1 Abichou et al 04 on SBM model

Network model for hydraulic conductivity of sandbentonite mixtures

Tarek Abichou, Craig H. Benson, and Tuncer B. Edil

Abstract: A network formulation was used to model the hydraulic conductivity of sandbentonite mixtures (SBMs) asa function of bentonite content. The sand particles were assumed to be spheres, and their arrangement was defined us-ing a discrete element model simulating sand particle interactions. Pores between the spheres were approximated as anetwork of straight capillary tubes. The space defined by the spheres was divided into a collection of neighboring tetra-hedrons, and the geometry of the tetrahedrons was used to define tube diameters and lengths in the network. Hydraulicheads throughout the network were computed by solving a system of equations describing flow through the tubes. Hy-draulic conductivity of the network was calculated as the rate of flow per unit area for a given network of tubes drivenby a one-dimensional hydraulic gradient. Bentonite was introduced into the network in several schemes to simulateSBMs. SBMs prepared with powdered bentonite were modeled as a packing of sand, where the sand particles arecoated with bentonite (grain coating model and pipe blocking model), whereas SBMs prepared with granular bentonitewere modeled as a packing of sand with bentonite occupying pores between the sand particles (junction blockingmodel). The relationship between hydraulic conductivity and bentonite content obtained from the grain coating modelwas similar to that measured on sand powdered bentonite mixtures. A comparable relationship was also obtained forhydraulic conductivities predicted with the junction blocking model using a size-based filling approach and hydraulicconductivities measured on sand granular bentonite mixtures.

Key words: sandbentonite mixtures, network models, hydraulic conductivity, degree of bentonation, bentonite distribu-tion.

Rsum : On a utilis une formulation de rseau pour modliser la conductivit hydraulique des mlanges de sable-bentonite (MBSs) en fonction de la teneur en bentonite. On a suppos que les particules de sable taient sphriques, etleur arrangement a t dfini au moyen dun modle dlments discrets simulant les interactions entre les particules desable. On a considr que les pores entre les sphres se rapprochaient approximativement dun rseau de tubes capillai-res droits. Lespace dfini par les sphres a t divis en une collection de ttradres voisins et la gomtrie des ttra-dres a t utilise pour dfinir les diamtres et les longueurs des tubes dans le rseau. Les charges hydrauliques dans lerseau ont t calcules en partant de la solution dun systme dquations dcrivant lcoulement travers les tubes.La conductivit hydraulique du rseau a t calcule comme tant le dbit par unit de surface pour un rseau donnde tubes soumis un gradient hydraulique unidimensionnel. On a introduit de la bentonite dans le rseau de diffrentesmanires pour simuler des MBSs. Des MBSs prpars avec de la bentonite en poudre ont t modliss comme unbourrage de sable o les particules de sable ont t enrobes de bentonite (Modle denrobage des grains et Modle deblocage des tubes), dans lesquels les MBSs prpars avec de la bentonite en grains ont t modliss comme un bour-rage de sable avec la bentonite occupant les pores entre les particules de sable (Modle de jonction de blocage). La re-lation entre la conductivit hydraulique et la teneur en bentonite obtenue du modle denrobage des grains taitsemblable celle mesure sur les mlanges de sable et de poudre de bentonite. On a aussi obtenu une relation compa-rable pour les conductivits hydrauliques prdites avec le modle de jonction de blocage en utilisant une approche deremplissage base sur les dimensions et les conductivits hydrauliques mesures sur des mlanges de sable et de bento-nite en grains.

Mots cls : mlanges sable-bentonite, modles de rseau, conductivit hydraulique, degr de caractre bentonitique, dis-tribution de bentonite.

[Traduit par la Rdaction] Abichou et al. 712

Can. Geotech. J. 41: 698712 (2004) doi: 10.1139/T04-016 2004 NRC Canada

698

Received 15 April 2003. Accepted 20 January 2004. Published on the NRC Research Press Web site at http://cgj.nrc.ca on25 August 2004.

T. Abichou.1 Department of Civil and Environmental Engineering, Florida State University, Tallahassee, FL 32310, USA.C.H. Benson and T.B. Edil. Department of Civil and Environmental Engineering, University of Wisconsin-Madison,2214 Engineering Hall, 1415 Engineering Drive, Madison, WI 53706, USA.1Corresponding author (e-mail: [email protected]).

Introduction

Almost all studies on the hydraulic conductivity of sandbentonite mixtures (SBMs) focus on their use as hydraulicbarriers (Lundgren 1981; Chapuis 1981, 1990; Abeele 1986;Daniel 1987; Garlanger et al. 1987; Kenney et al. 1992;OSadnick et al. 1995; Gleason et al. 1997; Howell andShackelford 1997; Kraus et al. 1997; and others). The objec-tive generally is to determine the quantity of bentonite re-quired to obtain suitably low hydraulic conductivity. Thehydraulic conductivity of SBMs typically ranges from 1 106 to 1 109 cm/s, with lower hydraulic conductivitiesgenerally associated with higher bentonite contents.

The traditional approach to determine the required ben-tonite content is experimental. A series of specimens is pre-pared, their hydraulic conductivity is measured, and then thebentonite content to achieve the target hydraulic conductivityis obtained from the experimental data. A second approachis to predict the relationship between hydraulic conductivityand bentonite content using a model that simulates pores fill-ing with bentonite. A limited number of tests are then con-ducted to check that the bentonite content predicted from themodel is appropriate. A model of this sort can also be usedto understand how the hydraulic conductivity of clean sandtransitions to that of clay as the pores in the sand fill withbentonite.

A network model is described in this paper that relates thehydraulic conductivity of SBMs prepared with clean uniformsands and powdered or granular bentonite to the bentonitecontent. The model was developed to understand how thehydraulic conductivity of SBMs is related to changes inmicrostructure that occur as the pores in the sand are filledwith bentonite. The model consists of extracting a pore net-work from packings of spheres, introducing bentonite intothe network using different schemes, and computing the hy-draulic conductivity of the network.

Models of SBMs

Models for predicting the hydraulic conductivity of SBMsgenerally assume that the mixture is ideal, i.e., bentonitefills all the pore spaces between the sand particles. The sandparticles are assumed to be nonconductive, and the hydraulicconductivity of the bentonite controls the hydraulic conduc-tivity of the mixture (Chapuis 1990; Kenney et al. 1992;Mollins et al. 1996; Sllfors and berg-Hgsta 2002).

