Explicit and implicit coupling during solute transport through clay membrane barriers

Transcription

1 Journal of Contaminant Hydrology 72 (2004) Explicit and implicit coupling during solute transport through clay membrane barriers Michael A. Malusis a,1, Charles D. Shackelford b, * a Sentinel Consulting Services, LLC, 14 Inverness Drive East, Suite G228, Englewood, CO 80112, USA b Department of Civil Engineering, Colorado State University, Fort Collins, CO 80523, USA Received 4 June 2003; received in revised form 17 November 2003; accepted 5 December 2003 Abstract Simulations of salt (KCl) flux through a 1-m-thick clay membrane barrier (CMB) based on coupled solute transport theory are compared to simulated fluxes based on traditional advective dispersive transport theory. The simulations are based on measured values for the effective saltdiffusion coefficient (D s *) and chemico-osmotic efficiency coefficient (x) for a bentonite-based barrier material subjected to KCl solutions. The results indicate that the exit salt flux is reduced due to both explicit coupling (hyperfiltration and chemico-osmotic counter-advection) and an implicit coupling effect resulting from the decrease in D s * due to a decrease in the apparent tortuosity factor, s a, with an increase in x. Implicit coupling is shown to be more significant than explicit coupling for reducing and retarding salt flux through a CMB under diffusion-dominated conditions. Failure to account for the implicit coupling effect may result in unrealistic results, such as the existence of salt flux through a perfect (ideal) clay membrane (i.e., x = 1). D 2003 Elsevier B.V. All rights reserved. Keywords: Advective dispersive transport; Chemico-osmosis; Clay membranes; Containment; Coupled solute transport; Diffusion; Hyperfiltration 1. Introduction Solute transport analyses for low-permeability earthen containment barriers, such as compacted clay liners, geosynthetic clay liners, and soil-bentonite vertical cutoff walls, * Corresponding author. Tel.: ; fax: addresses: mmalusis@sentinelteam.com (M.A. Malusis), shackel@engr.colostate.edu (C.D. Shackelford). 1 Tel.: ; fax: /$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi: /j.jconhyd

2 260 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) commonly are performed to demonstrate performance of the barrier. Such analyses typically are performed using solutions to the traditional advective dispersive transport equation (e.g., Shackelford, 1988, 1990; Rabideau and Khandelwal, 1998). However, in the case where the containment barrier acts as a semi-permeable membrane, referred to herein as a clay membrane barrier (CMB), the advective dispersive transport theory may not be appropriate due to the solute restriction and chemico-osmotic flow aspects associated with membrane behavior (e.g., Yeung, 1990; Shackelford, 1997). For example, coupled solute transport equations based on irreversible thermodynamics indicate that the solute flux through a CMB is explicitly reduced by hyperfiltration, whereby solutes are filtered out of solution as the solution passes through a CMB under an applied hydraulic gradient, as well as by counter-advection of solutes due to chemicoosmosis that occurs opposite to the direction of solute diffusion (Malusis and Shackelford, 2002a). These explicit coupling effects are a function of the chemico-osmotic efficiency coefficient, x, for the soil that theoretically ranges from zero for non-membrane soils to unity for ideal CMBs that completely prohibit solute migration (i.e., 0 V x V 1). In addition to the explicit coupling effects of hyperfiltration and chemico-osmotic counteradvection, an additional coupling effect characterized by a decrease in the effective saltdiffusion coefficient, D s *, with an increase in x also exists, since by definition D s * must approach zero in the limit as x approaches unity (i.e., D s *! 0asx! 1). This additional coupling effect is referred to as an implicit coupling effect because the correlation is not explicitly accounted for in the coupled solute transport equation (Malusis and Shackelford, 2002a,b). Both explicit and implicit coupling effects act to reduce solute flux through a CMB relative to an earthen containment barrier that does not behave as a semi-permeable membrane. Although the traditional approach for solute transport analyses associated with earthen containment barriers for geoenvironmental applications is based on advective dispersive transport theory, the potential limitations of advective dispersive transport theory for describing contaminant migration through CMBs have not been evaluated. Since advective dispersive theory represents a limiting case in which the potential influence of membrane coupling is ignored (i.e., x = 0), a direct comparison of solute transport based on advective dispersive transport theory versus coupled solute transport theory that accounts for both explicit and implicit coupling effects is needed. As a result of the aforementioned considerations, model simulations are performed in this study to evaluate the significance of explicit and implicit coupling effects on solute transport through CMBs. Simulations are performed using a coupled solute transport model that explicitly accounts for hyperfiltration and chemico-osmotic counter-advection in the absence of electrical current. The simulations also account for the implicit relationship between x and D s * based on previously reported results of laboratory chemico-osmotic/diffusion tests on a bentonite-based barrier material. Results of these simulations illustrate the factors affecting coupled solute transport through a CMB and, when compared to the results of simulations based on advective dispersive theory, also illustrate the differences between the coupled transport theory and the advective dispersive transport theory. The validity of using advective dispersive theory in lieu of the coupled transport theory to model solute transport in CMBs is also discussed.

