1. INTRODUCTION

Determination of hydraulic conductivity of montmoril-lonitic clays is required in many situations in geotechnicalengineering like problems related to waste contaminants,seepage flow, consolidation etc. Recently expansive soils arebeing used as a mixture backfill material to meet variousfunctional requirements e.g. backfill material for nuclearwaste disposal systems in which knowledge of hydraulic con-ductivity is essential. Among all the important properties ofclays for analysis of geotechnical problems, hydraulic con-ductivity is mostly influenced by pore fluid chemistry, voidratio and fabric of clay (Mitchell 1993).

Almost all engineering properties of clays are influencedquite significantly by the existence of diffuse double layer but

the effect is predominant for hydraulic conductivity (Achariet al. 1999; Sivapullaiah et al. 2000; Besq et al. 2003; Sridharanand Nagaraj 2005). Hydraulic conductivity will be higherwhen the double layer is depressed and on the contrary itwould be lower for extended diffuse double layer.

Due to isomorphous substitutions in the crystal lattice,in general the clay particles carry negative charges at surfacesof the platelets. Hence, exchangeable cation in the clay –porefluid media get attracted to the negative charges and thisattraction is opposed by the tendency of ions to get distrib-uted. So an electric “diffuse double layer” is formed (Gouy,1910). For the case of double clay platelet, interacting diffusedouble layer extends to a finite distance. Several researcherssuch as, Verwey and Overbeek (1948), van Olphen (1991),Mitchell (1993) and others made several contributions con-cerning the factors influencing the thickness of diffuse dou-ble layer. Bolt (1956), Sridharan and Jayadeva (1982),Mitchell (1993), to name a few, explained compressibility ofclays on the basis of diffuse double layer theory. RecentlyKomine and Ogata (2003, 2004) had predicted the swellingcharacteristics of bentonites and bentonite-based buffermaterials. Also, Sridharan and Choudhury (2002) presented a

International Journal of Geotechnical Engineering (2008) 1: (1-10)

DOI 10.3328/IJGE.2008.02.01.1-10

Asuri Sridharan1 and Deepankar Choudhury2*

Computation of hydraulic conductivity ofmontmorillonitic clays by diffuse doublelayer theory

ABSTRACT: Knowledge of hydraulic conductivity of clays is very important in many cases and in waste containment systemin particular. Several investigations have been carried out on hydraulic conductivity in the past and it has been brought out thatthe hydraulic conductivity is strongly influenced by the pore fluid chemistry apart from the soil type and void ratio. Based onthe theoretical considerations, the effect of dielectric constant of the pore fluid, valency and concentration of cations in the porefluid, it is surmised that the hydraulic conductivity is significantly influenced by diffuse double layer thickness, which reducesthe pore volume through which flow takes place. A study of Gouy-Chapman diffuse double layer theory show that the diffusedouble layer thickness is negligible for non-polar fluid with very low dielectric constant (e.g. CCl4). Considering the thicknessof diffuse double layer contributed by CCl4 as zero, the equivalent thickness of diffuse double layer as affected by dielectric con-stant, valency, concentration, hydrated radius of cation and specific surface has been estimated and presented for montmoril-lonitic clays. The effective void ratio has been evaluated and a unique relationship between effective void ratio and hydraulicconductivity has been brought out for montmorillonitic clays with different pore medium chemistry. The present study throwsmore light on the fundamental mechanisms controlling the hydraulic conductivity in clays.

KEYWORDS: Clay minerals; effective void ratio; semi-empirical; pore fluid characteristics.

*Corresponding Author

1Honorary Professor (Retd.), Department of Civil Engineering, IndianInstitute of Science, Bangalore – 560 012, India. Email: sridharana-suri@yahoo.com

2Associate Professor, Department of Civil Engineering, Indian Institute ofTechnology Bombay, Powai, Mumbai – 400 076, India. Email:dc@civil.iitb.ac.in

J. Ross Publishing, Inc. © 2008

1

quantitative expression for void ratio-pressure relationshipfor sodium montmorillonitic clays based on Gouy-Chapmandiffuse double layer theory. Detailed study of literature showsthat Gouy-Chapman double layer theory is found to predictwell the potential-distance relationship for montmorilloniticclays (e.g. Bolt 1955; Bolt and Miller 1955; Hunter 1989;Mitchell 1993).

