Aquitard contaminant storage and flux resulting from dense nonaqueous phase liquid source zone dissolution and remediation

Authors


Corresponding author: G. H. Brown, Interdisciplinary Program in Hydrologic Sciences, Department of Environmental Engineering Sciences, University of Florida, Gainesville, FL 32611, USA. (ghbrown@ufl.edu)

Abstract

[1] 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.

1. Introduction

[2] 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].

[3] Ball et al. [1997] 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.

[4] 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].

[5] In addition to aquitards, Parker et al. [2008] 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. [2008] 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.

[6] 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.

[7] 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.

[8] During the decades or centuries required for groundwater to completely dissolve a DNAPL spill, the source architecture may change because of local depletion in the higher hydraulic conductivity zones, potentially weakening the overall source strength (i.e., the mass discharge from the source zone) and plume concentration [National Research Council, 2004; Falta et al., 2005a]. Both laboratory and field studies have documented reductions in mass discharge 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 reduction in approximately a 1:1 relationship. Moreover, at most field sites monitoring and remedial activities initiate 10 to 40 years after the spill, and a recent study suggested that at aged sites a 1:1 mass discharge to DNAPL mass reduction relationship may be assumed, regardless of initial spill architecture [Chen and Jawitz, 2009].

[9] Thus, recent 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 analytical models that use either Lagrangian or Eulerian methods have been developed [Parker and Park, 2004; Falta et al., 2005b; Jawitz et al., 2005; Christ et al., 2006; Basu et al., 2008b; Zhang et al., 2008; Chen and Jawitz, 2009; DiFilippo and Brusseau, 2011]. These 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].

[10] 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

[11] 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 math formula [M L−3] leaving the source zone across a control plane with cross sectional area math formula [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:

display math

where math formula [M] is the mass of the contaminant in the source zone, and math formula [L T−1] is the groundwater flux. Zhu and Sykes [2004] and Falta et al., [2005a] provide solutions for equation (1) when math formula is related to math formula using a power law expression,

display math

where math formula [M L−3] is the initial, flux-averaged concentration crossing the source zone control plane, math formula [M] is the initial mass of the contaminant in the source zone, and math formula 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 math formula, which represents the constant concentration source; math formula which represents linear source decay; and math formula, which represents exponential source decay. Using the definitions provided in Table 1, equations (1) and (2) can be expressed in dimensionless form as

display math
display math

where the source to aquitard mass transfer coefficient is math formula [dimensionless] as defined in Table 1. The term math formula represents the relative extent to which mass is transferred from the source to the aquitard, and is the product of the source decay function math formula [T−1] given by Zhu and Sykes [2004], and a new variable defined here as the aquitard diffusion timescale math formula [T] (Table 1). Consequently, math formula couples source dissolution processes in math formula and effective diffusional processes in math formula. A low math formula represents slow source zone dissolution and/or low aquitard retardation where the plume concentration remains high for a longer time, while a high math formula represents rapid source dissolution and/or high aquitard retardation. The effective diffusion coefficient D* in the aquitard is used in math formula and math formula (Table 1), and will be discussed in more detail in section 2.2.

Figure 1.

Conceptual diagram of a dense nonaqueous phase liquid (DNAPL) source zone: the dissolved contaminant plume (gray) with a flux-averaged concentration C(t) crossing the source zone control plane math formula; and a 1-D diffusion domain (dashed box): with an underlying, near source, semi-infinite aquitard with a time variable upper boundary C(t).

Table 1. Definitions of Dimensionless Variables
 Definitions
Source zone mass and concentration math formula; math formula
Aquitard depth, time, and concentration math formula; math formula; math formula
Source to aquitard mass transfer coefficient math formula; math formula; math formula
Postremedial source mass and concentration math formula; math formula
Postremedial source to aquitard mass transfer coefficient math formula; math formula

[12] Substitution of equation (4) into equation (3), along with an initial condition of math formula, allows for solutions of math formula and math formula following methods similar to those presented by either Zhu and Sykes [2004] or Falta et al. [2005a]:

display math
display math

As math formula the limit of equations (5a) and (5b) approach [Abramowitz and Stegun, 1970]

display math

Detailed derivations for equations (5a), (5b), and (5c) can be found in the auxiliary material.