3 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Theoretical background The governing equations for coupled solute transport through a soil are based on principles of irreversible thermodynamics applied to non-equilibrium systems (e.g., Katchalsky and Curran, 1965; Olsen et al., 1990; Yeung, 1990; Yeung and Mitchell, 1993). The specific case involving the influence of soil membrane behavior on solute transport through a CMB under isothermal conditions is illustrated in this study by considering the condition in which no electrical current exists across the CMB. This condition is relevant for engineered containment applications that do not involve application of an electrical current across the clay barrier. Under this condition, the general expression for total flux of a solute j, J j, in a system containing N different solute species for the case of low-permeability soils and/or short transport distances in which mechanical dispersion is negligible can be written as follows (Shackelford et al., 2001; Malusis and Shackelford, 2002a): J j ¼ð1 xþq h C j þ q p C j þ nd sj * BC j fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} ffl{zffl} Bx J ha J p fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} J d where q h [=k h i h, where k h is hydraulic conductivity and i h is the hydraulic gradient] is the hydraulic flux of solution, q p is the chemico-osmotic flux of solution, C j is the molar concentration of solute j, n is the total porosity of the soil, D* sj is the effective saltdiffusion coefficient of solute j, J ha is the hyperfiltrated advective solute flux, J p is the counter-advective solute flux, and J d is the diffusive solute flux. The chemico-osmotic solution flux, q p, can be written as follows (Shackelford et al., 2001; Malusis and Shackelford, 2002a): ð1þ q p ¼ x k h Bp c w Bx ð2þ where c w is the unit weight of water, and p is chemico-osmotic pressure that is related to the concentrations of N solute species by the van t Hoff expression, or p ¼ RT XN C i i¼1 ð3þ where R is the universal gas constant [8.314 J/mol-K] and T is absolute temperature [K]. The hyperfiltrated advective flux, J ha, in Eq. (1) represents the advective solute flux term that is reduced by a factor of (1 x) due to the membrane behavior of the soil. In physical terms, the factor (1 x) is considered to represent the process of hyperfiltration whereby solutes are filtered out of solution as the solution passes through the membrane under a hydraulic gradient. The counter-advective solute flux, J p, in Eq. (1) is due to the chemico-osmotic flow, q p, which occurs opposite to the direction of solute diffusion. The diffusive solute flux, J d, in Eq. (1) is expressed in the form of Fick s first law for diffusion in soil as defined by Shackelford and Daniel (1991).

4 262 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) The governing partial differential equation for one-dimensional, coupled transport of solute species, j, is obtained by substituting Eq. (1) into a mass balance expression to yield (Malusis and Shackelford, 2002a) R dj BC j Bt ¼ D sj * B2 C j Bx 2 þ BDsj * BC j Bx Bx ð1 xþv BC j h Bx v BC j p Bx C Bv p j Bx ð4þ where v h is the hydraulic seepage velocity ( = q h /n), v p is the chemico-osmotic seepage velocity ( = q p /n), and R dj represents the retardation factor of species j (Freeze and Cherry, 1979), or R dj ¼ 1 þ q dk dj n ð5þ where q d is the dry mass density of the soil, K d is the distribution coefficient for linear, reversible, and instantaneous adsorption, and n is the porosity of the soil. For the case where the soil does not exhibit membrane behavior (i.e., x = 0), q p = 0 and the explicit coupling terms disappear from Eq. (1). Thus, for the limiting case in which x = 0, Eq. (1) reduces to the expression for the traditional advective diffusive solute flux, or J j A x¼0 ¼ q h C j nd sj * BC j Bx Similarly, Eq. (4) reduces to the following expression: R dj BC j Bt x¼0 ¼ D sj * B2 C j Bx 2 þ BD sj* BC j Bx Bx v BC j h Bx ð6þ ð7þ Note that D sj * is not considered to be a constant in either Eq. (4) or Eq. (7), since D sj * may vary with both time and space within a porous medium during transient transport (Malusis and Shackelford, 2002a). For example, D sj * for a particular solute in a multispecies solution is affected by interactions among solute species with different mobilities during salt diffusion that cause slower-moving solutes to diffuse more rapidly and fastermoving solutes to move more slowly (Robinson and Stokes, 1959; Shackelford and Daniel, 1991). In such an environment, D sj * for a solute j can be expressed in general terms as follows (Malusis and Shackelford, 2002a): D sj * ¼ D*F D P j*c j Az j A D *Az ABC P D þ *Az þ ABC þ j P BC j D *ÞAz A 2 C þ P D þ *Az þ A 2 C þ where D* values are effective self-diffusion coefficients related to the individual mobility of the solute species, z is ionic charge, and subscripts + and represent cations and anions, respectively. Eq. (8) is valid, for example, during transient transport of reactive (i.e., ion exchanging) solutes, since ion exchange results in desorption of exchangeable cations into the pore fluid. ð8þ