In this paper, the Gouy-Chapman theory is examined inparticular to montmorillonitic clays to obtain the variation ofpotential with the separation distance between the parallelclay platelets as affected by pore fluid chemistry. The param-eters of pore medium chemistry considered are dielectricconstant, valency, cation concentration and hydrated radiusof the cations. Both the specific surface and separation dis-tance of parallel clay platelets, 2d have been considered in thepresent analysis. Making use of results available in literatureas well as authors’ experimental results on void ratio versushydraulic conductivity for montmorillonitic clays with dif-ferent pore medium chemistry, with variation in dielectricconstant, valency and concentration of different cations, theequivalent thickness of diffuse double layer (includingadsorbed water) has been estimated. Herein after the equiva-lent thickness of diffuse double layer includes the adsorbedwater also. It may be mentioned that the thickness ofadsorbed water is relatively much lesser than the diffuse dou-ble layer water (Lambe and Whitman 1969). Knowing thethickness of diffuse double layer, effective void ratio is evalu-ated and related to hydraulic conductivity.

2. THEORETICAL CONSIDERATIONS

The Kozney-Carman equation works well to describe thehydraulic conductivity of uniformly graded sands and somesilts, but serious discrepancies are found in clays (Mitchell1993). The major factor responsible for this discrepancy isthat the fabric of clays does not contain uniform pore sizes(Olsen 1962). But it is well known that the hydraulic conduc-tivity of clays is the functions of effective pore space and sur-face area of the soils (Mitchell 1993). The pore fluidcharacteristics viz. dielectric constant, ε, valency, v, cationconcentration, n, significantly affect the thickness of diffusedouble layer (1/K) and hence, the effective pore space for flowto take place (Mesri and Olson 1971; Sridharan and Jayadeva1982).

To get the idea of effective pore space, the concept of dif-fuse double layer should be known (van Olphen 1991;Mitchell 1993). Based on Gouy-Chapman diffuse doublelayer theory for parallel clay platelets, Bolt (1956) has giventhe following equations,

(1)

(2)

(3)

where y = non-dimensional potential at distance x from clay surface = vQψ/kBT

u = non-dimensional mid-plane potential = vQψ/kBT

z = non-dimensional potential at clay surface= vQψ/kBT

ξ = Kxx = distance from the surface of clay plateletQ = elementary electric charge = 1.602 � 10-19 Cψ = electric potential at a distance x from the clay

surfaceψo = electric potential at clay surfaceψm = midplane electric potentialv = valency of cationT = temperature in KelvinkB = Boltzmann constant = 1.380 � 10-23 J/Kd = half the distance between parallel clay plateletsCEC = cation exchange capacity of the clay in

meq/100 gS = specific surface of clayε = dielectric constant of pore fluidn = concentration of ions in the bulk pore fluid

And the void ratio is related to the separation distance asgiven by Bolt (1956) for parallel clay plate model,

e = Gs γw S d (4)

where e = void ratio

Gs = specific gravity of clayγw = unit weight of water

Void ratio, e can be determined in the laboratory bydetermining the weight of the dry clay sample, Ws and vol-ume Vs = Ws/Gsγw; total volume of the clay sample, V isdetermined in the laboratory. Volume of voids, Vv is calcu-lated as, Vv = V – Vs; and then void ratio is given by, e = Vv/Vs.Now, from the analysis of potential-distance relationshipgiven by Eqs. (1) to (4) for parallel platelets and also consid-

KnQ v

k TB

=⎡

⎣⎢

⎤

⎦⎥

8 2 21 2

πε

/

−⎛

⎝⎜⎞

⎠⎟= −( )

=

dy

dz u

xξ

0

1 22 2cosh cosh

/

at= ( ) ( ) =CEC S nk T y zB/ /2π ε

2 21 2

0

cosh cosh/

y u dy d Kdd

z

u

−( ) = − = −−

∫∫ ξ

2 International Journal of Geotechnical Engineering

ering the results obtained by Sridharan and Jayadeva (1982),for four cases the potential versus distance is plotted in Figs.1a to 1d. For two different practical limits of void ratios ofmontmorillonitic clays viz. 2.5 and 4.5 are considered in theanalysis. The effect of pore medium chemistry, i.e. valency ofcation, v, cation concentration, n and dielectric constant, εare shown. In Fig. 1a, Case 1 deals with three types of fluid aspore medium viz. water (ε = 80.4), ethanol (ε = 24.3) andCCl4 (e = 2.18) along with v = 1, n = 0.0001 and e = 2.5.Similarly for all these three fluids as medium, Case 2 (Fig. 1b)deals with v = 1, n = 0.0001, e = 4.5; Case 3 (Fig. 1c) deals withv = 1, n = 0.001, e = 2.5; and Case 4 (Fig. 1d) deals with v =2, n = 0.0001, e = 2.5 respectively.

