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PAGE 1 1 A STOCHASTIC MODEL FOR ESTIMATI NG GROUNDWATER AND CONTAMINANT DISCHARGES FROM BOREHOLE MEASURES OF WATER AND CONTAMINANT FLUX By ZLEM ACAR 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 2011 PAGE 2 2 2011 zlem Acar PAGE 3 3 ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Kirk Hatfield for g ranting me the opportunity to do my research o n this innovative area and for his support and guidance during my graduate studies D valuable suggestions throughout this dissertation are highly appreciated. Also I would like to thank Drs. Michael Annable and Mark Newman for their support in this research. Lastly, I would like to express my appreciation to all of those who supported me in any respect during the completion of this degree. PAGE 4 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 3 LIST OF TABLES ................................ ................................ ................................ ............ 6 LIST OF FIGURES ................................ ................................ ................................ .......... 7 LIST OF ABBREVIATIONS ................................ ................................ ............................. 9 ABSTRACT ................................ ................................ ................................ ................... 10 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 12 Technology Description of Fractured Rock Passive Flux Meter .............................. 14 Dissertation Outline ................................ ................................ ................................ 17 2 FIELD SCALE FLOW AND TRANSPORT MODELS OF FRACTURE NETWORKS ................................ ................................ ................................ ........... 19 Equivalent Continuum Models ................................ ................................ ................ 20 Discrete Fracture Network Models ................................ ................................ .......... 21 Hybrid Models ................................ ................................ ................................ ......... 25 3 A STOCHASTIC MODEL FOR ESTIMATION OF DISCHARGES AT A VERTICAL TRANSECT OF MULTIPLE BOREHOLES IN FRACTURED ROCK: 1ST STEP: RANDOM NUMERICAL REALIZATIONS OF FRACTURE NETWORKS ................................ ................................ ................................ ........... 31 Main Assumptions for Generation of FRPFM Data ................................ ................. 32 Input f ................................ ................................ ................................ 34 Simulation of Fracture Locations ................................ ................................ ............ 35 Homogeneous Poisson Process ................................ ................................ ...... 35 Inhomogeneous Poisson Process ................................ ................................ .... 36 Poisson Cluster Process ................................ ................................ .................. 36 Cox Process ................................ ................................ ................................ ..... 37 Fracture Generation ................................ ................................ ................................ 37 Probability Distributions of Fracture Orientations ................................ ............. 38 Probability Distributions of Fracture Lengths ................................ .................... 39 Probabi lity Distributions of Fracture Apertures ................................ ................. 39 Flux Distributions Assigned to Each Fracture Set ................................ ............ 39 ................................ ................................ ................................ 40 PAGE 5 5 4 A STOCHASTIC MODEL FOR ESTIMATION OF DISCHARGES AT A VERTICAL TRANSECT OF MULTIPLE BOREHOLES IN FRACTURED ROCK: 2ND STEP: DISCHARGE ESTIMATIONS AND UNCERTAINTY ANALYSIS ........ 41 Theory ................................ ................................ ................................ ..................... 41 Lin ear Fracture Frequency and Related Sampling Bias Correction ........................ 43 Groundwater Discharge Estimation at the Simulated Transect ............................... 44 Contaminated Mass Discharge Estimation at the Simulated Transect ................... 47 Consideration of Truncation and Edge Effects for Simulations ............................... 48 Monte Carlo Simulations for Discharge Estimation and Uncertainty Analysis ......... 49 Assessment of Stability of Monte Carlo Simula tions to Model Real Discharges ..... 51 Reliability Analysis of Predicted Discharges ................................ ........................... 54 Simulations with Constant Length, Constant Orientation Traces ...................... 55 Simulations with Trace Length as Random Variable ................................ ........ 65 Simulations with exponentially distribute d trace lengths ............................ 65 Simulations with log normally distributed trace lengths .............................. 69 Simulations with Orientation as Random Variable ................................ ............ 72 Simulations with uniformly distributed orientations ................................ ..... 72 Simulations with normally distributed orientations ................................ ...... 74 Simulations with Flux as Random Variable ................................ ...................... 76 Simulations with Lengths, Orientations and Fluxes as Random Variables ....... 80 5 ILLUSTRATION OF DISCHARGE ESTIMATION AND UNCERTAINTY ANALYSIS METHODS ON A TRANSECT OF FRACTURED ROCK WITH MULTIPLE BOREHOLES ................................ ................................ ....................... 83 6 SUMMARY AND CONCLUSIONS ................................ ................................ .......... 88 LIST OF REFERENCES ................................ ................................ ............................... 92 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 101 PAGE 6 6 LIST OF TABLES Table page 4 1 Input parameters used in FRPFM data simulation for discharge estimation on a 1m x 1m transect with horizontal, constant length traces ................................ 56 4 2 Input parameters used in FRPFM data simulation to test the effect of transect with horizontal, consta nt length traces ................................ .................. 60 4 3 Input parameters used in FRPFM data simulation to test the effect of various constant orientations on disc harge estimations on a 1m x 1m transect with constant length traces ................................ ................................ ........................ 62 4 4 Input parameters used in FRPFM data simulation to anal yze prediction errors for discharge estimation on a 1m x 1m transect with horizontal, exponentially distributed length traces ................................ ................................ ..................... 66 4 5 Input parameters used in FRPFM data simulation to analyze prediction errors for discharge estimation on a 1m x 1m transect with horizontal, log normally distributed length traces ................................ ................................ ..................... 69 4 6 Input parameters used in FRPFM data simulation to analyze prediction errors for discharge estimation on a 1m x 1m transect of constant length traces with uniformly distributed orientations ................................ ................................ ........ 72 4 7 Input parameters used in FRPFM data simulation to analyze prediction errors for discharge estimation on a 1m x 1m transect of cons tant length traces with orientations following normal distribution ................................ ............................ 74 4 8 Input parameters used in FRPFM data simulation to analyz e prediction errors for discharge estimation on a 1m x 1m transect of constant length traces with fluxes following log normal distribution ................................ ............................... 76 4 9 Input parameters used in FRPFM data simulation to analyze prediction errors for discharge estimation on a 1m x 1m transect with lengths, orientations and fluxes as random variables ................................ ................................ ................. 80 5 1 Input parameters used in FRPFM data simulation to analyze prediction errors for discharge estima tion on a 1m x 1m transect for 5 fracture sets with lengths, orientations and fluxes as random variables ................................ ......... 85 PAGE 7 7 LIST OF FIGURES Figure page 1 1 A multiple well transect in fractured rock to measure groundwater and contaminant fluxes. ................................ ................................ ............................. 14 1 2 FRPFM and its horizontal cross section. ................................ ............................ 15 1 3 A profile view of an unscreened borehole containing FRPFM (ESTCP, 2008). .. 16 3 1 A) An illustration showing a vertical transect taken from a 3D fractured rock. B) A representing a possible realization of this vertical transect with one borehole in the middle. ................................ ................................ ................................ ...... 33 4 1 Fracture set intersected by a sampling line of general orientation (Priest, 1985). ................................ ................................ ................................ ................. 44 4 2 Zones ar ound each borehole considered for discharge estimations. .................. 45 4 3 A) One realization of a 10m by 10m total simulation area with hori zontal traces of 0.3m length. B) Focus on 1m by 1m vertical transect with one borehole placed in the middle (Areal density = 20). ................................ ............ 49 4 4 Convergence of the sample mean groundwater discharge to the real value as function of number of Monte Carlo simulations. ................................ .................. 53 4 5 Convergence of the sample standard deviation of groundwater discharge as function of number of Monte Carlo simulations. ................................ .................. 54 4 6 Results of Monte Carlo simulations with horizontal, constant length traces length/transect ................................ ................................ ............... 58 4 7 ................................ ................................ ............... 58 4 8 Mean predicted discharge errors as function of number of trace mid points within the transect (for horizontal, constant length traces). ................................ 60 4 9 Standard deviations of normalized absolute errors for Q as function of number of trace mid points within t he transect. ................................ .................. 61 4 10 One realization for a 1m by 1m transect with 0.3m constant length traces of 40 orientation. ................................ ................................ ................................ ... 62 PAGE 8 8 4 11 Results of simulations to test the effect of constant orientations on the discharge estimations for the basic transect with constant leng th traces for an areal density of 40. ................................ ................................ ............................. 63 4 12 Standard deviations of discharge prediction errors as function of orientation angles in the basic transect with constant length traces for areal density of 40. ................................ ................................ ................................ ...................... 64 4 13 One realization for a 1m by 1m transect of horizontal traces with exponentially distributed lengths of mean 0.3m (areal density = 20). ................. 65 4 14 Mean prediction errors fo r horizontal traces with exponentially distributed lengths. ................................ ................................ ................................ ............... 68 4 15 Standard deviations of errors for simulations with horizontal exponentially distributed trace lengths. ................................ ................................ .................... 68 4 16 Mean prediction errors for horizontal traces with log normally distributed leng ths. ................................ ................................ ................................ ............... 70 4 17 Standard deviations of errors for horizontal traces with log normally distributed lengths. ................................ ................................ ............................. 71 4 18 One realization for a 1m by 1m transect with 0.3m constant length traces of uniformly distributed orientation between 30 and 50 (areal density = 40). ....... 72 4 19 Results of simulations for constant length traces with uniformly distributed orientations for areal density of 40. ................................ ................................ ..... 73 4 20 Results of simulations for constant length traces with normally distributed orientations for areal density of 40. ................................ ................................ ..... 75 4 21 Discharge prediction errors for simulations with log normally distributed fluxes. ................................ ................................ ................................ ................. 77 4 22 Standard deviations of errors for log normally distributed fluxes. ....................... 78 4 23 One realization for a 1m by 1m transect with 5 fracture sets all variables assigned randomly. ................................ ................................ ............................ 81 4 24 Histogram of normal ized absolute errors of discharge estimations for 5 fracture sets with all variables assigned randomly. ................................ ............. 81 5 1 One realization of the hypothetical 60m by 20m transect with six boreholes. ..... 84 5 2 Mean prediction errors and standard deviations of errors for dis charge estimations as function of number of boreholes across the transect. ................. 