Corresponding author: G. H. Brown, Interdisciplinary Program in Hydrologic Sciences, Department of Environmental Engineering Sciences, University of Florida, Gainesville, FL 32611, USA. (email@example.com)
 A one-dimensional diffusion model was used to investigate the effects of dense nonaqueous 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 behavior on aquitard mass storage and release: the power law exponential term Γ, which reflects the influence of the source zone architecture, and a new variable defined herein as the source to aquitard mass transfer coefficient β, which reflects the influences of both the source characteristics and the aquitard media properties. As Γ increased or as β increased because of more rapid source dissolution, the aquitard concentrations, depth of penetration, and long-term back-diffusion flux decreased. However, when β increased because of increased sorption, concentrations and back diffusion increased but penetration 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 aquitard was 3 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 4 or more orders of magnitude less than the initial source zone flux. 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.
 The presence of aqueous contaminants in low-conductivity media down gradient of suspected or known dense nonaqueous 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].
Ball et al.  used high-resolution sampling of a soil core to investigate the distribution of contaminants in an aquitard down gradient 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 aquifer compared to those in 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 3 years [Liu and Ball, 2002]. Conditions to induce back diffusion within the sheet piling were maintained over the final 2 years of the study, and results were consistent with those expected with back diffusion as the dominant transport process. Field data were evaluated using a multilayer one-dimensional (1-D) 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 estimating 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 nonunique 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 loading 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 [Parker et al., 2004]. The source was isolated with sheet piling and 6 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 surficial aquifer just above the aquitard interface, suggested plume persistence due to back diffusion. A 1-D diffusion model was used to predict the measured concentration profiles in the aquitard, and the agreement with field data improved by using a stepped, declining 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 down gradient of a DNAPL source zone. Groundwater concentrations were observed to decrease by several orders of magnitude in the first 5 years after hydraulic isolation of the source, yet they remained above MCLs in the down-gradient 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 (2-D) modeling by Parker et al.  applied a constant concentration boundary condition in an aquifer with discrete low-conductivity layers. After 30 years, the source was terminated and TCE diffusion out of the low-conductivity layers was observed for up to 200 years.
 Finally, a recent study used laboratory and modeling efforts to elucidate the effect of reduced contaminant loading on down-gradient water quality [Sale et al., 2008]. 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 fitted with a two dimensional model with constant concentration boundary conditions. The investigators 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 summary, these studies have used field and lab data to identify and demonstrate the importance of back diffusion, and have used diffusion models to help interpret the data. These 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 the significance of diffusive storage and release have not been clearly established.
 The work reported here explored the relationship between source depletion and diffusive transport to low-conductivity media using one dimensional (1-D) analytical methods. An analytical SDM was utilized to provide a time-varying boundary condition for the 1-D diffusion model. 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, 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., the magnitude of contaminant flux from the aquitard). Thus, hydrogeologic and contaminant parameters that affect mass loading to and release from the aquitard were examined.
2. One-Dimensional Model
2.1. Dimensionless Source Depletion
 Our conceptual model begins with a DNAPL source zone in an aquifer, shown in Figure 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 [M L−3] leaving the source zone across a control plane with cross sectional area [L2]. 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:
where [M L−3] is the initial, flux-averaged concentration crossing the source zone control plane, [M] is the initial mass of the contaminant in the source zone, and is an 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 , which represents the constant concentration source; which represents linear source decay; and , which represents exponential source decay. Using the definitions provided in Table 1, equations (1) and (2) can be expressed in dimensionless form as
where the source to aquitard mass transfer coefficient is [dimensionless] as defined in Table 1. The term represents the relative extent to which mass is transferred from the source to the aquitard, and is the product of the source decay function [T−1] given by Zhu and Sykes , and a new variable defined here as the aquitard diffusion timescale [T] (Table 1). Consequently, couples source dissolution processes in and effective diffusional processes in . A low represents slow source zone dissolution and/or low aquitard retardation where the plume concentration remains high for a longer time, while a high represents rapid source dissolution and/or high aquitard retardation. The effective diffusion coefficient D* in the aquitard is used in and (Table 1), and will be discussed in more detail in section 2.2.
Table 1. Definitions of Dimensionless Variables
Source zone mass and concentration
Aquitard depth, time, and concentration
Source to aquitard mass transfer coefficient
Postremedial source mass and concentration
Postremedial source to aquitard mass transfer coefficient
 The simple screening level SDM (equations (4), (5a), (5b), and (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 groundwater 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, 2007], 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 therefore considered to represent the aquifer concentration above the aquitard. This assumption is considered appropriate in the region immediately down gradient of the source zone.