5 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Since the purpose of this study is to evaluate the influence of membrane coupling on solute transport, without consideration of additional effects that may result from the presence of multiple solute species (e.g., ion exchange, sorption, etc.), the simulations presented herein are performed based on the migration of the single cation and single anion of a strong (i.e., fully dissociating), binary salt (e.g., KCl). Also, the salt cation (e.g., K + ) and the salt anion (e.g., Cl ) are considered to be non-reactive solutes such that R d =1 for both species. Although, in reality, K + likely would be subjected to ion exchange with the negatively charged surfaces of clay particles, resulting in retardation of K + (i.e., R d >1) the assumption that R d = 1 for K + precludes the need to consider such effects thereby accentuating the influence of membrane coupling effects on the solute migration. Under these conditions, migration of both species must occur at the same rate in order to maintain electroneutrality in solution. Thus, the individual flux of either species is representative of the flux of the salt as a whole (e.g., KCl flux), and each solute migrates at the same rate in accordance with the effective salt-diffusion coefficient for the salt, D s *. In this case, the coupled solute flux equation (Eq. (1)) can be written as J s ¼ð1 xþq h C s q p C s nd s * BC s Bx ð9þ and the resulting solute transport equation becomes BC s Bt BC s Bx v BC s p Bx C Bv p s Bx ¼ D s * B2 C s Bx 2 ð1 xþv h ð10þ where the subscript s refers to the salt. In addition, the chemico-osmotic seepage velocity, v p, can be written as v p ¼ q p n ¼ xk h mrt nc w BC s Bx where v represents the number of ions per molecule of salt (e.g., v = 2 for KCl). Substitution of Eq. (11) into Eqs. (9) and (10) and rearrangement yields the following coupled salt flux and transport equations, respectively: J s ¼ð1 xþq h C s xk h x w mrt BC s Bx C s nd s * BC s Bx ð11þ ð12þ and BC s Bt ¼ xk h B 2 C s D s * C s mrt nc w Bx 2 ð1 xþv h xk h mrt nc w BC s Bx BCs Bx ð13þ For the limiting case in which x = 0 (i.e., for non-membrane soils), Eqs. (12) and (13) reduce to traditional advective dispersive flux and advective dispersive transport equations (i.e., in the absence of mechanical dispersion) as follows: J s A x¼0 ¼ q h C s nd s * BC s Bx ð14þ

6 264 and BC s Bt M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) ¼ D s * B2 C s Bx 2 v h x¼0 BC s Bx ð15þ 3. Implicit coupling Hyperfiltration and counter-advection are considered explicit coupling effects, since these effects are represented explicitly in Eq. (5). However, the coupled solute transport theory does not account for the inherent correlation between x and the effective saltdiffusion coefficient, D s *, as described by Malusis and Shackelford (2002b). This correlation is based on the fact that no solutes can enter the pores of an ideal semipermeable membrane (i.e., x = 1) and, therefore, D s * must approach zero as x approaches unity. This relationship between D s * and x represents an implicit coupling effect that must be considered when modeling solute transport through a clay membrane barrier (CMB). Otherwise, a concentration profile can exist in an ideal semi-permeable membrane (i.e., x = 1) due to diffusion, a condition that is physically impossible. The relationship between D s * and x for a CMB based on the conceptual considerations above is supported by the results of chemico-osmotic/diffusion tests reported by Malusis and Shackelford (2002b) on specimens of a bentonite-based containment barrier subjected to KCl solutions. Measured values of x and D s * at steady state for 10-mm-thick specimens of the barrier material are provided in Table 1 and are illustrated versus source KCl concentration, C o,infig. 1a and b, respectively. The results indicate that x decreases and D s * increases with increasing C o. Both of these trends are attributed to collapse of the diffuse double layers surrounding the clay particles. The decrease in x with increasing salt concentration in Fig. 1a is consistent with results of previous studies on clay soils (Kemper and Rollins, 1966; Olsen, 1969; Kemper and Quirk, 1972). When extrapolated to concentrations beyond the highest source concentration (i.e., C o >47 mm), the trend in Fig. 1a indicates virtually no chemico-osmotic membrane efficiency at concentrations beyond approximately 100 mm (i.e., x! 0asC o! 100 mm KCl). The relationship between D s * and x represents the implicit coupling effect described previously and is illustrated in Fig. 1c. The results show that implicit coupling apparently Table 1 Results of steady-state chemico-osmotic/diffusion tests for a clay membrane barrier (from Malusis and Shackelford, 2002b) Test no. Porosity, n Source KCl concentration, C o (mm) Hydraulic conductivity, k h ( m/s) a Chemico-osmotic efficiency coefficient, x Effective salt-diffusion coefficient, D s * ( m 2 /s) Apparent tortuosity factor, s a b a Values of k based on permeation with the source KCl solution. b s a = D*/D s so where D so = m 2 /s for KCl at 25 jc (Shackelford and Daniel, 1991).