87 PAGE 9 9 LIST OF ABBREVIATION S FRPFM Fractured Rock Passive Flux Meter PFM Passive Flux M eter DNAPL Dense Nonaqueous Phase Liquid PAGE 10 10 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 A STOCHASTIC MODEL FOR ESTIMATING GROUNDWATER AND CONTAMINANT DISCHARGES FROM BOREHOLE MEASURES OF WATER AND CONTAMINANT FLUX By zlem Acar May 2011 Chair: Kirk Hatfield Major: Civil Engineering The f ractured rock passive flux meter (FRPFM) is a new technology that measures t he magnitudes and directions of cumulative water and contaminant fluxes in fractured rock aquifers. Based on this novel flux measuring technique, the scope of this study is to formulate and demonstrate a method for estimating contaminant discharges from po int wise measurements of cumulative fluxes at boreholes in fractured media. For this purpose, a stochastic model was developed in which stochastic simulation s of discrete fracture networks with synthetic FRPFM data are conducted followed by water and conta minant discharge estimations and uncertainty analys e s over simulated transects of fractured rock. The m ain parameters considered for the uncertainty analysis of the predicted discharges were areal densities, trace lengths, orientations and fluxes. Hypothetical scenarios ha d been executed model separately t o observe the related isolated effects T he obtained discharge predictions were unbiased with uncertainties varying due to the probability of PAGE 11 11 intersection of the traces with the borings as function of the investigated joint characteristic Transects with multiple rock wells were also investigated to assess the reliability of estimated discharges as function of the number of wells. R esults sh ow ed that increasing the number of boreholes w ould reduce the uncertainty of the predict ions up to a certain optimum number of wells which was consider ed to be case specific. Thus, a decision to meet a certain accuracy level while considering the associated costs will require a pre optimization field plan with an exploratory borehole and further analysis of the measured data with the presented method. In terms of application to real field conditions, the developed stochastic model can be combined w ith an areal fracture density and/or trace length estimation technique for detailed analys i s of the in situ collected data. R esults will provide input to enhance the conceptual site model and form a foundation for the development of a robust contaminant tr ansport model in fractured media. PAGE 12 12 CHAPTER 1 INTRODUCTION Fractured rock formations are complex hydrogeological environments and predictive capabilities relating to flow and transport p henomena still remain severely limited in this media (Berkowitz, 2002) Particularly, economic and technical challenges conventional methods have restricted usage in terms of site depiction monitoring and simulation of flow and transport p rocesses in fractured rock groundwater systems (Faybishenko et al., 2000, 2005; Dietrich et al., 2005; The Interstate Technology & Regulatory Counci l, 2010). On the other hand, quantification of contaminant discharge near source zones is crucial for assessing long term risk, evaluating remedial performance and meeting regulatory compliance (ESTCP, 2008). The current state of the art technologies are b ased on using measurements/calculations of Darcy water fluxes and groundwater concentration samples in space and time to estimate contaminant discharges from a source zone. These indirect estimations of contaminant discharges are subject to high uncertaint y levels due to the temporal and spatial variability of hydrogeologic conditions in heterogeneous aquifers. In contrast mass flux estimates can better portray a contaminated site than typical monitoring networks (Feenstra et. al, 1996). The additional understanding of plume dynamics provided by mass flux leads to an improved conceptual site model and better remediation decisions (Nichols & Roth, 2004; Basu et a l., 2006). Direct measurement and interpretation of groundwater and contaminant fluxes (instead of concentrations) aims at better characterization of spatial and temporal variations in water and contaminant fluxes to improve development of predictive mode ls PAGE 13 13 for solute transport and attenuation processes. The passive flux meter (PFM) developed by Hatfield et al. (2002, 2004) and Annable et al. (2005) allows monitoring time integrated water and contaminant mass fluxes in the long term. Currently, the PFM is the only passive sampler which has proven to effectively measure mass fluxes near source zones (Verreydt et al., 2010). Derived from the same idea of passive flux meters, the fractured rock passive flux meter (FRPFM) is a new technology that measures the m agnitudes and directions of cumulative water and contaminant fluxes in fractured rock aquifers (Klammler et al., 2006). The FRPFM incorporates novel methods for measuring water and DNAPL fluxes in fractures whereas retaining the field tested concepts of th e PFM. The FRPFM has a closed hole passive flux sensor that directly measures: 1) the location of active or flowing fractures; 2) active fracture orientation i.e., strike, dip and dip orientation; 3 ) direction of groundwater flow in each fracture plane; 4 ) cumulative magnitude of groundwater flux in each fracture plane; 5 ) cumulative magnitude of contaminant flux in each fracture plane (ESTCP, 2008). Currently, there are available methods for determination of parameters (1) through ( 2 ). However, paramete rs through ( 3 ) to ( 5 ) can only be measured by FRPFM which makes this new sensor an innovative technology. The in situ measurements of direction and magnitude of water and contaminant fluxes in active fractures by FRPFM need to be interpreted for the estima tion of groundwater and contaminant discharges at a transect of multiple boreholes (Figure 1 1 ). Based on this novel flux measuring technique, the objective of this study is to formulate and demonstrate a method for interpreting contaminant discharges from PAGE 14 14 Groundwater and solute fluxes perpendicular to the transect point wise measurements of cumulative contaminant fluxes in fractured media. For this purpose, this dissertation elaborates a stochastic model predicated on the idea of using FRP FM data from boreholes. Figure 1 1. A multiple well transect in fractured rock to measure groundwater and contaminant fluxes. Technology Description of Fractured Rock Passiv e Flux Meter The FRPFM developed by Klammler et al. (2006) constitutes a new closed hole passive sensing technology for fractured media. The sensor is easily deployable at any depth and in any orientation depending on the flow system being monitored and pr ovided that the unit is placed in a saturated flow system (ESTCP, 2008). Cumulative contaminant fluxes have never been measured in fractured rock aquifers under ambient closed hole conditions. They are usually calculated using observed contaminant concentr ations from boreholes or screened wells and depth averaged groundwater flows are then calculated or measured under open hole conditions. However, open borehole techniques are not capable of capturing the true ambient flows as they induce PAGE 15 15 magnitude and dire ctional changes in water and contaminant fluxes in fractures. Instead, closed hole conditions are preferred since they eliminate the exchange of water and contaminants between fractures as occurs in open boreholes and thus restore (almost) natural flows i n fractures. Figure 1 2. FRPFM and its horizontal cross section. The FRPFM is essentially an inflatable (or mechanically expandable) packer or impermeable flexible liner that holds a reactive permeable fabric against the wall of the borehole and against any water filled fractures intersected by the borehole (ESTCP, 2008 ). Reactive fabrics intercept and retain target groundwater contaminants (i.e. TCE, DCE, VC) and release non toxic resident tracers (e.g. visible dyes and branch alcohols). Tracer loss is proportional to fracture flow and yield s measures of ambient flow. L eached visible tracers reveal location, orientation and aperture of flowing fractures and PAGE 16 16 direction of flow (Figure 1 2). Contaminant mass captured is proportional to flux and yields ambient contaminant flux. Figure 1 3. A profile view of an unscreened borehole containing FRPFM (ESTCP, 2008). As seen in Figure 1 3, the FRPFM is a device composed of an impermeable flexible liner and a permeable reactive sorbent layer sandwiched between the impermeable flexible liner and the borehole circumf erence (ESTCP, 2008). The sorbent is a permeable fabric derived from activated carbon, ion exchange resin, etc. The impermeable flexible liner is made of a fluid impermeable flexible material typically available in a tube or sock design that is easily fitt ed into a borehole or equivalent aperture in a formation. Once it is inserted it is inflated with a fluid to cause it to conform to the shape of the borehole so that the permeable sorbent layer is pressed against the well screen or borehole wall. Due to th e impermeability of the flexible liner, the flow within the fractures does not enter the borehole, but is instead diverted around the impermeable flexible liner. The sorptive layer around the flexible liner passively PAGE 17 17 intercepts portions of fractures in ord er to simultaneously measure local cumulative groundwater and solute fluxes. Dissertation Outline Based on the explained novel fractured rock passive flux meter technology, the scope of this dissertation is to present a method for estimation of groundwater and contaminant discharges at a transect of fractured rock using FRPFM data from multiple boreholes. Chapter 2 covers a general review of field scale flow and transport models in fractured media. The modeling technique applied in this research for simulat ion of fracture networks falls under the category of discrete fracture network models. The stochastic model developed for estimation of discharges using FRPFM data is explained through Chapters 3 and 4. The model employed as Monte Carlo simulations has mai nly two parts: First part, simulation of synthetic FRPFM data by random numerical realizations of discrete fracture networks and second part, discharge estimations and uncertainty analysis on th e s e simulated transect s Chapter 3 presents the first step of the stochastic model that is random numerical realizations of fracture developed by MATLAB. The output of the code is fictitious FRPFM data which is used as an input for th e second part of the stochastic model that is explained through Chapter 4 for discharge estimations and uncertainty analysis. Throughout Chapter 4, first the theory and formulae used for discharge estimations are presented and then application of this disc harge estimation technique using synthetic FRPFM data is illustrated on possible scenarios considering all the parameters included in the stochastic model. Estimation of discharges on the simulated transect is carried out by PAGE 18 18 Carlo simulation and after realizations are run many times, the results are stored in output files and statistical analysis is carr ied out on the ensemble results to evaluate the uncertainty of predicted discharges. Chapter 4 mainly elaborates on the conceptualization of the discharge estimation technique on hypothetical transects with one borehole and uncertainty analysis of the obta ined results. Chapter 5 illustrates application of the stochastic model on a vertical transect of fractured rock with multiple boreholes. Chapter 6 presents a summary of conclusions of this research study, discusses the application of the developed model t o real field data and also points to future directions that may follow the spirit of presented results. PAGE 19 19 CHAPTER 2 FIELD SCALE FLOW AND TRANSPORT MODELS OF FRACTURE NETWORKS In order to develop a mathematical model to represent fluid and contaminant transp ort in a fractured medium, all the information about fracture geometry has to be combined with the interpretation of field hydraulic and tracer test results from borehole measurements. Formulation of a hydrogeological simulation model involves mainly two s teps: first description of the geometrical features of the fracture network system and then conceptualization of fluid flow and solute transport on this defined joint network system. The m ain purpose in defining the geology of the fractured rock is to iden tify the primary pathways which will potentially dominate the hydrology of the rock under study. Fracture pathways can be placed in a wide spectrum between two extremes which are few major fractures in a relatively impermeable matrix or a network of highly connected fractures in a relatively permeable matrix. Thus, g eological investigation of borehole data is a necessary guide for defining the main pathways that will play the major role for conceptualization of the flow system. In addition to accurate description of the geology of the fractured rock under study, another important factor to be considered during model development is the scale of interest. A fracture network may be highly connected on the larger scale whereas boreho le measurements might show only a few connected joints or vice versa. Field scale measurements indicate small scale hydrogeological characteristics, though flow and transport models are aimed at for large scale predictions. For field scale flow and transpo rt models of fractured networks, there are three broad classes of mathematical conceptualization (National Research Council, 1996): PAGE 20 20 1) equivalent continuum models, 2) discrete network models, 3) hybrid models. These models differ in their representation of the heterogeneity of the fractured rock mass. Continuum models represent hydraulic properties of the rock mass by coefficients that express the volume averaged behavior of many fractures, individual fractures are not explicitly included in these models. O n the other hand, discrete network models focus on individual fractures and handle heterogeneity at a scale smaller than the continuum models. Since the in situ data of individual fractures are limited and their interconnection is unknown, discrete network models solve these problems through use of stochastic models and probabilistic approaches. In general, the basic concern for discrete fracture network approach is that it is based on the obser ved fracture geometry to predict flow and transport, whereas a majority of the fractures can be nonconductive. Hybrid models combine discrete network simulations with continuum approximations. Some of the recent innovations in modeling flow and transport p henomena in fractured modeling flow and transport in fractured formations is to develop large scale models from small scale measurements (i.e. limited number of borehole me asurements due to financial considerations). The modeling method used in this study is an extension of discrete fracture network modeling technique. FRPFM measures particularly active fractures A ccordingly stochastic discrete fracture network modeling se ems to be the most suitable technique for simulation of FRPFM data. Equivalent Continuum Models Equivalent continuum models are based on the idea that heterogeneity and hydraulic properties of the fractured medium can be represented by coefficients that PAGE 21 21 ex press the volume averaged behavior of many fractures. Depending on the availability of field data, these models can be either deterministic or stochastic. Helmig (1993) states that it is possible to describe a modeled fractured rock area as a heterogeneous anisotropic continuum if the representative elementary volume concept is valid and the scale of the investigated area is sufficiently large. This approach is applicable to cases where groundwater and solute flux measurements are not influenced by individ ual fractures but rather th e s e data are collected from a highly interconnected fracture network. Equivalent continuum models can be either single or dual porosity (National Research Council, 1996). Single porosity models assume all the flow occurs via frac tures whereas dual porosity models also consider the role of rock matrix in flow and transport. The advantage of continuum approach is its capability of addressing complex flow patterns in simple mathematical formulations especially from practical applicat