2.2. Dimensionless Diffusion
 Our diffusion model in the aquitard is a 1-D, semi-infinite domain with zero concentration at infinite depth, zero concentration initially throughout the domain and an imposed time variable concentration boundary condition at the surface (Figure 1). The location of the aquitard is immediately down gradient of the DNAPL source zone. The aquitard can be above or below the near source aquifer region; however, an underlying aquitard was chosen as defined by the direction (Figure 1). The upper boundary condition will be represented by the flux-averaged concentration of dissolved contaminant leaving the DNAPL source zone, equation (5a) or (5c). The system is governed by Fick's second law of diffusion,
where [M L−3] is the concentration in the aquitard, [L] is the depth in the aquitard, and [L2 T−1] is the effective diffusion coefficient:
where, is the molecular diffusion coefficient [L2 T−1], is the tortuosity [dimensionless], is the retardation factor [dimensionless], is the media bulk density [M L−3], is the porosity (dimensionless), and is the distribution coefficient [L3 M−1], assuming equilibrium, reversible, and linear partitioning of the contaminant between the aquitard pore water and solid media.
 Using the dimensionless framework proposed in Table 1, the dimensionless form of Fick's second law becomes
2.3. One-Dimensional Aquitard Diffusion With a Source Depletion Boundary Condition
The upper boundary condition of the aquitard can be represented by a function that imposes a time variable concentration, represented by the SDM in equation (5a) or (5c). The SDM equation is substituted for in equation (9) to convert the general solution to a specific one. The resulting specific solution was evaluated with MathCAD 14.0, an engineering calculations software program. Other analytical solutions of equation (8) for specific cases of have been previously published [Crank, 1976; Bear et al., 1994]. These solutions were used for model verification and are available in the auxiliary material.
Equation (9) is fully dimensionless, is applied to a single-layer semi-infinite aquitard, and employs the convolution theorem, as did Booker and Rowe  and Liu and Ball . Several 1-D 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 governing equation [Rubin and Rabideau, 2000; Foose et al., 2001]. In a recent study, Chen et al.  developed a 1-D analytical solution with a time changing boundary condition and applied it to diffusion and mass flux through a multilayer landfill liner. This solution used the separation of variable method instead of the Laplace transform method. While equation (9) also determines the concentration profiles like previous studies, this work moves into a risk framework by focusing on the mass storage and mass release out of the aquitard resulting from time variable DNAPL sources.
 The model for diffusion in the aquitard contains several simplifying assumptions. The groundwater flow is assumed to be large 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. 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 reentering the aquifer from the aquitard would be reduced by dispersion and biodecay.
2.4. Dimensionless Mass Storage and Dimensionless 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,
by integration of dimensionless concentration through dimensionless space,
or integration of dimensionless concentration in dimensionless space in the Laplace domain and application of the convolution method,
The integral in equations (10a) and (10c) were evaluated analytically and equation (10b) numerically. Equation (10a) 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 the relative error was zero for equations (10a) and (10c) and less than 5% for equation (10b).
 The flux into or out of the aquitard is given by Fick's first law of diffusion, and was evaluated in dimensionless form using
where equation (5a) or (5c) is substituted into the convolution for . The convolution solution does not allow an evaluation at . Thus, a very small value for (i.e., ) was used in equation (11) to approximate the flux at the aquifer/aquitard interface. For example, corresponds to mm for m2. In this work, an interface depth of 2.5 to 3.5 mm was used to evaluate the flux at the aquifer/aquitard interface.
 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 the auxiliary material). Finite difference methods were employed for verification purposes of determined by equation (11) at the stated depths for other cases . The finite difference approach used equation (5a) for the boundary condition and solved equation (8) using a central difference solution to obtain the concentration profiles, as shown in the auxiliary material [Grathwohl, 1998]. The flux into and out of the layer with equation (11) was verified with numerical differentiation serving as the 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%. Second, just before back diffusion initiates and there were larger than 5% relative errors in the model comparisons in that instant as well. However since the magnitude of the mass flux is so small, this 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 aquitard; 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 concentration 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 conditions similar to those modeled here is that no sequestration of contaminant is expected within the aquitard.
2.5. Dimensionless Time
 The range of for typical field sites needs to be established to define the modeling timescales. Given the definition of in Table 1, this requires an estimate of the ranges associated with , , and . While includes several solute and media properties in equation (7), it will seldom vary more than 1 to 2 orders of magnitude and tends to range from 10−5 to 10−7 m2 d−1. On the basis of previous work [Wood et al., 2009], a suggested range for is 10 m2 to 1000 m2. Thus, for 0.01 t 10,000 years, will range from 10−9 to 10 in most applications.
 There are two key events in time that were evaluated with the model: (1) , and (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/aquitard interface. Therefore, no additional mass will be loaded into the aquitard for and represents the time of maximum mass storage, . For the cases, the back-diffusion flux will peak at , where
For the cases, peak back-diffusion flux will occur at some time after .