7 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Fig. 1. Implicit coupling relationships among chemico-osmotic efficiency coefficient (x), source concentration (C o ), effective salt-diffusion coefficient (D s *), and apparent tortuosity factor (s a ) for a clay membrane barrier subjected to KCl solutions (data from Malusis and Shackelford, 2002b). diminishes as x decreases (i.e., as C o increases) such that D* s and the apparent tortuosity factor, s a [=D*/D s so, where D so is the free-solution salt-diffusion coefficient] approach a maximum value as x approaches zero (i.e., D*! s D s,max * and s a! s a,max as x! 0). Since membrane behavior does not exist when x = 0, the maximum apparent tortuosity factor, s a,max, represents the matrix tortuosity that accounts for the tortuous nature of the

8 266 Table 2 Input parameters for simulation scenarios Coupling scenario Case no. Source KCl concentration, C o (mm) Chemico-osmotic efficiency coefficient, x Apparent tortuosity factor, s a Hydraulic conductivity, k h ( m/s) Hydraulic gradient, i h Hydraulic seepage velocity, v h ( m/s) Explicit and implicit (EI) Explicit Only (E) Peclet no., P L M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004)

9 Implicit only (I) No coupling or non-membrane (NM) M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004)

10 268 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) solute migration pathway through a porous medium due solely to the geometry of the interconnected pores (Malusis and Shackelford, 2002b). As indicated in Fig. 1c, s a,max = 0.12 for the bentonite barrier material tested. Conversely, s a approaches zero as x approaches unity (i.e., s a! 0asx! 1) since, by definition, an ideal completely restricts the passage of solutes. 4. Model simulation scenarios and cases Simulations based on a numerical solution of Eq. (13) are performed for a 1-m-thick clay barrier (i.e., L = 1 m). The input parameter values for the different simulation scenarios and cases considered are given in Table 2. The input parameter values for the simulation scenarios and cases have been established based on the measured values given in Table 1 to provide self-consistency among the various parameter values. For example, a simulation using a hydraulic conductivity, k h, of 10 5 m/s could be performed for a range of membrane efficiencies (i.e., 0 < x < 1), but the results would not be representative since a k h of 10 5 m/s, which is representative of a sand, is not consistent with the existence of membrane behavior (i.e., x>0). Thus, the use of the measured parameter values shown in Table 1 ensures that consistent parameter values are used for the simulations performed in this study. The assumption inherent in this approach is that the parameter values in Table 1 for a 10-mm-thick bentonite barrier are equally applicable to a 1-m-thick barrier consisting of the same material. This assumption is not likely to be practical, since the 1-m-thick barrier would have to be constructed entirely with sodium bentonite, resulting in a barrier that would be cost prohibitive. Nonetheless, a 1-m-thick barrier is more common, and the trends in the results and the corresponding conclusions reported in this study should be the same as those for a 10-mm-thick barrier. As shown in Table 2, four different simulation cases are considered. Scenarios with prefix EI represent simulations in which both explicit and implicit coupling effects are included. The explicit coupling effects are included by using the appropriate value of x (>0) corresponding to the value of C o such that hyperfiltration and counter-advection both influence the salt flux. The implicit coupling effect is included by using the appropriate value of s a corresponding to the value of C o (Fig. 1b). For example, in EI case 3 (C o = 8.7 mm KCl), the appropriate values of x and s a are 0.49 and 0.063, respectively, based on the results in Table 1. Simulation cases with only the prefix E are cases in which only explicit coupling is included (i.e., x>0) by assuming s a = s a,max = For example, x = 0.14, but s a = 0.12 for E case 3. Scenarios with prefix I and NM represent simulations where the chemicoosmotic efficiency of the clay is ignored (i.e., x = 0) and, thus, explicit coupling effects (i.e., hyperfiltration and chemico-osmotic counter-advection) are neglected. For these cases, Eq. (13) reduces to the advective dispersive transport equation (Eq. (15)). The I scenario represents advective dispersive transport that includes implicit coupling (s a < s a,max ), whereas the NM scenario represents non-membrane simulations in which x = 0 and s a = s a,max. For example, x = 0 and s a = in I case 3, whereas x = 0 and s a = s a,max = 0.12 in NM case 3.