From all these results it is clear that the distance of cen-troid of potential vs. distance curve which can be assumed asthe equivalent thickness of diffuse double layer, is maximumfor fluid with higher dielectric constant viz. water. And com-pared to water for all the cases, the equivalent thickness ofdiffuse double layer for CCl4 which is about ~1/6th of that ofwater, can be assumed negligible for all practical purposes.Comparing Case 1 and 2, for higher value of void ratio, i.e.higher separation distance, the magnitude of potential is alsohigher as one can expect. Now comparing Case 1 and Case 3,the effect of cation concentration is seen. As concentration ofcations increases, potential decreases. And the effect of

valency can be seen clearly comparing Case 1 and Case 4.Potential decreases with increase in valency of cation.

3. EXPERIMENTAL RESULTS

3.1. From LiteratureAvailable experimental results in the literature and resultsobtained by authors are used in the present analysis forobtaining the equivalent diffuse double layer thickness andeffective void ratio. Results given by Mesri and Olson (1971),Sridharan et al. (1986), Sridharan et al. (1987) and Rao et al.(1988) are used for various montmorillonites, homoionizedbentonites, bentonite and naturally available montmoril-lonitic black soil. Results for hydraulic conductivity for differ-ent void ratios with different pore fluids as medium given byMesri and Olson (1971) are redrawn in Fig. 2. For clarity nopoints are shown. Both the results for sodium and calciummontmorillonites, used by Mesri and Olson (1971) are usedin the present analysis.

Fig. 3a and Fig. 3b show the results for hydraulic conduc-tivity for different void ratios, for homoionized bentoniteswith monovalent and multivalent cations (Sridharan et al.1986) respectively. It may be mentioned here that the rela-tionship between the hydraulic conductivity and void ratio isnot unique for monovalent ions because the variation in

Computation of hydraulic conductivity of montmorillonitic clays by diffuse double layer theory 3

4.30

4.55

4.80

5.05

5.30

5.55

5.80

6.05

6.30

6.55

6.80

7.05

7.30

7.55

0.00 11.80 23.60

Non

-dim

ensi

onal

pot

entia

l (x

10-

4 )

Non

-dim

ensi

onal

pot

entia

l (x

10-

4 )

Non

-dim

ensi

onal

pot

entia

l (x

10-

4 )

Water

Ethanol

CCl4CCl4

3.30

3.80

4.30

4.80

5.30

5.80

6.30

6.80

0.00 21.20 42.40

Water

Ethanol

Case 1: e = 2.5v = 1, n = 0.00012d = 23.6x10-8cm

Case2: e = 4.5v = 1, n = 0.0001d = 42.4x10-8cm

d (x 10-8) cm d (x 10-8) cm d (x 10-8) cm d (x 10-8) cm

2.30

2.80

3.30

3.80

4.30

4.80

5.30

5.80

0.00 11.80 23.601.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

0.00 11.80 23.60

Case 3: e = 2.5v=1, n = 0.00

2d=23.6x10-8cm

Case 4: e = 2.5v = 2, n = 0.0001d = 23.6x10-8cm

CCl4

Water

Ethanol

CCl4

Water

Ethanol

Non

-dim

ensi

onal

pot

entia

l (x

10-

4 )

Figure 1. Plot of potential vs. distance for diffused double layer (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4

A B C D

hydrated radius of cations (given in Table 1) change theequivalent diffuse double layer thickness (Fig. 3a). For diva-

lent ions, namely calcium and magnesium, the hydraulic con-ductivity and void ratio relationship is essentially same astheir hydrated radii are not very much different (10.8A° and9.6A° respectively). The points are scattered slightly and it isdifficult to draw two different lines. But for barium, the rela-tionship between the hydraulic conductivity and void ratio isdifferent from that of calcium and magnesium, although,they all are divalent. The reason may be attributed to thelower value of the size of the hydrated radius (8.8A°) of bar-ium. For both the trivalent ions, namely aluminum and iron,the hydraulic conductivity and void ratio relationship couldbe taken as a unique line.

Fig. 4 and Fig. 5 give the results of Sridharan et al. (1987)and Rao et al. (1988) for hydraulic conductivity with differ-ent void ratios for bentonite and montmorillonitic black soilrespectively.