[13] 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

[14] 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 math formula 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,

display math

where math formula [M L−3] is the concentration in the aquitard, math formula [L] is the depth in the aquitard, and math formula [L2 T−1] is the effective diffusion coefficient:

display math

where, math formula is the molecular diffusion coefficient [L2 T−1], math formula is the tortuosity [dimensionless], math formula is the retardation factor [dimensionless], math formula is the media bulk density [M L−3], math formula is the porosity (dimensionless), and math formula is the distribution coefficient [L3 M−1], assuming equilibrium, reversible, and linear partitioning of the contaminant between the aquitard pore water and solid media.

[15] Using the dimensionless framework proposed in Table 1, the dimensionless form of Fick's second law becomes

display math

2.3. One-Dimensional Aquitard Diffusion With a Source Depletion Boundary Condition

[16] Using the convolution theorem [Abramowitz and Stegun, 1970], a general analytical solution to equation (8) with boundary conditions of math formula and math formula, and an initial condition of math formula is obtained:

display math

The upper boundary condition of the aquitard can be represented by a function math formula that imposes a time variable concentration, represented by the SDM in equation (5a) or (5c). The SDM equation is substituted for math formula 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 math formula have been previously published [Crank, 1976; Bear et al., 1994]. These solutions were used for model verification and are available in the auxiliary material.

[17] Equation (9) is fully dimensionless, is applied to a single-layer semi-infinite aquitard, and employs the convolution theorem, as did Booker and Rowe [1987] and Liu and Ball [1998]. 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. [2009] 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.

[18] 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

[19] 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 math formula can be determined for any math formula or math formula through integration of dimensionless flux math formula through dimensionless time,

display math

by integration of dimensionless concentration through dimensionless space,

display math

or integration of dimensionless concentration in dimensionless space in the Laplace domain and application of the convolution method,

display math

The integral in equations (10a) and (10c) were evaluated analytically and equation (10b) numerically. Equation (10a) gives math formula as a result of loading when using the dimensionless time at which back diffusion starts math formula as the upper integration bound and zero as the lower bound. To determine the mass that has left the aquitard for math formula, the lower integration bound is math formula, and the upper is the final math formula of interest. Calculations of math formula were verified with previously published cases for math formula and the relative error was zero for equations (10a) and (10c) and less than 5% for equation (10b).

[20] The flux into or out of the aquitard is given by Fick's first law of diffusion, and was evaluated in dimensionless form using

display math

where equation (5a) or (5c) is substituted into the convolution for math formula. The convolution solution does not allow an evaluation at math formula. Thus, a very small value for math formula (i.e., math formula) was used in equation (11) to approximate the flux at the aquifer/aquitard interface. For example, math formula corresponds to math formula mm for math formula 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.

[21] This depth was assumed to be representative of the interface, and justification for this assumption was that math formula obtained at a depth of 2.5 to 3.5 mm varied less than 5% from math formula obtained at the interface using previously published analytical solutions for math formula (see the auxiliary material). Finite difference methods were employed for verification purposes of math formula determined by equation (11) at the stated depths for other cases math formula. 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, math formula and math formula, 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 math formula 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.

[22] 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

[23] The range of math formula for typical field sites needs to be established to define the modeling timescales. Given the definition of math formula in Table 1, this requires an estimate of the ranges associated with math formula, math formula, and math formula. While math formula 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 math formula is 10 m2 to 1000 m2. Thus, for 0.01 math formulat math formula 10,000 years, math formula will range from 10−9 to 10 in most applications.

[24] There are two key events in time that were evaluated with the model: (1) math formula, and (2) the dimensionless time that the source is exhausted by dissolution math formula. At any time after math formula, 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 math formula and math formula represents the time of maximum mass storage, math formula. For the math formula cases, the back-diffusion flux will peak at math formula, where

display math

For the math formula cases, peak back-diffusion flux will occur at some time after math formula.