11 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Time-dependent concentration profiles for the salt are obtained in this study by solving Eq. (13) iteratively using an implicit-in-time, centered-in-space finite difference algorithm. The soil of length L is divided into cells of thickness Dx, and the transit time is divided into discrete intervals of length Dt. Values of Dx and Dt were chosen in order to maintain compliance with stability requirements for the implicit-in-time, centered-in-space approach, and the algorithm was adjusted properly to account for numerical dispersion (Smith, 1985). Further details regarding the finite difference algorithm are described by Malusis (2001). The effect of hydraulic gradient, i h, is examined by considering three different values in the simulations (i h = 0, 10, 100). Values of the Peclet number, P L, for each case are related to i h and s a as follows: P L ¼ v hl D s * ¼ k hi h L ns a D so ð16þ These values of P L represent the relative significance of advection versus diffusion. For example, values of P L < 20 typically are representative of a diffusion-dominated system, whereas advection tends to be dominant for P L >50 (Shackelford, 1995). Since implicit coupling is characterized by a reduction in s a, values of P L for the cases in which implicit coupling is included in the analysis (i.e., the I and EI cases) are higher than P L values for cases with the same hydraulic gradient in which implicit coupling is ignored (i.e., the E and NM cases). However, explicit coupling does not affect P L, since Eq. (16) is independent of x. The boundary conditions considered in this study are a constant source entry boundary [C(x =0,t)=C o ] and a perfectly flushing exit boundary [C(x = L,t) = 0]. In addition, the porous medium is assumed to be initially free of the salt [C(x,t = 0) = 0]. Similar initial and boundary conditions were employed by Greenberg et al. (1973) to describe the coupled transport of NaCl through a clay aquitard below a salt-water aquifer. The problem geometry for this situation is analogous to the problem geometry associated with containment of miscible pollutants with engineered clay barrier materials (Shackelford, 1997). The perfectly flushing exit boundary condition also is recommended as a conservative approach for design of vertical barriers, such as cutoff walls (Rabideau and Khandelwal, 1998). Moreover, approximately the same conditions were utilized in the chemico-osmotic/diffusion tests from which the results shown in Table 1 and Fig. 1 were derived (Malusis and Shackelford, 2002b). 5. Results 5.1. Exit salt flux The influence of membrane coupling effects are examined in this study based on the exit mass flux of salt from the soil, J s (x = L). Simulated values for the exit salt (KCl) flux, J s (x = L), for C o = 47, 20, 8.7, and 3.9 mm KCl based on the cases given in Table 2 are illustrated versus time in Figs. 2 5, respectively. The theoretical condition corresponding

12 270 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Fig. 2. Simulated salt exit flux versus time for a 1-m-thick clay membrane barrier and a source KCl concentration, C o, of 47 mm as a function of hydraulic gradient [NM = no membrane behavior; E = membrane behavior with explicit coupling only; I = membrane behavior with implicit coupling only; EI = membrane behavior with both explicit and implicit coupling].

13 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) to J s (x = L) = 0 for an ideal membrane that completely restricts the passage of solutes also is shown for comparison. The results indicate that the highest values of exit flux for a given C o and hydraulic gradient, i h, typically occur for the non-membrane ( NM ) cases (i.e., x =0, s a = 0.12), whereas the lowest values of exit flux occur for EI cases in which both explicit and implicit membrane coupling are considered in the analysis. In addition, the reduction in exit flux due to combined explicit and implicit coupling relative to the non-membrane case is greater as C o decreases, since x increases and s a decreases with decreasing C o. Thus, as C o decreases, the values of exit flux corresponding to the cases that include both explicit and implicit coupling effects approach more closely the ideal membrane condition in which J s (x = L)=0. For the cases in which C o = 47 mm KCl (Fig. 2) and C o = 20 mm KCl (Fig. 3), the reduction in exit flux is primarily due to explicit coupling associated with x = 0.14 (for C o = 47 mm) and x = 0.32 (for C o = 20 mm). Implicit coupling has little or no effect on exit flux in Figs. 2 and 3, since the s a values corresponding to these source concentrations (0.119 and 0.112, respectively) are only slightly lower than the maximum tortuosity factor of For the lower source concentrations (i.e., C o = 8.7 mm KCl in Fig. 4 and C o = 3.9 mm KCl in Fig. 5), explicit and implicit coupling effects both contribute to the reduction in exit flux. The values of x corresponding to C o = 8.7 mm and C o = 3.9 mm are 0.49 and 0.63, respectively, while the s a values are and 0.039, respectively. However, the relative contribution of explicit and implicit coupling in these cases appears to be dependent upon the hydraulic gradient, i h, and the Peclet number, P L. For example, a majority of the reduction in exit flux in Figs. 4 and 5 appears to be due to implicit coupling when i h =0 (and thus, P L = 0), whereas the significance of explicit coupling relative to implicit coupling increases as i h (and P L ) increases. This trend is consistent with the increase in advection relative to diffusion with increasing P L. For i h = 100, the exit salt fluxes in Figs. 4 and 5 for the I cases that include only implicit coupling (i.e., s a < s a,max and x = 0) are as high or higher than the exit flux values for a non-membrane (i.e., s a = s a,max and x = 0). If we consider that C s approaches zero as x approaches L (i.e., C s! 0asx! L) due to the perfectly flushing condition established at the exit boundary of the clay barrier, the flux equation (Eq. (12)) at x = L can be approximated as follows: J s ðx ¼ LÞc nd s * BC s BC s ¼ ns Bx a D so ð17þ Bx x¼l Eq. (17) illustrates that the exit salt flux for a soil with a given porosity, n, and for a solute with a given free-solution salt-diffusion coefficient, D so, is governed by the apparent tortuosity factor, s a, and gradient in salt concentration, BC s /Bx, at the exit boundary. Therefore, the aforementioned results for i h = 100 in Figs. 4 and 5 may seem counterintuitive, since implicit coupling (i.e., a decrease in s a ) is expected to result in a lower exit flux based in Eq. (17). However, the lower s a values associated with implicit coupling also results in an increase in the Peclet number, P L, as shown in Fig. 6. The results in Fig. 6 also show that P L increases with increasing i h as indicated by Eq. (16). Higher P L values x¼l

14 272 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Fig. 3. Simulated salt exit flux versus time for a 1-m-thick clay membrane barrier and a source KCl concentration, C o, of 20 mm as a function of hydraulic gradient [NM = no membrane behavior; E = membrane behavior with explicit coupling only; I = membrane behavior with implicit coupling only; EI = membrane behavior with both explicit and implicit coupling].