ions point of view (Long et al., 1982) On the other hand, identifying the scale at which continuum approach will be valid is the main drawback of this method (Neuman, 1987, 1990) Discrete Fracture Network Models When the flow and transport processes in t he fractured media are dominated by fracture zones, it is feasible to describe these features specifically using a discrete fracture network model (Helmig, 1993). Discrete fracture networks are aimed to be representative of field scale. However the knowled ge of in situ individual joint characteristics is limited, and this fact has led discrete network models to be based on stochastic concepts. For this purpose, spatial statistics associated with a fracture network are measured as the first step and then PAGE 22 22 the se are used to generate realizations of fracture networks with similar spatial characteristics. Initially stochastic fracture models had unbounded discontinuities. Later bounded fractures were modeled as Baecher disks (Baecher et al., 1977; Barton, 1978) a nd Veneziano (1978) polygonals. However, these models still lacked the spatial dependence and statistical relationships specific to each fracture set. A stochastic model assumes that the locations of joints are purely independent of each other, random. How ever in reality, a spatial periodicity (which leads to inhomogeneity) is observed in fractured rocks. Geostatistical analysis adds spatial correlation (Isaaks and Srivastava, 1989) to a stochastic simulation which makes it more realistic. These spatial rel ationships were later incorporated into stochastic models with examples such as La Pointe (1980), Long and Billaux (1987), Villaescusa and Brown (1990), Dershowitz et al. (1991a) Desbarats (1996) and Rafiee and Vinches (2008) In discrete fracture network modeling approach, network geometry is defined by statistical probability distribution functions of trace mid point location, orientation, length and aperture for each fracture set. From this statistically defined fracture network model, multiple realizat ions are generated and solved for flow and transport through each realization. Since geostatistical simulations follow a stochastic approach and each realization is one possible representation of the fractured rock mass from which the data ha ve been gather ed, a large number of network realizations need to be carried out to get representative results. When this large number of realizations for flow and transport predictions is averaged in a Monte Carlo simulation, results can be drawn for the expected behavi or of the fractured system and also variability about the mean. PAGE 23 23 All the basic stochastic discrete fracture network models are based on Poisson processes (Chiles and de Marsily, 1993). To model the fracture network, the fractures in each fracture set are first located assuming a point Poisson process. Then at these points, fractures are modeled as either Poisson planes for infinite fractures or random discs for finite fractures. These models are called object based simulations, Boolean models. Chiles (1988 ), Billaux et al. (1989), Chiles and Guerin (1992), Chiles and de Marsily (1993), Chiles and Delfiner (1999) gave examples of disc based simulations to model various different fractured rock systems A further improvement for geostatistical simulations is constraining the results of the simulation to fit the observed data at sampled locations. Conditioning the simulated fracture network on the observed fractures facilitates creating realizations that are more similar to the fractured network where field mea surements are made. Discrete fracture networks have been used for modeling various fractured rock sites located at different sites of the world. Fanay Augeres uranium mine in France and Stripa research mine in Sweden are two sites that have been studied by many researchers ( Neretnieks et al., 1985; Rouleau and Gale, 198 5 ; Long and Billaux, 1987; Rouleau and Gale, 198 7 ; Chiles, 1988; Billaux et al., 1989; Dverstorp and Andersson, 1989; Cacas et al., 1990a, 1990b; Kulatilake et al., 1990; Tsang and Tsang, 199 0; Dverstorp, 1991; Long et al., 1991; Kulatilake et al., 1993; Witherspoon, 2000; Sie and Frape, 2002 ) particularly on discrete fracture network modeling, and also conceptualizing flow and transport on these models (Some of these investigations fall unde r the category of hybrid methods discussed in the next section ) PAGE 24 24 Chiles (1988) gave an example of fractal and geostatistical modeling of Fanay Augeres mine which lies in a granite massif. Long and Billaux (1987) worked on incorporating spatial structure of field data from Fanay Augeres mine into fracture network modeling. Billaux et al. (1989) developed a three dimensional stochastic model of fractured rock mass from the Fanay Augeres mine us ing parent daughter process for locating discs to model the discon tinuities. Two of the most detailed works on fractured rock in the Stripa mine were done by Rouleau and Gale, first on statistical characterization of the fracture system (1985) and then on simulation of groundwater flow (1987). A case study by Kulatilake et al. (1990) showed development of a three dimensional stochastic joint modeling of data from Stripa mine. Later in 1993, they performed validation techniques on that developed joint geometry model. A more recent example of discrete fracture network model ing can be found in Jones et al. (1999), where groundwater resources exploitation in carboniferous rocks in southwest Ireland was studied In another study, Hestir et al. (2001) generated simulations of a physically based stochastic model that are conditio ned on field observations from faults in granitic plutons of the central Sierra Nevada, California and Fort Riley limestone in Kay County, Oklahoma. The latest alternative to object based modeling is pixel based algorithms which bring the flexibility to model non planar fractures. The pixel based algorithms are also called process based models since this approach aims at modeling the fracturing process itself (Chiles, 2005). In these models, fractures are not located in their final states but instead after initial seeding of mid points of fracture traces, fracture tips are PAGE 25 25 propagated following geomechanical rules. Fracnet (Gringarten, 1998) uses a hybri d pixel object approach where fractures are constructed as objects which are themselves sets of connected pixels rather than parametric objects. Renshaw and Pollard (1994) proposed an approach for 2D pixel based fracture simulation using geomechanical prin ciples to propagate fracture tips. Srivastava et al. (2005) improve d their method to extend modeling to 3D. Since simulations based on geomechanical principles are computationally intensive, they suggest using geostatistical rules instead of geomechanical principles for propagating fracture tips. An important aspect of discrete fracture networks is that these models are practical tools for site specific simulations and this feature is demonstrated (in the next chapter) using these models for simulation of F RPFM data. A second and even more essential use of discrete network models is their role as a primary tool for concept evaluation (Long and Witherspoon, 1985). Following this idea, in later chapters of this dissertation stochastic discrete network models w ill be used to conceptualize flow and transport processes using FRPFM measurements. Hybrid Models The idea of hybrid models is based on using discrete fracture network models in building continuum approximations (National Research Council, 1996). For fluid flow, large scale permeability measurements are not practical. One way to solve this problem is to perform numerical flow experiments on fracture network models to build continuum approximations. combines borehole frequency and orientation measurements with estimates of aperture size from packer tests to calculate an equivalent permeability tensor. He modeled fractures as infinite PAGE 26 26 parallel plates based on the assumption that the fracture density is high enough that nearly all the fractures play role in effective permeability. Later studies ( Hudson and La Pointe, 1980; Long et al. 1982 ; Dershowitz 1984) analyze d fracture networks to estimate continuum properties by applying fluid flow on these disc rete fracture networks with numerical methods. Andersson et al. (1984) developed a conditioned stochastic model to evaluate the uncertainty of predicting the fluid flow paths in fractured rock. Later Andersson and Thunvik (1986) demonstrated that knowing t he details of fracture geometry would lower the uncertainty in predictions of solute transport. They concluded that introducing more cores would decrease the estimated uncertainty; and also for each core detected fracture lengths, fracture line density and spatial correlation of aperture along the fracture would influence the value of geometrical information. Similarly, Andersson and Dverstorp (1987) stated that combined effect of trace length and density would determine the uncertainty in flow through a di screte fracture network. They also used Stripa mine data for calibration and validation of their model (1989). assumption that these networks behave like porous media, did not ac count for the interconnection and heterogeneity effects. La Pointe and Hudson (1985) used electrical analog models to study finite fractures considering connectivity and proposed formulas for estimating areal and volumetric joint densities from borehole me asurements. Long and Witherspoon (1985) investigated whether permeability could be determined from the fracture frequency and individual fracture permeabilities as measured in a borehole without knowing the actual lengths and fracture densities. The basis for their length PAGE 27 27 density study is one of the concepts used in this dissertation and will be explained in detail in Chapter 4. Chiles and Guerin (1992) emphasized the importance of calibrating the flow and transport simulations with the field data in additi on to conditioning the geostatistical simulation of fracture network. Chiles and de Marsily (1993) noted that the calibrated flow model could be used either to calculate the large scale equivalent hydraulic conductivities of a continuum or to directly solv e flow problems. An interesting illustration of using the calibrated flow model to calculate large scale hydraulic conductivities was given by Cacas et al. (1990a, 1990b). In their study of modeling fracture flow with a stochastic discrete fracture network and its calibration and validation for flow and transport models, they first generated numerical networks of fractures represented by disks that had the same local geometric and hydraulic properties of field data (Fanay Augeres mine). However for flow and transport simulations, they used a different approach and assumed that the flow occurred through channels inside the fractures instead of taking place over the entire area of fractures. Then they used this three dimensional random network of pipes at an i ntermediate scale to predict global properties of the fractured medium. Their study also sets an important demonstration of hybrid modeling technique, since they start with local scale field data and use stochastic modeling to estimate global representativ e elementary volume hydraulic behavior. Tsang and Tsang (1990) worked on the permeability and dispersivity of variable al. (1992) used the idea that channeling could be e xplained by variation of apertures within a single fracture plane and proposed a variable aperture fracture network model PAGE 28 28 for flow and transport in fractured rocks. Instead of channeling, they used variable aperture fractures for flow and transport modelin g. In another study based on Cacas et in discrete channelized fracture networks, Guerin and Billaux (1994) studied the relationship between connectivity and the continuu m approximation in fracture flow and transport modeling. Chiles and de Marsily (1993) pointed out to the importance of the study of the geometry of the single fracture before running flow simulations on the generated fracture network. Since actual fracture s are not parallel plates or disks, they suggested replacing the network of disks by network of linear links and computing flow on this network of linear links. This is an idea very parallel to the work of Cacas et al. (1990a, 1990b). A second method for e stimating continuum behavior by analyzing fracture networks is percolation theory (National Research Council, 1996). In nature, rocks and soils exhibit correlations in their properties at all scales and these correlations are modeled by fractal geometry. S imilarly to fractal concepts, percolation theory explains how the interconnectivity of various regions of a system affects its overall properties (Sahimi, 1995). Depending on the percolation state of a fractured rock system and how far it is from critical point, the connectivity and permeability of the fractured medium can be expressed in terms of fractal power law relationships. Robinson (1983), Hestir and Long (1990) used this approach for Poisson line systems. One of the issues in characterizing fracture network systems based on field data measurements is that on any scale that is aimed to be modeled usually a small number PAGE 29 29 of large features dominate the behavior (Long et al., 1989). To overcome this problem, Long et al. (1991) proposed an inverse approach simulated annealing, for construction of fracture hydrology models conditioned by geophysical data. In their work, rock is represented by a three dimensional network of finite fractures. Then, simulated annealing is used to find an appropriate pattern of connected fractures given some experimental information about the studied rock system. The model is used to predict the inflow to the simulated drift experiment of Stripa mine. As summarized in this chapter, there have been a wide range of approaches to m odel discrete fracture networks in combination with different flow and transport conceptualizations. Global optimization methods are among the most recent computationally advanced methods in this area. La Pointe (2000) used various statistical pattern reco gnition methods including multivariate regression and neural net analyses for predicting hydrology of fractured rock masses from geology. Sirat and Talbot (2001) applied artificial neural networks to fracture analysis at the sp Hard Rock Laboratory, Swed en. Tran et al. (2009) proposed a new objective function formulation by fuzzy sensitivity analysis for modeling of fracture networks. This dissertation is based on the idea of an extended discrete fracture network modeling technique for discharge estimatio ns using the data of a new passive flux meter for fractured rock. One of the major drawbacks of discrete fracture network modeling technique is that some of the detected fractures through conventional borehole measurements can be nonconductive. Since FRPFM measures flow through only active fractures, this concern is automatically eliminated by the method followed here. In addition to that, FRPFM is an innovative technology for in situ measurements of PAGE 30 30 cumulative groundwater and contaminant fluxes in