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1 AQUITARD CONTAMINANT STORAGE AND FLUX By GORDON HITCHINGS BROWN 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 UN IVERSITY OF FLORIDA 2014
2 2014 Gordon Hitchings Brown
3 To my parents and my family
4 ACKNOWLEDGMENTS I would like to thank my family, including my wife, my daughter, and my parents for their encouragem ent in this endeavor. At the University of Florida, within the Engineering School of Sustainable Infrastructure and Environment (ESSIE) the Department of Environmental Engineering Sciences provided financial and technical sup port, in particular, my advis or Dr. Michael D. Annable and former ESSIE post doc Dr. Harold Klammer. Much of t he work upon which this dissertation is based was supported by the U.S. Environmental Protection Agency through its Office of Research and Development with funding provided by the Strategic Environmental Research and Development Program (SERDP), a collaborative effort involving the U.S. Environmental Protection Agency (EPA), the U.S. Department of Energy (DOE), and the U.S. Department of Defense (DoD). It has not been subjecte d to Agency review and, therefore, does not necessarily reflect the views of the Agency and no official endorsement should be inferred. EPA researchers Dr. Michael C. Brooks, Dr. A. Lynn Wood, and Dr. Junqi Huang were instrumental in guiding the model dev elopment and results analysis.
5 TABLE OF CONTENTS P age ACK NOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 LIST OF SYMBOLIC NOTATION ................................ ................................ .................. 10 ABSTRACT ................................ ................................ ................................ ................... 14 CHAPTER 1 REVIEW OF LITERATURE ................................ ................................ .................... 16 Overview ................................ ................................ ................................ ................. 16 Contemporary Related Work ................................ ................................ .................. 16 Focus of This Work ................................ ................................ ................................ 22 2 ONE DIMENSIONAL MODEL DEVELOPMENT ................................ ..................... 23 One Dimensional Model for Diffusion ................................ ................................ ...... 23 Dimensionless Source Depletion ................................ ................................ ...... 23 Dimensionless Diffu sion ................................ ................................ ................... 27 Dimensionless Mass Storage and Mass Flux ................................ ................... 31 Dimensionless Time ................................ ................................ ......................... 35 Source Removal ................................ ................................ ............................... 35 One Dimensional Model for Diffusion with Decay and Leakage ............................. 37 Dimensionless Diffusion with Decay and Leakage ................................ ........... 37 Dimensionless Mass Storage and Mass Flux with Leakage and Decay ........... 40 Source Removal with Leakage and Decay ................................ ....................... 43 3 CONCENTRATION, MASS, AND FLUX RESULTS ................................ ............... 45 Dimensionless Results for Diffusion ................................ ................................ ....... 45 Aquitard Concentration Profiles ................................ ................................ ........ 45 Aquitard Mass Storage ................................ ................................ ..................... 52 Lon gevity and Hysteresis ................................ ................................ ................. 53 Aquitard Source Functions ................................ ................................ ............... 55 Dimensionless Results for Diffusion with Decay ................................ ..................... 62 Aquitard Concentration Profiles ................................ ................................ ........ 62 Aquitard Source Funct ions for Diffusion with Decay ................................ ......... 71 Dimensionless Results for Diffusion with Leakage ................................ ................. 74 Aquitard Concentration Profiles ................................ ................................ ........ 74 Aquitard Source Functions for Diffusion with Leakage ................................ ..... 82
6 4 REMEDIATION RESULTS ................................ ................................ ..................... 86 Source Removal Effects on Diffusion ................................ ................................ ..... 86 Aquitard Conc entration Profiles with Source Remediation ............................... 86 Aquitard Source Functions with Source Remediation ................................ ...... 86 Source Removal Effects on Diffusion with Decay ................................ ................... 89 Aquitard Depth Profiles for Diffusion with Decay with Source Remediation ..... 89 Aquitard Source Functions for Dif fusion with Decay and Remediation ............. 89 Source Removal Effects on Diffusion with Leakage ................................ ............... 95 Aquitard Concentration Profiles for Diffusion with Leakage and Remediation .. 95 Aquitard Source Functions for Diffusion with Leakage and Remediation ......... 95 5 CONCLUSIO NS ................................ ................................ ................................ ... 101 1D Aquitard Diffusion ................................ ................................ ............................ 101 Source Removal ................................ ................................ ................................ ... 103 Future Work ................................ ................................ ................................ .......... 104 APPENDIX ................................ ................................ ................................ .................. 105 LIST OF REFERENCES ................................ ................................ ............................. 110 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 115
7 LIST OF TABLES Table page 2 1 Definitions of Dimensionless Variables ................................ ............................... 25 2 2 Definitions of Dimensionless Variables with Leakage and Decay ....................... 39 3 1 Aquifer parameters used in the model for DNAPL source dissolution ................ 47 3 2 Aquitard media values used in model for diffusion ................................ ............. 47 3 3 Relative reduction in maximum mass stored in aquitard, .................... 53 3 4 Dimensionless values used in decay mo deling ................................ .................. 63 3 5 Summary of results for diffusion with decay modeling, ........... 64 3 6 Summary of results for 1D decay modeling with sorption, ....... 68 3 7 Parameter values used in aquitard diffusion with leakage modeling .................. 74 3 8 Summary of results for diffusion modeling with leakage, ...................... 76 3 9 Summary of results fo r 1D leakage modeling with sorption, ................... 80 4 1 Summary of Aquitard Diffusion with Decay and Source Remediation, ................................ ................................ ................................ 94 4 2 Summary of Aquitard Diffusion with Leakage and Source Remediation, ................................ ................................ ................................ ............... 99
8 LIST OF FIGURES Figure page 2 1 Conceptual diagram of the model ................................ ................................ ....... 24 3 1 SDM and resultant depth profiles for and ..... 48 3 2 depth profiles when back diffusion begins and source is exhausted ............. 49 3 3 SDM and resultant depth profiles for ................................ .................... 50 3 4 Maximum mass stored in the aquitar d ................................ ................................ 54 3 5 Relative mass in the aquitard as a function of ................................ ................ 56 3 6 Aquitard source functions ................................ ................................ ................... 58 3 7 depth profiles for diffusion ( ) with decay ... 64 3 8 depth profiles for diffusion ( ) with decay for specific times ................................ ................................ ....... 66 3 9 depth profiles for diffusion wit h sorption ( ) and decay ................................ ................................ ................................ 68 3 10 depth profiles for diffusion with sorption ( ) and decay for specific times ................................ ................................ ...... 70 3 11 Aquitard source functions with decay and and for and ................................ ................................ ................................ ............. 72 3 12 depth profiles for diffusion ( ) with leakage ................................ ................. 76 3 13 depth profiles for diffusion ( ) with leakage ; for specific times ................................ ................................ ................................ ...... 78 3 14 depth profiles for diffusion with sorption ( ) and leakage ; ................................ ................................ ............................... 79 3 15 depth profiles for diffusion and sorption ( ) with leakage ; for specific times ................................ ................................ .... 81
9 3 16 Aquitard source functions for diffusion with leakage and and .......... 84 4 1 Remedial SDM and resultant depth profiles ................................ .................. 87 4 2 Aquitard source functions for diffusion with remediation ................................ ..... 88 4 3 depth profiles for diffusion ( ) with decay and 70% source remediation at 25 years ................................ 90 4 4 Aquitard source functions for diffusion and decay with remediation and without sorption ................................ ................................ ........ 91 4 5 Dimensioned aquitard source functions for diffusion and decay with and without remediation, all without sorption, ................................ 93 4 6 depth profiles for diffusion with sorption ( ) and leakage with 70% source remediation at 25 years ................................ ................................ .......... 96 4 7 Aquitard source functions for diffusion and leakage with remediation ............... 97 4 8 Dimensioned aquitard source functions for diffusion and leakage with remediation for specific leakages ................................ ................................ ....... 98
10 LIST OF SYMBOLIC NOTATION DNAPL D ense non aqueo us phase liquid TCE Trichloroethylene PCE Tetrachloroethylene SDM Source depletion model 1D One dimensional 2D Two dimensional Source zone initial flux averaged concentration [ML 3 ] Source zone flux a veraged concentration at time t [ML 3 ] Source zone dimensionless concentration at dimensionless time Source dimensionless flux average d concentration at time of remediatio n Source zone dimensionless flux averaged concentration in the Laplace domain Aquitard concentration [ML 3 ] Aquitard total concentration [ML 3 ] Aquitard solid phase concentration [M M 1 ] Aquitard aqueous phase concentration [ML 3 ] Peak concentration in aquitard [ML 3 ] Dimensionless aquitard concentration at dimensio nless time A quitard dimensionless concentration in the Laplace Domain Peak dimensionless concentration in aquitard Fraction of source mass remov ed Source zone cross sectional area [L 2 ]
11 Source zone initial mass [M] Source mass per unit area [ML 2 ] Source mass at time t [M] Source zone dimensionless mass at dimensionless time Source dimensionless mass remaining at time of remediation Stored mass per unit area in the aquitard at t ime t [M L 2 ] Dimensionless aquitard mass per unit area at dimensionless time Aquitard dimensionless mass per unit area at time back diffusion starts Maximum aquitard mass per unit Source zone initial flux [ML 2 T 1 ] Aquitard back diffusion flux at time t [ML 2 T 1 ] Peak back diffusion flux [ML 2 T 1 ] Total aquitard flux [ML 2 T 1 ] Dimensionless aquitard flux at dimensionless time Dimensionless time Dimensionless time back diffus ion starts Dimensionless time the source is exhausted or completely removed Post remedial dimensionless time Post remedial dimensionless time the source is exhausted or com pletely removed Groundwater flux through source zone [LT 1 ] Vertical groundwater flux through aquitard [LT 1 ]
12 Vertical seepage velocity [LT 1 ] D epth in aquitard below boundary [L] Dimensionless depth in the aquitard below boundary Aquitard retardation factor [ ] Aquitard porosity [ ] Aq uitard bulk density [ML 3 ] Volumetric water content [ ] Distribution coefficient [ LM 3 ] Aquitard w ater content [ ] Aquitard t ortuosity [ ] Aquitard dispersivity [L] Aqueous diffusion constant [L 2 T 1 ] Effective diffusion constant [L 2 T 1 ] Aquitard hydrodynamic dispersion coefficient [L 2 T 1 ] E mpirical parameter that accounts for flow field heterogeneity, DNAPL distribution, and the correlation between the two [ ] Source to aquitard mass transfer coefficient [ ] Post remedial source to aquitard mass transfer coefficient [ ] Source decay function [T 1 ] Post remedial source decay function [T 1 ] Aquitard diffusion timescale [T] Post remedial aquitard diffusion timescale [T] Reaction term in aquitard [M L 3 T 1 ]
13 F irst order aquitard degradation constant [T 1 ] Dimensionless de cay coefficient Peclet number in aquitard [ ] Fractional m ass reduction Fractional f lux reduction
14 Abstract of Dissertation Presented to the Graduate School of the Universi ty of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AQUITARD CONTAMINANT STORAGE AND FLUX By GORDON HITCHINGS BROWN May 2014 Chair: Michael D. Annable Major: Environmental Engineering Scienc es A one dimensional diffusion model with a semi infinite domain was used to investigate the effects of dense non aqueous phase liquid (DNAPL) source zone dissolution and remediation on the storage and release of contaminants from aquitards. Source zone dissolution was represented by a power law source depletion model, which served as a time variable boundary condition to the diffusion equation used to describe mass transport in the aquitard. Two key variables were used to assess source zone dissolution b ehavior on aqui tard mass storage and release: the power law exponential term which reflects the influence of the source zone architecture, and a new variable defined herein as the source to aquitard mass transfer coefficient which reflects the influences of both the source characteristics and the aquitard media properties. As increased or as increased due to more rapid source dissolution, the dimens ioned aquitard concentrations, depth of penetration, and long term back diffusion flux decreased. However, when increased due to increased sorption, the dimensioned concentrations and back diffusion increa sed but the depth of pen etration decreased. The duration of aquitard mass loading was found to be significantly less than the duration of mass release. Moreover, the mass per unit area stored in the
15 aquitard was three or more orders of magnitude less than the initial DNAPL source zone mass per unit area, and the back diffusion flux from the aquitard was typically four or more orders of magnitude less than the initial source zone mass flux. Contaminant decay in the aquitard was modeled at 1 3 orders of m agnitude lower than aquifer ranges for and as increased both the mass s t ored and the back diffusion flux decreased. At the upper range of decay was high enough such that no back diffusion occurred. Vertical l eakage in the aquitard was modeled at 4 orders of magnitude less than the groundwater velocity in the aquifer Downward increased dimensioned aquitard mass storage both the loading and back diffusion mass flux, and the de pth of penetration while upward decreased dimensioned storage, flux, and penetration. At high vertical upward leakage rates back diffusion from the aquitard was not obs erved. Additionally, the effects of partial source zone remediation were investigated, and the results suggest that source remediation can have a favorable effect on long term back diffusion risk. In the short term following remediation, back diffusion fro m the aquitard was elevated due to the increased concentration gradient and in the long term, back diffusion was reduced by remediation.