2.6. Source Removal

[25] 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 math formula from the remaining source mass by some remedial process at time math formula. The relative mass remaining in the source zone at math formula, prior to the remediation i.e., math formula, is calculated using equation (5b), and in turn is used to determine the postremedial initial relative mass math formula and relative concentration math formula (Table 1). Likewise, these are used to calculate a new postremedial source decay function math formula, and a postremedial source to aquitard mass transfer coefficient math formula (Table 1). Consequently, the postremedial relative concentration and the relative mass remaining in the source zone are

display math
display math

respectively, where math formula is the dimensionless elapsed time after remediation (i.e., math formula). For the math formula case, the remaining mass and postremedial source concentration are given by

display math

Finally, the dimensionless time at which the mass remaining in the source zone after remediation is exhausted by natural dissolution becomes

display math

3. Results and Discussion

3.1. Dimensionless Aquitard Concentration Profiles

[26] There are two main parameters that affect the model results: math formula and math formula. The effects of math formula on the model were investigated first, and then the effects of math formula 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.

Figure 2.

(a) Source depletion model (SDM) as a function of math formula for math formula and math formula. The math formula depth profiles in the aquitard are shown for (b) math formula, (c) math formula, and (d) math formula.

Figure 3.

(a) The math formula depth profiles are shown at the time back diffusion begins, math formula, as a function of math formula (solid lines) with math formula and as a function of math formula (dashed lines) with math formula. (b) The math formula depth profiles are shown at the time the source is exhausted, math formula, as a function of math formula.

Table 2. Dense Nonaqueous Phase Liquid Source Zone Values Used in the Model
 Aquifer Parameters for Source Dissolution
math formula (mg L−1) math formula (kg) math formula (m d−1) math formula (m2) math formula (d−1)
Figures 2, 3, 5a, 6, 7a, 7b, 7e, and 7f15016200.0548301.52 × 10−4
Figures 3, 4, 5b, 6, 7c, and 7d15016200.00782302.19 × 10−5
 15016200.0548301.52 × 10−4
 15016200.172304.78 ×10−4
Table 3. Aquitard Media Values Used in the Model
 Aquitard Parameters for Diffusion
math formula (m2 d−1) math formula math formula (L kg−1) math formula (g mL−1) math formula math formula math formula (m2 d−1) math formula (days)
Figures 2, 3, 4, 5a, 6, and 7a–7d1.46 × 10−51.41.22.60.458.01.30 × 10−62.30 × 107
Figures 3, 4, 5b, 6, 7e, and 7f1.46 × 10−51.41.250.040.351.149.13 × 10−63.28 × 106
 1.46 × 10−51.41.22.60.458.01.30 × 10−62.30 × 107
 1.46 × 10−51.41.112.10.5525.14.15 × 10−77.22 × 107

[27] One method to evaluate the effects of source dissolution on aquitard storage and release is to construct profiles of math formula as a function of depth using equation (9). Figure 2a illustrates the SDM for math formula and Figures 2b, 2c, and 2d illustrate the depth profiles of math formula for math formula, respectively. As math formula 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 math formula was reduced with less penetration as math formula increased. For example, at math formula, math formula for math formula, respectively, and these values of math formula occurred at depths of math formula, respectively.

[28] In Figure 3, the math formula depth profiles are shown at specific events in time (i.e., math formula and math formula) as a function of math formula and math formula. The highest concentration in the aquitard is achieved at math formula. Figure 3a shows that the penetration and magnitude of math formula at math formula decreased as math formula and math formula increased. For example, with math formula and at math formula, math formula for math formula, respectively. Likewise, with math formula and at math formula, math formula for math formula, respectively. In Figure 3b, the concentration profiles at math formula for cases with math formula are shown. As math formula increased, concentrations in the aquitard were reduced because of three factors: (1) a more rapid decline in math formula, (2) math formula decreased as math formula increased, resulting in more time for back diffusion to remove mass, and (3) math formula increased as math formula increased, resulting in more time for downward diffusion of mass. For math formula, math formula was maintained until math formula, which occurred at math formula (and hence math formula at math formula). For the other cases, math formula.