15 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Fig. 4. Simulated salt exit flux versus time for a 1-m-thick clay membrane barrier and a source KCl concentration, C o, of 8.7 mm as a function of hydraulic gradient [NM = no membrane behavior; E = membrane behavior with explicit coupling only; I = membrane behavior with implicit coupling only; EI = membrane behavior with both explicit and implicit coupling].

16 274 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Fig. 5. Simulated salt exit flux versus time for a 1-m-thick clay membrane barrier and a source KCl concentration, C o, of 3.9 mm as a function of hydraulic gradient [NM = no membrane behavior; E = membrane behavior with explicit coupling only; I = membrane behavior with implicit coupling only; EI = membrane behavior with both explicit and implicit coupling].

17 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Fig. 6. Peclet numbers as a function of hydraulic gradient based on non-membrane (NM) behavior and on membrane behavior with implicit coupling only (I). are representative of more advective-dominated solute transport, which results in a higher gradient in salt concentration at the exit boundary. Thus, the implicit effect is only significant for diffusion-dominated systems. For example, salt concentration profiles inside the clay barrier at steady state (i.e., t! l) are presented for each case in Fig. 7 (C o = 47 mm), Fig. 8 (C o = 20 mm), Fig. 9 (C o =8.7 mm), and Fig. 10 (C o = 3.9 mm). The results illustrate that the greater significance of advection relative to diffusion, due to either an increase in i h or a decrease in s a, results in a higher exit concentration gradient because the salt front extends further into the soil. The exit concentration gradients corresponding to i h = 100 in Figs. 4 and 5 for the cases that include only implicit coupling (i.e., s a < s a,max and x = 0) are higher than the exit concentration gradient for the a non-membrane case (i.e., s a = s a,max and x = 0). Thus, the similar exit flux values at steady state for these cases in Figs. 4 and 5 reflect the competing effects of a reduction in exit flux associated with a lower s a and an increase in exit flux due to a higher exit concentration gradient associated with an increase in P L. The concentration profiles in Figs also indicate that the exit concentration gradient is lower for the cases that include explicit coupling (i.e., x>0) relative to the cases in which explicit coupling is ignored (i.e., x = 0). As explained earlier, explicit coupling terms are negligible in the salt flux equation at x = L (Eq. (17)) because C s =0 at x = L in

18 276 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Fig. 7. Simulated steady-state concentration profiles within a 1-m-thick clay membrane barrier for a source KCl concentration, C o, of 47 mm as a function of hydraulic gradient [NM = no membrane behavior; E = membrane behavior with explicit coupling only; I = membrane behavior with implicit coupling only; EI = membrane behavior with both explicit and implicit coupling]. accordance with the perfectly flushing exit boundary condition. However, the influence of explicit coupling becomes more significant at locations up-gradient of the exit boundary (i.e., at x < L) where C s >0. In particular, the greatest influence of explicit coupling on solute migration occurs at the entry boundary of the soil barrier (x = 0) where C s is a

19 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Fig. 8. Simulated steady-state concentration profiles within a 1-m-thick clay membrane barrier for a source KCl concentration, C o, of 20 mm as a function of hydraulic gradient [NM = no membrane behavior; E = membrane behavior with explicit coupling only; I = membrane behavior with implicit coupling only; EI = membrane behavior with both explicit and implicit coupling].

20 278 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Fig. 9. Simulated steady-state concentration profiles within a 1-m-thick clay membrane barrier for a source KCl concentration, C o, of 8.7 mm as a function of hydraulic gradient [NM = no membrane behavior; E = membrane behavior with explicit coupling only; I = membrane behavior with implicit coupling only; EI = membrane behavior with both explicit and implicit coupling].

21 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Fig. 10. Simulated steady-state concentration profiles within a 1-m-thick clay membrane barrier for a source KCl concentration, C o, of 3.9 mm as a function of hydraulic gradient [NM = no membrane behavior; E = membrane behavior with explicit coupling only; I = membrane behavior with implicit coupling only; EI = membrane behavior with both explicit and implicit coupling].

