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CHAPTER 9

Conduction Phenomena

9.1 INTRODUCTION

Virtually all geotechnical problems involve soil or rockdeformations and stability and/or the flow throughearth materials of fluids, chemicals, and energy in var-ious forms. Flows play a vital role in the deformation,volume change, and stability behavior itself, and theymay control the rates at which the processes occur.Descriptions of these flows, predictions of flow quan-tities, their rates and changes with time, and associatedchanges in the properties and composition of both thepermeated soil and the flowing material are the sub-jects of this chapter.

Water flow through soil and rock has been most ex-tensively studied because of its essential role in prob-lems of seepage, consolidation, and stability, whichform a major part of engineering analysis and design.As a result, much is known about the hydraulic con-ductivity and permeability of earth materials. Chemi-cal, thermal, and electrical flows in soils are alsoimportant. Chemical transport through the ground is amajor concern in groundwater pollution, waste dis-posal and storage, remediation of contaminated sites,corrosion, leaching phenomena, osmotic effects in claylayers, and soil stabilization. Heat flows are importantrelative to frost action, construction in permafrost ar-eas, insulation, underground storage, thermal pollution,temporary ground stabilization by freezing, permanentground stabilization by heating, underground transmis-sion of electricity, and other problems. Electrical flowsare important to the transport of water and ground sta-bilization by electroosmosis, insulation, corrosion, andsubsurface investigations.

In addition to the above four flow types, each drivenby its own potential gradient, several types of coupled

flow are important under a variety of circumstances. Acoupled flow is a flow of one type, such as hydraulic,driven by a potential gradient of another type, such aselectrical.

This chapter includes a review of the physics of di-rect and coupled flow processes through soils and theirquantification in practical form, an evaluation of rele-vant parameters, their magnitudes, and factors influ-encing them, and some examples of applications.

9.2 FLOW LAWS AND INTERRELATIONSHIPS

Fluids, electricity, chemicals, and heat flow throughsoils. Provided the flow process does not change thestate of the soil, each flow rate or flux Ji (as shown inFig. 9.1) relates linearly to its corresponding drivingforce Xi according to

J � L X (9.1)i ii i

in which Lii is the conductivity coefficient for flow.When written specifically for a particular flow type andusing familiar phenomenological coefficients, Eq. (9.1)becomes, for cross section area A

Water flow q � k i A Darcy’s law (9.2)h h h

Heat flow q � k i A Fourier’s law (9.3)t t t

Electrical flow I � � i A Ohm’s law (9.4)e e

Chemical flow J � Di A Fick’s law (9.5)D c

In Eqs. (9.2) to (9.5) qh, qt, I, and JD are the water,heat, electrical, and chemical flow rates, respectively.

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Copyright © 2005 John Wiley & Sons Retrieved from: www.knovel.com

252 9 CONDUCTION PHENOMENA

Figure 9.1 Four types of direct flow through a soil porousmass. A is the total cross-section area normal to flow; n isporosity.

Coefficients kh, kt, �e, and D are the hydraulic, thermal,electrical conductivities, and the diffusion coefficient,respectively. Typical ranges of values for these prop-erties are given later. The driving forces for flow aregiven by the respective hydraulic, thermal, electrical,and chemical gradients, ih, it, ie, and ic, respectively.

The terms in Eqs. (9.2) through (9.5) are identifiedin Fig. 9.1 and in Table 9.1, which also shows analogsbetween the various flow types. As long as the flowrates and gradients are linearly related, the mathemat-ical treatment of each flow type is the same, and theequations for flow of one type may be used to solveproblems of another type provided the property valuesand boundary conditions are properly represented. Twowell-known practical illustrations of this are the cor-respondence between the Terzaghi theory for clay con-solidation and one-dimensional transient heat flow, andthe use of electrical analogies for the study of seepageproblems.

9.3 HYDRAULIC CONDUCTIVITY

Darcy’s law1 states that there is a direct proportionalitybetween apparent water flow velocity vh or flow rateqh and hydraulic gradient ih, that is,

v � k i (9.6)h h h

q � k i A (9.7)h h h

where A is the cross-section area normal to the direc-tion of flow. The constant kh is a property of the ma-terial. Steady-state and transient flow analyses in soilsare based on Darcy’s law. In many instances, moreattention is directed at the analysis than at the value ofkh. This is unfortunate because no other property ofimportance in geotechnical problems is likely to ex-hibit such a great range of values, up to 10 orders ofmagnitude, from coarse to very fine grained soils, orshow as much variability in a given deposit as doesthe hydraulic conductivity. Some soils exhibit 2 or 3orders of magnitude variation in hydraulic conductivityas a result of changes in fabric, void ratio, and watercontent. These points are illustrated by Fig. 9.2 inwhich hydraulic conductivity values for a number ofsoils are shown.

Different units for hydraulic conductivity are oftenused by different groups or agencies; for example, cen-timeters per second by geotechnical engineers, feet peryear by groundwater hydrologists, and Darcys by pe-troleum technologists. Figure 9.3 can be used to con-vert from one system to another. The preferred unit inthe SI system is meters/second.

Theoretical Equations for Hydraulic Conductivity

Fluid flow through soils finer than coarse gravel is lam-inar. Equations have been derived that relate hydraulicconductivity to properties of the soil and permeatingfluid. A usual starting point for derivation of suchequations is Poiseuille’s law for flow through a roundcapillary, which gives the average flow velocity, vave,according to

2 Rpv � i (9.8)ave h8�

where � is viscosity, R is tube radius, and p is unit

1 This ‘‘law’’ was established empirically by Darcy based on the re-sults of flow tests through sands. Its general validity for the descrip-tion of hydraulic flow through most soil types has been verified bymany subsequent studies. Historical accounts of the development ofDarcy’s law are given by Brown et al. (2003).

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HYDRAULIC CONDUCTIVITY 253

Table 9.1 Conduction Analogies in Porous Media

Fluid Heat Electrical Chemical

Potential Total head h (m) Temperature T(�C)

Voltage V (volts) Chemical potential � orconcentration c (molm�3)

Storage Fluid volume W(m3/m3)

Thermal energy u(J /m3)

Charge Q (Coulomb) Total mass per unit totalvolume, m (mol/m3)

Conductivity Hydraulicconductivity kh

(m/s)

Thermalconductivity kt

(W/m/ �C)

Electrical conductivity �(siemens/m)

Diffusion coeff. D(m2/s)

Flow qh (m3/s) qt (J /s) Current I (amp) jD (mol/s)Flux qh/A (m3/s /m2) qt /A (J /s /m2) I /A (amp/m2) JD � jD/A (mol s�1 m�2)

Gradient (m/m)�h

i � �h �x(�C/m)

�Ti � �t �x

(v/m)�V

i � �e �x(mol m�4)

�ci � �c �x

Conduction Darcy’s law�h

q � �k Ah h �x

Fourier’s law�T

q � �k At t �x

Ohm’s law�V V

I � �� A �e �x R

Fick’s law�c

J � �D AD �xCapacitance Coefficient of

volume changeVolumetric heat

C(J / �C/m3)Capacitance C (farads �

coul/volt)Retardation factor, Rd

(dimensionless)

�dW aw vM � �dh 1 � e

kh

cv

dQC �

dT

Continuity�W qh� � � 0� ��t A

�u qt� � � 0� ��t A

�Q I� � � 0� �

�t A�(m)

� �J � 0D�tSteady state �2qh � 0 �2qt � 0 �2I � 0 �2JD � 0

Diffusion2�h k � hh� 2�t M �x

2�T k � Tt� 2�t C �x

2�V � � V� 2�t C �x

2�c D* � c� 2�t R �xD

k� c� �vM

k� a� �C

weight of the flowing fluid. Because the flow channelsin a soil are of various sizes, a characteristic dimensionis needed to describe average size. The hydraulic ra-dius RH

flow channel cross-section areaR �H wetted perimeter

is useful.For a circular tube flowing full,

2�R RR � � (9.9)H 2�R 2

so Poiseuille’s equation becomes

1 p 2q � R i a (9.10)cir H h2 �

where a is the cross-sectional area of the tube. Forother shapes of cross section, an equation of the sameform will apply, differing only in the value of a shapecoefficient Cs, so

2 Rp Hq � C i a (9.11)s h�

For a bundle of parallel tubes of constant but irreg-ular cross section contributing to a total cross-sectionalarea A (solids plus voids), the area of flow passages Aƒ

filled with water is

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Figure 9.2 Hydraulic conductivity values for several soils. Soil identification code: 1, com-pacted caliche; 2, compacted caliche; 3, silty sand; 4, sandy clay; 5, beach sand; 6, compactedBoston blue clay; 7, Vicksburg buckshot clay; 8, sandy clay; 9, silt—Boston; 10, Ottawasand; 11, sand—Gaspee Point; 12, sand—Franklin Falls; 13, sand–Scituate; 14, sand–PlumIsland; 15, sand–Fort Peck; 16, silt—Boston; 17, silt—Boston; 18, loess; 19, lean clay; 20,sand—Union Falls; 21, silt—North Carolina; 22, sand from dike; 23, sodium Boston blueclay; 24, calcium kaolinite; 25, sodium montmorillonite; 26–30, sand (dam filter) (FromLambe and Whitman (1969). Copyright � 1969 by John Wiley & Sons. Reprinted withpermission from John Wiley & Sons.

Figure 9.3 Hydraulic conductivity and permeability conversion chart.

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A � SnA (9.12)ƒ

where S is the degree of saturation and n is the poros-ity. For this condition the hydraulic radius is given by

A A L volume available for flowƒ ƒR � � �H P PL wetted area

Vw� (9.13)V Ss 0

where P is the wetted perimeter, L is the length of flowchannel in the direction of flow, Vs is the volume ofsolids and S0 is the wetted surface area per unit volumeof particles. The wetted surface area depends on theparticle sizes and the soil fabric and may be consideredas an effective surface area per unit volume of solids.It is less than the total specific surface area of the soilsince flow will not occur adjacent to all particle sur-faces.

For void ratio e and volume of solids Vs, the volumeof water Vw is

V � eV S (9.14)w s

Equation (9.11) becomes

ep p2 2q � C R Sni A � C R S i A� � � � � �s H h s H h� � 1 � e

(9.15)

and substitution for RH using Eqs. (9.13) and (9.14)gives

31 ep 3q � C S i A (9.16)� �� s h2S � 1 � e0

By analogy with Darcy’s law,

3 1 ep 3k � C S (9.17)� � � �h s 2� S 1 � e0

For full saturation, S � 1, and denoting Cs by 1/(k0T

2), where k0 is a pore shape factor and T is a tor-tuosity factor, Eq. (9.17) becomes

3� 1 eK � k � (9.18)� � � �h 2 2 k T S 1 � ep 0 0

This is the Kozeny–Carman equation for the permea-bility of porous media (Kozeny, 1927; Carman, 1956).The hydraulic conductivity kh has units of velocity

(LT�1), and the absolute or intrinsic permeability K hasunits of area (L2).

The effects of permeant properties are accounted forby the � /p term, provided the fabric of the soil is thesame in the presence of different fluids. The pore shapefactor k0 has a value of about 2.5 and the tortuosityfactor has a value of about in porous media con-2taining approximately uniform pore sizes.

For equal size spheres, S0 becomes 6/D (�surfacearea/volume of a sphere), where D is the diameter. Ifa soil is considered to consist of spheres of differentsizes, an effective diameter Deff can be computed fromthe particle size distribution (Carrier, 2003) accordingto

100%D � (9.19)eff �(ƒ /D )i ave,i

where fi is the fraction of particles between two sizes(Dli and Dsi) and Dave,i is the average particle size be-tween two sizes (� ); S0 can also be estimated0.5 0.5D Dli si

from the specific surface area. Methods for nonplasticsoils and clayey soils are given in Chapter 3 and alsoare summarized by Chapuis and Aubertin (2003). Var-ious modifications for S0 are available to take irregularparticle shapes (Loudon, 1952; Carrier, 2003) into ac-count.

The Kozeny–Carman equation accounts well for thedependency of permeability on void ratio in uniformlygraded sands and some silts; however, serious discrep-ancies are often found when it is applied to clays. Themain reasons for these discrepancies are that most claysoils do not contain uniform pore sizes and changes inpore fluid type are often accompanied by changes inthe clay fabric. Particles in clays are grouped in clus-ters or aggregates that have large intercluster pores andsmall intracluster pores. The influences of fabric andnonuniform pore sizes on the hydraulic conductivity offine-grained soils are discussed further later in this sec-tion.

If comparisons are made using materials having thesame fabric, the influence of permeant on hydraulicconductivity is quite well accounted for by the p /�term. If, however, a fine-grained soil is molded or com-pacted in different permeants, then the fabrics may bequite different, and the hydraulic conductivities forsamples at the same void ratio can differ greatly.

If Cs in Eq. (9.17) is taken as a composite shapefactor, and noting that total surface area per unit vol-ume is inversely proportional to particle size, then

3 ew2 3k � CD S (9.20)� �h s � 1 � e

where Ds is a characteristic grain size.

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Like the Kozeny–Carman equation, Eq. (9.20) de-scribes the behavior of cohesionless soils reasonablywell, but it is inadequate for clays. For a uniform sandwith bulky particles and a given permeant, Eqs. (9.17)and (9.20) indicate that kh should vary directly withe3 /(1 � e) and , and experimental observations sup-2Ds

port this.Despite the inability of the theoretical equations to

predict the hydraulic conductivity accurately in manycases, they do reflect the overwhelming importance ofpore size. Flow velocity depends on the square of poreradius, and hence the flow rate depends on radius tothe fourth power. The specific surface in the Kozeny–Carman equation and the representative grain size termin Eq. (9.20) are both measures of pore size. All otherfactors equal, the hydraulic conductivity depends farmore on the fine particles than on the large. A smallpercentage of fines can clog the pores of an otherwisecoarse material and result in a manyfold lower hydrau-lic conductivity. On the other hand, the presence offissures, cracks, root holes, and the like can result inenormous increases in the rate of water flow throughan otherwise compact soil layer.

Equation (9.20) predicts that the hydraulic conduc-tivity should vary with the cube of the degree of sat-uration, and some, but not all, experimental datasupport this, even in the case of fine-grained soils.Consideration of flow through unsaturated soils isgiven in Section 9.4.

Validity of Darcy’s Law

A basic premise of Darcy’s law is that flow is laminarand steady through saturated porous media. If particleand pore sizes and flow rates are sufficiently great, thenflow is turbulent, and Darcy’s law no longer applies.Turbulent flow conditions are likely in flows throughgravel and rockfill (Ahmed and Sunada, 1969; Ar-bhabhirama and Dinoy, 1973; George and Hansen,1992; Hansen et al., 1995; Li et al., 1998).2 Some mod-ification of Darcy’s law is needed also to account fornonsteady and wave-induced flows through sands, silts,

2 Flow transitions from laminar to turbulent flow when the Reynoldsnumber Re, defined as the ratio of inertial to viscous forces, exceedsa critical value. For flow through soils the critical value of interstitialflow Re is in the range of 1 to 10, with Re defined as (Khalifa et al.,2002)

4��vRe �

�(1 � n)Avd

in which � is fluid density, � is tortuosity (ratio of flow path meanlength to thickness), v is flow velocity, n is porosity, and Avd is theratio of pore surface area exposed to flow to the volume of solid.

and clays (Khalifa et al., 2002). These nonsteady andturbulent flow conditions are not treated herein.

As early as 1898, instances were cited in which hy-draulic flow velocity in fine-grained materials in whichlaminar flow can be expected increased more than pro-portionally with increases in gradient (King, 1898).The absence of water flow at finite hydraulic gradientsin ceramic filters of 0.1-�m average pore diameterwas reported by Derjaguin and Krylov (1944). Oakes(1960) found no detectable flow through a 30-cm-longsuspension of 6 percent Wyoming bentonite subjectedto a 50-cm head of water. Experiments by Miller andLow (1963) led to the conclusion that there was athreshold gradient for flow through sodium montmo-rillonite. Flow rates through clay-bearing sandstoneswere found to increase more than directly with gradi-ent up to gradients of 170 by von Englehardt and Tunn(1955). Deviations from Darcy’s law in pure and nat-ural clays up to gradients of 900 were measured byLutz and Kemper (1959). Apparent deviations fromDarcy’s law for flow in undisturbed soft clay are shownin Fig. 9.4.

The reported deviations from linearity between flowrate and hydraulic gradient are most significant in thelower range of gradients. Hydraulic gradients in thefield are seldom much greater than one. Thus, de-viations from Darcy’s law, if real, could have veryimportant implications for the applicability ofsteady-state and transient flow analyses, including con-solidation, that are based on it. Furthermore, gradientstypically used in laboratory testing are high, commonlymore than 10, and often up to several hundred. Thisbrings the suitability of laboratory test results as indi-cators of field behavior into question.

Three hypotheses have been proposed to account fornonlinearity between flow velocity and gradient: (1)non-Newtonian water flow properties, (2) particle mi-grations that cause blocking and unblocking of flowpassages, and (3) local consolidation and swelling thatis inevitable when hydraulic gradients are appliedacross a compressible soil. The apparent existence ofa threshold gradient below which flow was not de-tected was attributed to a quasi-crystalline water struc-ture. It is now known, however, that many of theeffects interpreted as resulting from unusual waterproperties can be ascribed to undetected experimentalerrors arising from contamination of measuring sys-tems (Olsen, 1965), local consolidation and swelling,and bacterial growth (Gupta and Swartzendruber,1962). Additional careful measurements by a numberof investigators (e.g., Olsen, 1969; Gray and Mitchell,1967; Mitchell and Younger, 1967; Miller et al., 1969;Chan and Kenney, 1973) failed to confirm the exis-tence of a threshold gradient in clays. Darcy’s law was

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Figure 9.4 Dependence of flow velocity on hydraulic gradient. Undisturbed soft clay fromSka Edeby, Sweden (from Hansbo, 1973).

obeyed exactly in several of these studies. Thus it isunlikely that unusual water properties are responsiblefor non-Darcy flow behavior.

On the other hand, particle migrations leading tovoid plugging and unplugging, electrokinetic effects,and chemical concentration gradients can cause appar-ent deviations from Darcy’s law. Analysis of interpar-ticle bond strengths in relation to the magnitude ofseepage forces shows that particles that are not partic-ipating in the load-carrying skeleton of a soil mass canbe moved under moderate values of hydraulic gradient.Soils with open, flocculated fabrics and granular soilswith a relatively low content of fines appear particu-larly susceptible to the movement of fine particles dur-ing permeation.