16 CHAPTER 1 REVIEW OF LITERATURE Overview The presence of aqueous contaminants in low conductivity media downgradient of suspected or known dense non aqueous phase liquid (DNAPL) source zones has been attributed to diffusional transport [ Ball et al., 1997; Liu and Ball 2002; Chapman and Parker, 2005; Parker et al., 2008]. A groundwater plume will provide the concentration gradient required to move contaminants into an initially uncontaminated low conductivity layer. Over time the source mass will be depleted, the resulting plume concentrations will decrease, and the concentration gradient will reverse. Mass then diffuses out of the low conductivity layers (a phenomenon known as back diffusion) potentially leading to plume persistence [ Chapman and Parker, 2005 ; Parker et al., 2008]. Because of this, it has been suggested that remedial efforts may be ineffective at reducing risk, based on maximum contaminant limits (MCLs), since contaminant mass in low conductivity layers can serve as a secondary source to the plume long after the original DNAPL source mass has been removed or isolated [ Parker et al., 2008 ; Sale et al., 2008 ]. Contemporary Related Work Ball et al.  used high resolution sampling of a soil core to investigate the distribution of contaminants in an aquitard downgradient of contaminant sources at Dover Air Force Base, Delaware. Aqueous concentrations of tetrachloroethene (PCE) were higher in the aquifer compared to those in the underlying aquitard, suggesting forward diffusion of PCE into the aquitard. In contrast, trichloroethene (TCE) was detected at relatively lower aqueous concentrations in the aquif er compared to those in
17 the underlying aquitard, suggesting that TCE loading had terminated and that back diffusion had begun. Sheet piling was used to hydraulically isolate the test area, and three additional soil cores were collected over approximately three years [ Liu and Ball 2002]. Conditions to induce back diffusion within the sheet piling through water circulation were maintained over the final two years of the study, and results were consistent with those expected with back diffusion as the domin ant transport process. Field data were evaluated using a multilayer one dimensional (1D) analytical model of diffusional transport in the aquitard based on the convolution method. Inverse methods were utilized to fit the model to the field data by estima ting the historic concentration time series in the aquifer (i.e., upper boundary condition) using several different approaches [ Ball et al. 1997; Liu and Ball 1998]. The solutions provided reasonable fits to the aquitard data but were non unique in that very different boundary conditions generated very similar aquitard concentration profiles. Similar approaches were used to investigate aquitard profiles at a TCE contaminated site in Connecticut [ Parker et al., 2004; Chapman and Parker 2005]. The load ing of the TCE into the aquitard was likely due to both diffusion and pumping in the lower aquifer, which created a strong downward gradient across the aquitard and thus downward advective flow [ Parker et al., 2004]. The source was isolated with sheet pil ing and six years later, coring in the plume showed low concentrations near the aquifer/aquitard interface and higher concentrations with depth in the aquitard [ Chapman and Parker 2005]. These profiles, along with persistent concentrations in the surfici al aquifer just above the aquitard interface suggested plume persistence due to back diffusion. A 1D diffusion model was used to predict the measured concentration
18 profiles in the aquitard, and the agreement with field data improved by using a stepped, de clining source compared to a constant concentration boundary condition [ Parker et al., 2004; Chapman and Parker 2005]. In addition to aquitards, Parker et al.  investigated back diffusion from discrete, thin clay layers downgradient of a DNAPL sou rce zone. Groundwater concentrations were observed to decrease by several orders of magnitude in the first five years after hydraulic isolation of the source, yet they remained above MCLs in the downgradient sampling transects. Cores into the clay layers showed lower concentration in regions near the aquifer interface, and higher concentrations deeper into the clay, indicating back diffusion releases. Two dimensional (2D) modeling by Parker et al.  employed a constant concentration boundary conditi on in the aquifer with discrete low conductivity layers present After 30 years, the source was terminated and TCE diffusion out of the low conductivity layers was observed for up to 200 years. Finally, several laboratory and modeling methods were used to elucidate the effect of reduced contaminant loading on downgradient water quality [ Sale et al., 2008 ; Chapman et al., 2012; Seyedabbasi et al., 2012 ]. In the Sale et al  laboratory and modeling work, a two layer laboratory aquifer model, with an upper layer of sand and a lower layer of silt, was used to demonstrate back diffusion from the lower layer once the source was turned off. The results were fit with a two dimensional model with constant concentration boundary conditions. The investigato rs reported that 15 to 44% of contaminant mass was stored in the aquitard and the modeling projected that the stored mass would release for many years after the source was terminated. In another
19 study, a sand tank was packed with several clay lenses and s ubjected to a constant concentration of bromide and fluorescein for 25 days followed by clean water flow. Approximately 60 days after the influent concent ration was terminated and replaced by clean water flow, the contributions to the effluent due to bac k diffusion were 4 orders of magnitude lower than the initial influent concentrations [ Chapman et al., 2012]. A nother modeling study set up many small low conductivity lenses with pooled DNAPL and then looked at the contribution of both the dissolved DNAP L and matrix diffusion from the lenses as components in the plume [ Seyedabbasi et al., 2012]. Results showed that in the mass discharge rate, the contribution of matrix diffusion increased for the more soluble species whereas NAPL and mixed contributions were more dominant in the less soluble contaminants In summary, these studies have used field and lab oratory data to identify and demonstrate the importance of back diffusion, and have used diffusion models to help interpret the data. These experiments a nd models however, have often used constant concentration boundary conditions, or step changes in concentration as the boundary condition, which are idealized approximations of source zone dissolution. An exception is the work of Liu and Ball [1998, 1999, 2002], which considered complex functions for boundary conditions; however, their work focused on estimating the boundary condition from the measured concentration profile in the aquitard. Nevertheless, the field site characteristics that contribute to t he significance of diffusive storage and release have not been clearly established. During the decades or centuries required for groundwater to completely dissolve a DNAPL spill, the source architecture may change due to local depletion in the higher
20 hyd raulic conductivity zones, potentially weakening the overall source strength (i.e., the mass discharge from the source zone) and plume concentration [ NRC 2004; Falta et al., 2005a]. Both laboratory and field studies have documented reductions in mass dis charge as a result of active source depletion [ Fure et al., 2006; Page et al., 2007; Kaye et al., 2008; McGuire et al., 2006; Brooks et al., 2008 ; DiFilippo and Brusseau, 2008]. In addition, both laboratory [ Fure et al., 2006; Suchomel and Pennell 2006; Brusseau et al., 2008; Kaye et al., 2008; DiFilippo and Brusseau 2011] and field studies [ McGuire et al., 2006; Basu et al., 2009; DiFilippo and Brusseau 2008] suggest that in some cases mass discharge reduction is proportional to contaminant mass reduct ion in approximately a 1:1 relationship. Moreover, at most field sites monitoring and remedial activities were initiate d 10 to 40 years after the spill, and a nother study suggest s that at aged sites a 1:1 mass discharge to DNAPL mass reduction relationshi p may be assumed, regardless of initial spill architecture [ Chen and Jawitz 2009]. Thus, much work has focused on modeling DNAPL source zones to account for reductions in source strength as DNAPL mass is removed from the source zone. Several analy tical models that use either Lagrangian or Eulerian methods have been developed [ Parker and Park 2004; Falta et al., 2005b; Park and Parker 2005; Jawitz et al., 2005; Christ et al., 2006; Basu et al., 2008b; Zhang et al., 2008; Chen and Jawitz 2009; DiF ilippo and Brusseau 2011]. These screening level source depletion models (SDMs) offer methods to evaluate the effects of source depletion and incorporate basic parameters to describe heterogeneous DNAPL contamination, heterogeneous groundwater flow, and the potential correlation between the two [ Basu et al., 2008a].
21 The reducing conditions inherently present in aquifer s provide an environment conducive to contaminant reaction processes [ Alexander, 1999] These can be abiotic and biotic in nature, breaki ng down the parent compound to daughter products. Less work has been performed to assess the degradation potential or activity of these processes in aquitards. Most of this work has focused on aquifer decay rates [ Newell et. a l. 200 3 ] However, the reduc ing environment extends down into the aquitard and the presence of organic matter in the silt and clay may foster increased degradation. A recent study in Japan determined that the dechlorination of PCE by bacteria can occur in an organic rich clayey aquit ard, and that the aquitard may plan an important role in natural attenuation in the adjacent aquifer [ Takeuchi et al., 201 1 ]. Another potential component of aquitard hydrology is vertical advection through leakage Vertical gradients can cause very slow g roundwater flow through an aquitard either in downward or upward direction depending on the direction of the gradient. The gradient can be natural due to head pressure or induced by well pumping. A recent study in Australia determined groundwater flux th rough the regional aquitard to be ~ 8x10 9 8x10 6 m d 1 [ Gardner et al., 2012]. Groundwater leakage through confining units has also been studied in the context of lake bed losses [ Annable et al., 1996; Motz, 2010 ] however research on contaminant transpo rt through confining units has been less common. Most of the research has been focused on landfill liner leakage [ Johnson et al., 1989 ; Rubin and Rabideau 2000; Foose et al., 2003] Johnson et al., [19 8 9] determined that diffusion was the dominant transp ort mechanism for clay liner with an average downward linear velocity of a ~5x10 6 m d 1 More focus had been on the groundwater and contaminant flow through the fractures in clay aquitards [ Harrison
22 et al., 1992; Parker et al., 1996 ; Parker and Chapman 2005]. The comple xity of studying this phenomenon and the ability to sample in aqui tards without compromising their integrity to verify computer and laboratory studies with field data may be a reason. Focus of This Work The work reported here explores th e relationship between source depletion dynamics and diffusive transport in and out of low conductivity media using one dimensional analytical methods as a potential screening level too l for site assessment An analytical SDM was utilized to provide a tim e varying boundary condition for the 1D diffusion model with a semi infinite domain Specifically, the work reported herein used the power law model [ Rao et al., 2002 ; Rao and Jawitz, 2003 ; Zhu and Sykes, 2004 ; Falta et al., 2005a; Falta et al., 2005b] to create a temporally variable boundary condition. Since diffusion is a gradient driven phenomenon, the changing concentration in the overlying aquifer will influence the amount of contaminant mass driven into the aquitard and its subsequent release. The potential for plume persistence due to back diffusion is dependent on two factors: (1) the contaminant mass stored in the aquitard due to loading during forward diffusion, and (2) the rate at which the contaminant is released during back diffusion (i.e., t he magnitude of contaminant flux from the aquitard). Thus, hydrogeologic and contaminant parameters that affect ma ss loading to and release from the aquitard were examined. The initial results of aquitard diffusi on, mass storage, mass flux, with and with out source remediation were recently published [ Brown et al., 2012]. The parameters studied in this work include th ose influencing the source zone dissolution and those influencing contaminant transport in the aquitard including the diffusion coefficient, advection in the low permeability unit, decay rates in the aquitard, and the influence of remedial activities on the source zone
