would not exist.
Dr. Wendy Graham deserves special recognition for her tireless faith and support.
She has allowed me a free reign to pursue whatever interests I chose. In hindsight, this
method coupled, with my own personality quirks and spectrum of interests, may not
IV
PAGE 1
SOLUTE TRANSPORT IN HETEROGENEOUS POROUS FORMATIONS By GEORGE GARY DEMMY, JR. A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1999
PAGE 2
For my parents.
PAGE 3
ACKNOWLEDGMENTS Dissertations have but one name in the author slot, but anyone who has written one will readily testify that it represents the work and support of myriad people. Mary Hall freed my mind from many of the administrative hassles students encounter. Dawn Mendoza appointed, unappointed, and reappointed me numerous times throughout my tenure, and made the business end of things run as painlessly as possible. Ralph Hoffmann kept me in keys and desks and office space, and helped me get things wired when they needed wiring. Wally Hunter, system administrator and Unix hacker, deserves as much praise as I can heap upon him for keeping the network up and running and the data contained with in it safe and available. The quality of an institution and its faculty is reflected in the students they attract. I have enjoyed interacting with the students of the Hydrologic Sciences Academic Cluster, and this interaction has enhanced my efforts. Liyong Li, Yan Zhang, and Xavier Foussereau shared the same computer lab trenches as I, and my work cannot be separated from their influence. Many rewarding research experiences have been associated with collaborations made with Jim Jawitz. Jim is a colleague whose outlook and expertise is somewhat different than my own, but it is precisely these differences that have made the collaborations so rewarding. I shall seek to continue working with Jim through out my career. Dr. Andrew I. James provided access to computer codes he developed in the course of his own research and technical assistance. iii
PAGE 4
Dr. Kenneth Campbell deserves special recognition not only for his exemplary service on my committee, but also for the example he has set as an engineer, a scientist, and a professor. I have appreciated Dr. Mike Annable's wit and example through the years and have enjoyed interacting with him in a wide variety of settings, whether in the classroom, on the football field, or elsewhere. Dr. Rao has always challenged me with questions that require looking beyond a simple formula and that evade a fillintheblank answer. Should I become scientist of any repute, Dr. Rao would be the man to blame. Dr. Kirk Hatfield is my source. He has been a source of wisdom, sympathy, advice, and funding. I have greatly enjoyed working with Dr. Hatfield as both teaching and research assistant. Dr. Gia Destouni taught a few stochastic subsurface hydrology lectures in the spring of 1994 that would forever change my thinking. She introduced the Lagrangian approach to transport with which I have been obsessed ever since. Dr. Destouni pursues fruitful collaboration vigorously, and this aggressive, yet collaborative, style is one that I now seek to emulate. Dr. Vladimir Cvetkovic has generously hosted three visits to the Kungl Tekniska Hogskolan in Stockholm, Sweden. His considerable generosity flows from his intense interest in stochastic Lagrangian transport. Dr. Sten Berglund has been a friend and mentor through this arduous journey. It is no exaggeration to say that without his help and encouragement this document would not exist. Dr. Wendy Graham deserves special recognition for her tireless faith and support. She has allowed me a free reign to pursue whatever interests I chose. In hindsight, this method coupled, with my own personality quirks and spectrum of interests, may not iv
PAGE 5
be the most efficient for producing degrees and papers, but it has produced several interesting adventures. George and Ellen Demmy, my parents, have provided love and all manner of support throughout my life, academic and otherwise. They both instilled in me a love of learning and an insatiable curiosity about all things that serve me well, and will remain among my best traits. Finally, I'd like to extend my thanks and love to Celine Bufkin, for her love, encouragement, support, and vision.
PAGE 6
TABLE OF CONTENTS page ACKNOWLEDGMENTS iii LIST OF TABLES ix LIST OF FIGURES x ABSTRACT xv CHAPTERS 1 INTRODUCTION 1 1.1 Primary Contribution 1 1.2 Historical Context 1 1.3 Research Goals 3 1.3.1 Lagrangian Velocity Mean and Covariance 3 1.3.2 Injection Mode Analysis 4 1.3.3 Heterogeneous Reaction Parameter 4 1.3.4 Application to "Reality" 4 2 LAGRANGIANEULERIAN FLOW FIELD RELATIONSHIP ... 5 2.1 Introduction 5 2.2 Flow Field Partitioning Strategies 7 2.2.1 Simple Subdivision 8 2.2.2 General Subdivision 10 2.3 Descriptions of Fields 12 2.4 Numerical Experiments 13 2.4.1 The Log Conductivity Field 14 2.4.2 Velocity Field Generation 18 2.4.3 Trajectory Generation 19 2.4.4 Monte Carlo Simulation 19 2.4.5 Sampling Strategy 20 2.4.6 Simulation Results 22 2.4.7 Log Conductivity Statistics 23 2.4.8 Velocity Statistics 29 2.4.9 Head and Head Gradient 40 2.5 Theory 40 2.5.1 Effective Conductivity 40 vi
PAGE 7
2.5.2 Eulerian Velocity Field 42 2.5.3 Streamtube Weights 48 2.5.4 Control Plane Oriented Velocity Moments 51 2.5.5 Control Time Oriented Velocity Moments 55 2.5.6 Log Conductivity Moments 59 2.6 Evaluation of Simulation and Theoretical Results 60 2.6.1 Mean Velocities in Space 60 2.6.2 Velocity Variances in Space 60 2.6.3 Mean Velocities in Time 61 2.6.4 Velocity Variances in Time 62 2.6.5 Log Conductivity Statistics 62 2.7 General Conclusions 63 3 NONREACTIVE SOLUTE TRANSPORT 65 3.1 Introduction 65 3.2 Injection Mode and Streamtube Selection 65 3.2.1 Uniform Resident Injection 65 3.2.2 Injection in Flux 67 3.3 Integrated Streamtube Properties 67 3.3.1 Displacements 68 3.3.2 Travel Times 69 3.4 Solute Transport 70 3.5 Evaluation of Expressions 71 3.5.1 Displacements 71 3.5.2 Travel Times 72 3.6 General Conclusions 75 4 REACTIVE SOLUTE TRANSPORT 79 4.1 Introduction 79 4.2 Temporal Moments 79 4.3 Conceptual Overview of Solute Transport 80 4.4 Simulation of Reactive Solute Transport 83 4.4.1 Reaction Parameter 84 4.4.2 Reaction Flow Path 85 4.4.3 Reaction Flow PathTravel Time Cross Moment ... 93 4.5 Theory of Reactive Solute Transport 93 4.5.1 Mean Reaction Flow Path 96 4.5.2 Reaction Flow Path Variance 97 4.6 Discussion 98 4.7 Spatial Moments 99 4.7.1 Injection Mode 105 4.7.2 Discussion 106 vii
PAGE 8
5 APPLICATION OF CONCEPTS 108 5.1 Introduction 108 5.2 Theory 108 5.2.1 Spatial Moments 109 5.2.2 Relative Degradation 110 5.3 Application: Simulation Ill 5.3.1 Results 113 5.3.2 Discussion 113 5.4 Application: Field Data 115 5.4.1 Results and Discussion: Center of Mass 116 5.4.2 Results and Discussion: Degradation 117 5.5 General Conclusions 117 6 GENERAL CONCLUSIONS 119 APPENDICES A EULERIAN VELOCITY DISTRIBUTION PROPERTIES 125 B VELOCITY LOG CONDUCTIVITY CORRELATION 127 C CONSISTENT FIRST ORDER APPROXIMATION 131 C.l Travel Time 131 C.2 Displacements 133 REFERENCES 134 BIOGRAPHICAL SKETCH 138 viii
PAGE 9
LIST OF TABLES Table page 2.1 Comparison of target log conductivity values and observed values along areaweighted Eulerian trajectories 26 2.2 Comparison of target log conductivity values for areaweighted Eulerian trajectories in space and observed areaweighted Lagrangian trajectories in time 29 2.3 Comparison of a priori Eulerian velocity variance to observed values. The decimal representation of the exact fractional values of the predicted variance are provided for convenience 35 2.4 Comparison of observed log hydraulic conductivity variances to estimators. Observed value is zeroslope line regressed through data. The decimal representation of the fractional value is included for convenience 63 5.1 The center of mass in meters of steady plumes estimated by traditional trapezoidal integration of spatial concentration data and by the mean v/k estimator Equation (5.7) 116 5.2 Degradation rates relative to naphthalene and absolute degradation rates in reciprocal days as estimated by Equation (5.9) and as reported by Maclntyre et al. [1993]. NR denotes results not reported in Maclntyre et al. [1993]. Naphthalene degradation reported to be 0.0064 d1 117 ix
PAGE 10
LIST OF FIGURES Figure page 2.1 Schematic representation of an aquifer 6 2.2 Simple subdivision of the flow field by one of four strategies: A) equalarea straight line, B) equalarea streamline, C) equalflow streamline, D) equalflow straight line 9 2.3 Four different partitioning strategies for the same flow field: A) area weight Eulerian trajectory, B) area weight Lagrangian trajectory, C) flow weight Lagrangian trajectory, D) flow weight Eulerian trajectory 11 2.4 Log conductivity distribution from a 41400 node realization generated by the turningbands method compared to the standard normal distribution. Target distribution parameters are //i n fc = 0 and of nJb = 1. 16 2.5 Estimated log conductivity longitudinal covariance estimated from a realization generated by the turningbands method compared to an exponential covariance function. Variance of log conductivity 0f nfc = 1 and correlation length Ai n fc = 1. Spacing for log conductivity values was Ai nfc /5 17 2.6 Mean log conductivity as function of displacement in mean flow direction for the four different trajectory collections and af nk Â— 1/2 (a) an
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2.10 Comparison of positiondependent mean velocity estimators to data for the four collections of trajectories for of nfc = 1/2 (a) and af nk = 1 (b). Velocities are normalized by U 2.11 Comparison of the positiondependent velocity variance estimators to data for the four collections of trajectories for af nk = 1/2 (a) and af nk = 1 (b). The variances are normalized by U 2 baf nk 2.12 Comparison of spatial Eulerian (AW/ET) and Lagrangian trajectory (FW/LT) velocity covariance data to covariance model. Data are taken from the cr, 2 nfc = 1 set of realizations. Covariances presented as dimensionless lag correlation functions 2.13 Comparison of timedependent mean velocity estimators to data for the four collections of trajectories for af nk = 1/2 (a) and a 2 nk = 1 (b). Velocities are normalized by U 2.14 Comparison of timedependent velocity variance estimators to data for the four collections of trajectories for of nfc = 1/2 (a) and a 2 nk = 1 (b). The variances are normalized by U 2 ba 2 nk 2.15 Comparison of temporal Lagrangian trajectory (AW/LT) velocity covariance data to covariance model. Data are taken from the of n k = 1 set of realizations. Covariances presented as dimensionless lag correlation functions 2.16 Normalized log initial velocities for twodimensional model aquifer compared to the standard normal distribution. The random variable was normalized by the sample mean and standard deviation to illustrate the lognormality 2.17 Comparison of velocity covariance expression Equation (2.25) to the approximation Equation (2.13) for o 2 nk = 1. Covariances are presented as dimensionless correlation functions 2.18 Components of the velocity covariance function 3.1 Comparison of mean displacements for the two injection modes or streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT) statistics correspond to a uniform resident injection. The top figure contains the af nk Â— 1/2 data and the bottom figure contains the of nfc = 1 data xi
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Comparison of displacement variances for the two injection modes or streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT) statistics correspond to a uniform resident injection. The top figure contains the af nk = 1/2 data and the bottom figure contains the of nA = 1 data. Variances are normalized by \ 2 nk
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4.4 Mean reaction path observed values and estimators for different values of correlation parameter fi and different trajectory collections for the af nk = 1/2 set of simulations. Top figure is fi = {1,+1}. Middle figure is fi = {1/2, +1/2}. Bottom figure is fi = 0 (no correlation). Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF) 4.5 Reaction path variance observed values and estimators for different values of correlation parameter fi and different trajectory collections for the of nfc = 1 set of simulations. Top figure is fi = {1,+1}. Bottom figure is fi = {1/2, +1/2}. Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF). 91 4.6 Reaction path variance values and estimators for different values of correlation parameter fi and different trajectory collections for the fnfc = V 2 set of simulations. Top figure is fi = {1,+1}. Bottom figure is fi = {1/2, +1/2}. Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF) 92 4.7 Correlation between the reaction flow path and travel time for different values of correlation parameter fi and different trajectory collections for the of nfc = 1 set of simulations. Top figure is fi = {1,+1}. Bottom figure is fi = {1/2, +1/2}. Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF). 94 4.8 Correlation between the reaction flow path and travel time for different values of correlation parameter fi and different trajectory collections for the af nk = 1/2 set of simulations. Top figure is fi = {1,+1}. Bottom figure is fi Â— {Â—1/2, +1/2}. Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF). 95 4.9 Assumed timedependent log conductivity covariance function along Lagrangian trajectories (Equation (4.29)) compared to simulation data for variances of log conductivity af nk Â— {1/2,1}. The Lagrangian velocity covariance function is Equation (2.61) 103 4.10 Hypothesized timedependent Eulerian log conductivity covariance functions for 0\ nk = {1/2,1} compared to simulation data and Equation (4.29) 104 4.11 Mean reaction flow path as a function of nonreactive travel time compared to estimator Equation (4.25) for correlation parameter values fi = {1,0,1} 104 4.12 Reaction flow path variance a function of nonreactive travel time for correlation parameter values fi = { Â— 1,0, 1} 105 xiii
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4.13 Mean reaction flow path as a function of nonreactive travel time compared to estimator Equation (4.34) for correlation parameter values P = {1/2,0,1/2} (top figure) and f3 = {1,0,1} (bottom figure).
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SOLUTE TRANSPORT IN HETEROGENEOUS POROUS FORMATIONS By George Gary Demmy, Jr. December 1999 Chair: Wendy Graham Major Department: Agricultural and Biological Engineering This work quantifies relationships between the spatial, or Eulerian, distribution of the properties of a chemically and physically heterogeneous porous medium and those as observed along the natural, or Lagrangian, trajectories that a fluid particle traces in a steady and irrotational flow. From these relationships, expressions that relate the transport of solutes through the porous medium along the natural trajectories to the aforementioned Eulerian distributions are developed. The effects of injection mode upon global measures of transport as reflected by the temporal moments of breakthrough curves and spatial moments of a solute plume are developed. The coupled effects of correlation of a linear equilibrium sorption to the underlying log hydraulic conductivity field and injection mode on the evolving temporal moments of mass breakthrough curve and the coupled effects of correlation of a firstorder decay coefficient and injection mode upon the spatial moments of a solute plume are examined. xv
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CHAPTER 1 INTRODUCTION 1.1 Primary Contribution The primary contribution of this work is documenting a consistent approach to the development of estimators of solute transport in the Lagrangian framework. This consistent approach is to identify the collection of Lagrangian trajectories associated with a stationary Eulerian velocity field for which the Lagrangian velocities are also stationary. This is a modest extension of work pioneered by myriad hydrologists. The benefit derived from this contribution is enhanced understanding of transport of solutes in heterogeneous velocity fields. 1.2 Historical Context A grant to study the impact of heterogeneous source morphology upon subsequent solute transport initiated this research. In seeking a suitable framework and "toolset" with which to attack this problem, there appeared what seemed to be fundamental inconsistencies in the literature. Given the peculiar tendencies of the author, the "larger issues" could not be addressed until these apparent inconsistencies were resolved. Gedeon Dagan developed a robust theory of solute transport using a stochasticLagrangian framework [Dagan, 1982a, b]. In fact, an entire "school" of stochastic subsurface hydrology sprung from his work. Allen Shapiro and Vladimir Cvetkovic wrote Stochastic analysis of solute arrival time in heterogeneous porous media in 1988 [Shapiro and Cvetkovic, 1988]. The authors made an explicit assumption often implicitly made in the Lagrangian transport literature, namely that a fluid parcel deviates little from its mean trajectory in weakly 1
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2 heterogeneous conductivity fields, and that the Lagrangian and Eulerian velocities are approximately equal. They consistently applied this concept, developing expressions for arrival time means and variances for nonreactive solutes in the "Standard Model Aquifer," a hypothetical aquifer characterized by a heterogeneous conductivity field with a prescribed correlation structure and subject to certain prescribed boundary conditions (see Section 2.1). These equations predicted that the mean arrival time for solute subject to a uniform resident injection is given by where angle brackets <> denote an ensemble average operator and V h is the harmonic mean Lagrangian velocity. From the small deviation assumption, the Lagrangian and Eulerian velocities are approximately the same, thus the estimated harmonic mean Lagrangian velocity is equal to the estimated harmonic mean Eulerian velocity. However, the mean arrival time for uniform resident injection of solute after travelling several integral scales is given by where U is the arithmetic mean Eulerian velocity. This expression is also the "consistent firstorder approximation" (CFOA) of the travel time. Widespread adoption of the CFOA through out the stochasticLagrangian transport literature probably stems Feynman's requirement: we must reproduce what we already know. The harmonic mean of a positive definite process is always less than the arithmetic mean, so Equation (1.1) systematically overpredicts travel times for large displacements. Disturbing was that the estimators derived by Shapiro and Cvetkovic [1988] using clear, consistent, intuitive and logical methods seemed to violate Feynman's rule, especially in light of the success enjoyed by the Dagan school. = x/V h (1.1) = x/U (1.2)
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3 This harmonic/arithmetic mean discrepancy was explicitly noted by Dagan et al. [1992], who relegated the Shapiro and Cvetkovic [1988] result to applicability in areas "close enough to the input zone" [Dagan et al, 1992 p. 1374]. Cvetkovic et al. [1996] developed a semianalytical expression that described the transition from the "near field" harmonic mean velocity to the "far field" expression based upon the these known "endpoints" and an estimate of the Lagrangian velocity integral scale. Their detailed analysis of the Lagrangian velocity field revealed that the Lagrangian velocity is nonstationary in displacements in space, and resulted in a nonlinear propagation of the mean arrival time with distance. This was strange. Why was the Lagrangian velocity field nonstationary when the Eulerian was stationary? It was commonly assumed that the stationarity of one implied the stationarity of the other. What had started as a simple preliminary literature review resulted in a quandry. The paths followed in pursuit of this quandry led to the "research goals" of the dissertation. 1.3 Research Goals There were four broad objectives specified for this work. All centered around the flow of water and the transport of solutes in aquifers characterized by heterogeneous hydraulic conductivity fields. 1.3.1 Lagrangian Velocity Mean and Covariance A primary goal of this research was to develop a "Lagrangian covariance function" for the "Standard Model Aquifer". This was extended slightly to the mean and covariance of the spacestationary velocities along equal flowweight streamlines and timestationary velocities along equal areaweight streamlines. Quantification of this covariance function greatly simplifies the development of equations, or estimators, that describe the movement of water and solutes in the heterogeneous velocity fields associated with the "Standard Model Aquifer."
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4 1.3.2 Injection Mode Analysis The stochasticLagrangian framework is convenient for the analysis of the effects of the method by which solute is introduced into the flow field, or injection mode, upon the subeqent transport. These effects of injection mode upon nonreactive solute transport are characterized by the mean and variance of a breakthrough curve, or mass arrival distribution, and by the mean and variance of mass displacement. A goal of this research was to develop estimators for these quantities. 1.3.3 Heterogeneous Reaction Parameter Set as a goal were estimators for the mean and variance of a heterogeneous "reaction parameter" integrated along trajectories of stationary velocty. These estimators were used to evaluate the coupled effects of injection mode and a heterogeneous linear equilbirium sorption process in a stochasticLagrangian framework. 1.3.4 Application to "Reality" A further goal was to develop a "traditional" estimator for the center of mass of a continuously injected solute plume subject to firstorder decay and to evaluate data from a natural attenuation experiment and numerical experiments. Estimator performance was evaluated using the results of the prevous developments.
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CHAPTER 2 LAGRANGIANEULERIAN FLOW FIELD RELATIONSHIP Develop an intuitive judgment and understanding for everything. Miyamoto Musashi 1 2.1 Introduction Consider the steady, welldeveloped flow of water in an aquifer meeting Bear's [1972] definition of a porous medium. In most natural settings, gentle gradients induce a flow wellquantified by Darcy's law. Knowledge of the water fluid properties, relative permeability of porous medium, and potential gradients in space and time would assure a reasonable estimate corresponding water fluxes. 2 One way to record this knowledge would be to create maps of these different properties in space at different times. This framework is commonly called the Eulerian framework, and is the most common approach to hydrologic problems. Another way to record this knowledge is to map properties along moving coordinates in time. This approach is commonly called the Lagrangian framework, and its application may without controversy be described as less widespread than Eulerian approaches. Dagan [1989] and Gelhar [1993] review development of flow of fluids and transport of contaminants in porous media in both of these frameworks. 1 Musashi [1982 p. 49] 2 The concepts associated with the following discussion are quite general, and are applicable to a much larger variety of environmental transport problems. The emphasis upon groundwater follows from an interest in simplicity. 5
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6 Consider now flow of water in a more specific model aquifer: the steady, irrotational, and Darcian flow of water of homogeneous fluid properties in a finite rectangular twodimensional aquifer of heterogeneous hydraulic conductivity (see Figure 2.1). Mobile water 9 completely saturates the homogeneous pore space. An orthogonal \\\\\\\\\\\\\\\\\\\\\\\W & no flow boundary random conductivity field Â• A\\\\\\\\\\\\\\\\\\\\\\\ c B no flow boundary average head gradient Figure 2.1: Schematic representation of an aquifer. h2 Cartesian coordinate system lies parallel to the aquifer boundaries, and x and y denote its directions. Parallel and impervious boundaries lie at y = 0 and y Â— L y Constant potential boundaries located at x = 0 and x = L x maintain a constant average potential gradient J across the domain, such that water flows into the aquifer at x Â— 0 and out of the aquifer at x = L x at a volumetric discharge rate Q (dimensions [L 3 T 1 ]). For an intuitive dimensional consistency, allow the aquifer some small and
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7 uniform thickness in the vertical direction, say h. The total area of the vertical plane at x = 0 through which water enters the aquifer is A. Define the spatiallyaveraged groundwater seepage velocity at this inlet plane (IP) as jj I A ^ a \ da L da (2.1) = Q_ OA For a large class of hydrologic applications, water quantity is the sole concern. For these problems, the "route" taken by the water is of little interest, as is, say the distribution of ages or residence times. Spatial and temporal distributions of hydraulic properties hold some interest, but only in how they relate to effective bulk properties. However, a large class of practical hydrologic problems exist for which the trajectories of water parcels across the flow domain is of great importance. Many of these problems involve the fate and transport of contaminants through the aquifer system. The fundamental problem is this: to relate an Eulerian flow field to its Lagrangian, or trajectorybased, equivalent. 2.2 Flow Field Partitioning Strategies A streamline is a hydrodynamic entity that is at all points tangent to the velocity vector. For twodimensional flow, a streamtube is an object defined by two streamlines. Stream surfaces, the intersection of which form streamlines, define threedimensional streamtubes. Infinite collections of streamlines exist that subdivide the flow field into collections of streamtubes. Again, the interest of clarity suggests a focus upon twodimensional flow. For a lucid discussion of streamlines, stream functions, and stream surfaces in two and threedimensional flows, see Bear [1972]. Impervious boundaries are streamlines, thus the entire model domain is a streamtube by definition. Viewed as a "black box," Eulerian and Lagrangian distinctions have little meaning. This black box approach is the purview of reactor engineering,
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8 an important and welldeveloped discipline, from which a great deal of insight and useful mathematics may be drawn. For an excellent review, especially with respect to concepts related to transfer functions, see Jury and Roth [1990]. For a large domain, the scale at which chemical reactions takes place is generally much smaller than that of the domain. That is, processes such as sorption, microbial and chemical degradation, and the like, are dependent upon chemical potentials across distances on the order of the pore scale to perhaps that of a "microDarcy" scale, defined as the smallest volume for which Darcy's law is generally applicable. However, the scale of interest tends to be much larger, say that of a well, or its capture zone or radius of influence, or perhaps the interface across which groundwater discharges into a surface water body, or, perhaps the Floridan aquifer. In order to consider these reactions in detail, it is necessary to resolve the larger domain into smaller subdomains. For modeling real systems, the scale and degree to which the domain is subdivided should be based upon the available data. 2.2.1 Simple Subdivision Consider the water flowing into the model aquifer across the inlet, or injection, plane (IP). Assume it is of interest to divide the domain into two subdomains. There are many ways to do this. One would be to divide the domain "down the middle," that is, at a line parallel to the x axis located y = L y /2 (curve A Figure 2.2). This division creates two subdomains of equal volume and gives an equal "volume weight" to each subdomain. The division also creates two subIPs of equal area and gives an equal "area weight" to each subdomain. For a perfectly homogeneous porous medium, the groundwater velocity vectors will be parallel to this line, and the discharges through each subdomain will be the same as well. Moreover, there will be no advective transfer of water across the boundary. However, the introduction of conductivity heterogeneity in the transverse direction will almost certainly lead to deviations of the streamlines, and to streamlines crossing the boundary. This implies that while
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0 Figure 2.2: Simple subdivision of the flow field by one of four strategies: A) equalarea straight line, B) equalarea streamline, C) equalflow streamline, D) equalflow straight line. the volume of water contained in each subdomain is constant, the discharge through a plane perpendicular to the partition is a function of plane position in x. Consider now a streamline originating at (0,L y /2) (curve B on Figure 2.2). For a homogeneous medium, this corresponds to an equal volume partitioning. That is, the domain centerline is a streamline. However, transverse heterogeneity would likely induce changes in the velocity field such that the streamline would not remain on the centerline. For general transverse heterogeneity, the flow rate entering the two domains will not be equal, in general. Thus, the flow weight and the volume weight of the two subdomains are unequal in this case. Again, the magnitude of the areas through which water enters the subdomains are equal. Consider now a streamline originating at some coordinate (0, y 0 ) such that the volumetric flow rate is equal for both domains (curve C on Figure 2.2). This streamline corresponds to the stream function mv]= mL > ] : mo] (2.2)
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10 This is an equal partitioning with respect to the flow field, or the stream function. This sort of partitioning arises quite naturally in fluid mechanics and in numerical modeling of heterogeneous flow systems using streamtube approaches (see, e.g., Theile et al. [1996]). The flow weight of each subdomain is equal, although the volume weights are different. A fourth scheme would be to extend a line from (0,y 0 ) to (L x ,y 0 ) (curve D Figure 2.2). In this case, the subdomains share only equal flow rates at x Â— 0. In subsequent discussions, the phrase Eulerian trajectory will refer to such a straight line trajectory, and the phrase Lagrangian trajectory will refer to trajectory of a streamline. The phrase area weight will refer to an equal area partitioning scheme at a definition plane, and the phrase flow weight will refer to an equal discharge partitioning scheme. At this point, it is important to note that no mention has been made of boundary conditions as they relate to transport equations. These partitioning schemes are simply ways of subdividing a flow field. An area weight Eulerian trajectory scheme corresponds to traditional spectral approaches that assume small deviations about a mean trajectory. The use of trajectory is quite general, and does not imply a "natural" path or streamline. 2.2.2 General Subdivision Consider dividing the flow field into N partitions according to the strategies described (see Figure 2.3). Theile et al. [1996] demonstrated that streamlines may represent streamtubes. Therefore, the remainder of this discussion will focus upon streamline properties, with the implication that these properties may be extended to streamtubes as well. A primary purpose of subdividing the domain is to represent the continuum of domain properties with a discrete set of data. A regular finitedifference grid may be thought of as a set of points at equal spacing in space along a collection of areaweighted Eulerian trajectories. An alternative method would be to create spacings based upon travel times along the trajectory, where travel time is defined as
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11 B Figure 2.3: Four different partitioning strategies for the same flow field: A) area weight Eulerian trajectory, B) area weight Lagrangian trajectory, C) flow weight Lagrangian trajectory, D) flow weight Eulerian trajectory.
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12 the path integral of the inverse velocity over the trajectory r[l] = / Jo ds (2.3) V[C] Â• a where s is a unit vector that is at all points the parallel to the trajectory. It is clear that this sort of Lagrangian description has little meaning in a static system, and is in this sense less general than Eulerian descriptions. Thus, Lagrangian description is a tool reserved for dynamic systems and is a supplement to Eulerian descriptions. It is worthy to note the distinction between a description, a general approach, and a trajectory, a specific entity used in a description. To summarize, a primary objective is to approximate a steady and irrotational flow field continuum with a discrete set of trajectories. The trajectories follow either a straight line (Eulerian trajectory) or a streamline (Lagrangian trajectory). The trajectories carry either an equal flowweight or an equal injection areaweight. Consider once again the model aquifer depicted in Figure 2.1. As mentioned in the introduction of this chapter, the aquifer has certain features in which a hydrologist might be interested, e.g., the intrinsic permeability or the porosity. These features may be thought of as a field in which parameters vary continuously. This is a fundamental concept of continuum mechanics and is discussed in a hydrologic context by Bear [1972], Dagan [1989], and Gelhar [1993]. The ultimate description of any such field would be a map of parameter values valid at any point, time, or scale. In lieu of this "ultimate description," stochastic hydrologists characterize spatial heterogeneity with conditional probability density functions that reproduce the dominant characteristics of the field and observed values at given locations. The entire subject of geostatistics is devoted to this end (see Journel and Huijbregts [1978]). An Eulerian field refers to areaweighted Euleriantrajectory oriented field. A Lagrangian field refers to a flowweighted Lagrangiantrajectory oriented field. 2.3 Descriptions of Fields
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13 It is instructive to illustrate these concepts with an example and to interpret the results of this illustration. The illustration of these concepts requires a heterogeneous flow field and collections of trajectories, both Eulerian and Lagrangian, that correspond to this flow field. The requirement that the flow field be based upon "reference fields" of known properties enhances the possibility of theoretical interpretation drawn from the large body of literature concerning flow and solute transport in idealized porous media. A further requirement that our reference fields bear some resemblance to what might be expected in nature enhances the possibility of useful applicability. A numerical simulation code modeled steady groundwater flow in a hypothetical heterogeneous twodimensional aquifer. A particle tracking code traced Eulerian and Lagrangian trajectories across this aquifer, recording data that included the local groundwater velocity and local hydraulic conductivity at intervals in space and time along the different trajectories. Statistics, specifically mean, variance, and covariance, summarize the results of these "measurements." The conceptual view of the aquifer is that given by Figure 2.1. To generalize the system, correlation length, or integral scale, of the log conductivity field Ai n fc serves as a characteristic length with which to normalize length scales. The characteristic time \\nk/U serves to normalize time scales, where U is the arithmetic mean velocity defined in Equation (2.1). The definition of an integral scale is the integral from zero to infinity of the correlation function R, or, 2.4 Numerical Experiments (2.4)
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14 The design criteria for the model aquifer are as follows In A: correlation length X\ n k = 1 effective conductivity K e = 1 average gradient J = 1 /50 mobile water content 6 = 1/5 From these criteria and Darcy's law, the areaaveraged velocity is U = K e J/6 = 1/10. Clearly, the parameter values are artificial and selected for convenience, but values for K e in md~ l and 6 fall in ranges typical for sand [Domenico and Schwartz, 1990]. Two 5000 replicate Monte Carlo simulations of aquifers with log conductivity variances of 1/2 and 1, respectively, comprised the suite of aquifer simulations. The following sections describe the experimental design and the results. 2.4.1 The Log Conductivity Field The foundation upon which to build the illustration is a heterogenous hydraulic conductivity field. A commonlyused stationary lognormally distributed and exponentially correlated isotropic model serves as the Eulerian hydraulic conductivity reference field (see e.g., Freeze [1975] or Gelhar and Axness [1983]). Working from a hypothesized conductivity field is preferable to generating a hypothetical velocity field with specified characteristics (e.g., Bellin et al. [1994]) because an objective of this work is to describe the underlying conductivity field along Lagrangian trajectories and relate its Lagrangian description to its Eulerian properties. For completeness, it is pertinent to briefly review description of a heterogeneous field by treating the process as being "random" Measured values of hydraulic conductivity of virtually any natural porous formation will vary from place to place. In the absence of measurement error, this variability in the measured values reflects the heterogeneity of myriad physical factors and processes that control the hydraulic conductivity. As mentioned, measurements of hydraulic conductivity in the field often follow a lognormal distribution. In these cases, it is useful to conceive of the
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15 conductivity field as a realization of a lognormal random process. The process is said to be stationary if the ensemble mean and variance do not vary in space. The process is nonstationary if some trend exists in the mean or variance. In practice, only single realization is available, and the "ensemble" properties are ultimately unknowable. However, if the field is large compared to the scale of variability, one may assume ergodicity, or, that the ensemble parameters may be estimated from a sample drawn from the single realization since there exists a the possibility of a large number of statisticallyindependent observation points. For a more detailed development of these concepts from the hydrologist's perspective, see Gelhar [1993]. For a more formal development of these concepts, see Papoulis [1991]. The model of hydraulic conductivity variability is a stationary, exponentiallycorrelated isotropic lognormal distribution. A lognormal random variable is one for which its natural logarithm is normally distributed. As such, three parameters summarize the statistical properties of the adopted conductivity model, namely the mean of log k /link, the variance of log k af nfc and the correlation length of log k \\ n kThe covariance model is given by where r is the separation between two points in the log conductivity field. Use of this simple covariance model requires stationarity of variance, and usually implies a stationarity of the mean as well. The moments of the "real space" conductivity field are given by where n is the order of the moment. In the absence of a perfectly described physical aquifer, the method of turningbands generated equallylikely realizations of a hypothetical conductivity field with C\ nk [r] = af^expfr/Ainfc] (2.5) <& n >= exp[n//i nJt + n 2 af nk /2] (2.6)
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16 prescribed ensemble statistics [Tompson et al, 1989]. Turningbands was capable of producing two and threedimensional fields with the desired statistical properties and was available as a portable and easily modified code. Figure 2.4 illustrates the lognormality of the a typical twodimensional log conductivity field realization. Figure 2.5 compares an estimate of the longitudinal covariance to the hypothesized model. In order to capture the features of the log conductivity spatial variability, the 3210 1 2 3 log conductivity Figure 2.4: Log conductivity distribution from a 41400 node realization generated by the turningbands method compared to the standard normal distribution. Target distribution parameters are fj,\ n k = 0 and af nk = 1. turningbands algorithm generated five log conductivity observations per correlation length. This level of discretization falls in the range of other similar sets of simulations. Cvetkovic et al. [1998] used a discretization of 4 nodes per Ai n fc for simulations of fields with of nfc as large as 1. Bellin et al. [1992] used a discretization of 8 nodes per Ai n fc for simulations of fields with a^ nk as large as 1.6.
