Predicting Solute Flux through a Clay Membrane Barrier
Michael A. Malusis1 and Charles D. Shackelford2
Abstract: Measured solute flux breakthrough curves 共FBCs兲 from column tests performed on a semipermeable clay membrane subjected
to KCl solutions are compared with predicted FBCs using independently measured flow and transport properties. The predicted FBCs are
based on three scenarios: 共1兲 Advective–dispersive transport that neglects membrane behavior; 共2兲 advective–dispersive transport that
accounts for the concentration dependency of the effective salt-diffusion coefficient (D s* ) resulting from membrane behavior, referred to
as partially coupled transport; and 共3兲 fully coupled transport that includes both the explicit coupling terms 共e.g., hyperfiltration, chemicoosmosis兲 associated with clay membrane behavior and the concentration dependency of D s* . The FBCs predicted by fully coupled
transport agree best with the measured FBCs. However, for the diffusion-controlled conditions of the column tests, the steady-state solute
fluxes predicted by partially coupled transport are only 23– 69% higher than the measured steady-state fluxes. The results imply that the
advective–dispersive transport theory can be used to provide reasonably accurate, albeit somewhat conservative, estimates of steady-state
solute flux through clays that behave as semipermeable membranes, provided diffusion is a significant, if not dominant, solute transport
process and the concentration dependency of D s* are taken into account.
DOI: 10.1061/共ASCE兲1090-0241共2004兲130:5共477兲
CE Database subject headings: Barriers; Bentonite; Clay liners; Contaminant; Diffusion; Geosynthetics; Membrane processes;
Osmosis; Transport phenomena.
Introduction
Solute transport analyses for both natural and engineered barriers
consisting of low-permeability clays, such as aquicludes, vertical
cutoff walls 共e.g., soil–bentonite slurry walls兲, and waste containment 共e.g., landfill兲 liners, are typically performed using models
based on the advective–dispersive transport theory 共e.g., Rowe
1987; Shackelford 1988; 1990; Rabideau and Khandelwal 1998;
Rowe 1998; Katsumi et al. 2001; Kim et al. 2001; Foose et al.
2002; Toupiol et al. 2002兲. However, advective–dispersive transport represents the limiting case of the more general coupled
transport theory in that the coupling terms 共i.e., hyperfiltration,
chemico-osmosis兲 associated with semipermeable membrane behavior are assumed to be negligible 共e.g., Yeung 1990; Shackelford 1997兲. While this assumption is likely acceptable for the
relatively high flow rates commonly associated with coarsegrained soils 共e.g., aquifers兲, the use of advective–dispersive
transport theory for low-permeability clays may not be appropriate if the clays act as semipermeable membranes that restrict the
passage of solutes 共e.g., Kemper and Rollins 1966; Olsen 1969;
1
Sentinel Consulting Services, LLC, 14 Inverness Dr. E, Suite G228,
Englewood, CO 80112.
2
Dept. of Civil Engineering, Colorado State Univ., Fort Collins, CO
80523-1372
共corresponding
author兲.
E-mail:
shackel@
engr.colostate.edu
Note. Discussion open until October 1, 2004. Separate discussions
must be submitted for individual papers. To extend the closing date by
one month, a written request must be filed with the ASCE Managing
Editor. The manuscript for this paper was submitted for review and possible publication on March 17, 2003; approved on July 24, 2003. This
paper is part of the Journal of Geotechnical and Geoenvironmental
Engineering, Vol. 130, No. 5, May 1, 2004. ©ASCE, ISSN 1090-0241/
2004/5-477– 487/$18.00.
Kemper and Quirk 1972; Fritz and Marine 1983; Malusis and
Shackelford 2002a兲.
For example, the results of simulations based on models that
account for clay membrane behavior indicate that membrane behavior affects solute migration through soils that behave as semipermeable membranes 共e.g., Bresler 1973; Greenberg et al. 1973;
Barbour and Fredlund 1989; Yeung 1990; Yeung and Mitchell
1993; Manassero and Dominijanni 2003兲. However, little experimental evidence of the effects of membrane behavior on solute
transport through clay membrane barriers currently is available.
In addition, the ability of each transport theory to accurately predict measured solute fluxes through semipermeable clay membranes has not been evaluated.
As a result of the aforementioned considerations, the overall
objective of this study is to assess the accuracy of advective–
dispersive transport theory relative to coupled transport theory
that accounts for clay membrane behavior in predicting the solute
flux that occurs through a semipermeable clay membrane. This
assessment will be made by comparing measured solute fluxes of
a simple salt 共KCl兲 emanating from a column containing 10-mmthick specimens of a geosynthetic clay liner 共GCL兲 that has been
shown to behave as a semipermeable clay membrane with predicted solute fluxes based on both advective–dispersive transport
theory and coupled transport theory using independently measured flow and transport properties. This study differs from previous studies evaluating the same GCL as a semipermeable membrane 共i.e., Malusis and Shackelford 2002a, b兲 in that this study
includes both hydraulically driven solute transport 共advection兲
and diffusive solute transport in an open system, whereas the
previous studies were performed under purely diffusive 共i.e., no
flow兲 conditions in a closed system. Also, this study represents
the first attempt to evaluate the accuracy of the proposed coupled
solute transport theory in predicting solute transport through a
semipermeable clay membrane.
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2004 / 477
Theoretical Background
multiple solutes as a function of the effective self-diffusion coefficients of all species 共i.e., solute diffusion assuming no interaction with other species兲, D *j , can be written for one-dimensional
diffusion in soil as follows 共Malusis and Shackelford 2002c兲:
Total Coupled Solute Flux
The coupled solute transport theory used in this study is based on
the theory originally proposed by Yeung 共1990兲 and Yeung and
Mitchell 共1993兲, adapted for the special case of no applied electrical current, and modified to account for different ionic mobilities of all solute species as described by Malusis and Shackelford
共2002c兲. In this case, the total flux for one-dimensional migration
of a solute species j, or J j (mol L⫺2 t ⫺1 ), in a low-permeability
soil 共i.e., neglecting mechanical dispersion兲 for isothermal conditions may be represented as follows 共Malusis and Shackelford
2002c兲:
J j ⫽J ha, j ⫹J ␲, j ⫹J d, j ⫽ 共 1⫺␻ 兲 q h C j ⫹q ␲ C j ⫺nD *
s, j
⳵C j
(1)
⳵x
where
J ha⫽hyperfiltrated
advective
solute
flux;
J␲
⫽chemico-osmotic solute flux; J d ⫽diffusive solute flux; ␻
⫽chemico-osmotic efficiency coefficient (0⭐␻⭐1); q h
⫽Darcy flux (⫽k h i h , where k h ⫽hydraulic conductivity, i h
⫽hydraulic gradient兲; q ␲ ⫽chemico-osmotic flux; n⫽porosity;
D s* ⫽effective salt-diffusion coefficient 共e.g., Shackelford and
Daniel 1991兲; C⫽molar solute concentration; and x⫽direction of
transport.