[29] Shown in Figure 4a is the SDM for math formula and math formula; and in Figures 4b through 4d the resultant math formula depth profiles. Similar to the influence of math formula, math formula decreased as β increased as a result of the more rapid reduction in the concentration gradient. For example, at math formula, math formula for math formula, respectively. This resulted in a lower math formula for a given time and depth. At math formula and at a depth of math formula, math formula for math formula, respectively.

Figure 4.

(a) SDM as a function of math formula for math formula and math formula. The math formula depth profiles are shown for (b) math formula, (c) math formula, and (d) math formula.

[30] 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 math formula years math formula, math formula mg kg−1 which occurred at depths of math formula m for math formula, respectively. Evaluating the influence of math formula on concentration in dimensioned time and space, however, is not as straight forward. At math formula m math formula and math formula years math formula, the total soil concentration (aqueous and solid phase) math formula mg kg−1 for math formula, respectively, where values of math formula were obtained considering math formula m d−1, respectively, math formula, and all other parameters defined as shown in Tables 2 and 3. However, the same dimensionless profiles for math formula occur when math formula results from math formula, respectively, math formula m d−1, and all other parameters equal to the same previous values (Tables 2 and 3). These values of math formula may occur, for example, considering typical properties associated with sandy silt, silt, and silty clay aquitards. Because math formula is a function of math formula, the results at math formula in dimensioned time occur at math formula years for math formula, respectively. Consequently, at the same depth of math formula m math formula, math formula mg kg−1 for math formula, respectively, demonstrating an increase in total mass as math formula increased because of sorption. Thus, increasing math formula decreased the aquitard contaminant concentrations and the depth of contaminant penetration. Likewise, increasing math formula by increasing math formula, decreased aquitard contaminant concentrations and the depth of contaminant penetration. However, when math formula increased because of an increase in math formula (via increased math formula), solid phase concentrations increased (increasing total concentrations), but the contaminant penetration depth decreased.

3.2. Aquitard Mass Storage

[31] 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 math formula increased, the amount of mass stored in the aquitard decreased. Specifically, using equation (10b) to integrate the math formula depth profiles in Figure 2 for math formula, math formula for math formula, respectively. As math formula 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 math formula, math formula for math formula, respectively. Higher math formula in the latter two cases results from higher concentrations at the boundary for longer durations (e.g., Figure 2a for math formula), reducing the back-diffusion gradient, which in turn yields slightly more dimensionless mass at this point in time.

[32] To evaluate the maximum mass storage math formula, the math formula depth profiles at math formula (Figure 3a) were integrated using equation (10b), and the results are shown in Figure 5a as a function of math formula and in Figure 5b as a function of math formula. As math formula increased from 0 to 10, math formula decreased from 0.019 to 0.0039. Likewise, as math formula increased from 250 to 11,000, math formula decreased from 0.039 to 0.006. Figure 5a illustrates the error that might occur in estimates of math formula when assuming math formula, as is often done, for those sites with characteristics better represented by math formula. As further illustration of this point, Table 4 lists the relative reduction in math formula, defined as math formula, as a function of math formula. The relative reduction in maximum stored mass ranged from 36 to 60% for math formula. Thus, while the math formula 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.

Figure 5.

(a) Maximum mass stored in the aquitard ( math formula) and math formula as a function of math formula, math formula, and (b) math formula and math formula as a function of math formula, math formula.

Table 4. Relative Reduction in Maximum Mass Stored in Aquitard, math formula
Γ math formula
0.00.00
0.250.25
0.50.36
0.750.43
1.00.48
1.250.52
2.00.60
4.00.71
10.00.81

3.3. Longevity and Hysteresis

[33] Contaminated site diffusion processes are hysteretic because they are gradient driven, and loading occurs much more rapidly than release. To investigate this behavior, math formula is plotted as a function of math formula and math formula in Figure 5. Starting at math formula, math formula increased to its peak at math formula, then decreased as math formula increased (Figure 5a). In general, as math formula 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 math formula the source, in theory, is never completely exhausted by dissolution; consequently, back diffusion lasts for an infinitely long time. Similar to math formula, as math formula increases, back diffusion begins earlier (Figure 5b). When math formula was increased by more rapid source dissolution, back-diffusion flux initiated earlier and less mass was loaded into the aquitard. However, if math formula was increased because of increased aquitard sorption only, math formula decreased, but that value corresponded to the same dimensioned time since math formula is also a function of math formula (Table 1). Thus, aquitard sorption has no effect on the time when mass loading into the aquitard ends.