23 CHAPTER 2 ONE DIMENSIONAL MODEL DEVELOPMENT One Dimensional Model for Diffusion Dimensionless Source Depletion Our conceptua l model begins with a DNAPL source zone in an aquifer, shown in Figure 2 1. Media heterogeneity in the aquifer will cause high and low groundwater velocities and an uneven DNAPL distribution on a local scale. Thus, as a screening level approximation, we focus on the large scale, flux averaged concentration [ML 3 ] leaving the source zone across a control plane with cross sectional area [L 2 ]. The SDM chosen to represent this time variable concentration is based on a mass balance in the source zone with source depletion by dissolution only: (2 1 ) where is the mass of the contaminant in the source zone [M] is ti me [T], and is the groundwater flux [LT 1 ] Zhu and Sykes  and Falta et al., [2005a] provide solutions for E q 2 1 when is related to using a power law expression, ( 2 2) where is the initial, flux averaged concentration crossing the source zone control plane [ML 3 ] is the initial mass of the contaminant in the source zone [M ] a nd is a unit less empirical parameter that accounts for flow field heterogeneity, DNAPL distribution, and the correlation between the two [ Falta et al., 2005a]. Special cases of interest include : whic h represents the constant concentration source; which represents linear source decay; and which represents exponential source
24 Figure 2 1 Conceptual diagram of the model. Note: The diagram consists of a DNAPL source zone, the dissolved contaminant plume (gray) with a flux averaged concentration crossing the source zone control plane and 1D domain with diffusion, leakage, and decay (dashed box) an underlying, near source, semi infinite aquitard with a time variable upper boundary
25 decay. Using the definitions provided in Table 2 1, equations (1) and (2) can be expressed in dimensionless form as ( 2 3) and ( 2 4) where the source to aquitard mass transfer coefficient is [ ] as defined in Table 2 1. Table 2 1. Definitions of Dimensionless Variables Source zone ma ss and concentration ; Aquitard depth, time and concentration ; ; Source to aquitard m ass transfer coefficient ; ; Post remedial source mass and concentration ; Post remedial s ource to aquitard mass transfer coefficient ; The term represents the relative extent to which mass is transferred from the source to the aquitard, and is th e product of the source decay function [T 1 ] given by Zhu
26 and Sykes , and a new variable defined here as the aquitard diffusion timescale [T] (Table 1). Consequently, c ouples source dissolution processes in and effective diffusional processes in A low represents slow source zone dissolution and/or low aquitard retardation where the plume c oncentration remains high for a longer time, while a high represents rapid source dissolution and/or high aquitard retardation. The soil diffusion coefficient and the retardation factor in the aquitard are used in and (Table 2 1), and will be discussed in more detail in the next section Substitution of Eq. 2 4 into Eq. 2 3, along with an initial condition of allows for solutions of and following methods similar to those presented by either Zhu and Sykes  or Falta et al., [2005a]: ( 2 5a) and ( 2 5b) As the limit of Eq. 2 5a and Eq. 2 5b approach ( 2 5c) [ Abramowitz and Stegun 1970]. De tailed derivations for Eq. 2 5a Eq. 2 5b, and Eq. 2 5c can be found in t he Appendix T he simple screening level SDM (E q 2 4, 2 5a, 2 5b, and 2 5c) is a dimensionless form of the equations utilized in the Remediation Evaluation Model for Chlorinated Solvents (REMChlor) [ Falta et al., 2007; Falta 2008]. It assumes that
27 gro undwater velocity is one dimensional and uniform, that contaminant discharge is described as a power function of source mass, and that the power function exponent is invariant over time. While source zone contaminant decay could be incorporated into the SDM (e.g., Falta et al. [2005a] and Falta et al. ), we assume this factor is insignificant. Finally, changes in the flux averaged concentration leaving the source due to lateral or longitudinal dispersion and decay are neglected, and the SDM is ther efore considered to represent the aquifer concentration above the aquitard. This assumption is considered appropriate in the region immediately down gradient of the source zone. Dimensionless Diffusion The model represents a 1D aquitard semi infinite dom ain with zero concentration at infinite depth, zero concentration initially throughout the domain and an imposed time variable concentration boundary condition at the surface (Figure 2 1). The location of the aquitard is immediately down grad ient of the D NAPL source zone, above or below the near source aquifer region; however, an underlying aquitard was chosen as defined by the direction (Figure 2 1). The upper boundary condition is be represented by the flux averaged concentrati on of dissolved contaminant leaving the DNAPL source zone, Eq. 2 5a or Eq. 2 5c The system is nd law of diffusion, ( 2 6) where is the retardation factor in the aquitard [ ], the contaminant concentration [M L 3 ] is t he effective diffusion coefficient [L 2 T 1 ] : ( 2 7 )
28 where is the water diffusion coeffi cient and is the tortuosity of the medium [ ]. The value of was held constant for this study but there is likely considerable variability in the tortuosity of different aquitards and even within the same aquitard. To assess the worst case scenario, this value was kept low to model higher effective diffusion. The total aquitard concentration [M L 3 ] can be represented by ( 2 8 ) where is the aquitard bulk density [M L 3 ] is the aquitard solid phase concentration [M/M] is the aquitard volumetric water content [ ], and is the aquit ard water concentration [M L 3 ]. Assuming equilibrium, reversible, and linear partitioning of the contaminant between the aquitard pore water and solid media, with ( 2 9 a) ( 2 9b ) an d ( 2 9 c) w here i s the media bulk density [ML 3 ] is the porosity of the aquitard [ ], is the aquitard moisture content ( = 1.0 for a saturated medium) and distribution coefficient in the aquitard [ LM 3 ] f or saturated conditions, E q 2 2 becomes ( 2 10 ) Transforming E q 2 6 using the dimensionless term s in Table 2 1 nd Law becomes
29 ( 2 11 ) A solution for Eq. 2 11 in the Laplace domain is ( 2 12 ) W ith boundary conditions of ( 2 1 3a) and ( 2 13b) and an initial condition of ( 2 13c) Eq. 2 12 becomes ( 2 1 4 ) T he Lapla ce Inversion is in the form of ( 2 15) Abramowitz & Stegun . Thus, a general solution to E q 2 11 is ( 2 16 ) T he upper boundary condition of the aquitard can be represented by a function that imposes a time variable c oncentrat ion, represented by the SDM in Eq. 2 5a or Eq. 2 5c. The SDM equation is substituted for in E q 2 16 to convert the general solution s to a specific one. The dimensioned concentration is:
30 at ( 2 1 7 a) and ( 2 1 7 b) The resulting specific solution was evaluated with MathCAD TM 1 5 .0, an engineering calculations software program. Other analytical solutions o f E q 2 11 for specific cases of in E q 2 4 have been previously published [ Crank 1975; Bear et al. 1994] These solutions were used for model verification and are available in the Appendix This solution considers diffusion a nd sorption in the aquitard. More complex conditions will be explored in the next section Equation 2 16 is fully dimensionless; is applied to a single layer semi infinite aquitard; and employs the convolution theorem, as did Booker and Rowe  and Li u and Ball . Several 1D analytical solutions have been used in the study of finite thickness landfill liner contaminant transport, but these use constant concentration boundaries and usually include a leakage term for vertical flow in their governin g equation [ Rubin and Rabideau 2000; Foose et al. 2001]. In a recent study, Chen et al.  developed a 1D analytical solution with a time changing boundary condition and applied it to diffusion and mass flux through a multi layer landfill liner. T his solution used the separation of variable method instead of the Laplace tran sform method. While Eq 2 16 also determines the concentration profiles like previous studies, this work moves into a risk framework by focusing on the mass storage and mass re lease out of the aquitard resulting from time variable DNAPL sources and source remediation as demonstrated in the next few sections.
31 The model for diffusion in the aquitard contains several simplifying assumptions. The groundwater flow is assumed to be l arge enough that diffusion from the aquitard does not influence the aquifer concentration (i.e. no feedback to the concentration gradient). All aquitard properties were assumed to be homogeneous which is adequate for this screening level model, but in re ality properties like and thus may be variable Permanent sequestration and decay in the aquitard are not considered, but may both occur in a field setting. The results presented are conservative in that they represent the worst case risk due to flux from the aquitard. In a real system, the concentration re entering the aquifer from the aquitard would be reduced by aquifer dispersion and biodecay. Dimensionless Mass Sto rage and Mass Flux Aquitard contaminant mass and flux are important to assess the risk and significance of back diffusion from the aquitard. The dimensionless mass per unit area in the aquitard can be determined for any or through integration of dimensionless flux through dimensionless time, ( 2 1 8 a) by integration of dimensionless concentration through dimens ionless space, ( 2 1 8 b) or integration of dimensionless concentration in dimensionless space in the Laplace domain and application of the convolution method,
32 ( 2 1 8 c) The di mensioned mass is ( 2 19 ) The integral in Eq. 2 1 8 a and Eq. 2 1 8 c were evaluated analytically and E q 2 1 8 b numerically. Equation 2 1 8 a gives as a result of loading when using the dimensionless time at which back diffusion starts as the upper integration bound and zero as the lower bound. To determine the mass that has left the aquitard for the lower integration bound is and the upper is the final of interest. Calculations of were verified with previously published cases for and t he relative error was zero for Eq. 2 18a and Eq. 2 18c and less than 1 % for E q 2 1 8 b. The flux into or out of the aquitard was derived from the temporal derivative of dimensionless mass per unit area and st law of diffusion Integrating both sides of E q 2 11 in dimensionless space yields ( 2 20 a) which simplifies to ( 2 20 b ) given E q 2 1 8 b and the boundary conditions in E q 2 13a and Eq. 2 13c O n t he left hand side of E q 2 20 b is the temporal ch ange in dimen sionless mas s in the aquitard. At the beginning of time, the flux is instantaneously infinite so a Dirac Delta function is employed and the dimensionless flux is
33 ( 2 21 a) On the right hand side of E q 2 20 b is t he chan ge in concentration across the boundary , which is also the dimensionless flux ( 2 2 1 b ) The dimensioned flux is given by ( 2 22) In E q 2 21a and Eq. 2 21 b, E q 2 5a or Eq. 2 5c is substituted into the convolution for Equation 2 21 a evaluated the boundary flux throughout the time scale. However, t he convolution solution in E q 2 21 b does not allow an evaluation at Thus, a very small value for (i.e., ) was used in E q 2 2 1 b to approximate the flux at the aquifer/aquitard interface. For example, corresponds to mm for m 2 In this work, an interface depth of 2.5 to 3.5 mm was used in E q 2 20b This depth was assumed to be representative of the interface, and justification for this assumption was that obtained at a depth of 2.5 to 3.5 mm varied less than 5% from obtained at the interface using previously published analytical solutions for (see Appendix ). Furthermore, f inite difference methods were employed for verification purposes as well for determined by E q 2 20a and at the stated depth for E q 2 2 0 b for other cases The finite differ ence approach used Eq. 2 5a for the boundary condition
34 an d solved E q 2 11 using a central difference solution to obtain the concentration profiles, as shown in the Appendix as Eq. A 14 [ Grathwohl 1998]. The flux into and out of the layer with E q 2 20 a was verified with numerical differentiation serving as th e spatial derivative approximation. With two exceptions, the error was less than 5%. First, at the start of the model, and so the flux is infinitely large at early times thereafter. For a depth of 2 .5 to 3.5 mm in the aquitard, neither the convolution nor the finite difference solution could resolve an instantaneous, infinite flux so relative differences were greater than 5 % however Eq. 2 20a did not have this problem Second, just before back dif fusion initiates there were larger than 5 % relative error s in model Eq. 2 20b and the finite difference solution in the Appendix (Eq. A 14) in that instant as well. However since the magnitude of the mass flux is so small, thi s error is negligible. Finally, it is noteworthy that for this theoretical model, all of the mass that entered the aquitard eventually returned to the aquifer. This seems counter intuitive given the conceptual model of diffusion into a semi infinite aquit ard; however, once back diffusion began, transport of contaminant mass occurred only through back diffusion and downward migration. After the start of back diffusion, the imposed boundary condition at the aquifer/aquitard interface resulted in a concentra tion gradient that was larger than the concentration gradient at the bottom of the contaminant distribution. Therefore, the flux leaving the aquitard was larger than that driving it downward. The practical implication of this result for sites with condit ions similar to those modeled here is that no sequestration of contaminant is expected within the aquitard.