Chapuis (1990) describes an ideal-mixture model wherethe swollen bentonite is assumed to fill all of the pore spacebetween the sand particles as water becomes available. Aportion of the water is assumed to be fixed to the surface ofbentonite particles, rendering the water immobile. The re-maining water is also associated with the bentonite but ismobile. The mobile water represents the portion of the porespace available for flow, which is referred to as the efficientporosity. Chapuis proposed two empirical equations relatinghydraulic conductivity to efficient porosity for ideal mix-tures and compared hydraulic conductivities predicted by theequations with data for SBMs prepared with mostly uniformsoils having 2%15% nonplastic fines and bentonite.

Kenney et al. (1992) theorize that an ideal SBM is a two-component mixture of sand and saturated bentonite where

all of the water is associated with the bentonite. The matrixof the saturated bentonite is assumed to be continuous andthe sand particles are assumed to be impervious inclusionsin the bentonite matrix. The fabric of the bentonite is as-sumed to be unaffected by the presence of sand particles.Under these conditions, the hydraulic conductivity of themixture is controlled by the hydraulic conductivity of thebentonite fraction. Kenney et al. also indicate that the hy-draulic conductivity of the bentonite is a unique functionof its void ratio. Thus, the hydraulic conductivity of idealSBMs can be computed as the product of the total porosityof the mixture and the hydraulic conductivity of the benton-ite at its void ratio in the mixture.

Sllfors and berg-Hgsta (2002) investigate the effectsof degree of compaction on the hydraulic conductivity ofSBMs. They use the ideal-mixture concept to derive a newindex quantity, k1, which combines the dry unit weight of themixture and the bentonite content. SBMs were prepared withpowdered bentonite and well-graded medium sand. The pa-rameter k1 was calculated for all mixtures and correlatedwith hydraulic conductivity.

Mollins et al. (1996) developed empirical power law rela-tionships between the void ratio of the bentonite in an SBMand the vertical effective stress. The power laws were cali-brated using data from one-dimensional swelling and hy-draulic conductivity tests on SBMs containing 5%, 10%, and20% bentonite at varying effective stress. The SBMs wereprepared with a silty fine angular quartz sand and sodiumbentonite. Mollins et al. indicated that their model cannot beused for mixtures with low bentonite content because thebentonite is not evenly distributed in the SBM.

Network model

Ideal-mixture models can predict the hydraulic conductiv-ity of SBMs reasonably well when the SBMs containenough bentonite to satisfy the ideal-mixture assumptions.However, the hydraulic conductivity of SBMs with low ben-tonite content and uneven bentonite distribution cannot bepredicted using ideal-mixture models. In contrast, networkmodels can be used to simulate nonideal mixtures associatedwith low bentonite content because they explicitly accountfor filling of the pores with bentonite. Network models canalso be used to study how the hydraulic conductivity of anSBM changes as the pores in the sand are filled and how thehydraulic conductivity depends on the type of pore filling.

A network model was developed in this study to evaluatehow the hydraulic conductivity of SBMs transitions fromthat of sand to that of bentonite. The model was used to ex-amine mechanisms responsible for the decrease in hydraulicconductivity and to identify which of the mechanisms isassociated with powdered and granular bentonites. The net-work model developed in this study employs four steps:(i) generating packings of equal-sized spheres to simulatesand grains, (ii) extracting networks of capillary tubes repre-senting the pore space in the packing of spheres, (iii) calcu-lating the hydraulic conductivity of the entire network usinga capillary tube model, and (iv) evaluating how introducingbentonite into the network in different schemes affectshydraulic conductivity. The methods used to introduce the

2004 NRC Canada

Abichou et al. 699

bentonite are based on pore-scale observations reported byAbichou et al. (2002).

Sand pack generationA three-dimensional discrete element model (DEM) de-

veloped by Horner (1997) was used to generate packings ofspherical particles representing sand. The objective was todevelop a reasonably accurate representation of the packingof sand particles and the pore spaces between these particles.In the DEM, soil particles are modeled as points in space,and each carries with it an estimate of the state of the soilmass within its vicinity (Horner 1997). For the application inthis study, particles were randomly selected for placementon a lattice, with the spacing between the centers of particlesbeing large enough to minimize the initial interparticleforces. A simulation was then conducted where the particleswere rained into a rigid-wall container until the particlesachieved static equilibrium. When denser packings were de-sired, walls of the simulated container were pushed to newpositions, causing the particles to achieve a denser equilib-rium state.

There is a distinct difference between the DEM approachand an actual SBM which deserves mention. In an SBM, thesand and bentonite are mixed together and then compressed.In contrast, the collection of sand particles is compressed inthe DEM, and then bentonite is added. The presence of ben-tonite may affect friction that develops between the sandgrains and to some degree the packing of sand grains. Thiseffect is believed to be small, however, unless the bentonitecontent is so large that the sand grains become independentinclusions in the bentonite matrix. Data in Mollins et al.(1999) and Goodhue et al. (2001) support this supposition.Both studies show that the shear strength of SBMs remainsequal to that of the sand alone at low bentonite contents.This suggests that the intergranular friction is not affectedappreciably by the bentonite content until the bentonite con-tent is large.

For simplicity, the DEM code developed by Horner wasused to generate packings of equal-sized spheres, althoughthe code allows for up to five grain sizes. The output con-

sisted of the coordinates of the center of each sphere in thepacking. Packings were obtained for spheres with a diameterof 0.2 mm (fine sand), 1.2 mm (medium sand), and 3.4 mm(coarse sand). Porosities of the packings were 0.29 (densepacking), 0.36 (medium packing), and 0.44 (loose packing).