[34] 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 math formula for math formula and math formula, and as a function of math formula for math formula and math formula. Relative mass in Figure 6 was scaled to math formula for each specific case. As math formula 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, math formula had a minimal effect on the timescale of the loading and release of mass in the aquitard, while math formula had a stronger influence in the model results. As math formula decreased, both the loading and release times increased.

Figure 6.

Relative mass in the aquitard as a function of math formula in log scale for math formula and math formula (solid lines) and as a function of math formula in log scale for math formula and math formula (dashed lines). The inset is plotted in a linear time scale.

3.4. Aquitard Source Functions

[35] 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 math formula and math formula to dimensioned values:

display math
display math

where math formula is aquitard mass per unit area [M L−2] and math formula is back-diffusion flux [M L−2 T−1]. Aquitard source functions are shown in Figure 7 as a function of math formula and math formula, along with the corresponding dimensioned times series for math formula (dashed lines) and math formula (solid lines). The ratio math formula was used to normalize math formula, while math formula was used to normalize math formula. Temporal progression can also be followed in the aquitard source functions themselves, which begin at math formula on the right, and increases in time from there to the left.

Figure 7.

(a) Aquitard source functions for math formula and math formula. (b) Dimensioned mass storage and mass flux time series for Figure 7a. (c) Aquitard source functions for math formula and math formula, where math formula is varied by math formula (m d−1), shown as math formula (d−1). (d) Dimensioned mass storage and mass flux time series for Figure 7c. (e) Aquitard source functions for math formula and math formula, where math formula is varied by math formula, shown as math formula (d). (f) Dimensioned mass storage and mass flux time series for Figure 7e. Dashed lines are mass ( math formula, right y axis), and solid lines are flux ( math formula, left y axis).

[36] Figure 7a presents aquitard source functions for math formula and math formula, and illustrates three distinct shapes. For math formula, the boundary concentration jumped from math formula to math formula instantaneously, resulting in an infinite flux at math formula. In contrast, the back-diffusion flux peaked after math formula for cases with math formula because the concentration in the aquitard and the SDM boundary were equal at math formula, resulting in no flux. For cases with math formula (as illustrated by math formula), a sharp but finite peak occurred when the source was exhausted at math formula. For math formula (as illustrated by math formula and math formula), 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 math formula increased, both the relative mass and flux decreased, indicating reduced risk due to back diffusion. Specifically, as math formula 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 math formula case, was three or more orders of magnitude less than the initial source zone flux.

[37] 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 math formula increased the peak mass and mass flux decreased. The rates of decrease, however, varied as a function of math formula. For example, within the first 10 years after math formula, the order of math formula as a function of math formula was: math formula > math formula > math formula > math formula. By 250 years after math formula (off scale in Figure 7b), the order was reversed, and math formula g m−2 yr−1 for all cases of math formula. Additionally, comparing the math formula result to that of math formula, 10 years after math formula the back-diffusion flux was 68% greater for math formula, than for a site with math formula. However, 100 years after math formula, math formula resulted in 30% less flux than math formula. This is due to the more rapid removal of mass by the increased flux in the math formula case at earlier times.

[38] Since math formula, the effect of math formula on aquitard source functions was investigated by separately exploring the impacts of math formula and math formula. In Figure 7c, aquitard source functions are shown for math formula and math formula. These values of math formula correspond to math formula d−1, and were generated using math formula m d−1, and the other parameters as shown in Table 2. As math formula increased from math formula ( math formula d−1) to math formula ( math formula d−1), the maximum relative mass decreased from math formula to math formula, and the maximum relative flux decreased from math formula to math formula. 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 math formula and math formula that lead to an increase in math formula and hence math formula. Variations in math formula affect math formula, math formula, and math formula; and the resulting impacts on aquitard mass and flux are more complex and not shown here.