35 Dimensionless Time The range of for typical field sites needs to be established to define the modeling timescales. Giv en the definition of in Table 2 1, this requires an estimate of the ranges associated with , and Because includes and which contain several media properties shown in E q 2 7 and E q 2 9c and a suggested range for of 10 m 2 to 1000 m 2 for 0.01 t 1 ,000 years, will range from 10 9 to 10 1 in most applications. There are two key events in time that were evaluated with the model: (1) the dimensionless time that back diffusion begins a nd (2) the dimensionless time that the source is exhausted by dissolution At any time after contaminant mass leaves the aquitard due to the reversal in the concentration gradient at the aquifer/aquit ard interface. Therefore, no additional mass will be loaded into the aquitard for and represents the time of maximum mass storage, For the case s, the back diffusion flux will peak at where ( 2 2 3 ) For the cases, peak back diffusion flux will occur at some time after Source Rem oval The impact of remediation on diffusion in the aquitard can be investigated by m odifying the dimensionless SDM E q 2 5a and Eq. 2 5b to account for partial source removal following the method of Falta et al., [2005a]. This modification is based on the instantaneous removal of mass fraction from the remaining source mass by some
36 remedial process at time The relative mass remaining in the source zone at prior to the re mediation i.e. is calculated using Eq. 2 5b, and in turn is used to determine the post remedial initial relative mass and relative concentration (Table 2 1). Likewise, these are used to calculate a new post remedial source decay function and a post remedial source to aquitard mass transfer coefficient (Table 2 1). Consequently, the post remedial relative concentration an d the relative mass remaining in the source zone are ( 2 2 4 a) and ( 2 2 4 b) respectively, where is the dimensionless elapsed time after remedi ation (i.e., ). For the case, the remaining mass and post remedial source concentration are given by ( 2 24 c) T he dimensionless time at which the mass remaini ng in the source zone after remediation is exhausted by natural dissolution becomes ( 2 25 ) Because the solution for dimensionless flux, E q 2 21 a, takes the derivative of the boundary equation used in the convolution the f lux is infinite at the time of
37 instantaneous source depletion. Integrating the Dirac Delta leads to a Heaviside step function in the post remedial solution for flux ( 2 26 ) These results will be evaluated for parameter response in Chapter 3. One Dimensional Model for Diffusion with Decay and Leakage Dimensionless Diffusion with Decay and Leakage An aquitard may have vertical gradients causing leakage and may degrade aqueous phase contaminants th rough reaction processes The vertical gradient created by pumping can influence the amount of mass loaded into the aquitard compared to diffusion alone. Referring back to Figure 2 1 the elements of leakage and decay we re added and the 1D solutions for concentration, mass, and flux we re derived. The model development including these additional factors begins with conservation of mass: ( 2 2 7 ) where is the total soil concentration in the aqui tard [ML 3 ], is total mass flux in/out of the aquitard [M L 2 T 1 ] and is the reaction term in the aquitard [M L 3T 1 ] A ssuming that mass flux occurs only in the aqueous phase, the flux can be express ed by
38 ( 2 2 8 a ) where is the aquitard advection [L T 1 ] and is the hydrodynamic dispersion coefficient in the aquitard [L 2 T 1 ] : ( 2 2 8 b) where is the dispersivity of the medium [L] is the seepage velocity [ L T 1 ], and is the effective diffusion coefficient [ L 2 T 1 ]. The value was held at 0.5 to ensure a reasonably large effect of advection in the dispersion coefficient. The seepage velocity for a saturated medium is defined as ( 2 2 8 c ) Contaminant decay is often repre sented by a reaction term [M L 3 T 1 ]: ( 2 29 a ) where is t he effective degradation constant for decay in the aqueous phase only in a saturated medium and is defi ned as: ( 2 29b ) w ith representing the first order aqueous phase degradation constant in the aquitard [T 1 ]. Substituting E q 2 28a thru Eq. 2 29b into Eq. 2 27 the governing equati on for 1D transport of a contaminant in the aquitard considering storage, diffusion, advection, and decay is ( 2 30 )
39 This assumes equilibrium, reversible, and linear partitioning of the contaminant between the aquitard pore water and solid media. Transforming the variables to dimensionless forms in Table 2 2, the dimensionless governing equation for 1D transport of a contaminant in the aquitard with leakage and decay is ( 2 31 ) Table 2 2 Definitions of Dimensionless Variables with Leakage and Decay T ime Source to aquitard mass transfer coefficient ; ; Aquitard Decay Peclet Number where the definition of terms are provided in Table 2 2. A solution to in the Laplace domain is ( 2 32 ) W ith the boundary conditions and initial conditions from E q 2 13a c, the solution in the Laplace domain becomes ( 2 33 )
40 which is in the form of E q 2 15 U sing the Delay Theorem ( 2 34 ) [ Abr amowitz & Stegun, 1970], and applyin g the convolution principle, a general s olution can be written as: ( 2 35 ) Dimensionless Mass Storage and Mass Flux with Leakage and Decay We recognize that a soluti on for E q 2 31 in Laplace space i s E q 2 33. Integration over the entire domain gives ( 2 3 6 ) T he Lapla ce Inversion is in the form of ( 2 37 ) Abramowitz & Steg un , w ith Applying convolution and delay theorems gives the dimensionless mass per unit area including leakage and decay
41 ( 2 3 8 ) The dimensionless mass flux was obtained by taking the temporal derivative of mass by starting wi th the dimensionless governing E q 2 31 and m ov ing d ecay over to the left hand side o f the equation. Integrating bo th sides in dimensionless space yields ( 2 39 ) due to E q 2 18b and the boundary condition ( Eq. 2 13b), w here the terms on the left hand side are the temporal change in dimensionless mass, wh ich in the absence of decay, must be equal to what is coming in or leaving through the interface. In the presence of decay, the second term corrects for the amount degraded in dimensionless time. Using E q 2 36 the terms are group ed such that Eq. 2 37 remains applicable, and then the temporal derivative is identical to E q 2 38 with the exception that is substituted by its derivative For the aq uifer concentration makes a step change, such that i s a Dirac impulse at that time such that
42 ( 2 40 ) T herefore the left hand side of E q 2 39 is the dimensionless flux given by the change in dimensionless mass per unit area in the aq uitard in dimensionless time, ( 2 41 a) And the right hand side of E q 2 39 is the dimensionless flux given by the change in concentration ( 2 41 b) The dimensioned flux is gi ven by ( 2 4 2) The solution in E q 2 41a will evaluate the dimensionless mass flux over the complete timescale. However, a solution for E q 2 41b could not be obtained analytically so the finite difference solution for fl ux (see the Appendix ) was used to verify E q 2 41b and resulted in less than 5% error The solutions for dimensionless mass and flux were also verified with the previous soluti ons in Chapter 2 by setting and with zero error.
43 Source Removal with Leakage and Decay The source removal SDM is unaffected by the addition of leakage or decay. However, t he solution for dimensionless flux in E q 2 40 uses a derivative of the source function For any case where the source is eventually exhausted, or for any case with instantaneous source depletion (e.g. remediation or source isolation ) the derivative will be infinite at that time and a Dirac delta function is used and respectively. After integration in the convolution, the Dirac function becomes a Heaviside step function, and the post source depletion flux is ( 2 4 3 ) w here is the Heaviside step function, is the source function change due to source depletion These functions can be used to predict concentr ation profiles and mass flux from aquitards under vertical advection (leakage) and first order aqueous decay processes. T he sim plifying assumptions used in th e diffusion with decay and leakage model development included an assumed homogeneity of the aquita rd. In fact, a quitard parameters like , and can be highly variable but for a screening level
44 model this assumption is considered appropriate. Furthermore, the use of constant parameters elimina ted the potential flow variability due to fractures or highly interconnected pores which was not the focus of this work.
45 CHAPTER 3 CONCENTRATION, MASS, AND FLUX RESULTS Dimensionless Results for Diff usion Aquitard Concentration Profiles There are two main parameters that affect the model results: and The effects of on the model were investigated first, and then the effe cts of were explored. Aquifer and source properties for the set of s imulations featured in Figure 3 1 and Figure 3 2 are shown in Table 3 1 representing a hypothetical PCE spill and are similar to those used by Falta et al. [20 05a]. Aquitard parameters used in these simulations are presented in Table 3 2 One method to evaluate the effects of source dissolution on aquitard storage and release is to construct profiles of as a function of dep th using eq uation (9). Figure 3 1 a illustrates the SDM for and Figures 3 1 b, 3 1 c, and 3 1 d illustrate the depth profiles of for respectively. As incr eased, the profiles in Figures 3 1 b through 3 1 d show reduced concentrations in the aquitard for a given time and depth due to the reduced concentration at the aquifer/aquitard boundary. Likewise, the peak of the aquitard concentration was reduced with less penetration as increased. For example, at for respectively; and these values of occurred at dep ths of respectively. In Figure 3 2 the depth profiles are shown at specific events in time (i.e., and ) as a function of and The highest concentration in the aquitard is
46 achieved at Figure 3 2 a shows that the penetration and magnitude of at decreased as and increased. For example, with and at for respectively. Likewise, with and at for respectively. In Figure 3 2 b, the concentration profiles at for cases with are shown. As increased, concentrations in the aquitard were reduced due to three factors: (1) a more rapid decline in (2) decreased as increased, resulting in more time for back diffusion to remove mass, and (3) increased as increased, resulting in more time for downward diffusion of mass. For was maintained until which occurred at (and hence at ). For the other cases, Shown in Figure 4a is the SDM for an d ; and in Figures 3 b through 3 d the resultant depth profiles Similar to the influence of decreased as increased as a result of the more rapid re duction in the concentration gradient. For example, at for respectively. This resulted in a lower for a given time and depth. At and at a depth of for respectively. To add context to these results in terms of actual site conditions, the d imensionless results in Figure 3 1 were converted to dimensioned time and space using our hypothetical PCE site values (Tables 3 1 and 3 2 ).
47 Table 3 1. Aquifer p arameters used in the model for DNAPL source dissolution. Table 3 2. Aquitard media values used in model for diffusion. (mg l 1 ) (kg) (m d 1 ) (m 2 ) (d 1 ) Fig 2,3,5a,6,7a b,7e f 150 1620 0.0548 30 1.52 x 10 4 Fig 3,4,5b,6,7c d 150 1620 0.00782 30 2.19 x 10 5 0.0548 1.52 x 10 4 0.172 4.78 x 10 4 (m 2 d 1 ) (l kg 1 ) (g ml 1 ) (m 2 d 1 ) (d) Fig 2,3,4,5a,6,7a d 1.46 x 10 5 1.4 1.2 2.6 0.45 8.0 1.30 x 10 6 2.30 x 10 7 Fig 3,4,5b,6,7e f 1.46 x 10 5 1.4 1.25 0.04 0.35 1.14 9.13 x 10 6 3.28 x 10 6 1.2 2.6 0.45 8.0 1.30 x 10 6 2.30 x 10 7 1.1 12.1 0.55 25.1 4.15 x 10 7 7.22 x 10 7
48 Figure 3 1. SDM and resultant depth profiles for and Note: (a) SDM as a function of for and depth profiles in the aquitard are shown for (b) (c) (d)
49 Figure 3 2. depth profiles when back diffusion begins and source is exhausted. Note: (a) depth profiles ar e shown at the time back diffusion begins, as a function of (solid lines) with ; and as a function of (dashed lines) with (b) depth profiles are shown at the time the source is exhausted, as a function of Results are reported as soil concentrations similar to previous back diffusion field studies, which is the sum of the mass in solution and the sorbed mass. For illustration, at yr mg kg 1 which occurred
50 Figure 3 3. SDM and resultant depth profiles for Note: (a) SDM as a function of for and depth profiles are shown for (b) (c) and (d)
51 at depths of m for respectively. Evaluating the influence of on concentr ation in dimensioned time and space, however, is not as straight forward. At m and yr the total soil concentration (aqueous and solid phase) mg kg 1 for respectively; where values of were obtained considering m d 1 respectively, and all other parameters defined as shown in Table 3 1 and Table 3 2 However the same dimensionless profiles for occur when results from respectively, m d 1 and all other parameter s equal to the same previous values (Tables 3 1 and 3 2 ). These values of may occur, for example, considering typical properties associated with sandy silt, silt, and silty clay aquitards. Because is a function of the results at in dimensioned time occur at yr for respectively. Consequently, at the same depth of m mg kg 1 for respectively, demonstrating an increase in total mass as increased due to sorption. Thus, increasing decreased the aquitard contaminant concentrations and the depth of contaminant penetration. Likewise, increasing by increasing decreased aquitard contaminant concentrations and the depth of contaminant penetrat ion. However, when increased due to an increase in (via increased ), solid phase concentrations increased (increasing total concentrations), but the contaminant penetration de pth decreased.
52 Aquitard Mass Storage The risk of back diffusion is dependent on the contaminant mass present in the aquitard at any given point in time. As expected from the results in Figure 3 1 as increased, the amount of ma ss stored in the aquitard decreased. Specifically, using equation (10b) to integrate the depth profiles in Figure 3 1 for for respectively. As increased, less mass entered the aquitard due to the more rapid decrease in source concentration and thus a reduced diffusion gradient. At later times most of the mass in the aquitard had been released back to the aquifer, so t he profiles were flat and the dimensionless mass remaining was less variable. For example, at for respectively. Higher in the latter two cases results from higher concentrations at the boundary for longer durations (e.g., Figure 3 1 a for ), reducing the back diffusion gradient, which in turn yields slightly more dimensionless mass at this point in time. To evaluate the m aximum mass storage the depth profiles at (Figure 3 2 a) were integrated using E q 2 1 8c and the results are shown in Figure 3 4 a as a function of and in Figure 3 4 b as a function of As increased from 0 to 10, decreased from 0.019 to 0.0039. Likewise, as increased from 250 to 11000, decrease d from 0.039 to 0.006. Figure 3 4 a illustrates the error that might occur in estimates of when assuming as is often done, for those sites with characteristics better repre sented by As further illustration of this point, Table 3
53 3 lists the relative reduction in defined as as a function of The relative reduction in maximum stored mass ranged from 36 to 60% for Thus, while the case is often used as a conservative approach, it will likely over predict the aquitard stored mass, thus increasing the perceived ri sk of back diffusion, if this assumption is not valid. Table 3 3 Relative reduction in maximum mass stored in aquitard, 0.0 0.00 0.25 0.25 0.5 0.36 0.75 0.43 1.0 0.4 8 1.25 0.52 2.0 0.60 4.0 0.71 10.0 0.81 Longevity and Hysteresis Contaminated site diffusion processes are hysteretic because they are gradient driven, and loading occurs much more rapidly than release. To investigate this behavior, is plotted as a function of and in Figure 3 5 Starting at increased to its peak at then decreased as increased (Figure 3 4 a). In general, as increases, the source zone architecture results in a reduced forward concentration gradient at the aquifer aquitard interface, which in turn results in an earlier ba ck diffusion time. When the source, in theory, is never completely exhausted by dissolution; consequently, back diffusion lasts for an infinitely long time.