4 International Journal of Geotechnical Engineering

0.1 1 10 100

Water (Na-montmorillonite)

Water (Ca-montmorillonite)

Ethyl alcohol

Carbon tetrachloride and benzene

Void ratio, e

Hyd

raul

ic C

ondu

ctiv

ity x

10-1

0 , k (

cm/s

)

0.01

0.1

1

10

100

1000

10000

100000

1000000

Figure 2. Plot of hydraulic conductivity vs. void ratio for Na and Ca-montmorillonite (Results of Mesri and Olson, 1971 redrawn)

0.1

1

10

100

1000

0 2 4 6 8

Li-bentonite

Na-bentonite

NH4-bentonite

K-bentonite

Void ratio, e

Hyd

raul

ic C

ondu

ctiv

ity x

10-1

0 , k (

cm/s

)

1

10

100

1000

0.8 1.2 1.6 2 2.4

Fe-bentonite

Al-bentonite

Ba-bentonite

Ca-bentonite

Mg-bentonite

Void ratio, e

Hyd

raul

ic C

ondu

ctiv

ity x

10-1

0 , k (

cm/s

)

Figure 3. Plot of hydraulic conductivity vs. void ratio for homoionisedbentonites (Results of Sridharan et al., 1986 redrawn) (a) For monova-lent cations (b) For divalent and trivalent cations

0.1

1

10

100

0 1 2 3 4 5 6

Void ratio, e

Hyd

raul

ic C

ondu

ctiv

ity x

10-

10, k

(cm

/s)

Figure 4. Plot of hydraulic conductivity vs. void ratio for bentonite(Results of Sridharan et al., 1987 redrawn)

0.1

1

10

100

0 1 2 3 4 5 6

Void ratio, e

Hyd

raul

ic C

ondu

ctiv

ity x

10-

10, k

(cm

/s)

Figure 5. Plot of hydraulic conductivity vs. void ratio for montmorilloniticblack soil (Results of Rao et al., 1988 redrawn)

A

B

3.2. From Present ExperimentsOne-dimensional consolidation tests for homoionizedsodium and calcium bentonite and montmorillonitic blacksoil were carried out in standard fixed ring consolidometersusing stainless steel rings of 7.6cm in diameter and 2.5cmhigh. The tests were carried out with CCl4 as fluid medium.The physical properties of the soil used are given bySridharan et al. (1986) and Rao et al. (1988) and are reportedin Table 1. The experiments were carried out in a similar wayas given by Sridharan et al. (1986). Falling head hydraulicconductivity test at the end of each load increment with CCl4as pore fluid has been carried out. The fluid in the burette wasallowed to pass through the sample and time versus theheight of fall in the fluid level in the burette has been meas-ured to calculate the hydraulic conductivity. Fig. 6 shows theplot of hydraulic conductivity versus the void ratio forsodium bentonite, calcium bentonite and montmorilloniticblack soil with CCl4 as fluid [Table 1 shows the physical prop-erties] along with the experimental results available in litera-ture. It may be seen that there is not much variation betweenthe sodium and calcium bentonites. This is primarily becausethe valency effect will be essentially negligible for a non-polarpore fluid with very low dielectric constant like CCl4. Formontmorillonitic clays with varying specific surfaces, thehydraulic conductivity (k) versus void ratio (e) could betreated as unique and can be given by the following equation,

k (cm/sec) = 6 � 10-8 e6.58 (5)

And the regression coefficient is 0.998. Fig. 6 could betaken as hydraulic conductivity versus void ratio relationship

without any reduction in void space due to the diffuse doublelayer for montmorillonitic clays.

4. CONCEPT OF EFFECTIVE VOID RATIO

Due to the presence of diffuse double layer in a clay-fluid sys-tem, the effective pore space available for the flow of fluiddecreases drastically depending on the size of the thickness ofdiffuse double layer. Hence, the hydraulic conductivity is dueto the available pore space or effective void ratio and not dueto the total voids available. The effective void ratio is the totalvoid ratio devoid of equivalent volume of water present in thedouble layer. The total void ratio, e for a saturated system canbe given by,

Computation of hydraulic conductivity of montmorillonitic clays by diffuse double layer theory 5

Table 1. Physical properties of clays considered in the analysis

Specific Specific Liquid Plastic Hydratedgravity surface limit limit CEC radius

Clay type (Gs) (S) m2/g (%) (%) (meq/100g) (A0) Reference

Na-montmorillonite 2.65 500 880 – 1160 – – – Mesri and Olson (1971)

Ca-montmorillonite 2.80 680 189 – 220 31 – 35 – – Mesri and Olson (1971)

K-bentonite 2.72 ~300* 233 57.8 113.4 3.80 – 5.32 Sridharan et al. (1986)

