rate-limited cation exchange in thin bentonitic barrier layers...rate-limited cation exchange in...

Rate-limited cation exchange in thin bentoniticbarrier layers

Ho Young Jo, Craig H. Benson, and Tuncer B. Edil

Abstract: A three-compartment model was developed for simulating cation transport in bentonitic barrier layers thatincorporates diffusion-controlled cation exchange among the mobile intergranular water (bulk pore water), immobileinterparticle and interlayer water, and the montmorillonite mineral solid. Exchange on the external surfaces andinterlayer region of montmorillonite is included. The model was evaluated for divalent-for-monovalent cation exchangein bentonite with experiments. A parametric study was conducted using the model to investigate factors affecting thetime required to establish chemical equilibrium (i.e., completion of cation exchange) between the permeant liquid andthin layers of bentonite simulating geosynthetic clay liners (GCLs). Predictions obtained with the model were in gen-eral agreement with the data without calibration, except for Na concentrations in the effluent at very long times. Para-metric simulations conducted with the model show that the time required to establish chemical equilibrium in GCLs isaffected by the rate at which adsorbing cations are delivered to the pore space (affected by seepage velocity or influentconcentration), the rate of mass transfer between the mobile and immobile liquid phases (controlled primarily by gran-ule size of the bentonite), and the number of sites available for sorption (controlled by CEC and the dry density of thebentonite).

Key words: bentonite, montmorillonite, exchange complex, diffusion, immobile liquid, interlayer.

Résumé : On a développé un modèle à trois compartiments pour simuler le transport de cations dans des couchesd’étanchéité bentonitiques qui incorpore l’échange de cations contrôlé par la diffusion entre l’eau libre intergranulaire(eau interstitielle globale), l’eau interparticule et intercouche fixe, et le minéral solide de montmorillonite. L’échangesur les surface externes et dans la région intercouche de montmorillonite est inclus. Le modèle a été évalué pour unéchange de cations bivalents pour monovalents dans la bentonite au cours d’expériences. On a réalisé une étude para-métrique au moyen du modèle pour évaluer les facteurs affectant le temps requis pour établir l’équilibre chimique (i.e.,l’achèvement de l’échange de cations) entre le liquide percolant et les minces couches de bentonite simulant les mem-branes de géosynthétique d’argile (GCLs). Les prédictions obtenues avec le modèle étaient en concordance généraleavec les données sans calibrage, sauf pour les concentrations de Na dans l’effluent à des temps très longs. Des simula-tions paramétriques conduites avec le modèle montrent que le temps requis pour atteindre l’équilibre chimique dans lesGCLs est influencé par la vitesse à laquelle les cations adsorbants atteignent l’espace interstitiel (affectée par la vitessede l’infiltration ou la concentration de l’affluent), la vitesse de transfert de masse entre les phases libre et adsorbée(contrôlée principalement par la grosseur des granules de bentonite), et le nombre de sites disponibles pour la sorption(contrôlé par la CEC et la densité sèche de bentonite).

Mots clés : bentonite, montmorillonite, complexe d’échange, diffusion, liquide adsorbé, intercouche.

[Traduit par la Rédaction] Jo et al. 391

Introduction

Thin layers of bentonite encased between geosynthetics,referred to as geosynthetic clay liners (GCLs), are used ashydraulic barriers in liner systems for waste containment fa-cilities to minimize the discharge of contaminated liquids(Estornell and Daniel 1992; Koerner 1997). A typical GCLis 5–7 mm thick when dry and contains 3.5–4.5 kg/m2 of drybentonite sandwiched between two carrier geotextiles. In

some cases the bentonite is glued to a geomembrane or ageomembrane is laminated to one of the carrier geotextiles.The bentonite consists primarily of the mineral montmorillo-nite, which has high specific surface area, high cationexchange capacity (CEC), and low hydraulic conductivity(≈ 10–9 cm/s) to deionized (DI) water (Mitchell 1993;Shackelford et al. 2000). Most GCLs are filled with sodium(Na) bentonite, meaning that the exchange complex of themontmorillonite is dominated by Na+ ions.

Can. Geotech. J. 43: 370–391 (2006) doi:10.1139/T06-014 © 2006 NRC Canada

370

Received 4 August 2004. Accepted 14 November 2005. Published on the NRC Research Press Web site at http://cgj.nrc.ca on14 March 2006.

H.Y. Jo. Department of Earth and Environmental Sciences, Korea University, Anam-dong, Seongbuk-gu, Seoul, 136–713, Korea.C.H. Benson1 and T.B. Edil. Geological Engineering Program, University of Wisconsin-Madison, 1415 Engineering Drive,Madison, WI 53706, USA.

1Corresponding author: (e-mail: [email protected]).

The hydraulic conductivity of GCLs without a geomem-brane is directly related to the physical properties (e.g.,swelling) and chemical composition (e.g., exchangeable cat-ions) of the montmorillonite fraction of the bentonite(Egloffstein 1997; Shackelford et al. 2000; Jo et al. 2001). Aparticular concern is the gradual replacement of Na+ ionsinitially existing on the montmorillonite surface bymultivalent ions (e.g., Ca2+) in the permeant liquid, whichcan result in increases in hydraulic conductivity of an orderof magnitude or more. When the permeant solution is dilute(<25 mmol/L) and primarily contains divalent cations, theincrease in hydraulic conductivity occurs very slowly andcontinues until the exchange of divalent ions for Na+ ions iscomplete. Laboratory tests by Shackelford et al. (2000),Egloffstein (2001), Lee (2004), and Jo et al. (2005) indicatethat several years can be required to reach equilibrium insome cases. The slow rate of cation exchange is believed tobe due to the limited rate at which cations diffuse in and out

of the interlayer region of the montmorillonite mineral(Ogwada and Sparks 1986; McBride 1994; Jo et al. 2004).

The primary objective of this paper is to describe andevaluate a three-compartment model for simulating cationtransport in GCLs that incorporates diffusion-controlled cat-ion exchange. Experimental data are used to evaluate themodel. Predictions made with the model are used to illus-trate factors affecting cation exchange and the time requiredto establish chemical equilibrium between the permeant so-lution and bentonite in GCLs, as might be encountered inlaboratory column tests or field settings.

Background

Bentonite structure and ion exchangeIn general, GCLs consist of granular Na–bentonite

(Fig. 1a). Each granule is an aggregation of edge-to-facemontmorillonite particles (Fig. 1b) (van Olphen 1991;

© 2006 NRC Canada

Jo et al. 371

Fig. 1. (a) Schematic representation of a GCL (a), (b) flow and transport pathway in the intergranular pores of bentonite, (c) diffusionbetween mobile and immobile liquid phases in the intergranular and interparticle pore spaces, and (d) diffusion between immobile liq-uid phases in the interparticle and interlayer pore spaces. GT, geotextile.

Mitchell 1993; Pusch 1999), with each particle comprised ofstacks of a layered 2:1 aluminum silicate (Fig. 1c) (Sposito1984; Yong 1999; Bourg et al. 2003; Kröhn 2003). The porespace in saturated granular bentonite can be assumed to con-sist of intergranular (ig), interparticle (ip), and interlayer (il)spaces (Fig. 1). The pores between the granules constitutethe intergranular pore space. The interparticle pore space ex-ists within the granules and between the particles, but out-side the interlayer space between the montmorillonitelamella (i.e., the 2:1 octahedral and tetrahedral sheets). Wa-ter in the intergranular pore space is assumed to be hydrauli-cally mobile. Water in the interparticle and interlayer poresis assumed to be strongly bound by electrostatic forces andimmobile (Cheung et al. 1986; Pusch and Hokmark 1990;Pusch 1999).