54 Figure 3 4. Maximum mass stored in the aquitard. Note: (a) Maxim um mass stored in the aquitard ( ) and as a function of (b) and as a function of Similar to as increases, back dif fusion begins earlier ( Figure 3 4 b). When was increased by more rapid source dissolution, bac k diffusion flux initiated earlier
55 and less mass was loaded into the aquitard. However, if was increased due to increased aquitard sorption only, decreased, but that value corresponded to the same dimen sioned time since is also a function of (Table 2 1). Thus aquitard sorption has no effect on the time when mass loading into the aquitard ends The hysteresis of contaminant mass loading and release i s shown in Figure 3 5 where the relative mass in the aquitard is plotted as a function of for and and as a function of for and Relative mass in these figures was scaled to for each specific case. As increased the contaminant mass was loaded relatively rapidly into the aquitard due to the high concentration gradients, peaked at the start of back diffusion, and was slowly released thereafter. This hysteretic behavior is demonstrated in the linear scale overlay. Overall, had a minimal effect on the timescale of th e loading and release of mass in the aquitard, while had a stronger influence in the model results. As decreased, both the loading and release times increased. Aquitard Source Functions A convenient m eans to evaluate the risk of contaminant source mass in subsurface systems is through source functions, defined as the relative relationship between contaminant flux and mass (e.g., Rao et al ., ). Likewise, aquitard source functions can be defined a s the relationship between the relative back diffusion flux and the contaminant mass in the aquitard. Aquitard source functions were constructed by converting and to dimensioned values using Eq. 2 19 and Eq. 2 22, respectively where is aquitard mass per unit area [ML 2 ] and is back diffusion flux
56 Figure 3 5 Relative mass in the aquitard as a function of Note: Relative mass in the aquitard as a function of in log scale for (solid lines), ; and as a function of in log scale for and (dashed lines); overlay is plotted in a linear time scale.
57 [ML 2 T 1 ]. Aquitard source functions are shown in Figure 3 6 as a function of and along with the corre sponding dimensioned times series for (dashed lines) and (solid lines). The ratio was used to normalize while was used t o normalize Temporal progression can also be followed in the aquitard source functions themselves, which begin at on the right, and increases in time from there to the left. Figure 3 6 a presents aqu itard source functions for and and illustrates three distinct shapes. For the boundary concentration jumped from to in stantaneously, resulting in an infinite flux at In contrast, the back diffusion flux peaked after for cases with because the concentration in the aquitard and the SDM bounda ry were equal at resulting in no flux. For cases with (as illustrated by ), a sharp but finite peak occurred when the source was exhausted at Fo r (as illustrated by and ), aquitard source functions were more curvilinear paths resulting from the more gradual decay of the DNAPL source zone flux. An important feature evi dent in Figure 3 6 a is that as increased, the relative mass and flux decreased, indicating reduced risk due to back diffusion. Specifically, as increased from 0 to 2, the maximum relative mass decrease d from 1.04x10 3 to 4.3x10 4 respectively; while the 1.3x10 4 respectively. Moreover, in all cases, the mass per unit area stored in the aquitard was three or more orders of magnitude less than the initial source zone mass per unit area; a nd the back diffusion flux, with the exception of the case, was three or more orders of magnitude less
58 Figure 3 6. Aquitard source functions.
59 Note: (a) Aquitard source functions for and (b) Dimensioned mass storage and mass flux time series for (a). (c) Aquitard source functions for and where is varied by [m d 1 ], shown as [d 1 ]. (d) Dimensioned mass storage and mass flux time series for (c). Figure 3 6. Continued. Note: (e) Aquitard source functions for and where is varied by shown as [d]. (f) Dimensioned mass storage and mass flux time series for (e). Dashed lines are mass ( right y axis), solid lines are f lux ( left y axis). than the initial source zone flux. The results from Figure 3 6 a were converted to dimensioned flux (solid lines) and dimensioned mass per unit area
60 (d a shed lines), and plotted as a function of elapsed d imens ioned time from the s tart of back diffusion (Figure 3 6 b). Similar to the results in Fi gure 3 6 a, as increased the peak mass and mass flux decreased. The rates of decrease, however, varied as a function of For example, within the first ten years after the order of as a function of was: > > > By 250 years after (offscale in Figure 3 6b), the order was reversed, and g m 2 yr 1 for all cases of Additionally, comparing the result to that of 10 years after the back diffusion flux was 68 % greater for than for a site with However, 100 years after resulted in 30 % less flux than This is due to the more rapid removal of mass by the increased flux in the case at earlier times. Since the effect of on aquitard source functions was investigated by separately exploring the impacts of and In Figure 3 6 c, aquitard source functions are shown for and These values of correspond to d 1 and were generated using m d 1 and the other parameters as shown in Table 3 1 As increased from ( d 1 ) to ( d 1 ), the maximum relative mass decreased from to and t he maximum relative flux decreased from to Moreover, in all cases the aquitard mass per unit area and flux were at least three orders of magnitude less than the source mass per unit are a and flux. Similar results are expected for changes in and that lead to an
61 increase in and hence Variations in affect , and ; and the resulting impacts on aquitard mass and flux are more complex and not shown here. Figure 3 6 d shows the dimensioned time series for and corresponding to th e dimensionless data in Figure 3 6 c. Compariso n of Figures 3 6c and 3 6 d for ( d 1 ) indicated that while this case has the highest relative flux, it h ad the lowest dimensioned flux. This was due to the fact that is normalized to a low initial source flux resulting from the low groundwater velocity in this case. Comparing the case of ( d 1 ) to the others shown in Figure 3 6 d, the source dissolved more slowly, which generated a stronger forward diffusion gradient over a longer time and reduced the gradient for back diffusion. Thus, the ( d 1 ) case had the greatest and the lowest for This illustrates a case where there may be a large amount of mass in the aquitard, but the back diffusion flux is low, resulting in low, but perhaps prolonged risk. Conversely, ( d 1 ) represents a site with a high that dissolves the source more rapidly, resulting in less mass within the aquitard, but higher back diffusion flux (Figure 3 6 d). This represents a greater short term risk, but a lower long term risk. Finally, it is worth noting that the magnitude of the mass storage in Figure 3 6 d is in the same range as that calculated by Chapman and Parker  for an industrial site in Connecticut. Changes in retardation factors associated with changes in aquitard media are evaluated in the aquitard so urce functions shown in Figure 3 6 e for and These values of correspond to
62 d, and represent the dissolution of a PCE source over silty sand silt and silty clay aquitards, with the modeling parameters shown in Table 3 2 The results showed that as increased (and hence ), additional mass was stored in the aquitard (on the solid phase as expected). Per haps less obvious however is the result that the maximum relative flux likewise increased due to the additional so rbed mass. In Figure 3 6 f, the time series for and are shown f or the model results in F igure 3 6 e. Increasing (and hence ) due to increased retardation increased both and even though the penetration depth was reduced (see Figure 3 3 ) Dimensionless Results for Diffusion with Decay Aquitard Concentration Profiles To investigate the effect of aquitard contaminant decay, a conservative approach was used starting with values two to three orders of magnitude les s than the published range for TCE (1.0x10 2 to 1.0x10 3 d 1 ) in an aquifer [ Newell et al., 2002]. Furthermore, only aqueous phase was considered and the decay rate was assumed to be constant. The range of selected d 1 which yields dimensionless decay coefficients of for a low However, for sites with higher sorption, the required to exhibit similar parent reductio ns was zero to two order s of magnitude less than the published range, where d 1 yields dimensionless decay coefficients of for The cases with decay were compared to the original
63 baseline cases without decay ( ). The values of the parameters used in aquitard diffusion modeling with decay are shown in Table 3 4 Table 3 4 Dimensionless values used in decay model ing The starting value of is the same as Chapter 2 since the values of and are the same, but since there is a definition c hange in (T able 3 1 ) Three dimensionless variables , and now contain instead of As b efore, both and contain ( Table 2 1 ) Using E q 2 35, profiles were created to represent a site with out sorption ( ) with and without decay in the aquitard in Figure 3 7 Comparing the no decay case (Figure 3 7a) to those with decay (Figure s 3 7b c) the aquitard concentration s were reduced by decay especially in the later times In early tim es ( black lines 5 y rs) the effect of decay was not apparent. However, as time increased, the effect of decay increased in the later times ( progressively lighter gray lines 25, 30, 50 and 75 years after spill ) The concentrat ion reductions were slightly visible in the reduction in the peak magnitude (i.e. flattening of the peaks in the profiles ) in Figure s 3.7 b c However, the slight differences in these profiles can lead to more observable changes in mass and flux that will be explored in Table 3 5. In Figure 3 8, several time s for the different decay rates were isolated. The flattening of the concentration profiles as the decay rate increased wa s most visible at 25 (Figure 3 8b), 1 1 1 1 1 1 500 500 500 3500 3500 3500 0.35 0.35 0.35 0.45 0.45 0.45 1.14 1.14 1.14 8.0 8.0 8.0 (d 1 ) 0 1.0x10 5 2.5x10 5 0 5.0x10 4 1.0x10 3 0 72 180 0 400 800
64 Figure 3 7 depth profiles for diffusion ( ) with decay Note: depth profiles are shown for 5, 15, 25, 50, and 75 years with aquitard aqueous decay (a) (b) (c) Table 3 5. Summary of results for diffusion with decay modeling m g m 2 m g m 2 d 1 5 yrs 25 30 50 75 5 yrs 25 30 50 75 0 7272 8470 8037 6314 4850 3.67 0.63 0.71 0.59 0.35 72 7169 7725 7152 4988 3192 3.92 0.37 0.46 0.41 0.22 180 7019 6781 6070 3607 1799 4.25 0.03 0.15 0.21 0.11 5 yrs 25 30 50 75 5 yrs 25 30 50 75 72:0 0.014 0.088 0.110 0.210 0.342 0.417 0.346 0.305 0.362 180:0 0.035 0.199 0.245 0.429 0.629 0.956 0.782 0.645 0.689
65 Note: a quitard mass storage and mass flux for Negative flux ( ) is loading direction, positive flux is back diffusion flux. Fractional mass reduction ( ) and fractional flux reductio n ( ) are compared to results ( ) indicates back diffusion flux not occurring 50 (Figure 3 8c), and 75 years (Figure 3 8d) after source initiation. Furthermore, because an increase in decreased it also decreased mass storage and mass flux. The dimensioned m ass storage per unit area was calculated using Eq. 2 38 and Eq. 2 19, and the results are shown in Table 3 5. The mass was reduced by the effect of decay in the aquitard and the e ffect increased as time progressed T he fractional mass reduction was calculated and shown as and it increased accordingly. Table 3 5 summarizes the results of the decay modeling in the case of an aquitard without sorption. Fo r a site represented by with an aquitard decay represented by ( d 1 ), at years after the spill respectively Similarly, the dimensioned back diffusion flux was determined by Eq. 2 41a and Eq 2 42. As dimensionless decay i ncreased the effect on dimensionless mass flux was more complex however. The early time flux increased and the late flux dec reased. The increased rate of decay in the aquitard, acted to lower the concentration in the aquitard, thus increasing the loading gradient and loading flux, at 5 years Conversely, as the source boundary concentration decrease d over time and back diffusion began, the reduction of mass in the aquitard due to decay reduced the magnitude of the back diffusion flux, at 25 years and later. For a site where is representative with decay represented by a rate of ( d 1 ), at years after the initial spill respectively.