fracture d media. Taking advantage of this ground breaking feature of FRPFM the cumulative groundwater fluxes are assigned as random variables to the elements of the simulated discrete fracture networks to conceptualize the suggested discharge estimation method. T he next three chapters will elaborate on the details of this method. PAGE 31 31 CHAPTER 3 A STOCHASTIC MODEL F OR ESTIMATION OF DIS CHARGES AT A VERTICA L TRANSECT OF MULTIPLE BOREHOLES IN FRACTUR ED ROCK: 1ST STEP: RANDOM NUMERICAL REA LIZATIONS OF FRACTUR E NETWORKS Fr actured rock masses are composed of rock material and discontinuities. These discontinuities act as the main pathways that determine the performance of rock in terms of groundwater and solute transport. Despite the importance of joints as means of fluid tr ansport, most of the time it is not feasible to map fracture networks on a large scale and measurement of all discontinuities is impossible. Fractures are not accessible directly but only at their intersections with the boreholes. Therefore, for practical purposes stochastic models are formed from sparse data which come from borehole readings. Once the geometry of the joint network is specified, the flow through this network can be studied as the next step. For the purpose of stochastic modeling, a code is developed in MATLAB. The software developed includes two parts: In the first part of the code, FRPFM data are groundwater and contaminant discharges are estimated using th e s e cre ated FRPFM These stochastic simulations (including both parts of the code) are repeated many times in order to get a realistic picture of possible sets of fracture patterns whic h cover the range of possible fracture networks that would comply with the assigned probability distributions of joint characteristics (Monte Carlo method). In this way, an assessment is possible in terms of probabilities for the encountered contaminant pa thways. As the final step of the analysis, uncertainty evaluations are carried out on the ensemble results. PAGE 32 32 Simulation of a fracture network involves the following steps: First trace mid points are located according to a defined fracture density; then at t hese points orientation, fracture length and aperture values are assigned to each joint due to probability density distribution of these characteristics. For this study, groundwater and solute fluxes are also assigned to joints according to predefined prob ability distributions. The output of e s e data are used as input for discharge estimations for the second part of the code. Regarding FRPFM data, an important point that needs to be noted is that flux measurements by the passive flux meter actually represent discharges per unit trace length, which are flux times aperture values The fracture aperture s play a crucial role on the flow and transport processes in a fractured formation (Silberhorn Hemminger et al., 2005). Nonet heless the determination of the aperture size is not trivial (Chiles and de Marsily, 1993) and a perture measurements are circumstantial (Berkowitz, 2002) Therefore, it is a convenient fact that FRPFM measurements directly provide aperture integrated flux es and do not require separate measurement of fracture apertures for discharge estimation. Main Assumptions for Generation of FRPFM Data The main assumptions for generation of FRPFM data and discharge estimations using th ese data are summarized as follows: FRPFM measures fluxes through only active fractures. Thus parallel to this, for modeling purposes it is assumed that throughout the fractured medium under consideration, the flow occurs only through the active fractures. Hence the contribution of porous r ock material is not taken into account. The simulated fracture networks are composed of linear, constant aperture discontinuities embedded in an impermeable matrix. PAGE 33 33 Figure 3 1. A) An illustration showing a vertical transect taken from a 3D fractured rock. representing a possible realization of this vertical transect with one borehole in the middle. PAGE 34 34 It is assumed tha t the groundwater and solute fluxes are coming perpendicular to the simulated vertical transect of fractured rock under study. The simulated two vertical transect containing th e two dimensional, linear trace intersections of fractures with that sampled plane (Figure 3 1). In fact, the assigned input fracture data (probabilistic joint characteristics) might be the result of many different combinations of traces on that 2D plane. Since one simulation corresponds to only one possible representation of the reality, the model is run many times to get a realistic picture of the fractured pathways. Accordingly, the only component of the connectivity of the fractures that will play a rol e for the groundwater and contaminant flows is in the direction perpendicular to the simulated face of the rock. The fractured rock passive flux meter detects flow only through active fractures. Therefore, the simulated fracture network is composed of acti ve fractures only. Though at the two dimensional vertical transect they might not be connected, it is assumed that they are connected in the third perpendicular dimension through which the flow occurs. For purposes of flow and transport simulations, since the related connectivities for groundwater and solute flows are in the perpendicular direction to the simulated 2D vertical transect, it is assumed that these flux distributions are stochastic and can be described by a probability distribution function. Th e parameters of these The first step in stochastic modeling is simulation of FRPFM data and this has for simulation of fractures and location of boreholes from two different Excel files whose horizontal and vertical dimensions are determined by the user. The number of poin ts are located due to a point Poisson process in a way that will comply with the user defined areal fracture density (number of trace mid points per area). After that orientation, trace length distribution and aperture characteristics are assigned to each joint set. The distribution parameters for all these are user defined by the input file. PAGE 35 35 Finally groundwater and solute fluxes are assigned to each set of discontinuities b oreholes across the vertical transect. The number and location of boreholes are user Simulation of Fracture Locations As stated before, basic stochastic discrete fracture network models are based on P oisson processes (Chiles and de Marsily, 1993). It is common practice to group fractures into sets due to orientation based on the fact that in nature discontinuities formed by the same geological activity (stress strain relationship) will have similar cha racteristics i.e. similar orientations. To model a fracture network, the fractures in each fracture set are first located assuming a point Poisson process. The Poisson point process corresponds to the intuitive idea of randomly distributed points in space. Basically there are four types of point process models for locating trace mid points. These are: Homogeneous Poisson process, inhomogeneous Poisson process, Poisson cluster process, Cox process. Homogeneous Poisson Process This is the most fundamental poi nt process defined by the following properties: 1. In a finite planar region A (which is a subregion of R ), for a constant intensity function of the process ; the number of points falling inside A N(A) follows a Poisson distribution with mean ( A is the area of the planar region.) 2. For any disjoint subregions, A 1 A 2 A k within R N(A 1 ) N(A 2 ) N(A k ) are independent random variables. The Poisson point process has an important conditional property corresponding to the idea of random points that is, given that N(A) = n these n points are independently and uniformly distributed over A PAGE 36 36 points is achieved by generating n random numbers from Poisson distribution (3 1) with mean where is areal fracture density (defined as number of trace mid points per area) for each fracture set. Then for each n points in each joint set, 2D values are drawn from uniform distributions to be used as coordinates of the trace mid points simulated for that set. Inhomogeneous Poisson Process If the constant density within the region A (which is a subset of region R ), is no longer constant but is replaced by a location dependent variable intensity function then the Poisson distribution becomes inhomogeneous with the following properties: 1. In a finite planar region A (which is a subregion of R ), the number of points falling inside A N(A) follows a Poisson distribution with mean where X is the location variable within A 2. For any disjoint subregions, A 1 A 2 A k within R N(A 1 ) N(A 2 ) N(A k ) are independent random variables. Poisson Cluster Process This is also called parent daughter process. It describes a class of spatial point patterns with aggregated characteristics. Poisson cluster process is defined by the following properties: 1. Parent points are generated from a homogene ous Poisson process with constant density P 2. For each parent, p i a random number of daughters, d i ( i = 1, 2, is generated independently and identically for each parent from the same probability distribution. PAGE 37 37 3. The locations of dau ghters relative to their parents are independently and identically distributed according to a probability distribution function. In general, daughters form a cluster centered at their parent point. Cox Process The Cox point process, named after Cox (1955) is an underlying general model for most point processes. The three Poisson processes (homogeneous, inhomogeneous and cluster) can all be derived from the idea of Cox process (Diggle, 2003). The inhomogeneous Poisson process has a location dependent variabl e intensity function If this intensity function itself is the realization of a stochastic process, then the resulting point process is known as Cox process (Schabenberger and Gotway, 2005). Thus Cox process adds stochastic nature to location dependen t and this is why this method is also referred to as doubly stochastic process. An inhomogeneous Poisson process with intensity function creates aggregated patterns. Regions where is high will have apparent clusters of points compared to reg ions of low intensity. The source of this localized heterogeneity might be stochastic in nature and Cox process uses random intensity to explain this. Accordingly, is a particular realization. Diggle (2003) define Cox process with the following p roperties: 1. { R d d = 2 or 3 } is a non negative valued stochastic process. 2. Conditional on { R d d = 2 or 3 }, the events form an inhomogeneous Poisson process with intensity function (For each realized random sample of the random process the Cox process becomes an inhomogeneous process with intensity function ) Fracture Generation After location of trace mid points by one of the above explained point processes, traces are generated according to orientation and length values drawn from appropriate probability distributions. In the past, extensive research has been performed to PAGE 38 38 determine most typical probability distribution functions related to trace orientations, lengt hs and apertures. These are summarized as follows: Probability Distributions of Fracture Orientations One of the most commonly used probability distributions for modeling trace orientations is Fisher distribution. Fisher (1953) assumed that the discontinui ty normals would be distributed about some true value within a fracture set. Fisher distribution has the following density: 2) whe re is the angular deviation from the mean, or true orientation and K constant, which is a measure of the degree of clustering or preferred orientation within the population. Typical K values for rock joints within a set range from 20 to 300 (Kemeny and Post, 2003). For simulation purposes, the computer generated picks from Fisher distribution are calculated using the following approximate estimate (Priest, 1993): (3 3) where is the angular deviation from the mean and Random(0,1) is a uniform random number between 0 and 1. This is a symmetric distribution that is more suitable to apply for symmetric data. Einstein and Baecher (1983), Kelker and Langenberg (1976), Mardia (1972) and Watson (1966) describe numerous other models, such as Bingham distribution, that can better describe asymmetric data (Priest, 1993). Literature review shows that most preferred distributions for modeling joint orientations are uniform, normal and Fisher distributions. PAGE 39 39 Probability Distributions of Fract ure Lengths As other trace characteristics, f racture length is assumed to be the result of random processes. Dershowitz and Einstein (1988) state that a number of distributions have been proposed to describe probability distributions of trace lengths, thes e being exponential, log normal and hyperbolic. Among these, most commonly preferred distributions to explain joint lengths are exponential (e.g. the study by Priest and Hudson (1981)) and log normal (e.g. the study by Baecher et al. (1977)). Variations in the fitted distributions to field joint data originate from the fact that measured traces on the outcrops are limited to the size of the cutoff and do not reflect the actual joint lengths. Also, these differences in the observed trace length distributions are due to different geomechanical processes which lead to the formation of fractures. Dershowitz and Einstein (1988) suggest that uniform stresses result in exponentially distributed discontinuities whereas multiplicatory processes like breakage cause lo g normally distributed discontinuities. Probability Distributions of Fracture Apertures Statistical studies on fracture apertures report both log normal (e.g. Snow, 1965) and exponential (e.g. Barton, 1982) distributions. Flux Distributions Assign ed to Each Fracture Set e s e data include groundwater and solute discharge measurements per length of fractures (on the 2D vertical transect considered) at the intersection points of acti ve fractures with the boreholes. For this model, fractures are assumed to be connected in the perpendicular third dimension and the groundwater and solute fluxes are assumed to be random. Accordingly, for the purpose of generation of FRPFM data, probabilit y distributions are PAGE 40 40 assigned to both groundwater and solute fluxes and related parameters are entered as of the problem modeled, this will be illustrated in the next c hapter. e s e data include: number of joint intersections at each borehole, related X and Y coordinates, orientation angle, groundwater discharge per length of fracture (on the 2D vertical transect) [L2T 1] which is equal to { groundwater flux [LT 1] aperture [L] }, contaminant discharge per length of fracture (on the 2D vertical transect) [ML 1 T 1 ] which is equal to { cont aminant mass flux [ML 2 T 1 ] aperture [L] }, values detected at the intersection points of joints with the boreholes. PAGE 41 41 CHAPTER 4 A STOCHASTIC MODEL F OR ESTIMATION OF DIS CHARGES AT A VERTICA L TRANSECT OF MULTIPLE BOREHOLES IN FRACTUR ED ROCK: 2ND STEP: DIS CHARGE ESTIMATIONS A ND UNCERTAINTY ANALY SIS Theory One of the biggest challenges regarding borehole data is that neither areal fracture density (number of