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17 0.4 0.2 1 \ 1 1 estimated In k covariance + exponential covariance \ t i i 0 12 3 r/\\nk Figure 2.5: Estimated log conductivity longitudinal covariance estimated from a realization generated by the turningbands method compared to an exponential covariance function. Variance of log conductivity af nk Â— 1 and correlation length \\^ = 1. Spacing for log conductivity values was Ai nfc /5. Trajectories covered several characteristic lengths X\ n k and several characteristic times \\ n k/U to capture both nearfield and farfield behavior in space and Lagrangian time. The first requirement was easy to enforce by making the domain size equal to or greater than the required number of correlation lengths. The second requirement was somewhat more problematic, since the distance traveled in some number of \\ n k/U varies from trajectory to trajectory. Preliminary simulations indicated that a domain 40Ai n fc long was adequate to capture travel up to 10 characteristic time scales \\ n k/U in 2000 trajectory realizations generated in the most heterogeneous field considered (of njt Â— 1). Thus, the selected domain extent was 40Ai n jt. The conceptual model aquifer is a large, bounded, primarily twodimensional, steady flow field. As such, it should exhibit a variety of head, head gradient, and velocity nonstationarities due to the boundary conditions (see Figure 2.1). While
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18 researchers such as Rubin and Dagan [1988], [1989], and Osnes [1998] have investigated such nonstationarities, the additional complication of a rigorous consideration of these sorts of nonstationarities would unduly obfuscate the very simple concepts illustrated here. To minimize the effect of these nonstationarities, "solute transport" is constrained to take place some distance from the boundaries of the numerical domain. Following the example of Bellin et al. [1992], a 3Ai nfe "buffer" was established on both sides of the required 40 for a total longitudinal domain length of 46Ai nfc Preliminary simulations indicated that the maximum transverse displacement of a streamline in a af nk = 1 field was around one third the longitudinal displacement. For a streamline originating on the aquifer centerline parallel to the x axis, an aquifer width of 36Ai n fc was deemed adequate to contain large transverse displacements over a 40Ai nfc longitudinal displacement and provide a similar "buffer" to mitigate boundary effects. Combining the "physical" domain requirements with the discretization required to capture the log conductivity variability resulted in a numerical grid 230 by 180 for a twodimensional aquifer and a numerical domain with 41400 nodes. A threedimensional analysis under the same assumptions would have required a numerical domain of more than 7 million nodes. Simulations of this magnitude were untenable in light of the available computational resources. 2.4.2 Velocity Field Generation A mixed finiteelement scheme solved coupled velocity and pressure equations derived from Darcy's law and the continuity equation. The code took input parameters that matched the design criteria and the log conductivity field generated by turningbands. Andrew I. James wrote and tested the core of the code. James et al. [1997] used a derivative version of this code to generate flow fields for synthetic tracer tests. A mixed finiteelement solution to the flow field was specified, following the results and arguments of Mose et al. [1994]. The mixed finiteelement scheme is known
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19 to provide an accurate velocity field solution for groundwater flow in heterogeneous conductivity fields, one that is more accurate than other, more traditional, schemes [Mose et ai, 1994]. 2.4.3 Trajectory Generation An objective of this research was to record data, such as local velocity and and conductivity, along different collections of trajectories, both Eulerian and Lagrangian. These trajectories were traced using a "particletracking" scheme based upon a semianalytical method presented by Pollock [1988]. This method assumes that velocity varies linearly over an element, and that the velocity in some direction, say i, is independent of displacements in orthogonal directions. From these assumptions, it is a relatively simple exercise to determine the displacement of a "particle" in a linear velocity field for a given time, or the time required to make a certain displacement. A computer code recorded particle position, time, total path length along trajectory, velocity, local conductivity, and a value from another random field with the same statistical properties as, but uncorrelated to, the log conductivity field at several control planes equallyspaced along the mean flow direction and at several "control times" after injection along both trajectories. This uncorrelated random variable was recorded for reactive transport simulations, to be discussed in subseqent chapters. 2.4.4 Monte Carlo Simulation The possibility of theoretical interpretation of the results is enhanced by requiring nearergodic samples from the population. That is, the samples must reproduce the population statistics. To do this, the number of statistically independent replicates drawn from the velocity and conductivity fields must be quite large. Rather than attempt a massive singlerealization, the following method generates a large number of statistically independent realizations
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20 Â• generate a large, twodimensional conductivity field with the desired statistical properties Â• generate a flow field by imposing a constant mean gradient across the saturated conductivity field Â• trace the trajectories of two particles injected at a common "control location", one trajectory is a straight line or Eulerian trajectory and the other a streamline or Lagrangian trajectory, and record properties at equal lags in space and in time along trajectories that pass through "inner domains" that are relatively undisturbed by nonstationarities induced by the boundaries At this juncture, it is perhaps worth mentioning a minor "philosophical" point. These simulations do not represent a real aquifer, per se, but rather a hypothetical aquifer with known properties. The goal is understand the convolution of the physics of the groundwater flow and a known heterogeneity that has analogs in nature, and it is hoped that the reader finds some merit to this modest goal. The 41400 conductivity values needed for any single realization do not represent 41400 measurements taken from some real aquifer. The Monte Carlo aspect of this experiment reenforces this point. 2.4.5 Sampling Strategy Consider "dropping particles" into a flow field, with no particular regard to location. These particles move along streamlines without interaction with the porous medium. Any one location into which one is dropped is as equallylikely as any other. The trajectories associated with these particles carry an equal area weight, analogous to equalarea subdivision of an injection plane. Consider now, the same disposal of particles, but this time an area is assigned to each particle at the injection point such that an equal discharge is associated with each particle, and its associated trajectory. For a homogeneous mobile water content, this area is inversely proportional to the local velocity normal to the area. Thus, each particle may be thought to carry an equal
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21 flow weight. In this way, it is possible to "map" from one collection of trajectories to another. At a given control plane or time, the areaweighted mean and variance of some property over N realizations, say v was calculated using the familiar estimators 3 1 N i=i 1 N vark] = J^(Y1 v i ~ N ^ ( 2 8 ) i=i While these particular estimators seem obvious, it is important to realize that "dropping the particle" into the flow field assigns an equal area weight to each trajectory. For a single realization, this is equivalent to starting each particle with equidistant spacing. These estimators are based upon an assumption of equallylikely statisticallyindependent samples (see, e.g., Mood et al. [1974]). In order to draw even a few hundred independent samples from a single realization, the required flow domain would be enormous. This was a primary consideration for adopting Monte Carlo simulation using a minimal domain. The flowweighted mean property v was calculated via Ei=i<[o] 3 Press et al. [1988] have criticized the variance estimator in Equation (2.8) as being prone to accumulation of roundoff error for large data sets. The rather substantial size of the data sets are naturally a concern. Comparision of the variance calculated by a corrected twopass method presented by Press et al. [1988] and the "traditional" estimator yielded no difference between the two methods within the first five significant digits.
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22 where u;[0] is the initial velocity recorded for trajectory i. The variance by Ei=i w An alternative method for generating a collection of streamlines would be to identify the streamline corresponding to the "center" stream function expressed in Equation (2.2). In this case, the estimators of the equal flow statistics would be Equations (2.7) and (2.8). The equal area statistics would be given by = EÂ£Wm[o] (211) Et=i i/i[o] and varM = ^tM^lum EÂ£i.l/*[0] The "reference" collection of the trajectories selected uniformly in space was specified in the simulations for convenience. 2.4.6 Simulation Results The following sections present the results of two 5000 replicate Monte Carlo simulations. Detailed discussion of these results is deferred until after the development of some theoretical models with which to analyze these results. However, some heuristics are provided to familiarize the reader with some of the concepts developed in subsequent sections and chapters. A summary of the notation used with the following results provides clarity. AW denotes an areaweighted trajectory and FW denotes a flowweighted trajectory ET denotes an Eulerian trajectory and LT denotes a Lagrangian. A particular weighttrajectory combination is concatenated the with a /. Thus, an areaweighted Eulerian trajectory is denoted as AW/LT. For a given simulation, that is, for one set of 5000 replicates for a log conductivity field of some given a? nk the statistics of the log conductivity and velocity along the
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23 four different trajectories are calculated at intervals in time and in space. Thus, for any one property, say log conductivity, and any one statistic, say the mean, there will be two sets of results, namely that for af nk = 1/2 and that for af nk = 1. Associated with each set will be two subsets, namely statistics along the four trajectories recorded at space or time intervals. For convenience, the notation and the trajectories to which each notation corresponds as depicted in Figures 2.2 and 2.3 is summarized here: AW/ET areaweighted Eulerian trajectory (Figures 2.2 and 2.3 A) AW/LT areaweighted Lagrangian trajectory (Figures 2.2 and 2.3 B) FW/LT flowweighted Lagrangian trajectory (Figures 2.2 and 2.3 C) FW/ET flowweighted Eulerian trajectory (Figures 2.2 and 2.3 D) 2.4.7 Log Conductivity Statistics The log conductivity field serves as the "foundation" upon which the simulations are built, as well as the theory. The data for the areaweighted Eulerian trajectories correspond to those design criteria specified. The log conductivity mean and variance for AW/ET appear stationary in space (see Figures 2.6 and 2.7). That is, they appear to "bounce around" their "target" values with no apparent trend. Moreover, the statistics of FW/LT appear likewise stationary, but the mean is somewhat larger and the variance slightly smaller than AW/ET. The magnitude of the differences appear proportional to of nfc A comparison of the "target" values, e.g., the specified inputs into the turningbands algorithm, and the observed ensemble values along the trajectories is presented in Table 2.1. The "observed" entry in Table 2.1 represents the statistic average along the trajectory, as estimated by regressing a zeroslope line through the data using a generalized leastsquares method. There is close correspondence between the target and observed values, supporting an ergodic hypothesis for the log conductivity statistics. That is, sample statistics are an accurately estimation
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24 0.6 c =1 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 +__ + FW/LT + AW/LT x FW/ET AW/ET Eq. 2.72 reference + X ix, S + + + + ** 5* ** + x x X x I L. .I*.... JW.! A. ..S. "flJ.l X FW/LT AW/LT FW/ET AW/ET Eq. 2.72 reference 10 20 30 40 Figure 2.6: Mean log conductivity as function of displacement in mean flow direction for the four different trajectory collections and af nk = 1/2 (a) and af ak = 1 (b).
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25 FW/LT AW/LT FW/ET AW/ET reference Eq. 2.73 1.1 1.05 1 0.95 m 0.9 b 0.85 0.8 0.75 0.7 *x V< + + ** X* X $ + 10 20 30 FW/LT AW/LT FW/ET AW/ET reference Eq. 2.73 40 Figure 2.7: Log conductivity variance as function of displacement in mean flow direction for the four different trajectory collections and af nk = 1/2 (a) and a? ak = 1 (b). Variances are normalized by target variance of Eulerian log conductivity field of nk
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26 Table 2.1: Comparison of target log conductivity values and observed values along areaweighted Eulerian trajectories. of the population statistics. Again, this is important for a quantitative theoretical analysis of these results. The AW/LT and the FW/ET exhibit distinctly nonstationary behavior in the mean and less distinct behavior in variance as a function of displacement along the mean flow direction (see Figures 2.6 and 2.7). At the injection point, the AW/ET and AW/LT statistics are identical and the FW/ET and FW/LT statistics are identical, because these trajectories carry the same weight and the points in consideration are identical (see e.g., Figure 2.2). At very large displacements, one would expect the statistics of like trajectories to assume similar characteristics, regardless of weight. As the trajectory traverses several integral scales that characterize the heterogeneity in question, "information" about its starting location, is diminished. In the case of the FW/ET, the statistics "start" at the same place as FW/LT, and eventually "end" at the same place as the AW/ET. Similarly, the AW/LT statistics start where do the AW/ET and end where do the FW/LT. In fact, the data exhibit this behavior (see Figures 2.6 and 2.7). The log conductivity statistics in time show somewhat different behavior. Only the AW/LT appear stationary in time (see Figures 2.8 and 2.9). There is an important observation to be gleaned from these results. The AW/LT statistics, in addition to appearing stationary, also appear to have a similar value to the input AW/ET statistics (see Table 2.2). Recall, the AW/LT statistics are clearly nonstationary in space, and are clearly distinct from those of the AW/ET. This observation greatly parameter target observed 0 0.003 0 0.005 1/2 0.498 1 1.002
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27 FW/LT + AW/LT x FW/ET AW/ET o est AW/LT 0.4 0.2 0.2 0.4 m x x xx xx X x X xÂ„* ++ ++ + ++ ++V + ++ + + tt ++ + o xÂ„ 0! rx~ i^XXvx n *" x *xxxV "n xxx 13 Xxx xxxxx xx IK D *x B X xxxxx x xx x x 5! x x xx&x* *, FW/LT + AW/LT x FW/ET AW/ET est AW/LT 4 6 Ut/I 10 Figure 2.8: Mean log conductivity as function of travel time to control planes along mean flow direction for the four different trajectory collections and af nk = 1/2 (a) and a lk = 1 (b).
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28 1.1 1.05
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29 Table 2.2: Comparison of target log conductivity values for areaweighted Eulerian trajectories in space and observed areaweighted Lagrangian trajectories in time. parameter target observed MinfcK^ = 1/2) 0 0.000 HinkKk = 1) 0 0.017 1/2 0.497 1 1.000 simplifies relating the Eulerian and Lagrangian fields, as shall be shown. The "longtime" asymptotic mean log conductivity values for both Eulerian and Lagrangian trajectories are somewhat less than their "largedisplacement" counterparts, and the magnitude of the difference is proportional to o^ nk (compare Figures 2.6 and 2.8). This is consistent with the harmonic averaging implicit with a timeaveraged timedependent velocity and the arithmetic averaging implicit with a spaceaveraged spacedependent velocity. Recall, we map the quantities onto the trajectories by following velocities, whether along "natural" streamlines (Lagrangian trajectories) or not (Eulerian trajectories). A simple heuristic is as follows. Low conductivities correspond to low velocities. For an equaltime spacing (or equallylikely observations in time) more "measurements" will come from low velocities, since the particle tracing the trajectory spends more time going slow. For an equaldistance spacing more measurements will come from high velocities, since the particle "covers more ground" while going fast. This is perhaps more intuitive when discussed in terms of the velocity field itself. 2.4.8 Velocity Statistics The mean velocities along the different trajectories exhibit a similar behavior as do the mean log conductivities along the same trajectories (see Figure 2.10 and compare Figure 2.6). The AW/ET exhibits apparently stationary behavior in the mean and variance (see Figure 2.11) in accordance to established theory (e.g., Dagan [1989]). The AW/LT exhibits nonstationary behavior in the mean as observed by Cvetkovic et al. [1996]. Again, the AW/ET and FW/LT trajectories appear to be stationary
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30 1.45 1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05 1 0.95 1.45 1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05 1 0.95 ,x *** x ** x x x x FW/LT + AW/LT x FW/ET AW/ET Eq. 2.35 Eq. 2.43 Eq. 2.44 reference 10 20 30 40 ++ ++ +\ *J .+:. i.^.$a: x **, *x % x FW/LT + AW/LT x FW/ET AW/ET o Eq. 2.35 Eq. 2.43 Eq. 2.44 reference 10 20 30 40 Figure 2.10: Comparison of positiondependent mean velocity estimators to data for the four collections of trajectories for af nk = 1/2 (a) and af nk Â— 1 (b). Velocities are normalized by U.
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31 2 b 1.8 1.6 1.4 b 1.2 0.8 Â— *++*%++ ++X Â„ x x+ x *+ + + + x *+ X x + + X x x + *** 3 .:as.vV Â£3* !_^ FW/LT + AW/LT x FW/ET AW/ET Eq. 2.37 Â— Eq. 2.55 Eq. 2.57 Eq. 2.13 10 20 z/Alnfc 30 40 1.8 1.6 CM s b 1.4 1.2 .' x .' x Xx + x + x x x x *x +t 5 x x xx X *x + x x + X FW/LT + AW/LT x FW/ET AW/ET Eq. 2.37 Eq. 2.55 Eq. 2.57 Eq. 2.13 0.8 10 20 30 40 Figure 2.11: Comparison of the positiondependent velocity variance estimators to data for the four collections of trajectories for cr, 2 nfc = 1/2 (a) and af nk Â— 1 (b). The variances are normalized by U 2 baf nk
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32 in both the mean and variance. The velocity variances exhibit behavior to similar to that of the means for their corresponding trajectories and quite different than the behavior exhibited by the log conductivity variances (compare Figures 2.11 and 2.10 and compare Figures 2.11 and 2.7). There is a qualitative similarity between the nonstationary velocity means and the nonstationary velocity variances. A heuristic is that the velocity statistic "starts" at one value "in the nearfield" and transitions to an asymptotic value "in the farfield." The following sections demonstrate that these endpoints and the transition are predictable. The qualitative dissimilarity between the log conductivity variance and velocity variance is an artifact of examining the log conductivity values on the one hand and the "real space" velocity values on the other. Consider the F W/LT log conductivity mean and variance for the af nk = 1 simulation. The FW/LT mean log conductivity is larger than that of the AW/ET, but the variances are roughly the same value. In fact, regressing a zeroslope line through the data results in a "mean" FW/LT variance that is slightly less than that of the AW/ET. However, the "real space" conductivity mean and variance are both functions of both the mean and variance of the log conductivity process (recall Equation (2.6)). A cursory examination of the data in conjunction with Equation (2.6) reveal that the mean and variance of the "real space" conductivity along FW/LT are both higher than their AW/ET counterparts, and the magnitude of this difference is proportional to af nk The mean velocities in Figure 2.10 are normalized by the arithmetic mean velocity U predicted from the input parameters for the flow simulator. That the mean velocity along AW/ET closely follows 1 is a good indication that the effective conductivity expression is accurate. Notice that the FW/LT and farfield AW/LT values are higher than AW/ET, and the magnitude is proportional to the variance of the log conductivity (see Figure 2.10). This behavior is intuitively correct. Consider a
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33 particle injected into a flow field. Constraint of the trajectory to an Eulerian trajectory forces the trajectory through lowvelocity areas that a natural streamline, or Lagrangian trajectory, would have otherwise bypassed. Of course, some streamlines pass through even the lowest velocity areas, but these are few in an equalflow sense. Thus, equalarea weighting preferentially weights low velocity zones. That there are implications for sampling schemes in heterogeneous media, especially in context of multilevel samplers, should be intuitively obvious to even the most casual observer. A discussion of some of these implications is made in a subsequent section, after these concepts are developed further in a more quantitative and theoretical framework. It is appropriate to evaluate the adopted expression for the AW/ET longitudinal covariance function suggested by Dagan and Cvetkovic [1993], namely Cua[r) = al k b[e]U 2 exp [b[e)r / \ Xnk ] (2.13) where r is the separation between points along the ET, and b[e] is a function of the anisotropy ratio e. The anisotropy ratio itself is a function of the correlation lengths of the log conductivity in the different directions. The primarily consideration is that b may be derived from theoretical considerations and is not some ad hoc fitting parameter. The value of this function for a twodimensional isotropic aquifer is 3/8. Again, for further background on this function, see Dagan and Cvetkovic [1993]. In the limit r = 0, the covariance reduces to the point variance, which for the model case is a\ = 3[/ 2 of nfc /8. For the remainder of this discussion, a reference to b contains its dependence upon e implicitly The subscript ua may be decoded as follows: for a subscript tw, the t is the trajectory and property (u for Eulerian velocity, and v for Lagrangian velocity) and the w is the weight (a for area and / for flow). Thus, C ua is the covariance function for velocities along Eulerian trajectories. We present this covariance function and that covariance function estimated from the assumedstationary collection of AW/ETs in Figure 2.12 The perceptive observer will note
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34 AW/ET + FW/LT x Eq. 2.13 X \ + '. X \ + \ X '. + i\ + *. + 'x.. 0 5 10 15 r/Ai n fe Figure 2.12: Comparison of spatial Eulerian (AW/ET) and Lagrangian trajectory (FW/LT) velocity covariance data to covariance model. Data are taken from the of nfc = 1 set of realizations. Covariances presented as dimensionless lag correlation functions. that the analytical expression overestimates the covariance at short lags and underestimates the covariance at longer lags. This discrepancy and its implications are discussed in some detail in a subsequent section. The velocity variances in Figure 2.11 are normalized by the zerolag covariance predicted by placing the input simulation parameters into Equation 2.13. The AW/ET velocity variance appears wellpredicted by this expression, as evidenced by a relatively good fit to 1 for both values of o? nk in Figure 2.11. A regression of a zeroslope line through these data indicates that the a priori estimate under predicts the observed values by about five percent for the larger o^ nk (see Table 2.3) The apparent stationarity of the velocities along the FW/LTs suggest estimation of the covariance function for these velocities as for the AW/ET (see Figure 2.12). The correlation length of the FW/LT is very close to that of the AW/ET, although the actual value is probably slightly higher. Recall the definition of the correlation length as given by 0.8  06 'M n OJ 8 0.4 0.2
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35 Table 2.3: Comparison of a priori Eulerian velocity variance to observed values. The decimal representation of the exact fractional values of the predicted variance are provided for convenience. para meter predicted observed o 2 u {o'i k = 1/2) 3/1600=0.001875 0.001871 ^(^i 2 nfe = 1 ) 3/800=0.00375 0.00394 Equation (2.4). The correlation function at each considered lag along the estimated functions is systematically larger for F W/LT, even though the magnitude of the difference is small. Integrating these functions would result in a slightly larger value for FW/LT, and thus a larger correlation length. In the case there is no variability in the log conductivity, Eulerian and Lagrangian trajectories coincide. Moreover, an equalarea weight corresponds to an equal flow weight. Thus, the estimated correlation functions from "either set" would be identical. From this conclude that there is some dependence of the correlation length of the FW/LT trajectories upon af nk Cvetkovic et al. [1996] observed similar behavior in log velocity correlation functions estimated from AW/LTs with a stationarity assumption for the nonstationary AW/LT velocities. Thus, this is an slight extention of this work by identifying the stationary streamlinebased velocity field from which to work, and work from that field. Moreover, this supports their findings of an increasing correlation length with af nk Understanding the relationship between the variability of the log conductivity field and the correlation length of the associated velocity field is of fundamental importance to completely understanding the nature of flow in heterogeneous media. Sadly, these "secrets" remain uncovered and must be left this to future research efforts. The velocity means as a function of time exhibit qualitatively similar behavior as do the log conductivity means in time (see Figure 2.13 and compare Figure 2.8). Again, only the AW/LT exhibits stationary behavior in time for both the mean and variance (see Figure 2.14) as did the AW/LT log conductivity values. The mean and variance of the AW/LT in time are approximately those of the AW/ET in space
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36 1.4 1.3 1.2 1.1 0.9 0.8 0.7 1.4 1.3 1.2 1.1 FW/LT + AW/LT x FW/ET AW/ET o Eq. 2.63 Eq. 2.59 Eq. 2.67 Eq. 2.66 \ ^effect, ** 4 6 10 0.9 0.8 0.7 FW/LT AW/LT FW/ET AW/ET Eq. 2.63 Eq. 2.59 Eq. 2.67 Eq. 2.66 *x x xxxxx x 1 ; 3 D **w C =G :: 4 6 tu/x Xuk 8 10 Figure 2.13: Comparison of timedependent mean velocity estimators to data for the four collections of trajectories for af nk = 1/2 (a) and a^ nk = 1 (b). Velocities are normalized by U.