22 280 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) maximum. Steady-state concentration profiles in Figs for all cases in which x>0 are marked by a discontinuity in salt concentration at x = 0 such that C s is less than the source concentration, C o. Conversely, no discontinuity is observed at x = 0 for the cases in which explicit coupling is ignored (i.e., x = 0). In addition, salt-concentration profiles for i h =0 are linear for cases in which x = 0 but non-linear (i.e., concave) for cases in which x>0. Thus, although explicit coupling effects appear to be negligible at x = L based in Eq. (17), the influence of explicit coupling at x < L ultimately causes a decrease in exit concentration gradient and a reduced exit flux. The discontinuity in salt concentration at x = 0 for cases in which x>0 in Figs (i.e., E and EI cases) is consistent with evidence from previous studies indicating that non-ideal clay membranes partially restrict entry of solutes into the pore space (e.g., Kemper and Maasland, 1964; Kemper and van Schaik, 1966; Kharaka and Berry, 1973; Ishiguro et al., 1996; Sawatsky et al., 1997). For example, Kemper and van Schaik (1966) performed diffusion tests (i h = 0) on bentonite pastes (L = 20 mm) with NaCl and CaCl 2 salt solutions. After steady-state diffusion of the salt was achieved, the specimens were cut into f 1-mm sections and pore fluid concentrations of salt were measured in each section. Kemper and van Schaik (1966) observed that the salt concentration immediately inside the specimens (i.e., x = 0 to 1 mm) was considerably lower than the source concentration, C o, even in tests with C o as high as 1.0 M NaCl. Non-linear steady-state concentration profiles also were observed in some of these tests. Kemper and van Schaik (1966) attributed this behavior to osmotic flow and partial exclusion of the salt from the clay, but did not investigate the potential relationship between this behavior and chemico-osmotic efficiency coefficients for the bentonite. Similar results have been obtained in other experimental studies (e.g., Dutt and Low, 1962). Typically, restriction of solutes at the entry interface of clay membranes is attributed to hyperfiltration under a hydraulic gradient, in accordance with the reduction factor (1 x) in Eq. (12) (e.g., Kharaka and Smalley, 1976). However, the results in Figs indicate that the discontinuity in salt concentration at the entry boundary also occurs for the cases in which i h = 0 and, thus, (1 x)q h = 0, which is consistent with the aforementioned experimental results based on diffusion tests in which i h = 0. These results suggest that the apparent solute restriction may be related to the counter-advection term in Eq. (12) (e.g., Quigley et al., 1987). Reduction of salt flux due to membrane coupling effects also is illustrated in Fig. 11 based on the ratio of the steady-state exit flux with coupling included (explicit and/or implicit) to the steady-state exit flux for the non-membrane case (i.e., no coupling). With reference to Fig. 11, the steady-state exit fluxes with only explicit coupling, only implicit coupling, both explicit and implicit coupling, and no coupling (non-membrane) are denoted as J E, J I, J EI, and J NM, respectively. The results in Fig. 11 indicate that the greatest reduction in steady-state exit flux occurs when both explicit and implicit coupling are included in the analysis. Extrapolation of the data trends given by J EI /J NM beyond x = 0.63 indicates that the salt flux approaches zero (i.e., J EI /J NM! 0) as the chemico-osmotic efficiency coefficient approaches unity (i.e., as x! 1). Thus, these simulations show that the expected behavior for an ideal membrane that completely restricts entry of solutes into the pore space is approached provided that both explicit and implicit coupling are included in the analysis.

23 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Fig. 11. Normalized exit salt flux at steady state [NM = no membrane behavior; E = membrane behavior with explicit coupling only; I = membrane behavior with implicit coupling only; EI = membrane behavior with both explicit and implicit coupling]. When only explicit or only implicit coupling is included, the steady-state exit flux generally is reduced to a lesser degree. Moreover, some of the trends in the simulation results for J E /J NM and J I /J NM do not appear to approach zero as x approaches unity, suggesting that salt can be transported through an ideal membrane. Since solute transport through an ideal membrane is not physically possible, the results in Fig. 11 illustrate the importance of considering both explicit and implicit coupling effects when modeling salt flux through soil barriers that exhibit membrane behavior. The results in Fig. 11 also illustrate that, in the absence of a hydraulic gradient (i.e., i h = 0), implicit coupling is more significant than explicit coupling with respect to