Capillary tube networkOnce a sphere packing is generated, the space defined by

the spheres was divided into tetrahedrons using VoronoiDelaunay (VD) triangulation. Each tetrahedron is formedby the closest neighboring four spheres as illustrated inFig. 1a and has four faces (i.e., each side of the tetrahedron).Each tetrahedron has four windows (one on each face)formed by the space between the three spheres on each faceand one internal pore in the center of the tetrahedron. Thewindow in each face is equivalent to a pore throat, and theinternal pore in the center of the tetrahedron is equivalent toa pore. This arrangement can be seen as four tubes (onefrom each face) forming a junction at the center of the pore(Fig. 1b). This technique transforms the pore space definedby the packing into a network of capillary tubes where theinterior of a tetrahedron corresponds to a junction of tubes inthe network (Mason 1971), as shown in Figs. 1a and 1b. Acode developed by P.K. Sweby of the Department of Mathe-matics at Reading University (Reading, UK) and modifiedby Bryant et al. (1993a, 1993b) transforms the centers ofspheres into a network geometry.

Once the geometry of the network of tubes is defined, thediameter and length of each tube in the network are deter-mined. Bryant et al. (1993a) found that accurate estimates offlow are obtained when the area of each tube is assignedusing the arithmetic mean of the radius of the largest circlebetween the spheres surrounding a pore (rm in Fig. 2a) andthe radius of a circle having equivalent area as the porespace between the same adjacent spheres (req in Fig. 2b).This arithmetic mean is referred to as the effective radius(reff) and was used to define the tube radius in this study.

Since packings were obtained with equal-sized spheres,the tube diameters are similar in size to the distance betweenthe centroid and the edge of the tetrahedron. This condition

2004 NRC Canada

700 Can. Geotech. J. Vol. 41, 2004

Fig. 1. Typical tetrahedron (a) and branches of network (b).

leads to double counting of the tube length in the central re-gion where the tubes intersect if the tube length is assumedto be equal to the distance between the central pore and theedge of the tetrahedron. This overestimation was correctedby applying an algorithm developed by Bryant et al. (1993a)that shortens the tube length so that the volume of tubes inthe network equals the volume of pores in the packing. Thetube connecting two adjacent junctions of the network isthen comprised of two sections where each has a differentlength and radius.

Examples of the distributions of tube length and diameterare shown in Fig. 3. These distributions were obtained fromthe networks extracted from dense, medium, and loose pack-

ings of 1000 spheres with a diameter of 0.2 mm. The distri-bution of tube radius shifts in the direction of smaller tubesas the packing becomes denser (Fig. 3a). Also, denser pack-ings tend to have more short tubes (Fig. 3b).

Network hydraulic conductivityIf flow in the network is laminar, then the flow rate (Q) in

each tube can be described by the Hagen-Poiseuille equa-tion:

[1] Q HrL

=

eff4

8

where H is the difference in total head between the twoends of the tube, reff is the effective radius of the tube, L isthe length of the tube, and is the viscosity of the fluid.

The flow rate can also be written in terms of a hydraulicconductance Kt (units of L2/T) of the tube as[2] Q K H= t

2004 NRC Canada

Abichou et al. 701

Fig. 2. Schematic of largest circle (a) and equivalent circle(b) for pipe sizing.

Fig. 3. Distribution of tube radius (a) and length (b) in a net-work extracted from dense, medium, and loose packings of 1000spheres with a radius of 0.2 mm.

Hydraulic conductance was used instead of hydraulic con-ductivity because conductance is easier to work with num-erically. Equating eqs. [1] and [2] yields the hydraulicconductance Kt as a function of the tube geometry:

[3] K rLteff

=

4

8

Pathways between two neighboring tetrahedrons are repre-sented by a tube with two contiguous sections havingdifferent conductances (K1, K2). The conductance of the path-way is represented by an equivalent tube with hydraulic con-ductance Ke:

[4] KK Ke

= +

1 11 2

1

The hydraulic head at the end of each tube in the networkis determined by applying conservation of mass at eachjunction. Conservation of mass requires the net flow rate ateach junction to be zero, i.e.,[5] Qij + Qik + Qil + Qim = 0where Qij, Qik, Qil, and Qim are flow rates through the fourtubes (ij, ik, il, and in) that meet at junction i and emanatefrom junctions j, k, l, and n as shown conceptually in Fig. 4.The flow rate through tube Qij is[6] Qij = Kij(Hi Hj)where Hi is the total head at junction i, Hj is the total head atthe other end of the tube (junction j), and Kij is the conduc-tance of tube ij defined by eqs. [3] and [4]. Boundary condi-

tions are applied by assigning hydraulic heads at the ends ofthe tubes emanating from the influent and effluent faces ofthe network (upper and lower surfaces shown in Fig. 4). Thehead at each junction of the network is obtained by solvingthe system of equations defined by application of eq. [5] toeach junction. The system was solved iteratively by GaussSidel successive over-relaxation.

Once the heads are determined at each junction, the flowrate through the entire network (QT) is estimated by sum-ming the flows in the tubes emanating from the up-gradientor down-gradient faces of the network. The hydraulic con-ductivity of the entire network (Kn) is estimated by applyingDarcys law to the entire network:

[7] K Q LH AnT n

n

=

where Ln is the linear length of the network (influent face tothe effluent face), Hn is the drop in total head across thenetwork, and A is the gross area of the soil at the effluentface (see Fig. 4).

Introduction of bentonite

Grain coating model (GCM)Bentonite is introduced into the network in a manner con-

sistent with observations of the microstructure of mixtures ofglass beads and bentonite (powdered and granular) made byAbichou (1999) and Abichou et al. (2002) using optical andscanning electron micrographs. For mixtures prepared withpowdered bentonite, bentonite coats the grains at low ben-tonite content and, in the presence of water, the bentoniteswells and fills some of the pore spaces between the grains.

2004 NRC Canada


Fig. 4. Conceptual sketch of network with detail showing a junction, four intersecting tubes, and four adjacent junctions. Upper andlower horizontal surfaces are constant-head boundaries. Vertical surfaces are no-flow boundaries. For clarity, this sketch contains fewertubes and junctions than that of an actual network.

As the bentonite content increases, the thickness of the coat-ing increases and more of the area available for flow isblocked by swollen bentonite. The pores are filled by a con-tinuous matrix of bentonite only at relatively high bentonitecontents (>8%).