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37 2 1.8 1.6 Q 1.4 J Oi p 1 b 0.8 0.6 0.4 2 1.8 1.6 1? 1.4 c > 12 FW/LT + AW/LT x FW/ET AW/ET n Eq. 2.65 Eq. 2.61 Eq. 2.70 Eq. 2.69 x x x .. xx x++x + ^* ? x. x xx* b X X X Â£ x$*Â£* CM s b 4 6 tU/Kk Â— i 1 10 FW/LT AW/LT J. FW/ET AW/ET i Eq. 2.65 Â• \ Eq. 2.61 Eq. 2.70 Eq. 2.69 A, y V + V ++ \ \ ++ + Â• X + \ x xx v v + + + + + x, x x + + *v X ****x x X jx"SfÂ— *.; 5J 0.8 h b ''Â•Â•Â•....^B^ggn^ s 08 B 58 aB 8 B a SBfigg 8 8 B a b 0.4 4 6 tu/x lnk 10 Figure 2.14: Comparison of timedependent velocity variance estimators to data for the four collections of trajectories for af nk Â— 1/2 (a) and af nk = 1 (b). The variances are normalized by U 2 baf nk
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38 for both simulations. This result is important, because it implies the possibility of mapping from AW/ET to AW/LT. We estimated the covariance function for the stationary AW/LT velocities (see Figure 2.15). As an estimate of the covariance o V o u 4 6 tU/K k 10 Figure 2.15: Comparison of temporal Lagrangian trajectory (AW/LT) velocity covariance data to covariance model. Data are taken from the af nfc = 1 set of realizations. Covariances presented as dimensionless lag correlation functions. function, the substitution r = Ut was made in Equation 2.13 and the result compared to the estimated covariance function (see Figure 2.15). The closedform predictor behaves similarly to its spatial counterpart: earlytime correlation is underpredicted and longtime correlation is overpredicted (compare Figures 2.15 and 2.12). For completeness, one other aspect of the velocity field was examined, namely the distribution of initial velocities (see Figure 2.16). The logarithm of the initial velocity was rankordered and subject to the transformation In v Â— In v \nv' J \n v (2.14)
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39 3210123 log velocity Figure 2.16: Normalized log initial velocities for twodimensional model aquifer compared to the standard normal distribution. The random variable was normalized by the sample mean and standard deviation to illustrate the lognormality. where lni; is the sample mean and Ei n Â„ is the sample standard deviation. Working here with the logarithm, rather than the velocity itself, better demonstrates the apparent lognormality of the velocity fields. Recall, the logarithm of a lognormal random variable is itself normally distributed. Figure 2.16 indicates that the transformed log initial velocities are welldescribed by a standard normal distribution. The lognormality of the velocities seems logical since the conductivity field is itself lognormally distributed. However, the velocity is in general a product of random variables: the conductivity, the head gradient, and the mobile water content. The logarithm of the velocity is the sum of random variables. Sums of random variables tend toward normality by the central limit theorem (see Mood et al. [1974]), and thus, products of random variables tend towards lognormality. It would be interesting to test this
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40 hypothesis by examining the velocity characteristics of flow fields arising from different log conductivity distributions (e.g., fractal, uniform, white noise, etc.), and this is left to future work. 2.4.9 Head and Head Gradient Head and head gradient information along the trajectories was not recorded. This is a regrettable oversight, since much of the behavior of the flow water and the transport of solutes in heterogeneous media can be summarized as the interplay of head, head gradient and conductivity covariance functions. The literature concerning Eulerian head and head gradient fields is extensive and is summarized in large part by Gelhar [1993]. In fact, much is drawn from this body of literature for some of the theoretical developments in the following sections. That this information was not recorded detracts mainly from the completeness of this work, and further investigations are left to future works. 2.5 Theory 2.5.1 Effective Conductivity As with the simulations, the theoretical developments are founded on the bedrock of the log conductivity field. Section 2.4.7 demonstrates that the simulation results well match the "target" ensemble quantities. From this field is had directly a derived parameter, namely the a priori effective conductivity K e An expression for the effective conductivity, similar to that given by Gelhar and Axness [1983], is derived here to illustrate the mechanics of different techniques employed through out these derivations. Consider a onedimensional Darcy's law of the form (2.15) where q is the specific discharge, k is the hydraulic conductivity, and is the hydraulic head. Consider a large stationary conductivity field such that the log conductivity
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41 may be resolved into a mean, and a zeromean perturbation, e.g., In A; = fj,\ n k + /Â• Let the head gradient likewise be resolved into a mean and zeromean perturbation, e.g., d(p/dx = J(l + ;'). Inserting these expressions into Equation (2.15) yields q = exp[/i[ n k + f]J{l + j) (2.16) = K g Jexp[f](l+j) Expand the exponential term in this equation as a Taylor series about zero, truncating the series at secondorder, and expand the product of the perturbation terms q~K g J(l + f + f/2)(l+j) (2.17) = K g J(l + f + f 2 /2 + j+fj + f 2 j/2) Take the expected value of this expression, dropping terms higher than secondorder, resulting in an expression < q >= K g J(l + al k /2+ < fj >) (2.18) Consider this equation and the three terms in the parenthesis. The first is a "mean" term, the second is a "contribution" due to the variability of the conductivity field and the third is the log conductivity and head gradient crosscovariance. Gelhar [1993] evaluates this expression for isotropic log conductivity fields using spectral methods. The result, which is independent of the shape of the correlation function, is surprisingly simple [Gelhar, 1993] =olJd (2.19) where d is the dimension of the domain. This negative correlation between head gradient and log conductivity has a profound effect on the flow. The effective conductivity may be defined as that value when multiplied by the spatial mean gradient results in the observed specific discharge (flow rate divided by discharge area). The a priori effective conductivity is the effective conductivity estimated from Equations 2.18 and 2.19. With d = 2, we have for twodimensional conductivity fields, the a priori
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42 effective conductivity is simply K e = K g Notice that the effective conductivity can be significantly less than the arithmetic mean conductivity K a = K g exp[af nk /2]. The definition of the effective conductivity is quite general, and a general effective conductivity expression is denoted K e = K g c where c is the "correction" due to the aforementioned processes. Under this definition, there is always some effective conductivity value, regardless of nonstationarities in boundaries or flow domains, but its magnitude may depend upon several factors. There is no general a priori effective conductivity estimator. The estimator given here is limited to stationary head and conductivity fields. 2.5.2 Eulerian Velocity Field The relationship between the Eulerian conductivity field and its associated Eulerian velocity field has been one of the more thoroughly examined aspects of subsurface hydrology. For the example problem, closedform expressions are sought for the Eulerian, or spatial, mean velocity U, the Eulerian velocity covariance C ua [r], and its point distribution, given that the underlying conductivity field is large, stationary, lognormallydistributed, exponentially correlated, and characterized by parameters H\nk, cfnfc' and Af nfc and subject to the conditions prescribed for the model aquifer (see Section 2.1 and Figure 2.1). From the development of the effective conductivity relationship in the preceding section, an expression for the mean Eulerian velocity is K Q cJ U = ~T( 2 2 ) where c = 1 for a twodimensional isotropic aquifer. The c is retained to emphasize that this expression is not the "consistent firstorder approximation" of the velocity that would result from dropping the a? nk and < fj > terms in Equation (2.18). For the twodimensional isotropic aquifer, which is the model case, these two terms happen to be equal in magnitude and opposite in sign, and the consistent firstorder
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43 expression and effective expression are the same. This damping effect of the negative head gradient and log hydraulic conductivity is largely responsible for the robustness of these perturbation expansions. The truncated terms are not negligible in the sense their magnitude is small. For instance, for a 2 nk = 1, their magnitude is onehalf that of the mean. This brings us to an important philosophical point. A primary usefulness of these expansions is to identify the dominant "components" of the process. A purely "consistent" approach could lead to neglect of some important process. For instance, consider a strict, firstorder expansion of Equation (2.16). Equation (2.17) would contain no f 2 term, and Equation (2.18) would contain no af nk term. The resulting effective expression would seriously underestimate the specific discharge, since the contribution of the conductivity variability would "carry no weight" with respect to the other contributing processes. Analysis of the neglected higher order terms is necessary for a more complete understanding of the process. The general goal of the approach, however, is to identify the dominant subprocesses that contribute to the larger process, and the relative importance, or weight, of each subprocess. Expressions for the Eulerian velocity covariance for the model system have been given by various researchers, including Graham and McLaughlin [1989a] and Rubin [1990]. Shapiro and Cvetkovic [1988] develop an inverse velocity covariance function using a perturbation expansion of the inverse flow velocity and spectral methods. Consider once again the perturbed Darcy's law expression given by Equation (2.16). Dividing this by the homogeneous mobile water content, neglecting secondorder terms and higher, the meanremoved expression, or "velocity perturbation," is u' = ^(f + j) (2.21)
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44 If these processes are stationary stochastic processes, they can be represented represented by a FourierStieltjes integral (see e.g., Bakr et al. [1978]) poo u'= exp[iMx]dZ u [M] (2.22) J OO where i Â— \/T, M is the wave number vector, and dZ u < is the complex random amplitude of the velocity perturbation. The FourierStieltjes representation of Equation (2.21) is dZ u = ^(dZ f + dZj) (2.23) u Compare this expression to that derived by Shapiro and Cvetkovic [1988] (their Equation (A4)) dZ l/v = Yj{dZ f + dZ j ) (2.24) The spectra that result from these complex random amplitudes are structurally the same, differing by the square of the mean values by which they are multiplied (e.g., by (KgJ/9) 2 on the one hand and (6/(K g J)) 2 on the other). In general, the correlation structure of the inverse of a random function is not the same as that of the random function itself. The firstorder series approximation of the inverse velocity is linear in the velocity itself (e.g., l/v ~ 1 Â— v ), and thus the velocity correlation characteristics are recovered as an approximation of the inverse velocity correlation characteristics. For the case of the exponentially correlated isotropic conductivity field, the longitudinal velocity covariance of a twodimensional field is given by (Shapiro and Cvetkovic [1988]) C u [s] = 3al k (K g J/6) 2 (exp[s](s2 + 3*3 + 3s~ 4 ) + s" 2 /2 3s~ 4 ) (2.25)
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45 where s = x/\\ n kThe corresponding expression for a threedimensional field is C u [s] = Sal k (K g J/6) 2 (exp[s](s2 + 5s" 3 + 12s" 4 + 12s" 5 ) + s" 3 12s" 5 ) (2.26) Equation (2.25) provides a superior match to the data in comparison to Equation (2.13) (see Figure 2.17). However, the expression for the Eulerian covariance 1 0.8 c o u 0.4 _o > 0.2 0 0 2 4 6 8 10 r/Xlnk Figure 2.17: Comparison of velocity covariance expression Equation (2.25) to the approximation Equation (2.13) for of nfc Â— 1. Covariances are presented as dimensionless correlation functions. given in Equation (2.13) captures the dominant features of the velocity covariance structure, including the correlation length and the point velocity variance of Equation (2.25) in a simple and familiar form. As noted, however, Equation (2.13) exhibits some systematic differences from the observed behavior. The data show a rapid initial decay, followed by a longrange persistence (see Figure 2.17). This behavior is examined with an expression adapted
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46 from Dagan [1984] o r i 2 (r, M dR ]nk4> [r] d^r}\ Rua[r] = ^lÂ„fc ( R\nk[r\ ^ J l^'J where R\ nk is the log conductivity correlation function, R\ n k is the log conductivity and head crosscorrelation function, and 74, is the head variogram. From this expression, glean that the velocity covariance is a process comprised of three dominant subprocesses, namely R\ nk R\nk, and 7^. At small displacements, all three of these subcomponents exhibit rapid trend towards zero (see Figure 2.18). The R\ n k and 7^ o o 0.5 1 1 1 1 1 0 2 4 6 8 10 r/\\nk Figure 2.18: Components of the velocity covariance function. expressions exhibit effects that persist well beyond that of the log conductivity field. The longrange correlation effects of the head field in the mean direction of flow are cannot be precisely predicted with the simple exponential function Equation (2.13). In order to recover the proper correlation length, the approximation must systematically overpredict the velocity correlation at small separations. The "cost" of the
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47 approximation, however, will realize significant returns in the form of simplicity, as shown in subsequent sections. Previous works have focused upon the moments of the Eulerian velocity field that result from mean gradients imposed upon heterogeneous conductivity fields. These works shall be extended slightly with a hypothesis about a specific case that allows a variety of insights to be drawn. Hypothesis: The point velocity distribution of an Eulerian velocity field induced by an stationary mean gradient J across a lognormally distributed, exponentially correlated log hydraulic conductivity field characterized by parameters n\ n k,
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48 normal. This indicates that the moments n u = U and a\ Â— U 2 baf nk may require some further investigation. The adequacy of these estimates are discussed in subsequent sections. 2.5.3 Streamtube Weights The reference velocity field is now to be subdivided into streamtubes. Each streamtube may be considered to carry some "weight" in relation to other streamtubes for a given collection. Consider the partitioning of the flow field into N streamlines separated by an equal distance along the IP. This is an equalarea weighted partitioning scheme, and results in N subIPs of equal area, but dissimilar discharge. The discharge through subIP i is where Aa = A/N. The relative flowweight of this streamtube i is given by An equalflow partitioning scheme results in in N subIPs of equal discharge, but of dissimilar area. The discharge through subIP i is di = 9AaUi (2.30) Wfi = Ui/U (2.31) d = 9AciiVi (2.32) = Q/N Solving for the area yields (2.33) The relative areaweight of this streamtube i is Wai = U/Vi (2.34)
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49 The perceptive reader will notice a slight notational change between the two partitioning schemes. A lowercase u represents velocities drawn uniformly in space and a lowercase v represents velocities drawn uniformly in flow. The distinction is important. It is through these subIPs that water enters the streamtubes and a description of the subIP properties is essential to streamtube, and thus flow field, descriptions. If from a collection of N subIPs the random selection, or drawing, of one subIP is equally likely as any other, the weight assigned to any one observation from M samples would be 1/M. Assume that all of the subIPs are sampled. The average subIP discharge is the same for the equalarea and equalflow partitioning schemes, and equal to d = 8AU/N. However, the discharge variance is zero for equalflow and nonzero for equalarea, when the velocity field is heterogeneous. The weights expressed in Equations (2.34) and (2.31) allow the properties of one collection of streamtubes to be mapped to the other collection of streamtubes. These weights combined with the assumed values of the Eulerian velocity field permit derivation of properties of the initial velocities of the equalflow partitioning schemes. The expected value of v must equal the expected value flowweighted u. Thus, = V u =< Â—u> U = jjiA + l*u) (2.35) = U(l + bal k ) = ^(1 + WÂ„ fc )
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50 The variance of v is given by var[u] =< v 2 > 2 The expected value of v 2 must equal the expected value of flowweighted u 2 That is 2 ^2 =< Â—u l > = ^exp[3/ii nu + ^a 2 nu ] = exp[3(^, nu + a 2 nu /2) + Sa 2 n J (2.36) = U 2 (l + bcl k ) Thus, the variance of v is given by Â™[v} = V 2 bo 2 nk (2.37) It is worthy to note that these results do not follow from an assumption that v Â— u 2 /U. That is, the velocity along the Lagrangian trajectory is not simply the flowweighted velocity along an Eulerian trajectory. Hypothesis: The point velocity distribution along a flowweighted Lagrangian trajectory in the model stationary Eulerian velocity field is lognormal with approximate parameters Mint; = ^ and K g Jc e (2.38) a 2 nv = \n[l + ba 2 nk } (2.39) Moreover, the process is assumed secondorder stationary in displacements along the mean direction of flow, that is, at control planes, and is characterized by the covariance function C vf [r] = V 2 ba 2 nk exp[6r/A lnjfc ] (2.40)
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51 where r is the lag along the mean flow direction. Again this follows from the requirement that the distribution match the assumed "real space" v moments fi v = V = U(l + ba? nk ) and o 2 v = V 2 ba? nk The correlation structure of the FW/LT velocities is similar to that of AW/ET (see Figure 2.12). Employing these velocity distribution hypotheses, it is now possible to derive some relationships between the different flow field partitioning schemes. From observation, it is inferred that the velocities along areaweighted Euleriantrajectories and flowweighted Lagrangiantrajectories are secondorder stationary in displacements along the mean flow direction (recall Figures 2.10 and 2.11). Having adopted or postulated expressions for these first two moments, it is now possible to derive expressions for the nonstationary velocity means and variances of the areaweighted Lagrangian and the flowweighted Eulerian streamtubes. 2.5.4 Control Plane Oriented Velocity Moments Cvetkovic et al. [1996] postulated a semianalytical estimator for a nonstationary mean velocity along areaweighted Lagrangiantrajectory observed at control planes. We would like to work directly from the assumed stationary flowweighted Lagrangiantrajectory velocity field. The general expression for the expected value of the areaweighted Lagrangiantrajectory velocity is Resolve the stationary mean velocities into the mean of the flowweighted Lagrangiantrajectory field and zeromean perturbation. (2.41) v a [x] =< U V{l + v'[x})> (2.42) V(l+v'[0])
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52 Expand the denominator in a series to second order v a [x] =< U{1 v'[0] + v'[0] 2 ){l + v'[x}) > = U(l + & ^) (243) = U(l + ba 2 nk (lexp[br/X lnk })) In a similar fashion, an expression is derived for the nonstationary mean velocity along a flowweighted Euleriantrajectory. In this case, it is the stationary Euleriantrajectory velocity field that serves as a basis. The mean of the perturbation expansion is the Eulerian mean velocity. It is interesting to note that this result requires no series expansion. u f [x] =<^u[x}> = = 17(1 + Â£=M) = U(l + ba 2 nk exp[br/\ lnk ]) The nonstationary velocity variances along the respective crossweighted trajectories are derived in a similar fashion. The general expression for the areaweighted Lagrangiantrajectory velocity variance is var[u a [x]] =<^(v[x] v a [x]) 2 > (2.45) v[0\ This equation may be rewritten v&r[v a [x}] =< ^v[x] 2 > v a [x} 2 (2.46) v[0\ Resolve the first term of the preceding equation into perturbation expressions
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53 Expanding the denominator in a series, taking the expected value, and neglecting thirdorder and higher moments yields < ^Axf >= UV(1 + 2ba 2 nk {\ + exp] bx/X lak })) (2.48) v[0\ In order to improve the quality of the estimate, the righthand side (RHS) of the preceding equation is required to match the known extreme values of the lefthand (LHS) side. The LHS at X = 0 is <T7^[0] 2 > = U v[0] 1 J 1 J (2.49) = UV The RHS at X = 0 is RHS[0] = UV{\ + 2bal k (l exp[W/A lnfc ])) (2.50) = UV The statistical independence of the initial velocity and the velocities as x goes to infinity are exploited to calculate the LHS as x Â— > oo LHS[oo] =<^j> = ol + V 2 (251) = UV(l + bo? nk ) 2 The RHS as x > oo is RHS[oo] = UV(1 + 2baf nk (l exp[6oo/A lnfc ])) = 1^(1 + 2^) The RHS is deficient a term {ba 2 nk ) 2 Rewrite the RHS thus (2.52) RHS = UV(1 + (2ba 2 nk + (ba 2 nk ) 2 )(l exp[6r/A lnfc ])) (2.53)
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54 Substituting Equations (2.53) and (2.43) into Equation (2.45) yields vax[v a [x]] = UV(l + (2bal k + (kr 2 nfc ) 2 )(l exp[6r/A lnfc ])) (2.54) (U(l + ba? nk (lexp[br/\ lnk }))) 2 Further expansion of this expression offers no intuitively obvious simplifications or insights in its entirety. However, this expression may be approximated by dropping a few terms thus varK[x]] = V Vn* U 2 bal k {2bal k + (6a 2 nfc ) 2 ) exp[6r/A lnfc ] (2.55) Two unconventional techniques were employed in this derivation that shall now be defended. The first is requiring the series expansion to match the known endpoints. Simple expressions that condense the system behavior into an compact form that allows the relative contribution of the component processes to be easily grasped are sought. The series expansion and subsequent truncation is itself an approximation. The systematic error introduced by "missing" a known endpoint is distracting. In the case of the velocity covariance, the shape of the correlation function was "sacrificed" for simplicity, but the correlation length was retained. Had the simplified expression neither matched the shape of the data nor reproduced the correlation length, the expression would have been somewhat more questionable. In this case, the expression is required meet the observed endpoint values at the "cost" of the rigor of the series expansion and truncation. The second simplification is the ad hoc dropping of terms to simplify the expression. Again, simple expressions that allow the relative contribution of the different processes be easily grasped are sought. It would be difficult to hypothesize Equation (2.55) a priori as an empirical expression. A primary function of the series expansion is to indicate the dominant processes and give an estimate of their relative contributions. This hybrid approach is useful for the purposes here. These techniques
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55 are not a panacea, however, and the approximations have only been tested for the log conductivity variances tested in this work. The nonstationary flowweighted Euleriantrajectory velocity variance may be derived in a similar fashion. These results are presented directly. The complete expression is var [u, [*]] = t/ 2 (l + ba 2 Xnk + [2baf nk + 3 {ba 2 nk ) 2 + (ftojU) 3 ) (1 exp [6r/A lnfc ])) (L/(l + 6a 2 n ,exp[6r/A ln ,])) 2 (2.56) and an approximation of the above obtained by neglecting a few terms is var [u f [x\] = U 2 bo? nk (l + (2bo 2 nk + {bal k ) 2 ) exp [6r/A lnfc ]) (2.57) 2.5.5 Control Time Oriented Velocity Moments The displacementinvariant velocity covariances for u and v permitted derivation of expressions for the first two moments of the respective velocities for the four different trajectory collections. However, only the areaweighted Lagrangiantrajectory collection appears to be stationary in time. This is perhaps not intuitively obvious, as the velocity is clearly not stationary in displacements along the mean direction of flow (e.g., Equations (2.43) and (2.54)) because there is an implicit distribution of times associated with the establishment of a reference injection plane. After long travel distances and times, the areaweighted Lagrangiantrajectory point velocity statistics are equal to the flowweighted Lagrangiantrajectory point statistics. At the injection plane, the areaweighted Lagrangiantrajectory point velocity statistics are identically equal to the areaweighted Euleriantrajectory point velocity statistics. These relationships can be exploited to derive equations that provide a few insights about how properties change in Lagrangian time, and to derive an expression for a Lagrangiantime dependent velocity covariance function.
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56 The areaweighted velocity at the IP is U by definition. Since the areaweighted Lagrangiantrajectory velocity is stationary in time, its mean in time is U regardless of position. At long travel times, the point velocity is stationary in displacements as well. For constant means in space and time, the following relationship holds ?/>(*jr iif (2 58) This expression states that the harmonic mean velocity in space equals the arithmetic mean velocity in time. Returning now to the model aquifer, invoke the hypothesis of lognormality of the velocities along either trajectory. The harmonic mean of v[x] is given by V h = exp[/ii n Â„ (Tint,/ 2] = exp[/ii n Â„] Â• exp[<7i n Â„] 1/2 1 (2.59) = U Similarly, the harmonic mean of u[x] is given by Uh = 1^2 ( 2 6 ) Although derived by different means, this result is equal to the result given by Shapiro and Cvetkovic [1988]. These harmonic means are the asymptotic mean velocities of the two trajectories in time. The initial timedependent velocities are identical to the initial spacedependent velocities, an artifact of starting all of the trajectories at the injection plane at an implied time t = 0. This indicates that the velocity covariance in time along the areaweighted Lagrangian trajectory is C va [t = 0] = U 2 bo~\ nk As mentioned, at large times, the statistical properties of the Lagrangiantrajectories converge. Making a change of variables x = Vht Â— Ut'm Equation (2.40) and substituting the known C va [t = 0] for the value of C v f[r = 0] in yields an
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57 expression for the stationary areaweighted Lagrangiantrajectory velocity covariance C va [t] = U 2 bal k exp[bUt/X lnk ] (2.61) Similar expressions have been derived before by other researchers (see e.g., Dagan [1989]), but the conceptual development is quite different. Rather than seek to approximate the covariance function, an approach to quantify it directly from properties of the system is demonstrated. Moreover, the quantity U is perceived to be the harmonic mean of the Lagrangiantrajectory velocity, and that it is quantitatively equal to the areaaverage velocity of the flow field. This is an important point. The mean velocity and velocity variance along the flowweighted Lagrangiantrajectory is found in a manner similar to those in space, but there are some conceptual issues that should be addressed. It is the collection of areaweighted Lagrangiantrajectories that exhibits a stationary velocity. The mean velocity is Vft[t]=<^v[t}> (2.62) Expand the velocities into a mean and a zeromean perturbation. The mean here is the time average mean, or the harmonic mean of the space average. /<[<] = ^ = U (1 + bal k exp[bUt/\ lnk }) This expression requires no series expansion. For the control plane oriented velocity statistics, the mapping was from flowweighted to areaweighted trajectories. Equation 2.63 is identical in form to Equation 2.44 with the substitution r = Ut. Recognize
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58 the quantitative similarity and write velocity variance equations directly The complete expression is var [v f [t]} = U 2 (l + ba 2 nk + (26a, 2 n + 3 (ba 2 nk ) 2 + {ba 2 nk ) 3 ) (1 exp [bUt/ X lnk })) (U (l + ba 2 nk ex V [br/\ lnk \)) 2 (2.64) and the approximation is var[v/[t]] = U 2 baf nk (l + [2ba 2 nk + {ba 2 ak ) 2 ) exp[bUt/X lnk }) (2.65) Calculation of the mean velocities along the Eulerian trajectories is problematic in the absence of a stationary velocity covariance function. However, note that the areaweighted velocities start at U and that the flowweighted velocities start at V and both progress to U/(l + baf nk ). Recognizing the similarity shared by the different mean time estimators, write approximate expressions directly uat [t] = ,Ji 2 (1 + ba 2 nk exp [bUt/X {nk ]) (2.66) and W = T^TTI 1 + (2^infc + *lk) (^P [Mtt/Ain*])) (267) 1 + 0<7 ln fc Again, devoid of a stationary covariance function for the Eulerian velocity field in time, intuitive arguments lead us to anticipate an asymptotic u variance in time to be 2 t r2 l 2 a ut = U h ba \nk ( U \\ 2 ( 2 68 ) The initial velocity variances are known from the covariance functions in space. It is possible to anticipate forms of the nonstationary variance equations similar to those found previously, and include them for the sake of symmetry and completeness. The
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59 areaweighted Euleriantrajectory timedependent velocity variance approximation is varK[i]] = (y^J Wnfc (1 + {^< k + {b/c Expanding, retaining secondorder terms, and taking the expected value yields < f 1 />=4,(ll/^)/c (2.72) which, for twodimensional fields, is crf ak /2. Similarly, the variance is < ^o] /2> < !ffl />2= {
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60 Finally, note that the AW/LT log conductivity appears stationary in time as does the velocity. Moreover, it appears to have values equal in magnitude to its AW/ET counterpart, as does the velocity. Unlike the velocity, expressions for the trajectorybased crosscovariance of the velocity and log hydraulic conductivity that are necessary for developing the nonstationary log hydraulic conductivity moments were not developed. This is left for future work. 2.6 Evaluation of Simulation and Theoretical Results 2.6.1 Mean Velocities in Space Figure 2.10 presents the control plane oriented mean velocities for the four different collections of streamtubes. The mean velocities are normalized by U = K g J/6 = 1/10 and this value was used in the mean velocity expressions as well. That is, the parameter values are the input parameters to the simulations. Other input parameter values are 6 = 3/8 and of nfc = {1/2, 1} depending upon the set of simulations. The deviation of the mean of areaweighted Euleriantrajectory data from 1 are an indication of the quality of the effective conductivity expression. The areaweighted Euleriantrajectory mean velocity appears stationary, in accordance with established theory, and its value appears to be well predicted by the effective conductivity relationship. The fluxweighted Lagrangiantrajectory mean velocity appears stationary, supporting the stationarity hypothesis. Moreover, the value of the mean velocity appears to be well predicted for both variances of log conductivity. Additionally, the nonstationary mean velocity estimators appear to accurately reproduce the transition between the asymptotic velocities. 2.6.2 Velocity Variances in Space Figure 2.11 presents the control plane oriented velocity variances for the four different collections of streamtubes. The variances are normalized by the areaweighted
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61 Euleriantrajectory velocity variance estimated from the input parameters to the simulations. That is, by ^ = (V/)V hi (274) As with the mean velocity estimators, the variance estimators employ only the simulation input parameters. Once again, the areaweighted Euleriantrajectory variance exhibited its presumed behavior, and supports the well established fact that the Eulerian velocity field is fairly well understood in space. The hypothesized flowweighted Lagrangiantrajectory velocity variance is somewhat higher that the data for both af nk A possible reason is the negative correlation between log conductivity and head gradient is not weighted properly in the collection of trajectories skewed towards higher log conductivity values. The other two estimators also suffer from this bias in the asymptotic Lagrangiantrajectory velocity variance, although the transition between initial and final values for both seems to be well characterized. 2.6.3 Mean Velocities in Time Figure 2.13 presents the timedependent mean velocities for the four different collections of streamtubes. Again, the mean velocities are normalized by U as estimated from the simulation input parameters. Only the areaweighted Lagrangiantrajectory mean velocity exhibited a constant mean behavior in time. Its estimated harmonic mean is U, and the estimate seems good, although one might argue that there is a slight overpredictive bias. The estimated harmonic mean of Eulerian trajectories appear to have a stronger under predictive bias. The transition of the flowweighted Lagrangiantrajectory appears to be captured better than those of the Eulerian trajectories. It should be noted that the Lagrangian trajectory transition was based upon an expression for
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62 a stationary velocity field, whereas the Eulerian trajectories are ad hoc estimators, based upon a transition between known end points. 2.6.4 Velocity Variances in Time Figure 2.14 presents the timedependent velocity variances for the four different collections of streamtubes. The variances are normalized by the areaweighted Lagrangiantrajectory velocity variance estimated from the input parameters to the simulations. The areaweighted Lagrangianvelocity appears to be stationary in the variance. The data seem to lie slightly above the estimated value. The overprediction of the initial fluxweighted velocity variance has far less impact upon the timedependent velocity variances. This combined with what appears to be a good estimate of the longtime Euleriantrajectory velocity variance yields a better match to the simulations than one might expect from the means. However, there is a strong initial decline in the variance not captured by the simple exponential expressions. The areaweighted Lagrangiantrajectory covariance function exhibits a similar rapid decay. The early time behavior is likely dominated by the conductivity covariance. As the trajectory experiences a wide range of conductivity variability, this effect diminishes, and the longer range correlations related induced by the head field begin to dominate. 2.6.5 Log Conductivity Statistics The predictors of the stationary log conductivity statistics worked well for both the spacedependent and timedependent means although there seemed to be a slight overpredictive bias for the flowweighted trajectory (see Figures 2.6 and2.8). The AW/ET log conductivity variance was "well predicted" in that the turningbands method performed well, as discussed in Sections 2.4.1 and 2.4.7. The log conductivity variance of the FW/LT appeared to be lower than that of the AW/ET, as predicted by
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63 Equation (2.73), but the magnitude of under prediction was larger than that observed (see Table 2.4). Table 2.4: Comparison of observed log hydraulic conductivity variances to estimators. Observed value is zeroslope line regressed through data. The decimal representation of the fractional value is included for convenience. < k = 1/2 parameter predicted observed AW/ETai 2 nJt [x] 1/2 0497" FW/LTcrj^x] 7/16=0.4375 0.484 AW/LTajUM 1/2 0.498 parameter predicted observed M/ETal k [x] 1 L002~ FW/LTaf nk [x] 3/4 0.944 AW/LTagJi] 1 1.000 2.7 General Conclusions In an steady irrotational flow field, a fluid parcel trajectory follows a hydrodynamic streamline. The flow field properties as observed along streamlines can differ radically than those observed "uniformly in space." For the model system, a stationary Eulerian log conductivity field, this work identified the trajectory collections for which certain properties appear statistically stationary to second order. For displacements along the mean flow direction, point velocity and log conductivity appear stationary along streamlines separated by equal discharges. For displacements in travel time, these same properties appear stationary along streamlines that are selected uniformly in space. The stationarity of these properties facilitates the development of simple expressions for the statistics of displacement and travel times, as we shall show in following chapters. An immediate result of the preceding analysis is that the bulk of the flow field in the sense of total discharge travels at a velocity that is locally greater than the spatial
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64 average, and this disparity increases with heterogeneity of the underlying conductivity field. This might be described as "preferential flow." A practical implication for the transport of kinetically sorbing or interacting solutes is that the local velocity for the majority of the flow may occur in a regime significantly higher than that estimated as the flow field average. An understanding of the medium heterogeneity is essential if laboratory experiments are to be "scaled up." It is intuitively obvious, and has certainly been noted before, that the bulk of the flow passes through an increasingly small portion of the total "swept volume" as the medium heterogeneity increases. Again, the practical implications are likewise obvious. Consider some aquifer remediation effort that involves flushing a "contaminated zone" with some fluid. This might be as simple as the upgradient water in the case of pumpandtreat or as extravagant as an surfactantenhanced microemulsion. In heterogeneous flow systems, the bulk of the flushing fluid contacts a disproportionately small volume of the total swept volume. For contaminants that tend to reside in areas of high conductivity, the serves to enhance the efficacy of remediation efforts. Contaminants that tend to reside in areas of low permeability, conversely, would be more difficult to remove. While these conclusions may be drawn from "common sense," application of the concepts developed here may help to better understand these issues quantitatively.