Internal swelling and dispersion of clay particlesduring permeation can cause changes in flow rate andapparent non-Darcy behavior. Tests on illite–siltmixtures showed that the hydraulic conductivity de-pends on clay content, sedimentation procedure,compression rate, and electrolyte concentration. Sub-sequent behavior was quite sensitive to the type andconcentration of electrolyte used for permeation andthe total throughput volume of permeant. Changes inrelative hydraulic conductivity that occurred while the

electrolyte concentration was changed from 0.6 to 0.1N NaCl are shown in Fig. 9.5. The cumulative through-put is the ratio of the total flow volume at any time tothe sample pore volume. The hydraulic conductivitiesfor these materials ranged from more than 1 � 10�7 toless than 1 � 10�9 m/s.

Practical Implications Evidence indicates thatDarcy’s law is valid, provided that all system variablesare held constant. However, unless fabric changes, par-ticle migrations, and internal void ratio redistributionscaused by effective stress and chemical changes canbe shown to be negligible, hydraulic conductivity mea-surements in the laboratory should be made under con-ditions of temperature, pressure, hydraulic gradient,and pore fluid chemistry as closely approximatingthose in the field as possible. This is particularly im-portant in connection with the testing of clays as po-tential waste containment barriers, such as slurry wallsand liners for landfills and impoundments (Daniel,1994). Microbial activities may be important as well,as they can lead to formation of biofilms, pore clog-ging, and large reductions in hydraulic conductivity asshown, for example, by Dennis and Turner (1998).

Unfortunately, duplication of field conditions is notalways possible, especially as regards the hydraulic

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Figure 9.5 Reduction in hydraulic conductivity as a result of internal swelling (from Hard-castle and Mitchell, 1974).

gradient. If hydraulic gradients are low enough to du-plicate those in most field situations, then the labora-tory testing time usually becomes unacceptably long.In such cases, tests over a range of gradients are de-sirable in order to assess the stability of the soil struc-ture against changes due to seepage forces.

Similarly, the gradients that are developed in labo-ratory consolidation tests on thin samples are manytimes greater than exist in thick layers of the same clayin the field. The variation of hydraulic gradient i withtime factor T during one-dimensional consolidation ac-cording to the Terzaghi theory is shown in Fig. 9.6.The solution of the Terzaghi equation gives excesspore pressure u as a function of position (z/H) andtime factor

� 2u Mz 20 �M Tu � sin e (9.21)� � �M Hm�0

where M � �(2m � 1)/2. Thus, the hydraulic gradientis

�� u 2u Mz 20 �M Ti � � cos e (9.22)�� z H Hm�0w w

If a parameter p is defined by

� Mz 2�M Tp � 2 cos e (9.23)� � �Hm�0

Eq. (9.22) becomes

u0i � p (9.24) Hw

The real gradient for any layer thickness or loadingintensity can be obtained by using actual values of u0

and H and the appropriate value of p from Fig. 9.6.For small values of u0 /wH, as is the case in the

field, for example, for u0 � 50 kPa, H � 5m, thenu0 /wH � 1, and the field gradients are low throughoutmost of the layer thickness during the entire consoli-dation process. On the other hand, for a laboratorysample of 10 mm thickness and the same stress in-crease, u0 /wH is 500, and the hydraulic gradients arevery large. In this case a gradient-dependent hydraulicconductivity could be the cause of significant differ-ences between the laboratory-measured and field val-ues of coefficient of consolidation. Constant rate ofstrain or constant gradient consolidation testing of suchsoils is preferable to the use of load increments be-cause lower gradients minimize particle migration ef-fects.

Anisotropy

Anisotropic hydraulic conductivity results from bothpreferred orientation of elongated or platy particles andstratification of soil deposits. Ratios of horizontal-to-vertical hydraulic conductivity from less than 1 tomore than 7 were measured for undisturbed samplesof several different clays (Mitchell, 1956). These ratioscorrelated reasonably well with preferred orientation ofthe clay particles, as observed in thin section. Ratiosof 1.3 to 1.7 were measured for kaolinite consolidatedone dimensionally from 4 to 256 atm, and 0.9 to 4.0

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Figure 9.6 Hydraulic gradients during consolidation according to the Terzaghi theory.

were measured for illite and Boston blue clay consol-idated over a pressure range up to more than 200 atm(Olsen, 1962). A ratio of approximately 2 was mea-sured for kaolinite over a range of void ratios corre-sponding to consolidation pressures up to 4 atm(Morgenstern and Tchalenko, 1967b). Thus, an averagehydraulic conductivity ratio of about 2 as a result ofmicrofabric anisotropy may be typical for many clays.

Large anisotropy in hydraulic conductivity as a re-sult of stratification of natural soil deposits or in earth-work compacted in layers is common. Varved clayshave substantially greater hydraulic conductivity in thehorizontal direction than in the vertical direction owingto the presence of thin silt layers between the thin claylayers. The ratio of horizontal values to vertical valuesdetermined in the laboratory, rk, is 10 � 5 for Con-necticut Valley varved clay (Ladd and Wissa, 1970).Similar values were measured for the varved clay inthe New Jersey meadows. Values less than 5 were mea-sured for New Liskeard, Ontario, varved clay (Chanand Kenney, 1973).

The practical importance of a high hydraulic con-ductivity in the horizontal direction depends on the dis-tance to a drainage boundary and the type of flow. Forexample, the rate of groundwater flow will clearly beaffected, as will the rate of consolidation when vertical

drains are used. On the other hand, lateral drainagebeneath a loaded area may not be greatly influencedby a high ratio of horizontal to vertical conductivity ifthe width of loaded area is large compared to the thick-ness of the drainage layer.

Fabric and Hydraulic Conductivity

The theoretical relationships developed earlier in thissection indicate that the flow velocity should dependon the square of the pore radius, and the flow rate isproportional to the fourth power of the radius. Thus,fabrics with a high proportion of large pores are muchmore pervious than those with small pores. For ex-ample, remolding several undisturbed soft clays re-duced the hydraulic conductivity by as much as afactor of 4, with an average of about 2 (Mitchell,1956). This reduction results from the breakdown of aflocculated open fabric and the destruction of largepores.

An illustration of the profound influence of com-paction water content on the hydraulic conductivity offine-grained soil is shown in Fig. 9.7. All samples werecompacted to the same density. For samples compactedusing the same compactive effort, curves such as thosein Fig. 9.8 are typical. For compaction dry of optimum,

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Figure 9.7 Hydraulic conductivity as a function of compac-tion water content for samples of silty clay prepared to con-stant density by kneading compaction.

Figure 9.8 Influence of compaction method on the hydraulicconductivity of silty clay. Constant compactive effort wasused for all samples.

clay particles and aggregates are flocculated, the resis-tance to rearrangement during compaction is high, anda fabric with comparatively large pores is formed. Forhigher water contents, the particle groups are weaker,and fabrics with smaller average pore sizes are formed.Considerably lower values of hydraulic conductivityare obtained wet of optimum in the case of kneadingcompaction than by static compaction (Fig. 9.8) be-cause the high shear strains induced by the kneadingcompaction method break down flocculated fabricunits.

Three levels of fabric are important when consid-ering the hydraulic conductivity of finer-grained soils.The microfabric consists of the regular aggregations ofparticles and the very small pores, perhaps with sizesup to about 1 �m, between them through which verylittle fluid will flow. The minifabric contains these ag-gregations and the interassemblage pores betweenthem. The interassemblage pores may be up to severaltens of micrometers in diameter. Flows through thesepores will be much greater than through the intraag-gregate pores. On a larger scale, there may be a ma-crofabric that contains cracks, fissures, laminations, orroot holes through which the flow rate is so great asto totally obscure that through the other pore spacetypes.

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Figure 9.9 Contours of constant hydraulic conductivity forsilty clay compacted using kneading compaction.

Figure 9.10 Cluster model for permeability prediction (after Ol-sen, 1962).

These considerations are of particular importance inthe hydraulic conductivity of compacted clays used asbarriers for waste containment. The controlling unitsin these materials are the clods, which would corre-spond to minifabric units. Acceptably low hydraulicconductivity values are obtained only if clods and in-terclod pores are eliminated during compaction (Ben-son and Daniel, 1990). This requires that compactionbe done wet of optimum using a high effort and amethod that produces large shear strains, such as bysheepsfoot roller.

The wide range of values of hydraulic conductivityof compacted fine-grained soils that results from thelarge differences in fabric associated with compactionto different water contents and densities is illustratedby Fig. 9.9. The grouping of contours means that se-lection of a representative value for use in a seepageanalysis is difficult. In addition, if it is required thatthe hydraulic conductivity of earthwork not exceed acertain value, such as may be the case for a clay linerfor a waste pond, then specifications must be carefullydrawn. In so doing, it must be recognized also thatother properties, such as strength, also vary with com-paction water content and density and that the com-paction conditions that are optimal for one propertymay not be suitable for the other. A procedure for thedevelopment of suitable specifications for compactedclay liners is given by Daniel and Benson (1990).

The primary reason equations such as (9.18) and(9.20) fail to account quantitatively for the variation ofthe hydraulic conductivity of fine-grained soils withchange in void ratio is unequal pore sizes (Olsen,1962). A typical soil has a fabric composed of small

aggregates or clusters as shown schematically in Fig.9.10. These aggregates of N particles each have anintracluster void ratio ec. The spaces between the ag-gregates comprise the intercluster voids and areresponsible for the intercluster void ratio ep. The totalvoid ratio eT is equal to the sum of ec and ep. Theclusters and intracluster voids comprise the microfa-bric, whereas the assemblage of clusters comprises theminifabric. Fluid flow in such a system is dominatedby flow through the intercluster pores because of theirlarger size.

The sizes of clusters depend on the mineralogicaland pore fluid compositions and the formational proc-ess. Conditions that favor aggregation of individualclay plates produce larger clusters than deflocculating,dispersing environments. There is general consistencywith the interparticle double-layer interactions de-scribed in Chapter 6. When a fine-grained soil issedimented in or mixed with waters of differentelectrolyte concentration or type or with fluids of dif-ferent dielectric constants, quite different fabrics result.This explains why the � / term in Eqs. (9.18) and(9.20) is inadequate to account for pore fluid differ-ences, unless comparisons are made using sampleshaving identical fabrics. This will only be the casewhen a pore fluid of one type replaces one of anothertype without disturbance to the soil.

The cluster model developed by Olsen (1962) ac-counts for discrepancies between the predicted andmeasured variations in flow rates through differentsoils. The following equation can be derived for theratio of estimated flow rate for a cluster model, qCM tothe flow rate predicted by the Kozeny–Carman equa-tion (9.18) qKC:

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3q (1 � e /e )CM c T2 / 3� N (9.25)4 / 3q (1 � e )KC c

Application of Eq. (9.25) requires assumptions forthe variations of ec with eT that accompany compres-sion and rebound. Olsen (1962) considered the relativecompressibility of individual clusters and cluster as-semblages. The compressibility of individual clustersis small at high total void ratios, so compression isaccompanied by reduction in the intercluster poresizes, but with little change in intracluster void ratio.This assumption is supported by the microstructurestudies of Champlain clay by Delage and Lefebvre(1984) Thus, the actual hydraulic conductivity de-creases more rapidly with decreasing void ratio duringcompression than predicted by the Kozeny–Carmanequation until the intercluster pore space is comparableto that in a system of closely packed spheres, whenthe clusters themselves begin to compress. Further de-creases in porosity involve decreases in both ec and eT.As the intercluster void ratio now decreases less rap-idly, the hydraulic conductivity decreases at a slowerrate with decreasing porosity than predicted by theKozeny–Carman equation. During rebound increase inporosity develops mainly by swelling of the clusters,whereas the flow rate continues to be controlled pri-marily by the intercluster voids.

Recent attempts to quantify saturation and hydraulicconductivity of fine-grained soils containing a distri-bution of particle sizes and fabric elements in terms ofpore-scale relationships have given promising results(Tuller and Or, 2003). Expressions for clay plate spac-ing in terms of surface properties and solution com-position derived using DLVO theory (see Chapter 6),combined with assumed geometrical representations ofclay aggregates and pore space in combination withsilt and sand components, are used in the formulation.

9.4 FLOWS THROUGH UNSATURATED SOILS

Darcy’s law [Eq. (9.7)] also applies for flow throughunsaturated soils such as those in the vadose zoneabove the water table where pore water pressures arenegative. However, the hydraulic conductivity is notconstant and depends on the amount and connectivityof water in the pores. For instance, Eq. (9.20) predictsthat hydraulic conductivity should vary as the cube ofthe degree of saturation.3 This relationship has been

3 The hydraulic conductivity can also be a function of volumetricmoisture content � or matric suction �. These variables are relatedto each other by the soil–water characteristic curve as described inChapter 7.

found reasonable for compacted fine-grained soils anddegrees of saturation greater than about 80 percent.

Similarly to Eq. (9.7), the unsaturated flow equationin the direction i can be written as

�� zv � �k(S) � (9.26)� �i �x �xi i

where k(S) is saturation-dependent hydraulic conduc-tivity, � is the matric suction equivalent head (L), and�z /�xi is the unit gravitational vector measured upwardin direction z (1.0 if xi is the direction of gravity z).When percolating water infiltrates vertically into drysoil, the hydraulic gradient near the sharp wetting frontcan be very large because of a large value of the �� /�x term. However, the wetting front becomes less sharpas the infiltration proceeds and the gravity term thendominates. The hydraulic gradient then is close to oneand the magnitude of flux is equal to the hydraulicconductivity k(S).

Using Eq. (9.26), the equation of mass conservationbecomes

�(nS) � �� z R� k(S) � � (9.27) � ��

�t �x �x �x �i i i w

where n is the porosity, �w is the density of the water,and R is a source or sink mass transfer term such aswater uptake by plant roots (ML�3).

If the soil is assumed to be incompressible and thereis no sink/sources (R � 0), Eq. (9.27) becomes

�S �� zn � k(�) � � ��

�� t �x �x �xi i i

�� zor C(�) � k(�) � (9.28) � ��

�t �x �x �xi i i

where C(�) � n(�S /��) and k(S) is converted to k(�)using the soil–water characteristic curve (S–� relation-ships). Equation (9.28) is called the Richards equation(Richards, 1931). For given S–� and k(�) relationshipsand initial /boundary conditions, the nonlinear govern-ing equation can be solved for � (often numerically bythe finite difference or finite element method).

The hydraulic conductivity of unsaturated soils canbe a function of saturation, water content, matric suc-tion, or others. Measured hydraulic conductivities ofwell-graded sand and clayey sand as a function of (a)matric suction and (b) saturation ratio are shown inFig. 9.11. Both figures are related to each other, as thematric suction is a function of saturation ratio by thesoil moisture characteristic curve as described in Sec-

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FLOWS THROUGH UNSATURATED SOILS 263

Sand

Rel

ativ

e P

erm

eabi

lity

k r

Sand

Sand

0 100

Rel

ativ

e P

erm

eabi

lity

k r

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05Matric Suction (kPa)

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05Matric Suction (kPa)

Clayey Sand > Sand

Clayey Sand

Sand > Clayey Sand

1.E+01

1.E-01

1.E-05

1.E-07

1.E-09

1.E-11

1.E-13

1.E-15

1.E-17

1.E-191.E-21

1.E-03

Hyd

raul

ic c

ondu

ctiv

ity (

m/s

)

1.E+01

1.E-01

1.E-05

1.E-07

1.E-09

1.E-11

1.E-13

1.E-15

1.E-17

1.E-191.E-21

1.E-03

Hyd

raul

ic C

ondu

ctiv

ity (

m/s

)

Saturation (%)

1.E+00

1.E-02

1.E-04

1.E-06

1.E-08

1.E-10

1.E-12

1.E-14

1.E-16

1.E+00

1.E-02

1.E-04

1.E-06

1.E-08

1.E-10

1.E-12

1.E-14

1.E-16

Saturation (%)

Clayey Sand

Clayey Sand

Sand

Clayey Sand

20 40 60 80

0 10020 40 60 80

(a) (b)(a)

(d)(c)

Figure 9.11 Hydraulic conductivity of partially saturated sand and clayey sand as a functionof matric suction and degree of saturation (from Stephens, 1996).

tion 7.12. Various methods to measure the hydraulicconductivity of unsaturated soils are available (Klute,1986; Fredlund and Rahardjo, 1993). However, themeasurement in unsaturated soils is more difficult toperform than in saturated soils because the hydraulicconductivity needs to be determined under controlledwater saturation or matric suction conditions.

A general expression for the hydraulic conductivityk of unsaturated soils can be written as

�gk � k K � k k (9.29)r r s�

where ks is the saturated conductivity, K is the intrinsicpermeability of the medium (L2) such as given by Eq.(9.18), � is the density of the permeating fluid (ML�3),g is the acceleration of gravity (LT�2), � is the dynamicviscosity of the permeating fluid (MT�1L�1), and ks isthe conductivity under the condition that the pores arefully filled by the permeating fluid (i.e., full saturation).The dimensionless parameter kr is called the relativepermeability, and the values range from 0 (� zero per-

meability, no interconnected path for the permeatingfluid) to 1 (� permeating fluid at full saturation). Theequation can be used for a nonwetting fluid (e.g., air)by substituting the values of � and � of the nonwettingfluid.

The data in Fig. 9.11a and 9.11b can be replottedas the relative permeability against matric suction inFig. 9.11c and against saturation ratio in Fig. 9.11d.The two different curves in Fig. 9.11d clearly showthat kr � S3 derived from Eq. (9.20) is not universallyapplicable. At very low water contents, the water inthe pores becomes disconnected as described in Chap-ter 7. Careful experiments show that the movement ofwater exists even at moisture contents of a few percent,but vapor transport becomes more important at this drystate (Grismer et al., 1986). Therefore, Eq. (9.20) isnot suitable for low saturations. One reason for thisdiscrepancy is that soil contains pores of various sizesrather than the assumption of uniform pore sizes usedto derive Eq. (9.20).