The grain coating model (GCM) was developed to simu-late the pore structure of SBMs prepared with powderedbentonite. Radii of the spheres in the original packing wereincreased to account for the bentonite coating withoutchanging the location of the centers of the spheres, i.e.,

[8] R R Ri i ib = + where Ri is the radius of the ith sphere, Rib is the radius ofthe ith sphere coated with bentonite, and Ri is the thicknessof the swollen bentonite coating (Fig. 5a). As the sphere sizeincreases because of the bentonite coating, the area availablefor flow in each window of each tetrahedron decreases. Thesimilarity between the observed structure and the idealizedscenario assumed in the model is shown in Fig. 5b. After ad-justing the sphere radii using eq. [8], a new tube is fitted toeach window of each tetrahedron as described in the previ-ous section.

The total volume of coating in the network equals thetotal volume of hydrated bentonite in the mixture, and theratio of the total volume of hydrated bentonite to the volumeof the pore space in the original packing (i.e., pore spacewithout bentonite) is defined as the degree of bentonation.The degree of bentonation (B) is a measure of how much ofthe original pore space is occupied by hydrated bentonite,whereas conventional bentonite content is the ratio of weightof dry bentonite to weight of the sand particles. As grainsare coated with more bentonite, B increases. The upperbound on B is 1, indicating that the entire pore space in thesand matrix is occupied by bentonite.

The coated tubes are composed of two zones. An equiva-lent hydraulic conductance (Keq) of the coated tube is calcu-lated by equating the flow through the coated tube, Q, to thesum of the flow rate through the coating (Qc) and the flowrate through the remainder of the tube (Qp):

[9] Q k HL

A K H kL

A K H K H= + = + =c c c c eq

where kc and Ac are the hydraulic conductivity and cross-sectional area, respectively, of the bentonite coating; K is theconductance of the inner uncoated portion of the tube (ascalculated in eq. [4]); and H is the difference in total headacross the tube.

As the bentonite content increases, the area Ac increasesuntil it reaches the area of the original tube (i.e., withoutbentonite). When this occurs, the tube is said to beblocked with bentonite. When four intersecting tubes areblocked, the junction where they meet is blocked. Tube andjunction blocking are a direct result of the coating of thesand grains with bentonite and are not independent mecha-nisms, as is the case in other models described in the nextsections.

In the GCM, and in all other models, the hydraulic con-ductivity of bentonite was assumed to be constant. This as-sumption was made for simplicity. In reality, the hydraulicconductivity of bentonite will be affected by the size of the

pores and the degree of bentonation. Both will affect theamount of swelling that can occur in the pore space and thevoid ratio of bentonite. Given the lack of knowledge on howpore size and bentonite content affect the void ratio and thehydraulic conductivity of the bentonite fraction in SBMs, thesimplification of constant hydraulic conductivity was con-sidered to be warranted.

Tube blocking model (TBM)In the GCM, tube blocking is a direct result of increasing

the thickness of the bentonite coating around each sphere.Another alternative to model SBMs prepared with powderedbentonite is to assume that bentonite fills the window be-tween three adjacent spheres, resulting in tubes that are ei-ther blocked or unblocked. When a tube is blocked withbentonite, the tube is assumed to be entirely filled with ben-tonite.

2004 NRC Canada

Abichou et al. 703

Fig. 5. Illustration of method used to reduce area of flow ingrain coating model (a) and scanning electron micrograph show-ing coating of grains with powdered bentonite (b). Unshadedportion of sphere in (a) corresponds to original sand grain.Shaded ring around sphere corresponds to bentonite coating.

The tube blocking model (TBM) was developed to simu-late this condition. The probability of being blocked is as-sumed to be independent of the size and location of the tube,and a tube cannot be partially blocked. A Bernoulli randomnumber generator was used to define whether a tube wasblocked or unblocked. The Bernoulli generator returns a 0(empty tube) or 1 (blocked tube) depending on the probabil-ity of blockage (p) for a given tube. If a network has Nttubes and Ft is the number of tubes blocked with bentonite,then the probability of success (p) of the Bernoulli randomvariable is Ft/Nt. The volume of hydrated bentonite in thenetwork is calculated as the sum of the volume of the tubescontaining bentonite. The ratio of the volume of tubes con-taining bentonite to the volume of all tubes in the network isthe degree of bentonation (B).

Junction blocking model (JBM)The junction blocking model (JBM) was developed to

simulate SBMs prepared with granular bentonite. Based onobservations reported by Abichou et al. (2002), bentonitegranules occupy the space between the grains (Fig. 6) andthen swell to fill the available space when hydrated. As thebentonite content increases, the number of granules in-creases, leading to more spaces being filled with bentonite.The GCM and the JBM do not provide good physical repre-sentations of mixtures with granular bentonite because the

bentonite does not coat the sand grains as reported byAbichou et al.

The JBM simulates pore filling by granular bentonite byfilling the four tubes that meet at a junction with bentonite.Therefore the conductance of each tube connected to ablocked junction is assigned the hydraulic conductance of atube filled with bentonite. The degree of bentonation (B)equals the volume of the tubes filled with bentonite dividedby the total volume of tubes in the network, which is equiva-lent to the volume of blocked junctions divided by the origi-nal volume of pore space.

Two scenarios were considered. In the first scenario, thebentonite is assumed to occupy larger pores first and thenfills smaller and smaller pores. This scenario is referred to asthe size-based filling scenario (JBM-SB). This scenario canalso represent the case where the bentonite granules arelarge and cannot fit in smaller pores. The pore volume ineach tetrahedron is calculated, and then the pores are sortedbased on their volume. The largest pore (i.e., junctions in thenetwork) is blocked first, followed by sequentially smallerpores. The second scenario assumes that the distribution ofbentonite is independent of the size of the pore and that theprobability of a pore being filled with bentonite is equal forall pores. This scenario is referred to as the random fillingscenario (JBM-R). The Bernoulli random number generatorwas used for this model as well. Tubes meeting at the junc-

2004 NRC Canada


Fig. 6. Scanning electron micrograph of SBM prepared with glass beads (diameter = 1.2 mm) and granular bentonite (bentonite con-tent = 5%).

tions where the Bernoulli random number generator returnsa 1 were blocked with bentonite and their conductance wasassigned to that of a tube filled with bentonite. Tubes meet-ing at junctions where the Bernoulli random number genera-tor returns a 0 were free of bentonite.

Simulation results

Particles in the simulations were chosen to simulate fine,medium, and coarse sands based on the mean grain diameterfor these grain sizes in the Unified Soil Classification Sys-tem (Holtz and Kovacs 1981). Porosities of the packingswere chosen to simulate loose, medium, and dense sand asdescribed by Mitchell (1993).