The chemico-osmotic efficiency coefficient, ␻, commonly is
referred to as the ‘‘reflection coefficient’’ 共Staverman 1952;
Katchalsky and Curran 1965兲 or the ‘‘osmotic efficiency coefficient,’’ ␴ 共Kemper and Rollins 1966; Olsen et al. 1990兲. The term
chemico-osmosis is used here to distinguish chemico-osmosis
from other osmotic phenomena, such as electro-osmosis and
thermo-osmosis, and the chemico-osmotic efficiency coefficient is
designated herein as ␻ since ␴ is routinely used in engineering to
designate stress or electrical 共specific兲 conductance 共e.g., Mitchell
1993兲.
The chemico-osmotic liquid flux, q ␲ , in Eq. 共1兲 represents the
flux of liquid across a membrane from a lower solute concentration to a higher solute concentration 共i.e., opposite to the direction
of solute diffusion兲, and is given by
q ␲ ⫽k ␲ i ␲ ⫽
␻k h ⳵␲
␳ w g ⳵x
冋
D*
s, j ⫽␶ a D s, j ⫽ D *
j ⫾
⫻
* 兩 z ⫺ 兩 ⳵C ⫺ ⫺ 兺 D *⫹ 兩 z ⫹ 兩 ⳵C ⫹ 兲
共 兺D⫺
2
2
共 兺D*
⫺兩 z ⫺兩 C ⫺⫹ 兺 D *
⫹兩 z ⫹兩 C ⫹ 兲
册
(4)
where ␶ a ⫽dimensionless apparent tortuosity factor (0⭐␶ a ⭐1)
as defined by Shackelford and Daniel 共1991兲; D s, j
⫽salt-diffusion coefficients of an ionic species j in aqueous solution 共i.e., free of solid soil particles兲; z⫽ion valence; and subscripts ⫹ and ⫺ represent cations and anions, respectively. The
‘‘⫾’’ term in Eq. 共4兲 represents the interaction among multiple
ionic species during salt diffusion, and is negative for anions and
positive for cations. The value of D *j used in Eq. 共4兲 for each
species is calculated through the following relationship 共Shackelford and Daniel 1991兲:
D *j ⫽␶ a D o, j
(5)
where D o, j ⫽aqueous 共free-solution兲 self-diffusion coefficients for
each solute species j.
The hyperfiltrated advective solute flux, J ha , in Eq. 共1兲 represents the traditional advective solute flux, J a (⫽q h C) that is reduced by a factor of (1⫺␻) due to the membrane behavior of the
soil. The factor (1⫺␻) has been described physically as the process whereby solutes are filtered out of solution as the solution
passes through the membrane under an applied hydraulic gradient
共e.g., see Fritz and Marine 1983兲. If the soil does not exhibit
membrane behavior 共i.e., if ␻⫽0), then q ␲ ⫽0 关see Eq. 共2兲兴 and
Eq. 共1兲 reduces to the traditional expression for advective–
dispersive solute flux 共i.e., assuming negligible mechanical dispersion兲 modified to include the mobility effect inherent in Eq.
共4兲, or
J j 兩 ␻⫽0 ⫽J a, j ⫹J d, j ⫽q h C j ⫺nD *
s, j
⳵C j
⳵x
(6)
Transient Transport
The governing partial differential equation for one-dimensional,
coupled transport of an ionic species j under transient conditions
is obtained by substituting Eq. 共1兲 into the mass balance constraint for a representative elementary volume of saturated soil, or
⳵Q j ⳵ 共 nC j ⫹␳ d K d, j C j 兲
⫽
⫽⫺ⵜ•J j
⳵t
⳵t
N
兺 Ci
i⫽1
⳵C j
(2)
where k ␲ ⫽chemico-osmotic permeability coefficient (⫽␻k h );
i ␲ ⫽chemico-osmotic gradient; ␳ w ⫽density of water; g
⫽acceleration due to gravity; and ␲⫽chemico-osmotic pressure.
For relatively dilute solutions 共i.e., ⬍1 M for monovalent salts兲,
which are typically a prerequisite for the presence of membrane
behavior in clays, the chemico-osmotic pressure is related to the
concentration of solutes by the van’t Hoff expression 共Malusis
and Shackelford 2002a兲, or
␲⫽RT
D *j C j 兩 z j 兩
(3)
where R⫽universal gas constant 共8.314 J/mol K兲; T⫽absolute
temperature; and N⫽number of solute species in solution.
The subscript j on the effective salt-diffusion coefficient in Eq.
共1兲 indicates recognition of the potential influence of multiple
solute species in a chemical mixture on the diffusion of the jth
species due to the different ionic mobilities of all species in solution 共Robinson and Stokes 1959; Shackelford and Daniel 1991兲.
For example, the general expression for the effective saltdiffusion coefficient of an ionic species in soil, D s,* j , containing
⫽⫺
冋
⳵C j
⳵
共 1⫺␻ 兲 q h C j ⫹q ␲ C j ⫺nD 쐓s, j
⳵x
⳵x
册
(7)
where Q j ⫽total moles 关aqueous and solid 共adsorbed兲 phase兴 of
solute species j per unit total volume of porous medium 共i.e.,
solids and voids兲; ␳ d ⫽dry density of the soil; and K d, j
⫽distribution coefficient that accounts for linear, reversible, and
instantaneous partitioning of species j between the soil solids and
the aqueous solution. Assuming steady hydraulic liquid flux 共i.e.,
q h ⫽constant) through a homogeneous, incompressible soil 共i.e.,
n⫽constant), Eq. 共7兲 reduces to the following governing transport equation:
478 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2004
쐓
R d, j
⳵C j
⳵ 2 C j ⳵D s, j ⳵C j
⳵C j
⫽D 쐓s, j 2 ⫹
⫺ 共 1⫺␻ 兲v h
⳵t
⳵x
⳵x ⳵x
⳵x
⫺v␲
⳵C j
⳵v␲
⫺C j
⳵x
⳵x
Table 1. Measured Properties of Bentonite in Geosynthetic Clay
Liner 共GCL兲 Used in this Study 共from Malusis and Shackelford
2002a兲
(8)
where v h ⫽hydraulic seepage velocity (⫽q h /n); v ␲
⫽chemico-osmotic seepage velocity (⫽q ␲ /n); and R d, j represents the retardation factor of species j 共Freeze and Cherry 1979兲,
or
R d, j ⫽1⫹
␳ d K d, j
n
(9)
For the limiting case of no membrane behavior, ␻⫽0 and
v ␲ ⫽0 共i.e., since k ␲ ⫽0), and Eq. 共8兲 reduces to the following
form of the advective–dispersive transport equation
쐓
R d, j
⳵C j
⳵ 2 C j ⳵D s, j ⳵C j
⳵C j
⫽D 쐓s, j 2 ⫹
⫺vh
⳵t
⳵x
⳵x ⳵x
⳵x
(10)
that accounts for the potential influence of different ionic mobilities associated with each species in a chemical mixture. For the
common assumption that the ionic mobilities of each species are
independent of each other, Eq. 共10兲 reduces to the more traditional form of the one-dimensional advective–dispersive equation, or
⳵C j
⳵ 2C j
⳵C j
R d, j
⫺vh
⫽D 쐓j
⳵t
⳵x 2
⳵x
Property
Value
Specific gravity, G s
2.43
Principal minerals 共%兲:
Montmorillonite
Mixed-layer illite/smectite
Quartz
Other
71
7
15
7
Exchangeable metals 共meq/100 g兲:
Ca
Mg
Na
K
Sum
20.8
6.4
31.0
0.8
59.0
Soluble metals 共mg/kg兲:
Ca
Mg
Na
K
443
407
4,636
263
Soil pH
9.2
Electrical conductance 共mS/m兲 at 25°C
120
(11)
where D s쐓 from Eq. 共10兲 reduces to the effective self-diffusion
coefficient for each individual solute species D 쐓j without regard to
the effect of interactions resulting from the different mobilities
among all chemical species in the pore water 关i.e., see Eq. 共4兲兴.