Ion exchange is assumed to occur as cations in thepermeant solution pass through the intergranular pores andgradually diffuse into the bentonite granules (interparticlepores) and interlayer space (Ogwada and Sparks 1986;Cheung et al. 1986; McBride 1994). Ion exchange pro-gresses until equilibrium is established between ions in thepermeant solution and the montmorillonite surface (Grim1968; van Olphen 1991; Bourg et al. 2003; McBride 1994).Diffusion between the intergranular, interparticle, andinterlayer pore spaces is assumed to control the rate of ionexchange and to be responsible for the extensive tailing ofeluted cations that is often observed during column testsconducted on GCLs with inorganic solutions (Lee 2004; Joet al. 2004).

Aggregated soil modelThe conventional method of analyzing transport of

sorbing solutes through GCLs and other barrier systems con-sists of incorporating a retardation factor into the advection–diffusion equation. Linear, reversible, and instantaneoussorption of solutes is generally assumed (Acar and Haider1990; Shackelford 1993; Smith and Jaffe 1994; Rowe et al.2000; Foose et al. 2002). However, this approach ignores themechanisms affecting mass transfer into the interparticle andinterlayer pore spaces where the majority of sorption occurs,as well as competition for sorption sites. As a result, theconventional approach fails to describe the very long timesrequired to reach equilibrium, or the persistent tailing of ad-sorbing and desorbing cations that can occur during columntests on GCLs (Jo et al. 2004).

An alternative approach is to account explicitly for themechanisms involved in ion exchange in montmorillonite(i.e., diffusion in the immobile water and cation exchange inthe interparticle and interlayer pore spaces). These mecha-nisms are similar to those occurring in aggregated soils (i.e.,aggregates consisting of clay, silt, or sand particles bound to-gether), where diffusion in immobile water within the aggre-gates is known to influence solute transport (Green et al.1972; Skopp and Warrick 1974; van Genuchten andWierenga 1976; van Eijkeren and Loch 1984; Wood et al.1990). Transport in aggregated soils is assumed to occur intwo phases, namely the mobile liquid in the larger inter-aggregate pores and the immobile liquid in microporeswithin the aggregates. Water in the micropores is assumed tobe “immobile” because the small size of these pores makesadvection negligible. As a result, transport in the micropores

occurs primarily via diffusion. The influence of the slow rateof diffusive transport in the micropores is manifested at themicroscale as tailing in the breakthrough curve. As the ag-gregates become larger, the pathways for diffusion in themicropores become longer, and the breakthrough curve ex-hibits greater and more persistent tailing (van Genuchtenand Wierenga 1976; Rao et al. 1980; van Eijkern and Loch1984). Because of the similarity between the transport mech-anisms in GCLs and aggregated soils, the aggregated soilmodel was used as the basis for developing the model in thisstudy.

Methods to describe the transport processes in aggregatedsoils containing mobile and immobile water have been pro-posed and described by various investigators (Dean 1963;Coats and Smith 1964; Skopp and Warrick 1974; vanGenuchten and Wierenga 1976; van Eijkeren and Loch 1984).These studies have used the advection–diffusion equationwith an additional component describing transport betweenthe mobile and immobile water phases.

van Genuchten and Wierenga (1976) describe an analyti-cal solution for transport of solutes through sorbing aggre-gated soils containing immobile and mobile liquid phases(i.e., a two-compartment model). In their model, diffusionoccurs between the mobile and immobile liquids accordingto the concentration difference between the two liquids.Sorption occurs at sites in both the mobile and immobile re-gions, but competition for adsorption sites is ignored. Diffu-sion into the immobile liquid continuously attenuates solutesfrom the bulk solution at a slow rate, resulting in a break-through curve with extensive tailing. van Genuchten andWierenga (1976) show that an increase in the fraction of im-mobile liquid results in breakthrough or elution curves hav-ing greater and more persistent tailing. They also show that adecrease in the mass transfer coefficient between the immo-bile and mobile phases exacerbates tailing caused by theslower exchange of solutes between mobile and immobileliquids.

van Eijkeren and Loch (1984) describe a two-compartment numerical model for transport of solutesthrough aggregated soils that is based on a modification ofthe model by van Genuchten and Wierenga (1976). Themodel combines advective–diffusive transport in the inter-aggregate pores, mass transfer between the mobile and im-mobile liquid phases by diffusion, and a reaction formonovalent-for-monovalent cation exchange. Results of theirsimulations show that advective–diffusive transport com-bined with diffusion of cations between the mobile and im-mobile liquid phases is manifested at the macroscale as anapparent retardation of the solute in the effluent. Cationexchange also results in additional apparent retardation ofsolute and more extensive and persistent tailing of the break-through or elution curve.

Mathematical model

Mass transport and cation exchange in GCLs were mod-eled using an approach similar to that in van Eijkeren andLoch (1984). However, in the current study, the pore spacehas a three-component structure and divalent cations replacemonovalent cations on the solid surface. Transport ofcationic species in the intergranular pores is assumed to oc-

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372 Can. Geotech. J. Vol. 43, 2006

cur because of advective–diffusive transport under saturatedwater flux conditions. Liquid is assumed to flow onlythrough the intergranular pores (i.e., advective mass trans-port only occurs in the mobile liquid). Hydrodynamic dis-persion is assumed to be negligible because of the very lowseepage velocity and the small pore diameters in bentonite(Gillham and Cherry 1982; Malusis and Shackelford 2002).Osmotic barrier effects are also assumed to be negligible.Shackelford and Lee (2003) show that osmotic effectsquickly degrade (<0.1 year) when divalent cations diffuseinto a GCL, whereas very long times (e.g., several years) arerequired for exchange to reach completion (Egloffstein2001; Jo et al. 2005).

Diffusion of cationic species is assumed to occur withinthe mobile liquid (Fig. 1b), between the mobile and immo-bile liquids (e.g., as in van Genuchten and Wierenga 1976;Rao et al. 1980; Tan et al. 1981; van Eijkeren and Loch1984; Parker and Valocchi 1986) (Fig. 1c), and within theimmobile liquid (i.e., between the interparticle pores andinterlayer pores) (e.g., as in Wu and Gschwend 1986)(Fig. 1d). Diffusion of cationic species in or between liquidphases is assumed to be proportional to a difference in con-centration (van Genuchten and Wierenga 1976; Rao et al.1980; Tan et al. 1981; van Eijkeren and Loch 1984; Parkerand Valocchi 1986).

Sorption sites are assumed to exist on the surfaces in boththe interparticle pores and the interlayer pores (vanGenuchten and Wierenga 1976; Wu and Gschwend 1986).Instantaneous exchange of cations is assumed to occur be-tween the mineral solid and the immobile water in theinterparticle and interlayer pores. However, the overall rateof exchange is limited by the rate at which cations diffuseinto the interparticle and interlayer pores.

Mass conservationThe equations governing mass transport were derived as-

suming that the control volume (VT) can be divided into

three compartments: the mobile liquid phase in the intergra-nular (ig) pore space, the immobile liquid phase in theinterparticle (ip) pore space, and the immobile liquid phasein the interlayer (il) pore space (Fig. 2). Mass transfer be-tween the compartments only occurs by diffusion and therate of mass transfer is assumed to be proportional to thedifference in concentration between compartments. Diffu-sion between the mobile intergranular and the immobileinterparticle liquids is described by

[1] nC

tC Cig

iggp ig ip

∂∂

α= − −( )

Similarly, diffusion between the immobile interparticleand interlayer liquid phases is described by

[2] nC

tC Cip

ippl ip il

∂∂

α= − −( )

where nig is the intergranular porosity, nip is the interparticleporosity, Cig is the molar concentration in the mobile inter-granular liquid (mol/L), Cip is the molar concentration in theimmobile interparticle liquid (mol/L), Cil is the molar con-centration in the immobile interlayer liquid (mol/L), αgp isthe mass transfer coefficient for diffusion between the mo-bile intergranular and the immobile interparticle liquids(s–1), αpl is the mass transfer coefficient for diffusion be-tween the immobile interparticle and interlayer liquids (s–1),and t is time. A similar formulation was used by vanGenuchten and Wierenga (1976) in their two-compartmentmodel.