fractures/area) nor extent of encountered fractures can be directly determined from borehole measurements. However, linear fracture frequency (number of active fractures intersected per unit length of borehole) is directly measured at each borehole and available as FRPFM data (ESTCP, 2008) Linear fracture frequency is a measure of the product of areal fracture density and trace length since the probability of a discontinuity intersecting a borehole is proportional to this product (Long and Witherspoon, 1985). For a given set of fractures, Robertson (1970) and Baecher et al. (1977) express this rel ationship as follows: (4 1) where is linear fracture frequency (number of fractures in the given set per unit length of borehole), is the mean orientation angle between the borehole and the normal to the given fracture set, is measured fracture frequency corrected for orienta tion bias (explained in the next section), is the areal fracture density (expected number of fractures per area) and is the mean trace length for that fracture set. The above relationship between areal fracture density and linear frequency is infer red from the assumption that the center points of joints are randomly and independently distributed in space forming a Poisson field (Baecher et al., 1977). It is well known that the points of intersection of an arbitrary line with a set of random planes PAGE 42 42 ( i.e. Poisson flats) themselves form a Poisson process and thus have an exponential interarrival distribution (spacing of joints) (Kendall and Moran, 1963). For deriving the density of jointing from spacing data, Baecher et al. (1977) has three principal as sumptions. In addition to the main assumption that the trace mid points are randomly and independently distributed according to Poisson distribution, they also assume that joints are circular two dimensional disks and the radii of joints are log normally d istributed. Accordingly, for a joint of radius r to intersect a boring parallel to its pole, the joint center must be located within a cylinder of radius r having the boring as its axis. Therefore, for the case of a joint whose pole makes an angle with t he boring, it is possible for this joint to intersect the boring as long as its center lies within an elliptical cylinder of major axis r and minor axis The orientation of the major diameter of this elliptical cylinder is parallel to the joint strik e. For joints with constant orientation and constant radius r the number of intersections along a boring is distributed as (Baecher et al., 1977): (4 2) When the radii follow a distribution of f(r) and orientations follow a distribution of the above equation becomes (Baecher at al., 1977): (4 3) Assuming the radii of joints are log (1968) derivation of expectation of Equation 4 3 when f(r) is rep laced by log normal distribution, Baecher et al. (1977) arrive at the following result: (4 4) PAGE 43 43 where corresponds to the mea n volumetric joint density (expected number of trace mid points/volume), N is the number of joints intersected along the boring length of L Equation 4 4 is valid for three dimensional case, representing the mean joint area. When this equation is applied to two dimensions and also orientations are allowed to follow a distribution, Equation 4 4 becomes equal to Equation 4 1; this time A (areal fracture density for expected number of fractures per area) replacing volumetric joint density and (mean trace length) replacing the mean joint area in Equation 4 4. Linear Fracture Frequency and Related Sampling Bias Correction lowed. As in Figure 4 1, when a set of N parallel planar discontinuities is considered (based on set definition in Chapter 3), the frequency of discontinuities from this set intersected by a sampling line that is normal to the set is defined as gth of set normal If there is a sampling line of general orientation making an acute angle with the set normal, the discontinuity frequency along this sampling line ( sampling ) is expected to reflect the normal frequency for the discontinuity set; howev er it is dependent on the angle Accordingly, the observed fracture frequency along the sampling line ( sampling ) is given by: (4 5) The above formula shows that the number of discontinuities from a given set, intersected by a sampling line that makes an acute angle to the set normal, reduces with increasing values of and approaches zero when PAGE 44 44 So orie ntation data from linear sampling lines could be severely biased and needs to be corrected Terzaghi (1965) suggested to eliminate this bias by weighting the sample size N for each discontinuity set by such that N = N sampling Figure 4 1. Fracture set intersected by a sampling line of general orientation (Priest, 1985). Accordingly, following the correction by Terzaghi, (in Equation 4 1) represents the linear frequency measured along a borehole corrected for orientation bias. Groundwater Discharge Estimation at the Simulated Transect The definition of true discharge through a transect is the summation of discharges throu gh active fractures whose mid points are located inside the transect. (4 6) l ength of set normal N discontinuities sampling PAGE 45 45 Groundwater and solute fluxes perpendicular to the transect W H where Q true(transect) is the real discharge throu gh the transect [L 3 /T], N t stands for number of a ctive traces whose mid points are inside the transect (considering the truncation effects which is explained later in this chapter), and l t represents trace length [L] and q t represents discharge per length (velocity times aperture) [L 2 /T] for each trace. As mentioned in Chapter 3, FRPFM measures fluxes through only active fractures H ence the traces that will be simulated for this study are assumed to be only active fractures. The method of estimating groun dwater and contaminant discharges from FRPFM data is based on the idea of linear fracture frequency representing the product of areal fracture density and trace length. For the purpose of discharge estimations, a transect of fractured rock is divided into regions around each borehole and discharge estimations are carried out separately for each borehole considering the zone in the vicinity of that borehole (Figure 4 2). Figure 4 2. Zones around each borehole considered for discharge estimations. The groundwater discharge for a borehole intersecting m number of fracture sets can be calculated by the equation: PAGE 46 46 (4 7) where for a fracture set i ; v i is the mean groundwater flux [L/T], A i is the total joint cross section area [L 2 ], w i is the mean aperture [L], l i is the mean trace length [L] and A is the areal fracture density for that joint set. WH is that part of the transect area which is considered for discharge estimations in the vicinity of the borehole under study. Equation 4 7 is a general representation of estimation of discharges for an area of WH considering groundwater fluxes perpendicular to that area. In reality, the field measurements of groundwater fluxes will include three spatial components and discharge estimations fo r each spatial component will be carried out considering respective spatial component of fluxes and corresponding transect area which receives these fluxes perpendicularly. The purpose of this study is to estimate groundwater and contaminant discharges thr ough a transect of fractured rock using FRPFM data. FRPFM measurements include location and orientation of active (or flowing) fractures, cumulative groundwater and contaminant fluxes measured at the intersection of these active fractures (perpendicular to the transect). These flux measurements are actually groundwater discharges per unit length of the fracture corresponding to velocity times aperture values. On the other hand, trace length measurements and areal fracture density ( A ) are not available from borehole data. Hence, using the estimation of gives the following groundwater discharge equation for each borehole intersecting m number of fracture sets: (4 8) PAGE 47 47 where Q borehole is the groundwater discharge estimated for each borehole [L 3 T 1 ] and definitions for each fracture set representing mean values for that set; q i : groundwater dis charge per length of fracture (on the 2D vertical transect) [L 2 T 1 ] which is equal to { groundwater flux [LT 1 ] aperture [L] } ( q values are obtained from FRPFM measurements), Li i : measured linear fracture frequency corrected for orientation bias i : orientation angle between joint normal and borehole. Again, WH corresponds to the region of the fractured rock (or part of the transect area) around the borehole considered. For a transect of multiple boreholes, discharge estimations for separate boreholes are summed in order to get the total discharge across the transect. The case of multiple boreholes is studied in the next chapter. Contaminated Mass Discharge Estimation at the Simulated Transect Following the same methodology explained above, the contaminated mass discharge at a transect of fractured rock with multiple boreholes is calculated by using the following formula. For each borehole: (4 9) where M Q borehole is the contaminant mass discharge estimated for each borehole [MT 1 ] and definitions for each fracture set representing mean values for that set; J ci : contam inant discharge per length of fracture (on the 2D vertical transect) [ML 1 T 1 ] which is equal to { contaminant mass flux [ML 2 T 1 ] aperture [L] } ( J c values are obtained from FRPFM measurements), L i i : measured linear fracture frequency corrected for orientation bias, i : orientation angle between joint normal and borehole. PAGE 48 48 Consideration of Truncation and Edge Effects for Simulations For the simulations of vertical transects of fractured rock, two t ypes of truncation effects apply at the limits of the transect when traces are partially inside or outside. There might be some joints partly extending out of the transect though their mid points are located inside the transect. Also there might be some jo ints partly crossing through the transect though their mid points are located outside the transect. The true (or real) discharge definition across a transect applies to the discharge through those portions of the discontinuities that are inside the boundar ies of the transect. Hence, in order to handle truncation effects, for true discharge calculations across the transect (Equation 4 6), only the traces whose mid points are located inside the transect are taken into consideration. In this way, it is assumed that the discharges carried out of the transect by those joints with mid points inside the transect and partly extending outside the transect will compensate for the discharges carried into the transect by those joints with mid points located outside and partly entering into the transect. Another effect that is considered for the simulations carried out for this study is the edge effects. In practice, it is not possible to have a standalone volume of rock that contains all the fractures within. In order t o avoid edge effects, the small domain that is the vertical transect chosen should be representative of a large infinitely extending discrete fracture pattern. Otherwise, there will not be sufficient traces intersecting the transect to mimic the real distr ibution of trace lengths. In order to get rid of these edge effects, outside a larger region of 10m by 10m is considered as the total simulation area. Then a vertical transect of 1m by 1m is placed with one borehole in the middle of this transect and disch arge estimations are focused on this 1m by 1m transect (Figure PAGE 49 49 4 3). In this way, it is aimed to eliminate the errors due to edge effects on the discharge estimations over the simulated transect. Figure 4 3. A) One realization of a 10m by 10m total simulation are a with horizontal traces of 0.3 m length. B) Focus on 1m by 1m vertical transect with one borehole placed in the middle (Areal density = 20). Monte Carlo Simulations for Discharge Estimation and Uncert ainty Analysis Discharge estimations using hypothetical FRPFM data at a vertical transect of fractured rock is carried out by using Monte Carlo method. Monte Carlo technique, first proposed by John von Neumann and Stanislaw Ulam is a statistical method of approximating the solution of a mathematical or physical system by generating random numbers for an arbitrary probability density function (Fishman, 1996). The aim is studying the behaviour of random processes where the input i s stochastic and the results are uncertain (Baecher and Christian, 2003). In this research, Monte Carlo technique is used for two purposes: first for simulation of fictitious FRPFM data and second for uncertainty analysis of proposed discharge estimation e quations for each created scenario. Accordingly each Monte Carlo simulation includes two stages of stochastic solution: first, stochastic modeling of A B PAGE 50 50 using Equations 4 8 and 4 After Monte Carlo simulations are run many times, output is stored in files and statistical analysis is carried out on the ensemble results to evaluate the uncertainty of predicted discharges. In the first step of the simulatio ns, the method is used to generate pseudorandom numbers to model probabilistic fractured rock characteristics such as location of trace mid points due to Poisson distribution, and generation of other joint characteristics like length, orientation, aperture and assignment of flux readings to those fractured networks according to different probability distribution functions that might be are repeated many times, all th e corresponding real groundwater and contaminant discharges are calculated. Since the main purpose of this study is testing the proposed groundwater and contaminant discharge estimation Equations 4 8 and 4 9, predicted discharges using these equations are compared to those real discharges and quantification of uncertainties in flow and transport predictions originating from variability of input parameters are carried out on the ensemble results of Monte Carlo simulations. The parameters involved in this mod eling process are transect width ( W ) and height ( H ), number of wells placed inside the transect, areal fracture density ( A ) (number of fractures per unit area), orientations, lengths and fluxes assigned to traces, number of intersections of traces per len gth of each borehole, orientations and fluxes observed at these intersections. Among these parameters, location of trace mid points, trace lengths, orientations and fluxes are assigned as random variables through the simulation procedure. The stochastic na ture of these variables lead s to variations in the PAGE 51 51 true discharge distributions across the transect as well as variations in the hypothetical FRPFM data which results in fluctuations in discharge predictions (which will be illustrated throughout the rest o f this chapter). Assessment of Stability of Monte Carlo Simulations to Model Real Discharges Monte Carlo technique is essentially a repeated sampling of the stochastic process (Baecher and Christian, 2003) and deciding the appropriate number of mode l runs whose ensemble will give results with the maximum possible accuracy is the first step in this analysis. Usually the