Experimental Methods and Materials
Clay Membrane Barrier
Specimens of the same GCL used by Malusis et al. 共2001兲 and
Malusis and Shackelford 共2002a,b兲 were tested in this study, because this GCL was shown to exhibit membrane behavior for the
same test conditions 共i.e., GCL thickness, salt and boundary concentrations兲 imposed in this study. This GCL consists of a layer of
sodium bentonite sandwiched between two nonwoven polypropylene geotextiles held together by needle-punched fibers. The
GCL is approximately 6 mm thick in an air-dried condition, but
quickly swells to a thickness typically ranging from 10 to 15 mm
upon exposure to water due to the bentonite content. Selected
properties of the bentonite portion of the GCL are given in Table
1; further details on the GCL are given elsewhere 共Malusis et al.
2001; Malusis and Shackelford 2002a,b兲.
Liquids
The liquids used in the column tests consist of tap water that is
processed to remove ions by passage through three Barnstead ion
exchange columns in series 关electrical conductance (EC)⫽at
25°C⫽0.32 mS/m, pH⫽6.93], and solutions of potassium chloride 共KCl兲 containing measured KCl concentrations of either 8.7
mM 共650 mg/L, EC⫽123 mS/m, pH⫽6.73) or 47 mM 共3,500
mg/L, EC⫽682 mS/m, pH⫽6.91).
Column Testing Apparatus
A schematic illustration of the column testing apparatus is shown
in Fig. 1. The testing cell consists of a rigid acrylic cylinder, top
piston, and base pedestal. The top piston is locked in place to
prevent soil expansion and, therefore, to control the thickness
共porosity兲 of the test specimen. The top piston and base pedestal
are equipped with ports that enable circulation of separate electrolyte solutions through porous stones at the specimen boundaries to establish and maintain a constant concentration at each
boundary. A difference in concentration between the top and the
bottom of the specimen establishes the concentration gradient for
chemico-osmotic flow and diffusion. The circulated solutions can
be collected in separate reservoirs for subsequent chemical analysis. Further details regarding the testing cell are provided by
Malusis et al. 共2001兲.
The column testing apparatus used in this study is similar
to the chemico-osmotic/diffusion test apparatus described by
Malusis et al. 共2001兲, except that solutions are circulated at the
ends of the specimen using peristaltic pumps instead of syringe
pumps, and the column testing system is an open system as opposed to the closed 共no-flow兲 system used by Malusis et al.
共2001兲. Also, a hydraulic gradient can be established and maintained by applying a differential pressure across the specimen
through the sample collection reservoirs at the source and effluent
ends of the specimen.
As discussed by Malusis et al. 共2001兲, continuous circulation
of the two liquids at either end of the specimen mimics a
constant-source boundary condition at the top and a perfectly
flushing boundary condition at the bottom. These same boundary
conditions were employed by Greenberg et al. 共1973兲 to describe
the coupled transport of NaCl through a clay aquitard below a
salt–water aquifer, and the perfectly flushing exit boundary condition has been recommended as a conservative approach for design of vertical cutoff walls 共Rabideau and Khandelwal 1998兲.
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2004 / 479
Fig. 1. Column test apparatus
Specimen Preparation
Circular specimens of the GCL with nominal diameters of 71.1
mm were placed on the base pedestal inside the testing cell. The
cylinder then was filled with the processed tap water to submerge
the specimen, and the top piston was lowered into the cylinder to
compress the specimen to the desired thickness of 10 mm. After
completion of compression, the top piston was locked in place to
prevent further volume expansion of the specimen due to swelling
of the bentonite.
Column Testing Procedure
Each specimen was permeated under backpressure with processed
tap water before column testing to saturate the specimen, remove
excess soluble salts, and measure the initial hydraulic conductivity. After permeation, a column test was initiated by circulating
the source KCl solution (C 0 ⬎0) at the top of the specimen, while
simultaneously circulating process tap water (C⫽0) at the bottom of the specimen. Samples of the circulated process tap water
exiting the bottom were collected over time increments, ⌬t, and
subsequently analyzed for concentrations of Cl⫺ using ion chromatography and K⫹ using inductively coupled plasma in order to
determine the exit fluxes. The exit flux of a solute j, or J j , then
was calculated based on the measured change in moles of solute
exiting the specimen, ⌬M j , over a known time interval in accordance with the following expression:
J j 共 x⫽L 兲 ⫽
⌬M j C j ⌬V e
⫽
A⌬t
A⌬t
(12)
where A⫽total cross-sectional area of the specimen; and ⌬V e
⫽incremental effluent sample volume of liquid collected over the
time interval ⌬t. At the end of column testing, each specimen
was permeated with the source KCl solution to determine the final
hydraulic conductivity.
Testing Program
Four column tests were performed. Three of the column tests
were conducted with a source KCl concentration, C 0 , of 8.7 mM,
but with different applied hydraulic gradients, i h , of 0, 70.3, and
703. The fourth column test was conducted with an i h of 703 and
C 0 of 47 mM.
Modeling
Simulation Scenarios
As shown by Malusis and Shackelford 共2002b兲, the coupled solute transport theory described above does not account for correlation among the source salt concentration, C 0 , ␻ and D s쐓 , and
therefore ␶ a 关i.e., through Eqs. 共4兲 and 共5兲兴. For example, as
shown in Fig. 2共a兲, ␻ decreases with an increase in the source
KCl concentration, C 0 , such that the observed membrane behavior for the GCL is not apparent for C 0 greater than approximately
100 mM KCl 共i.e., based on extrapolation of the data兲. The observed decrease in chemico-osmotic efficiency with increasing
source KCl concentration shown in Fig. 2共a兲 is consistent with
480 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2004
⬎0) as well as the implicit coupling effect represented by the
relationships in Fig. 2 关i.e., ␶ a ⫽ f (C 0 )⫽ f (␻)]. The second simulation scenario, designated as ‘‘partially coupled’’ 共PC兲, neglects
membrane behavior in that ␻ is assumed to be zero such that
J ha⫽J a and J ␲ ⫽0 关i.e., Eqs. 共6兲 and 共10兲兴, but includes the implicit coupling effect in that the concentration dependency of ␶ a
共i.e., D s쐓 ) resulting from the existence of membrane behavior
关Fig. 2共b兲兴. The third, or ‘‘advective–dispersive’’ 共AD兲, simulation scenario ignores all membrane behavior such that ␻⫽0 共i.e.,
J ha⫽J a and J ␲ ⫽0) and only the maximum value of ␶ a 关⬃0.120;
see Fig. 2共c兲兴 corresponding to ␻⫽0 is used in the simulations.