Considering that the soil contains pores of randomsizes, Marshall (1958) derived the following equation

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for hydraulic conductivity as a function of pore sizesfor an isotropic material:

2 2 2 2 2n r � 3r � 5r � � � � � (2m � 1)r1 2 3 mK � 2m 8

(9.30)

in which K is the specific hydraulic conductivity (per-meability) (L2), n is the porosity, m is the total numberof pore classes, and ri is the mean radius of the poresin pore class i. Pore sizes can be measured from dataon the amount of water withdrawn as the suction onthe soil is progressively increased. Using the capillaryequation, the radius of the largest water-filled pore un-der a suction of � (L) is given by

2�r � (9.31)

� g�w

in which � is the surface tension of water, �w is thedensity of water, and g is the acceleration of gravity.As it is usually more convenient to use moisture suc-tion than pore radius, Eq. (9.29) can be rewritten as

2 2� n�2 �2 �2K � [� � 3� � 5�1 2 32 2 22� g mw

�2� � � � � (2m � 1)� ] (9.32)m

The permeability K can be converted to the hydraulicconductivity k by multiplying the unit weight (�wg) di-vided by the dynamic viscosity of water �. This gives

2 2� n�2 �2 �2k � [� � 3� � 5�1 2 322� g� mw

�2� � � � � (2m � 1)� ] (9.33)m

Following Green and Corey (1971), the porosity nequals the volumetric water content of the saturatedcondition �S, and m is the total number of pore classesbetween �S and zero water content � � 0.

A matching factor is usually used in Eq. (9.33) toequate the calculated and measured hydraulic conduc-tivities. Matching at full saturation is preferable tomatching at a partial saturation point because it is sim-pler and gives better results. Rewriting Eq. (9.33) andintroducing a matching factor gives

l2 2k � �s S �2k(� ) � [(2j � 1 � 2i)� ]�i j2k 2� g� m j�1sc w

(i � 1, 2, . . . , l) (9.34)

in which k(�i) is the calculated hydraulic conductivityfor a specified water content �i; is i the last water con-tent class on the wet end, for example, i � 1 denotesthe pore class corresponding to the saturated watercontent �S, and i � l denotes the pore class correspond-ing to the lowest water content �L for which hydraulicconductivity is calculated; ks /ksc is the matching factor,defined as the measured saturated hydraulic conductiv-ity divided by the calculated saturated hydraulic con-ductivity; and l is the total number of pore classes (apore class is a pore size range corresponding to a watercontent increment) between � � �L and �S. Thus

m �S� (9.35)l � � �S L

A constant value of l is used at all water contents, andthe value of l establishes the number of pore classesfor which terms are included in the calculation at�2�j

saturation. Other pore size distribution models for un-saturated soils are available, and an excellent reviewof these models is given by Mualem (1986).

Equation (9.34) can be written in an integration formas (after Fredlund et al., 1994)

�2 pk � � � � xs Sk(�) � � dx (9.36)2�k 2� g� � (x)Lsc w

where suction � is given as a function of volumetricwater content �, and x is a dummy variable. The hy-draulic conductivity for fully saturated condition is cal-culated by assigning � � �S. For generality, the term

in Eq. (9.34) is replaced by , where p is a constant2 p� �S s

that accounts for the interaction of pores of varioussizes (Fredlund et al., 1994).

From Eq. (9.36), the relative permeability kr is afunction of water content as follows:

� �S� � x � � xk (�) � � dx � dx (9.37)�r 2 2

�r �r� (x) � (x)

Herein, the lowest water content �L is assumed to bethe residual water content �r. If the moisture content�–suction � relationship (or the soil–water character-istic curve) is known, the relative permeability kr canbe computed from Eq. (9.37) by performing a numer-ical integration. The hydraulic conductivity k is thenestimated from Eq. (9.29) with the knowledge of sat-urated hydraulic conductivity ks.

The use of the soil–water characteristic curve to es-timate the hydraulic conductivity of unsaturated soilsis attractive because it is easier to determine this curve

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THERMAL CONDUCTIVITY 265

in the laboratory than it is to measure the hydraulicconductivity directly. Apart from Eq. (9.37), the fol-lowing relative permeability function proposed by Mu-alem (1976) is often used primarily because of itssimplicity:

q 2� �s� � � d� d�rk (�) � � � (9.38)� � � � �r� �� (�) �(�)r rs r

where q describes the degree of connectivity betweenthe water-conducting pores. Mualem (1976) states thatq � 0.5 is appropriate based on permeability measure-ments on 45 soils. van Genuchten et al. (1991) substi-tuted the soil–water characteristic equation (7.52) intoEq. (9.38) and obtained the following closed-form so-lution4:

p 1 / m m 2� � � � � �r rk (�) � 1 � 1 �� r � � � � � �S r S r

(9.39)

Both Eq. (9.39) as well as Eq. (9.37) using the soil–water characteristic curve by Fredlund and Xing(1994) give good predictions of measured data asshown in Fig. 9.12.

The two hydraulic conductivity–matric suctioncurves shown in Fig. 9.11a cross each other at a matricsuction value of approximately 50 kPa (or 5 m abovethe water table under hydrostatic condition). Belowthis value, the hydraulic conductivity of sand is largerthan that of the clayey sand. However, as the matricsuction increases, the water in the sand drains rapidlytoward its residual value, giving a very low hydraulicconductivity. On the other hand, the clayey sand holdsthe pore water by the presence of fines and the hy-draulic conductivity becomes larger than that of thesand at a given matric suction.

If the sand is overlain by the clayey sand, then thematric suction at the interface is larger than 50 kPa,and the water infiltrating downward through the finerclayey sand cannot enter into the coarser sand layerbecause the underlying sand layer is less permeablethan the overlying clayey sand. The water will insteadmove laterally along the bedding interface. This phe-nomenon is called a capillary barrier (e.g., Zaslavskyand Sinai, 1981; Yeh et al., 1985; Miyazaki, 1988).The barrier will be maintained as long as the lateraldischarge along the interface (preferably inclined) is

4 m � 1 � 1 /n is assumed (van Genuchten et al., 1991). See Eq.(7.52).

larger than the vertically infiltrating water flow. How-ever, if the matric suction is reduced by large infiltra-tion, the barrier breaks and water enters into theinitially dry coarse layer. Solutions are available toevaluate the amount of water flowing laterally acrossthe capillary barrier interface at the point of break-through for a given set of fine and coarse soil hydraulicproperties and interface inclination (Ross, 1990; Steen-huis et al., 1991; Selkar, 1997; Webb, 1997).

Capillary barriers have received increased attentionas a means for isolating buried waste from ground-water flow and as part of landfill cover systems in dryclimates (Morris and Stormont, 1997; Selkar, 1997;Khire et al., 2000). The barrier can be used to divertthe flow laterally along an interface and/or to storeinfiltrating water temporarily in the fine layer so thatit can be removed ultimately by evaporation and tran-spiration. Capillary barriers are constructed as simpletwo-layer systems of contrasting particle size or mul-tiple layers of fine- and coarse-grained soils. If thethickness of the overlying fine layer is too small, cap-illary diversion is reduced because of the confiningflow path in the fine layer. The minimum effectivethickness is several times the air-entry head of the finesoil (Warrick et al., 1997; Smersrud and Selker, 2001).Khire et al. (2000) stress the importance of site-specificmetrological and hydrological conditions in determin-ing the storage capacity of the fine layer. The soil forthe underlying coarse layer should have a very largeparticle size contrast with the fine soil, but finesmigrations into the coarse sand should be avoided.Smesrud and Sekler (2001) suggest the d50 particle sizeratio of 5 to be ideal. The thickness of the coarse sandlayer does not need to be great, as the purpose of thelayer is simply to impede the downward water migra-tion.

9.5 THERMAL CONDUCTIVITY

Heat flow through soil and rock is almost entirely byconduction, with radiation unimportant, except for sur-face soils, and convection important only if there is ahigh flow rate of water or air, as might possibly occurthrough a coarse sand or rockfill. The thermal conduc-tivity controls heat flow rates. Conductive heat flow isprimarily through the solid phase of a soil mass. Valuesof thermal conductivity for several materials are listedin Table 9.2. As the values for soil minerals are muchhigher than those for air and water, it is evident thatthe heat flow must be predominantly through the sol-ids. Also included in Table 9.2 are values for the heatcapacity, volumetric heat, heat of fusion, and heat ofvaporization of water. The heat capacity can be used

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Predicted coefficientof permeability(drying)

Predicted coefficientof permeability(wetting)

Measured coefficientof permeability(drying)

Measured coefficientof permeability(wetting)

10

1

0.1

0.01

0.001

0.0001

Volumetric Water Content

Hyd

raul

ic C

ondu

ctiv

ity, k

× 10

(m/s

)

20 30 40 50 60

(a)

Hyd

raul

ic C

ondu

ctiv

ity, k

(cm

/day

)Figure 9.12 Comparisons of predicted and measured relationships between hydraulic con-ductivity and volumetric water content for two soils. (a) By Eq. (9.37) with the measureddata for Guelph loam (from Fredlund et al., 1994) and (b) by Eq. (9.39) with the measureddata for crushed Bandelier Tuff (van Genuchten et al., 1991).

to compute the volumetric heat using the simple rela-tionships for frozen and unfrozen soil given in the ta-ble. Volumetric heat is needed for the analysis of manytypes of transient heat flow problems. The heat of fu-sion is used for analysis of ground freezing and thaw-ing, and the heat of vaporization applies to situationswhere there are liquid to vapor phase transitions.

The denser a soil, the higher is its composite thermalconductivity, owing to the much higher thermal con-ductivity of the solids relative to the water and air.Furthermore, since water has a higher thermal conduc-tivity than air, a wet soil has a higher thermal conduc-tivity than a dry soil. The combined influences of soilunit weight and water content are shown in Fig. 9.13,which may be used for estimates of the thermal con-ductivity for many cases. If a more soil-specific valueis needed, they may be measured in the laboratory us-ing the thermal needle method (ASTM, 2000). Moredetailed treatment of methods for the measurement ofthe thermal conductivity of soils are given by Mitchelland Kao (1978) and Farouki (1981, 1982).

The relationship between thermal resistivity (inverseof conductivity) and water content for a partly satu-rated soil undergoing drying is shown in Fig. 9.14. Ifdrying causes the water content to fall below a certainvalue, the thermal resistivity increases significantly.This may be important in situations where soil is usedas either a thermally conductive material, for example,

to carry heat away from buried electrical transmissioncables, or as an insulating material, for example, forunderground storage of liquefied gases. The water con-tent below which the thermal resistivity begins to risewith further drying is termed the critical water content,and below this point the system is said to have lostthermal stability (Brandon et al., 1989).

The following factors influence the thermal resistiv-ity of partly saturated soils (Brandon and Mitchell,1989).

Mineralogy All other things equal, quartz sandshave higher thermal conductivity than sands con-taining a high percentage of mica.

Dry Density The higher the dry density of a soil,the higher is the thermal conductivity.

Gradation Well-graded soils conduct heat betterthan poorly graded soils because smaller grainscan fit into the interstitial spaces between thelarger grains, thus increasing the density and themineral-to-mineral contact.

Compaction Water Content Some sands that com-pacted wet and then dried to a lower water contenthave significantly higher thermal conductivitythan when compacted initially at the lower watercontent.

Time Sands containing high percentages of silica,carbonates, or other materials that can develop ce-

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ELECTRICAL CONDUCTIVITY 267

Table 9.2 Thermal Properties of Materialsa

ThermalConductivity Material Btu/h/ft2 / �F/ft W/m/K

Air 0.014 0.024Water 0.30 0.60Ice 1.30 2.25Snow(100 kg m�3)(500 kg m�3)

0.030.34

0.060.59

Shale 0.90 1.56Granite 1.60 2.76Concrete 1.0 1.8Copper 225 389Soil 0.15–1.5 (�1.0) 0.25–2.5 (�1.7)Polystyrene 0.015–0.035 0.03–0.06

Heat Capacity Material Btu/ lb/ �F kJ/kg/K

Water 1.0 4.186Ice 0.5 2.093Snow(100 kg m�3)(500 kg m�3)

0.050.25

0.211.05

Minerals 0.17 0.710Rocks 0.20–0.55 0.80–2.20

Volumetric Heat Material Btu/ft3 / �F kJ/m3/K

Unfrozen Soil d(0.17 � w /100) d(72.4 � 427w /100)SoilFrozen soil d(0.17 � 0.5w /100) d(72.4 � 213w /100)Snow(100 kg m�3)(500 kg m�3)

3.1315.66

2101050

Heat of Fusion Water 143.4 Btu/ lb 333 kJ/kgSoil 143.4(w /100)d Btu/ft3 3.40 � 104(w /100)d kJ/m3

Heat of Vaporization Water 970 Btu/ lb 2.26 MJ/kgSoil 970(w /100)d Btu/ft3 230(w /100)d MJ/m3

ad � dry unit weight, in lb/ft3 for U.S. units and in kN/m3 for SI units; w � water content in percent.

mentation may exhibit an increased thermal con-ductivity with time.

Temperature All crystalline minerals in soils havedecreasing thermal conductivity with increasingtemperature; however, the thermal conductivity ofwater increases slightly with increasing tempera-ture, and the thermal conductivity of saturatedpore air increases markedly with increasing tem-perature. The net effect is that the thermal con-ductivity of moist sand increases somewhat withincreasing temperature.

9.6 ELECTRICAL CONDUCTIVITY

Ohm’s law, Eq. (9.4), in which �e is the electrical con-ductivity, applies to soil–water systems. The electricalconductivity equals the inverse of the electrical resis-tivity, or

1 L� � (siemens/meter; S/m) (9.40)e R A

where R is the resistance ( ), L is length of sample

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Figure 9.13 Thermal conductivity of soil (after Kersten,1949).

Figure 9.14 Typical relationship between thermal resistivityand water content for a compacted sand.

(m), and A is its cross-sectional area (m2). The valueof electrical conductivity for a saturated soil is usuallyin the approximate range of 0.01 to 1.0 S/m. The spe-cific value depends on several properties of the soil,including porosity, degree of saturation, composition(conductivity) of the pore water, mineralogy as it af-

fects particle size, shape, and surface conductance, soilstructure, including fabric and cementation, and tem-perature.

Electrical measurements found early applications inthe fields of petroleum engineering, geophysical map-ping and prospecting, and soil science, among others.The inherent complexity of soil–water systems and thedifficulty in characterizing the wide ranges of particlesize, shape, and composition have precluded develop-ment of generally applicable theoretical equations forelectrical conductivity. However, a number of empiri-cal equations and theoretical expressions based onsimplified models may provide satisfactory results,depending on the particular soil and conditions. Theydiffer in assumptions about the possible flow paths forelectric current through a soil–water matrix, the pathlengths and their relative importance, and whethercharged particle surfaces contribute to the total currentflow.

Nonconductive Particle Models

Formation Factor The electrical conductivity ofclean saturated sands and sandstones is directly propor-tional to the electrical conductivity of the pore water (Ar-chie, 1942). The coefficient of proportionality depends onporosity and fabric. Archie (1942) defined the formationfactor, F, as the resistivity of the saturated soil, �T, dividedby the resistivity of the saturating solution, �W, that is,

� �T WF � � (9.41)� �W T

where �W and �T are the electrical conductivities of thepore water and saturated soil, respectively.

An empirical correlation between formation factor andporosity for clean sands and sandstones is given by

�mF � n (9.42)

where n is porosity, and m equals from 1.3 for loose sandsto 2 for highly cemented sandstones. An empirical relationbetween formation factor at 100 percent water saturationand ‘‘apparent’’ formation factor at saturation less than100 percent is

�WpF � (S ) (9.43)atS �1 ww �T

where p is a constant determined experimentally. Archiesuggested a value of p � 2; however, other published val-ues of p range from 1.4 to 4.6, depending on the soil and

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whether a given saturation is reached by wetting or bydrainage.

Capillary Model In this and the theoretical modelsthat follow, direct current (DC) conductivity is as-sumed, although they may apply to low-frequency al-ternating current (AC) models as well. Consider asaturated soil sample of length L and cross-sectionalarea A. If the pores are assumed to be connected andcan be represented by a bundle of tubes of equal radiusand length Le and total area Ae, where Ae � porosity� A, and Le is the actual length of the flow path, thenan equation for the formation factor as a function ofthe porosity n and the tortuosity T � Le /L is

2TF � (9.44)

n

For S � 1, and assuming that the area available forelectrical flow is nSA, then F � T2 /nS. In principle, ifF is measured for a given soil and n is known, a valueof tortuosity can be calculated to use in the Kozeny–Carman equation for hydraulic conductivity.

Cluster Model As discussed earlier in connectionwith hydraulic conductivity, the cluster model (Olsen,1961, 1962) shown in Fig. 9.10 assumes unequal poresizes. Three possible paths for electrical current flowcan be considered: (1) through the intercluster pores,(2) through the intracluster pores, and (3) alternatelythrough inter- and intracluster pores. On this basis thefollowing equations for formation factor as a functionof the cluster model parameters can be derived (Olsen,1961):

1 � e 1T2F � T (9.45)� �� e � e 1 � XT c

X � Y � Z (9.46)

2[(1 � e ) /(e � e )]T T cY � (9.47)2 21 � (T /T) [(1 � e ) /e (e � e )]c c c T c

2e TcZ � a (9.48)� �� e � e TT c c

in which T is the intercluster tortuosity, Tc is the intra-cluster tortuosity, and a is the effective cluster ‘‘contactarea.’’ The cluster contact area is very small except forheavily consolidated systems.

This model successfully describes the flow of cur-rent in soils saturated with high conductivity water. Insuch systems, the contribution of the surface conduc-

tance of clayey particles to the total current flow wouldbe small.

Conductive Particle Models

In conductive particle models the contribution of theions concentrated at the surface of negatively chargedparticles is taken into account. Two simple mixturemodels are presented below; other models can befound in Santamarina et al. (2001).

Two-Parallel-Resistor Model A contribution of sur-face conductance is included, and the soil–water sys-tem is equivalent to two electrical resistors in parallel(Waxman and Smits, 1968). The result is that the totalelectrical conductivity �T is

� � X(� � � ) (9.49)T W s

in which �s is a surface conductivity term, and X is aconstant analogous to the reciprocal of the formationfactor that represents the internal geometry.

This approach yields better fits of �T versus �W datafor clay-bearing soils. However, it assumes a constantvalue for the contribution of the surface ions that isindependent of the electrolyte concentration in the porewater, and it fails to include a contribution for the sur-face conductance and pore water conductance in a se-ries path.

Three-Element Network Model A third path is in-cluded in this formulation that considers flow alongparticle surfaces and through pore water in series inaddition to the paths included in the two-parallel-resistor model. The flow paths and equivalent electricalcircuit are shown in Fig. 9.15. Analysis of the electricalnetwork for determination of �T gives

a� �W s� � � b� � c� (9.50)T s W(1 � e)� � e�W s

If the surface conductivity �s is negligible, the sim-ple formulation proposed by Archie (1942) for sandsis obtained; that is, �T � constant � �W. Some of thegeometric parameters a, b, c, d, and e can be writtenas functions of porosity and degree of saturation; oth-ers are obtained through curve regression analysis of�T versus �W data.