NH4-bentonite 2.59 ~300* 223 55.8 113.4 5.37 Sridharan et al. (1986)

Na-bentonite 2.81 ~300* 495 49.2 113.4 5.60 – 7.90 Sridharan et al. (1986)

Li-bentonite 2.61 ~300* 675 49.1 113.4 7.30 – 10.3 Sridharan et al. (1986)

Ca-bentonite 2.65 ~300* 125 40.6 113.4 9.60 Sridharan et al. (1986)

Ba-bentonite 2.73 ~300* 108 45.8 113.4 8.80 Sridharan et al. (1986)

Al-bentonite 2.43 ~300* 108 60.5 113.4 – Sridharan et al. (1986)

Na-bentonite 2.81 ~300* 495 49.0 113.4 – Sridharan et al. (1987)

Montmorillonitic 2.79 ~200* 132 41.0 79.8 – Rao et al. (1988)black soil

*Determined by authors

0.1

1

10

100

1000

10000

100000

1000000

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Void ratio, e

Na-bentonite - present investigationCa-bentonite - present investigationMontmorillonitic black soil - present investigationMontmorillonite - Mesri and Olson (1971)

Fluid type: CCl4

Hyd

raul

ic C

ondu

ctiv

ity x

10-1

0 , k

(cm

/s)

k = 6 x 10-8e6.58

R2 = 0.998

Figure 6. Generalised expression for hydraulic conductivity vs. voidratio for CCl4

e = eeff + eddl (6)

where eeff = effective void ratioeddl = void ratio contributed by diffuse double layer

Fig. 7 is a schematic diagram for a parallel clay platemodel, at separation distance 2d, the thickness of diffuse dou-ble layer and adsorbed water is denoted by t. Hence the thick-ness of available pore space is 2(d – t) for which the effectivevoid ratio is given by,

eeff = Gs γw S (d – t) (7)

It can be considered that for montmorillonitic clays, dif-fuse double layer is the primary factor to influence thehydraulic conductivity when different pore medium chem-istry exists. Assuming clay platelets are of uniform size andarranged parallely, knowing the void ratio, the separation dis-tance can be calculated using Eq. (4) from known soil prop-erties of specific gravity, Gs; specific surface, S and unit weightof water, γw.

5. RESULTS AND DISCUSSIONS

Considering the equivalent diffuse double layer thickness forCCl4 as pore medium is nearly equal to zero and the porespace required for flow at any particular hydraulic conductiv-ity is same for all the fluids, the equivalent thickness of diffusedouble layer of all clays with different exchangeable cationsare obtained in the following manner. From the hydraulicconductivity versus void ratio relationship for CCl4 as poremedium as well as other pore medium chemistry, the voidratios for these different mediums are obtained for any par-ticular value of the hydraulic conductivity. The equivalent

thickness of diffuse double layer, t for any pore medium isthus equal to,

t = (di – dCCl4) = (eeff(i) – eeff(CCl4))/Gs γw S (8)

where eeff(i) = void ratio in any pore medium at a particularhydraulic conductivity

eeff(CCl4) = void ratio CCl4 at the same coefficient ofhydraulic conductivity

di = half the separation distance between parallel clayplatelets related to ei

dCCl4 = half the separation distance between parallelclay platelets related to eCCl4

Equation (8) assumes that the clay platelets are parallelyplaced and Eq. (4) is valid.

To carry out the analysis, the hydraulic conductivity ver-sus void ratio data from literature as well as author’s experi-mental data, are used. In Fig. 2, the data of void ratios ofdifferent fluids for montmorillonite at a particular value ofhydraulic conductivity are obtained. For example, from Fig.2, for a hydraulic conductivity of 10-7 cm/s, the values of voidratios are 1.1 for CCl4, 44.0 for sodium montmorillonite withwater as pore fluid, 4.1 for calcium montmorillonite in water,2.1 for ethyl alcohol. Now from the physical properties of theclays given in Table 1 (Gs and S), the half space separation dis-tances, di are calculated using Eq. (4). Now using Eq. (8), theequivalent diffuse double layer thickness, t is calculated. Theeffective void ratio in different pore medium is given as,

eeff = Gs γw S (di – t) (9)

which equals to the void ratio of CCl4 for any givenhydraulic conductivity value since t is assumed to be zero.The results are shown in Tables 2a and 2b.