66 F igure 3 8 depth profiles for diffusion ( ) with decay for specific times
67 Note: depth profiles for , ; (a) (b) (c) (d) years. One of the most interesting outcomes here is the reduction of flux and the direction change of flux due to decay. For a site with decay represented by a rate of ( d 1 ), the loading flux continued longer than a site without this level of decay. A greater decay rate would in fact eliminate back diffusion entirely (not shown). F or a site represented by ( d 1 not shown), the degradation is large enough to remove enough mass in the aquitard such that the concentration gradient never reverses and back diffusion does not occur. Overall, flux reductio ns were achieved by decay in the aquitard proportional to the magnitude of the decay for s ites with little or no sorption. I n order to represent a site with sorption ( , ) with decay, the coefficients were an order of magnitude greater for d 1 respectively Using the same equations, profiles were created to represent a site with an d without dec ay in the aquitard in Figure 3 9. Comparing the no decay case (Figure 3 9a) t o those with decay (Figures 3 9b c) decay acted to reduce the aquitard concentration s, but to a lesser degree than the sit e without sorption presented in Figures 3 7 a nd 3 8. Aq ueous phase decay had a lesser effect for a site with sorption due to a reduction in availability, since more mass is stored on the solid phase. However, t he effect of decay on was still visible. Figure 3 9 shows that de creased in the later times (progressively lighter gray lines, 25, 30, 50 and 75 years after spill). Those peaks were flattened in Figures 3.9b and 3.9c. One thing that stands out compared to the no sorption cases is the magnitude of the y axis in Figure 3 9. It
68 Fig ure 3 9 depth profiles for diffusion with sorption ( ) and decay Note: depth profiles are shown for 5, 15, 25, 50, and 75 years with aquitard aqueous decay (a) (b) (c) Table 3 6. Summary of results for 1D decay modeling with sorption m g m 2 m g m 2 d 1 5 yrs 25 30 50 75 5 yrs 25 30 50 75 0 247 77 28886 27417 21548 16554 9.887 1.607 1.830 1.541 0.915 400 24498 26854 24991 17868 11875 10.343 1.077 1.330 1.164 0.644 800 24225 25029 22864 14959 8664 10.794 0.583 0.874 0.850 0.449 5 yrs 25 30 50 75 5 yrs 25 30 50 75 400:0 0.011 0.070 0.088 0.171 0.283 0.330 0.273 0.245 0.296 800:0 0.022 0.134 0.166 0.306 0.477 0.637 0.523 0.448 0.510
69 Note: A quitard mass storage and mass flux for cases with Negative flux ( ) is loading direction, positive is back diffusion flux. Fractional mass reduction ( ) and fractional flux reduction ( ) are compared to ( ) i ndicates bac k diffusion flux not occurring. goes from to whereas in Figures 3 7 and 3 8, the Y axis ranged from to This is due to reduction of penetration depth by sorption shown previously in Figure 3 3. When the later times were isolated for specific times for the different decay rates (Figure 3 10), reductions were more easily observable at 25 (Figure 3 10b), 50 (Figure 3 10c), and 75 years (Figure 3 10d). In Table 3 6, the dimensioned mass per unit area and dimensioned mass flux are shown for the representative site ( , ). The additional mass, when compared to the results in Table 3 5, was due to solid phase storage. The trends were similar to the cases without sorption As the decay rate increased and time increased, mass storage was reduced. Additionally, the loading m ass flux increased with increased decay and decreased during back diffusion. In Table 3 5, a were achieved at years after the spill, respectively for a site represented by , with decay of ( ). F or the same site with a decay rate of ( ), were achieved at years Similar to the no sorption case, the loading increased as increased ( shown at 5 yrs). Sorption acted to remove mass from th e aqueous phase as it penetrated the aquitard. This reduce d the aqueous concentration in the aquitard increasing the inward gradient. After back diffusion beg an decay act ed to reduce the mass flux from the
70 Figure 3 10 depth profi les for diffusion with sorption ( ) and decay for specific times
71 Note: depth profiles for ; (a) (b) (c) (d) years. aquitard. Looking at Table 3 5, for a decay rate of ( ), at years. Likewise, a site represented with a decay rate of ( ), the at years. This larger decay rate used in the site with sorption is still at the low end of the published range for TCE [ Newell et al., 2002] Aquitard Source Function s for Diffusion with Decay A quitard source functions were generated to compare the effect of decay on mass loading and back diffusion flux in Figure 3 11 a for with In Figure 3 11a, t he relative flu x and the relative mass in the aquitard were increasingly reduced as increased ( progressively lighter gray lines) compared t o the no decay site (black line) The maximum mass storage reduction was evident by the location of at B ack diffusion began at lower total mass as increased. Likewise, the peak of back diffusion flux was reduced as increased, and the slope of t he flux decline was reduced demonstrated by the flattening effect on the aquitard source function s as increased. As expected, similar to the previous aquitard source functions (Figure 3 6) the relative mass and flux in the aq uitard were very low values, shown by the magnitude of the x and y axes T he dimensioned mass storage and flux were 4 5 orders of magnitude less than the initial source zone mass and flux. The dimensioned mass per unit area and the dimensi o ned flux time se ries are shown in Figure 3 11 b for with The mass storage per unit area in the aquitard is shown on the right vertical axis The no decay site ( ) is the
72 Figure 3 11. Aquitard source functions with decay and and for and
73 Note: (a) Aquitard source functions for with Temporal progression is from right to left, from to (b) Dimensioned mass storage and back diffusion flux time series for (a) (c) Aquitard source functions for with ( d ) Dimensioned mass storage and back diffusion flux time series for ( c ). black dashed line and incre asing decay, is shown by progressively lighter gray dashed lines. The magnitude of the maximum mass storage, where the dashed lines meet the left vertical axis, decreased as increased ; g m 2 for Similarly, as decay increased, the remaining mass at the end of the simulation (200 years) was reduced. The flux (solid lines) peaked at years for at a magnitude of g m 2 yr 1 respectively. T he source dissolution was unchanged by different aquitard conditions, but increasing in the aquita rd increased the loading flux. However, it reduced mag nitude of the peak and slightly reduced the time of the peak. After peak flux, the slope of the flux decreased as increased (solid progressively lighter gray lines) compared to the no decay site (solid black line). For site s with aquitard sorption, the effect of is shown in Figure 3 11c with aquitard source functions for T he relative mass storage and t he relative back diffusion flux we re greater than the no sorption sit es shown in Figure 3 9, and demonstrated reduction in mass and flux as increased The maximum relative mass on the x axis at decreased as increased (lighter gray lines) an d the peak flux decreased accordingly compared to the no decay site (black line). Looking at the dimensioned mass and flux time series shown in Figure 3 11d,
74 g m 2 for The flux (solid lines) peaked at years for at a magnitude of g m 2 yr 1 respectively. The flux slope was steeper as flux approached its peak and during the early times after peak flux, compared to the no sorption model ing If the decay rate would have to be another order of magnitude larger ( ) to produce a similar result (not shown) Because the equations only consider aqueous phase decay, the e ffect of degradation is reduced in the model for sites with sorption. The model can be modified to add solid phase decay to calculate bulk decay as desired Dimensionless Results for Diffusion with Leakage Aquitard Concentration Profiles Some sites have a leaky a quitard where vertical groundwater gradients induce seepage thru the aquitard. The effects of leakage on aquitard storage and flux were investigated using the parameter values in Table 3 7. Table 3 7. Parameter values used in aquitard diffusio n with leakage model ing Note: Top section is aquifer parameters, bottom section is aquitard parameters. 1 1 1 1 1 1 500 296 1590 3500 2883 7500 0.35 0.35 0.35 0.45 0.45 0.45 1.14 1.14 1.14 8.0 8.0 8.0 (md 1 ) 0 5.0x10 6 5.0x10 6 0 5.0x10 6 5.0x10 6 (m 2 d 1 ) 1.04x10 5 1.60x10 5 4.86x10 6 1.04x10 5 1.76x10 5 3.72x10 6 0 4.5 23.9 0 3.8 12.5
75 A conservative value of 5.0x10 6 m d 1 for leakage was used in this effort which is in the range for clay landfills [ Johnson et al., 1998] and a regional aquitard in Australia [ Gardner et al., 2012] Because incorporates in (Table 2 2), the value of changes with magnitude and direction of In Figure 3 12, the profiles are shown for a site without significant sorption ( ) with leakage in both the upward and downward directions. In Figure 3 12a, a site with no leakage is used for comparison to a site wit h a downward leakage in Figure 3 12b, and a site with upward leakage in Figure 3 12c. The most noticeable difference is in the depth of penetration of mass into the aquitard. Downward leakage ( ) in Figure 3 12b results in at at 25 years and at 50 years For a site without leakage occurs at at 25 years and at 50 y ears ( Figure 3 12a ) Conversely, upward leakage ( ) in Figure 3 12c results in at at 25 years and at 50 years. Thus, upward leakage can reduce o r delay back diffusion. Figure 3 13, a comparison of each specific time for the three sites modeled in Figure 3 12 is presented for (solid lines), (dotted lines), and (dashe d lines). The differences in the depth profiles are highly visible in this format with the greatest differences occurring in the 15 through 50 year profiles. Fifteen years is good predictor of site back diffusion potential bec ause it is essentially a snapshot of the when the back diffusion flux is the greatest (Figure 3 13b). At other times, the flux is either loading (5 years, Figure 3 13a), releasing but very slowly (18 20
76 Figure 3 12 dept h profiles for diffusion ( ) with leakage Note: depth profiles are shown for 5, 15, 25, 50, and 75 years with aquitard leakage (a) ; (b) ; (c) Table 3 8 Summary of results for diffusion modeling with leakage m g m 2 m g m 2 d 1 5 yrs 25 30 50 75 5 yrs 25 30 50 75 0 7272 8470 8037 6314 4850 3.67 0.63 0.71 0.59 0.35 4.5 10055 12966 12578 10738 9034 4.01 1.04 1.08 0.80 0 .44 23.9 3516 3173 2837 1748 987 2.96 0.02 0.10 0.19 0.13 5 yrs 25 30 50 75 5 yrs 25 30 50 75 4.5:0 0.383 0.531 0.565 0.701 0.863 0.648 0.523 0.358 0.279 23.9:0 0.517 0.625 0.647 0.723 0.796 0.858 0.670 0.628
77 Note: aquitard mass storage and mass flux for Negative flux ( ) is loading direction, positive flux is back diffusion flux. Fractional mass reduction ( ) and fractional flux reduction ( ) are compared to results ( ) indicates an increase in stored mass, ( ) indicates back diffusion flux not occurring when no number follo ws the ( ) and increase in flux otherwise. years, not shown), or negligible (75 years, Figure 13e). For the example site modeled here, a greater upward leakage would prevent any back diffusion flux from occurring, and a lower le akage has a lesser effect where the non leaky model may be an appropriate approximation. A summary of the results of aquitard diffusion with leakage and without sorption are presented in Table 3 7. T he presence of downward leakage ( ) not only leads to deeper penetration, but also results in significantly more mass stored The mass increase was at years respectively, compared to the site. On the other hand, a site represented by upward leakage ( ) demonstrated at Mass flux showed a similar trend. The loading flux ( years) increased with and decreased with Similarly, during back diffusion, increased with downward leakage while it decreased with upward leakage. The at years compared to the site. Upward leakage resulted in significant decreases in back diffusion flux with b ack diffusion was not occurring at years and a t years To investigate the effects of aquitard diffusion with leakage on a site with sorption ( ), depth profiles were generated and are shown in Figure 3 14. Leakage did n ot have as great effect compared to the non sorption site (Figures 3 11 and
78 Figure 3 13 depth profiles for diffusion ( ) with leakage ; for specific times
79 Note: depth profiles for ; (a) (b) (c) (d ) (e) years. Figure 3 14. depth profiles for diffusion with sorption ( ) and leakage ; Note: depth profiles ar e shown for 5, 15, 25, 50, and 75 years with aquitard leakage (a) ; (b) ; (c) Figure 3 12) likely due to the loss of mass to solid phase storage as seen previously. Downward leakage ( ) in Figure 3 14b results in at at 25 years and at 50 years, compared to at at 25 years and at 50 years for a site without leakage in Figure 3 14a. Conversely, upward leakage ( ) in Figure 3 14c results in at at 25 years and at 50 years. In
80 comparing the sorption sites to the non sorption sites, we also see the y axis is scaled much closer to the surface to capture the profiles, ranging from instead of respectively due to the decreased depth of penetration. Figure 3 15 shows the depth profiles for the three sites at specific times (5, 15, 25, 50 and 75 years) for (solid lines), (dotted lines), and (dashed lines). The differences were reduced compare d to the sites shown in Figure 3 13, due to the effects of sorption. During the time after back diffusion began for these sites ( years), the downward leakage increased the depth of the peak but also the depth of the tailing in while upward leakage decreased penetration and the tailing in The mass storage and mass flux results of the aquitard diffusion modeling for the sites with leakage and sorption are summarized in Table 3 9. The effects of Table 3 9. Summary of results for 1D leakage modeling with sorption, Note: aquita rd mass storage and mass flux for . Negative flux ( ) is loading direction, positive flux is back diffusion flux. Fractional mass reduction ( ) and f ractional flux reduction ( ) are compared to results ( ) indicates back diffusion flux not occurring. m g m 2 m g m 2 d 1 5 yrs 25 30 50 75 5 yrs 25 30 50 75 0 24777 28886 27417 21548 16554 9.8 9 1.61 1.83 1.54 0.92 3.8 31283 37658 35995 29067 23025 11.59 2.20 2.43 1.96 1.14 12.5 16337 17949 16810 12535 905 4 7.43 0.85 1.06 0.98 0.60 5 yrs 25 30 50 75 5 yrs 25 30 50 75 3.8:0 0.26 0.30 0.31 0.35 0.39 0.37 0.33 0.27 0.25 12.5:0 0.34 0.38 0.39 0.42 0.45 0.47 0.42 0.36 0.35