Soil conductivity as a function of pore fluid con-ductivity is shown in Fig. 9.16 for a silty clay. Thethree-element model fits the data well over the fullrange, the two-element model gives good predictionsfor the higher values of conductivity, and the simpleformation factor relationship is a reasonable average

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Figure 9.15 Three-element network model for electrical conductivity: (a) current flow pathsand (b) equivalent electrical circuit.

for conductivity values in the range of about 0.3 to 0.6S/m.

Alternating Current Conductivity and DielectricConstant

The electrical response of a soil in an AC field is fre-quency dependent owing to the polarizability proper-ties of the system constituents. Several scale-dependentpolarization mechanisms are possible in soils, asshown in Fig. 9.17. The smaller the element size thehigher the polarization frequency. At the atomic andmolecular scales, there are polarizations of electrons[electronic resonance at ultraviolet (UV) frequencies],ions [ionic resonance at infrared (IR) frequencies], anddipolar molecules (orientational relaxation at micro-wave frequencies). A mixture of components (like wa-ter and soil particles) having different polarizabilitiesand conductivities produces spatial polarization bycharge accumulation at interfaces (called Maxwell–Wagner interfacial polarization). The ions in the Sternlayer and double layer are restrained (Chapter 6), andhence they also exhibit polarization. This polarizationresults in relaxation responses at radio frequencies.Further details of the polarization mechanisms aregiven by Santamarina et al. (2001).

The effective AC conductivity �eff is expressed as

� � � � !�"� (9.51)eff 0

where � is the conductivity, !� is the polarization loss(called the imaginary relative permittivity), " is the

frequency, and �0 is the permittivity of vacuum [8.85� 10�12 C2/(Nm2)]. The frequency-dependent effectiveconductivities of deionized water and kaolinite–watermixtures at two different water contents (0.2 and 33percent) are shown in Fig. 9.18a. The complicated in-teractions of different polarization mechanisms are re-sponsible for the variations shown.

A material is dielectric if charges are not free tomove due to their inertia. Higher frequencies areneeded to stop polarization at smaller scales. The di-electric constant (or the real relative permittivity !�5)decreases with increasing frequency; more polariza-tion mechanisms occur at lower frequencies. Thefrequency-dependent dielectric constants of deionizedwater and kaolinite–water mixtures are shown in Fig.9.18b. The value for deionized water is about 79 above10 kHz. Below this frequency, the values increase withdecrease in frequency. This is attributed to experimen-tal error caused by an electrode effect in which charges

5 To describe the out-of-phase response under oscillating excitation,the electrical properties of a material are often defined in the complexplane:

� � �� j��

where � is the complex permittivity, j is the imaginary number( ), and �� and �� are real and imaginary numbers describing the�1electrical properties. The permittivity � is often normalized by thepermittivity of vacuum �0 as

�! � � !� � j!�

�0

where ! is called the relative permittivity.

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Figure 9.16 Soil electrical conductivity as a function of porefluid conductivity and comparisons with three models.

Deionized Water

33%

0.2%

No DataAvailable

100

100

10–2

10–4

10–6

102 104 106 108 1010

Deionized Water33%

0.2% No DataAvailable

100

100

102

104

106

102 104 106 108 1010

σ eff

(S/m

)

Electrode Effect

Frequency (Hz)

Frequency (Hz)

κ�

(a)

(b)

Figure 9.18 (a) Conductivity and (b) relative permittivity asa function of frequency for deionized water and kaolinite atwater contents of 0.2 and 33 percent (from Santamarina etal., 2001).

Figure 9.17 Frequency ranges associated with different polarization mechanisms (from San-tamarina et al., 2001).

accumulate at the electrode–specimen interface (Kleinand Santamarina, 1997). Similarly to the observationsmade for the effective conductivities, the real permit-tivity values of the mixtures show complex trends offrequency dependency.

For analysis of AC conductivity and dielectric con-stant as a function of frequency in an AC field, Smithand Arulanandan (1981) modified the three-elementmodel shown in Fig. 9.15 by adding a capacitor inparallel with each resistor. The resulting equations canbe fit to experimental frequency dispersions of the con-

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Table 9.3 Self-Diffusion Coefficients for Ions atInfinite Dilution in Water

Anion(1)

D0 � 1010(m2/s)(2)

Cation(3)

D0 � 1010(m2/s)(4)

OH� 52.8 H� 93.1F� 14.7 Li� 10.3Cl� 20.3 Na� 13.3Br� 20.8 K� 19.6I� 20.4 Rb� 20.7HCO3

� 11.8 Cs� 20.5NO3

� 19.0 Be2� 5.98SO4

2� 10.6 Mg2� 7.05CO3

2� 9.22 Ca2� 7.92— — Sr2� 7.90— — Ba2� 8.46— — Pb2� 9.25— — Cu2� 7.13— — Fe2�a 7.19— — Cd2�a 7.17— — Zn2� 7.02— — Ni2�a 6.79— — Fe3�a 6.07— — Cr3�a 5.94— — Al3�a 5.95

aValues from Li and Gregory (1974). Reprinted withpermission from Geochimica et Cosmochimica Acta, Vol.38, No. 5, pp. 703–714. Copyright � 1974, PergamonPress.

ductivity and apparent dielectric constant by computeroptimization of geometrical and compositional param-eters. The resulting parameter values are useful forcharacterizing mineralogy, porosity, and fabric. Moredetailed discussions on electrical models, data inter-pretation, and correlations with soil properties aregiven by Santamarina et al. (2001).

9.7 DIFFUSION

Chemical transport through sands is dominated by ad-vection, wherein dissolved and suspended species arecarried with flowing water. However, in fine-grainedsoils, wherein the hydraulic flow rates are very small,for example, kh less than about 1 � 10�9 m/s, chemicaldiffusion plays a role and may become dominant whenkh becomes less than about 1 � 10�10 m/s. Fick’s law,Eq. (9.5), is the controlling relationship, and D(L2T�1),the diffusion coefficient, is the controlling parameter.Diffusive chemical transport is important in claybarriers for waste containment, in some geologicprocesses, and in some forms of chemical soilstabilization. Comprehensive treatments of the diffu-sion process, values of diffusion coefficients andmethods for their determination, and applications,especially in relation to chemical transport and wastecontainment barrier systems, are given by Quigley etal. (1987), Shackelford and Daniel (1991a, 1991b),Shincariol and Rowe (2001) and Rowe (2001).

Diffusive flow is driven by chemical potential gra-dients, but for most applications chemical concentra-tion gradients can be used for analysis. The diffusioncoefficient is measured and expressed in terms ofchemical gradients. Maximum values of the diffusioncoefficient D0 are found in free aqueous solution atinfinite dilution. Self-diffusion coefficients for a num-ber of ion types in water are given in Table 9.3. Usu-ally cation–anion pairs are diffusing together, therebyslowing down the faster and speeding up the slower.This may be seen in Table 9.4, which contains valuesof some limiting free solution diffusion coefficients forsome simple electrolytes.

Diffusion through soil is slower and more complexthan diffusion through a free solution, especially whenadsorptive clay particles are present. There are severalreasons for this (Quigley, 1989):

1. Reduced cross-sectional area for flow because ofthe presence of solids

2. Tortuous flow paths around particles3. The influences of electrical force fields caused by

the double-layer distributions of charges

4. Retardation of some species as a result of ionexchange and adsorption by clay minerals andorganics or precipitation

5. Biodegradation of diffusing organics6. Osmotic counterflow7. Electrical imbalance, possibly by anion exclusion

The diffusion coefficient could increase with time offlow through a soil as a result of such processes as(Quigley, 1989):

1. K� fixation by vermiculite, which would decreasethe cation exchange capacity and increase thefree water pore space

2. Electrical imbalances that act to pull cations oranions

3. The attainment of adsorption equilibrium, thuseliminating retardation of some species

In an attempt to take some of these factors, espe-cially geometric tortuosity of interconnected pores,into account, an effective diffusion coefficient D* is

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DIFFUSION 273

Table 9.4 Limiting Free Solution DiffusionCoefficients for Some Simple Electrolytes

Electrolyte(1)

D0 � 1010(m2/s)(2)

HCl 33.36HBr 34.00LiCl 13.66LiBr 13.77NaCl 16.10NaBr 16.25NaI 16.14KCl 19.93KBr 20.16KI 19.99CsCl 20.44CaCl2 13.35BaCl2 13.85

Reported by Shackelford and Daniel, 1991a after Rob-inson and Stokes, 1959. Reprinted from the Journal ofGeotechnical Engineering, Vol. 117, No. 3, pp. 467–484.Copyright � 1991. With permission of ASCE.

used. Several definitions have been proposed (Shack-elford and Daniel, 1991a) in which the different factorsare taken into account in different ways. Althoughthese relationships may be useful for analysis of theimportance of the factors themselves, it is sufficient forpractical purposes to use

D* � � D (9.52)a 0

in which �a is an ‘‘apparent tortuosity factor’’ that takesseveral of the other factors into account, and use valuesof D* measured under representative conditions. Theeffective coefficient for diffusion of different chemicalsthrough saturated soil is usually in the range of about2 � 10�10 to 2 � 10�9 m2/s, although the values canbe one or more orders of magnitude lower in highlycompacted clays and clays, such as bentonite, that canbehave as semipermeable membranes (Malusis andShackelford, 2002b). Values for compacted clays arerather insensitive to molding water content or methodof compaction (Shackelford and Daniel, 1991b), instark contrast to the hydraulic conductivity, which mayvary over a few orders of magnitude as a result ofchanges in these factors. This suggests that soil fabricdifferences have relatively minor influence on the ef-fective diffusion coefficient.

Whereas Fick’s first law, Eq. (9.5), applies forsteady-state diffusion, Fick’s second law describes

transient diffusion, that is, the time rate of change ofconcentration with distance:

2�c � c� D* (9.53)2�t �x

For transient diffusion with constant effective diffusioncoefficient D*, the solution for this equation is of ex-actly the same form as that for the Terzaghi equationfor clay consolidation and that for one-dimensionaltransient heat flow.

An error function solution for Eq. (9.53) (Ogata,1970; Freeze and Cherry, 1979), for the case of one-dimensional diffusion from a layer at a constant sourceconcentration C0 into a layer having a sufficiently lowinitial concentration that it can be taken as zero at t �0, is

C x x� erfc � 1 � erf (9.54)

C 2D*t 2D*t0

where C is the concentration at any time at distance �from the source.

Curves of relative concentration as a function ofdepth for different times after the start of chloride dif-fusion are shown in Fig. 9.19a (Quigley, 1989). Aneffective diffusion coefficient for chloride of 6.47 �10�10 m2/s was assumed. Also shown (Fig. 9.19b) isthe migration velocity of the C /C0 front within the soilas a function of time. As chloride is one of the morerapidly diffusing ionic species, Fig. 9.19 provides abasis for estimating maximum probable migration dis-tances and concentrations as a function of time thatresult solely from diffusion.

When there are adsorption–desorption reactions,chemical reactions such as precipitation–solution, ra-dioactive decay, and/or biological processes occurringduring diffusion, the analysis becomes more complexthan given by the foregoing equations. For adsorption–desorption reactions and the assumption that there islinearity between the amount adsorbed and the equi-librium concentration, Eq. (9.53) is often written as

2�c D* � c� (9.55)2�t R �xd

where Rd is termed the retardation factor, and it isdefined by

�dR � 1 � K (9.56)d d�

in which �d is the bulk dry density of the soil, � is the

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Figure 9.19 Time rate of chloride diffusion (from Quigley,1989). (a) Relative concentration as a function of depth afterdifferent times and (b) velocity of migration of the front hav-ing a concentration C /C0 of 0.5.

volumetric water content, that is, the volume of waterdivided by the total volume (porosity in the case of asaturated soil), and Kd is the distribution coefficient.The distribution coefficient defines the amount of agiven constituent that is adsorbed or desorbed by a soilfor a unit increase or decrease in the equilibrium con-centration in solution. Other reactions influencing theamount in free solution relative to that fixed in the soil(e.g., by precipitation) may be included in Kd, depend-ing on the method for measurement and the conditionsbeing modeled. Distribution coefficients are usually de-termined from adsorption isotherms, and they may beconstants for a given soil–chemical system or varywith concentration, pH, and temperature. More de-

tailed discussions of distribution coefficients and theirdetermination are given by Freeze and Cherry (1979),Quigley et al., (1987), Quigley (1989), and Shackel-ford and Daniel (1991a, b).

9.8 TYPICAL RANGES OF FLOWPARAMETERS

Usual ranges for the values of the direct flow conduc-tivities for hydraulic, thermal, electrical, and diffusivechemical flows are given in Table 9.5. These rangesare for fine-grained soils, that is, silts, silty clays,clayey silts, and clays. They are for full saturation;values for partly saturated soils can be much lower.

Also listed in Table 9.5 are values for electroosmoticconductivity, osmotic efficiency, and ionic mobility.These properties are needed for analysis of couplingof hydraulic, electrical, and chemical flows, and theyare discussed further later.

9.9 SIMULTANEOUS FLOWS OF WATER,CURRENT, AND SALTS THROUGHSOIL-COUPLED FLOWS

Usually there are simultaneous flows of different typesthrough soils and rocks, even when only one type ofdriving force is acting. For example, when pore watercontaining chemicals flows under the action of a hy-draulic gradient, there is a concurrent flow of chemicalthrough the soil. This type of chemical transport istermed advection. In addition, owing to the existenceof surface charges on soil particles, especially clays,there are nonuniform distributions of cations and ani-ons within soil pores resulting from the attraction ofcations to and repulsion of anions from the negativelycharged particle surfaces. The net negativity of clayparticles is caused primarily by isomorphous substitu-tions within the crystal structure, as discussed in Chap-ter 3, and the ionic distributions in the pore fluid aredescribed in Chapter 6. Because of the small pore sizesin fine-grained soils and the strong local electricalfields, clay layers exhibit membrane properties. Thismeans that the passage of certain ions and moleculesthrough the clay may be restricted in part or in full atboth microscopic and macroscopic levels.

Owing to these internal nonhomogeneities in iondistributions, restrictions on ion movements caused byelectrostatic attractions and repulsions, and the de-pendence of these interactions on temperature, a vari-ety of microscopic and macroscopic effects may beobserved when a wet soil mass is subjected to flow

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SIMULTANEOUS FLOWS OF WATER, CURRENT, AND SALTS THROUGH SOIL-COUPLED FLOWS 275

Table 9.5 Typical Range of Flow Parameters for Fine-Grained Soilsa

Parameter Symbol Units Minimum Maximum

Porosity n — 0.1 0.7Hydraulic

conductivitykh m s�1 1 � 10�11 1 � 10�6

Thermalconductivity

kt W m�1 K�1 0.25 2.5

Electricalconductivity

�e siemens m�1 0.01 1.0

Electro osmoticconductivity

ke m2 s�1 V�1 1 � 10�9 1 � 10�8

Diffusioncoefficient

D m2 s�1 2 � 10�10 2 � 10�9

Osmoticefficiencyb

" — 0 1.0

Ionic mobility u m2 s�1 V�1 3 � 10�9 1 � 10�8

aThe above values of flow coefficients are for saturated soil. They may bemuch less in partly saturated soil.

b0 to 1.0 is the theoretical range for the osmotic efficiency coefficient. Valuesgreater than about 0.7 are unlikely in most fine-grained materials of geotechnicalinterest.

gradients of different types. A gradient of one type Xj

can cause a flow of another type Ji, according to

J � L X (9.57)i ij j

The Lij are termed coupling coefficients. They are prop-erties that may or may not be of significant magnitudein any given soil, as discussed later. Types of coupledflow that can occur are listed in Table 9.6, along withterms commonly used to describe them.6

Of the 12 coupled flows shown in Table 9.6, severalare known to be significant in soil–water systems, atleast under some conditions. Thermoosmosis, which iswater movement under a temperature gradient, is im-portant in partly saturated soils, but of lesser impor-tance in fully saturated soils. Significant effects fromthermally driven moisture flow are found in semiaridand arid areas, in frost susceptible soils, and in expan-sive soils. An analysis of thermally driven moisture

6 Mechanical coupling also occurs in addition to the hydraulic, ther-mal, electrical, and chemical processes listed in Table 9.6. A commonmanifestation of this in geotechnical applications is the developmentof excess pore pressure and the accompanying fluid flow that resultfrom a change in applied stress. This type of coupling is usually mosteasily handled by usual soil mechanics methods. A few other typesof mechanical coupling may also exist in soils and rocks (U.S. Na-tional Committee for Rock Mechanics, 1987).

flow is developed later. Electroosmosis has been usedfor many years as a means for control of water flowand for consolidation of soils. Chemicalosmosis, theflow of water caused by a chemical gradient actingacross a clay layer, has been studied in some detailrecently, owing to its importance in waste containmentsystems.

Isothermal heat transfer, caused by heat flow alongwith water flow, has caused great difficulties in thecreation of frozen soil barriers in the presence of flow-ing groundwater. Electrically driven heat flow, the Pel-tier effect, and chemically driven heat flow, the Dufoureffect, are not known to be of significance in soils;however, they appear not to have been studied in anydetail in relation to geotechnical problems.

Streaming current, the term applied to both hydrau-lically driven electrical current and ion flows, has im-portance to both chemical flow through the ground(advection) and the development of electrical poten-tials, which may, in turn, influence both fluid and ionflows as a result of additional coupling effects. Thecomplete roles of thermoelectricity and diffusion andmembrane potentials are not yet known; however, elec-trical potentials generated by temperature and chemicalgradients are important in corrosion and in somegroundwater flow and stability problems.

Whether thermal diffusion of electrolytes, the Soreteffect, is important in soils has not been evaluated;

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Table 9.6 Direct and Coupled Flow Phenomena

Gradient X

Flow J Hydraulic Head Temperature ElectricalChemical

Concentration

Fluid HydraulicconductionDarcy’s law

Thermoosmosis Electroosmosis Chemicalosmosis

Heat Isothermal heattransfer orthermal filtration

ThermalconductionFourier’s law

Peltier effect Dufour effect

Current Streaming current ThermoelectricitySeebeck orThompson effect

ElectricconductionOhm’s law

Diffusion andmembranepotentials orsedimentationcurrent

Ion Streaming currentultrafiltration(also known ashyperfiltration)

Thermal diffusionof electrolyte orSoret effect

Electrophoresis Diffusion Fick’slaw

however, since chemical activity is highly temperaturedependent, it may be a significant process in somesystems. Finally, electrophoresis, the movement ofcharged particles in an electrical field, has been usedfor concentration of mine waste and high water contentclays.