Tables 2a and 2b present the equivalent thickness of dif-fuse double layer for various cases. For montmorillonite withethyl alcohol as pore fluid, the equivalent thickness of diffusedouble layer, t varied from 5.3 A° to 7.8 A°, depending uponthe specific surface, 680 m2/gm or 500 m2/gm respectively.Compared to montmorillonite with water as pore fluid, thesevalues of t are much less (t ranges from 15.8 A° to 324 A° forCa and Na-montmorillonite respectively). This is in qualita-tive agreement with the theoretical consideration as given inFig. 1. For Na-montmorillonite (used by Mesri and Olson,1971), t is 324 A° and for Na-bentonite (used by Sridharan etal., 1986, 1987), t ranges from 23.3 A° to 60.8 A°. The vast dif-ference in the equivalent thickness of diffuse double layer isprimarily due to the higher values of specific surface of 500m2/gm of the Na-montmorillonite (used by Mesri and Olson,1971) when compared to 300m2/gm of specific surface of Na-bentonite (used by Sridharan et al., 1986, 1987). The varia-

6 International Journal of Geotechnical Engineering

Figure 7. Schematic diagram for diffuse double layer thickness

Computation of hydraulic conductivity of montmorillonitic clays by diffuse double layer theory 7

Table 2a. Calculation for diffuse double layer thickness and effective void ratio for data of Mesri and Olson (1971)

Half ThicknessVoid Half space of diffuse

Hydraulic Initial ratio, space distance double layer, Effective conductivity, void eCCl4; distance dCCl4 (cm); t void

k ratio, (CCl4 di (cm); (CCl4 as (cm) ratio, k and ei Clay type (cm/s) ei as fluid) (10-8) fluid) (10-8) eeff data from

Montmorillonite with 10-7 2.1+ 1.1 15.8 8.0 7.8 1.1 Mesri and ethyl alcohol as Olson (1971)pore fluid 10-7 2.1# 1.1 11.0 5.7 5.3 1.1

Na-montmorillonite 10-7 44.0 1.1 332.0 8.0 324.0 1.1 Mesri andwith water as Olson (1971)pore fluid

Ca-montmorillonite 10-7 4.1 1.1 21.5 5.7 15.8 1.1 Mesri andwith water as pore Olson (1971)fluid

+ for specific gravity of clays, Gs = 2.65 and S = 500 m2/gm, # for specific gravity of clays, Gs = 2.8 and S = 680 m2/gm)

Table 2b. Calculation for diffuse double layer thickness and effective void ratio for all others data

Half ThicknessVoid Half space of diffuse

Hydraulic Initial ratio, space distance double layer, Effective conductivity, void eCCl4; distance dCCl4 (cm); t void

k ratio, (CCl4 di (cm); (CCl4 as (cm) ratio, k and ei Clay type (cm/s) ei as fluid) (10-8) fluid) (10-8) eeff data from

Na-bentonite 10-9 2.5 0.54 29.7 6.4 23.3 0.54 Sridharan et al. (1986)

Na-bentonite 2x10-9 3.1 0.60 36.8 7.1 29.7 0.60 Sridharan et al. (1986)

Na-bentonite 6x10-9 4.7 0.70 55.7 8.3 47.4 0.70 Sridharan et al. (1986)

Na-bentonite 4.5x10-9 5.8 0.67 68.8 8.0 60.8 0.67 Sridharan et al. (1987)

K-bentonite 10-9 1.4 0.54 17.2 6.6 10.6 0.54 Sridharan et al. (1986)

K-bentonite 2x10-9 1.8 0.60 22.1 7.4 14.7 0.60 Sridharan et al. (1986)

K-bentonite 6x10-9 2.5 0.70 30.6 8.6 22.0 0.70 Sridharan et al. (1986)

NH4-bentonite 10-9 1.7 0.54 21.9 7.0 14.9 0.54 Sridharan et al. (1986)

NH4-bentonite 2x10-9 2.0 0.60 25.7 7.6 18.1 0.60 Sridharan et al. (1986)

NH4-bentonite 6x10-9 2.7 0.70 34.7 9.0 25.7 0.70 Sridharan et al. (1986)

Li-bentonite 10-9 3.2 0.54 40.9 6.9 34.0 0.54 Sridharan et al. (1986)

Li-bentonite 2x10-9 4.5 0.60 57.5 7.7 49.8 0.60 Sridharan et al. (1986)

Li-bentonite 6x10-9 7.3 0.70 93.2 9.0 84.2 0.70 Sridharan et al. (1986)

Ca-bentonite 10-9 1.4 0.54 17.6 6.8 10.8 0.54 Sridharan et al. (1986)