mean and standard deviation of a specific output of interest is plotted as a function of number of model runs and convergence of thes e moments to single values is used as a criterion to decide the sufficient number of Monte Carlo simulations (Langevin, 2002). Here, the criteria used for this stability assessment is the real groundwater discharge at the simulated transect for the most cr itical case in terms of predicting discharge. In order to be able to use the simulated fractured rock transects to test the reliability of proposed discharge estimation Equations 4 8 and 4 9, first it has to be assured that Monte Carlo simulations are repe ated enough number of times to be able to represent the ground truth data values. Then comparison of estimated discharges to these ground truth values will be carried out for reliability analysis (explained in the next section). The simulations illustrated throughout the rest of this chapter include trace lengths from 0.1m to 1.5m with areal densities 20, 40 and 60. The number of intersections detected along the boring plays the key role in proposed discharge estimation equations. The lower the areal fractu re density and the shorter the trace length, the less becomes the probability of intersections with the boring. Hence, for discharge estimations, the case with the shortest trace length and the smallest areal fracture PAGE 52 52 density is expected to be critical. Fo r the case of A equals to 20 and trace length is 0.1m, some test simulations resulted in no intersections with the borehole. For that reason, as the worst case scenario to decide about the number of simulations that will suffice for the convergence, the c ase of 0.3m length traces with areal density of 20 is chosen. If the convergence is achieved for this most critical case, then it will guarantee the reliability of the simulations for the other cases. tensive 200,000 Monte Carlo simulations are performed on a total simulation area of 10m by 10m where the vertical transect of 1m by 1m is placed in the middle of this area with one borehole (like in Figure 4 3). Simulations have been run for horizontal, co nstant length traces of 0.3m with areal density of 20. Accordingly, for 20 traces inside the transect if constant apertures of 0.5 mm and constant groundwater fluxes of 100m/day are assigned, the mean groundwater discharge across the transect is expected t o be 0.3 m 3 /day. Mean and standard deviation values of real groundwater discharges are plotted as function of number of Monte Carlo simulations as shown in Figures 4 4 and 4 5. Results suggest that 30,000 simulations would suffice to avoid errors that coul d occur from Monte Carlo simulations. Both figures illustrate that oscillations stay within reliable limits after 30,000 realizations. The next step of stability test is quantification of the uncertainty involved at the end of 30,000 simulations. The relat ive error (also called fractional standard deviation) for a Monte Carlo estimate is defined as (European Commission, Joint Research Centre, 2008): (4 10) PAGE 53 53 where N is the number of simulations, is the mean Monte Carlo estimate and x is the estimate of each simulation. This fractional standard deviation is the measure of the statistical precision of the Monte Carlo result. At the end of 30,000 simulatio ns, the mean real groundwater discharge is 0.299963 m 3 /day with R (fractional standard deviation) estimated according to Equation 4 10 as 0.00129 corresponding to 0.129% which is an indication of reliable estimates from Monte Carlo simulations. ( R values less than 0.1 are accepted as indication of reliable results according to European Commission, Joint Research Centre, 2008). Therefore it was decided that 30,000 runs would be enough to carry out the calculations through modeling for discharge est imations. Figure 4 4. Convergence of the sample mean groundwater discharge to the real value as function of number of Monte Carlo simulations. PAGE 54 54 Figure 4 5. Convergence of the sample standard deviation of groundwater discharge as function of number of Monte Carlo simulations. Reliability Analysis of Predicted Discharges After checking the statistical precision of the Monte Carlo simulations to model discharges, the next step was comparison of the real discharges at simulated transe cts with the estimated discharges calculated by the proposed stochastic method. The rest of this chapter examines these discharge prediction errors. In all the simulations explained in the following sections, discharge estimations and uncertainty analys es are carried out together. For all types of transects considered, Monte Carlo simulations are run 30,000 times for each case in order to avoid errors that could occur from Monte Carlo simulations. All the simulations consist of two steps; first step, gen second step, application of the above explained discharge estimation formulas to that PAGE 55 55 r used in the simulations throughout this dissertation is a general term to define the fluxes which can be either groundwater or contaminant. Since FRPFM data will include both cumulative local groundwater and contaminant fluxes, the flux parameter is in cluded in the following simulations as one variable to represent fluxes in general. Accordingly, the discharge results obtained might apply to both groundwater and contaminant discharges. In order to test the discharge estimation formulas, different scenar ios are considered and errors are evaluated on these different cases as explained as follows. Simulations with Constant Length, Constant Orientation Traces Definition of a fracture network by Poisson distribution in two dimensions involves two fracture characteristics, length and orientation as random variables. On the other ass igned to the joints as being defined as random variables. For a fracture system as indicated in Figure 4 3, and as long as the trace lengths orientations apertures and perpendicular fluxes across the vertical transect are described as constants instead of being random variables, the only random variable to be tested will be the location of trace mid points by Poisson distribution. As explained by Baecher et al. (1977), at the end of Monte Carlo simulations, the expected value of will eventually ap proach the number of intersected joints per length of the borehole (corrected for orientation bias). Hence, the estimated discharge by Equation 4 8 and/or 4 9 would be expected to approximate to the real discharge across the vertical transect with unbiased ness. This case is studied as follows. PAGE 56 56 Table 4 1. Input parameters used in FRPFM data simulation for discharge estimation s on a 1m x 1m transect with horizontal, constant length traces Set 1 Areal density (number of fractures/area) 20, 40, 60 Orientation (degrees) 0 Length (m) 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3, 1.4, 1.5 Aperture (m) 0.0005 Flux 100 As the first step of testing discharge predictions, for constant length horizontal traces across a 1m by 1m transect, simulations have been run for various trace lengths with variable areal densities (Table 4 1). For each case, 30,000 realizations have bee n run in order to be able to get results with maximum accuracy. To analyze the results of Monte Carlo simulations, the error for each realization is normalized with the mean value of the real discharge across the transect for ensemble of realizations. The resulting dimensionless absolute error, its mean and standard deviation are defined as follows: (4 11) (4 12) (4 13) PAGE 57 57 where N r stands for number of realizations and represents the mean real discharge over all number of realizations. The uncertainty analysis and related figures throughout this dissertation are based on these formulae. As observed in Figure 4 6, mean norm alized absolute error for Q remains within system, the success of discharge prediction depends on the number of intersections of that trace length per length of the bor ing. The stochastic discharge estimation method proposed here is based on the idea of i.e. the probability of a joint intersecting a unit length of a boring is proportional to the areal density and mean fracture length. Due to the stocha stic nature of modeling, sometimes number of intersections per length of the boring are less than the expected corresponding value of which gives rise to negative errors, and sometimes these number of intersections are greater than the corresponding reality value of which gives rise to positive errors. This fact is observed as fluctuations of mean error in Figure 4 6. The plots also point out that as the areal density increases it is more likely that the number of intersections per length of t he boring inside the transect will approach more to the exact value of The method predicts exactly correct discharges with zero error when the length equals to the transect width, the number of intersections per length of the boring is actually equal to the number of trace mid points across the transect (or in other words linear fracture frequency is equal to the areal trace density for this special case). As the to the exact real value. PAGE 58 58 Figure 4 6. Results of Monte Carlo simulations with horizontal, constant length traces across a 1m by 1m transect for variou Figure 4 PAGE 59 59 Standard deviations of normalized absolute errors show a decreasing trend as the lengths at the same length with transect width, and then increasing in the range of 1 to 1.5. T rends evident in Figure 4 7 are based on how easily is approaching When the trace length is a lot smaller than the transect width, fewer intersections with the borehole would be enough to predict the corresponding real areal density times the mean trace length. Since the model is a stochastic model, on the average the whole ensemble of 30,000 simulations will approach to the expected real value. However some realizations will have more number of intersections whereas some othe rs will have less number of intersections than the exact real value. For smaller trace lengths, even one intersection with the boring that is in excess or short of the exact expected value will result in a larger deviation of the estimated discharge from t he sample discharge. This impact will diminish as the trace length size gets closer to the transect, the cases where greater number of intersections are needed to predict the areal density times the trace length. So for these cases, the contribution of eac h intersection with the boring will be less in error calculations. This explanation is also valid to describe another observation from Figure 4 7, that is, as areal density increases, standard deviations of prediction errors decrease Results of discharge estimations with horizontal, constant length traces have also been tested for various number of trace mid points in a 1m by 1m transect. The input data for these simulations are given in Table 4 2. As seen in Figure 4 8, all the mean e rrors remain within 0.3% range. Again t he fluctuations around zero are due to the random nature of simulations. PAGE 60 60 Table 4 2. Input parameters used in FRPFM data simulation to test the effect of 1m x 1m transect with horizontal, constant length traces Set 1 Areal density (number of fractures/area) 10, 15, 20, 25, 30, 35, 40, 45, 50 Orientation (degrees) 0 Length (m) 0.3, 0.5, 0.8, 1.3 Aperture (m) 0.0005 Flux 100 Figure 4 8. Mean predicted discharge errors as function of number of trace mid points within the transect (for horizontal, constant length traces). As seen in Figure 4 9, standard deviations of errors decrease with increasing number of points within the tr ansect. This means that for smaller areal densities, PAGE 61 61 discharge predictions have higher uncertainty. As the figure suggests, the most reliable discharge predictions will be for the cases with higher areal trace densities in combination with trace lengths as close as possible to the transect width. Figure 4 9. Standard deviations of normalized absolute errors for Q as function of number of trace mid points within the transect. As an additional way to test the estimation errors for traces of constant length and orientation, traces have been given certain orientations and Monte Carlo simulations have been run for a number of constant trace length options (Figure 4 10). Input parameters for these simulations are given in Table 4 3. The results f or mean prediction errors and standard deviation s of errors as function of orientation angles for constant length traces are shown in Figures 4 11 and 4 12. PAGE 62 62 Table 4 3. Input parameters used in FRPFM data simulation to test the effect of various constant o rientations on discharge estimations on a 1m x 1m transect with constant length traces Set 1 Areal density (number of fractures/area) 40 Orientation (degrees) 10, 20, 30, 40, 50, 60, 70, 80 Length (m) 0.3, 0.5, 0.8, 1.3 Aperture (m) 0.0005 Flux 100 Figure 4 10. One realization for a 1m by 1m transect with 0.3m constant length traces of 40 orientation. Mean normalized absolute errors as function of orientation angles (ranging from 10 to 80) remains within 0.5% as observed in Figure 4 11. As in previous cases, the fluctuations of error originate from the stochastic nature of the model. PAGE 63 63 Figure 4 11. Results of simulations to test the effect of constant orientations on the discharge estimations for the basic transect with constant length traces for an areal density of 40. As explained earlier in this chapter, linear sampling techniques like boreh ole measurements are subject to orientation bias. The ideal case for determining the joint frequency occurs when the discontinuities intersect the borehole perpendicularly, tion is already included in Equations 4 8 and 4 9 as division by Results in Figure 4 11 verify that the orientation correction applied to various angles (in the code) works very well and it is not possible to detect an obvious trend of errors as func tion of orientation angles. On the other hand, the standard deviation s of errors show an increasing trend with increasing orientation angles pointing out to the fact that in the ensemble of 30,000 Monte Carlo simulations; for smaller orientation angles, th e prediction errors are all closely distributed around the expected mean zero error whereas for higher angles, it is PAGE 64 64 possible to see cases among these ensemble of realizations that might have high prediction errors. For traces with higher orientation angle s, the chances of intersecting with the boring is less (compared to those traces with smaller orientation angles) and since discharge predictions are based on the number of intersections observed at the borehole, th is larger uncertaint y show s up as larger standard deviations of error in Figure 4 12. Figure 4 12. Standard deviations of discharge prediction errors as function of orientation angles in the basic transect with constant length traces for areal density of 40. In fu ture simulations, trace length, orientation and fluxes are not kept constant but are introduced to the simulation code as random variables one by one to isolate effect s of each fracture characteristic as a random variable on discharge estimation PAGE 65 65 Simulations with Trace Length as Random Variable Simulations with