The relative importance of chemically and electri-cally driven components of total hydraulic flow is il-lustrated in Fig. 9.20, based on data from tests onkaolinite given by Olsen (1969, 1972). The theory fordescription of coupled flows is given later. A practicalform of Eq. (9.57) for fluid flow under combined hy-draulic, chemical, and electrical gradients is

H log(C /C ) EB Aq � �k A � k A � k Ah h c eL L L(9.58)

in which kh, kc, and ke are the hydraulic, osmotic, andelectroosmotic conductivities, H is the hydraulic headdifference, E is the voltage difference, and CA and CB

are the salt concentrations on opposite sides of a claylayer of thickness L.

In the absence of an electrical gradient, the ratio ofosmotic to hydraulic flows is

q k log(C /C )hc c B A� � ( E � 0) (9.59)� �q k Hh h

and, in the absence of a chemical gradient, the ratio ofelectroosmotic flows to hydraulic flows is

q k Ehe e� ( C � 0) (9.59a)� �q k Hh h

The ratio (kc /kh) in Fig. 9.20 indicates the hydraulichead difference in centimeters of water required to givea flow rate equal to the osmotic flow caused by a 10-fold difference in salt concentration on opposite sidesof the layer. The ratio ke /kh gives the hydraulic headdifference required to balance that caused by a 1 Vdifference in electrical potentials on opposite sides ofthe layer. During consolidation, the hydraulic conduc-tivity decreases dramatically. However, the ratios kc /kh

and ke /kh increase significantly, indicating that the rel-ative importance of osmotic and electroosmotic flowsto the total flow increases. Although the data shown inFig. 9.20 are shown as a function of the consolidationpressure, the changes in the values of kc /kh and ke /kh

are really a result of the decrease in void ratio thataccompanies the increase in pressure, as may be seenin Fig. 9.20c.

These results for kaolinite provide a conservative es-timate of the importance of osmotic and electroosmoticflows because coupling effects in kaolinite are usuallysmaller than in more active clays, such as montmoril-lonite-based bentonites. In systems containing confinedclay layers acted on by chemical and/or electrical gra-dients, Darcy’s law by itself may be an insufficientbasis for prediction of hydraulic flow rates, particularlyif the clay is highly plastic and at a very low void ratio.Such conditions can be found in deeply buried clay

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QUANTIFICATION OF COUPLED FLOWS 277

Figure 9.20 Hydraulic, osmotic, and electroosmotic conductivities of kaolinite (data fromOlsen 1969, 1972): (a) consolidation curve, (b) conductivity values, and (c) conductivitiesas a function of void ratio.

and clay shale and in densely compacted clays. Formore compressible clays, the ratios kc /kh and ke /kh maybe sufficiently high to be useful for consolidation byelectrical and chemical means, as discussed later in thischapter.

9.10 QUANTIFICATION OF COUPLED FLOWS

Quantification of coupled flow processes may be doneby direct, empirical determination of the relevant pa-rameters for a particular case or by relationships de-rived from a theoretical thermodynamic analysis of thecomplete set of direct and coupled flow equations.

Each approach has advantages and limitations. It is as-sumed in the following that the soil properties remainunchanged during the flow processes, an assumptionthat may not be justified in some cases. The effects offlows of different types on the state and properties ofa soil are discussed later in this chapter. However,when properties are known to vary in a predictablemanner, their variations may be taken into account innumerical analysis methods.

Direct Observational Approach

In the general case, there may be fluid, chemical, elec-trical, and heat flows. The chemical flows can be sub-

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divided according to the particular chemical speciespresent. Each flow type may have contributions causedby gradients of another type, and their importancedepends on the values of Lij and Xj in Eq. (9.57). Acomplete and accurate description of all flows may bea formidable task.

However, in many cases, flows of only one or twotypes may be of interest, some of the gradients maynot exist, and/or some of the coupling coefficients maybe either known or assumed to be unimportant. Thematrix of flows and forces then reduces significantly,and the determination of coefficients is greatly simpli-fied. For example, if simple electroosmosis under iso-thermal conditions is considered, then Eq. (9.57) yields

q � L �(�H) � L �(�E) � L �(�C) (9.60)w HH HE HC

I � L �(�H) � L �(�E) � L �(�C) (9.61)EH EE EC

J � L �(�H) � L �(�E) � L �(�C) (9.62)C CH CE CC

where qw � water flow rateI � electrical current

Jc � chemical flow rateH � hydraulic headE � electrical potentialC � chemical concentrationLij � coupling coefficients; the first subscript

indicates the flow type and the second de-notes the type of driving force

If there are no chemical concentration differencesacross the system, then the last terms on the right-handside of Eqs. (9.60), (9.61), and (9.62) do not exist. Inthis case, Eqs. (9.60) and (9.61) become, when writtenin more familiar terms,

q � k i � k i (9.63)w h h e e

I � � i � � i (9.64)h h e e

where kh � hydraulic conductivityke � electroosmotic hydraulic conductivity�h � electrical conductivity due to hydraulic

flow�e � electrical conductivityih � hydraulic gradientie � electrical potential gradient

If permeability tests are done in the absence of anelectrical potential difference, then the hydraulic con-

ductivity coefficient kh is readily determined.7 The co-efficient of electroosmotic hydraulic conductivity isusually determined by measuring the hydraulic flowrate developed in a known DC potential field underconditions of ih � 0. The electrical conductivity �e isobtained from the same experiment through measure-ment of the electrical current.

The main advantage of this empirical, but direct, ap-proach is simplicity. It is particularly useful when onlya few of the possible couplings are likely to be im-portant and when some uncertainty in the measuredcoefficients is acceptable.

General Theory for Coupled Flows

When several flows are of interest, each resulting fromseveral gradients, a more formal methodology is nec-essary so that all relevant factors are accounted forproperly. If there are n different driving forces, thenthere will be n direct flow coefficients Lii and n(n �1) coupling coefficients Lij(i j). The determinationof these coefficients is best done within a frameworkthat provides a consistent and correct description ofeach of the flows. Irreversible thermodynamics, alsotermed nonequilibrium thermodynamics, offers a basisfor such a description. Furthermore, if the terms areproperly formulated, then Onsager’s reciprocal rela-tions apply, that is,

L � L (9.65)ij ji

and the number of coefficients to be determined is sig-nificantly reduced. In addition, the derived forms forthe coupling coefficients, when cast in terms of meas-urable and understood properties, provide a basis forrapid assessment of their importance.

The theory of irreversible thermodynamics as ap-plied to transport processes in soils is only outlinedhere. More comprehensive treatments are given byDeGroot and Mazur (1962), Fitts (1962), Katchalskyand Curran (1967), Greenberg, et al. (1973), Yeungand Mitchell (1992), and Malusis and Shackelford(2002a).

Irreversible thermodynamics is a phenomenological,macroscopic theory that provides a basis for descrip-

7 Note that unless the ends of the sample are short circuited to preventthe development of a streaming potential, there will be a small elec-troosmotic counterflow contributed by the keie term in Eq. (9.63).Streaming potentials may be up to a few tens of millivolts in soils.Streaming potential is one of four types of electrokinetic phenomenathat may exist in soils, as discussed in more detail in Section 9.16.

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SIMULTANEOUS FLOWS OF WATER, CURRENT, AND CHEMICALS 279

tion of systems that are out of equilibrium. It is basedon three postulates, namely,

1. Local equilibrium, a criterion that is satisfied iflocal perturbations are not large.

2. Linear phenomenological equations, that is,

n

J � L X ( j � 1,2, . . . , n) (9.66)�i ij jj�1

3. Validity of the Onsager reciprocal relations, acondition that is satisfied if the Ji and Xj are for-mulated properly (Onsager, 1931a, 1931b). Ex-perimental verification of the Onsager reciprocityfor many systems and processes has been ob-tained and is summarized by Miller (1960).

Both the driving forces and flows vanish in systemsthat are in equilibrium, so the deviations of thermo-dynamic variables from their equilibrium values pro-vide a suitable basis for their formulation. Thedeviations of the state parameters Ai from equilibriumare given by

0� � A � A (9.67)i i i

where is the value of the state parameter at equilib-0Ai

rium and Ai is its value in the disturbed state.Criteria for deriving the forces and flows are then

developed on the basis of the second law of thermo-dynamics, which states that at equilibrium, the entropyS is a maximum, and �i � 0. The change in entropy S that results from a change in state parameter givesthe tendency for a variable to change. Thus �S /��i isa measure of the force causing �i to change, and iscalled Xi.

The flows Ji, termed fluxes in irreversible thermo-dynamics, are given by ��i /�t, the time derivative of�i. On this basis, the resulting entropy production �per unit time becomes

ndS� � � J X (9.68)� i idt i�1

The entropy production can be related explicitly tovarious irreversible processes in terms of proper forcesand fluxes (Gray, 1966; Yeung and Mitchell, 1992). Ifthe choices satisfy Eq. (9.68), then the Onsager reci-procity relations apply.

It has been found more useful to use # � T�, thedissipation function, in which T is temperature, than �

in the formulation of the flow equations. And # is alsothe sum of products of fluxes and driving forces:

n

# � J X (9.69)� i ii�1

The units of # are energy per unit time, and it is ameasure of the rate of local free energy dissipation byirreversible processes.

Application of the thermodynamic theory of irre-versible processes requires the following steps:

1. Finding the dissipation function # for the flows2. Defining the conjugated flows Ji and driving

forces Xi from Eq. (9.69)3. Formulating the phenomenological equations in

the form of Eq. (9.66)4. Applying the Onsager reciprocal relations5. Relating the phenomenological coefficients to

measurable quantities

When the Onsager reciprocity is used, the numberof independent coefficients Lij reduces from n2 to[(n � 1)n] /2.

Application

The quantitative analysis and prediction of flowsthrough soils, for a given set of boundary conditions,depends on the values of the various phenomenologicalcoefficients in the above flow equations. Unfortunately,these are not always known with certainty, and theymay vary over wide ranges, even within an apparentlyhomogeneous soil mass. The direct flow coefficients,that is, the hydraulic, electrical, and thermal conduc-tivities, and the diffusion coefficient, exhibit thegreatest ranges of values. Thus, it is important to ex-amine these properties first before detailed analysis ofcoupled flow contributions. For many problems, it maybe sufficient to consider only the direct flows, providedthe factors influencing their values are fully appreci-ated.

9.11 SIMULTANEOUS FLOWS OF WATER,CURRENT, AND CHEMICALS

Use of irreversible thermodynamics for the descriptionof coupled flows as developed above is straightforwardin principle; however, it becomes progressively moredifficult in application as the numbers of driving forcesand different flow types increase. This is because of(1) the need for proper specification of the differentcoupling coefficients and (2) the need for independent

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Figure 9.21 Schematic diagram of system for analysis ofsimultaneous flows of water, electricity, and ions through asoil.

methods for their measurement. Thus, the analysis ofcoupled hydraulic and electrical flows or of coupledhydraulic and chemical flows is much simpler than theanalysis of a system subjected to electrical, chemical,and hydraulic gradients simultaneously. Relationshipsfor the volume flow rate of water for several cases andfor thermoelectric and thermoosmotic coupling in sat-urated soils are given by Gray (1966, 1969). The si-multaneous flows of liquid and charge in kaolinite andthe fluid volume flow rates under hydraulic, electric,and chemical gradients were studied by Olsen (1969,1972). The theory for coupled salt and water flows wasdeveloped by Greenberg (1971) and applied to flowsin a groundwater basin (Greenberg et al., 1973) and tochemicoosmotic consolidation of clay (Mitchell et al.,1973).

Equations for the simultaneous flows of water, elec-tricity, cations, and anions under hydraulic, electrical,and chemical gradients were formulated by Yeung(1990) using the formalism of irreversible thermody-namics as outlined previously. The detailed develop-ment is given by Yeung and Mitchell (1993). Theresults are given here. The chemical flow is separatedinto its anionic and cationic components in order topermit determination of their separate movements as afunction of time. This separation may be important insome problems, such as chemical transport through theground, where the fate of a particular ionic species, aheavy metal, for example, is of interest.

The analysis applies to an initially homogeneous soilmass that separates solutions of different concentra-tions of anions and cations, at different electrical po-tentials and under different hydraulic heads, as shownschematically in Fig. 9.21. Only one anion and onecation species are assumed to be present, and no ad-sorption or desorption reactions are occurring.

The driving forces are the hydraulic gradient �(�P),the electrical gradient �(�E), and the concentration-dependent parts of the chemical potential gradients ofthe cation ) and of the anion ). The fluxesc c�(� �(�c a

are the volume flow rate of the solution per unit areaJv, the electric current I, and the diffusion flow ratesof the cation and the anion per unit area relatived dJ Jc a

to the flow of water. These diffusion flows are relatedto the absolute flows according to

dJ � J � c J (9.70)i i i v

in which ci is the concentration of ion i. The set ofphenomenological equations that relates the four flowsand driving forces is

cJ � L �(�P) � L �(�E) � L �(�� )v 11 12 13 c

c� L �(�� ) (9.71)14 a

cI � L �(�P) � L �(�E) � L �(�� )21 22 23 c

c� L (�� ) (9.72)24 a

d cJ � L �(�P) � L �(�E) � L �(�� )c 31 32 33 c

c� L �(�� ) (9.73)34 a

d cJ � L �(�P) � L �(�E) � L �(�� )a 41 42 43 c

c� L �(�� ) (9.74)44 a

These equations contain 4 conductivity coefficients Lii

and 12 coupling coefficients Lij. As a result of Onsagerreciprocity, however, the number of independent cou-pling coefficients reduces because

L � L12 21

L � L13 31

L � L14 41

L � L23 32

L � L24 42

L � L34 43

Thus there are 10 independent coefficients neededfor a full description of hydraulic, electrical, anionic,

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SIMULTANEOUS FLOWS OF WATER, CURRENT, AND CHEMICALS 281

and cationic flows through a system subjected to hy-draulic, electrical, and chemical gradients. If three ofthe four forces can be set equal to zero during a mea-surement of the flow under the fourth force, then theratio of the flow rate to that force will give the valueof its corresponding Lij. However, such measurementsare not always possible or convenient. Accordingly,two forces and one flow are usually set to zero and theappropriate Lij are evaluated by solution of simultane-ous equations. For measurements of hydraulic conduc-tivity, electroosmotic hydraulic conductivity, electricalconductivity, osmotic efficiency, and effective diffusioncoefficients done in the usual manner in geotechnicaland chemical laboratories, the detailed application ofirreversible thermodynamic theory led Yeung (1990)and Yeung and Mitchell (1993) to the following defi-nitions for the Lij. It was assumed in the derivationsthat the solution is dilute and there are no interactionsbetween cations and anions.8

k L Lh 12 21L � � (9.75)11 n Lw 22

keL � L � (9.76)12 21 n

�"c k L Lc h 12 23L � L � � (9.77)13 31 n Lw 22

�"c k L La h 12 24L � L � � (9.78)14 41 n Lw 22

�eL � (9.79)22 n

L � L � c u* (9.80)23 32 c c

L � L � �c u* (9.81)24 42 a a

D* cc cL � (9.82)33 RT

L � L � 0 (9.83)34 43

D* ca aL � (9.84)44 RT

where kh � hydraulic conductivity as usually mea-sured (no electrical short circuiting)

ke � coefficient of electroosmotic hydraulicconductivity

8 The Lij coefficients in Eqs. (9.75) to (9.84) were derived in termsof the cross-sectional area of the soil voids. They may be redefinedin terms of the total cross-sectional area by multiplying each termon the right-hand side by the porosity, n.

�e � bulk electrical conductivity of the soil" � coefficient of osmotic efficiency

w � unit weight of watercc � concentration of cationca � concentration of anion

�u*c effective ionic mobility of the cation�u*a effective ionic mobility of the anion�D*c effective diffusion coefficient of the cation�D*a effective diffusion coefficient of the anion

n � soil porosityR � universal gas constant (8.314 J K�1 mol�1)T � absolute temperature (K)

Subsequently, Manassero and Dominijanni (2003)pointed out that the practical equations for diffusionL33 and L44 do not take the osmotic efficiency " (Sec-tion 9.13) into account, so Eqs. (9.82) and (9.84) moreproperly should be

2(1 � ")D* c k"c cL � c � (9.85) �33 c RT nw

2(1 � ")D* c k"a aL � c � (9.86) �44 a RT nw

This modification becomes important in clays whereinosmotic efficiency, that is, the ability of the clay torestrict the flow of ions, is high.

As the flows of ions relative to the soil are of moreinterest than relative to the water, Eq. (9.70) and Eqs.(9.73) and (9.74) can be combined to give

J � (L � c L ) �(�h) � (L � c L )�(�E)c 31 c 11 w 32 c 12

RT� (L � c L ) �(�c )33 c 13 ccc

RT� (L � c L ) �(�c )34 c 14 aca

(9.87)

J � (L � c L ) �(�h) � (L � c L )�(�E)a 41 a 11 w 42 a 12

RT� (L � c L ) �(�c )43 a 13 ccc

RT� (L � c L ) �(�c )44 a 14 aca

(9.88)

where �(�h) is the hydraulic gradient. In Eqs. (9.87)and (9.88) the gradient of the chemical potential hasbeen replaced by the gradient of the concentration ac-cording to

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RTc�(�� ) � �(�c ) (9.89)i ici

These equations reduce to the known solutions forspecial cases such as chemical diffusion, advection–dispersion, osmotic pressure according to the van’tHoff equation [see Eq. (9.98)], osmosis, and ultra-filtration. They predict reasonably well the distributionof single cations and anions as a function of time andposition in compacted clay during the simultaneous ap-plication of hydraulic, electrical, and chemical gradi-ents (Mitchell and Yeung, 1990).