Ca-bentonite 2x10-9 1.7 0.60 21.0 7.5 13.5 0.60 Sridharan et al. (1986)

Ca-bentonite 6x10-9 2.1 0.70 27.0 8.8 18.2 0.70 Sridharan et al. (1986)

Ba-bentonite 10-9 1.3 0.54 15.9 6.6 9.3 0.54 Sridharan et al. (1986)

Ba-bentonite 2x10-9 1.5 0.60 18.4 7.3 11.1 0.60 Sridharan et al. (1986)

Ba-bentonite 6x10-9 1.9 0.70 22.7 8.6 14.1 0.70 Sridharan et al. (1986)

Al-bentonite 2x10-9 0.98 0.60 13.4 8.2 5.2 0.60 Sridharan et al. (1986)

Al-bentonite 6x10-9 1.48 0.70 20.3 9.6 10.7 0.70 Sridharan et al. (1986)

Montmorillonitic 10-9 3.1 2.17 55.6 38.9 16.7 2.17 Rao et al. (1988)black soil

*Determined by authors

tion in the equivalent thickness of diffuse double layer forNa-bentonite itself (23.3 A° to 60.8 A°) is primarily due todifferent ei values used for the calculation. As ei increases talso increases which is in qualitative agreement with the the-oretical considerations shown in Fig. 1. One can expect thisbecause as void ratio increases, the equivalent thickness ofdiffuse double layer gets extended. The equivalent thicknessof diffuse double layer for K-bentonite ranges from 10.6 A° to22.0 A°, which is less than of that of Na-bentonite, primarilydue to reduced hydrated radius for K ion compared to Na ion(Table 1). For NH4-bentonite, value of t varies from 14.9 A°to 25.7 A° which is more than of K-bentonite and less thanthat of Na-bentonite. The reason is that the hydrated radiusof NH4 ion is in between that of K and Na ions. Li-bentonitehas higher value of t than other monovalent bentonites pri-marily because of higher value of hydrated radius of Li ion.

Value of t of Ca-montmorillonite (15.8 A°) is much lessthan that of Na-montmorillonite (324 A°). This is primarilybecause the diffuse double layer gets depressed due to divalentCa ion (Fig. 1). Ca and Ba-bentonite have t in the same range

of Ca-montmorillonite, t varying from 9.3 A° to 14.1 A°. Allthe divalent montmorillonites and bentonites have muchlesser t values ranging from 9.3 A° to 15.8 A° as compared tothe monovalent montmorillonites and bentonites rangingfrom 10.6 A° to 324 A°. The effect of hydrated radius ofcations on t is significant for clays with monovalent cations.

Al-bentonite has much lesser t value, as less as 5.2 A° asexpected. The naturally available montmorillonitic black soilhas t value equal to 16.7 A°. This soil has exchangable ionsnamely, Ca, 45.8 meq/100gm, Mg, 24.1 meq/100gm, Na, 8.3meq/100gm, K, 1.6 meq/100gm (Rao et al., 1988). Since thissoil has multivalent exchangable ions, value of t = 16.7 A° canbe taken as reasonable. All the above results are in qualitativeagreement with the theoretical consideration as brought outin Fig. 1.

Fig. 8 shows the effect of hydrated radius on the equiva-lent diffuse double layer thickness, t for montmorilloniticclays with monovalent and divalent cations for three differentranges of hydraulic conductivities. In the literature for K,NH4, Na, Li, ranges of hydrated radii (Table 1) have been

8 International Journal of Geotechnical Engineering

0

10

20

30

40

50

60

70

80

90

4 5 6 7 8 9 10

Hydrated radius, R (Ao)

Equ

ival

ent d

oubl

e la

yer

thic

knes

s, t

(Ao )

K-bentoniteNH4-bentoniteNa-bentoniteLi-bentoniteCa-bentoniteBa-bentonitek = 10 x 10-10 cm/sk = 20 x 10-10 cm/sk = 60 x 10-10 cm/s

Figure 8. Plot of equivalent double layer thickness vs. hydrated radius for homoionisedbentonites at different hydraulic conductivity levels

given. In Fig. 8, the averages of hydrated radii of cations havebeen considered. It is clearly seen that t increases as hydratedradius increases and the effect is significant. Fig. 8 also showsthe effect of the level of hydraulic conductivity on t. Higherhydraulic conductivity is realised at higher void ratio. Sowhen the void ratio (ei) is more, obviously t is more (Fig. 1).The values of k and ei are also been tabulated in Tables 2a and2b. Fig. 8 also shows the effect of hydrated radii for divalentions. Even though for divalent ion, the effect of hydratedradius is not significant, t increases as hydrated radiusincreases. Also as k or ei increases, t increases.