24 282 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) reduction of salt flux for the higher values of x (i.e., x z 0.49) in which s a is considerably lower than the maximum tortuosity factor, s a,max. The relative dominance of implicit coupling when i h = 0 is responsible for the convergence of the data trends given by J I /J NM and J EI /J NM as x increases beyond However, for x V 0.32, implicit coupling is negligible and the trends given by J I /J NM and J EI /J NM diverge. As stated earlier, measured values of s a corresponding to x V 0.32 are only slightly lower than s a,max. Nonetheless, the results for i h = 0 indicate that salt flux through soil membranes under diffusion-dominated conditions can be modeled conservatively over the range 0 < x < 1 by considering only implicit coupling and neglecting explicit coupling. The advantages of neglecting explicit coupling when evaluating solute transport through soil membranes are twofold. First, values of x are assumed to be zero and, therefore, do not have to be measured. Second, when x = 0, the coupled solute flux equation (Eq. (13)) reduces to the traditional advective dispersive flux equation (Eq. (15)), for which numerous analytical solutions are available (e.g., see Van Genuchten and Alves, 1982). However, while use of advective dispersive transport theory with implicit coupling seems reasonable for i h = 0, the results also indicate that the influence of implicit coupling becomes less significant as i h (and, thus, P L ) increases. At i h = 100, implicit coupling is essentially negligible and explicit coupling is responsible for nearly all of the reduction in solute flux. As discussed earlier, the reduction in exit flux due to a lower s a is negated by an increase in exit flux due to a higher exit concentration gradient associated with an increase in P L as i h increases and the system becomes more advection-dominated. Thus, use of advective dispersive transport theory with implicit coupling does not appear to be sufficient for modeling advection-dominated solute transport through a soil membrane Solute transit time Based on the time-dependent exit flux values in Figs. 2 5, membrane behavior also appears to result in a retardation effect such that a longer time is required to reach steadystate transport of the salt (KCl) relative to the non-membrane case. Longer transit times are particularly evident in Fig. 5 for C o = 3.9 mm KCl, presumably due to the high x ( = 0.63) and low s a ( = 0.039) for this case. To illustrate this effect, the time corresponding to steady-state transport of KCl was estimated for each case based in Figs These steady-state times, T, for solute transport with only explicit coupling, only implicit coupling, both explicit and implicit coupling, and no coupling (non-membrane) are denoted as T E, T I, T EI, and T NM, respectively. Normalized values of steady-state time relative to the non-membrane case (i.e., T E /T NM, T I /T NM, T EI /T NM ) are plotted as a function of chemico-osmotic efficiency in Fig. 12. The results show that the longer transit times for cases in which i h =0 or i h = 10 are primarily due to implicit coupling, as indicated by the similar trends for T I /T NM and T EI /T NM as well as the fact that T E /T NM is only slightly greater than unity for the range of x values considered in this study. Conversely, longer transit times for cases in which i h = 100 are primarily due to explicit coupling since implicit coupling has little effect on transit time when solute transport is dominated by advection. These results are consistent with the exit

25 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) Fig. 12. Normalized transit times [NM = no membrane behavior; E = membrane behavior with explicit coupling only; I = membrane behavior with implicit coupling only; EI = membrane behavior with both explicit and implicit coupling]. flux results presented earlier, in that explicit coupling is dominant at high values of P L and implicit coupling is dominant at low values of P L. 6. Conclusion The results of numerical simulations indicate that solute flux is reduced and solute transit time is increased in a clay membrane barrier (CMB) relative to a nonmembrane clay barrier due to both explicit and implicit (or empirical) coupling

26 284 M.A. Malusis, C.D. Shackelford / Journal of Contaminant Hydrology 72 (2004) effects. Explicit coupling effects include hyperfiltration, whereby solutes are filtered out as the solution passes through the membrane under a hydraulic gradient, and counter-advection of solutes due to chemico-osmotic liquid flux opposite to the direction of the hydraulic liquid flux. Implicit coupling is characterized by a decrease in the apparent tortuosity factor, s a, with an increase in the chemico-osmotic efficiency coefficient, x, such that s a is less than the maximum tortuosity factor for the soil, s a,max, that represents the tortuosity due solely to the geometry of the interconnected pores. The increase in x also is correlated with a decrease in solute concentration, C. The concept of implicit coupling is supported by results of steady-state chemicoosmotic/diffusion tests performed using a bentonite-based barrier material and potassium chloride (KCl) solutions. Traditional advective dispersive transport theory is based on the limiting assumption that x = 0 and, thus, inherently neglects explicit coupling effects. Implicit coupling may be included when using advective dispersive transport theory by using an appropriate value of s a (<s a,max ) that corresponds to the source concentration, C o, or may be neglected by assuming that s a = s a,max. Comparison of theoretical exit salt fluxes [i.e., J s (x = L)] at steady state obtained using the coupled solute transport theory with those obtained using advective dispersive transport theory indicates that implicit coupling is more significant than explicit coupling in reducing solute flux under diffusion-dominated conditions. Under such conditions, use of advective dispersive transport theory in lieu of the coupled transport theory may be sufficient for modeling solute transport through a CMB, provided that implicit coupling is included in the analysis by using the appropriate value of s a corresponding to the source salt concentration. Acknowledgements Financial support for this study, which is part of a joint research effort between Colorado State University and the Colorado School of Mines, was provided by the U. S. National Science Foundation (NSF), Arlington, VA, under Grant CMS The assistance of Prof. Harold (Hal) W. Olsen of the Colorado School of Mines is appreciated. The opinions expressed in this paper are solely those of the writers and are not necessarily consistent with the policies or opinions of the NSF. References Dutt, G.R., Low, P.F., Diffusion of alkali chlorides in clay water systems. Soil Science 93, Freeze, R.A., Cherry, J.A., Groundwater. Prentice-Hall, Englewood Cliffs, NJ. Greenberg, J.A., Mitchell, J.K., Witherspoon, P.A., Coupled salt and water flows in a groundwater basin. Journal of Geophysical Research 78 (27), Ishiguro, M., Matsuura, T., Detellier, C., A study on the solute separation and the pore size distribution of a montmorillonite membrane. Separation Science and Technology 31 (4), Katchalsky, A., Curran, P.F., Nonequilibrium Thermodynamics in Biophysics Harvard Univ. Press, Cambridge, MA. 248 p.

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