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CHAPTER 3 NONREACTIVE SOLUTE TRANSPORT 3.1 Introduction The previous chapter saw the derivation of the relationship between the velocities that lie along Lagrangian trajectories and the parameters that describe a simple model Eulerian flow field. These Lagrangian relationships are employed to derive a few functions that characterize global measures of solute transport: namely the statistics of mass displacement in time and the statistics of mass arrival times at control planes. The frameworks are are that of Dagan [1982b] and that of Shapiro and Cvetkovic [1988], respectively. The recovery of a few wellknown results are anticipated, in addition to some that are new. However, the approach is novel in that the work is based upon trajectorybased statistical properties of the flow field. 3.2 Injection Mode and Streamtube Selection Demmy et al. [1999] noted the relationship between boundary conditions and sampling of streamlines. A clarification of this relationship and a generalization the results of that work are sought. Unlike the previous chapter, the Eulerian trajectories are not considered in the context of an actual transport trajectory, as they are of little practical importance. However, the results of Shapiro and Cvetkovic [1988] correspond to areaweighted Euleriantrajectories, as will be shown. Therefore, trajectory, Lagrangian trajectory, streamtube, and streamline are used interchangeably. 3.2.1 Uniform Resident Injection Consider the application of mass at a constant density c 0 across the entire face of the injection plane (IP) of the model aquifer (see Figure 2.1). Let the mass occupy 65
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66 some small Ax that is invariant with displacements along the IP. Following the terminology of Kreft and Zuber [1978], this injection mode is defined to be a uniform resident injection. Uniform refers to the homogeneous mass density in space and resident refers to the volume of resident fluid into which the solute is introduced. Subsequent references to a resident injection will refer to a uniform resident injection unless otherwise noted. Now, this injection mode has nothing to do with the flow field per se. That is, the mass "magically appears" and occupies a fixed volume at a fixed concentration without being influenced by local water fluxes. Consider the flow field to be divided into some number of streamtubes, say N. Moreover, let these streamtubes carry an equal areaweight at the IP. For a small depth of injection Ax, this injection mode has assigned an equal mass weight to each streamtube with respect to the total solute mass. The total mass of solute is M r = c Q 6hLAx (3.1) The mass weight for any one streamtube i is ^ = 1 (3.2) M N v This value is identically equal to the areaweight of the streamtube, and constant for all in the collection. Consider now dividing the streamtube into N streamtubes that all carry the same flow weight. The mass in some streamtube i is given by = b (3.3)
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67 3.2.2 Injection in Flux Consider now maintaining the IP at c 0 for some brief interval At. A mass Mj = c 0 8hLUAt (3.4) enters the domain and the different streamtube collections. This injection mode is a uniform injection in flux. Uniform, again, refers to the uniform mass density and "in flux" refers to the influent water that carries the solute into the flow domain. A mass rrii = c o 0h^ViAt (3.5) enters equalarea streamtube i. This corresponds to a mass weight of A mass rrii = coOhciiViAt (3.6) Q (37) = c 0 9h^At enters equalflow streamtube i. This corresponds to a mass weight of 1/N. 3.3 Integrated Streamtube Properties Let us return to the notion of the streamtubes as pure hydrodynamic entities. It is possible to discuss properties of these streamtubes with no mention of injection modes. Two properties hold some hydrodynamic interest. The first is the velocity integrated over time. For for some streamtube i this is Xi[t]= [ Vi[t]dt (3.8) Jo A velocity integrated over time is a "displacement" with a dimension of length. The second the inverse velocity integrated over some distance. For some streamtube i this
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68 is (3.9) This integral has a dimension of time. These integrals are given in terms of the x velocity component as a function of longitudinal displacement or time. It is suggested that future works examine the details of a more general approach, e.g., displacements along the trajectory. 3.3.1 Displacements Consider a model flow field such as the one given in the previous chapter. Quantification of the displacement and travel time statistics for areaweighted and flowweighted trajectories is sought. Previous results indicate that the velocity for a collection of areaweighted streamtubes is stationary in time, characterized by mean U and approximate covariance U 2 baf nk exp[Â— bUt/X\ n k\The expected value of Equation (3.8) is Although the velocity of the flowweighted streamtube collection is not stationary in time, the results of the previous chapter are employed to map the properties of the areaweighted collection to the flowweighted collection. Recognize that this equation may be rearranged to be the integral of Equation (2.63) in time. Substitute the RHS of Equation (2.63) into the preceding equation and =< / v[t]dt> (3.10) = Ut (3.11)
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69 integrate to yield = Ut + X ]nk al k {lexp[bUt/X lnk }) (3.12) The displacement variance for the areaweighted streamtube collection is given by var[x a [t]] = 2 3.3.2 Travel Times Expressions for the travel time statistics are sought, in analogy to those sought for displacement. The reference collection of velocities from which these expressions will be derived is the stationary velocities of the equal flow streamtubes, or the equalflow weight Lagrangian trajectories. The mean travel time for these streamtubes is given (3.13) 2 f \t s)C lat [s}ds (Ut) 2 by (3.14) x U The mean travel time for the areaweighted streamtubes is (3.15) = (x + X lnk a 2 nk {l exp[bx/X lnk }))
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70 The arrival time variance for the flowweighted collection of streamtubes is Jo v[x'\ Jo v[x"\ Approximate the inverse velocity covariance contained in the first term of the RHS of Equation 3.16 with the a series expansion of the perturbation equation 1 1 > = u[x']u[a;"] V 2 {1 + v'[x']){l + v'[x"]) =< 4(1 v'[x'] + v'[xf ...)(!v'[x"\ + v'lx'f ...)> r 1 V2 (3.17) = ^(l + 2a 2 v + C vf [x',x"}+UOT) 1 (, C vf [x',x"Y U2 y L + y 2 where HOT is the collection of higher order terms. Inserting this expression into Equation 3.16 yields an expression for the travel time variance for fluxweighted streamlines var[t/N] = ~f (exp[WA 2 nfc ] 1) j (318) 3.4 Solute Transport Expressions for the statistics of the integrated trajectory properties of displacement and travel time have been derived in the absence of an explicit reference to solute transport to demonstrate that these properties exist simultaneously in the same flow field. However, they are intimately related to solute transport. Consider the transport of an infinitesimal particle whose movement is constrained to follow a streamline. Moreover, at any time or position on the streamline, the particle velocity is identically equal to that of the underlying velocity field. Its displacement from x = 0 in t = T is given by Equation (3.8) and the time required to travel from x = 0 to x = X is given by Equation (3.9). For the same flow field, two collections of streamtubes have been discussed, namely areaweighted and flowweighted. Placing
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71 one particle into each areaweighted streamtube at x = 0 and t = 0 is analogous to the a uniform resident mass injection. Equations (3.10) and (3.13) express the mean particle displacement and particle displacement variance as a function of time. Equation (3.15) expresses the mean particle travel time at different control planes. Similarly, placing one particle into each flowweighted streamtube at x = 0 and t = 0 is analogous to the a uniform injection in flux. Equation (3.12) expresses the mean particle displacement as a function of time. Equations (3.14) and (3.18) express the mean particle travel time and particle travel time variance at different control planes. In addition to these two injection modes, there are hybrid injection modes. For example, consider if the introduction were somewhat random, or particle mass were some deterministic function of the injection velocity, or some combination. In fact, it is left to the imaginative reader to conceive of different injection mode possibilities and the physical systems to which these correspond. 3.5 Evaluation of Expressions In the previous chapter, a numerical experiment used to test certain hypothesis about the relationship between Lagrangian and Eulerian descriptions of flow fields was described. Those data are employed here. In the comparison of the derived expressions to the data, the functions are based upon the simulation input parameters, and all normalization is carried out using these same parameters. Therefore, the expressions represent a priori estimates of the simulation results, based, of course, on known input parameters. 3.5.1 Displacements Figure 3.1 presents the model estimates of mean mass displacements for the two injection modes. The flux injection preferentially selects, or mass weights, high velocity areas of the flow domain, and the propagation of the center of mass is characterized
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72 by a higher velocity. The timevarying mean velocity expression for the flowweighted Lagrangiantrajectories (Equation (2.63)) is, in fact, an expression for the mean velocity of a plume resulting from a uniform flux injection. The longtime asymptotic slope of the fluxweighted mean displacement is U, as is easily verified by taking limit of Equation (2.63) as t > oo. The mean displacement functions predict the observed behavior, and also provide an excellent fit to the data. The displacement variance data are interesting (see Figure 3.2). Injection mode seems to have little effect upon the displacement variance in time. This is not to say that injection mode has little effect upon the spatial distribution of the plume in time. At a given time t, a plume injected in flux will have traveled a greater distance, on average. After traveling the same distance, the displacement variance of the fluxinjected plume will be less than that of the resident injection. Thus, the "spreading as function of mean displacement" is less in the case of the fluxinjected solute. The areaweighted trajectory displacement variance overpredicts the observed displacement variances, and the longtime slopes appear to have different values. After 10 characteristic times, the exponential term in Equation (3.13) retains a little more that two percent of its original value. This difference does not seem large enough to account for the observed discrepancy. The variance of the stationary areaweighted Lagrangiantrajectory velocity field in time appears to be well predicted. Thus, the problem may lie with the rather poor description of the shorttime and longtime correlation exhibited by the observed velocity covariance given by the assumed model. 3.5.2 Travel Times Figure 3.3 shows good agreement between the estimated and observed mean travel times. The injection influx shows earlier arrival than that of the uniform resident injection. Uniform resident injection weights all areas of the IP equally. Large areas that contribute little to the overall flow receive and equal mass weighting as those
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73 10 AW/LT FW/LT Eq. 3.12 Eq. 3.10 Â•X* ..+ .Â•X .+ .x ..+ .X .+ .X 4* X .+ Â•x + .* .+Â• .X .Â•x .X ,+ 4 6 tu/x lnk 10 Figure 3.1: Comparison of mean displacements for the two injection modes or streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT) statistics correspond to a uniform resident injection. The top figure contains the a ink = 1/2 data and the bottom figure contains the af nk = 1 data.
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74 15 AW/LT + FW/LT x Eq. 3.13 6 b 10 0* 4 6 tU/\ lnk 10 15 AW/LT + FW/LT x Eq. 3.13 10 +x +x +x +x +x +x +x +x +x t CS H b / ** 4 6 tu/x lnk 10 Figure 3.2: Comparison of displacement variances for the two injection modes or streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT) statistics correspond to a uniform resident injection. The top figure contains the fnfc = V2 data and the bottom figure contains the of nA = 1 data. Variances are normalized by Xi k af nk
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75 areas that contribute more significantly to the overall flow. However, as the solute moves through the domain and experiences the full range of velocity variability along its trajectory, the "slow start" is damped out and the instantaneous average point velocity for all for the particles is V and the harmonic mean V h = U. The travel time variances appear to be better matched than the displacement variances (see 3.4). This is reasonable in light of the better apparent match of the assumed velocity covariance function in space than that in time. The uniform injection appears to exhibit less nonlinear behavior than the injection in flux. The uniform injection variance is reasonably expected to be larger than that of the flux injection, due to the skewed massweighting of lowvelocity areas by the uniform resident injection. 3.6 General Conclusions It is clear that the injection mode effects are most prevalent in the "near field" where the initial velocities and travel times or displacements are well correlated. After the plume travels several correlation scales, these injection mode effects diminish. The near field and the neartofar transitional scales are quite interesting, as these intermediate scales are on the order of sites associated with certain agricultural settings such as some feed lots and waste holding facilities, urban land uses such as gas stations and dry cleaners, and their associated remediation efforts, and a plethora of "field scale" experiments. These scales are examined in the perspective of the "error" in travel time estimate associated with assuming a resident injection when the "real" injection is in flux e[x] = Xi nk af nk (l exp[bx/\ lnk ])/x (3.19) given by subtracting Equation (3.15) by Equation (3.14) and dividing the difference by Equation (3.14) (see Figure 3.5) The "near field" might be characterized as the first five to ten correlation lengths, where the injection mode effects are quite prevalent, even for modest log conductivity variability. These effects diminish less rapidly over
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76 40 30 J? AW/LT + FW/LT x Eq. 3.15 Eq. 3.14 c 20 10 10 40 20 30 40 30 20 10 AW/LT FW/LT Eq. 3.15 Eq. 3.14 J210 20 30 40 Figure 3.3: Comparison of mean arrival times for the two injection modes or streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT) statistics correspond to a uniform resident injection. The top figure contains the a \nk = 1/2 data and the bottom figure contains the o^ nk = 1 data.
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77 80 60 Ji 6 Â£ 40 AW/LT + FW/LT x Eq. 3.18 tx .+>< Â•fx .fx ,4x + x ...fx ., + x b 20 + x 10 80 20 Z/Alnfc 30 40 + x ..Â• + X .+ xy 60 a 6 li 5 40 Hi b AW/LT FW/LT x Eq. 3.18 + x Â• + x .Â• + x/ + x.^ *' + x.+ xy + x20 + Â„x 5** 10 20 X/Xlnk 30 40 Figure 3.4: Comparison of arrival time variances for the two injection modes or streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT) statistics correspond to a uniform resident injection. The top figure contains the a ink = 1/2 data and the bottom figure contains the
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78 0 i 1 1 J 0 10 20 30 40 Figure 3.5: Travel time estimate error as function of displacement associated with assuming a uniform resident injection when injection is in flux for 0f nfc = {1/2, 1}. a "transition" region between 10 and 30 correlation lengths. Beyond 30 correlation lengths, the difference between the two modes is less than a few percent. Much of the Lagrangian transport literature is predicated upon a uniform resident injection, but the socalled consistent firstorder travel time estimate that is regularly employed is quantitatively that for a flux injection. For large displacements, this error is small, and the simplification afforded by this approximation is considerable (compare Equation (3.14) and Equation (3.15)). For many practical applications, an error of ten, or even more, percent might be considered negligible. Therefore, it is not suggested that this approximation be abandoned, but rather phrasing such as "inconsistent" firstorder approximation be applied in future works, since this Lagrangian expression is inconsistently based upon the Eulerian mean velocity.
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CHAPTER 4 REACTIVE SOLUTE TRANSPORT 4.1 Introduction The concepts developed in the previous chapters are employed to analyze reactive solute transport. The focus is upon global measures of transport, rather than predictions of local quantities. Particular processes of interest are injection mode effects, the effects of correlation of the local "reaction parameter" to the log conductivity field, and the interplay of these injection mode and correlation effects. A suite of numerical experiments illustrates these effects. Additionally, expressions are developed for temporal moments of mass breakthrough curves for a solute subject to linear equilibrium sorption for the different injection modes that reflect the observed behavior in relatively simple closedforms. Spatial moments of sorbing solutes are not directly treated in this work. The experimental design was targeted directly towards a temporal moment analysis, and this design decision has some implications for spatial moment analysis. These decision and implications are discussed, in hopes that future works might be guided by this analysis. Some of the properties of the travel time dependent reaction parameter are discussed, as are some of the implications for solute transport. 4.2 Temporal Moments The concepts developed in the preceding chapters are applied to examine the effect of injection mode upon the temporal moments of the breakthrough curve of a solute subject to linear equilibrium sorption. A uniform mean gradient induces a steady irrotational flow in a uniformly saturated porous medium characterized by an exponentially correlated lognormal conductivity distribution. This flow field is divided 79
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80 into collections of noninteracting streamtubes. Dispersion at the streamtube scale is considered to be negligible in its effect on the aggregatescale temporal moments (e.g., Dagan [1989]). The sorption process is related to a generic reaction parameter that is heterogeneous and may be correlated to the log conductivity field. The previous chapters demonstrated the relationship between streamtube selection strategies (equalarea or equalflow) and injection mode (uniform resident and influx). The reference collection is equalflow streamtubes, since the associated velocity and log conductivity statistics are stationary in displacements along the mean flow direction. The primary objective of this chapter is to examine the effect of injection mode upon the temporal moments of the mass breakthrough curve of a solute subject to linear equilibrium sorption in a heterogeneous flow and sorption field. A brief review the conceptual model of solute transport is given. General expressions are presented for the first two temporal moments of the breakthrough curves associated with the uniform injection of a solute both in the influent fluid flux and in the resident fluid. The results of the reactive solute transport simulations are presented, followed by the development of some analytical expressions that help explain the observed behavior. 4.3 Conceptual Overview of Solute Transport Consider a general multidimensional flow field that is resolved into a collection of noninteracting streamtubes. Transport along an individual streamtube is characterized with a transfer function 7[Â£; x] (see Jury and Roth [1990] and Cvetkovic et al. [1998]). For a conservative and nonreactive solute, the transfer function is simply j[t;x] = S[tT[x}] (4.1) where 8 is the Dirac delta function and r is the travel time. The physical interpretation of this transfer function is this: a particle located at x = 0 "arrives" at x Â— X at time t = r[X}. We conceive of this travel time as not only the time required to
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81 travel to some point x, but as an intrinsic integrated streamtube property, namely the inverse velocity integrated along the trajectory (e.g., Equation (3.9)). Recall reaction flow path concept of Cvetkovic et al. [1998]. The reaction flow path, parameterized in space, is defined as where P[x] is some reaction parameter of interest. For linear equilibrium sorption, P is a linear partitioning coefficient. The reaction flow path is also an integrated streamtube property. As such, any reference to /x properties in general are broadly applicable to any distributed reaction parameter, perhaps a firstorder decay coefficient or a Langmuir sorption parameter. However, the focus here is upon linear equilibrium sorption to promote a ready and intuitive grasp of the pertinent concepts. For this case, /i may be thought of as the time that the solute is sorbed. Thus the arrival time of a particle undergoing a linear equilibrium sorption process is the sum of the nonreactive travel time r and the time sorbed [i. The transfer function characterizing this transport and reaction process is (see Cvetkovic et al. [1998]) j[t ] x} = S[tr[x}fi[x}} (4.3) Following the work of Cvetkovic et al. [1998], the adopted reaction parameter model is P = P g exp[w] exp[/? In k] (4.4) where P g is the geometric mean of P, w is a zeromean, exponentiallycorrelated normal random variable with variance cr^ and correlation length X w = X\ n k that is uncorrelated to the log conductivity field, and )3 is a strength of correlation parameter relating P to the underlying log conductivity field. The equality X w = Ai nfc was specified for convenience.
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82 Two specific injection modes are considered: uniform injection influx and uniform resident injection. For the uniform injection influx, each streamtube receives an equal amount of solute mass (Equation (3.5)), and thus mass weight (Equation (3.6)). Uniform resident injection distributes the mass uniformly in space, but each streamtube receives a different mass and mass weight (Equation (3.3)). The mass breakthrough for some streamtube i is given by m i [t;x]=m i [0;0)j[tx] (4.5) An ergodic condition is assumed such that aggregate properties may be thought of as ensemble properties in the statistical sense. The aggregate mass breakthrough is given by taking the expected value of Equation (4.5). For injection influx, the expected solute breakthrough curve is given by = M <7M> ( 4 6 ) For uniform resident injection, the expected solute breakthrough curve is given by < S u [t] >= MU < 7[t; x]/v[0] > (4.7) where U is the arithmetic mean, or spatially averaged, velocity and v[0] is the velocity at the injection point. The temporal moment of this breakthrough curve is sought. The n temporal moment is given by t n = lim(l) n ^<,SM> (4.8) s>0 QS n where S is the Laplace transform of S and s the corresponding Laplace space variable. The Laplace transform of Equation (4.3) is given by j[s; x] = exp [s (r[x] + fi[x})} (4.9)
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83 Two temporal moments of interest for the influx breakthrough curve are the mean t lf = + (4.10) and the variance t 2f = a 2 T + al + 2a T ^ (4.11) For uniform resident injection, the moments are t lf =< Ut/v[0] > + < Un/v[0] > (4.12) and the variance t 2f =< U{t + n) 2 /v[0] > < Ut/v[Q) > 2 < Ufi/v[0] > 2 (4.13) From these equations, it is seen that the mean and variance of the expected reactive travel time distribution is comprised of the first two moments of the nonreactive travel time and the reaction flow path and the cross moment of the travel time and reaction flow path. 4.4 Simulation of Reactive Solute Transport The results of the simulations described in Chapter 2 are employed to simulate the transport of a reactive solute. As mentioned in that chapter, information was recorded along two particle trajectories: the first was a natural trajectory corresponding to a hydrodynamic streamline, and the second was a straight line parallel to the mean flow direction. Data were recorded at equal increments in time after injection, and at "control planes" perpendicular to the mean flow at equal displacements from the injection plane. Among these data were the velocity, the hydraulic conductivity, and an observation of a lognormal random field with the same statistical and correlation properties as the log hydraulic conductivity field.
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84 4.4.1 Reaction Parameter From these data, trajectorybased P fields were constructed using Equation (4.4). In this manner, any variety of correlation relationships may be analyzed using the same set of simulation data. There is a dearth of information about the structure of sorption fields in natural systems (see, e.g. Tompson [1993] or Jawitz [1999]). Thus, numerically convenient parameter values were selected. This suite of experiments examined correlation parameter values of /3 = {1, 1/2, 0, 1/2, 1} in log conductivity fields of variances of nJt = {1/2, 1}. The extreme correlation parameters {3 = {Â—1,1} are arguably physically implausible, yet they are included to demonstrate the effect of correlation without the "noise" associated with an uncorrelated component. The remaining three values, f) = {Â—1/2, 0, 1/2} might be considered a range of more plausible values. For consistency, the variance of point log reaction parameter was held constant and equal to that of the log conductivity field by requiring a 2 w = (1 Â— /3 2 )of nA .. The expected value of P was 1. This is a slightly different approach than that taken with the log hydraulic conductivity, where the geometric mean was fixed at K g = 1. In the case of the twodimensional isotropic log conductivity field, the effective conductivity is equal to the geometric mean. The expected value of the reaction value was fixed = 1, because in some practical applications P is assumed to be linearly related to some property of interest, and the "amount" of that property is to be estimated based upon the temporal moments of linearly sorbing solutes, or tracers (see e.g., Jin et al. [1995], Annable et al. [1998], and Jawitz [1999]). The point average of reaction parameter P as observed along a Lagrangian trajectory is a function of injection mode, correlation to the log conductivity and the variance of the log conductivity. As with the log conductivity, the equalflux streamtubes exhibit an apparently stationary mean reaction parameter in displacement along the mean flow direction, whereas the equalarea streamtubes exhibit distinctly nonstationary behavior that is analogous to that of the log conductivity (see Figures 4.1 and
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85 4.2). These observations, of course, are not surprising, since the reaction parameter is a function of the log conductivity. Positive correlation between the log conductivity and the reaction parameter (/3 > 0) results in a reaction parameter trajectory mean that is larger than the uniform spatial mean. Negative correlation (/3 < 0) results in a reaction parameter trajectorymean that is smaller than the uniform spatial mean. No correlation (/? = 0) results in equal trajectorybased and uniform spatial mean reaction parameter estimates. A decrease in the system variability (e.g., the variance of log conductivity of nk ) results in a smaller difference between the volume average and the trajectory averages. 4.4.2 Reaction Flow Path A simple trapezoidal rule was used to calculate the reaction flow path fi along a solute particle trajectory. For a collection of n trajectories, there were n \i "observations" at each control plane. For an intuitive grasp of what the statistics of the reaction path "means" recall that for linear equilibrium sorption, the reaction path corresponds to the time that the solute is sorbed. The sum of the the nonreactive travel time and the sorbed time equals the total reactive travel time of the solute from the point of injection to the point of observation. See Cvetkovic et al. [1998] for further application of the statistics of the reaction flow path. The mean and the variance of the reaction flow path are presented for Lagrangian trajectories that carry either an equal flow weight or an equal area weight at a reference plane. Recall that the equal flow collection can be thought to correspond to a uniform injection influx and that the equal area collection can be thought to correspond to a uniform injection into the resident fluid. The coupled effect of injection mode and reaction parameter correlation to the log conductivity are illustrated by examining reactive flow fields with five different correlation characteristics. The first case is that in which the reaction parameter is completely determined by the
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86 1.8 1.6 1.4 1.2 08 0 6 0.4 1.8 1.6 1.4 X +FL +AL +AE AE o AL FL o 10 20 30 40 1.2 ft. 08 06 04 x i S ** x i xx Â„ 1 1 iv^^^/^w* +FL +AL x +AE > AE AL FL o 10 20 30 40 10 20 i/A]Â„jt Figure 4.1: Mean reaction parameter values for different values of correlation parameter P and different trajectory collections for the af nk = 1 set of simulations. Top figure is /? = {1,+1}. Middle figure is f3 = {1/2, +1/2}. Bottom figure is /3 Â— 0 (no correlation). Trajectory collections are areaweighted Lagrangian (AL), flowweighted Lagrangian (AF) and areaweighted Eulerian (AE).
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87 18 1.6 Â• 14 12 tu Â•*, **** *** 0.8 0.6 0.4 ( 1.8 16 1.4 1.2 +FL +AL +AE AE o AL FL o 10 20 30 40 ** +FL +AL +AE AE AL Â• FL Figure 4.2: Mean reaction parameter values for different values of correlation parameter (3 and different trajectory collections for the of nfc = 1/2 set of simulations. Top figure is P = {1,+1}. Middle figure is 0 = {1/2, +1/2}. Bottom figure is P Â— 0 (no correlation). Trajectory collections are areaweighted Lagrangian (AL), flowweighted Lagrangian (AF) and areaweighted Eulerian (AE).
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88 conductivity field (ft = {1,1}). Considered are the two subcases, namely that of "perfect" negative correlation (/5 = 1) and "perfect" positive correlation (/5 Â— 1). For /3 = Â— 1, the equal area collection exhibits nonlinear behavior analogous to that of travel time (see Figure 4.3 and 4.4). However, this nonlinearity is enhanced by the correlation of high reaction parameter values in the low conductivity areas. For the equal flow collection, however, the nonlinearity is mitigated, and \i appears to be a linear function of displacement. For /3 = 1, the both trajectory collections exhibit the same linear behavior. Thus positive correlation appears to mitigate the effect of injection mode on the mean behavior. The reaction path variance is strongly affected by the sign of the correlation (see Figures 4.5 and 4.6). For negative correlation, there is a distinct difference between the injection mode. However, positive correlation again mitigates the difference in modes. Moreover, the positive correlation greatly reduces the reaction path variance. For linear equilibrium sorption, this implies a reduction of the reactive travel time variance. The next case considered is that of no correlation between the reaction parameter and the log conductivity (/? = 0). The reaction path mean exhibits the same behavior as does nonreactive travel time. The nonlinearity in the mean for the equal area streamtubes is attributable to the nonstationarity in the associated velocity field. However, the combined effects of heterogeneous velocity and sorption manifest themselves in the reaction flow path variance. The reaction flow path variance is higher than that of the nonreactive travel time, even though the scaled means are the same. The /3 Â— 0 reaction flow path variance "lies in between" that of the positive and negative correlation. For the lower "total system variability" case of af nk = 1/2, the magnitude of the effect of correlation appears nearly symmetric (see Figure 4.6). For cinfc Â— 1 5 however, there the effect of negative correlation seems to have a greater impact upon both total variability and the effect of injection mode (see Figure 4.5).
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89 Figure 4.3: Mean reaction path observed values and estimators for different values of correlation parameter ft and different trajectory collections for the crf nk = 1 set of simulations. Top figure is ft = {1, +1}. Middle figure is /3 = {1/2, +1/2}. Bottom figure is ft = 0 (no correlation). Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF).
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90 0 10 20 30 40 0 10 20 30 40 l/An* o JÂ£ 1 1 1 0 10 20 30 40 i/AiÂ„jt Figure 4.4: Mean reaction path observed values and estimators for different values of correlation parameter and different trajectory collections for the af nk = 1/2 set of simulations. Top figure is P = {1,+1}. Middle figure is /? = {1/2, +1/2}. Bottom figure is /3 = 0 (no correlation). Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF).
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91 Figure 4.5: Reaction path variance observed values and estimators for different values of correlation parameter @ and different trajectory collections for the af nk = 1 set of simulations. Top figure is /3 = {Â— 1,+1}. Bottom figure is /3 = {Â—1/2, +1/2}. Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF).
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92 Figure 4.6: Reaction path variance values and estimators for different values of correlation parameter /3 and different trajectory collections for the af nk = 1/2 set of simulations. Top figure is /3 = {1,+1}. Bottom figure is /3 = {1/2,11/2}. Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF).
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93 4.4.3 Reaction Flow PathTravel Time Cross Moment The reaction flow pathnonreactive travel time cross moment a^ T contributes to the variance of the expected solute flux (see Equations (4.11) and (4.13)). For linear equilibrium sorption, this reflects the correlation between the time sorbed and the time traveling. The correlation of the reaction flow path and the travel time is little affected by injection mode (see Figures 4.7 and 4.8). It is, however, sensitive to the correlation parameter @. As with the reaction flow path statistics, the deviation of from the /? = 0 is greater in the case of positive correlation for the same value of /?. The "total system variability" seems to diminish the effect of correlation upon a MT This is most readily apparent from a comparison of the two ft = {Â—1,1} cases for the different values of af nk (compare the top figures in Figures 4.7 and 4.8). The correlations for the correlated case "shift" towards that of the /3 Â— 0 in the higher log conductivity variance field, and the f3 = 0 case is slightly lower for the greater af nk In all cases except /3 = l,
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94 0.2 1 1 1 0 10 20 30 40 Figure 4.7: Correlation between the reaction flow path and travel time for different values of correlation parameter /? and different trajectory collections for the of nJfc = 1 set of simulations. Top figure is j3 = {1,+1}. Bottom figure is (3 = {1/2, +1/2}. Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF).
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95 0.8 0.6 0.4 0.2 0.2 AL FL x OAL OFL a +AL +FL o 10 20 30 40 b b 3. b 0.8 06 0.4 0.2 0.2 ^ 'Â•' AL FL x OAL OFL +AL +FL 10 20 30 40 Figure 4.8: Correlation between the reaction flow path and travel time for different values of correlation parameter /3 and different trajectory collections for the a \nk Â— 1/2 set of simulations. Top figure is /3 = {Â— 1,+1}. Bottom figure is ft Â— {Â—1/2, +1/2}. Trajectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian (AF).
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96 conductivity in the calculation of the mean, working from Lagrangian properties that are stationary in displacements along the mean flow direction, and examination of the effects of injection mode. 4.5.1 Mean Reaction Flow Path Substitution of Equation (4.4) into Equation (4.2) and taking the expected value yields < The stationary Lagrangian serves as a reference field. The log conductivity field is decomposed into its mean and mean removed perturbation / and the velocity into its mean and mean removed perturbation thus < Define P g = P* exp[/xi nfc ], and expand the nonlinear terms of the previous equation into series, take the expected value retaining second order terms thus = P exp ^ /2] (1 fa, + fo}/2 + al)x (4.16) Recall that aj = af nk The / subscript is used for clarity. This expression captures the linearity of the observed mean \x (see Figure 4.3), but shows a heavy dependence upon the sign of the correlation parameter f3 that is not apparent in the data. The mean reaction flow path for the equal flow streamtubes appears to be dependent only upon the mean value of the reaction parameter, e.g. <>= P >^ Kl l+^>l\ +
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97 For the hypothesized Lagrangian distribution V = U(l + baj) and a\ = baj, thus the Equation reduces to <>= P g ex V [a 2 j2 + P 2 a 2 /2]x/U (4.18) The equal area streamtube mean [i is = p;e Xp[4 /2,,f(2^)^ (4,9) Manipulations similar to that which yielded Equation (4.16) yield < Uv/v[0] >= ^^Hf f (1 + 2aÂ„ 2 + P 2 o 2 /2 + C v [x] 2(3C vf [x})dx (4.20) 1/(1+00" j) J Q Substituting the simplified relationship C vf [x] = a 2 exp[a;/(2A)]/2 yields 40^< 1+2 ^ + ^ 2) (4 21) + 2/3AaJ(exp[x/(2A)] 1) + \o){\ exp[ox/A]) Notice that the first collection of terms, or the linear portion, of the preceding equation is the same as that in Equation (4.16). Similar arguments are employed to yield the equal area streamtube mean // estimator P g exp[^/2 + /?V;/2] < U,/v[0] >= f Â—x (4 22) + 2^AaJ(exp[x/(2A)] 1) + Xa){\ exp[6x/A]) 4.5.2 Reaction Flow Path Variance Adapting an expression for the reaction flow path variance given by Cvetkovic et al. [1998] (their Equation (B4)) to the simplified velocity and velocitylog conductivity functions central to this work, write al\x] = 2P 2 exp[a 2 w }/U 2 / (x s)(C w [s] + p 2 C f [s] 4pC fv [s] + C v [s})ds (4.23) Jo
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98 For the special case of a\ = (4.24) + (24P)x/\ f } A simple and intuitive reaction flow path variance estimator for the equalarea streamtube collection is untenable in this framework. This is due in part to the nonlinear reaction flow path mean. Figures 4.3 and 4.4 demonstrate a close match between Equations (4.18) and (4.22) and simulation data. The parameters placed into the estimators are the target values selected for the simulation, and thus represent an a priori estimate of the mean reaction flow path. Equation (4.22) captures not only the nonlinearity of the nonreactive travel time, but the nonlinearities induced by reaction parameter and log conductivity correlation. While the "absolute difference" between Equations (4.18) and (4.22) appear small, consider that the scale of the simulations is fairly large. The solute has traveled over forty correlation scales of the log conductivity. In settings where the the scale of conductivity heterogeneity are large compared to the overall scale, the relative impact of injection mode and reaction parameter correlation to the log conductivity will have a relatively greater effect. The equalflux streamtube reaction flow path variance estimator (Equation (4.24)) shows relatively poor performance in the more heterogeneous system (Figure 4.5), compared to that in the less heterogeneous system (Figure 4.6). Moreover, this expression does not work well at all for values of positive correlation, yielding physically implausible reaction flow path variances (cr^ < 0) for correlation parameter values greater that 1/2. The lack of "robustness" is perhaps attributable to an inadequate 4.6 Discussion
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99 representation of the myriad interacting processes by the firstorder perturbation expansions. Injection of solute in the influent fluid mitigates the effect of correlation on the mean and the nonlinearity of mean nonreactive travel time. For tracer experiments that determine the volume or area of some property based upon mean arrival times of reactive and nonreactive tracers, correlation will have little effect upon the first temporal moment of the solute breakthrough curve if and only if the solute is uniformly injected in flux. However, injections that preferentially favor lowflux areas may exhibit strong nonlinearities, and be particularly sensitive to negative correlations between the local conductivity and reaction parameter. The contribution of the fj, variance to the second moment of the reactive solute travel time is extremely sensitive to correlation between the reaction parameter and the conductivity. Experiments that rely upon higher order moments of tracer experiments are, therefore, questionable if there is no pln k correlation information. Whereas injection mode dominates the correlation behavior in the mean, correlation dominates injection mode in the reactive travel time variance. 4.7 Spatial Moments Symmetry would dictate proceeding with a corresponding spatial moment analysis for transport of solute subject to a linear equilibrium sorption in a heterogeneous reaction parameter field. However, this process presents a particular complication that thwarts the development of relatively simple expressions following the same methods those previously employed. In the case of the travel time statistics, the "control structures" are two control planes. A fluid and solute parcel travels between these two control planes, sweeping out a fixed and constant volume, without regard to correlation properties, reaction parameters, or processes. The time required to sweep this volume, of course, is highly dependent upon these factors. For a "clock time"
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100 control structure, a solute parcel will trace different volumes for different processes and correlation structures. The reaction parameter model is comprised of two random processes. The first is that of the log conductivity field and the second is a random component that has the same distributional properties as, but is otherwise uncorrected to, the log conductivity field. In the case of the temporal moments, the weighted covariance functions of the log conductivity and this random component were employed directly. These functions can be parameterized, of course, in terms of the nonreactive travel time. However, these travel time parameterized covariance functions time are not immediately applicable to sorption analysis in the way that covariance parameterized in clock time would be, because the displacement for a given clock time is highly dependent upon the correlation of the reaction parameter to the log conductivity field. This is not to say that sorptiondependent spatial moment analysis is not viable in this framework. In fact, Cvetkovic et al. [1998] analyzed the spatial moments of a solute subject to nonlinear equilibrium sorption in a heterogeneous sorption field in this framework. However, the sorption parameter and the underlying log conductivity field were uncorrelated. Further clock time oriented are developments left for future work. Expressions involving the travel time dependent reaction flow path are developed and applications to and implications for first order decay are discussed. Continuing the theme of this dissertation, expressions based upon the trajectorybased covariance functions are sought. The timedependent reaction flow path for the selected reaction parameter model is related to the log conductivity covariance function, as shall be demonstrated in the following. As in the previous chapter, transport along Lagrangian trajectories is considered. Assumed are secondorder stationary velocities along areaweighted trajectories in time and secondorder stationary velocities along flowweighted trajectories in space.