81 Figure 3 1 5 depth profiles for diffu sion and sorption ( ) with leakage ; for specific times.
82 Note: depth profiles for ; (a) (b) (c) (d) (e) years. ( ) resulted in decreased with at years compared to the site. downward leakage ( ) resulted in increased compared to the site. The at years. On the other hand, a site represented by upward leakage Mass flux showed the same trend. The loading flux was increased by and decreased by Additionally after back diffusion began, increased with downward leakage while it decreased with upward leakage. The at years c ompared to the site. For the site at year s Aquitard Source Functions for Diffusion with Leakage A quitard source functions were generated to compare the effect of leakage on mass loading and back diffusion. In Figure 3 16a, these are shown for with The relative flux and the relative mass in the aquitard shown in Figure 3 16a, were both greater for a site with downward leakage (medium gray line) compared to the non leaky site (black line), and were both reduced for a site with upward leakage (light gray line). The mass storage increase for and decrease for were demonstrated by the location of the maximum relative mass at T he peak of back diffusion flux increa sed with and decreased with The most interesting result of the downward leakage modeling was the very slight rightward lean of the curve in Figure 3 16a. Mass storage in creased very slightly after back diffusion began. This is counterintuitive, but knowing that for any site
83 represented by back diffusion will begin at about and that the downward flow is unchanged t hroughout the timescale, it is likely that the downward leakage overwhelms the very low back diffusion flux in the early stages of release, such that some loading continues from the plume by advection even though stored mass is being released through back diffusion. The dimensioned mass per unit area and the dimensioned flux time series are shown in Figure 3 1 6 b for with The mass storage per unit area in the aquitard is shown on the right vertical a xis for the three sites with the non leaky site (black dashed line), the downward leakage site (medium gray dashed line), and the upward leakage site ( l ight gray dashed line). Where the meets the left vertical axis is and g m 2 for Similarly, the remaining mass at the end of the simulation (200 years) showed the same pattern with g m 2 The flux (solid lines) peaked at years for at a magnitude of g m 2 yr 1 respectively. After 100 years, the flux was below 0.075 g m 2 yr 1 and by 200 years it was below 0.025 g m 2 yr 1 in all cases. Fo r the s ite s with aquitard sorption and leakage the aquitard source functions are shown in Figure 3 16c The relative mass storage and t he relative back diffusion flux displayed similar patterns to those in the no sorption case in Figure 3 16a, but the di fferences from the site to those with leakage were slightly reduced d ue to sorption. T he magnitude of each axis was greater from additional solid phase mass
84 Figure 3 16. Aquitard source functions for diffusion with leakage and and
85 Note: (a) Aq uitard source functions for and Temporal progression is from right to left, from to (b) Dimensioned mass storage and back diffusion flux time series for (a). (c) Aquitard source functions for and ( d ) Dimensioned mass storage and back diffusion flux time series for ( c ) stored. The maximum relative mass on the x axis is at It increased for compared to and decreased for T he maximum relative mass was 7.5x10 4 for (x axis in Figure 3 16c) and it was 2. 5x10 4 for (x axis in Figure 3 16a) Flux also increased due to sorption. The maximum relative flux was 3.0x10 4 for (y axis in Figure 3 16c) and 1.4x10 4 for (y axis in Figure 3 16a). In Figure 3 16d, the dimensioned mass and flux time series are presented for the sites with leakage and sorption ( ), with (dashed lines) and ( solid lines ) Maximum mass occurs where the mass line intersects the y axis on the left side of the figure and for these sites g m 2 for At the end of simulation, the mass storage remaining in the aquitard was g m 2 The flux peaked at years for at a magnitude of g m 2 yr 1 respectively. By 100 years the flux was below 0.2 g m 2 yr 1 and by 200 years it was below 0.1g m 2 yr 1 in all cases Similar to the sites without sorption, the simulation of sites with sorption demonstrated the same trends but the differences were reduced.
86 CHAPTER 4 REMEDIATION RESULTS Source Removal Effects on Diffusion Aquitard Concentration Profiles with Source Remediation The effects of source zone mass removal on aquitard storage and release for diffusion alone are illu strated in Figure 4 1 and Figure 4 2. The remedial SDM (Figure 4 1 a) and the prof iles were generated using Eq. 2 5 c and Eq. 2 13c, and represent a site where under three cases: (1) no remediation (Figure 4 1 b), (2) 70 % DNAPL mass reduction (i.e., X = 0.7 ) at (12.6 yrs) (Figure 4 1 c), and (3) 70 % DNAPL mass reduction at (28.5 yrs) (Figure 4 1 d) Compared to Figure 4 1b, Figures 4 1c and 4 1 d show a reduction in concentration in the aquita rd due to remed iation. In Figure 4 1c remediation occurs prior to the start o f ba ck diffusion but, in Figure 4 1d remediation occurs after the start of back diffusion. Thus, the maximum potential mass storage in th e aquitard is attained in Figure 4 1 b and Figure 4 1 d but not in Figure 4 1 c Aquitard Source Functions with Source Remediation Figure 4 2 a shows the impact of partial source remediation on the aquitard source function. The earlier remediation occurs, the greater the reduction in the aquifer concentration and hence reduced concentration distributions in the aquitar d as shown previously (Figure 4 1 ). While remediation had a lesser impact on the mass that diffused into the aquitard (i.e. the starting point on the x axis), the relative flux out of the aquit ard increased as a result of remediation. In Figure 4 2 b, the dimensioned aquitard mass storage and mass flux time series are shown. Abrupt changes in flux and mass due to the remedial events are evident.
87 Figur e 4 1. Remedial SDM and resultant depth profiles. Note: (a) Remedial SDM and resultant depth profiles are shown for (b) no remediation, (c) remediation at (12.6 yrs) and (d) remediation at (28.5 yrs); , and for all cases
88 Figure 4 2. Aquitard source functions for diffusion with remediation Note: (a) Aquitard source functions for remediation at (28.5 yrs) and (12.6 yrs); , and for all cases. (b) Dimensioned mass sto rage and mass flux time series for (a). While remediation reduces the source strength, it may increase the in itial back diffusion flux. However, as illustrated in Figure 4 2 b the rate of decline of both mass and flux out of the aquitard were greater for the remediation cases than the non remedial case. Therefore, the risk of long term back diffusion flux was reduced as a result of remediation. While the increased initial mass flux may be an impediment to near term site closure, longer term reductions i n back diffusion can help achieve remedial goals.
89 Source Removal Effects on Diffusion with Decay Aquitard Depth Profiles for Diffusion with Decay with Source Remediatio n The effects of source zone remediation on sites with decay in the aquitard are shown in Figures 4 3, 4 4, and 4 5. The profiles for a site with and and are presented in Figure 4 3 for a partial source removal of at 25 years after the contaminant release. Figures 4 3 a c show the profiles with increasing As expected, as increased, the pea k concentrations decreased. Thi s effect was increased in the post remedial profiles ( years) for The concentration s were reduced and the peak s flattened. In Figures 4 3 d f, the same profiles are shown i n the post remedial years only (solid lines) with the no remediation profiles for the same In the earlier post remedial years (30 and 50 years after spill; i.e. and ), the reduction in compared to the no remediation lines was more pronounced tha n the late time profiles (75 years). The significance of this is that the early post remedial times capture the higher stored mass and the greater back d iffusion flux, so remediation will manifest as a reduction in stored mass and long term back diffusion flux, and therefore a reduction in overall site risk. However, as Figure 4 1 and 4 2 demonstrated, there will be an initial increase in flux due to the sudden change in the diffusion gradient and the increase in flux compared to the no remediation case may last for 15 to 20 years after remediation. Then, the post remedial flux will be low er than the no remediation case. Aquitard Source Functions for Diff usion with Decay and Remediation Aquitard source functions were generated to assess the mass storage and flux response to remediation for site s with aquitard decay ( shown in Figure 4 4 )
90 Figure 4 3. depth profiles fo r diffusion ( ) with decay and 70% source remediation at 25 years.
91 Note: depth profiles with remediation. (a) (b) (c) (d) comparison of post remedial profiles in (a) (solid lines) to no remediation (dashed lines), (e) comparison of (b) to no remediation, and (f) comparison of (c) to no remediatio n; all cases are , at years. Figure 4 4 Aquitard source functions for diffusion and decay with r emediation and without sorption Note: (a) Aquitard source functions for remediation at y ea rs ; , and for all cases. (b) Dimensioned mass storage and mass flux time series for (a). On the x axis of Figure 4 4a the maximum relative mass was unchanged compared to the relative mass in Figure 3 11a since the remediation occurred after The remedial lines demonstrate a marked increase in relative flux c ompared to the aquitard source functions for diffusion with d ecay and without remediation (Figure 3 11). The maximum relative ( 25
92 flux increased almost two orders of magnitude with remediation from 4.5x1 0 3 (Figure 3 11a) to 9.0x10 5 (Figure 4 4a). In Figure 4 4b, the dimensioned times series are presented for the curves in Figure 4 4a. Looking at the mass (dashed lines), there is an steep decrease in mass due to the source removal effects (i.e. the larg e decrease in concentration gradient) The flux (solid lines) go es off scale instantaneously at the time of source mass removal (25 years) due to the large change in the diffusion gradient. As a result, the flux increase removes mass from the aquitard ra pidly, leading to a large reduction in mass. This effect tapers off in the first five years, evident in the reduced slopes of both the mass and flux lines after 30 years ( ) The dimensioned time series for each in Figure 4 4b were p lotted separatel y in Figure 4 5, adding the non remedial lines for each for comparison. Due to the flux decrease as increased, the left hand y axis for flux (solid l ines) is scaled differently for each figure The right hand y axis remained the same scale for mass (dashed lines). The reduction was highly visible in the divergence of the remedial and non remedial lines. Five years after r emediation ( 30 years ), Because flux was increased by the sharp gradient drop from remediation, it took a few years before the benefits of source removal were realized in back diffusion flux. Howev er, from then on the flux will be lower than without remediation. The initial due to remediation occurred at for Five years later at A summary of the mass and flux for a site with and and is shown in Table 4 1for partial source re moval ( at years).
93 Figur e 4 5 Dimensioned aquitard source functions for diffusion and decay with and without remediation all without sorption ( 25 ( 25 ( 25
94 Note: Remediation at y ea rs ; , and for all cases. ( a ) Dimensioned mass storage and mass flux time series for (a) (b) (c) Thin solid line is flux without remediation; thin dotted line is aquitard mass storage without remediation. Table 4 1. Summary of Aquitard Diffusion with Decay and Source Remediation, Note: aquitard mass storage and mass flux for remediation with with an R designates remediation results. Fractional mass red uction ( ) and fractional flux reduction ( ) are compared to without remediation results ( ) indicates an increase in back diffusion flux Twent y five years after the remedial effort ( 50 years), and compared to the no remediation case for respectively. By (7 5 years), and At the end of the simulations (200 years), and for and the back diffusion flux is extremely low ( g m 2 yr 1 ) at this time so it is likely indistinguishable from the remaining source zone flux. m g m 2 m g m 2 d 1 30 yrs 40 5 0 75 200 30 yrs 40 5 0 75 200 0 R 6760 5564 4835 3746 2119 1.340 0.694 0.476 0.175 0.045 72 R 3773 3107 2601 1745 392 1.141 0.536 0.344 0.150 0.010 180 R 2124 1583 1192 598 30 0.890 0.353 0.203 0.071 0.001 30 yrs 40 50 75 200 30 yrs 40 5 0 75 200 0 R : 0 0.16 0.22 0.23 0.23 0.19 0.91 0.02 0.19 0. 30 0.22 72 R :72 0.27 0.28 0.28 0.26 0.23 1.46 0.12 0.16 0.32 0.27 180 R : 180 0.34 0.35 0.35 0.35 0.32 4.75 0.55 0.02 0.34 0.37
95 Source Removal Effects on Diffusion with Leakage Aquitard Concentration Profiles for Diffusion with Leakage and Remediation Source z one remediation affected aquitard diffusion with leakage as well. The profiles for a site with and and are present ed in Figure 4 6 for a partial source removal of at 25 years after the contaminant release. Figures 4 6a c show the profiles with source remediation. Figures 4 6d f show both the no remediation cases (dotted lines), and the post remediation cases (solid lines) profiles for comparison. The earliest post remedial line s at years (30 years ) in Figures 4 6a c demonstrate the effect s of the drop in bou ndary concentration from source removal. In particular, t he upper portion of the profile (near interface) drops considerably in the 30 year solid lines In Figures 4 6 d f, the post remedial profiles show a reduction in peak conc entration at all times for the remedial lines compared to the no remediation lines. Aquitard Source Functions for Diffusion with Leakage and Remediation Presented in Figure 4 7 are the aquitard source functions generated for sites with and and and remediation ( at years). The relative mass and relative flux are shown in Figure 4 7a and the d imensioned mass storage and back diffusion flux time series are shown in Figure 4 7b for ( black lines ), ( darker gray lines ), and ( light gray lines ). Looking along the x axis, there was a large increase in maximum relative mass due to (at ) and a decrease for Sim ilarly, the flux was increased due to but there was not a