exponentially distributed trace lengths As previously stated in Chapter 3, one of the most commonly preferred distributions to describe the probability distribution of joint lengths is exponential distr ibution. In order to test the reliability of discharge estimation equations for case where trace length is a random variable, Monte Carlo simulations have been executed assuming exponentially distributed trace lengths and keeping all the other trace chara cteristics constant (Figure 4 13) Input parameters for this case are given in Table 4 4. Figure 4 13. One realization for a 1m by 1m transect of horizontal traces with exponentially distributed lengths of mean 0.3m (areal density = 20). The resul ts graphed in Figure 4 14 are very similar to those results in Figure 4 6 for horizontal, constant length traces. In both figures, the mean of random variations of error (originating from stochastic nature of simulations) approach to zero ; a result that is anticipated from Equation 4 1. For exponentially distributed trace lengths, random PAGE 66 66 variations of error due to the stochastic nature of the model is more apparent, something that would be expected naturally when trace lengths are distributed randomly inste ad of being kept constant. The increased chance of getting more exact results with higher areal densities is another observation that is common for both cases. Table 4 4. Input parameters used in FRPFM data simulation to analyze prediction errors for discharge estimation s on a 1m x 1m transect with horizontal, exponentially distributed length traces Set 1 Areal density (number of fractures/area) 20, 40, 60 Orientation (degrees) 0 Mean length (m) Exponential distribution 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5 Aperture (m) 0.0005 Flux 100 Priest and Hudson (1981) define the general form of an exponential probability distribution for trace lengths as: (4 14) The corresponding cumulative probability function is: (4 15) where x is the measured length or spacing value and is the mean value of that variable. Exponential distribution favors more frequent occurrence of shorter and closely spaced joints than longer, widely spaced joints (Wyllie, 1999). R egarding trace lengths another point to be considered is that borehole mea surements are subject to length bias ; that is the longer the trace, the more likely it will be intersected. FRPFM data do PAGE 67 6 7 not include any trace length measurements, however the number of intersections per length of the borehole is the key passive flux mete r measurement that leads to estimation of product of areal density and mean trace lengths. From this point of view, f or the hypothetical case considered here the choice of the transect dimensions with respect to the mean trace lengths will play a role for reliability of discharge estimations in Monte Carlo simulations. When the studied transect area is large enough to include sufficient amount of discontinuitie s that will represent the true distribution of the fracture network, the number of intersections detected by the borehole will reflect a more realistic estimation of mean joint length times areal density resulting in more accurate discharge estimates. In o rder to avoid all these related biases, mean trace lengths are chosen in the range of 0.1m to 1.5m for 1m by 1m study area in Monte Carlo simulations. The purpose of keeping parameters of joint distributions within close range of transect dimensions is to be able to observe the true nature of the distribution of trace lengths within the studied transect. Standard deviations of error for exponentially distributed trace lengths is an indication of how is approaching In fact Figure 4 15 is quite similar to Figure 4 7 showing the same trends with only one main exception that for variable length traces it is not possible to observe zero standard deviation. In both figures it is closer to 1, the reliability of discharge predictions gets higher. Accordingly, in real field applications if mean trace length is known, th is information can be used as a clue to size the transect to ensure the reliability of discharge predictions PAGE 68 68 Figure 4 14. Mean prediction errors for horizontal traces with exponentially distributed lengths. Figure 4 15. Standard deviations of errors for simulations with horizontal, exponentially distributed trace lengths. PAGE 69 69 Simulations with log normally distributed trace lengths Log normal distribution is another commonly used probability distribution for description of joint lengths. The discharge estimations using FRPFM data for the case of log normally distributed fractures is studied for a hypothetical scenario whose parameters are explained in Table 4 5. Table 4 5. Input parameters used in FRPFM data simulation to analyze prediction errors for discharge estimation s on a 1m x 1m transect with horizontal, log normally distributed length tra ces Set 1 Areal density (# of fractures/area) 20, 40, 60 Orientation (degrees) 0 Mean length (m) 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3, 1.4, 1.5 St. deviation (m) Aperture (m) 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75 0.0005 Flux 100 Baecher et al. (1977) use log normal probability function for describing distribution of joint lengths as follows: (4 16) where and are standard deviation and mean of ln(x) In description of joint lengths, the log normal distribution is similar to the exponential distribution in the way that shorter discontinuities have higher frequencies compared to longer discontinuities. The actual shape of the distribution depends on the choice of the parameter standard deviation with respect to the mean, being more PAGE 70 70 skewed for larger standard dev iations. Here, for this case study, the standard deviation values are chosen as half values of the means (coefficient of variation = 0.5) giving the possibility of having more equally distributed joints around the mean rather than a strongly skewed fractur e distribution. For higher standard deviations, the log normal distribution will be more alike to the exponential distribution. (A property of exponential distribution is that its mean and standard deviation are equal.) Figure 4 16. Mean prediction errors for horizontal traces with log normally distributed lengths. As Equation 4 1 anticipates that the number of intersections per length of the boring will approach to the areal density times the mean fracture length, the predicted discharges approach to the real discharge values with very small prediction errors. Like in previous cases, the fluctuations of error in Figure 4 16 are due to the random nature PAGE 71 71 trace zero. Figure 4 17. Standard deviations of errors for horizontal traces with log normally distributed lengths. Standard deviations of errors for discharge predictions for log normally distributed trace lengths (in Figure 4 17) are very similar to those standard deviation s of errors in Figure 4 15 for exponentially distributed trace lengths. Since here the parameter s of the log normal distribution are chosen to have more equally distributed lengths, standard deviations of errors in Figure 4 ns of exponentially distributed lengths. PAGE 72 72 Simulations with Orientation as Random Variable Simulations with uniformly distributed orientations Figure 4 18. One realization for a 1m by 1m transect with 0.3m constant length traces of uniformly distributed orientation between 30 and 50 (areal density = 40). Table 4 6. Input parameters used in FRPFM data simulation to analyze prediction errors for discharge estimation s on a 1m x 1m transect of constant length traces with uniformly distributed or ientations Set 1 Areal density (number of fractures/area) 40 Orientation (degrees) Uniform distribution 0 20, 10 30, 20 40, 30 50, 40 60, 50 70, 60 80, 70 80 Length (m) 0.3, 0.5, 0.8, 1.3 Aperture (m) 0.0005 Flux 100 Uniform distribution is one of the probability distributions used to describe the in situ distribution of joint poles in a fractured rock mass. Related parameters used in PAGE 73 73 Monte Carlo simulations for this case are given in Table 4 6 and one representative realization is shown i n Figure 4 18 Figure 4 19. Results of simulations for constant length traces with uniformly distributed orientations for areal density of 40. PAGE 74 74 As explained throughout this chapter, discharge estimations for this research are based on the relationship Orientation readings are directly available as FRPFM data from field measurements as well as number of intersections with the bor ings. Accuracy of discharge predictions depend on the extent to which estimation of areal density times mean joint length is represented by the number of intersections per unit length of the borehole. Linear fracture frequencies recorded along the borehole s are subject to orientation bias and corrected by dividing with Equations 4 8 and 4 9, orientation as a random variable is not expected to be a source of errors for the predictions. On the other hand, the standard deviations of errors reflect the probability of intersection of the traces with the borings as function of mean orientation angles. The results graphed in Figure 4 19 parallel findings i n Figures 4 11 and 4 12. Simulations with normally distributed orientations Table 4 7. Input parameters used in FRPFM data simulation to analyze prediction errors for discharge estimation s on a 1m x 1m transect of constant length traces with orientations f ollowing normal distribution Set 1 Areal density (number of fractures/area) 40 Mean orientation (degrees) Standard deviation (degrees) 10, 20, 30, 40, 50, 60, 70, 75 2, 5, 5, 5, 5, 5, 5, 2 Length (m) 0.3, 0.5, 0.8, 1.3 Aperture (m) 0.0005 Flux 100 PAGE 75 75 As mentioned in Chapter 3, normal distribution is another option for modeling trace orientations. Related input parameters for this case are given in Table 4 7. Figure 4 20. Results of simulations for constant length traces with normally distributed orientations for areal density of 40. PAGE 76 76 All the prior comments relating to uniformly distributed angles also appl y to normally distributed orientations. For both constant and variable orientation cases, as the orientation angle gets closer to 90, the estimation error standard deviation becomes worse. (Results in Figure 4 20 validate this conclusion.) This is explaine d by the fact that as traces become more parallel to the boring, the probability of intersection decreases and this affects the predicted value and thus discharge. Simulations with Flux as Random Variable Simulations with log normally distributed fl uxes The purpose of simulations with fluxes assigned as random variable is to study the uncertainties that would arise from heterogeneities in flux distribution across the vertical transect. Since the flux data from FRPFM measurements represent discharges per unit trace length, which are flux times aperture values the simulations for this case include both apertures and fluxes to be assigned as random variables to the discontinuities. Table 4 8. Input parameters used in FRPFM data simulation to analyze p rediction errors for discharge estimation s on a 1m x 1m transect of constant length traces with fluxes following log normal distribution Set 1 Areal density (number of fractures/area) 40 Orientation (degrees) 0 Length (m) 0.3, 0.5, 0.8, 1.3 Mean aperture (m) (log normal) St. deviation of aperture (m) 0.0005 0.000125, 0.00025, 0.0005, 0.00075, 0.001, 0.0015 Mean flux (log normal) St. deviation of flux 100 25, 50, 100, 150, 200, 300 PAGE 77 77 Previous studies have indicated that fractur e aperture sizes conform to log normal distribution. Accordingly, log normal distribution is used to simulate apertures in the code. Considering that groundwater flow follows cubic law through fractures (for laminar flow between parallel plates assumption) fluxes are also assigned log normal distribution. The p roduct of apertures and fluxes is treated as a single random variable to represent the hypothetical FRPFM data. Like in the previous cases in this chapter, in order to observe the isolated effects of only fluxes being random variables, all other variables are kept constant. (Detailed input parameters used for the Monte Carlo simulations are given in Table 4 8.) Figure 4 21. Discharge prediction errors for simulations with log normally distributed fluxes. Since the focus is to observ e the effects of the variability of flux times aperture values on predicted discharges, different coefficient of variation values (0.25, 0.5, 1.0, 1.5, 2.0, 3.0) are tested keeping the mean apert ure and fluxes the same for log normal PAGE 78 78 distributions. In reality, water and contaminant fluxes can have quite different degrees of variability and the results of simulations in this section also serve to illustrate this fact such that the variability of water fluxes would be expected to be less (corresponding to coefficient of variation values up to 2 in Figures 4 21 and 4 22) than the variability of contaminant fluxes (corresponding to higher coefficient of variation values such as 3). Figure 4 22. Standard deviations of errors for log normally distributed fluxes. As seen in Figure 4 21, assigning apertures and fluxes as random variables does not cause any obvious change in mean prediction errors. As in previous cases, the mean e rrors observed are random fluctuations due to the stochastic nature of the model. On the other hand, the standard deviations of errors show an increasing trend as the coefficient of variations increase. For the previous cases when fluxes are kept constant discharge estimations depend on the extent to which number of intersections per length of the boring (which is equal to the estimated discharge by Equation 4 8 PAGE 79 79 and/or 4 9) approach e s (the expected real discharge). For the case of variable fluxes, the situation changes to how (that is the estimated discharge from intersected fluxes per length of boring) approach es the expected real discharge that is (where and correspond to the mean trace length and flux over the tr ansect area WH ). Standard deviations of errors are still under 50% until the coefficient of variation is equal to 1.0. The uncertainty of predictions increase s with increasing coefficients of variation as log normal distribution becomes more skewed Hence, as the variability of fluxes increase s, it is more difficult to estimate the mean fluxes over the transect b ased on me asured fluxes at the intersection points with in boreholes. The uncertainty levels for contaminant fluxes (corresponding to the range of coefficient of variation values greater than 2) are much higher than those for water fluxes showing that predicting solute transport is more sensitive to heterogeneiti es in the fractured medium than groundwater flow. Figures 4 21 and 4 on the predicted discharges as a function of variability of fluxes. Both prediction errors and standard deviations of errors are higher for trace lengths 0.3m and 0.5m whereas for the case of trace lengths that are close to the transect width (0.8m and 1.3m) discharge predictions are much more reliable even for high coefficient of variation values for fluxes. One final observation regarding Figure 4 22 is that discharge predictions are more susceptible