For the intergranular pore space (i.e., mobile liquid phase),the rate of cation accumulation is

[3]∂

∂∂

∂∂

∂∂∂ 2

M

tV n

C

tV q

C

tn D

C

xig

T igig

Tig

ig eig= = − +

⎛

⎝⎜⎜

⎞

⎠⎟

2

⎟

− −V C CT gp ig ipα ( )

© 2006 NRC Canada

Jo et al. 373

Fig. 2. Schematic representation of the control volume.

where Mig is the number of moles in the mobile intergranularliquid (mol), q is the Darcy velocity (cm/s), and De is the ef-fective diffusion coefficient for each cation species in theintergranular pores (cm2/s). For the interparticle pore space(i.e., immobile liquid phase), the rate of cation accumulationis

[4]∂

∂∂

∂α

M

tV n

C

tV C Cip

T ipip

T gp ig ip= = −( )

− − −V C CM

tT pl ip il

s, ipα∂

∂( )

where Mip is the number of moles in the immobileinterparticle liquid (mol) and Ms,ip is the number of molessorbed on the solid in the interparticle pore space (mol).Similarly, for the interlayer (il) pore space (i.e., immobileliquid phase), the rate of cation accumulation is

[5]∂

∂∂∂

α∂

∂M

tV n

Ct

V C CM

til

T ilil

T pl ip ils, il= = − −( )

where Mil is the number of moles in the immobile interlayerliquid (mol), Ms,il is the number of moles sorbed on the solidin the interlayer pore space (mol), and nil is the interlayerporosity.

The right-most term of eqs. [4] and [5] is the rate ofchange of the cationic species (in moles) on the solid surfacein the interparticle and interlayer pore spaces, respectively.The term for the interparticle pore space can be rewritten as

[6]∂

∂∂∂

∂∂

∂∂

M

t

M

M

M

tR

M

ts, ip s, ip

ip

ipip

ip= =

Similarly, for the interlayer pore space

[7]∂

∂∂∂

∂∂

∂∂

M

t

M

MM

tR

Mt

s, il s, il

il

ilil

il= =

In eqs. [6] and [7], Rip is a partition coefficient betweenthe interparticle liquid and solid surface, and Ril is a partitioncoefficient between the interlayer liquid and solid surface.Equations [1]–[7] are written for the adsorbing cation (AC)and desorbing cation (DC) (i.e., two sets of equations).Electroneutrality in each of the phases is implicitly assumed.

Liquid–solid cation exchangeCation exchange between the solid and the immobile

interparticle liquid can be written as a cation exchange reac-tion for a system with monovalent desorbing cations (DC)and divalent adsorbing cations (AC) (i.e., as the case for Na–bentonite permeated by solutions containing divalent cat-ions). This reaction is written as (Allison et al. 1991;McBride 1994; Parkhurst and Appelo 1999)

[8] 2M M Ms, ip,DC ip,AC s, ip,ACSO− + ⇔ + 2 ip,DCM

where SO designates the solid, Ms,ip,AC is the number ofmoles of the AC species on the solid in the interparticlespace (mol), Ms,ip,DC is the number of moles of the DC spe-cies on the solid in the interparticle space (mol), Mip,AC isthe number of moles of the AC species in the interparticleliquid (mol), and Mip,DC is the number of moles of the DC

species in the interparticle liquid (mol). Electroneutrality inthe liquid phase is implicitly assumed when eq. [8] is ap-plied.

When the activity coefficients are assumed to be unity, theequilibrium constant for the reaction (Keq, ip* ) in eq. [8] canbe written as

[9] Kn

X M

X Meq, ip

ip

ip,AC ip,DC

ip,DC ip,AC

*( )( )

( ) ( )= 1 2

2

where Xip,AC and Xip,DC are the mole fractions of the AC andDC, respectively, on the solid in the interparticle space. Theformulation for the equilibrium constant in eq. [9] is thesame formulation used by Fletcher and Sposito (1989) andTang and Sparks (1993).

A similar reaction can be written for the cation exchangebetween the solid and immobile interlayer liquid. For unitactivity coefficients, the equilibrium coefficient for this reac-tion (Keq, il* ) is

[10] Kn

X M

X Meq, il

il

il,AC il,DC

il,DC il,AC

* ( )( )

( ) ( )= 1 2

2

where Xil,AC and X il,DC are the mole fractions of the AC andDC, respectively, on the solid in the interlayer pore spaceand Mip,AC and Mip,DC are the number of moles of the ACand DC, respectively, in the interlayer liquid.

The total number of moles sorbed on the solid in theinterparticle pore space (Ms, ip* ) is given by

[11] M f M M Ms, ip ip s, t s, ip,AC s, ip,DC* *= = +2

where fip is the fraction of sorption sites on the solid in theinterparticle pore space and Ms, t* is the total number of molessorbed on the solid (mol). The term Ms, t* is equal to the productof the CEC and the dry density of the bentonite, ρd (Mg/m3).Similarly, the total number of moles in the interlayer porespace (Ms, il* ) is

[12] M f M M Ms, il ip s, t s, il,AC s, il,DC* ( ) *= − = +1 2

Combining eqs. [9] and [11] yields

[13] 4 2 2 2K M M M M Meq, ip ip,AC s, t s, ip,AC ip,DC s, t( * ) ( ) [( ) ( * )−

+ 4 2K f M M Meq, ip ip s, t ip,AC s, ip,AC( * ) ]( )

+ =K f M Meq, ip ip s, t ip,AC( * )2 0

© 2006 NRC Canada


where Keq,ip = nipKeq, ip* . The number of moles of AC species sorbed on the solid interparticle pore space (Ms,ip,AC) is obtainedby re-arranging eq. [13] as follows:

[14] MK f M M M K f

s, ip,ACeq, ip ip s, t ip,AC ip,DC eq, ip=

+ +4 82* ( ) [ ip s, t ip,AC ip,DC ip,DC

eq, ip s, t i

M M M M

K M M

* ( ) ( ) ]

*

/2 4 1 2

8

+

p,AC

Equation [14] can be differentiated with respect to Mip,AC to obtain the partition coefficient between interparticle liquid andsolid surface in the interparticle pore space for the AC (Rip,AC)

[15] RM

Mip,AC

s, ip,AC

ip,AC

=∂∂

=+ +4 8K f M K f M M Meq, ip ip s, t eq, ip ip s, t ip,AC ip,D0.5* {[ * ( C ip,DC eq, ip ip s, t ip,DC

eq,

) ( ) ] [ * ( ) ]}/2 4 1 2 28

8

+ −M K f M M

K ip ip,AC s, tM M*

−+ +8 4 82K K f M M M K feq, ip eq, ip ip s, t ip,AC ip,DC eq, ip{ * ( ) [ ip s, t ip,AC ip,DC ip,DC

eq, ip ip,A

M M M M

K M

* ( ) ( ) ] }

(

/2 4 1 2

8

+

C s, t) *2 M

A similar procedure is used for cation exchange between the solid and the immobile interlayer liquid for the AC species.This procedure yields

[16] MK f M M M M

s, il,ACeq, il ip s, t il,AC il,DC il=

− + +4 1 2( ) * ( ) [( ,DC eq, il ip s, t il,AC il,DC

eq, i

) ( ) * ( ) ] /4 2 1 28 1

8

+ −K f M M M

K l s, t il,ACM M*

and

[17] RM

Mil,AC

s, il,AC

il,AC

=∂∂

=− + − +4 1 8 1K f M K f M Meq, il ip s, t eq, il ip s, t il,0.5( ) * {[ ( ) * AC il,DC il,DC eq, il ip s, t il( ) ( ) ] [ ( ) * (/M M K f M M2 4 1 2 8 1+ −−

,DC

eq, il il,AC

) ]}2

8K M

−− + +8 4 1 82K K f M M M Keq, il eq, il ip s, t il,AC il,DC eq{ ( ) * ( ) [ , il ip s, t il,AC il,DC il,DC

eq

( ) * ( ) ( ) ] }

(

/1

8

2 4 1 2− +f M M M M

K , il il,AC s, tM M) *2

where Keq,il = nilKeq, il* . The rate of accumulation of the AC species in the immobile interparticle liquid can be obtained bycombining eq. [4] and eq. [6] as follows:

[18]∂

∂∂

∂α α

M

tR

M

tV C C Vip,AC

ip,ACip,AC

T gp ig,AC ip,AC T+ = − −( ) pl ip,AC il,AC( )C C−

Similarly, the rate of accumulation of the AC species in the immobile interlayer liquid can be obtained by

[19]∂

∂∂

∂α

M

tR

M

tV C Cil,AC

il,ACil,AC

T pl ip,AC il,AC+ = −( )

The rate of accumulation of the DC species on the solid in the interparticle pore space can be obtained by differentiatingeq. [11] with respect to time (t) as follows:

[20]∂

∂∂

∂∂∂

M

t

M

t

M

Ms, ip,DC s, ip,DC s, ip,AC

ip,

= −⎛

⎝⎜⎜

⎞

⎠⎟⎟ = −2 2

AC

ip,ACip,AC

ip,AC⎛

⎝⎜⎜

⎞

⎠⎟⎟ = −

∂∂

∂∂

M

tR

M

t2

Combining eqs. [4] and [20] yields

[21]∂

∂∂

∂α

M

tR

M

tV C C Vip,DC

ip,ACip,AC

T gp ig,DC ip,DC T− = − −2 ( ) αpl ip,DC il,DC( )C C−

Similarly, the rate of accumulation of the DC species in the immobile interlayer liquid can be obtained by

© 2006 NRC Canada

Jo et al. 375

[22]∂

∂∂

∂α

M

tR

M

tV C Cil,DC

il,ACil,AC

T pl ip,DC il,DC− = −2 ( )

Concentration formulaThe rate of cation accumulation in the liquid phases in eqs. [3], [4], and [5] (i.e., the time derivative on the left hand side)

can be rewritten in terms of molar concentrations.For the AC species:

[23] nC

tn D

C

xq

C

xig

ig,ACig e,AC

ig,AC ig,AC∂∂

∂∂

∂∂

= −⎛

⎝⎜⎜

⎞

⎠⎟

2

2 ⎟ − −αgp,AC ig,AC ip,AC( )C C

[24] n RC

tC Cip ip,AC

ip,ACgp,AC ig,AC ip,AC pl,A( ) ( )1 + = − −

∂∂

α α C ip,AC il,AC( )C C−

and

[25] n RC

tC Cil il,AC

il,ACpl ip,AC il,AC( ) ( )1 + = −

∂∂

α

Similarly, for the DC species:

[26] nC

tn D

C

xq

C

xig

ig,DCig e,DC

ig,DC ig,DC∂∂

∂∂

∂∂

= −⎛

⎝⎜⎜

⎞

⎠⎟

2

2 ⎟ − −αgp,DC ig,DC ip,DC( )C C

[27] nC

tR n

C

tC Cip

ip,DCip,AC ip

ip,ACgp,DC ig,DC ip

∂∂

∂∂

α− = −2 ( ,DC pl,DC ip,DC il,DC) ( )− −α C C

and

[28] nC

tR n

C

til

il,DCip,AC il

il,AC= −∂

∂∂

∂2

= −αpl ip,DC il,DC( )C C

Solving this set of six coupled equations provides a de-scription of AC and DC transport in GCLs when divalent-for-monovalent cation exchange is occurring. This formula-tion implicitly assumes that electroneutrality exists in eachof the liquid phases. This assumption was satisfied by usingsalt diffusion coefficients as input, as described in Malusisand Shackelford (2002).

Equations [23]–[28] were solved by applying the mass-conserving third type flux (Dankwerts) boundary conditionsat the upper and lower surface of the GCL (x = 0 and x = L)(Shackelford 1994; Khandelwal et al. 1998; Rabideau andKhandelwal 1998). For the AC species, the boundary condi-tions are

[29] qC t n Dx

C t qCig,AC ig e,AC ig,AC AC, in( , ) ( , )0 0+ +− =∂∂

and

[30] qC L t n Dx

C L t qCig,AC ig e,AC ig,AC AC,e( , ) ( , )− −− =∂∂

where CAC,in is the influent concentration of the AC species(mol/L), CAC,e is the effluent concentration of the ACspecies (mol/L), and + and – indicate limits from above andbelow. Similarly, the boundary conditions for the DC speciesare

[31] qC t n Dx

C t qCig,DC ig e,DC ig,DC DC, in( , ) ( , )0 0+ +− =∂∂

and

[32] qC L t n Dx

C L t qCig,DC ig e,DC ig,DC DC,e( , ) ( , )− −− =∂∂

where CDC,in is the influent concentration of the DC species(mol/L) and CDC,e is the effluent concentration of the DCspecies (mol/L).

Solution method and verificationThe transport and cation exchange equations were solved

sequentially by using an operator-splitting scheme (Valocchiand Malmstead 1992; Miller and Rabideau 1993; Parkhurstand Appelo 1999). A finite-difference method was used tosolve the governing equations. A block-centered scheme wasused for spatial discritization and a Crank–Nicholsonscheme was used for temporal discritization (Peaceman1977; Wang and Anderson 1982; Zheng and Bennet 1995).Peaceman (1977) shows that the Crank–Nicholson schemefor the advection–diffusion equation is accurate to the sec-ond order and is unconditionally stable. The nodal spacingwas set at 0.1 mm and the time step at 0.001 d, which wereadequate to minimize dispersion, prevent oscillation, andprovide accuracy when using the operator-splitting scheme(Jo 2003). The finite-difference equations were solved by us-ing the successive over-relaxation (SOR) method, which issimilar to the Gauss–Seidel iteration method except the re-sidual is multiplied by a relaxation factor to accelerate con-vergence (Wang and Anderson 1982; Strikwerda 1989).

© 2006 NRC Canada


A check of the model was made by comparing predictedbreakthrough curves to breakthrough curves reported by vanEijekern and Loch (1984). Because the van Eijekern andLoch (1984) model consists of only two liquid phases (mo-bile and immobile), the comparison was made assuming thatthe bentonite consisted only of intergranular pores filledwith the mobile liquid and interparticle pores filled with theimmobile liquid (i.e., nt = nig + nip and nil = 0). The ex-change reaction was excluded because van Eijekern andLoch’s model simulates monovalent-for-monovalent cationexchange, whereas the model in this study simulates diva-

lent-for-monovalent cation exchange. Breakthrough curvesfor both models are shown in Fig. 3 along with the input pa-rameters that were used. The predicted breakthrough curvesclosely resemble those reported by van Eijeken and Loch(1984), and both models show that breakthrough occurs ear-lier as the mass transfer coefficient between the mobile andimmobile liquid (αAC) decreases because more of thesorbing solute bypasses the immobile phase.

A comparison was also made between breakthroughcurves predicted by the model and breakthrough curves foran analytical solution of the advection–diffusion equation

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Jo et al. 377

Fig. 3. Relative AC effluent concentration obtained from simulations with the model compared with predicted data reported by vanEijkeren and Loch (1984) when the mass transfer coefficients of AC are (a) 1.2 × 10–4 s–1 and (b) 1.2 × 10–5 s–1.

(van Genuchten and Alves 1982) by setting αgp = αpl = 0.Nearly identical breakthrough curves were obtained for thiscomparison as well (Jo 2003).

Experiments

BentoniteThe bentonite used in the study was obtained from a GCL

that contained 4.5 kg/m2 of air dry granular sodium benton-ite. The average initial gravimetric water content of the ben-tonite was 9% and the specific gravity of solids was 2.65.X-ray diffraction showed that the bentonite consisted pri-marily of montmorillonite (80%), but also contained quartz,opaline silica, feldspars, calcite, illite, and mica (Jo et al.2005). The granule sizes ranged from 0.1 mm to 0.4 mm,and sedimentation analysis (conducted per ASTM D 422 asdescribed in ASTM 2002) indicated that the fraction of clay-size particles in the bentonite (<0.002 mm) was 88%.