Packings without bentoniteSimulations were first performed on packings with identi-

cal porosities but comprised of 10, 100, and 1000 spheres.The hydraulic conductivity predicted using the model wasthe same for each of these packings, indicating that a pack-ing with 1000 spheres was more than adequate to representthe network. All subsequent simulations were conductedwith packings of 1000 spheres.

Hydraulic conductivities of 1000 sphere packings withoutbentonite are shown in Fig. 7 for several sphere diameters.Also shown in Fig. 7 are hydraulic conductivities of glassspheres (measured by Chu and Ng 1989) and uniform finesand (measured by Abichou 1999). Hydraulic conductivitiespredicted by the Hazen, Harleman, and KozenyCarmenequations are also shown in Fig. 7 as smooth curves. Atlarge grain size the model predictions are very similar to

measured hydraulic conductivities (Fig. 7). As the size ofthe particles gets smaller, however, hydraulic conductivitiespredicted by the model are slightly lower than the measuredhydraulic conductivities and the hydraulic conductivitiespredicted using the equations from the literature.

Grain coating modelThe general relationship between hydraulic conductivity

and degree of bentonation for a packing with a sphere diam-eter of 0.2 mm and a porosity of 0.36 (i.e., porosity ofspheres without bentonite) is shown in Fig. 8a. The percent-age of blocked tubes and junctions in the network is alsoshown in Fig. 8a as a function of degree of bentonation B.The same relationships for packings with sphere diametersof 0.2 mm (fine sand) and 3.4 mm (coarse sand) and porosi-ties ranging from 0.29 (dense sand) to 0.44 (loose sand) areshown in Fig. 8b.

There are three distinct zones in Fig. 8a. In zone 1, as thevolume of bentonite coating increases, the effective size ofthe tubes in the network decreases, leading to a slight reduc-tion in hydraulic conductivity. All the tubes are still un-blocked in zone 1, however, and the hydraulic conductivityremains high. In zone 2, tubes start to become blocked atB = 0.5, and the hydraulic conductivity begins decreasingappreciably as more tubes become completely filled withbentonite. Zone 3 corresponds to the region where B > 0.8.At this high degree of bentonation, tubes continue to getblocked with bentonite at a higher rate. At the same time,the additional coating starts to block entire junctions, i.e.,four tubes get blocked at the same time.

2004 NRC Canada

Abichou et al. 705

Fig. 7. Hydraulic conductivity of clean sands (no bentonite) versus median sand grain diameter. Predictions made with network modelare shown as solid circles. Hydraulic conductivities computed with Hazen, Harleman, and KozenyCarmen equations are shown aslines.

2004 NRC Canada


The faster rate of tube blocking and the start of junctionblocking (both caused by grain coating) cause the hydraulicconductivity to drop dramatically in zone 3 as more andmore of the permeable pathways through the network areeliminated. When B > 0.8, only a few continuous flow pathsexist in the network that are not blocked with bentonite andthe hydraulic conductivity becomes very low (dropping to7.1 1010 cm/s) when complete filling occurs (B = 1.0).This hydraulic conductivity is lower than that of the benton-

ite alone because a portion of the gross area of flow isblinded by the spheres.

Figure 8b shows similar trends of decreasing hydraulicconductivity with increasing degree of bentonation B. Pack-ings of smaller spheres and lower porosity have lower hy-draulic conductivities for a given B. Denser packings alsoreach low hydraulic conductivity at a slightly lower degreeof bentonation. Packings of smaller spheres also reach lowhydraulic conductivity at a lower degree of bentonation. At

Fig. 8. (a) Hydraulic conductivity, percent blocked tubes, and percent blocked junctions predicted with grain coating model (GCM)versus degree of bentonation for a packing with porosity of 0.36 and sphere diameter of 0.2 mm. (b) Variation of hydraulic conductiv-ity with degree of bentonation for packings with porosity varying from 0.29 to 0.44 and sphere diameter of 0.2 and 3.4 mm.

full bentonation (B = 1.0), all hydraulic conductivities areequal to that of bentonite corrected for the blinding spheres.Figure 8b also shows that the hydraulic conductivities ofSBMs with coarse packings are almost four orders of magni-tude higher than those of fine packings, when the degree ofbentonation is below 0.6. The difference in hydraulic conduc-tivity between coarse and fine packings decreases, however, as

the degree of bentonation increases beyond 0.6. The hydraulicconductivities of all packings are in the same order of magni-tude when the degree of bentonation approaches 1.0.

Tube blocking modelThe relationship between hydraulic conductivity and de-

gree of bentonation predicted with the TBM is shown in

2004 NRC Canada

Abichou et al. 707

Fig. 9. (a) Hydraulic conductivity and percent blocked tubes predicted with tube blocking model (TBM) versus degree of bentonationfor a packing with porosity of 0.29 and sphere diameter of 0.2 mm. (b) Variation of hydraulic conductivity with degree of bentonationfor packings with porosity varying from 0.29 (dense packing) to 0.44 (loose packing) and sphere diameter of 0.2 (simulating fine sand)and 3.4 mm (simulating coarse sand).

Fig. 9a for spheres with a diameter of 0.2 mm and a porosityof 0.36. The percent blocked tubes in the network is alsoshown in Fig. 9a as a function of degree of bentonation B.The same relationships for packings with sphere diametersof 0.2 mm (fine sand) and 3.4 mm (coarse sand) and porosi-ties ranging from 0.29 (dense sand) to 0.44 (loose sand) areshown in Fig. 9b.

Three distinct zones exist in Fig. 9a. In zone 1 (B = 0.00.25), tubes in the network begin to become blocked withbentonite, leading to a gradual decrease in hydraulic conduc-tivity. Most of the pathways through the network are un-blocked, however. As B increases in zone 2 (B = 0.250.45),more tubes are blocked with bentonite and the number ofunblocked flow paths decreases dramatically, leading to asharp decrease in hydraulic conductivity. For B > 0.45 (zone3), the hydraulic conductivity approaches that of bentonitebecause only a few flow paths through the network exist thatare unblocked. For B > 0.45, the hydraulic conductivities areof the same order of magnitude and gradually decrease tothat of bentonite corrected for blinding by the spheres (7.1 1010 cm/s). In the limiting case (B = 1.0), all the flow pathsare filled with bentonite and the hydraulic conductivity is7.1 1010 cm/s. Similar trends of decreasing hydraulicconductivity with increasing B are shown in Fig. 9b forpackings with different porosities and grain sizes. Denserpackings or packings with smaller spheres reach low hydrau-lic conductivities at slightly lower degrees of bentonationthan the other packings.