[39] Figure 7d shows the dimensioned time series for math formula and math formula corresponding to the dimensionless data in Figure 7c. Comparison of Figures 7c and 7d for math formula ( math formula d−1) indicated that while this case has the highest relative flux, it had the lowest-dimensioned flux. This was because math formula is normalized to a low initial source flux resulting from the low groundwater velocity in this case. Comparing the case of math formula ( math formula 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 math formula ( math formula d−1) case had the greatest math formula and the lowest math formula for math formula. 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, math formula ( math formula d−1) represents a site with a high math formula, 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 [2005] for an industrial site in Connecticut.

[40] Changes in retardation factors associated with changes in aquitard media are evaluated in the aquitard source functions shown in Figure 7e for math formula and math formula. These values of math formula correspond to math formula days, and represent the dissolution of a PCE source over silty sand math formula, silt math formula, and silty clay math formula aquitards, with the modeling parameters shown in Table 3. The results showed that as math formula increased (and hence math formula), 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 math formula and math formula are shown for the model results in Figure 7e. Increasing math formula (and hence math formula) due to increased retardation increased both math formula and math formula, even though the penetration depth was reduced (see Figure 4).

3.5. Source Removal

[41] 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 math formula profiles were generated using equations (5c) and (13c), and represent a site where math formula and math formula under three cases: (1) no remediation (Figure 8b), (2) 70% DNAPL mass reduction (i.e., X = 0.7) at math formula (12.6 years) (Figure 8c), and (3) 70% DNAPL mass reduction at math formula (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.

Figure 8.

(a) Remedial SDM. Resultant math formula depth profiles are shown for (b) no remediation, (c) remediation at math formula (12.6 years), and (d) remediation at math formula (28.5 years); math formula, math formula, and math formula for all cases.

Figure 9.

(a) Aquitard source functions for remediation at math formula (28.5 years) and math formula (12.6 years); math formula, math formula, and math formula for all cases. (b) Dimensioned mass storage and mass flux time series for Figure 9a.

Table 5. Values of Variables Used in the Remedial Model
 Parameters Used in Partial Source Remediation Modeling
math formula math formula (days) math formula (d−1) math formula math formula math formula (years)
  • a

    NR = no remediation.

Figures 8 and 91.02.30 × 1071.52 × 10−435000NRa
1.02.30 × 1071.52 × 10−435000.712.6
 1.02.30 × 1071.52 × 10−435000.728.5

[42] 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.

4. Conclusions

[43] 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 math formula, which reflects the source zone architecture, and 2) the source to aquitard mass transfer coefficient math formula, which reflects the influence of both the source characteristics ( math formula) and the aquitard properties ( math formula). The amount of contaminant mass in the aquitard, the back-diffusion flux magnitude, and their longevity were used as measures of back-diffusion risk.

[44] The greatest potential for back diffusion occurs when the source strength is constant until source mass is exhausted (i.e., math formula), as is often assumed in back-diffusion assessments. For sites where mass discharge decreases in time as source mass is depleted (i.e., math formula), less risk is expected. Specifically, the aquitard mass per unit area math formula, the dimensionless start time for back diffusion math formula, the depth of penetration, and the magnitude of back-diffusion flux math formula all decreased as math formula increased, indicating reduced risk due to back diffusion.

[45] Additional source zone characteristics that were investigated include the initial source mass math formula, the initial source concentration math formula, the groundwater flux math formula, and the source zone control plane area math formula. These terms were combined to represent the ratio of initial source zone mass discharge to initial DNAPL mass (i.e., math formula, the source decay function). In general, site conditions that lead to an increased math formula, and hence math formula (i.e., large initial mass discharge and small math formula), 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 math formula (i.e., small initial mass discharge and large math formula) 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.

[46] Increasing aquitard sorption, represented by a larger math formula, and hence math formula, had no effect on the loading concentration gradient, but increased both the mass storage math formula and the back-diffusion flux math formula. 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.

[47] 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.

[48] 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.

Acknowledgments

[49] 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.