2.6. Source Removal
 The impact of remediation on diffusion in the aquitard can be investigated by modifying the dimensionless SDM equations ((5a) and (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 remedial process at time . The relative mass remaining in the source zone at , prior to the remediation i.e., , is calculated using equation (5b), and in turn is used to determine the postremedial initial relative mass and relative concentration (Table 1). Likewise, these are used to calculate a new postremedial source decay function , and a postremedial source to aquitard mass transfer coefficient (Table 1). Consequently, the postremedial relative concentration and the relative mass remaining in the source zone are
respectively, where is the dimensionless elapsed time after remediation (i.e., ). For the case, the remaining mass and postremedial source concentration are given by
Finally, the dimensionless time at which the mass remaining in the source zone after remediation is exhausted by natural dissolution becomes
 There are two main parameters that affect the model results: and . The effects of on the model were investigated first, and then the effects of were explored. Aquifer and source properties for the set of simulations featured in Figures 2 and 3 are shown in Table 2, representing a hypothetical PCE spill and are similar to those used by Falta et al. [2005a]. Aquitard parameters used in these simulations are presented in Table 3.
Table 2. Dense Nonaqueous Phase Liquid Source Zone Values Used in the Model
 One method to evaluate the effects of source dissolution on aquitard storage and release is to construct profiles of as a function of depth using equation (9). Figure 2a illustrates the SDM for and Figures 2b, 2c, and 2d illustrate the depth profiles of for , respectively. As increased, the profiles in Figures 2b–2d 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 depths of , respectively.
 In Figure 3, 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 achieved at . Figure 3a 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 3b, the concentration profiles at for cases with are shown. As increased, concentrations in the aquitard were reduced because of 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 and ; and in Figures 4b through 4d the resultant depth profiles. Similar to the influence of , decreased as β increased as a result of the more rapid reduction 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 dimensionless results in Figure 2 were converted to dimensioned time and space using our hypothetical PCE site values (Tables 2 and 3). 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 years , mg kg−1 which occurred at depths of m for , respectively. Evaluating the influence of on concentration in dimensioned time and space, however, is not as straight forward. At m and years , 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 Tables 2 and 3. However, the same dimensionless profiles for occur when results from , respectively, m d−1, and all other parameters equal to the same previous values (Tables 2 and 3). 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 years for , respectively. Consequently, at the same depth of m , mg kg−1 for , respectively, demonstrating an increase in total mass as increased because of 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 penetration. However, when increased because of an increase in (via increased ), solid phase concentrations increased (increasing total concentrations), but the contaminant penetration depth decreased.
3.2. 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 2, as increased, the amount of mass stored in the aquitard decreased. Specifically, using equation (10b) to integrate the depth profiles in Figure 2 for , for , respectively. As increased, less mass entered the aquitard because of 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 the 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 2a for ), reducing the back-diffusion gradient, which in turn yields slightly more dimensionless mass at this point in time.
 To evaluate the maximum mass storage , the depth profiles at (Figure 3a) were integrated using equation (10b), and the results are shown in Figure 5a as a function of and in Figure 5b as a function of . As increased from 0 to 10, decreased from 0.019 to 0.0039. Likewise, as increased from 250 to 11,000, decreased from 0.039 to 0.006. Figure 5a illustrates the error that might occur in estimates of when assuming , as is often done, for those sites with characteristics better represented by . As further illustration of this point, Table 4 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 risk of back diffusion, if this assumption is not valid.
Table 4. Relative Reduction in Maximum Mass Stored in Aquitard,
3.3. 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 5. Starting at , increased to its peak at , then decreased as increased (Figure 5a). 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 back-diffusion time. When the source, in theory, is never completely exhausted by dissolution; consequently, back diffusion lasts for an infinitely long time. Similar to , as increases, back diffusion begins earlier (Figure 5b). When was increased by more rapid source dissolution, back-diffusion flux initiated earlier and less mass was loaded into the aquitard. However, if was increased because of increased aquitard sorption only, decreased, but that value corresponded to the same dimensioned time since is also a function of (Table 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 is shown in Figure 6, 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 Figure 6 was scaled to for each specific case. As increased, the contaminant mass was loaded relatively rapidly into the aquitard because of 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 the 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.
3.4. Aquitard Source Functions
 A convenient means 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., 2002]. Likewise, aquitard source functions can be defined as 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:
where is aquitard mass per unit area [M L−2] and is back-diffusion flux [M L−2 T−1]. Aquitard source functions are shown in Figure 7 as a function of and , along with the corresponding dimensioned times series for (dashed lines) and (solid lines). The ratio was used to normalize , while was used to 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 7a presents aquitard source functions for and , and illustrates three distinct shapes. For , the boundary concentration jumped from to instantaneously, 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 boundary 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 . For (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 evident in Figure 7a is that as increased, both the relative mass and flux decreased, indicating reduced risk due to back diffusion. Specifically, as increased from 0 to 2, the maximum relative mass decreased from 1.04 × 10−3 to 4.3 × 10−4, while the maximum relative flux decreased from ∞ to 1.3 × 10−4. 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; and the back-diffusion flux, with the exception of the case, was three or more orders of magnitude less than the initial source zone flux.