Initial and Boundary Conditions
The initial conditions assumed for the simulations are that the
specimen is initially free of both potassium and chloride, or
C j 共 x,0兲 ⫽0
(13)
This initial condition is consistent with the flushing of soluble
salts in the pore water of the specimen by permeation with processed tap water prior to the introduction of the source KCl solution. The boundary conditions considered in this study are the
constant source, perfectly flushing conditions consistent with the
column test setup. These boundary conditions are written as follows:
C j 共 0,t 兲 ⫽C 0 ;
C j 共 L,t 兲 ⫽0
(14)
Electroneutrality Constraint
Fig. 2. Correlations among source KCl concentration (C 0 ),
chemico-osmotic efficiency coefficient 共␻兲, and apparent tortuosity
factor (␶ a ) for geosynthetic clay liner: 共a兲 C 0 versus ␻; 共b兲 C 0 versus
␶ a ; and 共c兲 ␻ versus ␶ a 共data from Malusis and Shackelford 2002b兲
Electroneutrality in the pore water is taken into account by assuming that the primary cation exchange process in the soil is
potassium-for-sodium (K⫹ – Na⫹ ) exchange, as follows 共Malusis
and Shackelford 2002c兲:
C Na⫹ ⫽C Cl⫺ ⫺C K⫹
(15)
⫹
expected behavior based on diffuse double layer 共Gouy–
Chapman兲 theory, in that the thickness of the diffuse double layers of adjacent clay particles responsible for ion restriction inside
the pores of a clay membrane decreases as the ion concentration
in the pore water increases. Similarly, the increase in ␻ and,
therefore, solute restriction that occurs with decreasing source
KCl solution is accompanied by a corresponding decrease in D s쐓
and, therefore, the apparent tortuosity factor, ␶ a , as shown in Fig.
2共b兲. This decrease in ␶ a with decreasing C 0 is consistent with an
increase in solute restriction as reflected by an increase in ␻ with
a decrease in C 0 关Fig. 2共a兲兴.
The overall effect, shown in Fig. 2共c兲, is that ␶ a decreases
from a maximum value at ␻⫽0 关i.e., ␶ a,max⬃0.12 in Fig. 2共c兲兴 to
zero at ␻⫽1 since, in the limit as ␻ approaches unity 共i.e., perfect
membrane兲, no solute diffusion through a clay membrane can
occur. This correlation between ␻ and ␶ a is an implicit correlation, since the correlation is not explicitly included in the coupled
transport theory given by Eqs. 共1兲 and 共8兲 关i.e., D s쐓 , ␶ a ⫽ f (␻)]
and, therefore, must be determined experimentally 共Malusis and
Shackelford 2002b兲.
Based on these considerations, simulated flux breakthrough
curves 共FBCs兲 are predicted in this study for both chloride (Cl⫺ )
and potassium (K⫹ ) for three different simulation scenarios. The
first simulation scenario is designated as ‘‘fully coupled’’ 共FC兲,
since this scenario incudes both the explicit coupling effects inherent in the J ha and J ␲ terms in Eqs. 共1兲 and 共8兲 共i.e., since ␻
where C Na⫹ , C Cl⫺ , and C K⫹ ⫽molar concentrations of Na , Cl⫺ ,
and K⫹ , respectively 共i.e., since each ion has the same equivalents per mole兲. Although the exchange complex of the bentonite
also contains an appreciable amount of Ca2⫹ 共see Table 1兲, this
Ca2⫹ tends to be attracted more strongly to the soil surface than
Na⫹ due to a higher valence and, as a result, typically is not as
readily exchangeable as Na⫹ 共Mitchell 1993兲. As noted by Malusis and Shackelford 共2002c兲, a more rigorous analysis that includes the potential contribution of all exchangeable cations could
be performed by expanding the expressions for q ␲ and the electroneutrality constraint to include the additional species, and by
incorporating an ion exchange model to account for competition
among all migrating ions for the exchange sites. However, the
approach based on a single exchangeable cation is relatively easy
to implement, and represents a balance between the simplest approach in which multiple cations in solution are assumed to migrate independently of each other, and the most rigorous approach
in which knowledge of all chemical species is required. In addition, the effect of different ionic mobilities of chemical species is
potentially significant only during the transient portion of the
simulations, because this effect vanishes once steady-state transport is reached. Therefore, since the primary emphasis on the
evaluation in this study is with respect to the steady-state solute
fluxes, the assumption of a single exchangeable cation is not considered significant.
The appropriate system of equations for each simulation scenario, subject to Eqs. 共13兲, 共14兲, and 共15兲, are solved iteratively
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2004 / 481
Table 2. Input Values for Physical Parameters Used in Model
Simulations
Column
test No.
Source KCl
concentration
共mM兲
Hydraulic
gradient,
ih
Hydraulic
conductivity,
k h a (⫻10⫺11 m/s)
Specimen
porosity,
n
8.7
8.7
8.7
47
0
70.3
703
703
1.50
2.01
1.43
2.37
0.79
0.81
0.79
0.78
1
2
3
4
a
Values based on permeation with the source KCl solution after column
testing.
using an implicit-in-time, centered-in-space, finite difference algorithm. The cell and time increments are chosen in order to
maintain compliance with stability requirements for the implicitin-time, centered-in-space approach, and the algorithm is adjusted
properly to account for numerical dispersion 共Smith 1985兲. All
details regarding the finite difference algorithm and the associated
program are described by Malusis 共2001兲.
Input Parameters
The parameter values used as input for each of the model simulations are summarized in Tables 2 and 3. The values for n and k h
in Table 2 are measured values for each column test specimen,
and the values for ␻, ␶ a , and R d in Table 3 are taken from the
results of the steady-state chemico-osmotic/diffusion tests for
identical conditions 共i.e., salt and boundary concentrations, GCL,
and specimen thickness兲 reported by Malusis and Shackelford
共2002b兲.
The k h values shown in Table 2 are based on permeation of the
column test specimens with the source KCl solution 共i.e., 0.0087
M KCl or 0.047 M KCl兲 after completion of the column tests.