The analysis of multicomponent systems is morecomplex. The use of averaged chemical properties andthe assumption of composite single species of anionsand cations may yield reasonable approximate solu-tions in some cases. Malusis and Shackelford (2002a)present a more general theory for coupled chemicaland hydraulic flow, based on an extension of the Yeungand Mitchell (1993) formulation, which accounts formulticomponent pore fluids and ion exchange proc-esses occurring during transport.9

The flow equations can be incorporated into numer-ical models for the solution of transient flow problems.Conservation of mass of species i requires that

�ci � ��J � G (9.90)i i�t

in which Gi is a source–sink term describing the ad-dition or removal rate of species i from the solution.As commonly used in groundwater flow analyses ofcontaminant transport, Gi is given by

�K �K �cd d iG � 1 � � c � (9.90a) �i i in n �t

where �i is the decay constant of species i, � is thebulk dry density of the soil, Kd is the distribution co-efficient, and n is the soil porosity. As defined previ-ously, the distribution constant is the ratio of theamount of chemical adsorbed on the soil to that insolution. The quantity in the brackets on the right-handside of Eq. (9.90) is the retardation factor Rd definedby Eq. (9.56).

Advection rather than diffusion is the dominantchemical transport mechanism in coarse-grained soils.

9 Malusis and Shackelford (2002a) defined parameters in terms of thetotal cross-sectional area for flow rather than the cross-sectional areaof voids as used in the development of Eqs. (9.75) through (9.84).

At the pore scale level, the fluid particles carrying dis-solved chemicals move at different speeds because oftortuous flow paths around the soil grains and variablevelocity distribution in the pores, ranging from zero atthe soil particle surfaces to a maximum along the cen-terline of the pore. This results in hydrodynamic dis-persion and a zone of mixing rather than a sharpboundary between two flowing solutions of differentconcentrations. Mathematically, this is accounted forby adding a dispersion term to the diffusion coefficientin the L33 and L44 terms to account for the deviationof actual motion of fluid particles from the overall oraverage movement described by Darcy’s law. More de-tails can be found in groundwater and contaminationtextbooks such as Freeze and Cherry (1979) and Dom-inico and Schwartz (1997).

Numerical models are available for groundwaterflow and contaminant transport into which the aboveflow equations can be introduced (e.g., Anderson andWoessner, 1992; Zheng and Bennett, 2002). The mostwidely used groundwater flow numerical code isMODFLOW developed by the United States Geolog-ical Survey (USGS); various updated versions areavailable (e.g., Harbaugh et al., 2000). To solvesingle-species contaminant transport problems ingroundwater, MT3DMS (Zheng and Wang, 1999) canbe used. The code utilizes the flow solutions fromMODFLOW. More complex multispecies reactions canbe simulated by RT3D (Clement, 1997). POLLUTE(Rowe and Booker, 1997) provides ‘‘one- and one-half-dimensional’’ solution to the advection–dispersionequation and is widely used in landfill design. A va-riety of public domain groundwater flow and contam-inant transport codes is available from the web sites ofthe USGS, the U.S. Environmental Protection Agency(U.S. EPA), and the U.S. Salinity Laboratory.

9.12 ELECTROKINETIC PHENOMENA

Coupling between electrical and hydraulic flows andgradients can generate four related electrokinetic phe-nomena in materials such as fine-grained soils, wherethere are charged particles balanced by mobile coun-tercharges. Each involves relative movements of elec-tricity, charged surfaces, and liquid phases, as shownschematically in Fig. 9.22.

Electroosmosis

When an electrical potential is applied across a wetsoil mass, cations are attracted to the cathode and an-ions to the anode (Fig. 9.22a). As ions migrate, they

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ELECTROKINETIC PHENOMENA 283

Figure 9.22 Electrokinetic phenomena: (a) electroosmosis, (b) streaming potential, (c) elec-trophoresis, and (d) migration or sedimentation potential.

carry their water of hydration and exert a viscous dragon the water around them. Since there are more mobilecations than anions in a soil containing negativelycharged clay particles, there is a net water flow towardthe cathode. This flow is termed electroosmosis, andits magnitude depends on ke, the coefficient of elec-troosmotic hydraulic conductivity and the voltage gra-dient, as considered in more detail later.

Streaming Potential

When water flows through a soil under a hydraulicgradient (Fig. 9.22b), double-layer charges are dis-placed in the direction of flow. This generates an elec-trical potential difference that is proportional to thehydraulic flow rate, called the streaming potential, be-tween the opposite ends of the soil mass. Streamingpotentials up to several tens of millivolts have beenmeasured in clays.

Electrophoresis

If a DC field is placed across a colloidal suspension,charged particles are attracted electrostatically to one

of the electrodes and repelled from the other. Nega-tively charged clay particles move toward the anode asshown in Fig. 9.22c. This is called electrophoresis.Electrophoresis involves discrete particle transportthrough water; electroosmosis involves water transportthrough a continuous soil particle network.

Migration or Sedimentation Potential

The movement of charged particles such as clay rela-tive to a solution, as during gravitational settling, forexample, generates a potential difference, as shown inFig. 9.22d. This is caused by the viscous drag of thewater that retards the movement of the diffuse layercations relative to the particles.

Of the four electrokinetic phenomena, electroos-mosis has been given the most attention in geotechni-cal engineering because of its practical value fortransporting water in fine-grained soils. It has beenused for dewatering, soft ground consolidation, groutinjection, and the containment and extraction of chem-icals in the ground. These applications are consideredin a later section.

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9.13 TRANSPORT COEFFICIENTS AND THEIMPORTANCE OF COUPLED FLOWS

To assess conditions where coupled chemical, electri-cal, and hydraulic flows will be significant relative todirect flows, it is necessary to know the values of theLij relative to the Lii. Estimates can be made by con-sidering the probable values of the soil state parametersand the several flow and transport coefficients given inEqs. (9.75) to (9.84). Typical ranges are given in Table9.5.

In Table 9.5 the diffusion coefficients and ionic mo-bilities for cations and anions are considered togethersince they lie within similar ranges for most species.Values of ionic mobility for specific ions in dilute so-lution are given in standard chemical references, forexample, Dean (1973), and values of diffusion coeffi-cients are given in Tables 9.3 and 9.4. Ionic mobilityis related to the diffusion coefficient according to

D �z �Fi iu � (9.91)i RT

in which zi is the ionic valence and F is Faraday’sconstant. Similarly to the diffusion coefficients, theionic mobilities are considerably less in a soil than ina free solution, especially in a fine-grained soil.

The importance of coupled flows to fluid, electricalcurrent, and chemical transport through soil under dif-ferent conditions can be examined by study of the con-tributions of the different terms in Eqs. (9.71), (9.72),(9.87), and (9.88). For this purpose, the equations havebeen rewritten in one-dimensional form and in termsof the hydraulic, electrical, and chemical concentrationgradients: ih � �dh /dx, ie � �dV /dx, and ic � �dc /dx, respectively. In addition, the chemical flows havebeen represented by a single equation. This assumesthat all dissolved species are moving together. Termsinvolving the ionic mobility u do not exist in such aformulation because the cations and anions move to-gether, with the effects of electrical fields assumed toaccelerate the slower moving ions and to retard thefaster moving ions. Thus there is no net transfer ofelectric charge due to ionic movement. The Lij coeffi-cients have been replaced by the physical and chemicalquantities that determine them, as given by Eqs. (9.74)through (9.85). The resulting equations are the follow-ing. For fluid flow:

2k k k "kh e e hJ � � i � i � RT � i � � �v w h e cn � n n ne w

(9.92)

For electrical current flow:

k �e w eI � i � i (9.93) � �h en n

For chemical flow relative to the soil:

2(1 � ")ck ck ckh e w eJ � � i � i � �c h en n� ne

"ckh� D* � RT i (9.94) � cnw

Coupling Influences on Hydraulic Flow

In the absence of applied electrical and chemical gra-dients, flow under a hydraulic gradient is given by thefirst bracketed term on the right-hand side of Eq.(9.92). It contains the quantity , which com-2k /n�e w e

pensates for the electroosmotic counterflow generatedby the streaming potential, which causes the measuredvalue of kh to be slightly less than the true value ofL11.

As it is not usual practice to short-circuit betweenthe ends of samples during hydraulic conductivity test-ing, the second bracketed term on the right-hand sideof Eq. (9.92) is not zero. This term represents an elec-troosmotic counterflow that results from the streamingpotential and acts in the direction opposite to the hy-draulically driven flow. Analysis based on the valuesof properties in Table 9.5, as well as the results ofmeasurements, for example, Michaels and Lin (1954)and Olsen (1962) show that this counterflow is negli-gible in most cases, but it may become significant rel-ative to the true hydraulic conductivity for soils of verylow hydraulic conductivity, for example, kh � 1 �10�10 m/s. For example, for a value of ke of 5 � 10�9

m2/s-V, an electrical conductivity of 0.01 mho/m, anda porosity of 35 percent, the counterflow term is 0.7� 10�10 m/s.

In the presence of an applied DC field the secondbracketed term on the right-hand side of Eq. (9.92) canbe very large relative to hydraulic flow in soils finerthan silts, as ke, which typically ranges within onlynarrow limits, is large relative to kh; that is, kh is lessthan 1 � 10�8 m/s in these soils. The relative effect-iveness of hydraulic and electrical driving forces forwater movement can be assessed by comparing gra-dients needed to give equal flow rates. They will beequal if

k i � k i (9.95)e e h h

The hydraulic gradient required to balance the elec-troosmotic flow then becomes

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TRANSPORT COEFFICIENTS AND THE IMPORTANCE OF COUPLED FLOWS 285

Figure 9.23 Theoretical values of osmotic pressure as afunction of concentration difference across a clay layer fordifferent values of osmotic efficiency coefficient, ". (T �20�C).

kei � i (9.96)h ekh

As the hydraulic conductivity of soils in which elec-troosmosis is likely to be used is usually of the orderof 1 � 10�9 m/s or less, whereas ke is in the range of1 � 10�9 to 1 � 10�8 m2/s � V, it follows that evensmall electrical gradients can balance flows caused bylarge hydraulic gradients. Because of this, and becauseke is insensitive to particle size while kh decreases rap-idly with decreasing particle size, electroosmosis is ef-fective in fine-grained soils, as discussed further inSection 9.15.

Chemically driven hydraulic flow is given by the lastterm on the right-hand side of Eq. (9.92). It dependsprimarily on the osmotic efficiency ". Osmotic effi-ciency has an important influence on the movement ofchemicals through a soil, the development of osmoticpressure, and the effectiveness of clay barriers forchemical waste containment.

Osmotic Efficiency The osmotic efficiency of clay,a slurry wall, a geosynthetic clay liner (GCL), or otherseepage and containment barrier is a measure of thematerial’s effectiveness in causing hydraulic flow un-der an osmotic pressure gradient and of its ability toact as a semipermeable membrane in preventing thepassage of ions, while allowing the passage of water.The osmotic pressure concept can be better appreciatedby rewriting the last term in Eq. (9.92):

k k RT c 1h h" RTi � " (9.97)c n n xw w

This form is analogous to Darcy’s law, with the quan-tity RT c /w being the head difference. The osmoticefficiency is a measure of the extent to which this the-oretical pressure difference actually develops. Theo-retical values of osmotic pressure, calculated using thevan’t Hoff equation, as a function of concentration dif-ference for different values of osmotic efficiency areshown in Fig. 9.23.

The van’t Hoff equation for osmotic pressure is

� � kT (n � n ) � RT(c � c ) (9.98)� iA iB iA iB

where k is the Boltzmann constant (gas constant permolecule), R is the gas constant per molecule, T is theabsolute temperature, ni is concentration in particlesper unit volume, and ci is the molar concentration. Thevan’t Hoff equation applies for ideal and relatively di-lute solution concentrations (Malusis and Shackelford,2002c). According to Fritz (1986) the error is low(�5%) for 1�1 electrolytes (e.g., NaCl, KCl) and con-centrations �1.0 M.

Values of osmotic efficiency coefficient, ", ormembrane efficiency (" expressed as a percentage),have been measured for clays and geosynthetic clayliners; for example, Kemper and Rollins (1966), Leteyet al. (1969), Olsen (1969), Kemper and Quirk (1972),Bresler (1973), Elrick et al. (1976), Barbour and Fred-lund (1989), and Malusis and Shackelford (2002b,2002c). Values of membrane efficiency from 0 to 100percent have been determined, depending on the claytype, porosity, and type and concentration of salts insolution. The results of many determinations weresummarized by Bresler (1973) as shown in Fig. 9.24.The efficiency is shown as a function of a normalizingparameter, the half distance between particles b timesthe square root of the solution concentration .c

To put these relations into more familiar terms foruse in geotechnical studies, the half spacings were con-verted to water contents on the assumption of uniformwater layer thicknesses on all particles, using specificsurface areas corresponding to different clay types andnoting that volumetric water content equals surfacearea times layer thickness. The relationship betweenspecific surface area and liquid limit (LL) obtained byFarrar and Coleman (1967) for 19 British clays

LL � 19 � 0.56A (�20%) (9.99)s

in which the specific surface area As is in square metersper gram, was then used to obtain the relationshipsshown in Fig. 9.25. The computed efficiencies shownin Fig. 9.25 should be considered upper bounds be-cause the assumption of uniform water distributionover the full surface area underestimates the effectiveparticle spacing in most cases. In most clays, espe-

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Figure 9.24 Osmotic efficiency coefficient as a function ofb where c is concentration of monovalent anion in nor-cmality and 2b is the effective spacing between particle sur-faces (from Bresler, 1973).

cially those with divalent adsorbed cations, individualclay plates associate in clusters giving an effective spe-cific surface that is less than that determined by mostmethods of measurement. This means that the curvesin Fig. 9.25 should in reality be displaced to the left.

High osmotic efficiencies are developed at low watercontents, that is, in very dense, low-porosity clays, andin dilute electrolyte systems. Malusis and Shackelford(2002a, 2002b, 2002c) found that the osmotic effi-ciency decreases with increasing solute concentrationand attribute this to compression of the diffuse doublelayers adjacent to the clay particles.

Water flow by osmosis can be significant relative tohydraulically driven water flow in heavily overconsol-idated clay and clay shale, where the void ratio is lowand the hydraulic conductivity is also very low. Suchflow may be important in geological processes (Olsen1969, 1972). Densely compacted clay barriers forwaste containment, usually composed of bentonite,possess osmotic membrane properties. As the chemical

concentrations on the inside of a lined repositoryshould be greater than on the outside, osmoticallydriven water flow should be directed from the outsidetoward the inside. The greater the osmotic efficiencythe greater the driving force for this flow. Furthermore,if the efficiency is high, then outward diffusion of con-tained chemicals is restricted (Malusis and Shack-elford, 2002b). In diffusion-dominated containmentbarriers, the effect of solute restriction on reducing sol-ute diffusion is likely substantially more significantthan the effect of osmotic flow (Shackelford et al.,2001).

Coupling Influences on Electrical Flow

Substitution of values for the parameters in Eq. (9.93)indicates, as would be expected, that electrical currentflow is dominated completely by the electrical gradientie. In the presence of an applied voltage difference, theother terms are of little importance, even if the move-ments of anions and cations are considered separatelyand the contributions due to ionic mobility are takeninto account. On the other hand, when a soil layer be-haves as an open electrical circuit, small electrical po-tentials, measured in millivolts, may exist if there arehydraulic and/or chemical flows. This may be seen bysetting I � 0 in Eq. (9.93) and solving for ie, whichmust have value if ih has value. These small potentialsand flows are important in such processes as corrosionand electroosmotic counterflow.

Coupling Influences on Chemical Flow

Equation (9.94) provides a description of chemicaltransport relative to the soil. It contains two terms thatinfluence chemical flow under a hydraulic gradient;one for chemical transport under an electrical gradient,and one for transport of chemical under a chemicalgradient. The first term in the first bracket of the right-hand side of Eq. (9.94) describes advective transport.As would be expected, the smaller the osmotic effi-ciency, the more chemical flow through the soil is pos-sible. The second term in the same bracket simplyreflects the advective flow reduction that would resultfrom electroosmotic counterflow caused by develop-ment of a streaming potential. As noted earlier, thisflow will be small, and its contribution to the total flowwill be small, except in clays of very low hydraulicand electrical conductivities. Advective transport is thedominant means for chemical flow for soils having ahydraulic conductivity greater than about 1 � 10�9

m/s.The importance of an electrical driving force for

chemical flow depends on the electrical potential gra-dient. For a unit gradient, that is, 1 V/m, chemical flow

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TRANSPORT COEFFICIENTS AND THE IMPORTANCE OF COUPLED FLOWS 287

Figure 9.25 Osmotic efficiency of clays as a function of water content.

quantities are comparable to those by advective flowunder a unit hydraulic gradient in a clay having a hy-draulic conductivity of about 1 � 10�9 m/s. Electri-cally driven chemical flow is relatively less importantin higher permeability soils and more important insoils with lower kh. In cases where the electricallydriven chemical transport is of interest, as in electro-kinetic waste containment barrier applications, anion,cation, and nonionic chemical flows must be consid-ered separately using expanded relationships such asgiven by Eqs. (9.87) and (9.88).

The last bracketed quantity of Eq. (9.94) representsdiffusive flow under chemical gradients. The quantityD*ic gives the normal diffusive flow rate. The secondterm represents a restriction on this flow that dependson the clay’s osmotic efficiency, "; that is, if the clayacts as an effective semipermeable membrane, diffu-sive flow of chemicals is restricted. However, even un-

der conditions where the value of " is low such thatthe second term in the bracket is negligible, chemicaltransport by diffusion is significant relative to advec-tive chemical transport in soils with hydraulic conduc-tivity values less than about 1 � 10�9 to 1 � 10�10

m/s for chemicals with diffusion coefficients in therange given by Table 9.7, that is, 2 � 10�10 to 2 �10�9 m2/s.

This is illustrated by Fig. 9.26 from Shackelford(1988), which shows the relative importance of advec-tive and diffusive chemical flows on the transit timethrough a 0.91-m-thick compacted clay liner having aporosity of 0.5 acted on by a hydraulic gradient of1.33. A diffusion coefficient of 6 � 10�10 m2/s wasassumed. The transit time is defined as the time re-quired for the solute concentration on the dischargeside to reach 50 percent of that on the upstream side.For hydraulic conductivity values less than about 2 �

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Figure 9.26 Transit times for chemical flow through a 0.91-m-thick compacted clay liner having a porosity of 50 percentand acted on by a hydraulic gradient of 1.33 (from Shack-elford, 1988).

10�9 m/s the transit time in the absence of diffusionwould be very long. For diffusion alone the transit timewould be about 47 years.

Most compacted clay barriers and geosynthetic clayliners are likely to have hydraulic conductivity valuesin the range of 1 � 10�11 to 1 � 10�9 m/s, with thelatter value being the upper limit allowed by the U.S.EPA for most waste containment applications. In thisrange, diffusion reduces the transit time significantlyin comparison to what it would be due to advectionalone. This is shown by the curve labeled advection–dispersion in Fig. 9.26. The calculations were doneusing the well-known advection–dispersion equation(Ogata and Banks, 1961) in which the dispersion termincludes both mechanical mixing and diffusion. Me-chanical mixing is negligible in low-permeability ma-terials such as compacted clay.