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101 The log hydraulic conductivity is assumed to be similarly stationary. Moreover, the point random process P g exp[w] is assumed likewise stationary, since it is uncorrelated to the random log conductivity. From these assumptions, the expected value of the timedependent reaction flow path is < h[t] > = ^ < P g exp[w[t}] exp[p In k[t}} > dt = P g exp[a 2 j2 + /3 2 al k /2}r The variance is given by var[/x[r]] =< f P g exp [w[t]] exp [0 In k[t}} dt Jo (426) / P g exp [w[t\] exp [P In k[t]} dt'< h[t] > 2 J o Consider the integrals in the previous equation. The exact expression is denoted by I and is approximated by the truncated series expansions of the exponential terms thus J ~ p g f [ T < {I + w[t}) {I + w[t']) (1+P\n k[t}) (1+ 13 In k[t'])>dtdt' (4.27) Jo Jo Note that a possibly more elegant approach would be to explore the Lagrangian features of the conductivity field, rather than the log conductivity field, and work from a stationary conductivity field covariance function and a similar uncorrelated field. This would obviate the need for the expansions. This suggested inquiry is left to future works. Expanding the above expression, taking the expected value, and dropping products of covariances yields I = P 2 g f f (1 + C w [t,t'} + (3 2 C lnk [t,t'})dtdt' (4.28) 7o Jo A few observations allow the adoption of an approximate expression for the timedependent Lagrangian covariance function C\ n k[t, t']. Relatively simple expressions that capture the behavior of the dominant processes are sought here. These expressions might facilitate future works that deemphasize simplicity for greater rigor and
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102 perhaps a better fit to the data. The Eulerian log conductivity covariance is the foundation upon which the rest of this work has been built. The model field is exponentially correlated in space, with correlation length Ai nfc The Eulerian velocity covariance exhibits a much longer correlation length due to the dissipation of the head across the field (see previous chapters). It is reasonable, therefore, to expect that the Lagrangian log conductivity covariance correlation length will lie somewhere in between that of the Eulerian log conductivity and the Lagrangian velocity. At zero lag, the areaweighted Lagrangiantrajectory log conductivity covariance is equal to the variance of the Eulerian field. This value is af nfc by definition. Thus, an exponential approximation of the Lagrangian log conductivity covariance function is C lDk i at [t] = al k exv[b'Ut/\ lnk } (4.29) where b' is a function similar to b that relates the correlation length to a correlation time. The approximation b' Â— 1 is made, which is equivalent to assuming that the Lagrangian and Eulerian correlation lengths are equal is space and substituting x = Vht = Ut into the spacedependent Eulerian log conductivity covariance function. Comparison of Equation (4.29) with b' = 1 to covariance functions estimated from log conductivity observations taken along areaweighted Lagrangiantrajectories generated in the numerical experiments described in Chapter 2 shows good qualitative agreement (see Figure 4.9). The data exhibit an apparent dependence of the Lagrangian correlation length upon the variance of the log conductivity field that is not captured by Equation (4. 29). One might expect similar behavior in the Eulerian field a priori, since the appropriate substitution into the Eulerian covariance function would be x = Uht = Ut/(1 + baf nk ). The higher variance of log conductivity results in a lower harmonic mean average velocity, implying it takes longer to to travel a unit length in more heterogeneous fields and a larger Lagrangian correlation time. In
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103 0.8 0.6 c o Â£ 0.4 o u 0.2 1/2 1 Lagrangian In k Lagrangian V X 0.2 *Â•++ X + + x xx xx x x xx,. 4 6 Ut/X lnk 10 Figure 4.9: Assumed timedependent log conductivity covariance function along Lagrangian trajectories (Equation (4.29)) compared to simulation data for variances of log conductivity af nk = {1/2,1}. The Lagrangian velocity covariance function is Equation (2.61). fact, this is the behavior observed in the simulations (see Figure 4.10) The data are compelling enough to suggest further work. However, we shall accept Equation (4.29) with b' = 1 as a suitable approximation. Substituting Equation (4.29) with t = \t t'\ into Equation (4.28), integrating and inserting the result into Equation (4.26) yields var[/i[r]] = U 2 iUT/K k (lexp[Ur/X lnk }) P?exp[al + p 2 ol k ]T 2 (4.30) Neither this equation nor Equation (4.25) show a dependence upon the sign of (3. For the mean reaction flow path, the simulations show a weak dependence upon correlation parameter (see Figure 4.11). The linear propagation of the mean reaction flow path in time is due to the stationarity of the local reaction parameter along the areaweighted Lagrangian trajectories. Again, it should be noted that "time"
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104 0.8 1/2 1 Lagrangian In k Eulerian In k 1/2 Eulerian In k 1 c o o u 0.6 0.4 0.2 0 +++++ + ++ .0.2 1 1 1 1 0 2 4 6 8 10 Figure 4.10: Hypothesized timedependent Eulerian log conductivity covariance functions for of nJfc = {1/2, 1} compared to simulation data and Equation (4.29). Figure 4.11: Mean reaction flow path as a function of nonreactive travel time compared to estimator Equation (4.25) for correlation parameter values /5 = { Â— 1,0, 1}.
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105 refers to nonreactive travel time. Reaction flow path variance, however, is strongly affected by correlation, and Equation (4.30) is completely inappropriate (see Figure 4.12). Moreover, this equation can give physically implausible values at large 60 50 40 20 10 0 0 2 4 6 8 10 Ut/\ lnk Figure 4.12: Reaction flow path variance a function of nonreactive travel time for correlation parameter values /3 = {Â—1,0, 1}. values of time. This is a result of the series approximations required for the "raw second moment" term in the variance, and the lack of such approximations in the "first moment squared" term. 4.7.1 Injection Mode The coupled effects of injection mode and correlation on the mean travel time dependent reaction flow path are examined. Using the relationships developed in Chapter 3, the hypothesis that the statistics reaction flow path corresponding to the equalflow streamtubes are given by flowweighting the equalarea properties is made.
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106 Thus, the mean reaction flow path for a flux injection is given by < u frtT)>=< u f[pm> (4.3i) In taking the expected value, the crosscorrelation of the initial velocity and the that portion of the reaction parameter correlated to the log conductivity field must be considered. Using nowfamiliar expansions, the expected value of the flowweighted reaction flow path is < ^W] >= ^^^^ V + PC fu [t]/U)dt (4.32) U Jo where C fu is the covariance of the initial velocity perturbation and the log conductivity perturbation. Once again, this covariance shall be approximated with an exponential function of the form (see Appendix B) Ut (4.33) Substituting this term into Equation (4.32) and integrating yields < f ,H >= 5^^! (ur + ^.expIJ^]) (4.34) 4.7.2 Discussion Similar to its space dependent counterpart, the mean time dependent reaction flow path is sensitive to injection mode. However, it is the uniform resident injection that appears to mitigate correlation effects upon the mean reaction flow path. Correlation has a strong effect of [i variance, and it is quite likely that these effects dominate any injection mode effects, as was the case for the space dependent \x.
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107 10 A a. v +1/2 0 1/2 est +1/2 est 0 est 1/2 ..Â•'+ sk i .Â•Â•+ x W* At* ,r A x. >, A J* 7^7 .Â•+ .x 4 X .+ X .+ x ,+ *Â• *Â•' A J? J* + X X,4*** /Si*' Â•Â£XH4 6 10 10 A a. V X/' /+ X' /+J* *' 4 6 Ut/X ]nk ^ X ,+ .
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CHAPTER 5 APPLICATION OF CONCEPTS 5.1 Introduction An illustration of a practical application of trajectory based flow and transport modeling is sought. Useful and fairly robust relationships, namely predictors of spatial moments of a continuously injected plume subject to a first order decay, using simpler streamtube concepts (e.g. Jury and Roth [1990]). However, the observed bias in prediction shall be explained using the more complete trajectory based approach. In light of the robustness of the simple relationships developed here, they are used to estimate relative degradation rates for different solute species and applied to field data collected at a fieldscale macrodispersion and natural attenuation experiment conducted in Columbus, Mississippi (see Stauffer et al. [1994] and Stauffer et al. [1997]). 5.2 Theory Consider a subsurface source of groundwater contamination that releases solute into the aqueous phase at a constant concentration. An example of such a source is a poorly mobile nonaqueous phase liquid, comprised of relatively insoluble organic compounds, distributed at some nearresidual saturation through out a portion of some aquifer. The "constant" assumption applies when the rate of change of a property of interest, such as the aqueous phase concentration at the source, is small in comparison to the temporal scales associated with transport and decay. The actual groundwater flow field is modeld as a steady irrotational flow field that may be resolved into an aggregation of streamtubes. Neglecting the effect of local 108
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109 dispersive processes, an advectiondecay equation with streamtube specific parameters is assumed to describe the fate and transport of the solute along the streamtube trajectory. This advectiondecay equation is S* Â™ subject to c[x, t < 0] = 0 and c[x = 0, t => 0] = Cq, where v is an effective advective velocity [LT _1 ], and A: is a firstorder decay constant [T 1 ]. The assumption of effective streamtube parameters is the simplification that separates this method from the more general consideration of properties varying along the trajectory. Although much of this language is similar to that used in earlier chapters, the fundamental difference is that the streamtube has constant effective parameters. This is analgous to an assuption of stationarity of properties, but is different in significant ways, as shall be shown. For advective transport in a single streamtube, the resident concentration is equal to the fluxaveraged concentration. However, the difference between these detection modes manifests itself when concentrations are averaged over several streamtubes. For the remainder of this discussion, any reference to a concentration will implicitly imply a reference to a resident concentration. The solution to this equation for the given initial and boundary conditions is Â— = exp [Â— kx/v] h[vt Â— x] (5.2) Co where h[x] is the Heaviside step function, with the properties h[x] = 0 for x < 0 and h[x] = 1 for x > 0. 5.2.1 Spatial Moments The defintion of the raw spatial moment of order n of curve c is fj! n = J oox n c[a;](ix (5.3)
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110 A normalized moment of order n is a raw moment of order n divided by the zeroth moment, and is denoted by dropping the prime (i.e., /i n ). The zeroth spatial moment of this concentration curve is given by ^[t] = l[l exp[fct]] (5.4) The zeroth spatial moment of a contaminant plume related to the total mass in solution. The first normalized spatial moment, or center of mass, is given by rtn 1 exp[kt](kt + 1) Â™ = 1* J 1 exp[!fct] ( ] In the limit t Â— > oo n\ Â— v/k. The concentration given in Equation (5.2) is a function of this asymptotic center of mass v/k. Solving Equation (5.2) with h[] = 1 for v/k yields (5.6) k In [c/co] Thus, an observation of the concentration at any point of a steady streamtube concentration curve is an observation of its center of mass, given that cq and x are known. Assume that independent samples can be drawn uniformly from the collection of streamtubes that originate from a source Cq located at x Â— 0. From the linearity of first moments, the mean center of mass may be estimated from N samples by the simple estimator 1 N x*">mT^ (5 ?) 5.2.2 Relative Degradation Consider two solutes a and b subject to a firstorder decay process, characterized by rate constants A; a and k b respectively. The relative degradation rate of solute a to
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Ill b is defined as K = k a /k b (5.8) Let these solutes be released into a flow field at a steady rate, and allow the solutes to reach a "steady state" in which the concentration profiles for each do not vary in time. If within a streamtube i, the streamtube effective velocity Vi is a approximately equal for solutes a and b, the relative degradation rate of a to b in streamtube i is given by kri = (vi/k b )/(vi/k a ) (5.9) That is, the relative degradation rate is given by the ratio of the first normalized spatial moment of the steadystate concentration curve of solute b to that of a. The average relative degradation rate is estimated as the arithmetic mean of k ri Notice that this is not the ratio of the mean first spatial moments. 5.3 Application: Simulation The estimator was tested by simulating the release of a solute into a heterogeneous twodimensional aquifer using the United States Geological Survey flow and transport code MOC3D [Konikow et al, 1996]. The length units used herein are a dimensionless quantity resulting from the normalization of the fundamental length by that of the correlation length of the exponentially correlated log hydraulic conductivity field. In the interest of clarity, however, this dimensionless quantity will be referred to as a correlation length A. Let the reference Cartesian coordinate system be oriented such that the ^direction is parallel with the mean direction of flow, and the ydirection is orthogonal to that and lies within the plane formed by the aquifer. Let the origin lie at the lower left corner of the aquifer. The rectangular aquifer extended 15 A x and 25 A in y, and was
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112 subdivided into 10 nodes per A in both direction. Constant heads maintained at the x = OA and x = 15A boundaries imposed a steady mean gradient of J = 0.01. Noflow conditions were maintained at the y = OA and y = 25A boundaries. To simulate the release of a contaminant at constant concentration, the influent water concentration was assumed to be a constant concentration c 0 = 1. This "source" was centered on the x = 0 boundary, and was 10A wide. A turningbands algorithm generated a synthetic heterogeneous log conductivity field used in the simulations [Tompson et ai, 1989]. The output of the turningbands generator was a realization of an exponentiallycorrelated standard normal process (/x = 0,cr = 1). This field was subsequently converted to fields with variances of 0.1, 0.5, and 1.0 via the transformation / = VlnkZ (5.10) where a\ a is the target standard deviation of the log conductivity field / and z is the standard normal process. The aquifer porosity was assumed to be equal to the mobile water content and equal to a constant value n = 0.2. The effective conductivity of a twodimensional exponentially correlated log conductivity field is, to a good approximation, the geometric mean conductivity K g For \x Â— 0, K g = 1, in units of (dimensionless) correlation lengths per characteristic time. The independence of the effective conductivity from its variance convieniently allows the variance to be changed without substantially changing the bulk flow properties. Thus, the anticipated Darcy flux for the aquifer, regardless of o\ nk was = 0.01A per characteristic time, and the characteristic filtration velocity was vi, = 0.05A per characteristic time. The subscript b refers to a bulk property, to be distinguished from streamtube properties discussed earlier. The solute was assumed to undergo a constant decay process characterized by a firstorder rate coefficient k Â— 0.05 in units of an inverse characteristic time defined
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113 by Vf/X. This value yields what might be termed a "bulk" first spatial moment Vb/k = 1. For a homogeneous conductivity field, this is numerically equivalent to the first longitunial spatial moment of the steadystate plume. The steadystate concentration field was "sampled" by taking the reported numerical concentration values as the local resident concentration. The plume centerofmass was estimated by the estimator presented in Equation (5.7). The "actual" first longitudinal spatial moment was calculated using a trapezoidal rule integration. A fate and transport simulations were carried out in 6 sets of hydraulic conductivity fields corresponding to 6 standard normal process realizations. 5.3.1 Results For the case of a homogeneous conductivity field, any one nonzero concentration measurement used with the estimator given in Equation (5.7) returned a value for the first longitudinal spatial moment that for all practical purposes was equal to Vb/k, and five percent less than that predicted by the numerical integration. As the variance of log conductivity increased, however, both the numerical integration and the estimator tended to systematically predict values greater than Vb/k. Additionally, the estimator systematically underpredicted the value given by the numerical integration, and by an amount which generally increased with variance of the log conductivity. 5.3.2 Discussion This systematic increase in the the first longitudinal spatial moment of the plume with increasing log hydraulic condutivity variance, despite a fairly constant v b is explained by considering the nature of the underlying Lagrianian velocity field. The steadystate concentration plume is, in essence, a map of travel times from the source. As previously stated, Cvetkovic et al. [1996] demonstrated that the statisics of such a field are nonstationary. For small displacements from the source, the mean travel
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114 Figure 5.1: Resident concentration profile averaged along a transect orthogonal to the mean flow direction for plumes for cr, 2 nfc = {0.0, 0.1, 0.5, 1.0}. time propagates dr/dx = l/v h (5.11) where Vh is the harmonic mean velocity. For regions distant from the source region, the mean travel time propagates dr/dx = l/v a (5.12) where v a is the arithmetic mean velocity, and equal in value to Vb, the bulk filtration velocity. The harmonic mean velocity is necessarily less than or equal to the arithmetic mean velocity, which in turn indicates that the mean travel time propagates as a slower rate near the source. Additionally, the correlation of the Lagrangian velocity persists over greater distances than that of the underlying conductivity field. Solute which is introduced in
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115 areas of high local flux tend to move away from the source at relatively high velocities. The net effect of what might be termed a "preferential flow" of solute in these higher velocity streamtubes, is to induce a "tail" on a transverselyaveraged resident concentration profile, which increases with variance of log conductivity. These effects are illustrated by comparing concentration profile created by averaging concentrations along planes normal to the mean direction of flow to concentration profiles given by c/c 0 = exp[kx/v] and c = /c 0 exp[/cr[:r]], where r is given by t[x] = Â— (x + \inklk (! exp[te/Ai nfc ])) (5.13) "a Here Ai n is the correlation length of the log conductivity distribution and b is a shape factor for the conductivity anisotropy. 5.4 Application: Field Data Data from an elaborate experiment conducted at Columbus Air Force Base, Mississippi were analyzed using the results dervied in the previous section. The objective the experiment was to characterize the natural attenuation of certain hydrocarbons in groundwater emanating from subsurface nonaqueous phase liquid (NAPL) sources Stauffer et al. [1997]. A NAPL hydrocarbon source comprised of decane, benzene, toluene, ethyl benzene, para xylene (pxylene), and naphthalene was emplaced at the site of the MADE [Boggs et a/., 1992] and MADE2 [Maclntyre et al, 1993] experiments. This source released these poorlysoluble organic solutes into the groundwater, and the resultant plumes were monitored at downgradient multilevel sampler locations. The center of mass of the steadystate plume was estimated using a trapezoidal rule integration of concentration data in space and by a simple averaging of pointwise observations of concentration via Equation (5.7). Relative degradation and absolute degradation rates of the different constituents are estimated using Equation (5.9).
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116 5.4.1 Results and Discussion: Center of Mass The center of mass of the steady state plumes emanating from the emplaced contaminant source was estimated by two methods. The first might be considered a traditional approach, namely simple trapezoidal rule integrations of the data. A known initial concentration for each species was assumed known. This concentration was calculated from initial source mass fractions and aqueous solubilities given in Stauffer et al. [1997] and assuming Raoult's law (see Table 5.1). The second approach was to use Equation (5.7) (see Table 5.1). The same initial concentration used in the spatial integration was used with the estimator approach. Displacements were taken as the distance from the multilevel sampler to the source zone. benzene toluene ethyl benzene pxylene naphthalene integration 4l) 21) 3l 2^3 3^8 v/k 5.3 1.7 2.5 1.9 3.1 Table 5.1: The center of mass in meters of steady plumes estimated by traditional trapezoidal integration of spatial concentration data and by the mean v/k estimator Equation (5.7). Equation (5.7) gives results similar to those of the trapezoidal rule integration, while offering some compelling computational advantages. While the spatial integration is conceptually simple, it is computationally demanding for unequally spaced data in that the different "trapezoids" must be calculated and summed for each integration. New data require recalculation of these trapezoids. Of course, these calculations are trivial for computer codes, but tedious for "back of the envelope" calculations. Equation (5.7) is a fairly simple calculation for any scientific calculator, and provides an excellent tool for a rapid estimate of the center of mass from sparse and scattered local concentration. New data are easily incorporated into existing sets without the need of a global recalculation.
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117 5.4.2 Results and Discussion: Degradation The relative degradation and absolute degradation of the different contaminants were estimated using Equation (5.9) and degradation rates reported by Maclntyre et al. [1993]. These reported benzene, pxylene, and naphthalene degradation rates were estimated from another experiment conducted at the site in which a "cocktail" of hydrocarbons was injected into the groundwater and the fate of the resultant plume was monitored (see Table 5.2). This experiment was conducted before the NAPL benzene toluene ethyl benzene pxylene relative 087 3A L5 2l) estimated 0.0056 0.022 0.0094 0.013 reported 0.0070 NR NR 0.011 Table 5.2: Degradation rates relative to naphthalene and absolute degradation rates in reciprocal days as estimated by Equation (5.9) and as reported by Maclntyre et al. [1993]. NR denotes results not reported in Maclntyre et al. [1993]. Naphthalene degradation reported to be 0.0064 source was emplaced, and is in that sense independent of the concentration data associated with the natural attenuation experiment. Nevertheless, the rates estimated from the natural attenuation data were similar to those reported by Maclntyre et al. [1993]. The technique used here is quite simple, and does not require regressions or nonlinear fits to transport models. 5.5 General Conclusions These estimators, while fairly simple and apparently robust, are limited in applicability to quasisteady plumes, and it is not clear the extent to which such plumes exist. Sites where poorly mobile NAPLs have been known to exist for long times are likely candidates, especially if decay rates are known to be large in comparsion to transport time scales.
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118 Local dispersion was neglected, and local dispersion certainly has a significant effect on the local concentration variability. However, its neglect does not appear to diminish the results here to the point of uselessness. The nonstationarity of equalarea Lagrangian trajectories in a steady flow field was neglected. For highly heterogeneous conductivity fields, low velocity areas are disproportionately sampled with respect to the flow average velocity. For a spatially uniform decay process, this would tend to underestimate the plume extent, since the "observed values" will tend to be associated with long travel times. It should be noted that this sampling bias is intrinsic to multilevel samplers. The correlation of velocity and decay coefficient was neglected. A strong negative correlation could result in a significant underprediction of the center of mass in a highly heterogeneous field. Again, it should be noted that this is partially due to the sampling bias assumed by the equalweight multilevel sampler sampling strategy that will preferentially sample low velocity areas with respect to the flow weighted mean.
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CHAPTER 6 GENERAL CONCLUSIONS Any discipline in which you can publish a paper on boundary conditions is an immature discipline. Sten Berglund 1 Many of the results of this study are quantitatively and qualitatively the same as those of previous studies. For instance, chemical engineers have long known that the "mean residence time" of the "swept volume" of a reactor for a flux injection is L/U, where L is the reactor length and U is the mean filtration velocity. However, hydrologists seek to understand the effects of the heterogeneity of the "reactor" itself and the interplay of the processes as they manifest themselves "between x = 0 and x = L." An old premise served as the conceptual foundation of this work, namely that a system, in this case a heterogeneous flow field, may be observed in two ways: along "straightline," or Eulerian, trajectories and along "natural" or Lagrangian trajectories along hydrodynamic streamlines. The flow field was decomposed into an aggregation of elements, either as area elements that lie between Eulerian trajectories or hydrodynamic streamtubes that lie in between streamlines. Two criteria defined four different collections of the flow field elements, namely that at some definition plane, each element has either an equal area through which water enters the element, or an equal volume of water enters the element per unit time. The imaginative 1 From a conversation with the author about the acceptance of Demmy et al. [1999] for publication. 119
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120 reader might contrive a plethora of trajectory and trajectory weighting schemes. For compactness, the following notation summarizes the four different trajectories AW/ET equalarea weighted Eulerian, or straight line, trajectory AW/LT equalarea weighted Lagrangian, or streamline, trajectory FW/ET equalflow weighted Eulerian, or straight line, trajectory FW/LT equalflow weighted Lagrangian, or streamline, trajectory Myriad methods exist for mapping properties observed along the different trajectories. That is, the value of some property may be observed along the trajectory and recorded at different intervals in space or time. A traditional method is a regular Eulerian gridding that divides the flow field into equallyspaced trajectories and "records" information at regular intervals along the trajectory. Two specific parameterizations are considered: equal intervals in displacement along the mean flow direction and equal intervals in advective travel time. Employing these concepts, the behavior of different flow field properties was examined. These properties, such as log conductivity and local velocity, were "observed" along the different trajectories at "control planes," or planes equally spaced in the mean flow direction, or "control times," or at the displacement that an indivisible fluid parcel would be after traveling for some reference time. For a steady and irrotational flow field resulting from a constant mean gradient applied across a stationary Eulerian log conductivity field of uniform and constant mobile water content, the trajectories for which the statistics certain properties appeared to be secondorder stationary were identified. That is, the mean and variance of the property appeared to have a constant value at all observation points along the trajectory. The log conductivity and velocity observed at control planes appeared stationary for the equallyspaced Eulerian trajectories (AW/ET) and Lagrangian trajectories separated by an
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121 equal discharge (FW/LT). For equallyspaced control times, only Lagrangian trajectories equallyspaced at a reference plane (AW/LT) appeared to exhibit stationary log conductivity and velocity statistics. The stationarity of the properties greatly simplifies the development of predictors of solute transport. When taking the expected value of the integral of some property along a trajectory, say the integral of velocity in time, the timevarying velocity function may be replaced with its time average. As implied by the stationarity of the statistics along some trajectory collections, but not others, the numerical statistics of properties varied between the different collections. Adopting simple and accurate models for the statistical properties of some property in a reference collection, the statistics of the properties along other collections, including the observed nonstationarities, were well predicted. This had significant application in the evaluation of the effect of injection mode upon solute transport. Two methods by which solute might be uniformly introduced into a system were considered. The first was a resident injection were a fixed volume at the system inlet was uniformly filled with solute at some reference concentration. The second was to allow influent water at some reference concentration to carry the mass into the system. The displacement and travel time statistics of AW/LT correspond to solute transport associated with a uniform resident injection. The transport of solute injected in flux is described by the displacement and travel time statistics of the FW/LT. The socalled consistent firstorder approximations of the inverse velocity were shown to be built upon a flawed premise, but are nonetheless quantitatively correct and robust. The premise is that the Lagrangian flow field may be accurately approximated by the Eulerian flow field. While this may be true for the weakest of heterogeneities, this approximation breaks down rapidly with increasing heterogeneity if it is consistently applied. As inconsistently formulated, the approximation
PAGE 137
122 < 1/v >~ 1/U is quantitatively equal to the inverse of the harmonic mean of the Lagrangian velocity < 1/v >= l/V h Moreover, the timeaverage Lagrangian velocity is quantitatively equal to the space average Eulerian velocity, and these equalities contribute to the robustness of other approximations. Were the Lagrangian field to be consistently approximated by the Eulerian, travel times would be greatly overpredicted and displacements greatly underpredicted. It is, in fact, the consistent inconsistent approximation of the Lagrangian by the Eulerian that contributes to much of the observed robustness of the Lagrangian transport theory. While not many "new" equations were proposed or so many new phenomena were unearthed, it is hoped that a modicum of understanding was fostered. This is the primary contribution of this work. These concepts were illustrated with an analysis of the transport of an solute experiencing a linear equilibrium sorption in a field characterized by a variable sorption coefficient. While the process has been studied before, it is important to illustrate the new concepts and approaches here with familiar examples. The sorption study was restricted to transport as characterized by mass breakthrough at control planes. An interesting new result is that injection mode strongly affects the propagation of the mean breakthrough time. A uniform resident injection exhibits a nonlinearity in the mean, and this nonlinearity is predictably enhanced by negative correlation between the sorption coefficient and the underlying log conductivity field. Injection in flux, however, results in a linear propagation of mean arrival time in space, and mitigates the effect upon correlation. Arrival time variance, however, is dominated by the correlation between sorption coefficient and the log conductivity field. Injection mode has comparatively little effect. This has profound implications for tracer test analyses based upon the arrival time variance of breakthrough curves. The concepts developed in this work are further illustrated with the development and analysis of an estimator of the center of mass of a plume resulting from
PAGE 138
123 continuously injected solute subject to a firstorder decay. The assumption of noninteracting streamtubes with constant "effective" properties allows solution of the advectiondecay equation for a single streamtube. An ergodiclike hypothesis allows this solution to a distribution of streamtubes. This is what might be termed a "classic" streamtube approach. While surprisingly robust and perhaps worthy of application in field settings, the estimator shows a bias explained using the trajectory based analysis developed in this work. One measure of quality of a work is the number of questions that it answers. Perhaps another is the number of interesting questions that it begs. Several issues are left for further study. A few specific questions are addressed first, then a few more general. The relationship between an Eulerian "reference" system and its Lagrangian counterpart remains incompletely understood. A primary area that requires attention is the increasing Lagrangian velocity correlation length with increasing log conductivity variance. A great deal of work remains for predicting the spatial moments of a reactive solute. Numerical constraints limited this study to relatively mild heterogeneity. However, a highly heterogeneous medium, might be replaceable by two or more media characterized by lesser heterogeneity, and a systematic "replacement system" would greatly enhance computational efforts. The turning bands method employed in the generation of the random fields was computationally expensive, in comparison to, say, the particle tracking algorithms or data analysis programs, and to the flow solver for weak heterogeneity. The method requires the generation of several independent normal line processes. This problem is wellsuited to parallel and distributed computational techniques, and an easytouse implementation would certainly find wide use. Several research groups are working on parallel and distributed flow codes, and the research community would benefit greatly from a wider dissemination of or greater access to these codes.
PAGE 139
124 Thinking more broadly, some of the concepts employed here might find application in other fields. The flow of water through a wetland system might be reasonably approximated as a twodimensional aquiferlike object. Transport of contaminants might be modeled using similar techniques, especially if the transport time scales are relatively short compared to system variability. In general, heterogeneity will imply some sort of "preferential flow," and the identification of preferential flows in heterogeneous systems should be paramount consideration in the design of monitoring systems.
PAGE 140
APPENDIX A EULERIAN VELOCITY DISTRIBUTION PROPERTIES Consider lognormally distributed random variable V with moments < V >= U and var(V) = S. These moments are related to the parameters \i and a 2 via U = exp(At +
PAGE 141
126 The effective conductivity of a porous medium may be denned as the numerical conductivity value that satisfies the Darcy relationship Q = K e J (A.9) where Q is the specific discharge and J is the difference in hydraulic head across the formation. There are several expressions for the effective conductivity of a stationary heterogenous conductivity fields (see Gelhar [1993]). The general form of the effective conductivity is K e = K g f(al k ,g) (A.10) where K g and af nk are the geometric mean and the variance of the log conductivity field, respectively, g is a function of aquifer anisotropy, and / is a function that scales the geometric mean to the proper effective value. For a twodimensional isotropic aquifer, f{cr 2 nk ) = 1. Thus, for an aquifer of homogeneous mobile water content 6, an estimate of the spatial mean seepage velocity U is given by U= I Y (A.11) The point velocity variance for such a field is S = bU 2 a 2 nk where b = 3/8. Substitution of Equation (A. 11) into this expression yields bK 2 J 2 o 2 S= 9 Q2 (A.12) Substitution of Equations (A. 11) and (A.12) into Equations (A. 7) and (A. 8) 0y/TTbal k a 2 = ln(l + ba 2 nk ) (AAA) I* = ]n \ Â—jÂ£sL= ] (A.13)
PAGE 142
APPENDIX B VELOCITY LOG CONDUCTIVITY CORRELATION A simplified correlation relationship between the velocity field and the log conductivity is sought. Graham and McLaughlin [1989b] and Rubin and Dagan [1992] derived expressions based upon perturbation expansions of Darcy's law and spectral methods. Graham and McLaughlin [1989b] assume a "holetype" covariance function for the log conductivity field, in order to assure a complementary stationary head gradient field. Rubin and Dagan [1992] assume an exponentially correlated anisotropic log conductivity field. The velocitylog conductivity covariance functions for both analytical cases shows qualitatively similar behavior (see Figure B.l), and an simplified estimator would be applicable for both log conductivity correlation models. Although these rigorouslydeveloped estimators are attractive theoretically, the mundane and oftneglected task of mapping theoretical results into application often requires simplifying assumptions. Presented for your consideration is the covariance function of Graham and McLaughlin [1989b] (their Equation (B5) where a = 7r/(4A/) and K { is the modified Bessel function of order i. The integral scale of this covariance function is A/. An exponential approximation to this function that reproduces the integral scale, general trend, and point covariance of that given by Graham and McLaughlin [1989b] is sought. A function of the form (a/2) (C#i[aC]aC 2 ffo[aC])} (B.l) 29 exp[x/(2A)] (B.2) 127
PAGE 143
128 Figure B.l: Comparison of velocity and log conductivity correlation functions from Graham and McLaughlin [1989b] and Rubin and Dagan [1992] and a simplified exponential model. All functions are normalized by K g Jaf nk /9. The apparent "roughness" of the Rubin and Dagan [1992] function is due to estimation errors associated with inferring their function from graphical data.