These measured k h values are very close to similarly measured k h
values previously reported by Malusis and Shackelford 共2002a兲
for the same GCL of 1.33⫻10⫺11 m/s for C 0 of 8.7 and 1.48
⫻10⫺11 m/s for C 0 of 47 mM KCl, indicating good reproducibility of specimen properties. A similar comparison of these k h values with the values based on the initial permeation of identical
specimens with the processed tap water reported by Malusis and
Shackelford 共2002a兲 also shows very little difference 共21% for C 0
of 8.7 mM KCl and 70% for C 0 of 47 mM KCl兲, suggesting that
the relatively dilute KCl concentrations used in this study are not
sufficient to result in significant changes in the hydraulic conductivity of the GCL during the column tests 共e.g., see Shackelford
et al. 2000兲.
The values for ␻ and ␶ a in Table 3 are the same as those
shown in Figs. 2共a and b兲, respectively, for the two source KCl
concentrations considered in this study 共i.e., 8.7 and 47 mM KCl兲.
Since these values for ␻ and ␶ a are based on steady-state test
conditions, the predicted solute fluxes were expected to be relatively accurate only after steady-state conditions in the column
tests had been achieved.
The values of R d for Cl⫺ and K⫹ reported in Table 3 were
determined from the measured time lags associated with the
steady-state diffusion tests reported by Malusis and Shackelford
共2002b兲. The reasons for measured values of R d ⬎1 for Cl⫺ are
unknown, but may be related, in part, to counter diffusion of Cl⫺
associated with the exchangeable cations 共e.g., Na⫹ ) into the
source solution, i.e., due to the concentration gradient in the exchangeable cation established between the pore water and the
source solution. Also, the time-lag expression used to evaluate R d
does not account for slower Cl⫺ migration during transient transport with Na⫹ relative to steady-state transport with K⫹ due to
the lower mobility of Na⫹ relative to K⫹ . Regardless of the reasons for R d ⬎1 for Cl⫺ , the actual measured values of R d for Cl⫺
shown in Table 3 were used in the simulations.
In addition to the input parameter values shown in Tables 2
and 3, the values for D 0 for Cl⫺ , K⫹ , and Na⫹ required to calculate D * using Eq. 共5兲 and subsequently D s* for each species in
accordance with Eq. 共4兲 were 2.03⫻10⫺9 , 1.96⫻10⫺9 , and
1.33⫻10⫺9 m2 /s, respectively 共Shackelford and Daniel 1991兲.
The column Péclet numbers PL for the conditions in the four
column tests are relatively low (PL ⭐1.0), even for the cases in
which i h ⫽703, due to the thinness 共i.e., L⫽10 mm) and relatively low hydraulic conductivities (k h ⭐2.37⫻10⫺11 m/s) of the
specimens. Therefore, since PL ⬍20 typically represents
diffusion-dominated conditions 共Shackelford 1994兲, solute diffusion likely was the dominant transport process for the column
tests reported in this study.
Table 3. Input Values for Chemical Parameters Used in Model Simulations
Column
test Number
Source KCl
concentration
共mM兲
Chemico-osmotic
efficiency coefficient,
␻a
Apparent tortuosity
factor, ␶ a a
Fully coupled
1
2
3
4
8.7
8.7
8.7
47
0.49
0.49
0.49
0.14
Partially coupled
1
2
3
4
8.7
8.7
8.7
47
Advective–dispersive
1
2
3
4
8.7
8.7
8.7
47
Model simulation scenario
a
Retardation factor, R d b
Cl⫺
K⫹
0.063
0.063
0.063
0.119
1.53
1.53
1.53
1.94
12.6
12.6
12.6
17.5
0
0
0
0
0.063
0.063
0.063
0.119
1.53
1.53
1.53
1.94
12.6
12.6
12.6
17.5
0
0
0
0
0.120
0.120
0.120
0.120
1.53
1.53
1.53
1.94
12.5
12.5
12.5
17.5
Values at steady state from results of combined chemico-osmotic/diffusion tests as reported by Malusis and Shackelford 共2002b兲.
Values based on time lags associated with tests reported by Malusis and Shackelford 共2002b兲.
b
482 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2004
Fig. 4. Measured and predicted flux breakthrough curves for hydraulic gradient, i h , of 703 as function of KCl source concentration, C 0
共AD兲 advective–dispersive transport model; 共FC兲 fully coupled transport model
Fig. 3. Measured and predicted flux breakthrough curves for 47 mM
KCl source concentration, as function of hydraulic gradient, i h 共AD兲
advective–dispersive transport model; 共FC兲 fully coupled transport
model
Results and Discussion
Flux Breakthrough Curves
Measured FBCs for Cl⫺ and K⫹ are shown in Fig. 3 for Column
Tests 1–3 (C 0 ⫽8.7 mM KCl;i h ⫽0,70.3,703) along with the predicted FBCs based on the FC transport simulations 共i.e., ␻
⫽0.49, ␶ a ⫽0.063) and the AD transport simulations 共i.e., ␻⫽0,
␶ a ⫽0.120). The results indicate that the FC transport simulations,
in general, provide better matches to the measured fluxes for both
Cl⫺ and K⫹ than the AD transport simulations. The use of AD
transport theory (␻⫽0) with the maximum value for the apparent
tortuosity factor (␶ a ⫽0.120) results in overestimation of the
steady-state flux and underestimation of the transit time required
to reach steady-state conditions. Although neither simulation scenario is particularly effective at matching the transient portion of
the measured FBCs for K⫹ for Column Test 1 (i h ⫽0), the fully
coupled transport simulations provide reasonably good matches to
the transient portions of the measured K⫹ FBCs for the tests in
which i h ⫽70.3 and 703. These results suggest that the R d of 12.6
for K⫹ obtained from the chemico-osmotic/diffusion test probably is not accurate with respect to the results of Column Test 1
with i h ⫽0. Overall, the match between the measured FBC and
the FBC predicted by the fully coupled transport simulation appears to improve with increasing hydraulic gradient.
The results of Column Test 4 (C 0 ⫽47 mM KCl;i h ⫽703)
are compared with the results of Column Test 3 (C 0
⫽8.7 mM KCl;i h ⫽703) in Fig. 4. Less discrepancy is apparent
between the predicted FBCs given by the fully coupled transport
simulation and the AD transport simulation for C 0 of 47 mM KCl
relative to C 0 of 8.7 mM KCl. The similarity of the results from
the two simulation scenarios for C 0 of 47 mM KCl is attributed to
the fact that ␻ of 0.14 for this case is relatively low reflecting less
membrane behavior, and implicit diffusive coupling is not a factor
in Column Test 4 because the apparent tortuosity factor, ␶ a , of
0.119 used for the fully coupled transport simulation is almost the
same as the ␶ a of 0.120 used for the AD transport simulation.