The CEC of the bentonite was determined by the ammo-nium acetate method (Rhoades 1982a). Electrical conductiv-ity (EC) and pH of the bentonite were measured on pastesprepared with deionized (DI) water (Rhoades 1982b). TheCEC ranged between 53 and 75 mequiv./100g (seven tests),the paste pH was 8.9 ± 0.1 (four tests), and the paste EC was300 ± 16 mS/m (four tests). The exchange complex of thebentonite was determined by the ammonium acetate method(Thomas 1982) with extract analyzed by using a flameatomic absorption spectrometer (FAA) in accordance withUSEPA (1994). In the initial state, the exchange complexof the bentonite was dominated by sodium (eight tests:Na = 43.8 ± 16.8 mequiv./100g, Ca = 16.5 ±5.4 mequiv./100g, Mg = 6.0 ± 3.4 mequiv./100g, and K =0.8 ± 0.4 mequiv./100g).

Permeation proceduresColumn tests were conducted on thin layers of bentonite

simulating GCLs. A series of six replicate tests was con-ducted using a 20 mmol/L CaCl2 solution (pH 6.5 ± 0.1,EC = 437 ± 7 mS/m) as the permeant liquid, which has beenshown to cause gradual replacement of Na on bentonite(Egloffstein 2001; Jo et al. 2004; 2005; Lee 2004). The testswere terminated sequentially after 2, 5, 10, 15, 40, and 110pore volumes of flow (PVF) to investigate the temporal be-havior of the exchange complex of the bentonite. Becausetests to determine the exchange complex are destructive, thespecimens could not be permeated again after being termi-nated to characterize the bentonite. Thus, the number of rep-licate tests decreased as the PVF increased.

The tests were conducted in flexible-wall permeametersfollowing the procedures described in Jo et al. (2004) andASTM (2002). A specimen was prepared by placing a thinlayer of bentonite inside an acrylic ring fixed to the bottomcap of a flexible-wall permeameter. Each specimen had a drythickness equal to 7 mm and mass per unit area of 4.5 kg/m2

so as to simulate the GCL from which the bentonite was ob-tained. After placement in the ring, the bentonite was moist-ened lightly using a spray bottle to provide some cohesion.Then the acrylic ring was removed and the top cap and latexmembrane were installed.

No backpressure was applied during the tests so that efflu-ent samples for chemical analyses (i.e., pH, EC, and cationconcentrations) could be collected conveniently. Cell and in-fluent pressures were applied using gravity reservoirs tominimize fluctuations caused by variable gas pressure. Theaverage effective stress was 16.7 kPa and the average hy-draulic gradient (i) was 130. The specimens were hydratedwith the 20 mmol/L CaCl2 solution in the permeameter for48 h without application of a hydraulic gradient to allowsome swelling of the bentonite before permeation. Perme-ation with the 20 mmol/L CaCl2 solution began immediatelyafter the hydration period. Some cation exchange probablyoccurred during the initial hydration period, however, theamount of exchange that occurred is unknown because theexchange complex was not examined immediately afterhydration. The amount of exchange is believed to be smallgiven the very slow rate at which exchange occurs and thelow hydraulic conductivity of the bentonite that existed atthe onset of permeation.

Effluent samples were regularly collected for chemicalanalyses (i.e., pH, EC, and Na and Ca concentrations). Wheneach test was terminated, the exchange complex of the ben-tonite was determined using the method described previ-ously. The pH and EC of the effluent samples weremeasured using a pH meter and an electrical conductivityprobe. The Na and Ca concentrations were measured usingFAA.

The hydraulic conductivity, ratio of effluent to influent EC(ECout/ECin), effluent pH, and Ca and Na effluent concentra-tions are shown in Fig. 4. The hydraulic conductivities dif-fered by at most a factor of 1.3 at a given PVF, indicatinggood reproducibility. In addition, the effluent concentration(i.e., Ca and Na), ratio of effluent to influent EC, and efflu-ent pH obtained from all tests conducted with the samepermeant solution are similar at a given PVF (Fig. 4). Thus,properties (e.g., hydraulic conductivity and exchange com-plex) of the bentonite specimens obtained from the sequen-tially terminated tests were assumed to be representative ofthe general changes in the properties of the bentonite as afunction of PVF.

Comparison of predictions andexperimental data

Model inputsA comparison was made between predictions obtained

with the model and data obtained from the column tests.Material and solute properties used as input to the model aresummarized in Table 1. Physical properties of the bentonite(thickness, dry density, CEC, and initial exchange complex)were specified using values obtained from the experiments.Initial concentrations of Ca2+ and Na+ in the pore water ofthe test specimens were not measured. Thus, the Ca2+ andNa+ concentrations were set to zero in all phases of the porewater in an attempt to avoid an un-intended bias in themodel. The equilibrium constant (K) was set at 1.5 based onrecommendations in Tang and Sparks (1993) for Na–Ca ex-change on montmorillonite, and Keq, ip* and Keq, il* were as-sumed to be the same. The Darcy velocity was varied in timeaccordance with the temporal variations in hydraulic con-

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ductivity obtained from the hydraulic conductivity tests(Fig. 4). The fraction of sorption sites in the interparticlepore space (fip) was assumed to be equal to the ratio of thesurface area in the interparticle space relative to the totalsurface area. Each montmorillonite layer was assumed to be100 nm × 10 nm × 1 nm (Mitchell 1993), and each particlewas assumed to consist of three montmorillonite layers(Yong 1999). For this geometry, fip equals 0.4. Effective dif-fusion coefficients for Ca2+ and Na+ were estimated from thesalt diffusion coefficients in free water (Dso) and an apparent

tortuosity (τa) equal to 0.14 for salt diffusion in GCLs, as re-ported by Shackelford and Lee (2003).

The intragranular porosity (ng = nip + nil) of the bentonitespecimen permeated with the 20 mmol/L solution was com-puted by the following formula (Mitchell 1993; Achari et al.1999):

[33] ndG S

dG Sg

s L a

s L a

=+

ρρ1

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Jo et al. 379

Fig. 4. Data from tests conducted on bentonite with the 20 mmol/L CaCl2 solution as permeant liquid: (a) hydraulic conductivity,(b) ratio of effluent to influent EC, (c) effluent pH, and (d) effluent Ca and Na concentrations. T1, T2, T3, T4, T5, and T6 designatepoints of sequential termination.

where d is the half-distance between two layers (nm), Gs isthe specific gravity of montmorillonite, ρL is the density ofthe solution in the pores (Mg/m3), and Sa is the specific sur-face area of hydrated montmorillonite (750 m2/g, Mitchell1993). The half-distance (d) was estimated from free swelldata following the method suggested by Smalley (1994)

[34] daVV

L= −⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

s

cm

where Lm is the thickness of montmorillonite layer (≈ 0.9 nm,McBride 1994), a is the basal spacing (d001) of Ca–montmorillonite layers (≈1.96 nm, McBride 1994) in water,Vc is the free swell volume of Ca–montmorillonite in water(≈ 8.0 mL/2 g, Jo et al. 2004), and Vs is the free swell vol-ume of montmorillonite in the permeant solution(18.0 mL/2 g in the 20 mmol/L solution, Jo et al. 2004). Ahalf distance (d) of 1.4 nm (20 mmol/L CaCl2 solution) wascomputed using eq. [34], and a corresponding intragranularporosity (ng) of 0.24 was computed with eq. [33]. The inter-granular porosity (nig = nt – ng) was set at 0.53 (= 0.77 –0.24). The intragranular porosity was assumed to be equallydivided between the interparticle porosity and the interlayerporosity.