 The results from Figure 7a were converted to dimensioned flux (solid lines) and dimensioned mass per unit area (dashed lines), and plotted as a function of elapsed dimensioned time from the start of back diffusion (Figure 7b). Similar to the results in Figure 7a, as increased the peak mass and mass flux decreased. The rates of decrease, however, varied as a function of . For example, within the first 10 years after , the order of as a function of was: > > > . By 250 years after (off scale in Figure 7b), 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 7c, 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 2. As increased from ( d−1) to ( d−1), the maximum relative mass decreased from to , and the 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 area and flux. Similar results are expected for changes in and that lead to an 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 7d shows the dimensioned time series for and corresponding to the dimensionless data in Figure 7c. Comparison of Figures 7c and 7d for ( d−1) indicated that while this case has the highest relative flux, it had the lowest-dimensioned flux. This was because 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 7d, 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 7d). 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 7d 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 source functions shown in Figure 7e for and . These values of correspond to days, and represent the dissolution of a PCE source over silty sand , silt , and silty clay aquitards, with the modeling parameters shown in Table 3. The results showed that as increased (and hence ), additional mass was stored in the aquitard (on the solid phase as expected). Perhaps less obvious however is the result that the maximum relative flux likewise increased because of the additional sorbed mass. In Figure 7f, the time series for and are shown for the model results in Figure 7e. Increasing (and hence ) due to increased retardation increased both and , even though the penetration depth was reduced (see Figure 4).
3.5. Source Removal
 The effects of source mass removal on aquitard storage and release are illustrated in Figures 8 and 9, using the parameter values in Table 5. The remedial SDM (Figure 8a) and the profiles were generated using equations (5c) and (13c), and represent a site where and under three cases: (1) no remediation (Figure 8b), (2) 70% DNAPL mass reduction (i.e., X = 0.7) at (12.6 years) (Figure 8c), and (3) 70% DNAPL mass reduction at (28.5 years) (Figure 8d). Compared to Figure 8b, Figures 8c and 8d show a reduction in concentration in the aquitard due to remediation. In case 2, remediation occurs prior to the start of back diffusion but, in case (3) remediation occurs after the start of back diffusion. Thus, the maximum potential mass storage in the aquitard is attained in cases 1 and 3, but not in case 2.
Table 5. Values of Variables Used in the Remedial Model
Parameters Used in Partial Source Remediation Modeling
Figure 9a shows the impact of partial source remediation on the aquitard source function. While remediation has a relatively small impact on the mass diffusing into the aquitard (i.e., the starting point on the x axis), the relative flux out of the aquitard spiked noticeably as a result of remediation. The earlier remediation occurs, the greater the reduction in the aquifer concentration, and hence reduced concentration distributions in the aquitard as shown previously (Figure 8). In Figure 9b, 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. While remediation reduces the source strength, it may increase the initial back-diffusion flux. However, as illustrated in Figure 9b the rate of decline of both mass and flux out of the aquitard were greater for the remediation cases than the nonremedial 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 in back diffusion can help achieve remedial goals.
 A one-dimensional aquitard diffusion model, which used a source depletion model (SDM) as a boundary condition, was used to investigate the effects of DNAPL source architecture 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 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. For sites where mass discharge decreases in 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 diffusion , the depth of penetration, and the magnitude of back-diffusion flux all decreased as increased, indicating reduced risk due to back diffusion.
 Additional source zone characteristics 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 flux 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 highly sorbing contaminants are at greater risk for back diffusion.
 By employing 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.
 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 dispersion and degradation in the aquifer were neglected and degradation in the aquitard was neglected. Future two- and three-dimensional modeling efforts should incorporate advection, dispersion and degradation in the aquifer. Likewise, degradation in the aquitard should be incorporated, as the highly reducing environment of the aquitard will likely transform and degrade some of the stored mass. Daughter products formed during decay could be accounted in further model developments. Additionally, the impact of increasing interfacial areas associated with distributed lenses of high- or low-conductivity media on mass storage and flux should be explored.
 The work upon which this paper 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 subjected to agency review and therefore does not necessarily reflect the views of the agency and no official endorsement should be inferred. The authors are grateful to Martha A. Williams of SRA for assistance provided with the graphics. Comments from Tissa H. Illangasekare and three anonymous reviewers were appreciated and were used to improve the manuscript.