Comparison of Figs. 8 and 9 shows that bentonite is usedmore efficiently in the TBM than in the GCM. The hydraulicconductivity predicted using the TBM reaches low values atB = 0.45, whereas B > 0.8 was required to reach this condi-tion with the GCM. The presence of bentonite always in-duces pipe blocking in the TBM. In contrast, the bentonitecoats the grains in the GCM but does not block tubes onlyuntil the coating becomes adequately thick.

Junction blocking modelHydraulic conductivity, percent blocked junctions, and de-

gree of bentonation obtained from the JBM with size-basedfilling are shown in Fig. 10a for spheres with a diameter of0.2 mm and a porosity of 0.36. The percentage of blockedtubes and junctions in the network is also shown in Fig. 10aas a function of B. A similar graph for random filling isshown in Fig. 10b. Results obtained using packings with dif-ferent porosities and grain sizes are not shown for this modelbut are presented in Abichou (1999). The results for otherporosities and grain sizes follow the same general trends asthose shown in Figs. 8 and 9.

As with the TBM, the relationships between hydraulicconductivity and degree of bentonation B have three zones.The hydraulic conductivity decreases slightly in zone 1 as Bincreases to 0.5. Blocking the large pores does not affect thehydraulic conductivity greatly because the hydraulic conduc-tivity is controlled by smaller tubes in the network. In zone2 (B = 0.50.8), the hydraulic conductivity decreases morerapidly with increasing B (about two and one half orders ofmagnitude). In zone 3 (B > 0.8), an abrupt decrease in hy-draulic conductivity occurs as enough junctions controllingthe hydraulic conductivity are blocked to prevent continuouspathways without bentonite through the network. Ultimately,

at B = 1.0, the hydraulic conductivity reaches that of theideal mixture (7.1 1010 cm/s).

Three zones exist for random filling as shown in Fig. 10b.In zone 1 (B = 0.00.32), the hydraulic conductivitydecreases more rapidly than in size-based filling becausesmaller junctions in the network, which are bottlenecks toflow, are being blocked at lower B with random filling. Withsize-based filling, B must be >0.5 before these bottlenecksare filled. In zone 2 (B = 0.51), the hydraulic conductivityabruptly drops four orders of magnitude. It appears that, atB = 0.51, the network becomes devoid of pathways that arenot blocked with bentonite. In zone 3, the hydraulic conduc-tivity stays constant until the degree of bentonation reaches0.8 and then gradually reaches the limiting case (B = 1.0)where the hydraulic conductivity is equal to 7.1 1010 cm/s.

Comparison with experimental data

A comparison was made between hydraulic conductivitypredicted with the model and measured hydraulic conductiv-ities of SBMs to ascertain whether the trends observed inFigs. 810 are realized for SBMs prepared with powderedand granular bentonite. The mixtures were prepared withCETCO SS-100 (American Colloid Co., Arlington Heights,Ill.), a powdered sodium bentonite ground to 70% passingthe No. 200 sieve (0.075 mm), and Benseal (Baroid Corp.,Houston, Tex.), a granular sodium bentonite with a medianparticle size of 1.1 mm. The free swell, as defined by Amer-ican Society for Testing and Materials (ASTM) StandardD5890, is 25 mL for the powdered bentonite and 32 mL forthe granular bentonite. Two uniformly graded sands wereused. One was a fine sand with a median particle diameter of0.2 mm and a coefficient of uniformity (Cu) of 4. The otherwas a medium sand with a median grain diameter of 1.2 mmand Cu of 4.5.

Prior to mixing the sands and bentonite, the sands werelightly sprayed with water to simulate the natural moisturethat commonly exists in the field. This moisture also causedthe bentonite to stick to the surfaces of the sand grains. Aknown mass of bentonite (powdered or granular) was thenadded to achieve the desired bentonite content (ratio of theweight of dry bentonite to that of dry sand). The bentonitewas added in increments to the sand as the mixture wasblended in a large container using a hand trowel. Once all ofthe bentonite was added, the entire mixture was mixed thor-oughly in the same container until it appeared uniform. Thespecimens were compacted in rigid-wall permeameters(100 mm in diameter and 50 mm high). The desired dry unitweight (sand porosity of 0.36) was achieved by compactinga known weight of the mixture into the permeameter mold(having known volume) in two equal lifts using a standardProctor hammer.

The hydraulic conductivity tests were performed directlyin the compaction mold using rigid-wall permeameters. Thefalling-head method was used in accordance with ASTMStandard D5856. The inflow and outflow burettes were cov-ered with thin plastic film to minimize evaporation. Testswere terminated when no trend existed in the hydraulic con-ductivity data, the last four hydraulic conductivity valueswere within 25% of the mean, and inflow equaled outflow.Hydraulic gradients were maintained between 28 and 32,

2004 NRC Canada


and tap water was used as the permeant liquid. The per-meameters did not include swell rings, so bentonite in thesemixtures could only swell into the pore space in the sandmatrix. Details of the hydraulic conductivity tests can befound in Abichou et al. (2002).

After the hydraulic conductivity tests were terminated, a100 cm suction was applied by a hanging column to drainoff excess water held by capillarity (i.e., water not adsorbed

by either sand or bentonite). The remaining volume of waterwas assumed to be associated with the swollen bentonite.Degree of bentonation was calculated as the sum of the vol-ume of dry bentonite and the volume of water remaining inthe specimen divided by the original volume of pores in thesand matrix. This assumption tends to overestimate theamount of swell because some of the water that remains inthe specimen may be associated with the sand matrix and

2004 NRC Canada

Abichou et al. 709

Fig. 10. Hydraulic conductivity and percent blocked junctions versus degree of bentonation predicted with junction blocking model(JBM) for a packing having and original porosity of 0.36 and sphere diameter of 0.2 mm: (a) size-based filling scenario; (b) randomfilling scenario.

not available for swelling of bentonite. Additionally, some ofthe water may also be adsorbed onto the bentonite without in-ducing any volume change. The magnitude of the overestima-tion is believed to be small, however.