The parameter αgp was estimated using the following for-mula (van Genuchten and Dalton 1986):

[35] αgpg

e,g=15

2

n

rD

where De,g is the effective diffusion coefficient for the gran-ule (cm2/s) and r is the radius of the granule (mm). Equation[35] was developed for diffusion into porous spheres filledwith immobile water and has been used to estimate masstransfer coefficients in the immobile phase of aggregatedsoils (Gerke and van Genuchten 1993; Gwo et al. 1995;Schwartz et al. 2000). The radius in eq. [35] was set at0.3 mm to capture the expected granule size in hydrated ben-tonite (Jo et al. 2001). The effective diffusion coefficient forthe granule was set between 1.0 × 10–8 and 1.0 × 10–9 cm2/sbased on measurements reported by Cheung et al. (1986) forimmobile water in bentonite. Because of the very small sizeof interlayer pores, αpl was assumed to be approximatelytwo orders of magnitude lower than αgp. Wu and Gschwend(1986) also report similar differences in mass transfer coeffi-cients for two-phase immobile water systems.

For simplicity, the mass transfer coefficients were assumedto be the same regardless of the type of cation (i.e., adsorb-ing and desorbing cations), even though different diffusioncoefficients could be used in eq. [35] to represent differencesin ionic mobility (Li and Gregory 1974; Shackelford andDaniel 1991). The error incurred with this assumption issmall (<15%) because of the similarity in the salt diffusioncoefficients for the chloride salts of Na and Ca (Malusis andShackelford 2002).

Model predictionsComparisons between model predictions and data from

the hydraulic conductivity tests are shown in Fig. 5. Predic-tions are shown for three cases. The heavy lines in Fig. 5correspond to the input parameters described previously (re-ferred to as “base case” parameters), which are summarizedin Table 1.

The predictions obtained with the model for the base caseparameters are generally comparable to the data, eventhough the parameters were estimated independently (i.e.,the parameters are not based on a calibration). The predictedCa breakthrough curve for the base case is comparable to themeasured Ca breakthrough curve (Fig. 5a). The model alsopredicts the changes in the exchange complex with reason-able accuracy (Fig. 5c). However, the predicted Na concen-tration is comparable to the measured Na concentration onlyuntil 45 PVF. Subsequently, the predicted Na concentrationdrops rapidly, whereas the measured Na concentration con-tinues to decrease slowly (Fig. 5b).

The mass transfer coefficients αgp and αpl were varied todetermine if the tailing observed after 45 PVF could be pre-dicted with different parameters. However, as in previoussimulations, αgp and αpl for Na and Ca were the same(αgp, Ca = αgp, Na and αpl, Ca = αpl, Na). An initial analysis wasconducted by setting αgp = αpl = 4.0 × 10–5 s–1 or 6.0 ×10–8 s–1. This analysis showed that tailing in the Na elutioncurve past 45 PVF could be simulated by reducing αpl, butalso showed that capturing the shape of the breakthroughand Na elution curves as well as the changes in the exchangecomplex required different values be assigned to αgp and αpl(with αpl < αgp). That is, the analysis suggests that bothphases of immobile water need to be included to capture the

© 2006 NRC Canada


Co (mmol/L) 20

Thickness (mm) 10Radius of granulea (mm) 0.3ρd (Mg/m3) 0.65

Keq* (= Keq,ip* = Keq,il* )b 1.5

K (cm/s) varying with timeHydraulic gradient (i) 100DSO, CaCl 2

c (cm2/s) 1.3×10–5

DSO,NaClc (cm2/s) 1.6×10–5

τad 0.14

De,Ca (cm2/s) 1.8×10–6

De,Na (cm2/s) 2.2×10–6

nte 0.77

nig 0.53

nip 0.12

nil 0.12

fip 0.40

αgp,Ca (s–1) 4.0×10–5

αgp,Na (s–1) 4.0×10–5

αpl,Ca (s–1) 4.0×10–7

αpl,Na (s–1) 4.0×10–7

CECe (mequiv./100g) 60.0SCa

e (mequiv./100g) 10.0

SNae (mequiv./100g) 50.0

aJo et al. (2001).bTang and Sparks (1993).cShackelford and Daniel (1991).dShackelford and Lee (2003).eJo et al. (2004).

Table 1. Summary of material and chemical inputsto model.

temporal trends associated with the slow exchange processin bentonite.

Subsequent analyses were conducted with αpl initially setat 4.0 × 10–7 s–1, which provided reasonably good fits to theearly portion of the elution curve (Fig. 5b), and αpl between1.0 × 10–7 and 6.0 × 10–8 s–1 for times greater than 45 PVF.As shown in Fig. 5b, decreasing αpl for times greater than45 PVF results in more persistent tailing of Na and reducing

αpl to 6.0 × 10–8 s–1 (less than one order of magnitude) re-sults in very similar tailing as was observed experimentally.Similar curves are not shown in Fig. 5a because the highconcentration of Ca in the effluent at 45 PVF (and thereaf-ter) obscured the small differences in Ca concentrationcaused by reducing αpl. The better match obtained for Naelution with lower αpl may reflect collapse of the interlayerspace due to replacement of Na+ by Ca2+. A smaller

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Jo et al. 381

Fig. 5. Comparison between data from experiments and model predictions. (a) Ca concentrations in the effluent, (b) Na concentrationsin the effluent, and (c) mole fractions of Ca (XCa) and of Na (XNa) as a function of time for the 20 mmol/L CaCl2 solution (αgp,Ca =αgp,Na and αpl Ca = αpl,Na).

interlayer space will make the transport pathways narrowerand more tortuous, which should cause a reduction in themass transfer rate. Also, molecules remaining in this narrowspace are more strongly bound, which may further restrictdiffusion.

Parametric analysis

Simulations were conducted with the model to investigatefactors affecting the time required to establish chemicalequilibrium (i.e., complete cation exchange) during columntests on bentonite where divalent-for-monovalent exchangeoccurs. The influent was assumed to contain a single speciesof divalent cations (e.g., Ca2+) and the GCL was assumed tocontain a 10 mm thick layer of hydrated Na–bentonite. Atthe start of a test, soluble salts were assumed to be absentfrom the liquid phases, and the exchange complex was as-sumed to be homoionic. The hydraulic conductivity (K), ef-fective diffusion coefficient (De), the mass transfercoefficients between the different water phases (αgp and αpl),the initial mole fraction of Na on the exchange complex(XNa), the dry density of the bentonite layer (ρd), the CEC,and the initial Ca concentration (Co) were varied within ex-pected ranges to determine how they affect effluent concen-trations and the exchange complex.

Mass transport in intergranular porosityThe effect of mass transport in the mobile water in the

intergranular pore space was evaluated by varying theadvective and diffusive transport components in the intergra-nular (bulk) pore water. Sensitivity to advection was evalu-ated by varying the hydraulic conductivity between 3.8 ×10–8 cm/s and 3.8 × 10–10 cm/s, which is a typical range forthe hydraulic conductivity of bentonite permeated with di-lute solutions with divalent cations (Jo et al. 2005). The po-rosities nt and nig were varied along with the hydraulicconductivity (i.e., as the hydraulic conductivity increased, ntdecreased and nig increased) because nig controls the hydrau-lic conductivity of bentonite. The influence of diffusivetransport was evaluated using De = 2.0 × 10–5, 2.0 × 10–6,and 2.0 × 10–7 cm2/s for both the AC and DC.

Increasing advection causes earlier breakthrough of theAC (Fig. 6a) and more rapid reduction in the effluent con-centration for the DC (Fig. 6b) because the AC is deliveredmore rapidly to the pore space. The more rapid elution of Nais also evident in the exchange complex (i.e., ion exchangeoccurs more rapidly as advection increases) (Fig. 6c) and di-minished tailing of the elution curve for the DC. However,even when advection is high, tailing still exists in the break-through and elution curves because of the slow rate at whichthe AC and DC diffuse in the immobile interparticle andinterlayer liquid phases.

The influence of bulk diffusive transport is shown inFig. 7. For the typical and high De (i.e., De ≥ 2.0 × 10–6 cm2/s),diffusion in the mobile liquid has only a modest effect onthe breakthrough and elution curves and the rate of changein the exchange complex (Fig. 7). However, decreasing De to2.0 × 10–7 cm2/s causes an apparent retardation of the AC inthe breakthrough curve and more tailing of the DC in theelution curve. These changes occur because reducing therate of bulk diffusion causes the limiting step in mass trans-

fer to shift from the immobile water to the mobile water.However, given that bulk diffusion coefficients for divalentcations are typically greater than 10–6 cm2/s for the bentonitein GCLs, diffusion in the immobile water should control therate of exchange in most practical applications.