Measured FBCs for Cl⫺ and K⫹ for Column Tests 1–3 (C 0
⫽8.7 mM KCl; i h ⫽0, 70.3, and 703兲 are shown again in Fig. 5
along with predicted FBCs based on both the FC simulation scenario 共i.e., ␻⫽0.49, ␶ a ⫽0.063) and the PC simulation scenario
that assumes ␻⫽0 and, therefore, neglects the explicit coupling
effects 共i.e., J ha⫽J a , J ␲ ⫽0), but includes the implicit coupling
effect represented by Fig. 2共b兲 共i.e., ␶ a ⫽0.063). The predicted
FBCs based on the partially coupled transport simulations that
include implicit diffusive coupling appear to match the measured
data better than was previously shown in Fig. 3 for the AD transport simulation that neglects implicit diffusive coupling. This improved agreement suggests that the effect of implicit diffusive
coupling 关i.e., Fig. 2共b兲兴 is more significant than the combined
effects of hyperfiltration (J ha) and chemico-osmotic counter advection (J ␲ ) associated with the fully coupled transport simulations, at least for the diffusion-dominated conditions in these column tests.
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2004 / 483
Fig. 6. Measured and predicted flux breakthrough curves for hydraulic gradient, i h , of 703 as function of KCl source concentration, C 0
共PC兲 partially coupled transport model; 共FC兲 fully coupled transport
model
Fig. 5. Measured and predicted flux breakthrough curves for 47 mM
KCl source concentration, as function of hydraulic gradient i h 共PC兲
partially coupled transport model; 共FC兲 fully coupled transport model
Comparisons of the FBCs predicted using the PC transport
simulation that includes the implicit coupling effect and the FC
transport simulations with the measured FBCs from Column Tests
3 (C 0 ⫽8.7 mM KCl;i h ⫽703) and 4 (C 0 ⫽47 mM KCl;i h ⫽703)
are shown in Fig. 6. For the lower source concentration of 8.7
mM KCl, less discrepancy is apparent between the measured and
predicted FBCs based on the partially coupled transport simulation than was previously evident in Fig. 4 for the case when the
predicted FBCs were based on the AD transport simulation that
neglects implicit diffusion coupling. This better agreement apparently results from the almost twofold decrease in the apparent
tortuosity factor, ␶ a , of 0.063 relative to the maximum ␶ a of
0.120. However, for the higher source concentration of 47 mM
KCl, no distinction can be made between the predicted FBCs
based on the partially coupled transport simulation shown in Fig.
6 and the predicted FBCs based on the AD transport simulation
shown in Fig. 4, since the values of ␶ a are almost identical 共0.119
versus 0.120兲. Therefore, this comparison highlights the significance of the correlation between C 0 and ␶ a 共i.e., implicit diffusive
coupling兲 in terms of predicting the measured FBCs for the
diffusion-dominated conditions of the column tests.
Steady-State Fluxes
A comparison of the measured and predicted FBCs based solely
on the steady-state fluxes is more appropriate than the comparison
based on the entire FBCs because the values of ␶ a and ␻ used in
simulations are steady-state values and, as a result, may not be
entirely accurate for the transient stage of the solute transport. In
addition, a comparison based on steady-state transport precludes
the need to consider the appropriateness of the retardation factors
used in the simulations, since retardation pertains only to the transient stage of transport. Finally, from a practical viewpoint, an
evaluation based on the steady-state fluxes is appropriate in the
case of long-term containment applications.
Accordingly, the measured and predicted values of steady-state
flux for each column test are summarized in Table 4. Only one
value for the predicted steady-state flux is shown in Table 4 for
each simulation scenario and each column test because, by definition, the molar fluxes of Cl⫺ and K⫹ must be equal at steady
state due to electroneutrality. This electroneutrality constraint is
evident by the merging of the predicted FBCs for Cl⫺ and K⫹ to
a single flux value as shown in Figs. 3– 6.
The ratio of the measured steady-state flux for Cl⫺ relative to
the measured steady-state flux for K⫹ , or J Cl⫺ /J K⫹ , for each
column test is given in Table 4. These ratios are close to unity for
Column Tests 1 and 3, but somewhat higher than unity for Column Tests 2 and 4. Thus, steady-state transport was achieved for
both Cl⫺ and K⫹ in Column Tests 1 and 3, but only for Cl⫺ in
Column Tests 2 and 4. As a result, the predicted steady-state
fluxes are compared only with the measured steady-state fluxes
for Cl⫺ in the subsequent discussion.
The ratios of the predicted steady-state fluxes based on fully
coupled (J FC), partially coupled (J PC), and advective–dispersive
(J AD) simulation scenarios to the measured steady-state flux for
Cl⫺ (J Cl⫺ ) also are given in Table 4. For all tests, predicted
steady-state fluxes based on the fully coupled transport simulation
are closest to the measured steady-state fluxes. For Column Tests
484 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2004
Table 4. Measured and Predicted Steady-State Molar Fluxes
Molar fluxes at steady state, J (⫻10⫺8 mol/m2 s)
Column
test
number
1
2
3
4
a
Flux ratios
Hydraulic
gradient,
ih
Source KCl
concentration,
C 0 共mM兲
J Cl⫺
J K⫹
J AD
J PC
J FC
J Cl⫺ /J K⫹
J AD /J Cl⫺
J PC /J Cl⫺
J FC /J Cl⫺
0
70.3
703
703
8.7
8.7
8.7
47
5.01
7.53
9.66
108
5.08
4.82
9.60
91.8
16.3
17.3
21.0
134
8.49
9.33
13.4
133
6.50
6.34
8.60
93.9
0.986
1.56
1.01
1.18
3.25
2.30
2.17
1.24
1.69
1.24
1.39
1.23
1.30
0.842
0.890
0.869
Measureda
Predictedb
J Cl⫺ , J K⫹ ⫽measured fluxes for chloride and potassium, respectively.
J AD , J FC , J PC⫽solute fluxes based on advective–dispersive, fully coupled, and partially coupled transport models, respectively.
b
1–3 (C 0 ⫽8.7 mM KCl;i h ⫽0,70.3,703), advective–dispersive
transport theory provides a relatively poor match to the measured
steady-state fluxes when implicit diffusive coupling is neglected
共i.e., J AD), but a much closer match to the measured steady-state
fluxes when implicit diffusive coupling is included 共i.e., J PC). For
Column Test 4 (C 0 ⫽47 mM KCl,i h ⫽703), predicted steadystate fluxes based on advective–dispersive theory with implicit
diffusive coupling (J PC) and without implicit diffusive coupling
(J AD) are essentially the same. As stated earlier, implicit diffusive
coupling is not a factor in Column Test 4 because of the similarity
between the apparent tortuosity factors, ␶ a , for these two simulation scenarios 共i.e., 0.119 versus 0.120兲, and the dominance of
diffusion in the column tests.
The predicted-to-measured steady-state flux ratios given in
Table 4 are plotted in Fig. 7 as a function of hydraulic gradient,
i h , for the same source KCl concentration 共8.7 mM兲 and as a
function of source KCl concentration for the same hydraulic gradient (i h ⫽703). The results in Fig. 7 indicate that the predicted-
to-measured steady-state flux ratios for all three simulation scenarios tend to be relatively independent of the hydraulic gradient,
and that the best predictions of the steady-state chloride flux are
provided by the fully coupled transport simulations (J FC), which
slightly overestimate J Cl⫺ at i h ⫽0 共i.e., J FC /J Cl⫺ ⫽130%) and
slightly underestimate J Cl⫺ for i h of 70.3 and 703 共i.e., 84.2%
⭐J FC /J Cl⫺ ⭐89.0%), and by the partially coupled transport simulations (J PC), which slightly overestimate J Cl⫺ at all evaluated
hydraulic gradients 共i.e., 124%⭐J PC /J Cl⫺ ⭐169%). The
advective–dispersive transport simulation that neglects any membrane behavior (J AD) significantly overestimates J Cl⫺ at all evaluated hydraulic gradients 共i.e., 217%⭐J AD /J Cl⫺ ⭐325%).