96 Figure 4 6. depth profiles for diffusion with sorption ( ) and leakage with 70% source remediation at 25 ye ars.
97 Note: Note: depth profiles with remediation: (a) ;(b) (c) (d) comparison of post remedial profiles in (a) (solid lines) to no remediation (dashed lines), (e) comparison of (b) to no remediation, and (f) comparison of (c) to no remediation; all cases are , at years. Figure 4 7 Aquitard source functions for diffusion and leakage with remediation. Note: (a) Aquitard source functions for remediation at y ea rs ; , and for all cases. (b) Dimensioned mass storage and mass flux time series for (a). noticeable change in initial post remedial flux for compared to the in itial post remedial flux for In the dimensioned time series (Figure 4 7b), the changes in mass due to leakage are visible in the dashed lines, and the decreases in mass as result of source remediation are evident by the decrea se in slope beginning at (25 years). ( 25
98 Figure 4 8 Dimensioned a quitard source functions for diffusion and leakage with remediation for specific leakages. ( 25 ( 25 ( 25
99 Note: Solid lines are remediation at y ea rs ; , and for all cases. ( a ) Dimensioned mass storage and mass flux time series for (a) ;(b) (c) Thin solid lines are flux without remediation; thin dotted line is aquitard mass storage without remediation. Table 4 2. Summary of Aquitard Diffusion with Leakag e and Source Remediation, Note: aquitard mass storage and mass flux for remediation with with and R designates remediation results. Fractional mass reduction ( ) and fractional flux reduction ( ) are compared to witho ut remediation results ( ) indicates an increase in back diffusion flux The dimensioned time series for each are shown in Figures 4 8a c. The post remediation (thick lines) and the corresponding no remediation (thin lines) cases are plotted together for comparison. The dashed lines are and the solid lines are In these cases, the maximum mass storage was different for each value of such that the right hand y axis is scaled accordingly. Likewise, the left hand y axis for flux is also scaled accordingly but the maximum flux (25, ) was offscale in each case. Compared to the scale for each y axis for was greater and was reduced, as expected. The divergence of the mass lines for the post remedial and non m g m 2 m g m 2 d 1 30 yrs 40 5 0 75 200 30 yrs 40 5 0 75 200 0 R 23054 18979 16487 12771 7237 3.542 1.835 1.258 0.644 0.119 3.8 R 30513 25524 22402 17912 11145 4.450 2.298 1.561 0.794 0.145 12.5 R 13940 11108 9390 6853 3171 2.352 1.208 0.831 0.427 0.077 30 yrs 40 50 75 200 30 yrs 40 5 0 75 200 0 R : 0 0.16 0. 22 0.23 0.23 0.19 0.98 0.04 0.17 0.29 0.22 3.8 R :3.8 0.15 0.21 0.23 0.22 0.19 0.83 0.01 0.20 0.30 0.22 12.5 R : 12.5 0.17 0.23 0.25 0.24 0.19 1.22 0.09 0.15 0.29 0.22
100 remedial lines was highly visible but the change is flux was less evident. Five years after remediation ( ), The initial due to remediation occurred at for Five years later at At the end of the simulations (200 years), and for and the back diffusion flux is extremely low ( g m 2 t 1 ) at this time so it is likely indistinguishable from the remaining source zone flux.
101 CHAPTER 5 CONCLUSIONS 1D Aquitard Diffusion A one dimensional aquitard diffusion model with a semi infinite domain which used a sou rce depletion model (SDM) as a boundary condition, was used to investigate the effects of DNAPL source architecture and aquitard properties on the risk of back diffusion. Two key variables used in this assessment were: 1) the power law exponential term which reflects the source zone architecture, and 2) the source to aquitard mass transfer coefficient which reflects the influence of both the source characteristics ( ) and the aquitard properties ( ). The amount of contaminant mass in the aquitard, the back diffusion flux magnitude, and their longevity were used as measures of back diffusion risk. The use of a semi infinite domain presents an inherent limitation to applying the model to finite thickness aquitards, particularly if the aquitard is quite thin. If sorption is high, the model may be applicable to ~ 1m thick aquitards. It is not appropriate to thin lenses but could be utilized if the lens i s several meters thick and superposition is used. The greatest potential for back diffusion occurs when the source strength is constant until source mass is exhausted (i.e., ), as is often assumed in back diffusion assessments. Fo r sites where mass discharge decreases with time as source mass is depleted (i.e., ), less risk is expected. Specifically, the aquitard mass per unit area the dimensionless start time for back diffusio n the depth of penetration, and the magnitude of back diffusion flux all decreased as increased, indicating reduced risk due to back diffusion.
102 Additional source zone char acteristics that were investigated include the initial source mass the initial source concentration the groundwater flux and the source zone control plane area These terms were combined to represent the ratio of initial source zone mass discharge to initial DNAPL mass (i.e., the source decay function). In general, site conditions that lead to an increased and hence (i.e., large initial mass discharge and small ) result in higher, short term back diffusion flux due to more rapid source dissolution. However, they also result in a lower long term risk due to the reduced stored mass and penetration depth in the aquitard. Conversely, sites with a decreased (i.e. small initial mass discharge and large ) generate a lower back diffusion flu x due to slower source depletion. These sites present a greater long term risk due to back diffusion from the additional mass stored in the aquitard. Increasing aquitard sorption, represented by a larger and hence had no effect on the loading concentration gradient, but increased both the mass storage and the back diffusion flux However, increased sorption also decreased the depth of penetration of contaminants. This suggests that sites with greater sorption of contaminants are at greater risk for back diffusion, particularly in the short term. If contaminant decay occurs in the aquitard, the mass storage and back d iffusion flux decrease proportionally with the magnitude of the decay. The reduction in mass and flux will also decrease site longevity if back diffusion is the constraint on reaching MCL. Sites with sorption demonstrate less decay due to the reduction in available contaminant from solid phase storage. For a site with a greater the loading flux direction continues longer compared to a site without decay or one with low decay. In fact, i f the decay rate is great enough, back d iffusion never occurs due to the
103 increase of degradation of contaminant in the aquitard which never allows the concentration gradient to reverse. If the aquitard is leaky, downward vertical flow increases mass storage, depth of penetration, back diffusion flux, and site longevity, whereas upward leakage reduces mass storage, penetration, back diffusion flux, and longevity. Sorption tended to reduce the effects of leakage compared to sites without sorption. The modeling effort presented here represents the worst case scenario for back diffusion for a simplified aquitard with a time varying boundary immediately down gradient of the DNAPL source because advection, dispersion and degradation in the aquifer were neglected The as sumption of a homogenous aquitard also leads to increased diffusion, storage and mass flux. D egradation in the aquitard was mode led conservatively, up to 3 orders of magnitude less than the literature range for aquifers and only included aqueous phase decay Source Removal By emp loying an SDM as the time variable boundary condition, the effects of remediation on aquitard diffusion were simulated. A delay in DNAPL source zone remediation increases long term, back diffusion risk, which is similar to the conclusion made by Falta et al. [2005b] regarding the impact of delays in source zone remediation and the resulting increase in plume mass. Source removal can decrease the amount of mass loaded into the aquitard by reducing the aquifer concentration and hence the diffusion gradient Even after back diffusion has started, source zone remediation will provide long term reductions in back diffusion risk by accelerating the rate at which mass is removed from the aquitard. For
104 sites with decay in the aquitard, remediation will reduce t he mass storage and site longevity, but will increase the back diffusion flux in the early years after remediation. In the long term back diffusion flux will be reduced as well. Similarly, for sites with leakage, partial source removal will decrease mass storage and longevity, but also increase the back diffusion flux after remediation. In the long term, remediation demonstrates benefits by reducing the back diffusion flux and the site longevity. Future Work Future modeling could investigate the combined effects on diffusion for a site with aquitard decay and leaka ge. Decay could be modeled as a time changing decay rate due to remedial enhancements and the effects of solid phase decay could be modeled as well D aughte r product formation was neglected in this effort and for many DNAPL species may be significant, an d could be accounted for in further model developments. Leakage was held constant in this work and could be altered through the site lifespan to s imulate the effects of pumping through injection of extraction in an underlying aquifer. This could be furth er explored with 2D models as well. Additionally, the impact of increasing interfacial areas associated with distributed lenses of high or low conductivity media on mass stor age and flux should be explored in 2D models.
105 APPENDIX Zhu and Sykes  pro vide a de tailed derivation of Eq. 2 5a and Eq. 2 5b First, E q 2 2 can be represented by (A 1) Then, differentiating both sides with respect to twice leads to ( A 2) which has a general solution of ( A 3) Similarly, Zhu and Sykes  provide the specific case of where E q 2 2 can represented by ( A 4) Again differentiating both sides with respect to leads to (A 5) The solution to E q A 5 with an initial condition is (A 6) This be comes E q 2 5c in dimensionless form. The model verification utilized several existing analytical models for specific and boundary conditions. First, for the simple, infinite constant concentration boundary
106 case ( ) in E q 2 5a, for the conceptual model with time invariant boundary conditions of (A 7a) and ( A 7b) and an initial condition of (A 7c) an analyt ical solution [Crank, 1975] to E q 2 8 is (A 8) The diffusive flux and dimensionless mass storage per unit area are given by Crank  and have been transformed to dimensionless forms: (A 9) and (A 10) respectively. For the case, a new analytical solution was derived and used to verify the general solution for that case. This model incorporates linear decay of the source mass due to dissolution, but is only valid for the time prior to the initiation of back diffusion. (A 11 )
107 where For the case, Bear et al.  develope d an analytical solution that models exponential decay of the boundary concentration through the time period of layer diffusive loading, (A 1 2 ) The flux for this case is (A 1 3 ) Finite difference methods were employed to verify the new analytical model. The general analytical solution, equation (9), was verified with the central difference approximation for the second derivative [ Grathwolh, 1998]. for any (A 1 4 ) The finite difference solution is valid for any imposed boundary changes but is subject to a stability criteria of The flux into and out of the layer was verified with the spatial derivative approximatio n (A 1 5 ) comparing the boundary concentration to that of a very small location below the interface. Verification of the remediation model was performed for the case. Previous investigations have c onsidered the impact of source treatment on aquitard diffusion by considering a step change in concentration at the upper boundary of the aquitard [ Bear
108 et al. 1994; Chapman and Parker, 2005; Sale et al ., 2008]. To model complete source depletion, a cons tant concentration would be applied to the upper boundary of the aquifer for a period of time, and then the boundary condition concentration would be set to zero thereafter. In practice, this corresponds to the time at which the source was exhausted eithe r through natural dissolution or remedial treatment, or isolated by physical or hydraulic barriers. In the dimensionless framework of this work, a boundary condition of (i.e., Eq. 2 5a with ) would be a pplied to the initially clean aquitard (i.e., ) for a time period If the case source exhaustion occurred through natural dissolution, then in E q 2 12, and t he solution for the aquitard concentration is E q A 8 for This represents concentration resulting from the mass loaded into the aquitard. When complete source exhaustion or isolation occurred, the boundary would ch ange to zero The analytical solution for the concentration distribution in the aquitard for was given by Bear et al.  and the dimensionless version is ( A 1 6 ) This represents concentration resulting from the contaminant mass in the aquitard after loading is complete. It includes the diffusion of mass back to the aquifer due to the concentration gradient reversal at the interface, and the continued downward mig ration due to diffusion within the aquitard. The diffusive flux after source exhaustion was given by Bear et al.  and has been modified to the dimensionless form: (A 1 7 )
109 This solution was also utilized for remediatio n cases where 100% of the source mass is removed at However, rarely does perfect remediation or complete source exhaustion occur. Thus, the effects of partial source removal were investigated with ou r new model.
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115 BIOGRAPHICAL SKETCH Gordon Hitchings Brown was born on June 19, 1969, in Rochester, NY. He was raised by two scientists and grew up in and around the Helmer Nature Center that his mother directed. After moving to Northern California in 1985, he graduated from Palo Alto Senior High School, Palo Alto, California, in 1987. He enrolled at the State University of New York College at Brockport in 1996 and received a Bachelor of Science de gree in chemistry and earth science in 2003. He graduated as a departmental scholar in earth sciences, and received both analytical chemistry and sigma xi awards for excellence in undergraduate research at Brockport. He continued his education at the Unive rsity of Florida in the Department of Environmental Engineering Sciences and received a Master of Science degree in 2006. Gordon was a National Science Foundation Science Partners In Collaborative Inquiry based Education (SPICE) fellow from August 2005 thr ough August 200 8 He has served Resources Association (AWRA) from 2003 to 200 9 and served on the AWRA Florid a Section Board of Directors from 2006 2009 He pursued this re Laboratory Ground Water and Ecosystem Restora tion Division center in Oklahoma from 2009 2011 as a student services contractor. He has been working in health, safety, and environmental compliance in the oil and gas industry in Oklahoma since. He desires to return to Florida to work in environmental c onsulting and eventually to teach at the university level.