Fig. 9 shows the unique relationship between thehydraulic conductivity and effective void ratio. As one wouldexpect for a particular type of clay, say montmorilloniticclays, the relationship between k and eeff is unique and givenby,

k (cm/sec) = 6 � 10-8 eeff6.58 (10)

with regression coefficient of 0.998.

The variation in hydraulic conductivity with void ratiofor different clays with different exchangeable cations isshown in Fig. 10a. The exchangeable cations have significantinfluence on the thickness of diffuse double layer and henceon hydraulic conductivity. The same results are plotted witheffective void ratio, falls on a unique line with same equation(10) with regression coefficient of 0.998 as shown in Fig. 10b.

6. CONCLUSIONS

Estimation of hydraulic conductivity of clays has assumedgreater importance in recent days because of clay liners arebeing used in waste containment systems. In this paper aneasy empirical relationship between hydraulic conductivityand effective void ratio, based on diffuse double layer based

theory is proposed for computation of hydraulic conductiv-ity of montmorillonitic clays using present experimental dataand data from literature. The hydraulic conductivity is greatlyinfluenced by pore medium chemistry. The primary reasonfor this significant influence has been attributed to the reduc-tion in effective pore volume available for the flow to takeplace. Assuming that the equivalent diffuse double layerthickness, t for non-polar — low dielectric fluid like CCl4 tobe negligible and clay platelets are of uniform size and paral-lely placed, t has been determined for various pore mediumchemistry for montmorillonitic clays. The parameters con-sidered are (i) different montmorillonitic clays, (ii) valency ofexchangeable cations, (iii) dielectric constant of the poremedium and (iv) hydrated radius of the cation. As one wouldexpect, t is generally reduced for divalent and trivalentcations, say 324 A° for Na montmorillonite to 5.2 A° for Albentonite. Larger the hydrated radius of the cation, larger willbe the t value. For example, Li bentonite 84.2 A° as against

Computation of hydraulic conductivity of montmorillonitic clays by diffuse double layer theory 9

k = 6x10-8e eff6.584

1

10

100

1000

10000

100000

1000000

0.4 0.8 1.2 1.6 2.0 2.4

Effective void ratio, e eff

Hyd

raul

ic C

ondu

ctiv

ity x

10 -1

0 , k

(cm

/s)

Mesri and Olson (1971)Sridharan et al. (1986)Sridharan et al. (1987)Rao et al. (1988)

Figure 9. Relationship between hydraulic conductivity vs. effective voidratio for bentonites using published data

1

10

100

1000

10000

100000

1000000

0.1 1 10 100

Void ratio, e

Na-montmorillonite of Mesri and Olson (1971)Ca-montmorillonite of Mesri and Olson (1971)Na-bentonite of Sridharan et al. (1986)K-bentonite of Sridharan et al. (1986)NH4-bentonite of Sridharan et al. (1986)Li-bentonite of Sridharan et al. (1986)Ca-bentonite of Sridharan et al. (1986)Ba-bentonite of Sridharan et al. (1986)Al-bentonite of Sridharan et al. (1986)Na-bentonite of Sridharan et al. (1987)Montmorillonitic black soil of Rao et al. (1988)

Hyd

raul

ic C

ondu

ctiv

ity x

10-1

0 , k (

cm/s

)

Unique line for effective void ratiousing data given in Fig. 10a

0.1 1 10

Effective void ratio, eeff

Hyd

raul

ic c

ondu

ctiv

ity x

10-1

0 , k (

cm/s

)

1

10

100

1000

10000

100000

1000000

k = 6x10-8eeff6.58

R2 = 0.998

Figure 10. Plot of hydraulic conductivity vs. (a) void ratio and (b) effec-tive void ratio

A

B

10.6 A° for K bentonite. The equivalent diffuse double layerthickness also decreases as dielectric constant decreases (5.3A° — 7.8 A· for ethyl alcohol to 324 A° for water with Nacation) and the same increases as void ratio increases for anysystem. Even with the same pore medium chemistry, differentmontmorillonitic clays can have variation in t value to theextent of 10 to 15 times because of variation in surface areaand plasticity characteristics. A unique relationship betweenthe hydraulic conductivity and effective void ratio (i.e. voidratio devoid of void contributed by diffuse double layer) forvarious montmorillonitic clays with various pore mediumchemistry is obtained.

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