Interphase mass transferThe effect of the interphase mass transfer coefficients was

evaluated by varying αpl between 2.0 × 10–5 and 4.0 × 10–7 s–1

while holding αgp constant at 2.0 × 10–5 s–1 and varying αgpbetween 2.0 × 10–4 and 4.0 × 10–6 s–1 while holding αpl con-stant at 4.0 × 10–7 s–1. The mass transfer coefficients for theAC and DC were assumed to be the same. Results of thesimulations are shown in Fig. 8.

Decreasing αpl causes earlier breakthrough of the AC andgreater tailing in the DC elution curve because the rate ofmass transfer to and from the interlayer pore space is lower,resulting in less exchange at the solid surface (Figs. 8a and8b). As a result, more mass of the AC passes through thebentonite without interacting with the mineral surface result-ing in slower ion exchange (Fig. 8c). Decreasing αgp causesearlier breakthrough of the AC and more tailing of the DC inthe elution curve. However, the influence of αgp is small inthe range between 2.0 × 10–4 and 4.0 × 10–6 s–1 because con-trol on mass transport in the interlayer exchange sites is con-trolled primarily by the lower αpl (4.0 × 10–7 s–1).

Availability of sorption sitesThe availability of sorption sites depends on the dry den-

sity (ρd) and CEC of the bentonite as well as on the molefraction of Na initially on the exchange complex (XNa). Eachof these variables affects the exchange complex in a similarmanner (Jo 2003). Thus, only results for XNa are reported(Fig. 9). As XNa increases, breakthrough of the AC occursmore slowly (Fig. 9a) and tailing in the elution curve be-comes more significant (Fig. 9b). Ion exchange also occursover a longer period when XNa is larger (Fig. 9c) as moresites are available for sorption. Increasing ρd or CEC causesa similar effect, i.e., delayed breakthrough of AC, more tail-ing of the DC in the elution curve, and slower ion exchange(Jo 2003).

Practical implications

Establishing chemical equilibrium is one of the most im-portant criteria in column tests conducted to evaluate thecompatibility between a liquid to be contained and a barriersoil such as the bentonite in a GCL. The parametric studyhas shown that the rate at which divalent-monovalent cationexchange occurs in GCLs depends on the rate at which theAC is delivered to the pore space via advection, the rate ofmass transfer between the liquid phases (primarily αpl), andthe number of sites available for sorption (i.e., ρd, CEC, orXNa). The effect that each of these factors has on the time re-quired to establish chemical equilibrium during a columntest on a GCL with a Ca solution as the permeant liquid isshown in Figs. 10 and 11. Equilibrium was defined as XNa =0, XCa = 1.0, and negligible concentration of the DC in theeffluent (less than 0.2 mg/L as defined by Jo et al. 2005).

Simulations were conducted using K = 3.0 × 10–8, 3.0 ×10–9, and 1.0 × 10–9 cm/s and Co = 10, 20, and 40 mmol/L to

© 2006 NRC Canada


assess the effect of the rate at which the AC is delivered tothe pore space. The parameter αpl was set at 4.0 × 10–7 s–1

(less than 45 PVF) and 6.0 × 10–8 s–1 (less than 45 PVF) asshown in Fig. 5. Results of these simulations are shown inFigs. 10a and 10b. The time required to establish chemical

equilibrium increases as the hydraulic conductivity or influ-ent concentration decreases because both affect the rate of atwhich AC becomes available for exchange. This effect hasbeen observed experimentally as well (Jo et al. 2005). Testsconducted by Jo et al. (2005) on a GCL containing granular

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Jo et al. 383

Fig. 6. Effect of advection in the intergranular pores on effluent concentrations of the (a) adsorbing cation (AC), (b) desorbing cation(DC), and (c) mole fractions of AC and DC on the exchange complex.

sodium bentonite using a hydraulic gradient of 130 showedthat establishing chemical equilibrium required 780 d for a5 mmol/L CaCl2 solution, 680 d for a 10 mmol/L CaCl2 so-lution, and 560 d for a 20 mmol/L CaCl2 solution.

The most significant factor affecting the mass transfer co-efficients for the immobile liquid phases in a GCL is the

granule size (eq. [35]). Thus, simulations were conductedwith αgp = 2.0 × 10–4, 2.0 × 10–5, and 4.0 × 10–6 s–1 corre-sponding to granule sizes of 0.1, 0.4, and 0.7 mm (eq. [35]),respectively. The parameter �pl was set at 4.0 × 10–7 s–1 forall simulations because resistance to mass transfer in theinterlayer is not expected to be influenced by granule size.

© 2006 NRC Canada


Fig. 7. Effect of effective diffusion coefficient (De) on effluent concentrations of (a) adsorbing cation (AC), (b) desorbing cation (DC),and (c) mole fractions of AC and DC on the exchange complex.

Results of these simulations are shown in Fig. 10c. The timerequired to establish chemical equilibrium increases as thegranule size increases (i.e., as αgp decreases). However, thesensitivity to granule size is modest because the time toequilibrium is controlled more by αpl than αgp (Fig. 8). Be-cause most GCLs contain bentonite granules ranging in size

between 0.1 and 1.0 mm (Shackelford et al. 2000), granulesize should only have a modest effect on the time required toestablish equilibrium. However, the effect may be more pro-nounced than shown in Fig. 10c because GCLs with smallergranule sizes tend to be less permeable when permeated withsolutions containing divalent cations (Gleason et al. 1997;

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Jo et al. 385

Fig. 8. Effect of mass transfer rate coefficients on effluent concentrations of (a) adsorbing cation (AC), (b) desorbing cation (DC), and(c) mole fractions of AC and DC on the exchange complex.

Katsumi et al. 2002). This synergistic effect of granule sizeand hydraulic conductivity was not included in the simula-tions.

The time required to establish chemical equilibrium alsoincreases with increasing CEC, initial XNa, and ρd of the ben-tonite (Fig. 11) because each is proportional to the number

of available sites for exchange. Thus, GCLs that containgreater mass per area of bentonite or higher grade bentonite(larger CEC or XNa) will require more time to reach equilib-rium during a hydraulic conductivity test. Experiments de-scribed by Egloffstein (2001) illustrate this behavior.Egloffstein (2001) conducted hydraulic conductivity tests

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Fig. 9. Effect of initial mole fraction of Na (XNa) on the exchange complex on effluent concentrations of (a) adsorbing cation (AC),(b) desorbing cation (DC), and (c) mole fractions of AC and DC on the exchange complex.

using a 12.5 mmol/L CaCl2 solution on two GCLs with dif-ferent mass per unit area. Both GCLs contained granular so-dium bentonite, but the exchange complex was not reported.

Approximately 1200 d was required to establish equilibriumfor the GCL having a mass per unit area of 8.0 kg/m2,whereas approximately 700 d was required to establish equi-

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Jo et al. 387

Fig. 10. Effect of (a) hydraulic conductivity (K), (b) influent concentration, Co, and (c) granule size on the time required to establishchemical equilibrium.

librium for the GCL having a mass per unit area of4.0 kg/m2.

Summary and conclusions

A three-compartment model was developed for simulatingcation transport and exchange in GCLs that incorporates dif-fusion-controlled cation exchange. The model was evaluatedwith experimental measurements and a parametric study wasconducted to determine those factors having a significant ef-fect on the rate of exchange and the time required to reachequilibrium during column tests on GCLs. Predictions fromthe model using uncalibrated input parameters were gener-ally comparable to data, except at long times. Reducing themass transfer coefficient for the interlayer water at timegreater than 45 PVF resulted in better agreement at longtimes. Results of the parametric simulations showed that themost important variables affecting the rate of cation ex-change are advection in the intergranular pore space, theinterlayer mass transfer coefficient, and the number of sorp-tion sites on the mineral surface.