In terms of the source KCl concentration, the advective–
dispersive transport simulation that neglects any membrane behavior (J AD) provides a better estimate of J Cl⫺ at the higher of the
two concentrations evaluated, because the significance of membrane behavior decreases with increasing source concentration
关see Fig. 2共a兲兴. At a hydraulic gradient of 703, the predicted
steady-state flux based on the fully coupled transport simulation
(J FC) slightly underestimates J Cl⫺ at both concentrations 共i.e.,
86.9%⭐J FC /J Cl⫺ ⭐89.0%), and the predicted steady-state flux
based on the partially coupled transport simulation (J PC) slightly
overestimates J Cl⫺ at both concentrations 共i.e., 123%⭐J DC /J Cl⫺
⭐139%).
In general, the results summarized in Table 4 and shown in
Fig. 7 suggest that use of advective–dispersive theory in lieu of
the coupled solute transport theory results in reasonably accurate,
albeit somewhat conservative, estimates of the steady-state chloride flux provided that implicit diffusive coupling is included in
the analysis, even though ␻ is relatively high in three of these
tests 共i.e., ␻⫽0.49). These results are attributed primarily to
diffusion-dominated conditions in the column tests. Full coupling
effects 共i.e., hyperfiltration and chemico-osmotic counter advection兲 associated with ␻⬎0 may be more significant in advectivedominated systems, although advection is not likely to be very
significant relative to diffusion for the GCL specimens used in
this study, due to the low hydraulic conductivities and thinness of
the specimens. For example, a minimum hydraulic head difference across a GCL, ⌬h, of 96 m 共assuming ␶ a ⫽0.120, n⫽0.8,
k h ⫽2⫻10⫺11 m/s, D s0 ⫽10⫺9 m2 /s) would be required to increase the PL values in this study from ⭐1 to 20.
Summary and Conclusions
Fig. 7. Ratio of predicted-to-measured fluxes at steady-state transport as function of hydraulic gradient and KCl source concentration
Predicted FBCs obtained using both the fully coupled solute
transport theory and the advective–dispersive transport theory are
compared with measured FBCs for Cl⫺ and K⫹ from column
tests using a GCL that behaves as a semipermeable clay membrane subjected to KCl solutions. The predicted fluxes are based
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2004 / 485
on independently measured transport parameters from previous
chemico-osmotic/diffusion tests conducted on specimens of the
same GCL under the same test conditions. The results indicate
that the fully coupled transport theory provides better agreement
with the experimental data than the advective–dispersive theory.
The advective–dispersive theory, which neglects solute restriction
due to membrane behavior, tends to overestimate the steady-state
solute fluxes and underestimate the transit times relative to the
measured fluxes. However, advective–dispersive theory provides
reasonably accurate, albeit somewhat conservative, estimates of
the measured steady-state chloride flux when the appropriate
value of the apparent tortuosity factor, ␶ a , is used to account for
implicit diffusive coupling, primarily due to diffusion-dominated
conditions in the column tests. Under such conditions, use of
advective–dispersive transport theory in lieu of fully coupled
transport theory may be sufficient for modeling solute transport
through clay membranes, provided that implicit diffusive coupling is included in the analysis.
Acknowledgments
Financial support for this study, a joint research effort between
Colorado State University and the Colorado School of Mines, was
provided by the U.S. National Science Foundation 共NSF兲, Arlington, Va., under Grant No. CMS-9616854. The assistance of Professor Harold 共Hal兲 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.
Notation
The following symbols are used in this paper:
A ⫽ total cross-sectional area of specimen;
C j ⫽ molar concentration of solute species j;
D *j ⫽ effective self-diffusion coefficient of solute
species j;
D o, j ⫽ aqueous 共free-solution兲 self-diffusion
coefficients of solute species j;
D s, j ⫽ aqueous 共free-solution兲 salt-diffusion coefficient
of solute species j;
D s,* j ⫽ effective salt-diffusion coefficient of solute
species j;
g ⫽ acceleration due to gravity;
i h ⫽ hydraulic gradient;
i ␲ ⫽ chemico-osmotic gradient;
J AD ⫽ predicted advective–dispersive solute flux
assuming no membrane behavior;
J a, j ⫽ advective flux of solute species j (⫽q h C j );
J Cl⫺ , J K⫹ ⫽ measured fluxes for chloride and potassium,
respectively;
J d, j ⫽ diffusive flux of solute species j;
J FC ⫽ predicted fully coupled solute flux including
both explicit and implicit coupling;
J ha, j ⫽ hyperfiltrated advective flux of solute species
j;
J j ⫽ total flux of solute species j;
J PC ⫽ predicted partially coupled solute flux
including only implicit diffusive coupling;
J ␲, j ⫽ chemico-osmotic advective flux of solute
species j;
K d, j ⫽ distribution coefficient of solute species j;
k h ⫽ hydraulic conductivity;
k ␲ ⫽ chemico-osmotic permeability coefficient
(⫽␻k h );
M j ⫽ moles of solute species j;
N ⫽ number of solute species in solution;
n ⫽ porosity;
Q j ⫽ total moles of solute species j per unit total
volume of porous medium;
q h ⫽ Darcy liquid flux (⫽k h i h );
q ␲ ⫽ chemico-osmotic liquid flux;
R ⫽ universal gas constant 共8.314 J/mol K兲;
R d, j ⫽ retardation factor of solute species j;
T ⫽ absolute temperature;
t ⫽ time;
V e ⫽ effluent sample volume of liquid;
v h ⫽ hydraulic seepage velocity (⫽q h /n);
v ␲ ⫽ chemico-osmotic seepage velocity (⫽q ␲ /n);
x ⫽ direction of transport;
z ⫽ ion valence;
␲ ⫽ chemico-osmotic pressure;
␳ d ⫽ dry density of the soil;
␳ w ⫽ density of water;
␶ a ⫽ apparent tortuosity factor (0⭐␶ a ⭐1); and
␻ ⫽ chemico-osmotic
efficiency
coefficient
(0⭐␻⭐1).
References
Barbour, S. L., and Fredlund, D. G. 共1989兲. ‘‘Mechanisms of osmotic
flow and volume change in clay soils.’’ Can. Geotech. J., 26, 551–
562.
Bresler, E. 共1973兲. ‘‘Simultaneous transport of solutes and water under
transient unsaturated flow conditions.’’ Water Resour. Res., 9共4兲, 975–
986.