PAGE 144
129 meets these criteria (see Figure B.l). The covariance function presented by Rubin and Dagan [1992] is the sum of three twofunction products that involve integrations of Bessel functions. For twodimensional porous systems, the simplified function condenses that monumental work into an elegant and concise form. The simplified expression is tested against correlation functions estimated from the velocity and log conductivity data taken from the simulations (see Figure B.2). The simplified estimator gives a reasonable a priori estimate of the velocity and 1 1/2 x Eq. B.2 'xt + <* + x x + I I I 0 5 10 15 20 r/X } Figure B.2: Comparison of the simplified velocitylog conductivity correlation function to simulation data for aj = {1/2, 1}. log conductivity correlation observed along the equalflux Lagrangian trajectories. The reasonable match to analytical expressions and the observed data enhance our confidence in this simplified estimator. The results of Rubin and Dagan [1992] (their Figure 7) indicate that anisotropy decreases the point covariance and increases the long range correlation effects. Yet 0.8 0.6 b ^ 0.4 b 0.2 _n o
PAGE 145
130 the functional form appears to remain exponential. Therefore, it is suggested that future work focus upon generalizing the simple results found here to threedimensions and general anisotropy.
PAGE 146
APPENDIX C CONSISTENT FIRST ORDER APPROXIMATION C.l Travel Time Usage of a consistent firstorder approximation for travel time permeates the stochastic Lagrangian literature. This approximation is quantitatively correct for collections of equalflux streamtubes. However, the premise upon which it is built is incorrect. Consider a stationary lognormal process V. The lognormal distribution is selected for convenience, as a random variable with values guaranteed to be greater than zero is required. Since V is stationary by definition, then so is 1/V. Consider the expected value of the integral The order of evaluation may be interchanged, taking the expected value of the argument of the integral thus: where Vh is the point harmonic mean of the random process V. Consider now a perturbation expression for the random process V = V(l + v) where V is the mean of V, and v is a zeromean random process representing the perturbation of the process about its mean. Inserting this expression into Equation (C.l) yields (C.l) DO (C.2) = x/V h (C.3) 131
PAGE 147
132 Expand (1 + v) in a series, 1 f = T j <{1v[x] + v[x] 2 )>dx (C.4) v Jo Should this expression be truncated at first order, the leading term < r x/V is left. However, from Equation (C.2), the exact expression is < r >= x/V h < r >= x/V is a wellestablished result in the literature for the first temporal moment of a solute plume injected in flux Kreft and Zuber [1978]. The answer to this riddle lies in the examination of the properties of the flow field. Even if the deviations are relatively small from the straightline, or Eulerian, trajectory, the statistics of the properties recorded along the actual, or Lagrangian, trajectory can be quite different. For instance, the mean velocity recorded along equalflow weight streamlines in an quasiinfinite constant mean gradient flow in a exponentially correlated lognormal log conductivity field is approximately V = (1 + baf nk )U where U is the spatially averaged Eulerian velocity. The variance of the the velocity perturbation along these trajectories is approximately o\ = baf nk Evaluating the expected value in Equation (C.4) and retaining the second order term yields =t(l+ol) (C.5) Substituting our approximate values into the preceding equation yields \ l + h(J Lk) u (C.6) = x/U The socalled smalldeviation assumption is a poor one. The approximation of the Lagrangian field by the Eulerian field works because the correlation properties are quite similar and the "coincidence" that the harmonic mean Lagrangian velocity is approximately equal to the arithmetic mean Eulerian velocity.
PAGE 148
133 C.2 Displacements The center of mass of a solute plume uniformly injected in space is known to move = Ut, where U, again, is the spatially averaged Eulerian velocity. If the solute parcels move along Lagrangian trajectories, and the statistics of Lagrangian and Eulerian fields are known to be quite different, why does this work? The answer is that the temporally averaged velocity of equalarea Lagrangian streamtubes is stationary in time and equal to the spatially averaged Eulerian velocity. In fact, the temporally averaged Eulerian velocity is lower than its spatially averaged counterpart. So once again, the smalldisplacement assumption, if consistently applied, would result in a very poor result. What is remarkable is that the correlation properties of the two fields are so similar, and herein lies the strength of the approximations of Lagrangian fields with Eulerian fields.
PAGE 149
REFERENCES Annable, M. D., J. W. Jawitz, P. S. C. Rao, D. P. Dai, H. K. Kim, and A. L. Wood. Field evaluation of interfacial and partitioning tracers for characterization of effective naplwater contact areas. Ground Water, 36(3):495502, 1998. Bakr, A. A., L. W. Gelhar, A. L. Gutjahr, and J. R. MacMillan. Stochastic analysis of spatial variability in subsurface flows 1. Comparison of oneand threedimensional flows. Water Resour. Res., 14(2):263271, 1978. Bear, J. Dynamics of Fluids in Porous Media. Dover Publications, Inc., New York, 1972. Bellin, A., Y. Rubin, and A. Rinaldo. EulerianLagrangian approach for modeling of flow and transport in heterogeneous geological formations. Water Resour. Res., 30 (ll):29132924, 1994. Bellin, A., P. Salandin, and A. Rinaldo. Simulation of dispersion in heterogeneous porous formations: Statistics, firstorder theories, convergence of computations. Water Resour. Res., 28(9):22112227, 1992. Boggs, J. M., S. C. Young, L. M. Beard, L. W. Gelhar, K. R. Rehfeldt, and E. E. Adams. Field study of dispersion in a heterogeneous aquifer 1. Overview and site description. Water Resour. Res., 28(12):32813291, 1992. Cvetkovic, V. D., H. Cheng, and X.H. Wen. Analysis of nonlinear effects on tracer migration in heterogeneous aquifers using Lagrangian travel time statistics. Water Resour. Res., 32(6):16711680, 1996. Cvetkovic, V., G. Dagan, and H. Cheng. Contaminant transport in aquifers with spatially variable hydraulic and sorption properties. Proc. R. Soc. London Ser. A, 454:21732207, 1998. Dagan, G. Stochastic modeling of groundwater flow by unconditional and conditional probabilites 1. Conditional simulation and the direct problem. Water Resour. Res., 18:835848, 1982a. 134
PAGE 150
135 Dagan, G. Stochastic modeling of groundwater flow by unconditional and conditional probabilites 2. The solute transport. Water Resour. Res., 18:813833, 1982b. Dagan, G. Solute transport in heterogeneous formations. J. Fluid Mech., 145:151177, 1984. Dagan, G. Flow and Transport in Porous Formations. SpringerVerlag, New York, 1989. Dagan, G. and V. Cvetkovic. Spatial moments of a kinetically sorbing solute plume in a heterogeneous aquifer. Water Resour. Res., 29(12):40534061, 1993. Dagan, G., V. Cvetkovic, and A. Shapiro. A solute flux approach to transport in heterogeneous formations 1. The general framework. Water Resour. Res., 28(5): 13691376, 1992. Demmy, G., S. Berglund, and W. Graham. Injection mode implications for nonreactive solute transport in porous media: Analysis in a stochastic Lagrangian framework. Water Resour. Res., 35:19651974, 1999. Domenico, P. A. and F. W. Schwartz. Physical and Chemical Hydrogeology. John Wiley and Sons, New York, 1990. Freeze, R. A. A stochasticconceptual analysis of onedimensional groundwater flow in nonuniform homogeneous media. Water Resour. Res., 11(5):725741, 1975. Gelhar, L. W. Stochastic Subsurface Hydrology. PrenticeHall, Inc., Englewood Cliffs, New Jersey, 1993. Gelhar, L. W. and C. L. Axness. Threedimensional stochastic analysis of macrodispersion in aquifers. Water Resour. Res., 19(1):161Â— 180, 1983. Graham, W. and D. McLaughlin. Stochastic analysis of nonstationary subsurface solute transport 1. Unconditional moments. Water Resour. Res., 25(2):215232, 1989a. Graham, W. and D. McLaughlin. Stochastic analysis of nonstationary subsurface solute transport 2. Conditional moments. Water Resour. Res., 25(11):2331Â— 2355, 1989b. James, A. I., W. Graham, K. Hatfield, P. S. C. Rao, and M. D. Annable. Optimal estimation of residual nonaqueous phase liquid saturations using partitioning tracer concentration data. Water Resour. Res., 33(12) :26212636, 1997.
PAGE 151
136 Jawitz, J. W. Aquifer contaminant source zone characterization with partitioning tracers and remediation with singlephase microemulsion flushing. PhD thesis, University of Florida, Gainesville, Florida, 1999. Jin, M., M. Delshad, V. Dwarakanath, D. C. McKinney, G. A. Pope, K. Sepehrnoori, and C. E. Tilburg. Partitioning tracer test for detection, estimation, and remediation performance assessment of subsurface nonaqueous phase liquids. Water Resour. Res., 31(5):12011211, 1995. Journel, A. and C. Huijbregts. Mining Geo statistics. Academic Press, New York, 1978. Jury, W. A. and K. Roth. Transfer Functions and Solute Movement Through Soil. Birkhauser Verlag, Basel, 1990. Konikow, L. F., D. J. Goode, and G. Z. Hornberger. A threedimensional methodofcharacteristics solutetransport model (MOC3D). Waterresources investigations report 964267, United States Geological Survey, Restion, Virginia, 1996. Kreft, A. and A. Zuber. On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions. Chem. Eng. Sci., 33: 14711480, 1978. Maclntyre, W. G., M. Boggs, C. Antworth, and T. B. Stauffer. Degradation kinetics of aromatic organic solutes introduced into a heterogeneous aquifer. Water Resour. Res., 29(12):40454051, 1993. Mood, A. M., F. A. Graybill, and D. C. Boes. Introduction to the Theory of Statistics. McGrawHill Publishing Company, New York, 1974. Mose, R., P. Siegel, P. Ackerer, and G. Chavent. Application of the mixed hybrid finite element approximation in a groundwater flow model: Luxury or necessity? Water Resour. Res., 30(11):30013012, 1994. Musashi, M. A Book of Five Rings. The Overlook Press, Woodstock, New York, 1982. Osnes, H. Stochastic analysis of velocity variability in bounded rectangular heterogeneous aquifers. Adv. Water Resour., 21:203215, 1998. Papoulis, A. Probability, Random Variables, and Stochastic Processes. McGrawHill Publishing Company, New York, 1991. Pollock, D. W. Semianalytical computation of path lines for finitedifference models. Ground Water, 26(6):743750, 1988.
PAGE 152
137 Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes in C. Cambridge University Press, Cambridge, England, 1988. Rubin, Y. Stochastic modeling of macrodispersion in heterogeneous porous media. Water Resour. Res., 26(7):16891697, 1990. Rubin, Y. and G. Dagan. Stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers 1. Constant head bounary. Water Resour. Res., 24(10):16891697, 1988. Rubin, Y. and G. Dagan. Stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers 2. Impervious boundary. Water Resour. Res., 25(4):707712, 1989. Rubin, Y. and G. Dagan. A note on the head and velocity convariances in threedimensional flow through heterogeneous anisotropic porous media. Water Resour. Res., 28(5):14631470, 1992. Shapiro, A. M. and V. D. Cvetkovic. Stochastic analysis of solute arrival time in heterogeneous porous media. Water Resour. Res., 24(10): 171 11718, 1988. Stauffer, T. B., C. P. Antworth, R. G. Young, W. G. Maclntyre, J. M. Boggs, and L. M. Beard. Degradation of aromatic hydrocarbons in an aquifer during a field experiment demonstrating the feasibility of remediation by natural attenuation. Technical report, Armstrong Laboratory, United States Air Force, Tyndall Air Force Base, Florida, 1994. Stauffer, T. B., J. M. Boggs, and W. G. Maclntyre. Ten years of research in groundwater transport studies at Columbus Air Force Base, Mississippi. In Sayler, G., J. Sanseverino, and K. L. Davis, editors, Biotechnology in the Sustainable Environment, Plenum Press, New York, 1997. Theile, M. R., R. P. Batycky, M. J. Blunt, and F. M. Orr, Jr. Simulating flow in heterogeneous systems using streamtubes and streamlines. SPERE, 11:512, 1996. Tompson, A. F. B. Numerical simulation of chemical migration in physically and chemically heterogeneous porous media. Water Resour. Res., 29(ll):37093726, 1993. Tompson, A. F. B., R. Ababou, and L. W. Gelhar. Implementation of the threedimensional turning bands random field generator. Water Resour. Res., 25(10): 22272243, 1989.
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BIOGRAPHICAL SKETCH George Demmy was born to George and Ellen Demmy 23 November 1966 in Lakeland, Florida. He received a Bachelor of Arts in Physics and German Literature from the Florida State University in 1990, a Master of Engineering in Agricultural Engineering from the University of Florida in 1993, and Doctor of Philosophy in Agricultural and Biological Engineering from the University of Florida in 1999. 138
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Wendy Professor of A^ricitltura^and Biological Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. P.S.C Rao/ CoChair Lee A. Rieth Distinguished Professor of Environmental Engineering Purdue University I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Kenneth L. Campbell Professor of Agricultural and Biological Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Kirk Hatfield Associate Professor of Civil Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Michael D. Annable Associate Professor of Environmental Engineering Sciences
PAGE 155
This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1999 M. J. Ohanian Dean, College of Engineering Winfred M. Phillips Dean, Graduate School
CHAPTER 6
GENERAL CONCLUSIONS
Any discipline in which you can publish a paper on boundary conditions
is an immature discipline. Sten Berglund1
Many of the results of this study are quantitatively and qualitatively the same
as those of previous studies. For instance, chemical engineers have long known that
the mean residence time of the swept volume of a reactor for a flux injection is
L/U, where L is the reactor length and U is the mean filtration velocity. However,
hydrologists seek to understand the effects of the heterogeneity of the reactor itself
and the interplay of the processes as they manifest themselves between x = 0 and
x = L.
An old premise served as the conceptual foundation of this work, namely that
a system, in this case a heterogeneous flow field, may be observed in two ways:
along straightline, or Eulerian, trajectories and along natural or Lagrangian
trajectories along hydrodynamic streamlines. The flow field was decomposed into an
aggregation of elements, either as area elements that lie between Eulerian trajectories
or hydrodynamic streamtubes that lie in between streamlines. Two criteria defined
four different collections of the flow field elements, namely that at some definition
plane, each element has either an equal area through which water enters the element,
or an equal volume of water enters the element per unit time. The imaginative
1 From a conversation with the author about the acceptance of Demmy et al. [1999]
for publication.
119
31
FW/LT
AW/LT
FW/ET
AW/ET
Eq. 2.37
Eq. 2.55
Eq. 2.57
Eq. 2.13
FW/LT
AW/LT
FW/ET
AW/ET
Eq. 2.37
Eq. 2.55
Eq. 2.57
Eq. 2.13
^/^ln k
x
Figure 2.11: Comparison of the positiondependent velocity variance estimators to
data for the four collections of trajectories for afnk = 1/2 (a) and c>fnA. = 1 (b). The
variances are normalized by U2bafnk.
27
FW/LT
AW/LT
FW/ET
AW/ET
est AW/LT
X
X
FW/LT +
AW/LT x
FW/ET *
AW/ET
est AW/LT 
0 2 4 6 8 10
Ut/1
>
o
13
T3
C
o
o
O)
o
c
CO
CD
E
0.4 
0.2 
* +
+
X
X
X
X
t&tSZ&SM ++.+.+..:..T+t^x+*++.H;..
fTr *XXxx xxxxxXxXxxxxxx**?
xxxxxxxXxx x
0.2 
w
X
a_
0.4 b
**
qd
Q.
dQqb *******x
xPa?DOorf^^95fi^gg<
Figure 2.8: Mean log conductivity as function of travel time to control planes along
mean flow direction for the four different trajectory collections and afnk = 1/2 (a)
and a\nk = 1 (b)
100
control structure, a solute parcel will trace different volumes for different processes
and correlation structures.
The reaction parameter model is comprised of two random processes. The first
is that of the log conductivity field and the second is a random component that
has the same distributional properties as, but is otherwise uncorrelated to, the log
conductivity field. In the case of the temporal moments, the weighted covariance
functions of the log conductivity and this random component were employed directly.
These functions can be parameterized, of course, in terms of the nonreactive travel
time. However, these travel time parameterized covariance functions time are not
immediately applicable to sorption analysis in the way that covariance parameterized
in clock time would be, because the displacement for a given clock time is highly
dependent upon the correlation of the reaction parameter to the log conductivity
field. This is not to say that sorptiondependent spatial moment analysis is not viable
in this framework. In fact, Cvetkovic et al. [1998] analyzed the spatial moments
of a solute subject to nonlinear equilibrium sorption in a heterogeneous sorption
field in this framework. However, the sorption parameter and the underlying log
conductivity field were uncorrelated. Further clock time oriented are developments
left for future work. Expressions involving the travel time dependent reaction flow
path are developed and applications to and implications for first order decay are
discussed.
Continuing the theme of this dissertation, expressions based upon the trajectory
based covariance functions are sought. The timedependent reaction flow path for
the selected reaction parameter model is related to the log conductivity covariance
function, as shall be demonstrated in the following.
As in the previous chapter, transport along Lagrangian trajectories is considered.
Assumed are secondorder stationary velocities along areaweighted trajectories in
time and secondorder stationary velocities along flowweighted trajectories in space.
61
Euleriantrajectory velocity variance estimated from the input parameters to the sim
ulations. That is, by
u2 f< = (KgJ/e)2bolh (2.74)
As with the mean velocity estimators, the variance estimators employ only the sim
ulation input parameters.
Once again, the areaweighted Euleriantrajectory variance exhibited its pre
sumed behavior, and supports the well established fact that the Eulerian velocity
field is fairly well understood in space. The hypothesized flowweighted Lagrangian
trajectory velocity variance is somewhat higher that the data for both
reason is the negative correlation between log conductivity and head gradient is not
weighted properly in the collection of trajectories skewed towards higher log conduc
tivity values.
The other two estimators also suffer from this bias in the asymptotic Lagrangian
trajectory velocity variance, although the transition between initial and final values
for both seems to be well characterized.
2.6.3 Mean Velocities in Time
Figure 2.13 presents the timedependent mean velocities for the four different col
lections of streamtubes. Again, the mean velocities are normalized by U as estimated
from the simulation input parameters. Only the areaweighted Lagrangiantrajectory
mean velocity exhibited a constant mean behavior in time. Its estimated harmonic
mean is U, and the estimate seems good, although one might argue that there is a
slight overpredictive bias.
The estimated harmonic mean of Eulerian trajectories appear to have a stronger
under predictive bias. The transition of the flowweighted Lagrangiantrajectory
appears to be captured better than those of the Eulerian trajectories. It should be
noted that the Lagrangian trajectory transition was based upon an expression for
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Wendy JJrailam,
Professor of Agrie
gineermg
and Biological En
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
P.S.C Rao/ CoChair
Lee A. Rieth Distinguished Professor of
Environmental Engineering
Purdue University
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Kenneth L. Campbell
Professor of Agricultural and Biological En
gineering
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Kirk Hatfield
Associate Professor of Civil Engineering
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Michael D. Annable
Associate Professor of Environmental En
gineering Sciences
94
*r/^ln k
Figure 4.7: Correlation between the reaction flow path and travel time for different
values of correlation parameter Â¡3 and different trajectory collections for the ofnA. = 1
set of simulations. Top figure is ft = {1,41}. Bottom figure is /3 = { 1/2, 41/2}.
Trajectory collections are areaweighted Lagrangian (AL) and flowweighted La
grangian (AF).
124
Thinking more broadly, some of the concepts employed here might find application
in other fields. The flow of water through a wetland system might be reasonably
approximated as a twodimensional aquiferlike object. Transport of contaminants
might be modeled using similar techniques, especially if the transport time scales
are relatively short compared to system variability. In general, heterogeneity will
imply some sort of preferential flow, and the identification of preferential flows in
heterogeneous systems should be paramount consideration in the design of monitoring
systems.
LIST OF FIGURES
Figure page
2.1 Schematic representation of an aquifer 6
2.2 Simple subdivision of the flow field by one of four strategies: A) equal
area straight line, B) equalarea streamline, C) equalflow streamline,
D) equalflow straight line 9
2.3 Four different partitioning strategies for the same flow field: A) area
weight Eulerian trajectory, B) area weight Lagrangian trajectory,
C) flow weight Lagrangian trajectory, D) flow weight Eulerian
trajectory 11
2.4 Log conductivity distribution from a 41400 node realization generated
by the turningbands method compared to the standard normal dis
tribution. Target distribution parameters are fi\nk = 0 and ofnk = 1. 16
2.5 Estimated log conductivity longitudinal covariance estimated from a
realization generated by the turningbands method compared to an
exponential covariance function. Variance of log conductivity =
1 and correlation length X\nk = 1. Spacing for log conductivity values
was Ainfc/5 17
2.6 Mean log conductivity as function of displacement in mean flow direc
tion for the four different trajectory collections and ofnfc = 1/2 (a)
and lk = 1 (b) 24
2.7 Log conductivity variance as function of displacement in mean flow
direction for the four different trajectory collections and = 1/2
(a) and
Eulerian log conductivity field ofnfc 25
2.8 Mean log conductivity as function of travel time to control planes along
mean flow direction for the four different trajectory collections and
\nk = V2 (a) and afnk = 1 (b) 27
2.9 Log conductivity variance as function of travel time to control planes
along mean flow direction for the four different trajectory collections
and ofnA. = 1/2 (a) and a?nk = 1 (b). Variances are normalized by
target variance of Eulerian log conductivity field cr?nk 28
x
110
A normalized moment of order n is a raw moment of order n divided by the zeroth
moment, and is denoted by dropping the prime (i.e., (j,n). The zeroth spatial moment
of this concentration curve is given by
lilt] = j[l exp[kt\]
(5.4)
The zeroth spatial moment of a contaminant plume related to the total mass in
solution. The first normalized spatial moment, or center of mass, is given by
Hi[t] =
'nil exp[kt](kt + 1)
.hi 1 exp[/ci]
(5.5)
In the limit t oo fix = v/k.
The concentration given in Equation (5.2) is a function of this asymptotic center
of mass v/k. Solving Equation (5.2) with h[] = 1 for v/k yields
v x
k ln[c/c0]
(5.6)
Thus, an observation of the concentration at any point of a steady streamtube con
centration curve is an observation of its center of mass, given that c0 and x are
known.
Assume that independent samples can be drawn uniformly from the collection of
streamtubes that originate from a source c0 located at x = 0. From the linearity of
first moments, the mean center of mass may be estimated from N samples by the
simple estimator
x
1 N
= ~y
/V
Xi
N jy/ In [cj/co]
(5.7)
5.2.2 Relative Degradation
Consider two solutes a and b subject to a firstorder decay process, characterized
by rate constants ka and kb, respectively. The relative degradation rate of solute a to
104
c
o
o
o
0.8
0.6
2 0.4
0.2
0.2
%
\\\
+\\
W\
V?*
1/2
1
Lagrangian In k
Eulerian In k 1/2
Eulerian In k 1
0 
i'4x_*xxxxxxXxxx>
*4:+^=E i
.xx xxxxxxxvxx ~ v,.
4 6
Ut/X\nk
10
Figure 4.10: Hypothesized timedependent Eulerian log conductivity covariance func
tions for ofnk {1/2,1} compared to simulation data and Equation (4.29).
A
a.
V
Ut/X\nk
Figure 4.11: Mean reaction flow path as a function of nonreactive travel time com
pared to estimator Equation (4.25) for correlation parameter values /3 = { 1,0,1}.
10
This is an equal partitioning with respect to the flow field, or the stream function.
This sort of partitioning arises quite naturally in fluid mechanics and in numerical
modeling of heterogeneous flow systems using streamtube approaches (see, e.g., Theile
et al. [1996]). The flow weight of each subdomain is equal, although the volume
weights are different.
A fourth scheme would be to extend a line from (0,yo) to (Lx,y0) (curve D Fig
ure 2.2). In this case, the subdomains share only equal flow rates at x 0. In
subsequent discussions, the phrase Eulerian trajectory will refer to such a straight
line trajectory, and the phrase Lagrangian trajectory will refer to trajectory of a
streamline. The phrase area weight will refer to an equal area partitioning scheme
at a definition plane, and the phrase flow weight will refer to an equal discharge par
titioning scheme. At this point, it is important to note that no mention has been
made of boundary conditions as they relate to transport equations. These partition
ing schemes are simply ways of subdividing a flow field. An area weight Eulerian
trajectory scheme corresponds to traditional spectral approaches that assume small
deviations about a mean trajectory. The use of trajectory is quite general, and does
not imply a natural path or streamline.
2.2.2 General Subdivision
Consider dividing the flow field into N partitions according to the strategies de
scribed (see Figure 2.3). Theile et al. [1996] demonstrated that streamlines may
represent streamtubes. Therefore, the remainder of this discussion will focus upon
streamline properties, with the implication that these properties may be extended to
streamtubes as well. A primary purpose of subdividing the domain is to represent the
continuum of domain properties with a discrete set of data. A regular finitedifference
grid may be thought of as a set of points at equal spacing in space along a collection
of areaweighted Eulerian trajectories. An alternative method would be to create
spacings based upon travel times along the trajectory, where travel time is defined as
99
representation of the myriad interacting processes by the firstorder perturbation ex
pansions.
Injection of solute in the influent fluid mitigates the effect of correlation on the
mean and the nonlinearity of mean nonreactive travel time. For tracer experiments
that determine the volume or area of some property based upon mean arrival times
of reactive and nonreactive tracers, correlation will have little effect upon the first
temporal moment of the solute breakthrough curve if and only if the solute is uni
formly injected in flux. However, injections that preferentially favor lowflux areas
may exhibit strong nonlinearities, and be particularly sensitive to negative correla
tions between the local conductivity and reaction parameter.
The contribution of the /x variance to the second moment of the reactive solute
travel time is extremely sensitive to correlation between the reaction parameter and
the conductivity. Experiments that rely upon higher order moments of tracer ex
periments are, therefore, questionable if there is no plnfc correlation information.
Whereas injection mode dominates the correlation behavior in the mean, correlation
dominates injection mode in the reactive travel time variance.
4.7 Spatial Moments
Symmetry would dictate proceeding with a corresponding spatial moment analysis
for transport of solute subject to a linear equilibrium sorption in a heterogeneous
reaction parameter field. However, this process presents a particular complication
that thwarts the development of relatively simple expressions following the same
methods those previously employed. In the case of the travel time statistics, the
control structures are two control planes. A fluid and solute parcel travels between
these two control planes, sweeping out a fixed and constant volume, without regard to
correlation properties, reaction parameters, or processes. The time required to sweep
this volume, of course, is highly dependent upon these factors. For a clock time
135
Dagan, G. Stochastic modeling of groundwater flow by unconditional and conditional
probabilites 2. The solute transport. Water Resour. Res., 18:813833, 1982b.
Dagan, G. Solute transport in heterogeneous formations. J. Fluid Mech., 145:151
177, 1984.
Dagan, G. Flow and Transport in Porous Formations. SpringerVerlag, New York,
1989.
Dagan, G. and V. Cvetkovic. Spatial moments of a kinetically sorbing solute plume
in a heterogeneous aquifer. Water Resour. Res., 29(12):40534061, 1993.
Dagan, G., V. Cvetkovic, and A. Shapiro. A solute flux approach to transport in
heterogeneous formations 1. The general framework. Water Resour. Res., 28(5):
13691376, 1992.
Demmy, G., S. Berglund, and W. Graham. Injection mode implications for non
reactive solute transport in porous media: Analysis in a stochastic Lagrangian
framework. Water Resour. Res., 35:19651974, 1999.
Domenico, P. A. and F. W. Schwartz. Physical and Chemical Hydrogeology. John
Wiley and Sons, New York, 1990.
Freeze, R. A. A stochasticconceptual analysis of onedimensional groundwater flow
in nonuniform homogeneous media. Water Resour. Res., 11 (5):725741, 1975.
Gelhar, L. W. Stochastic Subsurface Hydrology. PrenticeHall, Inc., Englewood Cliffs,
New Jersey, 1993.
Gelhar, L. W. and C. L. Axness. Threedimensional stochastic analysis of macrodis
persion in aquifers. Water Resour. Res., 19(1):161180, 1983.
Graham, W. and D. McLaughlin. Stochastic analysis of nonstationary subsurface
solute transport 1. Unconditional moments. Water Resour. Res., 25(2):215232,
1989a.
Graham, W. and D. McLaughlin. Stochastic analysis of nonstationary subsurface
solute transport 2. Conditional moments. Water Resour. Res., 25(11):23312355,
1989b.
James, A. L, W. Graham, K. Hatfield, P. S. C. Rao, and M. D. Annable. Opti
mal estimation of residual nonaqueous phase liquid saturations using partitioning
tracer concentration data. Water Resour. Res., 33(12):26212636, 1997.
LIST OF TABLES
Table page
2.1 Comparison of target log conductivity values and observed values along
areaweighted Eulerian trajectories 26
2.2 Comparison of target log conductivity values for areaweighted Eulerian
trajectories in space and observed areaweighted Lagrangian trajec
tories in time 29
2.3 Comparison of a priori Eulerian velocity variance to observed values.
The decimal representation of the exact fractional values of the pre
dicted variance are provided for convenience 35
2.4 Comparison of observed log hydraulic conductivity variances to estima
tors. Observed value is zeroslope line regressed through data. The
decimal representation of the fractional value is included for conve
nience 63
5.1 The center of mass in meters of steady plumes estimated by traditional
trapezoidal integration of spatial concentration data and by the mean
v/k estimator Equation (5.7) 116
5.2 Degradation rates relative to naphthalene and absolute degradation
rates in reciprocal days as estimated by Equation (5.9) and as re
ported by MacIntyre et al. [1993]. NR denotes results not reported
in MacIntyre et al. [1993]. Naphthalene degradation reported to be
0.0064 d1 117
IX
52
Expand the denominator in a series to second order
va[x] =
(2.43)
= U( 1 + exp[6r/AiJ))
In a similar fashion, an expression is derived for the nonstationary mean velocity
along a flowweighted Euleriantrajectory. In this case, it is the stationary Eulerian
trajectory velocity field that serves as a basis. The mean of the perturbation expan
sion is the Eulerian mean velocity. It is interesting to note that this result requires
no series expansion.
(2.44)
= [/(! + ba?nkexp[br/\ink})
The nonstationary velocity variances along the respective crossweighted trajec
tories are derived in a similar fashion. The general expression for the areaweighted
Lagrangiantrajectory velocity variance is
(2.45)
This equation may be rewritten
(2.46)
Resolve the first term of the preceding equation into perturbation expressions
(2.47)
106
Thus, the mean reaction flow path for a flux injection is given by
<^Â§a>=<^Â§[ nm> (i)
In taking the expected value, the crosscorrelation of the initial velocity and the that
portion of the reaction parameter correlated to the log conductivity field must be
considered. Using nowfamiliar expansions, the expected value of the flowweighted
reaction flow path is
< >= 'W\1+pClu[t)/v)dt
U Jo
(4.32)
where Cfu is the covariance of the initial velocity perturbation and the log con
ductivity perturbation. Once again, this covariance shall be approximated with an
exponential function of the form (see Appendix B)
Ur
Cfu[r) = ^
exp
2A
In k
(4.33)
Substituting this term into Equation (4.32) and integrating yields
< ^H^[r] >=
U
U
(ut + /3a12nfcAinfcexp[
]
2A]nfc
(4.34)
4.7.2 Discussion
Similar to its space dependent counterpart, the mean time dependent reaction flow
path is sensitive to injection mode. However, it is the uniform resident injection that
appears to mitigate correlation effects upon the mean reaction flow path. Correlation
has a strong effect of variance, and it is quite likely that these effects dominate any
injection mode effects, as was the case for the space dependent //.
91
O 10 20 30 40
*^/^ln k
%/'bn k
Figure 4.5: Reaction path variance observed values and estimators for different values
of correlation parameter f3 and different trajectory collections for the
simulations. Top figure is Â¡3 = { 1,+1}. Bottom figure is ^ {1/2,+1/2}. Tra
jectory collections are areaweighted Lagrangian (AL) and flowweighted Lagrangian
(AF).