9.14 COMPATIBILITY—EFFECTS OFCHEMICAL FLOWS ON PROPERTIES

Chemical Compatibility and Hydraulic Conductivity

The compatibility between waste chemicals, especiallyliquid organics, and compacted clay liners and slurrywall barriers constructed to contain them must be con-sidered in the design of waste containment barriers.Numerous studies have been done to evaluate chemicaleffects on clay hydraulic conductivity because of fearsthat prolonged exposure may compromise the integrityof the liners and barriers and because tests have shownthat under some conditions clay can shrink and crackwhen permeated by certain classes of chemicals. Sum-maries of the results of chemical compatibility studiesare given by Mitchell and Madsen (1987) and Quigleyand Fernandez (1989), and factors controlling the long-

term stability of clay liners are discussed by Mitchelland Jaber (1990).

Rigid wall, flexible wall, and consolidometer per-meameters are used for compatibility testing in the lab-oratory. These three types of test apparatus are shownschematically in Fig. 9.27. Tests done in a rigid wallsystem overestimate hydraulic conductivity wheneverchemical–clay interactions cause shrinkage and crack-ing; however, a rigid wall system is well suited forqualitative determination of whether or not there maybe adverse interactions. In the flexible wall system thelateral confining pressure prevents cracks from open-ing; thus there is risk of underestimating the hydraulicconductivity of some soils. The consolidometer per-meameter system allows for testing clays under a rangeof overburden stress states that are representative ofthose in the field and for quantitative assessment of theeffects of chemical interactions on volume stability andhydraulic conductivity. More details of these perme-ameters are given by Daniel (1994).

The effects of chemicals on the hydraulic conduc-tivity of high water content clays such as used in slurrywalls are likely to be much greater than on lower watercontent, high-density clays as used in compacted clayliners. This is because of the greater particle mobilityand easier opportunity for fabric changes in a higherwater content system. A high compactive effort or aneffective confining stress greater than about 70 kPa canmake properly compacted clay invulnerable to attackby concentrated organic chemicals (Broderick andDaniel, 1990). However, it is not always possible toensure high-density compaction or to maintain highconfining pressures, or eliminate all construction de-fects, so it is useful to know the general effects ofdifferent types of chemicals on hydraulic conductivity.

The influences of inorganic chemicals on hydraulicconductivity are consistent with (1) their effects on thedouble-layer and interparticle forces in relation to floc-culation, dispersion, shrinkage, and swelling, (2) theireffects on surface and edge charges on particles andthe influences of these charges on flocculation and de-flocculation, and (3) their effects on pH.

Acids can dissolve carbonates, iron oxides, and thealumina octahedral layers of clay minerals. Bases candissolve silica tetrahedral layers, and to a lesser extent,alumina octahedral layers of clay minerals. Removalof dissolved material can cause increases in hydraulicconductivity, whereas precipitation can clog pores andreduce hydraulic conductivity.

The most important factors controlling the effects oforganic chemicals on hydraulic conductivity are (1)water solubility, (2) dielectric constant, (3) polarity,and (4) whether or not the soil is exposed to the pureorganic or a dilute solution. Exposure of clay barriers

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COMPATIBILITY—EFFECTS OF CHEMICAL FLOWS ON PROPERTIES 289

Figure 9.27 Three types of permeameter for compatibility testing: (a) rigid wall, (b) flexiblewall, and (c) consolidometer permeameter (from Day, 1984).

to water-insoluble pure or concentrated organics islikely only in the case of spills, leaking tanks, and withdense non-aqueous-phase liquids (DNAPLs) or ‘‘sink-ers’’ that accumulate above low spots in liners. Somegeneral conclusions about the influences of organicson the hydraulic conductivity are:

1. Solutions of organic compounds having a lowsolubility in water, such as hydrocarbons, haveno large effect on the hydraulic conductivity. Thisis in contrast to dilute solutions of inorganic com-pounds that may have significant effects as aresult of their influence on flocculation and dis-persion of the clay particles.

2. Water-soluble organics, such as simple alcoholsand ketones, have no effect on hydraulic conduc-tivity at concentrations less than about 75 to 80percent.

3. Many water-insoluble organic liquids (i.e., non-aquoues-phase liquids, NAPLs) can cause shrink-age and cracking of clays, with concurrentincreases in hydraulic conductivity.

4. Hydraulic conductivity increases caused by per-meation by organics are partly reversible whenwater is reintroduced as the permeant.

5. Concentrated hydrophobic compounds (likemany NAPLs) permeate soils through cracks andmacropores. Water remains within mini- and mi-cropores.

6. Hydrophilic compounds permeate the soil moreuniformly than NAPLs, as the polar moleculescan replace the water in hydration layers of thecations and are more readily adsorbed on particlesurfaces.

7. Organic acids can dissolve carbonates and ironoxides. Buffering of the acid can lead to precip-

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Figure 9.28 (a) Hydraulic conductivity and (b) intrinsic permeability of compacted Sarniaclay permeated with leachate–dioxane mixtures. Initial tests run using water (●) followed byleachate–chemical solution (�). (from Fernandez and Quigley, 1988). Reproduced with per-mission from the National Research Council of Canada.

itation and pore clogging downstream. However,after long time periods these precipitates may beredissolved and removed, thus leading to an in-crease in hydraulic conductivity.

8. Pure bases can cause a large increase in the hy-draulic conductivity, whereas concentrations at orbelow the solubility limit in water have no effect.

9. Organic acids do not cause large-scale dissolu-tion of clay particles.

The combined effects of confining pressure and con-centration, as well as permeant density and viscosity,are illustrated by Fig. 9.28 (Fernandez and Quigley,1988). The data are for water-compacted, brown Sarniaclay permeated by solutions of dioxane in domesticlandfill leachate. Increased hydrocarbon concentrationcaused a decrease in hydraulic conductivity up to con-centrations of about 70 percent, after which the hy-draulic conductivity increased by about three orders ofmagnitude for pure dioxane (Fig. 9.28a), for samplesthat were unconfined by a vertical stress ( � 0). On��vthe other hand, the data points for samples maintainedunder a vertical confining stress of 160 kPa indicatedno effect of the dioxane on hydraulic conductivity rel-

ative to that measured with water. The decreases inhydraulic conductivity for dioxane concentrations upto 70 percent can be accounted for in terms of fluiddensity and viscosity, as may be seen in Fig. 9.28bwhere the intrinsic values of permeability are shown.As noted earlier in this chapter, the intrinsic permea-bility is defined by K � k� /.

Although many chemicals do not have significanteffect on the hydraulic conductivity of clay barriers,this does not mean that they will not be transportedthrough clay. Unless adsorbed by the clay or by or-ganic matter, the chemicals will be transported by ad-vection and diffusion. Furthermore, the actual transittime through a barrier by advection, that is, the timefor chemicals moving with the seepage water, may befar less than estimated using the conventional seepagevelocity. The seepage velocity is usually defined as theDarcy velocity khih, divided by the total porosity n. Insystems with unequal pore sizes the flow is almost to-tally through mini- and macropores, which comprisethe effective porosity ne, which may be much less thanthe total porosity. Thus effective compaction of claybarriers must break down clods and aggregates to de-crease the effective pore size and increase the propor-

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ELECTROOSMOSIS 291

Table 9.7 Coefficients of Electroosmotic Permeability

No. MaterialWater

Content (%)ke in 10�5

(cm2/s-V)Approximate kh

(cm/s)

1. London clay 52.3 5.8 10�8

2. Boston blue clay 50.8 5.1 10�8

3. Kaolin 67.7 5.7 10�7

4. Clayey silt 31.7 5.0 10�6

5. Rock flour 27.2 4.5 10�7

6. Na-Montmorillonite 170 2.0 10�9

7. Na-Montmorillonite 2000 12.0 10�8

8. Mica powder 49.7 6.9 10�5

9. Fine sand 26.0 4.1 10�4

10. Quartz powder 23.5 4.3 10�4

11. As quick clay 31.0 20.0–2.5 2.0 � 10�8

12. Bootlegger Cove clay 30.0 2.4–5.0 2.0 � 10�8

13. Silty clay, West Branch Dam 32.0 3.0–6.0 1.2 � 10�8–6.5 � 10�8

14. Clayey silt, Little Pic River,Ontario

26.0 1.5 2 � 10�5

ke and water content data for Nos. 1 to 10 from Casagrande (1952). kh estimated by authors; no. 11from Bjerrum et al. (1967); no. 12 from Long and George (1967); no. 13 from Fetzer (1967); no. 14from Casagrande et al. (1961).

tion of the porosity that is effective porosity, therebyincreasing the transit time.

9.15 ELECTROOSMOSIS

The coefficient of electroosmotic hydraulic conductiv-ity ke defines the hydraulic flow velocity under a unitelectrical gradient. Measurement of ke is made by de-termination of the flow rate of water through a soilsample of known length and cross section under aknown electrical gradient. Alternatively, a null indicat-ing system may be used or it may be deduced from astreaming potential measurement. From experience itis known that ke is generally in the range of 1 � 10�9

to 1 � 10�8 m2/s V (m/s per V/m) and that it is ofthe same order of magnitude for most soil types, asmay be seen by the values for different soils and afreshwater permeant given in Table 9.7.

Several theories have been proposed to explain elec-troosmosis and to provide a basis for quantitative pre-diction of flow rates.

Helmholtz and Smoluchowski Theory

This theory, based on a model introduced by Helm-holtz (1879) and refined by Smoluchowski (1914), isone of the earliest and most widely used. A liquid-filled capillary is treated as an electrical condenser with

charges of one sign on or near the surface of the walland countercharges concentrated in a layer in the liquida small distance from the wall, as shown in Fig. 9.29.10

The mobile shell of counterions is assumed to dragwater through the capillary by plug flow. There is ahigh-velocity gradient between the two plates of thecondenser as shown.

The rate of water flow is controlled by the balancebetween the electrical force causing water movementand friction between the liquid and the wall. If v is theflow velocity and � is the distance between the walland the center of the plane of mobile charge, then thevelocity gradient between the wall and the center ofpositive charge is v /�; thus, the drag force per unitarea is � dv /dx � �v /�, where � is the viscosity. Theforce per unit area from the electrical field is � E / L, where � is the surface charge density and E / Lis the electrical potential gradient. At equilibrium

v E� � � (9.100)

� L

or

10 A derivation using a Poisson–Boltzmann distribution of counter-ions adjacent to the wall gives the same result.

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Figure 9.29 Helmholtz–Smoluchowski model for electrokinetic phenomena.

L�� v (9.101)

E

From electrostatics, the potential across a condenser� is given by

�� (9.102)

D

where D is the relative permittivity, or dielectric con-stant of the pore fluid. Substitution for �� in Eq.(9.102) gives

�D Ev � (9.103)� �� L

The potential � is termed the zeta potential. It is notthe same as the surface potential of the double-layer�0 discussed in Chapter 6, although conditions thatgive high values of �0 also give high values of zetapotential. A common interpretation is that the actualslip plane in electrokinetic processes is located somesmall, but unknown, distance from the surface of par-ticles; thus � should be less than �0. Values of � in therange of 0 to �50 mV are typical for clays, with thelowest values associated with high pore water salt con-centrations.

For a single capillary of area a the flow rate is

�D Eq � va � a (9.104a)a � L

and for a bundle of N capillaries within total cross-sectional area A normal to the flow direction

�D Eq � Nq � Na (9.104b)A a � L

If the porosity is n, then the cross-sectional area ofvoids is nA, which must equal Na. Thus,

�D Eq � n A (9.105)A � L

By analogy with Darcy’s law we can write Eq.(9.105) as

q � k i A (9.106)A e e

in which ie is the electrical potential gradient E / Land ke the coefficient of electroosmotic hydraulic con-ductivity is

�Dk � n (9.107)e �

According to the Helmholtz–Smoluchowski theoryand Eq. (9.107), ke should be relatively independent ofpore size, and this is borne out by the values listed inTable 9.7. This is in contrast to the hydraulic conduc-tivity kh, which varies as the square of some effectivepore size. Because of this independence of pore size,electroosmosis can be more effective in moving waterthrough fine-grained soils than flow driven by a hy-draulic gradient.

This is illustrated by the following simple example.Consider a fine sand and a clay of hydraulic conduc-tivity kh of 1 � 10�5 m/s and 1 � 10�10 m/s, respec-tively. Both have ke values of 5 � 10�9 m2/s V. Forequal hydraulic flow rates khih � keie, so

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ELECTROOSMOSIS 293

kei � i (9.108)h ekh

If an electrical potential gradient of 20 V/m is used,substitution in Eq. (9.108) shows that ih is 0.01 for thefine sand and 1000 for the clay. This means that ahydraulic gradient of only 0.01 can move water as ef-fectively as an electrical gradient of 20 V/m in finesand. However, for the clay, a hydraulic gradient of1000 would be needed to offset the electroosmoticflow.

However, it does not follow that electroosmosis willalways be an efficient means to move water in claysbecause the above analysis does not take into accountthe power requirement to develop the potential gradientof 20 V/m or energy losses in the system. These pointsare considered further later.

Schmid Theory

The Helmholtz–Smoluchowski theory is essentially alarge-pore theory because it assumes a negligible ex-tension of the counterion layer into the pore. Also, itdoes not account for an excess of ions over thoseneeded to balance the surface charge. A model thatovercomes the first of these problems was proposed bySchmid (1950, 1951). It can be considered a small-pore theory.

The counterions are assumed to be distributed uni-formly throughout the fluid phase in the soil. The elec-trical force acts uniformly over the entire pore crosssection and gives the same velocity profile as shownby Fig. 9.29. The hydraulic flow rate through a singlecapillary of radius r is given by Poiseuille’s law:

4�rq � i (9.109)w h8�

The hydraulic seepage force per unit length causingflow is

2F � �r i (9.110)H w h

so

2rq � F (9.111)H8�

The electrical force per unit length FE is equal to thecharge times the potential, that is,

E2F � A F �r (9.112)E 0 0 L

where A0 is the concentration of wall charges in ionicequivalents per unit volume of pore fluid, and F0 is theFaraday constant. Replacement of FH by FE in Eq.(9.111) gives

4�r E F A0 0 2q � A F � r i a (9.113)a 0 0 e8� L 8�

so for a total cross section of N capillaries and area A

2A F r0 0q � ni A (9.114)A e8�

This equation shows that ke should vary as r2, whereasthe Helmholtz–Smoluchowski theory leads to ke in-dependent of pore size, as previously noted. Of the twotheories, the Helmholtz gives the better results forsoils, perhaps because most clays have a cluster or ag-gregate structure with electroosmotic flow controlledmore by the larger pores than by the intracluster pores.

Spiegler Friction Model

A completely different concept for electrokinetic proc-esses takes into account the interactions of the mobilecomponents (water and ions) on each other and of thefrictional interactions of these components with porewalls (Spiegler, 1958). This theory provides insightinto conditions leading to high electroosmotic effi-ciency.

The assumptions include:

1. Exclusion of coions,11 that is, the medium be-haves as a perfect perm-selective membrane,admitting ions of only one sign

2. Complete dissociation of pore fluid ions

The following equation for electroosmotic transportof water across a fine-grained porous material contain-ing adsorbed and free ions can be derived:

C3 � (W � H) � (9.115)C � C (X /X )1 3 34 13

in which is the true electroosmotic water flow(moles/faraday), W is the measured water transport

11 Ions of the opposite sign to the charged surface are termed coun-terions. Ions of the same sign are termed coions.

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Figure 9.30 Schematic prediction of water transport by elec-troosmosis in various clays according to the Donnan concept(from Gray, 1966).

(moles/faraday), H is the water transport by ion hy-dration (moles/faraday), C3 is the concentration of freewater in the material (mol/m3), C1 is the concentrationof mobile counterions m2, X34 is the friction coefficientbetween water and the solid wall, and X13 is the frictioncoefficient between cation and water.

Concentrations C1 and C3 are hypothetical and prob-ably less than values measured by chemical analysisbecause some ions may be immobile. Evaluation of X13

and X34 requires independent measurements of diffu-sion coefficients, conductance, transference numbers,and water transport. Thus Eq. (9.115) is limited as apredictive equation. Its real value is in providing a rel-atively simple physical representation of a complexprocess.

From Eq. (9.115),

1 � (W � H) � (9.116)

(C /C � X /X )1 3 34 13

At high water contents and for large pores, X34 /X13 →0 because X34 becomes negligible. Then

� C /C (9.117)X 3 134→0

This relationship indicates that a high water-to-cationratio implies a high rate of electroosmotic flow. At lowwater contents and for small pores, X34 will not bezero, thus reducing the flow. An increase in C1 reducesthe flow of water per faraday of current passed becausethere is less water per ion. An increase in X13 increasesthe flow because there is greater frictional drag on thewater by the ions.

Ion Hydration

Water of hydration is carried along with ions in a directcurrent electric field. The ion hydration transport H isgiven by

H � t N � t N (9.118)� �

where t� and t are the transport numbers, that is, num-bers that represent the fraction of current carried by aparticular ionic species. The numbers N� and N arethe number of moles of hydration water per mole ofcation and anion, respectively.

9.16 ELECTROOSMOSIS EFFICIENCY

Electroosmotic water flow occurs if the frictional dragbetween the ions of one sign and their surroundingwater molecules exceeds that caused by ions of the

opposite sign. The greater the difference between theconcentrations of cations and anions, the greater thenet drag on the water in the direction toward the cath-ode. The efficiency and economics of the process de-pend on the volume of water transported per unitelectrical charge passed. If the volume is high, thenmore water is transported for a given expenditure ofelectrical energy than if it is low. This volume mayvary over several orders of magnitude depending onsuch factors as soil type, water content, and electrolyteconcentration.

In a low exchange capacity soil at high water contentin a low electrolyte concentration solution, there ismuch more water per cation than in a high exchangecapacity, low water content soil having the same porewater electrolyte concentration. This, combinedwith cation-to-anion ratio considerations, leads tothe predicted water transport–water content–soiltype–electrolyte concentration relationships shownschematically in Fig. 9.30, where increasing electrolyteconcentration in the pore water results in a much

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ELECTROOSMOSIS EFFICIENCY 295

Figure 9.31 Electroosmotic water transport versus concentration of external electrolyte so-lution for homoionic kaolinite and illite at various water content (from Gray, 1966).

greater decrease in efficiency for inactive clay thanmore plastic, active clay. Tests on sodium kaolinite (in-active clay) and sodium illite (more active clay) gavethe results shown in Fig. 9.31, which agree well withthe predictions in Fig. 9.30.