SBMs prepared with powdered bentoniteA comparison of the predicted and measured relationships

between hydraulic conductivity and degree of bentonation isshown in Fig. 11. The results are presented in terms of a hy-draulic conductivity ratio (Kr), which is defined as the ratioof the hydraulic conductivity of the SBM to the hydraulicconductivity of the sand without bentonite. The measuredhydraulic conductivity of the samples prepared with fine sandvaried from 0.045 cm/s when B = 0.0 to 1.5 108 cm/s whenB = 1.0. The measured hydraulic conductivity of the samplesprepared with medium sand varied from 0.68 cm/s when B =0 to 2 108 cm/s when B = 1.0. Predictions are only shownfor the GCM and TBM because these network models weredeveloped for powdered bentonite.

Similar relationships exist between hydraulic conductivityand degree of bentonation for the actual SBMs and the pre-dictions made with the GCM. The degree of bentonation atwhich the hydraulic conductivity reaches lower values isslightly overestimated by the GCM for the fine sand and isestimated closely for medium sand. In contrast, the hydrau-lic conductivity predicted with the TBM reaches a low valueat a degree of bentonation significantly lower than occurs forthe actual SBMs. The closer agreement obtained with theGCM is consistent with the microstructural observationsmade by Abichou et al. (2002), i.e., powdered bentonite

coats the grains in an SBM rather than directly blocks thepores as assumed in the TBM.

The effect of sand size in both models may reflect an arti-fact of the model formulation, which assumes that bentonitehas constant hydraulic conductivity regardless of the size ofthe pores or the degree of bentonation. In reality, the benton-ite may have lower conductivity in fine sand because smallerpores will restrict swelling, resulting in a bentonite fractionwith lower hydraulic conductivity. As a result, the fine andmedium sands could have comparable Kr at a high degree ofbentonation. Moreover, if higher bentonite contents had beenused in the tests with fine sand, the measured Kr of the finesand may have dropped below that of the medium sand at asimilar degree of bentonation.

SBMs prepared with granular bentonitePredicted hydraulic conductivities obtained with the JBM

(size-based and random filling) and the measured hydraulicconductivities of the SBMs prepared with fine and mediumsand mixed and granular bentonite are shown in Fig. 12 ver-sus degree of bentonation. The measured hydraulic conduc-tivity of the samples prepared with fine sand varied from0.045 cm/s when B = 0 to 1.9 108 cm/s when B = 1.0.The measured hydraulic conductivity of the samples pre-pared with medium sand varied from 0.68 cm/s when B = 0to 1.8 108 cm/s when B = 1.0. The relationship betweenhydraulic conductivity and degree of bentonation predictedwith the JBM using size-based filling is similar to thosemeasured on the SBMs (especially the SBM with fine sand).The comparison is poorer for the medium sand. Neverthe-

2004 NRC Canada


Fig. 11. Hydraulic conductivity ratio versus degree of bentonation predicted with grain coating model (GCM) and tube blocking model(TBM) and measured for SBMs prepared with powdered bentonite and fine and medium sands (sand porosity = 0.36).

less, the similarity of the predicted and measured hydraulicconductivity ratios suggests that the JBM with size-basedfilling captures key mechanisms controlling the hydraulicconductivity of SBMs prepared with granular bentonite.

Much poorer agreement is obtained with random filling.The JBM with random filling predicts a large drop in hy-draulic conductivity at B = 0.51, whereas the hydraulic con-ductivity of the SBMs only dropped appreciably for B >0.75. Apparently, assuming an independent distribution ofbentonite does not reflect the bentonite distribution in SBMsprepared with granular bentonite. Rather, the distribution ofbentonite in SBMs prepared with granular bentonite proba-bly depends on the size of the granules, the size of the grainsof the sand, and maybe the bentonite content.

Summary and conclusions

A network model was developed to predict the hydraulicconductivity of SBMs prepared with powdered and granularbentonite. The model is based on a packing of equal-sizespheres created using a three-dimensional discrete elementmodel. The space occupied by the spheres is divided into acollection of neighboring tetrahedrons, and the geometry ofthe tetrahedrons is used to define tube diameters and lengthsin a pore network. The hydraulic conductivity of the networkis calculated by applying conservation of mass at each junc-tion in the network. Bentonite was introduced into the net-work in several schemes to simulate SBMs prepared withpowdered and granular bentonite. The quantity of bentonitein the network is characterized by the degree of bentonation,

B, which is the volume of swollen bentonite divided by thevolume of pores between the sand grains.

The schemes used to introduce the bentonite were selectedto mimic microstructural observations reported previously.Mixtures of sand and powdered bentonite were modeled as apacking of sand with each sand grain coated with a layer ofbentonite (grain coating model or GCM) or as a random dis-tribution of bentonite occupying the pore throats between thesand particles (tube blocking model or TBM). Mixtures ofsand and granular bentonite were modeled as a packing ofsand with bentonite granules occupying the pores betweenthe sand particles (junction blocking model or JBM).

Predictions made with the GCM show that bentonite re-duces the hydraulic conductivity of an SBM in three steps.Initially, the tube diameters decrease as the thickness of thebentonite coating increases, leading to a gradual reductionin hydraulic conductivity. However, all tubes remain un-blocked. In the second step (B > 0.5), the hydraulic conduc-tivity decreases appreciably as B increases because the graincoatings are thick enough to cause blockage of some tubes.In the third step (B > 0.8), additional coating induces block-ing of entire junctions (blocking four tubes at a time), whichdecreases the hydraulic conductivity abruptly. Predictionsmade with the TBM also show that bentonite reduces the hy-draulic conductivity of an SBM in three similar steps. First,the hydraulic conductivity decreases slightly as B increasesto 0.5. As the degree of bentonation increases from 0.5 to0.8, the hydraulic conductivity suddenly decreases by twoand a half orders of magnitude. At a degree of bentonationgreater than 0.8, enough junctions controlling the hydraulic

2004 NRC Canada

Abichou et al. 711

Fig. 12. Hydraulic conductivity ratio versus degree of bentonation predicted with junction blocking model (JBM) with size-based fill-ing and random filling along with measured hydraulic conductivities of SBMs prepared with granular bentonite and fine and mediumsand (porosity of sand = 0.36).

conductivity are blocked to prevent continuous pathways(without bentonite), leading to a steep decrease in hydraulicconductivity.