CHAPTER 4
REACTIVE SOLUTE TRANSPORT
4.1 Introduction
The concepts developed in the previous chapters are employed to analyze reac
tive solute transport. The focus is upon global measures of transport, rather than
predictions of local quantities. Particular processes of interest are injection mode
effects, the effects of correlation of the local reaction parameter to the log con
ductivity field, and the interplay of these injection mode and correlation effects. A
suite of numerical experiments illustrates these effects. Additionally, expressions are
developed for temporal moments of mass breakthrough curves for a solute subject to
linear equilibrium sorption for the different injection modes that reflect the observed
behavior in relatively simple closedforms.
Spatial moments of sorbing solutes are not directly treated in this work. The
experimental design was targeted directly towards a temporal moment analysis, and
this design decision has some implications for spatial moment analysis. These decision
and implications are discussed, in hopes that future works might be guided by this
analysis. Some of the properties of the travel time dependent reaction parameter are
discussed, as are some of the implications for solute transport.
4.2 Temporal Moments
The concepts developed in the preceding chapters are applied to examine the effect
of injection mode upon the temporal moments of the breakthrough curve of a solute
subject to linear equilibrium sorption. A uniform mean gradient induces a steady
irrotational flow in a uniformly saturated porous medium characterized by an expo
nentially correlated lognormal conductivity distribution. This flow field is divided
79
32
in both the mean and variance. The velocity variances exhibit behavior to similar
to that of the means for their corresponding trajectories and quite different than the
behavior exhibited by the log conductivity variances (compare Figures 2.11 and 2.10
and compare Figures 2.11 and 2.7). There is a qualitative similarity between the
nonstationary velocity means and the nonstationary velocity variances. A heuristic
is that the velocity statistic starts at one value in the nearfield and transitions
to an asymptotic value in the farfield. The following sections demonstrate that
these endpoints and the transition are predictable.
The qualitative dissimilarity between the log conductivity variance and velocity
variance is an artifact of examining the log conductivity values on the one hand and
the real space velocity values on the other. Consider the FW/LT log conductivity
mean and variance for the ofnfc = 1 simulation. The FW/LT mean log conductivity
is larger than that of the AW/ET, but the variances are roughly the same value.
In fact, regressing a zeroslope line through the data results in a mean FW/LT
variance that is slightly less than that of the AW/ET. However, the real space
conductivity mean and variance are both functions of both the mean and variance of
the log conductivity process (recall Equation (2.6)). A cursory examination of the
data in conjunction with Equation (2.6) reveal that the mean and variance of the real
space conductivity along FW/LT are both higher than their AW/ET counterparts,
and the magnitude of this difference is proportional to ofnfc.
The mean velocities in Figure 2.10 are normalized by the arithmetic mean veloc
ity U predicted from the input parameters for the flow simulator. That the mean
velocity along AW/ET closely follows 1 is a good indication that the effective con
ductivity expression is accurate. Notice that the FW/LT and farfield AW/LT values
are higher than AW/ET, and the magnitude is proportional to the variance of the
log conductivity (see Figure 2.10). This behavior is intuitively correct. Consider a
84
4.4.1 Reaction Parameter
From these data, trajectorybased P fields were constructed using Equation (4.4).
In this manner, any variety of correlation relationships may be analyzed using the
same set of simulation data. There is a dearth of information about the structure of
sorption fields in natural systems (see, e.g. Tompson [1993] or Jawitz [1999]). Thus,
numerically convenient parameter values were selected. This suite of experiments ex
amined correlation parameter values of /3 = {1, 1/2,0,1/2,1} in log conductivity
fields of variances a^nk = {1/2,1}. The extreme correlation parameters /3 = {1,1}
are arguably physically implausible, yet they are included to demonstrate the effect
of correlation without the noise associated with an uncorrelated component. The
remaining three values, Â¡3 = {1/2,0,1/2} might be considered a range of more plau
sible values. For consistency, the variance of point log reaction parameter was held
constant and equal to that of the log conductivity field by requiring cr[} = (1 /32)afnk.
The expected value of P was 1. This is a slightly different approach than that taken
with the log hydraulic conductivity, where the geometric mean was fixed at Kg = 1.
In the case of the twodimensional isotropic log conductivity field, the effective con
ductivity is equal to the geometric mean. The expected value of the reaction value
was fixed < P >= 1, because in some practical applications P is assumed to be lin
early related to some property of interest, and the amount of that property is to
be estimated based upon the temporal moments of linearly sorbing solutes, or tracers
(see e.g., Jin et al. [1995], Annable et al. [1998], and Jawitz [1999]).
The point average of reaction parameter P as observed along a Lagrangian trajec
tory is a function of injection mode, correlation to the log conductivity and the vari
ance of the log conductivity. As with the log conductivity, the equalflux streamtubes
exhibit an apparently stationary mean reaction parameter in displacement along the
mean flow direction, whereas the equalarea streamtubes exhibit distinctly nonsta
tionary behavior that is analogous to that of the log conductivity (see Figures 4.1 and
35
Table 2.3: Comparison of a priori Eulerian velocity variance to observed values. The
decimal representation of the exact fractional values of the predicted variance are
provided for convenience.
parameter predicted observed
^Kfc = l/2) 3/1600=0.001875 0.001871
crl((7\nk = 1) 3/800=0.00375 0.00394
Equation (2.4). The correlation function at each considered lag along the estimated
functions is systematically larger for FW/LT, even though the magnitude of the dif
ference is small. Integrating these functions would result in a slightly larger value
for FW/LT, and thus a larger correlation length. In the case there is no variability
in the log conductivity, Eulerian and Lagrangian trajectories coincide. Moreover, an
equalarea weight corresponds to an equal flow weight. Thus, the estimated correla
tion functions from either set would be identical. From this conclude that there
is some dependence of the correlation length of the FW/LT trajectories upon afnk.
Cvetkovic et al. [1996] observed similar behavior in log velocity correlation functions
estimated from AW/LTs with a stationarity assumption for the nonstationary AW/LT
velocities. Thus, this is an slight extention of this work by identifying the station
ary streamlinebased velocity field from which to work, and work from that field.
Moreover, this supports their findings of an increasing correlation length with ofnfc.
Understanding the relationship between the variability of the log conductivity field
and the correlation length of the associated velocity field is of fundamental impor
tance to completely understanding the nature of flow in heterogeneous media. Sadly,
these secrets remain uncovered and must be left this to future research efforts.
The velocity means as a function of time exhibit qualitatively similar behavior
as do the log conductivity means in time (see Figure 2.13 and compare Figure 2.8).
Again, only the AW/LT exhibits stationary behavior in time for both the mean and
variance (see Figure 2.14) as did the AW/LT log conductivity values. The mean
and variance of the AW/LT in time are approximately those of the AW/ET in space
83
Two temporal moments of interest for the influx breakthrough curve are the mean
tif = + < fx> (410)
and the variance
h f + 2 oTil (411)
For uniform resident injection, the moments are
t\f =< Ut/v[ 0] > + < Ufi/v[ 0] > (4.12)
and the variance
hf 2 [0]>2 (413)
From these equations, it is seen that the mean and variance of the expected reactive
travel time distribution is comprised of the first two moments of the nonreactive travel
time and the reaction flow path and the cross moment of the travel time and reaction
flow path.
4.4 Simulation of Reactive Solute Transport
The results of the simulations described in Chapter 2 are employed to simulate the
transport of a reactive solute. As mentioned in that chapter, information was recorded
along two particle trajectories: the first was a natural trajectory corresponding to a
hydrodynamic streamline, and the second was a straight line parallel to the mean
flow direction. Data were recorded at equal increments in time after injection, and
at control planes perpendicular to the mean flow at equal displacements from the
injection plane. Among these data were the velocity, the hydraulic conductivity, and
an observation of a lognormal random field with the same statistical and correlation
properties as the log hydraulic conductivity field.
This dissertation was submitted to the Graduate Faculty of the College of En
gineering and to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
December 1999
.
M. J. Ohanian
Dean, College of Engineering
Winfred M. Phillips
Dean, Graduate School
80
into collections of noninteracting streamtubes. Dispersion at the streamtube scale
is considered to be negligible in its effect on the aggregatescale temporal moments
(e.g., Dagan [1989]). The sorption process is related to a generic reaction parameter
that is heterogeneous and may be correlated to the log conductivity field.
The previous chapters demonstrated the relationship between streamtube selec
tion strategies (equalarea or equalflow) and injection mode (uniform resident and
influx). The reference collection is equalflow streamtubes, since the associated ve
locity and log conductivity statistics are stationary in displacements along the mean
flow direction.
The primary objective of this chapter is to examine the effect of injection mode
upon the temporal moments of the mass breakthrough curve of a solute subject to
linear equilibrium sorption in a heterogeneous flow and sorption field. A brief review
the conceptual model of solute transport is given. General expressions are presented
for the first two temporal moments of the breakthrough curves associated with the
uniform injection of a solute both in the influent fluid flux and in the resident fluid.
The results of the reactive solute transport simulations are presented, followed by the
development of some analytical expressions that help explain the observed behavior.
4.3 Conceptual Overview of Solute Transport
Consider a general multidimensional flow field that is resolved into a collection
of noninteracting streamtubes. Transport along an individual streamtube is charac
terized with a transfer function y[t; x\ (see Jury and Roth [1990] and Cvetkovic et al.
[1998]). For a conservative and nonreactive solute, the transfer function is simply
7[t, x\ = 5 [t t[x]} (4.1)
where 6 is the Dirac delta function and r is the travel time. The physical interpreta
tion of this transfer function is this: a particle located at x = 0 arrives at x X
at time t = r[X]. We conceive of this travel time as not only the time required to
3.2 Comparison of displacement variances for the two injection modes or
streamtube collections. The flowweighted Lagrangian trajectory
(FW/LT) statistics correspond to a flux injection, and the area
weighed Lagrangian trajectory (AW/LT) statistics correspond to a
uniform resident injection. The top figure contains the a?nk = 1/2
data and the bottom figure contains the ofnfc = 1 data. Variances
are normalized by A2n kk 74
3.3 Comparison of mean arrival times for the two injection modes or stream
tube collections. The flowweighted Lagrangian trajectory (FW/LT)
statistics correspond to a flux injection, and the areaweighed La
grangian trajectory (AW/LT) statistics correspond to a uniform res
ident injection. The top figure contains the ofnA. = 1/2 data and the
bottom figure contains the ofnfc = 1 data 76
3.4 Comparison of arrival time variances for the two injection modes or
streamtube collections. The flowweighted Lagrangian trajectory
(FW/LT) statistics correspond to a flux injection, and the area
weighed Lagrangian trajectory (AW/LT) statistics correspond to a
uniform resident injection. The top figure contains the a?nk = 1/2
data and the bottom figure contains the ofnfc = 1 data. Variances
are normalized by (U/(\ink&ink))2 77
3.5 Travel time estimate error as function of displacement associated with
assuming a uniform resident injection when injection is in flux for
4.1 Mean reaction parameter values for different values of correlation pa
rameter (3 and different trajectory collections for the ofnfc = 1 set
of simulations. Top figure is (3 {1,+1}. Middle figure is Â¡3 =
{1/2,+1/2}. Bottom figure is (3 0 (no correlation). Trajec
tory collections are areaweighted Lagrangian (AL), flowweighted
Lagrangian (AF) and areaweighted Eulerian (AE) 86
4.2 Mean reaction parameter values for different values of correlation pa
rameter Â¡3 and different trajectory collections for the ofnfc = 1/2
set of simulations. Top figure is Â¡3 = {1,+1}. Middle figure is
f3 = { 1/2,+1/2}. Bottom figure is Â¡3 0 (no correlation). Trajec
tory collections are areaweighted Lagrangian (AL), flowweighted
Lagrangian (AF) and areaweighted Eulerian (AE) 87
4.3 Mean reaction path observed values and estimators for different values
of correlation parameter Â¡3 and different trajectory collections for
the cr2n k 1 set of simulations. Top figure is Â¡3 = {1, +1}. Middle
figure is (3 = {1/2, +1/2}. Bottom figure is Â¡3 = 0 (no correlation).
Trajectory collections are areaweighted Lagrangian (AL) and flow
weighted Lagrangian (AF)
xii
89
107
Ut/X infc
Figure 4.13: Mean reaction flow path as a function of nonreactive travel time
compared to estimator Equation (4.34) for correlation parameter values Â¡3 =
{ 1/2,0,1/2} (top figure) and /3 = { 1,0,1} (bottom figure). afDk = 1.
98
For the special case of = (1 (32)cr2f, the expression integrates to
ofa] =2A2ct2P2 exp[(l p2)a}]/U2{exp[x/\f] 1
+ ^(exp[6s/A/] 1) + 8/3(1 exp[/(2A/)]) (4.24)
+ (2 4/3)x/A/}
A simple and intuitive reaction flow path variance estimator for the equalarea
streamtube collection is untenable in this framework. This is due in part to the
nonlinear reaction flow path mean.
4.6 Discussion
Figures 4.3 and 4.4 demonstrate a close match between Equations (4.18) and
(4.22) and simulation data. The parameters placed into the estimators are the target
values selected for the simulation, and thus represent an a priori estimate of the
mean reaction flow path. Equation (4.22) captures not only the nonlinearity of the
nonreactive travel time, but the nonlinearities induced by reaction parameter and log
conductivity correlation. While the absolute difference between Equations (4.18)
and (4.22) appear small, consider that the scale of the simulations is fairly large. The
solute has traveled over forty correlation scales of the log conductivity. In settings
where the the scale of conductivity heterogeneity are large compared to the overall
scale, the relative impact of injection mode and reaction parameter correlation to the
log conductivity will have a relatively greater effect.
The equalflux streamtube reaction flow path variance estimator (Equation (4.24))
shows relatively poor performance in the more heterogeneous system (Figure 4.5),
compared to that in the less heterogeneous system (Figure 4.6). Moreover, this ex
pression does not work well at all for values of positive correlation, yielding physically
implausible reaction flow path variances (cP < 0) for correlation parameter values
greater that 1/2. The lack of robustness is perhaps attributable to an inadequate
59
areaweighted Euleriantrajectory timedependent velocity variance approximation is
varK[t)] = (l+U ) bol, (1 + {2bolt+ (bait))exP[bUt/Kt]) (2.69)
The fluxweighted Euleriantrajectory timedependent velocity variance approxima
tion is
var[u/[t]] =
(t+TkT5) b(J'nk 1 + (36CTl2nfc + 3 + (b0^kY) exP [bUt/Xlfc])
(2.70)
2.5.6 Log Conductivity Moments
The first two moments of the log conductivity along those trajectories for which
the moments are stationary at control planes located along the mean direction of
flow or at control times are sought. Working from a stationary Eulerian reference
field, the mean and variance along the AW/ET are fi\nk and afnk, respectively. Infer
that the FW/LT is stationary as well, analogous to the velocity field results. The
flowweighted mean log conductivity perturbation is given by
[0] f ^ ^gexp[/]J(1+ j)9
u 1 9KeJ 1 (2.71)
=<(l + / + /2/2 + ..)(l+j)/>/c
Expanding, retaining secondorder terms, and taking the expected value yields
<^f>=olk(\l/d)/c (2.72)
which, for twodimensional fields, is ofnfc/2. Similarly, the variance is
<^f> <^/>2= Kt(l 1 (2.73)
which, for twodimensional fields, is afnfc(l
68
is
(39)
This integral has a dimension of time. These integrals are given in terms of the x
velocity component as a function of longitudinal displacement or time. It is suggested
that future works examine the details of a more general approach, e.g., displacements
along the trajectory.
3.3.1 Displacements
Consider a model flow field such as the one given in the previous chapter. Quan
tification of the displacement and travel time statistics for areaweighted and flow
weighted trajectories is sought. Previous results indicate that the velocity for a
collection of areaweighted streamtubes is stationary in time, characterized by mean
U and approximate covariance U2bo\nk exp[bUt/X\nk] The expected value of Equa
tion (3.8) is
(3.10)
= Ut
Although the velocity of the flowweighted streamtube collection is not stationary in
time, the results of the previous chapter are employed to map the properties of the
areaweighted collection to the flowweighted collection.
(3.11)
Recognize that this equation may be rearranged to be the integral of Equation (2.63)
in time. Substitute the RHS of Equation (2.63) into the preceding equation and
51
where r is the lag along the mean flow direction. Again this follows from the require
ment that the distribution match the assumed real space v moments /i = V =
U(1 + bo\nk) and a\ = V2bafnk. The correlation structure of the FW/LT velocities is
similar to that of AW/ET (see Figure 2.12).
Employing these velocity distribution hypotheses, it is now possible to derive some
relationships between the different flow field partitioning schemes. From observation,
it is inferred that the velocities along areaweighted Euleriantrajectories and flow
weighted Lagrangiantrajectories are secondorder stationary in displacements along
the mean flow direction (recall Figures 2.10 and 2.11). Having adopted or postulated
expressions for these first two moments, it is now possible to derive expressions for
the nonstationary velocity means and variances of the areaweighted Lagrangian and
the flowweighted Eulerian streamtubes.
2.5.4 Control Plane Oriented Velocity Moments
Cvetkovic et al. [1996] postulated a semianalytical estimator for a nonstation
ary mean velocity along areaweighted Lagrangiantrajectory observed at control
planes. We would like to work directly from the assumed stationary flowweighted
Lagrangiantrajectory velocity field. The general expression for the expected value
of the areaweighted Lagrangiantrajectory velocity is
Va[x] =< v[z] > (241)
Resolve the stationary mean velocities into the mean of the flowweighted Lagrangian
trajectory field and zeromean perturbation.
U
V{l+v'[Q])
va[x] =<
V(1 T i/[x]) >
(2.42)
14
The design criteria for the model aquifer are as follows
In A: correlation length Aifc = 1
effective conductivity Ke = 1
average gradient J = 1/50
mobile water content 6 = 1/5
From these criteria and Darcys law, the areaaveraged velocity is U = KeJÂ¡9 = 1/10.
Clearly, the parameter values are artificial and selected for convenience, but values
for Ke in md~l and 6 fall in ranges typical for sand [Domenico and Schwartz, 1990].
Two 5000 replicate Monte Carlo simulations of aquifers with log conductivity
variances of 1/2 and 1, respectively, comprised the suite of aquifer simulations. The
following sections describe the experimental design and the results.
2.4.1 The Log Conductivity Field
The foundation upon which to build the illustration is a heterogenous hydraulic
conductivity field. A commonlyused stationary lognormally distributed and expo
nentially correlated isotropic model serves as the Eulerian hydraulic conductivity
reference field (see e.g., Freeze [1975] or Gelhar and Axness [1983]). Working from a
hypothesized conductivity field is preferable to generating a hypothetical velocity field
with specified characteristics (e.g., Beilin et al. [1994]) because an objective of this
work is to describe the underlying conductivity field along Lagrangian trajectories
and relate its Lagrangian description to its Eulerian properties.
For completeness, it is pertinent to briefly review description of a heterogeneous
field by treating the process as being random Measured values of hydraulic con
ductivity of virtually any natural porous formation will vary from place to place.
In the absence of measurement error, this variability in the measured values reflects
the heterogeneity of myriad physical factors and processes that control the hydraulic
conductivity. As mentioned, measurements of hydraulic conductivity in the field of
ten follow a lognormal distribution. In these cases, it is useful to conceive of the
62
a stationary velocity field, whereas the Eulerian trajectories are ad hoc estimators,
based upon a transition between known end points.
2.6.4 Velocity Variances in Time
Figure 2.14 presents the timedependent velocity variances for the four differ
ent collections of streamtubes. The variances are normalized by the areaweighted
Lagrangiantrajectory velocity variance estimated from the input parameters to the
simulations. The areaweighted Lagrangianvelocity appears to be stationary in the
variance. The data seem to lie slightly above the estimated value.
The overprediction of the initial fluxweighted velocity variance has far less impact
upon the timedependent velocity variances. This combined with what appears to
be a good estimate of the longtime Euleriantrajectory velocity variance yields a
better match to the simulations than one might expect from the means. However,
there is a strong initial decline in the variance not captured by the simple exponential
expressions. The areaweighted Lagrangiantrajectory covariance function exhibits a
similar rapid decay. The early time behavior is likely dominated by the conductivity
covariance. As the trajectory experiences a wide range of conductivity variability,
this effect diminishes, and the longer range correlations related induced by the head
field begin to dominate.
2.6.5 Log Conductivity Statistics
The predictors of the stationary log conductivity statistics worked well for both
the spacedependent and timedependent means although there seemed to be a slight
overpredictive bias for the flowweighted trajectory (see Figures 2.6 and2.8). The
AW/ET log conductivity variance was well predicted in that the turningbands
method performed well, as discussed in Sections 2.4.1 and 2.4.7. The log conductivity
variance of the FW/LT appeared to be lower than that of the AW/ET, as predicted by
88
conductivity field (0 = {1,1}). Considered are the two subcases, namely that of
perfect negative correlation (/3 = 1) and perfect positive correlation (0 = 1).
For Â¡3 = 1, the equal area collection exhibits nonlinear behavior analogous to
that of travel time (see Figure 4.3 and 4.4). However, this nonlinearity is enhanced by
the correlation of high reaction parameter values in the low conductivity areas. For
the equal flow collection, however, the nonlinearity is mitigated, and // appears to be
a linear function of displacement. For 0 = 1, the both trajectory collections exhibit
the same linear behavior. Thus positive correlation appears to mitigate the effect of
injection mode on the mean behavior. The reaction path variance is strongly affected
by the sign of the correlation (see Figures 4.5 and 4.6). For negative correlation, there
is a distinct difference between the injection mode. However, positive correlation
again mitigates the difference in modes. Moreover, the positive correlation greatly
reduces the reaction path variance. For linear equilibrium sorption, this implies a
reduction of the reactive travel time variance.
The next case considered is that of no correlation between the reaction param
eter and the log conductivity (/3 = 0). The reaction path mean exhibits the same
behavior as does nonreactive travel time. The nonlinearity in the mean for the equal
area streamtubes is attributable to the nonstationarity in the associated velocity field.
However, the combined effects of heterogeneous velocity and sorption manifest them
selves in the reaction flow path variance. The reaction flow path variance is higher
than that of the nonreactive travel time, even though the scaled means are the same.
The 0 = 0 reaction flow path variance lies in between that of the positive and
negative correlation. For the lower total system variability case o oÂ¡nk = 1/2, the
magnitude of the effect of correlation appears nearly symmetric (see Figure 4.6). For
CTinjt = 1, however, there the effect of negative correlation seems to have a greater
impact upon both total variability and the effect of injection mode (see Figure 4.5).
112
subdivided into 10 nodes per A in both direction. Constant heads maintained at the
x = 0A and x = 15A boundaries imposed a steady mean gradient of J = 0.01. Noflow
conditions were maintained at the y = 0A and y = 25A boundaries. To simulate the
release of a contaminant at constant concentration, the influent water concentration
was assumed to be a constant concentration Co = 1. This source was centered on
the x = 0 boundary, and was 10A wide.
A turningbands algorithm generated a synthetic heterogeneous log conductivity
field used in the simulations [Tompson et al., 1989]. The output of the turningbands
generator was a realization of an exponentiallycorrelated standard normal process
(/x = 0,
0.5, and 1.0 via the transformation
/ =
where o\nk is the target standard deviation of the log conductivity field / and 2 is the
standard normal process. The aquifer porosity was assumed to be equal to the mobile
water content and equal to a constant value n = 0.2. The effective conductivity
of a twodimensional exponentially correlated log conductivity field is, to a good
approximation, the geometric mean conductivity Kg. For p = 0, Kg = 1, in units of
(dimensionless) correlation lengths per characteristic time. The independence of the
effective conductivity from its variance convieniently allows the variance to be changed
without substantially changing the bulk flow properties. Thus, the anticipated Darcy
flux for the aquifer, regardless of crin*, was q= 0.01 A per characteristic time, and the
characteristic filtration velocity was uÂ¡, = 0.05A per characteristic time. The subscript
b refers to a bulk property, to be distinguished from streamtube properties discussed
earlier.
The solute was assumed to undergo a constant decay process characterized by a
firstorder rate coefficient k = 0.05 in units of an inverse characteristic time defined
APPENDIX C
CONSISTENT FIRST ORDER APPROXIMATION
C.l Travel Time
Usage of a consistent firstorder approximation for travel time permeates the
stochastic Lagrangian literature. This approximation is quantitatively correct for
collections of equalflux streamtubes. However, the premise upon which it is built is
incorrect. Consider a stationary lognormal process V. The lognormal distribution is
selected for convenience, as a random variable with values guaranteed to be greater
than zero is required. Since V is stationary by definition, then so is 1/V.
Consider the expected value of the integral
roo
]>=< / dx/V[x]>
Jo
(C.l)
The order of evaluation may be interchanged, taking the expected value of the argu
ment of the integral thus:
roo
[x] > = / < dx/V[x] >
Jo
= x/Vh
(C.2)
where 14 is the point harmonic mean of the random process V. Consider now a
perturbation expression for the random process V = (1 + v) where V is the mean of
V, and v is a zeromean random process representing the perturbation of the process
about its mean. Inserting this expression into Equation (C.l) yields
y{l + v[x})>
(C.3)
131
37

a
s
b
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
FW/LT +
AW/LT x
FW/ET *
AW/ET
Eq. 2.65
Eq. 2.61
Eq. 2.70
Eq. 2.69
+4',
+. +
*+ +7+ + +
XX _X. "xx"x
.5
a _ x** **& ****
b "SSgg!. ** *
t^QSS. B mm
sftv.:##Baeae??
10
tU/X\nk
a
Ji
c
o S
b
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0 2 4 6 8 10
++* V
FW/LT +
AW/LT x
FW/ET *
AW/ET
Eq. 2.65
Eq. 2.61
Eq. 2.70
Eq. 2.69
V M ^
'+ + +
+ \ +
\ xx xxvv++ +
* x\ X xV.V+ +* *
+ + *
A* *
x x.\. x.
\ \ *
9 \x
g w** \ x
. V. *'*
XX )K ^ X
'"'DCIlbt3QQQSoF5DD*8Da"88a H8BB85fiSSS*8HB83
tU /\\n k
Figure 2.14: Comparison of timedependent velocity variance estimators to data for
the four collections of trajectories for a2nk = 1/2 (a) and ofnfc = 1 (b). The variances
are normalized by U2ba2nk.
97
For the hypothesized Lagrangian distribution V = U(1 + ba2) and a\ = ba2, thus the
Equation reduces to
= Pgexp[al/2 + /32a2/2]x/U (4.18)
dx
(4.19)
The equal area streamtube mean /x is
Manipulations similar to that which yielded Equation (4.16) yield
< Uy.lv[0] >= j\ 1 + 2al + /?VJ/2 + C,[l] 2/3C/[x])di (4.20)
Substituting the simplified relationship CvÂ¡[x] = a2 exp[x/(2X)]/2 yields
< Vp/v[0] >=^Â¡^1(1 + 2ba) + /}V,/2)x ^
+ 2(3\crf(exp[x/(2\)] 1) + Acr^(l exp[6x/A])
Notice that the first collection of terms, or the linear portion, of the preceding equa
tion is the same as that in Equation (4.16). Similar arguments are employed to yield
the equal area streamtube mean /x estimator
< Un/v[0\ >
Pgexp[al/2 + p2aj/2]
U
x
(4.22)
+ 2/?Aa^(exp[:r/(2A)] 1) + Ac^(l exp[bx/\})
4.5.2 Reaction Flow Path Variance
Adapting an expression for the reaction flow path variance given by Cvetkovic
et al. [1998] (their Equation (B4)) to the simplified velocity and velocitylog conduc
tivity functions central to this work, write
[x] = 2P2 exp[al\/U2 [ {x s)(Ctu[s] + P2Cf[s] 4pCfv[s\ + C^ds (4.23)
Jo
25
10
20
T'/'^ln k
30
FW/LT
AW/LT
FW/ET
AW/ET
reference
Eq. 2.73
40
FW/LT
AW/LT
FW/ET
AW/ET
reference
Eq. 2.73
Figure 2.7: Log conductivity variance as function of displacement in mean flow direc
tion for the four different trajectory collections and aÂ¡nk = 1/2 (a) and a^nk = 1 (b).
Variances are normalized by target variance of Eulerian log conductivity field
REFERENCES
Annable, M. D., J. W. Jawitz, P. S. C. Rao, D. P. Dai, H. K. Kim, and A. L.
Wood. Field evaluation of interfacial and partitioning tracers for characterization
of effective naplwater contact areas. Ground Water, 36(3):495502, 1998.
Bakr, A. A., L. W. Gelhar, A. L. Gutjahr, and J. R. MacMillan. Stochastic analysis of
spatial variability in subsurface flows 1. Comparison of one and threedimensional
flows. Water Resour. Res., 14(2):263271, 1978.
Bear, J. Dynamics of Fluids in Porous Media. Dover Publications, Inc., New York,
1972.
Beilin, A., Y. Rubin, and A. Rinaldo. EulerianLagrangian approach for modeling of
flow and transport in heterogeneous geological formations. Water Resour. Res., 30
(11) :29132924, 1994.
Beilin, A., P. Salandin, and A. Rinaldo. Simulation of dispersion in heterogeneous
porous formations: Statistics, firstorder theories, convergence of computations.
Water Resour. Res., 28(9):22112227, 1992.
Boggs, J. M., S. C. Young, L. M. Beard, L. W. Gelhar, K. R. Rehfeldt, and E. E.
Adams. Field study of dispersion in a heterogeneous aquifer 1. Overview and site
description. Water Resour. Res., 28(12):32813291, 1992.
Cvetkovic, V. D., H. Cheng, and X.H. Wen. Analysis of nonlinear effects on tracer
migration in heterogeneous aquifers using Lagrangian travel time statistics. Water
Resour. Res., 32(6):16711680, 1996.
Cvetkovic, V., G. Dagan, and H. Cheng. Contaminant transport in aquifers with
spatially variable hydraulic and sorption properties. Proc. R. Soc. London Ser. A,
454:21732207, 1998.
Dagan, G. Stochastic modeling of groundwater flow by unconditional and conditional
probabilites 1. Conditional simulation and the direct problem. Water Resour. Res.,
18:835848, 1982a.
134
50
The variance of v is given by var[i>] = 2. The expected value of v2 must
equal the expected value of flowweighted u2. That is
=
1 ro 9 2 l
= exp[3/iinu + alnu]
= y exp[3(x,nu + ofnu/2) + 3ofnJ
= U*(l + bo&k)
U 2
(2.36)
Thus, the variance of v is given by
var[u] = V2ba]
lnfc
(2.37)
It is worthy to note that these results do not follow from an assumption that v u2/U.
That is, the velocity along the Lagrangian trajectory is not simply the flowweighted
velocity along an Eulerian trajectory.
Hypothesis: The point velocity distribution along a flowweighted Lagrangian
trajectory in the model stationary Eulerian velocity field is lognormal with approxi
mate parameters
(2.38)
and
= M1 + ba\n k\
(2.39)
Moreover, the process is assumed secondorder stationary in displacements along
the mean direction of flow, that is, at control planes, and is characterized by the
covariance function
CVf[r\ = V2ba2nkexp[br/Xlnh\
(2.40)
78
Figure 3.5: Travel time estimate error as function of displacement associated with
assuming a uniform resident injection when injection is in flux for = {1/2,1}
a transition region between 10 and 30 correlation lengths. Beyond 30 correlation
lengths, the difference between the two modes is less than a few percent.
Much of the Lagrangian transport literature is predicated upon a uniform resident
injection, but the socalled consistent firstorder travel time estimate that is regularly
employed is quantitatively that for a flux injection. For large displacements, this
error is small, and the simplification afforded by this approximation is considerable
(compare Equation (3.14) and Equation (3.15)). For many practical applications,
an error of ten, or even more, percent might be considered negligible. Therefore,
it is not suggested that this approximation be abandoned, but rather phrasing such
as inconsistent firstorder approximation be applied in future works, since this
Lagrangian expression is inconsistently based upon the Eulerian mean velocity.