Foose, G. J., Benson, C. H., and Edil, T. B. 共2002兲. ‘‘Comparison of
solute transport in three composite liners.’’ J. Geotech. Geoenviron.
Eng., 128共5兲, 391– 403.
Freeze, R. A., and Cherry, J. A. 共1979兲. Groundwater, Prentice–Hall,
Englewood Cliffs, N.J.
Fritz, S. J., and Marine, I. W. 共1983兲. ‘‘Experimental support for a predictive osmotic model of clay membranes.’’ Geochim. Cosmochim.
Acta, 47, 1515–1522.
Greenberg, J. A., Mitchell, J. K., and Witherspoon, P. A. 共1973兲.
‘‘Coupled salt and water flows in a groundwater basin.’’ J. Geophys.
Res., 78共27兲, 6341– 6353.
Katchalsky, A., and Curran, P. F. 共1965兲. Nonequilibrium thermodynamics
in biophysics, Harvard University Press, Cambridge, Mass.
Katsumi, T., Benson, C. H., Foose, G. J., and Kamon, M. 共2001兲.
‘‘Performance-based design of landfill liners.’’ Eng. Geol. (Amsterdam), 60共1– 4兲, 139–148.
Kemper, W. D., and Quirk, J. P. 共1972兲. ‘‘Ion mobilities and electric
charge of external clay surfaces inferred from potential differences
and osmotic flow.’’ Soil Sci. Soc. Am. Proc., 36, 426 – 433.
Kemper, W. D., and Rollins, J. B. 共1966兲. ‘‘Osmotic efficiency coefficients across compacted clays.’’ Soil Sci. Soc. Am. Proc., 30, 529–
534.
Kim, J. Y., Edil, T. B., and Park, J. K. 共2001兲. ‘‘Volatile organic compound 共VOC兲 transport through compacted clay.’’ J. Geotech. Geoenviron. Eng., 127共2兲, 126 –134.
Malusis, M. A. 共2001兲. ‘‘Membrane behavior and coupled solute transport
through a geosynthetic clay liner.’’ PhD dissertation, Colorado State
Univ., Fort Collins, Colo.
Malusis, M. A., and Shackelford, C. D. 共2002a兲. ‘‘Chemico-osmotic efficiency of a geosynthetic clay liner.’’ J. Geotech. Geoenviron. Eng.,
128共2兲, 97–106.
486 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2004
Malusis, M. A., and Shackelford, C. D. 共2002b兲. ‘‘Coupling effects during
steady-state solute diffusion through a semipermeable clay membrane.’’ Environ. Sci. Technol., 36共6兲, 1312–1319.
Malusis, M. A., and Shackelford, C. D. 共2002c兲. ‘‘Theory for reactive
solute transport through clay membrane barriers.’’ J. Contam. Hydrol.,
59共3– 4兲, 291–316.
Malusis, M. A., Shackelford, C. D., and Olsen, H. W. 共2001兲. ‘‘A laboratory apparatus to measure chemico-osmotic efficiency coefficients for
clay soils.’’ Geotech. Test. J., 24共3兲, 229–242.
Manassero, M., and Dominijanni, A. 共2003兲. ‘‘Modeling the osmosis effect on solute migration through porous media.’’ Geotechnique, 53共5兲,
481– 492.
Mitchell, J. K. 共1993兲. Fundamentals of soil behavior, 2nd Ed., Wiley,
New York.
Olsen, H. W. 共1969兲. ‘‘Simultaneous fluxes of liquid and charge in saturated kaolinite.’’ Soil Sci. Soc. Am. Proc., 33, 338 –344.
Olsen, H. W., Yearsley, E. N., and Nelson, K. R. 共1990兲. ‘‘Chemicoosmosis versus diffusion-osmosis.’’ Transportation Research Record
1288 Transportation Research Board, Wasington, D.C., 15–22.
Rabideau, A., and Khandelwal, A. 共1998兲. ‘‘Boundary conditions for
modeling transport in vertical barriers.’’ J. Environ. Eng., 124共11兲,
1135–1139.
Robinson, R. A., and Stokes, R. H. 共1959兲. Electrolyte solutions, 2nd Ed.,
Butterworth Scientific Publications, London.
Rowe, R. K. 共1987兲. ‘‘Pollutant transport through barriers.’’ Geotechnical
practice for waste disposal, GSP No. 26, R. D. Woods, ed., ASCE,
New York, 159–181.
Rowe, R. K. 共1998兲. ‘‘Geosynthetics and the minimization of contaminant migration through barrier systems beneath waste.’’ Proc., 6th Int.
Conf. on Geosynthetics, Atlanta, Industrial Fabrics Association Inter-
national, St. Paul, Minn., 27–102.
Shackelford, C. D. 共1988兲. ‘‘Diffusion as a transport process in finegrained barrier materials.’’ Geotech. News, 6共2兲, 24 –27.
Shackelford, C. D. 共1990兲. ‘‘Transit time design of earthen barriers.’’ Eng.
Geology, (Amsterdam), 29共2兲, 79–94.
Shackelford, C. D. 共1994兲. ‘‘Critical concepts for column testing.’’ J.
Geotech. Eng., 120共10兲, 1804 –1828.
Shackelford, C. D. 共1997兲. ‘‘Modeling in environmental geotechnics: applications and limitations.’’ Proc., 2nd Int. Congress on Environmental Geotechnics, IS-Osaka ’96, M. Kamon, ed., Vol. 3, Balkema, Rotterdam, The Netherlands, 1375–1404.
Shackelford, C. D., and Daniel, D. E. 共1991兲. ‘‘Diffusion in saturated soil:
I. Background.’’ J. Geotech. Eng., 117共3兲, 467– 484.
Shackelford, C. D., Benson, C. H., Katsumi, T., Edil, T. B., and Lin, L.
共2000兲. ‘‘Evaluating the hydraulic conductivity of GCLs permeated
with non-standard liquids.’’ Geotext. Geomembr., 18共2– 4兲, 133–161.
Smith, G. D. 共1985兲. Numerical solutions of partial differential equations: finite difference methods, 3rd Ed., Oxford University Press, Oxford, U.K.
Staverman, A. J. 共1952兲. ‘‘Nonequilibrium thermodynamics of membrane
processes.’’ Trans. Faraday Soc., 48共2兲, 176 –185.
Toupiol, C., Willingham, T. W., Valocchi, A. J., Werth, C. J., Krapac, I.
G., Stark, T. D., and Daniel, D. E. 共2002兲. ‘‘Long-term tritium transport through field-scale compacted soil liner.’’ J. Geotech. Geoenviron. Eng., 128共8兲, 640– 650.
Yeung, A. T. 共1990兲. ‘‘Coupled flow equations for water, electricity and
ionic contaminants through clayey soils under hydraulic, electrical,
and chemical gradients.’’ J. Non-Equil. Thermodyn., 15, 247–267.
Yeung, A. T., and Mitchell, J. K. 共1993兲. ‘‘Coupled fluid, electrical, and
chemical flows in soil.’’ Geotechnique, 43共1兲, 121–134.
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2004 / 487