Predictions made using the GCM compare reasonablywell with hydraulic conductivities measured on SBMs pre-pared with fine and medium sand and powdered bentonite.In contrast, the TBM predicted decreases in hydraulic con-ductivity at much lower B than was measured. Predictionsobtained with the JBM with size-based filling compare rea-sonably well with hydraulic conductivities measured onSBMs prepared with granular bentonite. The favorable com-parisons obtained with the GCM and the JBM (with size-basedfilling) suggest that these models capture key mechanisms con-trolling the hydraulic conductivity of SBMs prepared withgranular and powdered bentonite.

Acknowledgments

The DEM simulations were performed by Dr. David Hor-ner. Dr. Steve Bryant provided assistance in developing thecodes for the original network model (no bentonite). Theirefforts are greatly appreciated. Financial support for thestudy described in this paper was provided by the State ofWisconsin Solid Waste Research Program (SWRP). Thefindings described in this paper are solely those of the au-thors. Endorsement by the SWRP is not implied and shouldnot be assumed.

References

Abeele, W.V. 1986. The influence of bentonite on the permeability ofsandy silts. Nuclear and Chemical Waste Management, 6: 8188.

Abichou, T. 1999. Hydraulic properties of foundry sands and theiruse as hydraulic barriers. Ph.D. thesis, Department of Civil andEnvironmental Engineering, University of Wisconsin, Madison,Wisc.

Abichou, T., Benson, C., and Edil, T. 2002. Micro-structure andhydraulic conductivity of simulated SBMs. Clays and Clay Min-erals Journal, 50(5): 537545.

Bryant, S., King, P., and Mellor, D. 1993a. Network model evalua-tion of permeability and spatial correlation in a real randomsphere packing. Transport in Porous Media, 11: 5370.

Bryant, S., Mellor, D., and Cade, C. 1993b. Physically representa-tive network models of transport in porous media. AIChE Jour-nal, 39(3): 387396.

Chapuis, R. 1981. Permeability testing of soilbentonite mixtures.In Proceeding of the 10th International Conference on Soil Me-chanics and Foundation Engineering, Stockholm, 1519 June1981. A.A. Balkema, Rotterdam, The Netherlands. pp. 744745.

Chapuis, R. 1990. Sandbentonite liners: predicting permeabilityfrom laboratory tests. Canadian Geotechnical Journal, 27: 4757.

Chu, C., and Ng, K. 1989. Flow in packed tubes with a small parti-cle diameter ratio. AIChE Journal, 35: 148158.

Daniel, D. 1987. Earthen liners for land disposal facilities. InGeotechnical Practice for Waste Disposal 87. Edited by R.D.Woods. ASCE Geotechnical Special Publication 13, pp. 2139.

Garlanger, J., Cheung, F., and Bishar, S. 1987. Quality control test-ing for sandbentonite liners. In Geotechnical Practice for WasteDisposal 87. Edited by R.D. Woods. ASCE Geotechnical Spe-cial Publication 13, pp. 488499.

Gleason, M., Daniel, D., and Eykholt, G. 1997. Calcium and so-dium bentonite for hydraulic containment applications. Journalof Geotechnical and Geoenvironmental Engineering, ASCE,123(5): 438445.

Goodhue, M., Benson, C., and Edil, T. 2001. Interaction of foundrysands with geosynthetics. Journal of Geotechnical and Geo-environmental Engineering, ASCE, 127(4): 353362.

Holtz, R., and Kovacs, W. 1981. An introduction to geotechnicalengineering. Prentice-Hall, Inc., Englewood Cliffs, N.J.

Horner, D. 1997. Application of DEM to micro-mechanical theoryfor large deformation of granular media. Ph.D. thesis, Universityof Michigan, Ann Arbor, Mich.

Howell, J., and Shackelford, C. 1997. Hydraulic conductivity ofsand admixed with processed clay mixtures. In Proceedings ofthe 14th International Conference on Soil Mechanics and Foun-dation Engineering, Hamburg, 612 September 1997. A.A.Balkema, Rotterdam, The Netherlands. pp. 307310.

Kenney, T.C., Van Veen, M.A., Swallow, M.A., and Sungaila, M.A.1992. Hydraulic conductivity of compacted bentonitesand mix-tures. Canadian Geotechnical Journal, 29: 364374.

Kraus, J., Benson, C., Erickson, A., and Chamberlain, E. 1997.Freezethaw cycling and the hydraulic conductivity of ben-tonitic barriers. Journal of Geotechnical and GeoenvironmentalEngineering, ASCE, 123(3): 229238.

Lundgren, T. 1981. Some bentonite sealants in soil mixed blankets.In Proceedings of the 10th International Conference on Soil Me-chanics and Foundation Engineering, Stockholm, 1519 June1981. A.A. Balkema, Rotterdam, The Netherlands. pp. 349354.

Mason, G. 1971. A model for the pore space in a random packingof equal spheres. Journal of Colloidal Interface Science, 35:279287.

Mitchell, J. 1993. Fundamentals of soil behavior. John Wiley andSons, Inc. New York.

Mollins, L., Stewart, D., and Cousens, T. 1996. Predicting the hy-draulic conductivity of bentonitesand mixtures. Clay Minerals,31: 243252.

Mollins, L.H., Stewart, D.I., and Cousens, T.W. 1999. Drainedstrength of bentonite enhanced sands. Gotechnique, 49(4):523528.

OSadnick, D., Simpson, B., and Kasel, G. 1995. Evaluation andperformance of a sandbentonite liner. In Geoenvironment 2000.Edited by Y.B. Acar and D.E. Daniel. ASCE Geotechnical Spe-cial Publication 46, pp. 688701.

Sllfors, G., and berg-Hgsta, A. 2002. Determination of hydrau-lic conductivity of sandbentonite mixtures for engineering pur-poses. Geotechnical and Geological Engineering, 20: 6580.

2004 NRC Canada


Documents

1 Abichou et al 04 on SBM model