126
The effective conductivity of a porous medium may be defined as the numerical
conductivity value that satisfies the Darcy relationship
Q = KeJ
(A.9)
where Q is the specific discharge and J is the difference in hydraulic head across the
formation. There are several expressions for the effective conductivity of a stationary
heterogenous conductivity fields (see Gelhar [1993]). The general form of the effective
conductivity is
Kc = KJ{alt,g)
(A.10)
where Kg and crfnk are the geometric mean and the variance of the log conductivity
field, respectively, g is a function of aquifer anisotropy, and / is a function that scales
the geometric mean to the proper effective value. For a twodimensional isotropic
aquifer, /(ofnJt) = 1. Thus, for an aquifer of homogeneous mobile water content 6,
an estimate of the spatial mean seepage velocity U is given by
(All)
The point velocity variance for such a field is S bU2afnk, where b 3/8.
Substitution of Equation (A. 11) into this expression yields
02
(A.12)
Substitution of Equations (A.11) and (A.12) into Equations (A.7) and (A.8)
fi = In
KgJ
0\/l + b(jfni
(A.13)
a2 = ln(l + ba2nk)
(A.14)
19
to provide an accurate velocity field solution for groundwater flow in heterogeneous
conductivity fields, one that is more accurate than other, more traditional, schemes
[Mos et al., 1994].
2.4.3 Trajectory Generation
An objective of this research was to record data, such as local velocity and and
conductivity, along different collections of trajectories, both Eulerian and Lagrangian.
These trajectories were traced using a particletracking scheme based upon a semi
analytical method presented by Pollock [1988]. This method assumes that velocity
varies linearly over an element, and that the velocity in some direction, say i, is
independent of displacements in orthogonal directions. From these assumptions, it is
a relatively simple exercise to determine the displacement of a particle in a linear
velocity field for a given time, or the time required to make a certain displacement.
A computer code recorded particle position, time, total path length along trajectory,
velocity, local conductivity, and a value from another random field with the same
statistical properties as, but uncorrelated to, the log conductivity field at several
control planes equallyspaced along the mean flow direction and at several control
times after injection along both trajectories. This uncorrelated random variable was
recorded for reactive transport simulations, to be discussed in subseqent chapters.
2.4.4 Monte Carlo Simulation
The possibility of theoretical interpretation of the results is enhanced by requiring
nearergodic samples from the population. That is, the samples must reproduce the
population statistics. To do this, the number of statistically independent replicates
drawn from the velocity and conductivity fields must be quite large. Rather than
attempt a massive singlerealization, the following method generates a large number
of statistically independent realizations
63
Equation (2.73), but the magnitude of under prediction was larger than that observed
(see Table 2.4).
Table 2.4: Comparison of observed log hydraulic conductivity variances to estimators.
Observed value is zeroslope line regressed through data. The decimal representation
of the fractional value is included for convenience.
parameter
predicted
observed
AW/ETffjU*]
1/2
0.497
FW/LTfffni[i]
7/16=0.4375
0.484
AW/LTafnt[tl
1/2
0.498
\nk = 1
parameter
predicted
observed
AW/ETffjUM
1
1.002
FW/LToU*]
3/4
0.944
AW/LTVfnt[i]
1
1.000
2.7 General Conclusions
In an steady irrotational flow field, a fluid parcel trajectory follows a hydrodynamic
streamline. The flow field properties as observed along streamlines can differ radically
than those observed uniformly in space. For the model system, a stationary Eu
lerian log conductivity field, this work identified the trajectory collections for which
certain properties appear statistically stationary to second order. For displacements
along the mean flow direction, point velocity and log conductivity appear stationary
along streamlines separated by equal discharges. For displacements in travel time,
these same properties appear stationary along streamlines that are selected uniformly
in space. The stationarity of these properties facilitates the development of simple
expressions for the statistics of displacement and travel times, as we shall show in
following chapters.
An immediate result of the preceding analysis is that the bulk of the flow field in
the sense of total discharge travels at a velocity that is locally greater than the spatial
87
O 10 20 30 40
%/^In k
Figure 4.2: Mean reaction parameter values for different values of correlation pa
rameter /3 and different trajectory collections for the crfnk 1/2 set of simulations.
Top figure is Â¡3 {1,+1}. Middle figure is Â¡3 = {1/2,+1/2}. Bottom figure is
/3 = 0 (no correlation). Trajectory collections are areaweighted Lagrangian (AL),
flowweighted Lagrangian (AF) and areaweighted Eulerian (AE).
53
Expanding the denominator in a series, taking the expected value, and neglecting
thirdorder and higher moments yields
<^]2>= UV(l + 2ba2nk(l + exp] bx/\\nk])) (2.48)
In order to improve the quality of the estimate, the righthand side (RHS) of the
preceding equation is required to match the known extreme values of the lefthand
(LHS) side. The LHS at X = 0 is
<^rn[0]2> = U
n[0] 1 J 1 J
= UV
(2.49)
(2.50)
The RHS at X = 0 is
RHS[0] = UV( 1 + 2&a2nfc(l exp[60/Aln*]))
= uv
The statistical independence of the initial velocity and the velocities as x goes to
infinity are exploited to calculate the LHS as x > oo
U
LHS[oo] =<  >< n[oo]2 >
= I + v2
= UV(l + bo2nk)2
The RHS as x > oo is
RHS[oo] = UV( 1 + 25cT]2nfc(l exp[5oo/Alnfc]))
= UV{ 1 + 2 ba2nk)
The RHS is deficient a term (5cr2nfc)2. Rewrite the RHS thus
(2.51)
(2.52)
RHS UV(1 + (2ba2nk + (6a2nfc)2)(l exp[br/\lnk}))
(2.53)
56
The areaweighted velocity at the IP is U by definition. Since the areaweighted
Lagrangiantrajectory velocity is stationary in time, its mean in time is U regardless
of position. At long travel times, the point velocity is stationary in displacements as
well. For constant means in space and time, the following relationship holds
1
T
(2.58)
This expression states that the harmonic mean velocity in space equals the arithmetic
mean velocity in time. Returning now to the model aquifer, invoke the hypothesis of
lognormality of the velocities along either trajectory. The harmonic mean of u[x] is
given by
Vh = exp[/qn crinu/2]
= exp[pin] exp[crin]1/2
= u^l + bol
= U
;ln k
V1 + ba\i
In k
(2.59)
Similarly, the harmonic mean of u[x] is given by
Uh
u
1 + b(Jlk
(2.60)
Although derived by different means, this result is equal to the result given by Shapiro
and Cvetkovic [1988]. These harmonic means are the asymptotic mean velocities of
the two trajectories in time. The initial timedependent velocities are identical to
the initial spacedependent velocities, an artifact of starting all of the trajectories
at the injection plane at an implied time t = 0. This indicates that the velocity
covariance in time along the areaweighted Lagrangian trajectory is Cva[t = 0] =
U2bafnk As mentioned, at large times, the statistical properties of the Lagrangian
trajectories converge. Making a change of variables x = V^t Ut in Equation (2.40)
and substituting the known Cva[t = 0] for the value of Cvj[r 0] in yields an
4.4 Mean reaction path observed values and estimators for different values
of correlation parameter /3 and different trajectory collections for the
a?nk = 1/2 set of simulations. Top figure is f3 = {1,+1}. Middle
figure is Â£ {1/2, +1/2}. Bottom figure is Â¡3 = 0 (no correlation).
Trajectory collections are areaweighted Lagrangian (AL) and flow
weighted Lagrangian (AF) 90
4.5 Reaction path variance observed values and estimators for different
values of correlation parameter Â¡3 and different trajectory collections
for the ofale = 1 set of simulations. Top figure is /3 = {1,+1}.
Bottom figure is /3 (1/2, +1/2}. Trajectory collections are area
weighted Lagrangian (AL) and flowweighted Lagrangian (AF). . 91
4.6 Reaction path variance values and estimators for different values of
correlation parameter /3 and different trajectory collections for the
ofnk = 1/2 set of simulations. Top figure is Â¡3 = { 1,+1}. Bottom
figure is 13 = {1/2, +1/2}. Trajectory collections are areaweighted
Lagrangian (AL) and flowweighted Lagrangian (AF) 92
4.7 Correlation between the reaction flow path and travel time for different
values of correlation parameter Â£ and different trajectory collections
for the cr,2nfe = 1 set of simulations. Top figure is /3 = { 1,+1}.
Bottom figure is (3 = {1/2,+1/2}. Trajectory collections are area
weighted Lagrangian (AL) and flowweighted Lagrangian (AF). . 94
4.8 Correlation between the reaction flow path and travel time for different
values of correlation parameter (3 and different trajectory collections
for the a?nk = 1/2 set of simulations. Top figure is Â¡3 = { 1, +1}
Bottom figure is Â¡3 = { 1/2, +1/2}. Trajectory collections are area
weighted Lagrangian (AL) and flowweighted Lagrangian (AF). . 95
4.9 Assumed timedependent log conductivity covariance function along
Lagrangian trajectories (Equation (4.29)) compared to simulation
data for variances of log conductivity afnfc = {1/2,1}. The La
grangian velocity covariance function is Equation (2.61) 103
4.10 Hypothesized timedependent Eulerian log conductivity covariance func
tions for \nk = {1/2,1} compared to simulation data and Equa
tion (4.29) 104
4.11 Mean reaction flow path as a function of nonreactive travel time com
pared to estimator Equation (4.25) for correlation parameter values
Â£ = {1,0,1} 104
4.12 Reaction flow path variance a function of nonreactive travel time for
correlation parameter values Â£ = { 1,0,1}
xiii
105
24
FW/LT
AW/LT
FW/ET
AW/ET
Eq. 2.72
reference
FW/LT
AW/LT
FW/ET
AW/ET
Eq. 2.72
reference
+
X
X
X
X
Figure 2.6: Mean log conductivity as function of displacement in mean flow direction
for the four different trajectory collections and a?nk = 1/2 (a) and rfnk = 1 (b).
30
FW/LT
AW/LT
FW/ET
AW/ET
Eq. 2.35
Eq. 2.43
Eq. 2.44
reference
+
FW/LT
AW/LT
FW/ET
AW/ET
Eq. 2.35
Eq. 2.43
Eq. 2.44
reference
D
Figure 2.10: Comparison of positiondependent mean velocity estimators to data for
the four collections of trajectories for afnJfc = 1/2 (a) and ofnJfc = 1 (b). Velocities are
normalized by U.
89
O 10 20 30 40
Â£/^ln k
Figure 4.3: Mean reaction path observed values and estimators for different values
of correlation parameter Â¡3 and different trajectory collections for the afnk = 1 set of
simulations. Top figure is Â¡3 = {1,+1}. Middle figure is [3 = { 1/2, +1/2}. Bottom
figure is (3 = 0 (no correlation). Trajectory collections are areaweighted Lagrangian
(AL) and flowweighted Lagrangian (AF).
2.5.2 Eulerian Velocity Field 42
2.5.3 Streamtube Weights 48
2.5.4 Control Plane Oriented Velocity Moments 51
2.5.5 Control Time Oriented Velocity Moments 55
2.5.6 Log Conductivity Moments 59
2.6 Evaluation of Simulation and Theoretical Results 60
2.6.1 Mean Velocities in Space 60
2.6.2 Velocity Variances in Space 60
2.6.3 Mean Velocities in Time 61
2.6.4 Velocity Variances in Time 62
2.6.5 Log Conductivity Statistics 62
2.7 General Conclusions 63
3 NONREACTIVE SOLUTE TRANSPORT 65
3.1 Introduction 65
3.2 Injection Mode and Streamtube Selection 65
3.2.1 Uniform Resident Injection 65
3.2.2 Injection in Flux 67
3.3 Integrated Streamtube Properties 67
3.3.1 Displacements 68
3.3.2 Travel Times 69
3.4 Solute Transport 70
3.5 Evaluation of Expressions 71
3.5.1 Displacements 71
3.5.2 Travel Times 72
3.6 General Conclusions 75
4 REACTIVE SOLUTE TRANSPORT 79
4.1 Introduction 79
4.2 Temporal Moments 79
4.3 Conceptual Overview of Solute Transport 80
4.4 Simulation of Reactive Solute Transport 83
4.4.1 Reaction Parameter 84
4.4.2 Reaction Flow Path 85
4.4.3 Reaction Flow PathTravel Time Cross Moment ... 93
4.5 Theory of Reactive Solute Transport 93
4.5.1 Mean Reaction Flow Path 96
4.5.2 Reaction Flow Path Variance 97
4.6 Discussion 98
4.7 Spatial Moments 99
4.7.1 Injection Mode 105
4.7.2 Discussion 106
vii
70
The arrival time variance for the flowweighted collection of streamtubes is
(316)
var
Approximate the inverse velocity covariance contained in the first term of the RHS
of Equation 3.16 with the a series expansion of the perturbation equation
1 1
<
v[x']v[x"}
> =<
>
V2(l + u'[a:'])(l 1 v'[x"])
=< T(i + vyf AA + V? )>
= lj(l + 2 al + C,[*',A + HOT)
(3.17)
_1_ / Cvf[x',x"}\
u2 V v2 )
where HOT is the collection of higher order terms. Inserting this expression into
Equation 3.16 yields an expression for the travel time variance for fluxweighted
streamlines
var[t/[x]] + ^p(exp[kr/A?nJt] 1)^J (3.18)
3.4 Solute Transport
Expressions for the statistics of the integrated trajectory properties of displace
ment and travel time have been derived in the absence of an explicit reference to
solute transport to demonstrate that these properties exist simultaneously in the
same flow field. However, they are intimately related to solute transport. Consider
the transport of an infinitesimal particle whose movement is constrained to follow a
streamline. Moreover, at any time or position on the streamline, the particle velocity
is identically equal to that of the underlying velocity field. Its displacement from
x = 0 in t = T is given by Equation (3.8) and the time required to travel from x = 0
to x = X is given by Equation (3.9). For the same flow field, two collections of
streamtubes have been discussed, namely areaweighted and flowweighted. Placing
APPENDIX A
EULERIAN VELOCITY DISTRIBUTION PROPERTIES
Consider lognormally distributed random variable V with moments < V > U
and var(V) = S. These moments are related to the parameters fx and a2 via
U = exp(/r + c2/2) (A.l)
S exp(2/x + 2cr2) exp(2/r + a2) (A.2)
Let M = exp(/r) and Z exp(cr2). Rewrite the previous equations as
U = MZ1/2 (A.3)
S = M2(Z2 Z) (A.4)
Solving for M and Z yields
M =
U2
VsTTp
(A.5)
z =
s + u2
u2
(A.6)
Expanding the definitions of M and Z and taking the natural log of both sides of the
previous two equations yields
H In
t2 = In
u2 \
Vs + u2)
(A7)
fS + U2\
(A.8)
( u2 )
125
43
expression and effective expression are the same. This damping effect of the negative
head gradient and log hydraulic conductivity is largely responsible for the robustness
of these perturbation expansions. The truncated terms are not negligible in the sense
their magnitude is small. For instance, for a2nk = 1, their magnitude is onehalf that
of the mean. This brings us to an important philosophical point. A primary usefulness
of these expansions is to identify the dominant components of the process. A purely
consistent approach could lead to neglect of some important process. For instance,
consider a strict, firstorder expansion of Equation (2.16). Equation (2.17) would
contain no f2 term, and Equation (2.18) would contain no o2nk term. The resulting
effective expression would seriously underestimate the specific discharge, since the
contribution of the conductivity variability would carry no weight with respect to
the other contributing processes. Analysis of the neglected higher order terms is
necessary for a more complete understanding of the process. The general goal of the
approach, however, is to identify the dominant subprocesses that contribute to the
larger process, and the relative importance, or weight, of each subprocess.
Expressions for the Eulerian velocity covariance for the model system have been
given by various researchers, including Graham, and McLaughlin [1989a] and Rubin
[1990]. Shapiro and Cvetkovic [1988] develop an inverse velocity covariance function
using a perturbation expansion of the inverse flow velocity and spectral methods.
Consider once again the perturbed Darcys law expression given by Equation (2.16).
Dividing this by the homogeneous mobile water content, neglecting secondorder
terms and higher, the meanremoved expression, or velocity perturbation, is
(2.21)
129
meets these criteria (see Figure B.l).
The covariance function presented by Rubin and Dagan [1992] is the sum of
three twofunction products that involve integrations of Bessel functions. For two
dimensional porous systems, the simplified function condenses that monumental work
into an elegant and concise form.
The simplified expression is tested against correlation functions estimated from
the velocity and log conductivity data taken from the simulations (see Figure B.2).
The simplified estimator gives a reasonable a priori estimate of the velocity and
Figure B.2: Comparison of the simplified velocitylog conductivity correlation func
tion to simulation data for aj {1/2,1}.
log conductivity correlation observed along the equalflux Lagrangian trajectories.
The reasonable match to analytical expressions and the observed data enhance our
confidence in this simplified estimator.
The results of Rubin and Dagan [1992] (their Figure 7) indicate that anisotropy
decreases the point covariance and increases the long range correlation effects. Yet
48
normal. This indicates that the moments /zu = U and o\ = U2ba2nk may require some
further investigation. The adequacy of these estimates are discussed in subsequent
sections.
2.5.3 Streamtube Weights
The reference velocity field is now to be subdivided into streamtubes. Each
streamtube may be considered to carry some weight in relation to other streamtubes
for a given collection. Consider the partitioning of the flow field into N streamlines
separated by an equal distance along the IP. This is an equalarea weighted parti
tioning scheme, and results in N subIPs of equal area, but dissimilar discharge. The
discharge through subIP i is
di 9/\aUi (2.30)
where A a = A/N. The relative flowweight of this streamtube i is given by
wfi = Ui/U
(2.31)
An equalflow partitioning scheme results in in N subIPs of equal discharge, but
of dissimilar area. The discharge through subIP i is
d = 9/\a,iVi
Q/N
Solving for the area yields
(2.32)
Aa' = fe
The relative areaweight of this streamtube i is
(2.33)
Wai ~ U/Vi
(2.34)
76
40
30
Vi
c
20
t
10
0
0 10 20 30 40
^/^ln k
40
30
Vi
G
^ 20
t
10
0
0 10 20 30 40
E / ^ln k
Figure 3.3: Comparison of mean arrival times for the two injection modes or stream
tube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics cor
respond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT)
statistics correspond to a uniform resident injection. The top figure contains the
a\nk = 1/2 data and the bottom figure contains the afnk = 1 data.
AW/LT
FW/LT
Eq. 3.15
Eq. 3.14
+*
jyr
d?
&
+v*
A'
X
d?
&
&
+y
AW/LT
FW/LT
Eq. 3.15
Eq. 3.14
+x
J
J?
+X
Jr
J
Jc
**
Jr
ft
jr
Jk
J
16
prescribed ensemble statistics [Tompson et al., 1989]. Turningbands was capable
of producing two and threedimensional fields with the desired statistical proper
ties and was available as a portable and easily modified code. Figure 2.4 illustrates
the lognormality of the a typical twodimensional log conductivity field realization.
Figure 2.5 compares an estimate of the longitudinal covariance to the hypothesized
model. In order to capture the features of the log conductivity spatial variability, the
log conductivity
Figure 2.4: Log conductivity distribution from a 41400 node realization generated
by the turningbands method compared to the standard normal distribution. Target
distribution parameters are /in* = 0 and ofnfc = 1.
turningbands algorithm generated five log conductivity observations per correlation
length. This level of discretization falls in the range of other similar sets of simula
tions. Cvetkovic et al. [1998] used a discretization of 4 nodes per Ain for simulations
of fields with
per Aink for simulations of fields with ofnJfc as large as 1.6.
86
Figure 4.1: Mean reaction parameter values for different values of correlation pa
rameter /? and different trajectory collections for the a^nk = 1 set of simulations.
Top figure is Â¡3 = {1,+1}. Middle figure is /3 = { 1/2,+1/2}. Bottom figure is
Â¡3 = 0 (no correlation). Trajectory collections are areaweighted Lagrangian (AL),
flowweighted Lagrangian (AF) and areaweighted Eulerian (AE).
93
4.4.3 Reaction Flow PathTravel Time Cross Moment
The reaction flow pathnonreactive travel time cross moment cr^ contributes to
the variance of the expected solute flux (see Equations (4.11) and (4.13)). For linear
equilibrium sorption, this reflects the correlation between the time sorbed and the
time traveling. The correlation of the reaction flow path and the travel time is little
affected by injection mode (see Figures 4.7 and 4.8). It is, however, sensitive to the
correlation parameter /3. As with the reaction flow path statistics, the deviation of
from the Â¡3 = 0 is greater in the case of positive correlation for the same value
of /?. The total system variability seems to diminish the effect of correlation upon
o^T. This is most readily apparent from a comparison of the two (3 {1,1} cases
for the different values of ofnA. (compare the top figures in Figures 4.7 and 4.8). The
correlations for the correlated case shift towards that of the (3 0 in the higher log
conductivity variance field, and the Â¡3 0 case is slightly lower for the greater
all cases except {3 = 1, Lh = ! the correlations appear to plateau at nonzero values.
This is not surprising, since both r and /i are integrals of the inverse Lagrangian
velocity.
4.5 Theory of Reactive Solute Transport
The concepts of the previous chapters are applied to extend the work of researchers
including Cvetkovic et al. [1998] and Demmy et al. [1999]. These results are used
to interpret the results of the numerical experiments. Expressions are sought for
the mean and variance of the reaction flow path for the two different collections
of streamtubes. Cvetkovic et al. [1998] have developed expressions for equal area
streamtubes in unbounded multidimensional flow fields. Their mean // expression
neglects the correlation between the reaction parameter and the log conductivity,
and is a linear function of displacement. Their variance expression is based upon
a series expansion of the inverse Eulerian velocity perturbation. This work extends
this work with consideration the correlation between the reaction parameter and log
132
Expand (1 + v) in a series,
1 r
=^r / < (1 u[x] + v[x]2 )> dx
VJ o
(C.4)
Should this expression be truncated at first order, the leading term < r x/V is left.
However, from Equation (C.2), the exact expression is < r >= x/Vh. < r >= x/V
is a wellestablished result in the literature for the first temporal moment of a solute
plume injected in flux Kreft and Zuber [1978].
The answer to this riddle lies in the examination of the properties of the flow
field. Even if the deviations are relatively small from the straightline, or Eulerian,
trajectory, the statistics of the properties recorded along the actual, or Lagrangian,
trajectory can be quite different. For instance, the mean velocity recorded along
equalflow weight streamlines in an quasiinfinite constant mean gradient flow in
a exponentially correlated lognormal log conductivity field is approximately V
(1 + bafnk)U where U is the spatially averaged Eulerian velocity. The variance of
the the velocity perturbation along these trajectories is approximately a% bafnk.
Evaluating the expected value in Equation (C.4) and retaining the second order term
yields
=1(1 + a2)
(C.5)
Substituting our approximate values into the preceding equation yields
(C.6)
= x/U
The socalled smalldeviation assumption is a poor one. The approximation of the
Lagrangian field by the Eulerian field works because the correlation properties are
quite similar and the coincidence that the harmonic mean Lagrangian velocity is
approximately equal to the arithmetic mean Eulerian velocity.
77
60
C
b
Â£ 40
S
b
20
AW/LT +
FW/LT x
Eq. 3.18
+ X'
Ay
+ x .
+ X
+ X *'
+ X.'
. X/
+ vxy
V
+ X..*
+ X.*
+ X/
+ X.'
+ *
+ X'
+ x
+ X'
, + X
. + X
++ X'
+ x*
+ x*
X
+ X'
+
+y
10
20
% / X\n fc
30
40
Figure 3.4: Comparison of arrival time variances for the two injection modes or
streamtube collections. The flowweighted Lagrangian trajectory (FW/LT) statistics
correspond to a flux injection, and the areaweighed Lagrangian trajectory (AW/LT)
statistics correspond to a uniform resident injection. The top figure contains the
inJt = 1/2 data and the bottom figure contains the ofnfc = 1 data. Variances are
normalized by {U/ (\\nkoXnk))2.
4.13 Mean reaction flow path as a function of nonreactive travel time com
pared to estimator Equation (4.34) for correlation parameter values
(3 {1/2,0,1/2} (top figure) and /3 = {1,0,1} (bottom figure).
5.1 Resident concentration profile averaged along a transect orthogonal to
the mean flow direction for plumes for a^nk = {0.0,0.1,0.5,1.0}. . 114
B.l Comparison of velocity and log conductivity correlation functions from
Graham and McLaughlin [1989b] and Rubin and Dagan [1992] and
a simplified exponential model. All functions are normalized by
KgJafnk/9. The apparent roughness of the Rubin and Dagan
[1992] function is due to estimation errors associated with inferring
their function from graphical data 128
B.2 Comparison of the simplified velocitylog conductivity correlation func
tion to simulation data for aj {1/2,1} 129
xiv
12
the path integral of the inverse velocity over the trajectory
rV\=f ^
Jo
(2.3)
o V[Cl S
where s is a unit vector that is at all points the parallel to the trajectory. It is clear
that this sort of Lagrangian description has little meaning in a static system, and is
in this sense less general than Eulerian descriptions. Thus, Lagrangian description
is a tool reserved for dynamic systems and is a supplement to Eulerian descriptions.
It is worthy to note the distinction between a description, a general approach, and a
trajectory, a specific entity used in a description.
To summarize, a primary objective is to approximate a steady and irrotational
flow field continuum with a discrete set of trajectories. The trajectories follow either
a straight line (Eulerian trajectory) or a streamline (Lagrangian trajectory). The
trajectories carry either an equal flowweight or an equal injection areaweight.
2.3 Descriptions of Fields
Consider once again the model aquifer depicted in Figure 2.1. As mentioned in the
introduction of this chapter, the aquifer has certain features in which a hydrologist
might be interested, e.g., the intrinsic permeability or the porosity. These features
may be thought of as a field in which parameters vary continuously. This is a funda
mental concept of continuum mechanics and is discussed in a hydrologic context by
Bear [1972], Dagan [1989], and Gelhar [1993]. The ultimate description of any such
field would be a map of parameter values valid at any point, time, or scale. In lieu of
this ultimate description, stochastic hydrologists characterize spatial heterogeneity
with conditional probability density functions that reproduce the dominant charac
teristics of the field and observed values at given locations. The entire subject of
geostatistics is devoted to this end (see Journel and Huijbregts [1978]). An Eulerian
field refers to areaweighted Euleriantrajectory oriented field. A Lagrangian field
refers to a flowweighted Lagrangiantrajectory oriented field.
ACKNOWLEDGMENTS
Dissertations have but one name in the author slot, but anyone who has writ
ten one will readily testify that it represents the work and support of myriad peo
ple. Mary Hall freed my mind from many of the administrative hassles students
encounter. Dawn Mendoza appointed, unappointed, and reappointed me numerous
times throughout my tenure, and made the business end of things run as painlessly
as possible. Ralph Hoffmann kept me in keys and desks and office space, and helped
me get things wired when they needed wiring. Wally Hunter, system administrator
and Unix hacker, deserves as much praise as 1 can heap upon him for keeping the
network up and running and the data contained with in it safe and available.
The quality of an institution and its faculty is reflected in the students they
attract. 1 have enjoyed interacting with the students of the Hydrologic Sciences
Academic Cluster, and this interaction has enhanced my efforts. Liyong Li, Yan
Zhang, and Xavier Foussereau shared the same computer lab trenches as I, and my
work cannot be separated from their influence.
Many rewarding research experiences have been associated with collaborations
made with Jim Jawitz. Jim is a colleague whose outlook and expertise is somewhat
different than my own, but it is precisely these differences that have made the col
laborations so rewarding. 1 shall seek to continue working with Jim through out my
career.
Dr. Andrew I. James provided access to computer codes he developed in the course
of his own research and technical assistance.
in
115
areas of high local flux tend to move away from the source at relatively high veloci
ties. The net effect of what might be termed a preferential flow of solute in these
higher velocity streamtubes, is to induce a tail on a transverselyaveraged resident
concentration profile, which increases with variance of log conductivity.
These effects are illustrated by comparing concentration profile created by averag
ing concentrations along planes normal to the mean direction of flow to concentration
profiles given by c/cq = exp[kx/v] and c = /c0 exp[/cr[x]], where r is given by
t\x] = (x 4 Kk^ink (1 exp[kr/AInfc])) (5.13)
va
Here Ah,*, is the correlation length of the log conductivity distribution and b is a shape
factor for the conductivity anisotropy.
5.4 Application: Field Data
Data from an elaborate experiment conducted at Columbus Air Force Base, Mis
sissippi were analyzed using the results dervied in the previous section. The objective
the experiment was to characterize the natural attenuation of certain hydrocarbons
in groundwater emanating from subsurface nonaqueous phase liquid (NAPL) sources
Stauffer et al. [1997]. A NAPL hydrocarbon source comprised of decane, benzene,
toluene, ethyl benzene, para xylene (pxylene), and naphthalene was emplaced at the
site of the MADE [Boggs et al., 1992] and MADE2 [MacIntyre et al, 1993] experi
ments. This source released these poorlysoluble organic solutes into the groundwater,
and the resultant plumes were monitored at downgradient multilevel sampler loca
tions. The center of mass of the steadystate plume was estimated using a trapezoidal
rule integration of concentration data in space and by a simple averaging of pointwise
observations of concentration via Equation (5.7). Relative degradation and absolute
degradation rates of the different constituents are estimated using Equation (5.9).
21
flow weight. In this way, it is possible to map from one collection of trajectories to
another.
At a given control plane or time, the areaweighted mean and variance of some
property over N realizations, say v was calculated using the familiar estimators3
= (2'7)
varM = jyiry(vi Nvl) (28)
21
While these particular estimators seem obvious, it is important to realize that drop
ping the particle into the flow field assigns an equal area weight to each trajectory.
For a single realization, this is equivalent to starting each particle with equidis
tant spacing. These estimators are based upon an assumption of equallylikely
statisticallyindependent samples (see, e.g., Mood et al. [1974]). In order to draw
even a few hundred independent samples from a single realization, the required flow
domain would be enormous. This was a primary consideration for adopting Monte
Carlo simulation using a minimal domain.
The flowweighted mean property v was calculated via
vf
EZ=ii[0fo
EÂ£it*[o]
(2.9)
3 Press et al. [1988] have criticized the variance estimator in Equation (2.8) as
being prone to accumulation of roundoff error for large data sets. The rather sub
stantial size of the data sets are naturally a concern. Comparision of the variance
calculated by a corrected twopass method presented by Press et al. [1988] and the
traditional estimator yielded no difference between the two methods within the
first five significant digits.
46
from Dagan [1984]
7> r i 2 r> ri dR\nktt>[r] d 7tf>[r]N\ ^ o7\
Rua[r] = ffinfc ^lnfcM j (227)
where RÂ¡nk is the log conductivity correlation function, Rink is the log conductivity
and head crosscorrelation function, and 7^ is the head variogram. From this ex
pression, glean that the velocity covariance is a process comprised of three dominant
subprocesses, namely R\nk, R\nk, and 7^. At small displacements, all three of these
subcomponents exhibit rapid trend towards zero (see Figure 2.18). The R\nk
Figure 2.18: Components of the velocity covariance function.
expressions exhibit effects that persist well beyond that of the log conductivity field.
The longrange correlation effects of the head field in the mean direction of flow are
cannot be precisely predicted with the simple exponential function Equation (2.13).
In order to recover the proper correlation length, the approximation must system
atically overpredict the velocity correlation at small separations. The cost of the
118
Local dispersion was neglected, and local dispersion certainly has a significant
effect on the local concentration variability. However, its neglect does not appear to
diminish the results here to the point of uselessness.
The nonstationarity of equalarea Lagrangian trajectories in a steady flow field
was neglected. For highly heterogeneous conductivity fields, low velocity areas are
disproportionately sampled with respect to the flow average velocity. For a spatially
uniform decay process, this would tend to underestimate the plume extent, since the
observed values will tend to be associated with long travel times. It should be
noted that this sampling bias is intrinsic to multilevel samplers.
The correlation of velocity and decay coefficient was neglected. A strong negative
correlation could result in a significant underprediction of the center of mass in a
highly heterogeneous field. Again, it should be noted that this is partially due to the
sampling bias assumed by the equalweight multilevel sampler sampling strategy that
will preferentially sample low velocity areas with respect to the flow weighted mean.