to variations in fluxes than variations in other random variables (that is lengths and orientations) since standard deviations in Figure 4 22 are greater than the standar d deviations plotted in the previous sections of this chapter. PAGE 80 80 Simulations with Lengths, Orientations and Fluxes as Random Variables Up to this point the random variables, which are location of trace mid points by Poisson distribution, lengths, orientations and fluxes, are introduced to the model one by one keeping all the other variables constant to observe the isolated effects of the studied parameter on reliability of discharge predictions. However under field conditions, none of these parameters are expected to be constant but rather be random. Therefore, in order to test the proposed discharge estimation technique for a more realist ic case, five fracture sets have been simulated with lengths, orientations and fluxes assigned following different probability distributions. Related input parameters are given in Table 4 9. As in the previous cases, Monte Carlo simulations were executed f or 30,000 simulations for a 1m by 1m transect. One realization is exemplified in Figure 4 23. Table 4 9. Input parameters used in FRPFM data simulation to analyze prediction errors for discharge estimation s on a 1m x 1m transect with lengths, orientations and fluxes as random variables Set 1 Set 2 Set 3 Set 4 Set 5 Areal density 20 20 40 20 20 Orientation Uniform distribution Range (degrees) 10 30 35 55 60 80 100 120 130 160 Length Exponential distribution Mean (m) 0.3 0.5 0.8 1.3 1.5 Aperture Log normal distribution Mean (m) Standard deviation (m) Flux 0.0005 0.00025 0.0005 0.0005 0.001 0.0005 0.001 0.001 0.002 0.001 Log normal distribution Mean Standard deviation 100 50 100 100 200 100 200 200 300 150 PAGE 81 81 Figure 4 23. One realization for a 1m by 1m transect with 5 fracture sets all variables assigned randomly. Figure 4 24. Histogram of normalized absolute errors of discharge estimation s for 5 fracture sets with all variables assigned randomly. PAGE 82 82 The error histogram shown in Figure 4 24 illustrates that the mean normalized absolute error for discharge estimation is 0.016% with a standard deviation of 26.11%. This find is in agreement with results obtained in the previous sections. The results from the stochastic model would be expected to be unbiased with standard deviations representing uncertainties originating from variability of random variables. Throughout this chapter, all the numerical simulations have been carried out on a 1m by 1m basic transect with one borehole placed in the middle of this transect to observe the effects of variability of different random variables on the reliability of discharge estimates. In all s representation of fluxes which might be either groundwater or contaminant. Thus all related discharge calculations and uncertainty analyses apply to both groundwater and solute discharges. The next chapter will focus on uncertainty assessments of discharge predictions on a larger transect with multiple boreholes. PAGE 83 83 CHAPTER 5 ILLUSTRATION OF DISC HARGE ESTIMATION AND UNCERTAINTY ANALYSIS METHODS ON A TRANSEC T OF FRACTURED ROCK WITH MULTIPLE BOREHOLES This chapter illustrates application of the method on a vertical transect of fractured rock with multiple boreholes. For this purpose, a transect of 60m (in horizontal direction) by 20m (in vertical direction) is chosen with 5 fracture sets for which the input parameters for the model are listed in Table 5 1. In order to test discharge prediction errors as a function of number of wells across the transect, Monte Carlo simulations have been executed f toge ther. Ten different scenarios have been studied for number of wells increasing from one to ten. In all of the cases, the borings run across the entire vertical length of the fractured rock cross ed at the way that the transect is divided into equal area zones ( as described in Figure 4 2) with each boring placed in the middle of this zone receiving fluxes from this region. The discharge estimations are carried out for each borehole using Equation 4 8 (and/or 4 9) considering that part of the transect area in the vicinity of that borehole. In order to calculate the total discharge across the transect, discharges estimated according to Equation 4 8 (and/or 4 9) for separate boreholes are summed. On the other hand, the real discharge across the studied area is computed using Equation 4 6 considering only the trace mid points that are inside the transect. To avoid ed ge effects, a larger simulation area is chosen as 250m by 250m. PAGE 84 84 Mean prediction errors for the ensemble of 30,000 Monte Carlo simulations have been determined using Equation 4 12. Also for assessment of reliability of predictions, the standard deviations o f errors (calculated using Equation 4 13) have been plotted (Figure 5 2). As fluxes which might be either groundwater or contaminant. In the same way, all the error and uncertainty a nalyses performed would be valid for either groundwater or contaminant discharges. An illustration of the investigated hypothetical transect with six boreholes is given in Figure 5 1. Figure 5 1. One realization of the hypothetical 60m by 20m transect with six boreholes. PAGE 85 85 Table 5 1. Input parameters used in FRPFM data simulation to analyze prediction errors for discharge estimation s on a 1m x 1m transect for 5 fracture sets with lengths, orientati ons and fluxes as random variables Set 1 Set 2 Set 3 Set 4 Set 5 Areal density 0.009 0.09 0.1 0.05 0.001 Orientation Uniform distribution Range (degrees) 125 140 30 75 10 25 100 120 150 170 Length Exponential distribution Mean (m) 25 8 5 10 18 Aperture Log normal distribution Mean (m) Standard deviation (m) Flux 0.002 0.001 0.0005 0.0005 0.0005 0.00025 0.001 0.0005 0.001 0.001 Log normal distribution Mean Standard deviation 300 150 100 100 100 50 200 100 200 200 The results of the Monte Carlo simulations are shown in Figure 5 2. The stochastic nature of the model is observed in the plot of the mean errors as the mean prediction errors fluctuate between 0.25% and 0.25% with changing number of wells. This is due to the randomness of intersection of traces with the borings. Conversely standard deviations of normalized errors show that the uncertainty of discharge predictions with only one well can be as high as 41% decreasing with each boring added for data collecti on. Figure 5 2 indicates that between six to ten wells, the standard deviations of normalized errors do not change much and stay around 20% implying that addition of any extra borings after six will not bring further reliability to discharge estimations. PAGE 86 86 F or every case, the standard deviation of the prediction error will depend on the areal fracture density and trace lengths of the fracture network contained in the studied transect with respect to the transect dimensions. Additionally variabilities in orie ntations and fluxes will play a role in the uncertainty of predicted discharges. Thus, b y looking at the results of the case studied here, it is not possible to make a generalization about the optimum number of wells that should be placed across the transe ct to get the most reliable results while also maintaining the cost of additional wells within reasonable limits. All the factors studied in Chapter 4, including areal fracture densities, trace lengths, orientations and fluxes, will contribute to the stand ard deviations of errors depending on their variability. When the heterogeneity of the fractured rock studied is high, with only one well placed across the transect, it will be more difficult to capture the true characteristics of the fracture network from field measurements and this will show up as high standard deviations of errors in the uncertainty analysis. On the other hand, if there exists a specified uncertainty level that is aimed at for a study; placing one well in the middle of the transect for e xploratory purposes and then analyzing and simulating th e s e observed data in Monte Carlo simulations will be the best option from practical applications point of view. If the results of the numerical uncertainty analysis point to the necessity of more numb er of wells, placing additional wells may be valuable, after taking into account associated costs. PAGE 87 87 Figure 5 2. Mean prediction errors and standard deviations of errors for discharge estimations as function of number of boreholes across the transect. PAGE 88 88 CHAPTER 6 SUMMARY AND CONCLUSIONS This dissertation presents a stochastic model for estimation of discharges across a transect of fractured rock using synthetic fractured rock passive flux meter data from boreholes. The method and the code developed for this purpose were applied to different hypothetical scenarios and reliability analysis of predicted discharges was ca rried out Since FRPFM data will basically include the number of active intersections with the borings and orientations and fluxes dete c te d at these points ; the key factor to be considered wh en interpreting the results is the probability of intersection o f the joints with the wells simulations were executed for horizontal traces with constant lengths The most reliable predictions occurred when this ratio was in the range of 0.5 to 1.5 due to the high probability of intersection of traces with the borings. The predictions were 100% correct without any uncertainty for trace lengths equal to transect width since for this special case linear fracture frequency eq uals areal density and all the fractures with mid poi n ts inside the transect intersect the boring. These simulations were conduc ted for two different assumed probability distrib utions of trace lengths : exponential and log normal. For both distributions, results were similar and also parallel ed results using constant trace length s As for the case of constant length traces, the most reliable estimations were mea n trace was close to 1. PAGE 89 89 Discharge predictions as function of number of trace mid points within the transect (which is equal to areal density*transect area): The most reliable discharge predictions were obtained for the cases with higher areal trace densities Discharge predictions as function of constant orientation angles: T he orientations of traces ha d an effect on the uncertainty of discharge predictions such that the most reliable predictions w ould be for the case of trace s intersecting the boreholes perpendicularly. For traces oriented more parallel to the boring, the probability of intersection with the borehole decreased which added more uncertainty in the discharge estimations. Discharge predictions as function of varia ble orientation angles: These simulations were conducted for uniformly and normally distributed orientation angles with mean values ranging from 10 to 75. R esults for both cases were parallel to results of simulations with constant orientations Discharg e predictions as function of variable fluxes: T he simulations with log normally distributed fluxes show ed that the reliability of discharge predictions was more susceptible to variations in fluxes than variations in trace lengths and orientations. As the variability of fluxes increase d it was more difficult to estimate the mean fluxes over the transect looking at the fluxes measured at the intersection points with the boreholes. Also another conclusion was that the uncertainty levels for contaminant fluxes were much higher than those for water fluxes showing that solute transport would be more sensitive to heterogeneities in the fractured medium than groundwater flow. Discharge predictions as function of number of wells: Increasing the number of wells after one well increase d reliability of discharge predictions up to a certain optimum PAGE 90 90 number of wells which was considered to be case specific. For every case, the uncertainty of predicted discharges will depend on the variability of areal fracture densities, t race lengths, orientations and fluxes within the studied transect. When the heterogeneity of the fractured rock is highly variable, with only one well placed across the transect, it will be more difficult to capture the true characteristics of the fracture network in the measured field data and this will cause high levels of uncertainty in predicted discharges. On the other hand, decision to meet a certain accuracy level while m onitoring the cost of the number of boreholes within acceptable limits will requ ire pre optimization field plan with an exploratory borehole and further analysis of th e s e collected FRPFM data with Monte Carlo simulations to see the reliability of predictions as function of number of wells. In the near future, when real field data ar e obtained from fractured rock flux meters deployed at investigation sites, there exist several possibilities of using th e s e data in combination with the developed stochastic model. After the data are collected, the first step of analysis will be arranging the discontinuities into sets. For this purpose, either the classical method of grouping fractures according to orientation angles or (as an alternative method) a clustering technique that will take into account both orientation angles and fluxes can be u sed. The next step will be estimation of trace length distributions. Unfortunately joint lengths can not be measured directly from boreholes and it is not possible to know where traces end between borings from FRPFM data. One possible method to be consider ed for estimation of trace lengths would be geostatistical analysis of FRPFM data along each borehole and between neighbouring boreholes. Another possibility is that the data obtained from fractured rock passive flux PAGE 91 91 meters c ould be used in an inverse meth od to define the probability distribution functions of trace lengths. As a different alternative, a fractal based method can be employed for determining distribution of lengths of discontinuities. Based on these estimated trace length distributions, areal fracture densities for each set can be density and mean trace length for each fracture set will enable detailed uncertainty analysis using the stochastic model developed in this dissertation. The probability distribution functions of fracture characteristics that are estimated (areal densities and trace lengths) and measured (orientations and fluxes) can be employed as input for the presented model to mimic the real field conditions, calculate discharges and carry out related reliability analysis. T he detailed analys i s of field collected FRPFM data when used together with the stochastic method will provide input to improve the conceptual site model and form a foundation for the development of a robust contaminant transport model in fractured media. PAGE 92 92 LIST OF REFERENCES Andersson, J., Dverstorp, B., 198 7 Conditional simulations of fluid flow in three dimensional networks of discrete fractures. W ater Resources Research 2 3 (1 0 ) 1 876 1 886 Andersson, J., Shapiro, A.M., Bear, J., 1984. A stochastic model of a fractured rock conditioned by measured information. Water Resources Research 20 (1), 79 88. Andersson, J., Thunvik R., 1986. 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