Parametric simulations conducted with the model illus-trate that long testing times may be required when conduct-ing column tests on GCLs to evaluate compatibility with asolution containing divalent cations. The most significanttesting variables affecting the time required to establishequilibrium include the flow rate and concentration of thepermeant liquid because both affect the rate at which diva-lent cations are delivered to the pore space. Quality andmass per unit area of the bentonite also have a significant ef-fect on the time required to reach equilibrium. Thus, fortests conducted with typical hydraulic gradients (100–200for GCLs), longer test times should be expected when the

GCL has lower hydraulic conductivity (e.g., GCLs that areprehydrated or GCLs with higher quality bentonite) or theGCL has more sites for cation exchange (higher grade ben-tonite with larger CEC or more Na on the exchange com-plex).

Acknowledgements

Support for this study was provided by the United StatesNational Science Foundation (NSF) under grant no. CMS-9900336. The findings, opinions, and conclusions reportedin this paper are solely those of the authors. Endorsement byNSF is not implied and should not be assumed. CharlesShackelford is acknowledged for his thoughtful contribu-tions to the study.

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Fig. 11. Effect of dry density of bentonite (ρd), CEC, and initial mole fraction of Na (XNa) on the time required to establish chemicalequilibrium.

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List of symbols

a basal spacing (d001) of Ca-montmorillonite layers(~1.96 nm) in water

AC adsorbing cationC effluent concentration (mol/L)

CAC,e effluent concentration of AC species (mol/L)CAC,in influent concentration of AC species (mol/L)CDC,e effluent concentration of DC species (mol/L)

CDC,in influent concentration of DC species (mol/L)CEC cation exchange capacity (mequiv./100g)

Cig concentration of cation species the mobile intergranularliquid (mol/L)

Cig,AC concentration of AC species in the mobile intergranularliquid (mol/L)

Cig,DC concentration of DC species in the mobile intergranularliquid (mol/L)

Cil concentration of cation species in the immobileinterlayer liquid (mol/L)

Cil,AC concentration of AC species in the immobile interlayerliquid (mol/L)

Cil,DC concentration of DC species in the immobile interlayerliquid (mol/L)

Cin influent concentration (mol/L)Cip concentration of cation species in the immobile

interparticle liquid (mol/L)Cip,AC concentration of AC species in the immobile

interparticle liquid (mol/L)Cip,DC concentration of DC species in the immobile

interparticle liquid (mol/L)Co influent concentration (mol/L)

d half-distance between two montmorillonite layers (nm)DC desorbing cationDe effective diffusion coefficient for each cation species in

intergranular pores (cm2/s)De,AC effective diffusion coefficient for AC species in intergra-

nular pores (cm2/s)De,Ca effective diffusion coefficient for Ca in intergranular

pores (cm2/s)De,DC effective diffusion coefficient for DC species in intergra-

nular pores (cm2/s)De,g effective diffusion coefficient in the granule (cm2/s)

© 2006 NRC Canada


De,Na effective diffusion coefficient for Na in intergranularpores (cm2/s)

Dso salt diffusion coefficient in free water (cm2/s)EC electrical conductivity

ECin influent ECECout effluent EC

fip fraction of sorption sites on the solid in the interparticlepore space

GCL geosynthetic clay linerGs specific gravity of montmorillonite

i hydraulic gradientig intergranularil interlayerip interparticleK hydraulic conductivity

Keq* equilibrium constantKeq,il nil Keq,il*Keq,il* equilibrium constant for reaction between immobile liq-

uid and solid in interlayer poresKeq,ip nip Keq,ip*Keq,ip* equilibrium constant for reaction between immobile liq-

uid and solid in interparticle poresL thickness of the soil specimen (cm)

Lm thickness of montmorillonite layer (≈ 0.9 nm)Ms,il* total number of moles of cation species sorbed on the

solid in the interlayer pore space (mol)Ms,ip* total number of moles of cation species sorbed on the

solid in the interparticle pore space (mol)Ms,t* total number of moles of cation species sorbed on the

solid (mol)Mig number of moles of cation species in the mobile inter-

granular liquid (mol)Mil number of moles of cations species in the immobile

interlayer liquid (mol)Mil,AC number of moles of AC species in the immobile

interlayer liquid (mol)Mil,DC number of moles of DC species in the immobile

interlayer liquid (mol)Mip number of moles of cation species in the immobile

interparticle liquid (mol)Mip,AC number of moles of AC species in the immobile

interparticle liquid (mol)Mip,DC number of moles of DC species in the immobile

interparticle liquid (mol)Ms,il number of moles of cations sorbed on the solid in the

interlayer pore space (mol)Ms,il,AC number of moles of AC species sorbed on the solid in

the interlayer pore space (mol)Ms,il,DC number of moles of DC species sorbed on the solid in

the interlayer pore space (mol)Ms,ip number of moles of cations sorbed on the solid in the

interparticle pore space (mol)Ms,ip,AC number of moles of AC species sorbed on the solid in

the interparticle pore space (mol)Ms,ip,DC number of moles of DC species sorbed on the solid in

the interparticle pore space (mol)ng intragranular porositynig intergranular porositynil interlayer porositynip interparticle porositynt total porosity

q Darcy’s velocity (cm/s)r radius of granule (mm)

Ril partition coefficient of cation species between interlayerliquid and solid surface

Ril,AC partition coefficient of AC species between interlayerliquid and solid surface

Rip partition coefficient of cation species between inter-particle liquid and solid surface

Rip,AC partition coefficient of AC species between interparticleliquid and solid surface

Sa specific surface area of hydrated montmorillonite (cm2/g)SCa sorbed concentration of Ca on the solid (mequiv.uiv./100g)SNa sorbed concentration of Na on the solid (mequiv.uiv./100g)SO solid

t timeVc free swell volume of Ca-montmorillonite in water

(≈ 8.0 mL/2g)Vs free swell volume of montmorillonite in the permeant

solution (mL/2g)VT control volume

x distanceXAC mole fraction of AC species on the exchange complexXCa mole fraction of Ca on the exchange complexXDC mole fraction of DC species on the exchange complex

Xil,AC mole fraction of AC species on the solid in the inter-layer pore space

Xil,DC mole fraction of DC species on the solid in the inter-layer pore space

Xip,AC mole fraction of AC species on the solid in the inter-particle pore space

Xip,DC mole fraction of DC species on the solid in the inter-particle pore space

XNa mole fraction of Na on the exchange complexαAC mass transfer coefficient of AC species (s–1)αgp mass transfer coefficient for diffusion between mobile

intergranular and the immobile interparticle liquids (s–1)αgp,AC mass transfer coefficient of AC species for diffusion

between mobile intergranular and the immobile inter-particle liquids (s–1)

αgp,Ca mass transfer coefficient of Ca for diffusion betweenmobile intergranular and the immobile interparticle liq-uids (s–1)

αgp,DC mass transfer coefficient of DC species for diffusionbetween mobile intergranular and the immobile inter-particle liquids (s–1)

αgp,Na mass transfer coefficient of Na for diffusion betweenmobile intergranular and the immobile interparticle liq-uids (s–1)

αpl mass transfer coefficient for diffusion between immo-bile interparticle and interlayer liquids (s–1)

αpl,AC mass transfer coefficient of AC species for diffusion be-tween immobile interparticle and interlayer liquids (s–1)

αpl,Ca mass transfer coefficient of Ca for diffusion betweenimmobile interparticle and interlayer liquids (s–1)

αpl,DC mass transfer coefficient of DC species for diffusion be-tween immobile interparticle and interlayer liquids (s–1)

αpl,Na mass transfer coefficient of Na for diffusion betweenimmobile interparticle and interlayer liquids (s–1)

ρd dry density of the bentonite (Mg/m3)ρL density of the pore solution (Mg/m3)τa apparent tortuosity

© 2006 NRC Canada

Jo et al. 391

rate-limited cation exchange in thin bentonitic barrier layers...rate-limited cation exchange in...

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