The slopes and locations of the curves can be ex-plained more quantitatively in the following way. Al-ternatively to the double-layer theory given in Chapter6, the Donnan (1924) theory can be used to describeequilibrium ionic distributions in fine-grained materi-als. The basis for the Donnan theory is that at equilib-rium the potentials of the internal and externalsolutions are equal and that electroneutrality is re-quired in both phases. It may be shown (Gray, 1966;Gray and Mitchell, 1967) that the ratio R of cations toanions in the internal phase for the case of a symmet-rical electrolyte (z� � z�) is given by

� 2 1 / 2C 1 � (1 � y )R � � (9.119)

� 2 1 / 2C �1 � (1 � y )

where

2C �0y � (9.120)A �0

The concentration C0 is in the external solution, isthe mean molar activity coefficient in the external so-lution, is the mean activity coefficient in the doublelayer, and A0 is the surface charge density per unit porevolume. The parameter A0 is related to the cationexchange capacity (CEC) by

(CEC)�wA � (9.121)0 w

where �w is the density of water and w is the watercontent. The higher R, the greater is the electroosmoticwater transport, all other things equal.

From Eqs. (9.119) to (9.121) it may be deduced thatexclusion of anions is favored by a high exchangecapacity (active clay), a low water content, and lowsalinity in the external solution. However, the concen-tration of anions in the double layer builds up more

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rapidly as the salinity of the external solution increasesin inactive clays than in active clays. As a result theefficiency, as measured by volume of water per unitcharge passed, decreases much more rapidly with in-creasing electrolyte concentration than in the more ac-tive clay.

The results of electroosmosis measurements on anumber of different materials are summarized in Fig.9.32, which shows water flow rate as a function ofwater content. This figure may be used as a guide forprediction of electroosmotic flow rates. The flow ratesshown are for open systems, that is, solution was ad-mitted at the anode at the same time it was extractedfrom the cathode. Electrochemical effects (Section9.18) and water content changes were minimized inthese tests. Thus, the values can be interpreted as upperbounds on the flow rates to be expected in practice.

Values of water content, electrolyte concentration inthe pore water, and type of clay are required for elec-troosmosis efficiency estimation. Water content is read-ily measured, the electrolyte concentration is easilydetermined using a conductivity cell, and the clay typecan be determined from plasticity and grain size in-formation if mineralogical data are not available. Elec-troosmotic flow rates of 0.03 to 0.06 gal/h/amp arepredicted using Fig. 9.32 for soils 11, 13, and 14 inTable 9.7. Electrical treatment for consolidation andground strengthening was effective in these soils. Forsoil 12, however, a flow rate of 0.008 to 0.012 gal/h/amp was predicted, and electroosmosis was not effec-tive.

Saxen’s Law Prediction of Electroosmosis fromStreaming Potential

Streaming potential can be measured directly during ameasurement of hydraulic conductivity by using ahigh-impedance voltmeter and reversible electrodes.Equivalence between streaming potential and elec-troosmosis may be derived. Expansion of Eq. (9.57)for coupled hydraulic and current flows gives

q � L P � L E (9.122)h HH HE

I � L P � L E (9.123)EH EE

in which qh is the hydraulic flow rate, I is the electriccurrent, LHH and LEE are the direct flow coefficients,LHE and LEH are the coupling coefficients for hydraulicflow due to an electrical gradient and electrical flowdue to a hydraulic gradient, P is the pressure drop,and E is the electrical potential drop.

In a usual hydraulic conductivity measurement,there is no electrical current flow, so I � 0, and E isthe streaming potential. Equation (9.123) then becomes

E LEH� � (9.124) P LEE

In electroosmosis P � 0, so Eq. (9.122) is

q � L E (9.125)h HE

and Eq. (9.122) becomes

I � L E (9.126)EE

so

q Lh HE� (9.127)I LEE

By Onsager’s reciprocity theorem LEH � LHE so

q Eh � � (9.128)� � � �I P P�0 I�0

This equivalence between streaming potential and elec-troosmosis was first shown experimentally by Saxen(1892) and is known as Saxen’s law. It has been ver-ified for clay–water–electrolyte systems. Care must betaken to ensure consistency in units. For example, theelectroosmotic flow rate in gallons per hour per ampereis equal to 0.0094 times the streaming potential in mil-livolts per atmosphere.

Energy Requirements

The preceding analysis leads to a prediction of theamount of water moved per unit charge passed, forexample, gallons or cubic meters of water per hour perampere or moles per faraday. If this quantity is denotedby ki, then

q � k I (9.129)h i

Unlike ke, ki varies over a wide range, as may be seenin Fig. 9.32. The power consumption P is

EqhP � E � I � (in W) (9.130)ki

for E in volts and I in amperes. The power consump-tion per unit volume of flow is

P E�3� � 10 (in kWh) (9.131)

q kh i

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297

Figure 9.32 Electroosmotic water transport as a function of water content, soil type, andelectrolyte concentration: (a) homoionic kaolinite and illite, (b) illitic clay and collodionmembrane, and (c) silty clay, illitic clay, and kaolinite.

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Relationship Between ke and ki

From Eqs. (9.108) and (9.129), the electroosmotic flowrate is given by

Eq � k I � k A (9.132)h i e L

Because E /I is resistance and L / (resistance � A) isspecific conductivity �, Eq. (9.132) becomes

kek � (9.133)i �

As ke varies within relatively narrow limits, Eq. (9.133)shows that the electroosmotic efficiency, measured byki, is a sensitive function of the electrical conductivityof the soil. For soils 11, 13, and 14 in Table 9.7, � isin the range of 0.02 to 0.03 S. For soil 12, in whichelectroosmosis was not effective, � is 0.25 S. In es-sence, a high value of electrical conductivity meansthat the current required to develop the voltage is toohigh for economical movement of water. In addition,if high current is used, the generation of gas, heat, andelectrochemical effects become excessive.

9.17 CONSOLIDATION BY ELECTROOSMOSIS

If, in a compressible soil, electroosmosis draws waterto a cathode where it is drained away and no water isallowed to enter at the anode, then consolidation of thesoil between the electrodes occurs in an amount equalto the volume of water removed. Water movementaway from the anode causes consolidation in the vi-cinity of the anode. The effective stress must increaseconcurrently. Because the total stress in the vicinity ofthe anode remains essentially unchanged, the pore wa-ter pressure must decrease. Water drains at the cathodewhere there is no consolidation. Therefore, the total,effective, and pore water pressures at the cathode re-main unchanged. As a result, hydraulic gradient de-velops that tends to cause water flow from cathode toanode. Consolidation continues until the hydraulicforce that drives water back toward the anode exactlybalances the electroosmotic force driving water towardthe cathode.

The usefulness of consolidation by electroosmosisas a means for soil stabilization was established by anumber of successful field applications, for example,Casangrande (1959) and Bjerrum et al. (1967). Twoquestions are important: (1) How much consolidationwill there be? and (2) How long will it take? Answersto these questions are obtained using the coupled flow

equations in place of Darcy’s law in consolidationtheory.

Assumptions

The following idealizing assumptions are made:

1. There is homogeneous and saturated soil.2. The physical and physicochemical properties of

the soil are uniform and constant with time.12

3. No soil particles are moved by electrophoresis.4. The velocity of water flow by electroosmosis is

directly proportional to the voltage gradient.5. All the applied voltage is effective in moving wa-

ter.13

6. The electrical field is constant with time.7. The coupling of hydraulic and electrical flows

can be formulated by Eqs. (9.63) and (9.64).8. There are no electrochemical reactions.

Governing Equations

For one-dimensional flow between plate electrodes(Fig. 9.33a), Eq. (9.63) becomes

k �u �Vhq � � � k (9.134)h e �x �xw

for the flow rate per unit area. For radial flow for theconditions shown in Fig. 9.33b and a layer of unitthickness

k �u �Vhq � � � 2�r � k � 2�r (9.135)h e �r �rw

Introduction of Eq. (9.134) in place of Darcy’s lawin the derivation of the diffusion equation governingconsolidation in one dimension leads to

2 2k � u � V �uh � k � m (9.136)e v2 2 �x �x �tw

and

12 Flow of water away from anodes toward cathodes causes a non-uniform decrease in water content along the line between electrodes.This leads to changes in hydraulic conductivity, electroosmotic hy-draulic conductivity, compressibility, and electrical conductivity withtime and position. To account for these effects, which are discussedby Mitchell and Wan (1977) and Acar et al. (1990), would greatlycomplicate the analysis because it would be highly nonlinear. Similarproblems arise in classical consolidation theory, but the simple lineartheory developed by Terzaghi is adequate for most cases.13 In most cases some of the electrical energy will be consumed bygeneration of heat and gases at the electrodes. To account for thoselosses, an effective voltage can be used (Esrig and Henkel, 1968).

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CONSOLIDATION BY ELECTROOSMOSIS 299

Figure 9.33 Electrode geometries for analysis of consoli-dation by electroosmosis: (a) one-dimensional flow and (b)radial flow.

2 2� u k � V 1 �ue� � (9.137)w2 2�x k �x c �th v

where mv is the compressibility and cv is the coefficientof consolidation. For radial flow, the use of Eq. (9.135)gives

2 2� u k � V 1 �u k �V 1 �ue e� � � �� w w2 2�r k �r r �r k �r c �th h v

(9.138)

Both V and u are functions of position, as shown inFig. 9.34; V is assumed constant with time, whereas uvaries.

Amount of Consolidation

When the hydraulic gradient that develops in responseto the differing amounts of consolidation between theanode and cathode generates a counterflow (kh /w) /(�u /�x) that exactly balances the electroosmotic flowke(�V /�x) in the opposite direction, consolidation iscomplete. As there then is no flow, qh in Eqs. (9.14)and (9.135) is zero. Thus Eq. (9.134) is

k �u �Vh � �k (9.139)e �x �xw

or

kedu � � dV (9.140)wkh

The solution of this equation is

keu � � V � C (9.141)wkh

At the cathode, V � 0 and u � 0; therefore, C � 0,and the pore pressure at equilibrium at any point isgiven by

keu � � V (9.142)wkh

where the values of u and V are those at any point ofinterest. A similar result is obtained from Eq. (9.135)for radial flow.

Equation (9.142) indicates that electroosmotic con-solidation continues at a point until a negative porepressure, relative to the initial value, develops that isproportional to the ratio ke /kh and to the voltage at thepoint. For conditions of constant total stress, there mustbe an equal and opposite increase in the effectivestress. This increase in effective stress causes the con-solidation. For the one-dimensional case, consolidationby electroosmosis is analogous to the loading shownin Fig. 9.35.

For a given voltage, the magnitude of effective stressincrease that develops depends on ke /kh. As ke onlyvaries within narrow limits for different soils, the totalconsolidation that can be achieved depends largely onkh. Thus, the potential for consolidation by electroos-mosis increases as soil grain size decreases because thefiner grained the soil, the lower is kh. However, theamount of consolidation in any case depends on thesoil compressibility as well as on the change in effec-tive stress. For linear soil compression with increase ineffective stress, the coefficient of compressibility av is

de dea � � � (9.143)v d�� du

or

de � a du � �a d�� (9.144)v v

in which d�� is the increase in effective stress.

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Figure 9.34 Assumed variation of voltage with distance during electroosmosis: (a) one-dimensional flow and (b) radial flow.

Thus, the more compressible the soil, the greaterwill be the amount of consolidation for a given stressincrease, just as in the case of consolidation under ap-plied loads. It follows, also, that electroosmosis will beof little value in an overconsolidated clay unless theeffective stress increases are large enough to bring thematerial back into the virgin compression range.

The consolidation loading of any small element ofthe soil is isotropic, as it is done by increasing theeffective stress through reduction in the pore waterpressure. The entire soil mass being treated is not con-solidated isotropically or uniformly, however, becausethe amount of consolidation varies with position, de-

pendent on the voltage at the point. Accordingly, prop-erties at the end of treatment vary along a line betweenthe anode and cathode, as shown, for example, by theposttreatment variations in shear strength and watercontent shown in Fig. 9.36. Values of these propertiesbefore treatment are also shown for comparison. Moreuniform property distributions between electrodes canbe obtained if the polarity of electrodes is reversedafter partial completion of consolidation (Wan andMitchell, 1976).

The results shown in Fig. 9.36 were obtained at asite in Norway where electroosmosis was used for theconsolidation of quick clay (Bjerrum et al., 1967). The

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CONSOLIDATION BY ELECTROOSMOSIS 301

Figure 9.35 Consolidation by electroosmosis and by direct loading, one-dimensional case:(a) electroosmosis and (b) direct loading.

variations in strength and water content after treatmentare consistent with the patterns to be expected basedon the predicted variation of pore pressure decreaseand vertical strain stress increase with voltage and po-sition shown in Fig. 9.35.

Rate of Consolidation

Solutions for Eqs. (9.137) and (9.138) have been ob-tained for several cases (Esrig, 1968, 1971). For theone-dimensional case, and assuming a freely draining(open) cathode and a closed anode (no flow), the porepressure is

k 2k Ve e w mu � V(x) �w 2k k �h h

� n(�1) (n � 1/2)�x� sin� �2(n � 1/2) Ln�0

1 2 2� exp � n � � T (9.145) � � �V2

where V(x) is the voltage at x, Vm is the maximum

voltage, and TV is the time factor, defined in terms ofthe distance between electrodes L and real time t as

c tvT � (9.146)V 2L

where cv is the coefficient of consolidation, given by

khc � (9.147)v m v w

The average degree of consolidation U as a functionof time is

2� n4 (�1) 1 2U � 1 � exp � n � � T� � � �V3 3� (n � 1/2) 2n�0

(9.148)

Solutions for Eqs. (9.145) and (9.148) are shown inFigs. 9.37 and 9.38. They are applied in the same wayas the theoretical solution for classical consolidationtheory.

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Figure 9.36 Effect of electroosmosis treatment on properties of quick clay at As, Norway(from Bjerrum et al., 1967): (a) Undrained shear strength, (b) remolded shear strength, (c)water content, and (d) Atterberg limits.

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ELECTROCHEMICAL EFFECTS 303

Figure 9.37 Dimensionless pore pressure as a function ofdimensionless time and distance for one-dimensional consol-idation by electroosmosis.

Figure 9.38 Average degree of consolidation versus dimen-sionless time for one-dimensional consolidation by electroos-mosis.

Figure 9.39 Average degree of consolidation as a functionof dimensionless time for radial consolidation by electroos-mosis (from Esrig, 1968). Reprinted with permission ofASCE.

A numerical solution to Eq. (9.138) gives the resultsshown in Fig. 9.39 (Esrig, 1968, 1971). For the caseof two pipe electrodes, a more realistic field conditionthan the radial geometry of Fig. 9.33b, Fig. 9.39 cannotbe expected to apply exactly. Along a straight line be-tween two pipe electrodes, however, the flow patternis approximately the same as for the radial case for aconsiderable distance from each electrode.

A solution for the rate of pore pressure buildup atthe cathode for the case of no drainage (closed cath-ode) is shown in Fig. 9.40. This condition is relevant

to pile driving, pile pulling, reduction of negative skinfriction, and recovery of buried objects. Special solu-tions for in situ determination of soil consolidationproperties by electroosmosis measurements have alsobeen developed (Banerjee and Mitchell, 1980).

One of the most important points to be noted fromthese solutions is that the rate of consolidation dependscompletely on the coefficient of consolidation, whichvaries directly with kh, but is completely independentof ke. Low values of kh, as is the case in highly plasticclays, mean long consolidation times. Thus, whereas alow value of kh means a high value of ke /kh and thepotential for a high effective consolidation pressure, italso means longer required consolidation times for agiven electrode spacing. The optimum situation iswhen ke /kh is high enough to generate a large porewater tension for reasonable electrode spacings (2 to 3m) and maximum voltage (50 to 150 V DC), but kh ishigh enough to enable consolidation in a reasonabletime. The soil types that best satisfy these conditionsare silts, clayey silts, and silty clays. Most successfulfield applications of electroosmosis for consolidationhave been in these types of materials. As noted earlier,the electrical conductivity of the soil is also important;if it is too high, as in the case of high-salinity porewater, adverse electrochemical effects and unfavorableeconomics may preclude use of electroosmosis forconsolidation.

9.18 ELECTROCHEMICAL EFFECTS

The measured strength increases in the quick clay atAs, Norway (Fig. 9.36), were some 80 percent greater

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Figure 9.40 Dimensionless pore pressure at the face of a cylindrical electrode as a functionof dimensionless time for the case of a closed cathode (a swelling condition) (from Esrigand Henkel, 1968).

than can be accounted for solely by reduction in watercontent. Also, the liquid and plastic limits werechanged as a result of treatment. Consolidation aloneshould have no effect on the Atterberg limits becausechanges in mineralogy, particle characteristics, and/orpore solution characteristics are needed to do this.

In addition to movement of water when a DC volt-age field is applied between metal electrodes insertedinto a wet soil, the following effects may develop: iondiffusion, ion exchange, development of osmotic andpH gradients, desiccation by heat generation at theelectrodes, mineral decomposition, precipitation ofsalts or secondary minerals, electrolysis, hydrolysis,oxidation, reduction, physical and chemical adsorption,and fabric changes. As a result, continuous changes insoil properties that are not readily accounted for by thesimplified theory developed previously must be ex-pected. Some of them, such as electrochemical hard-ening of the soil that results in permanent changes inplasticity and strength, may be beneficial; others, suchas heating and gas generation, may impair the effi-ciency of electroosmosis. For example, heat and gasgeneration were so great that a field test of consoli-dation by electroosmosis for foundation stabilization ofthe leaning Tower of Pisa was unsuccessful.

A simplified mechanism for some of the processesduring electroosmosis is as follows. Oxygen gas isevolved at the anode by hydrolysis

� �2H O � 4e → O ↑ � 4H (9.149)2 2

Anions in solution react with freed H� to form acids.

Chlorine may also form in a saline environment. Someof the exchangeable cations on the clay may be re-placed by H�. Because hydrogen clays are generallyunstable, and high acidity and oxidation cause rapiddeterioration of the anodes, the clay will soon alter tothe aluminum or iron form depending on the anodematerial. As a result, the soil is usually strengthenedin the vicinity of the anode. If gas generation at theanode causes cavitation and heat causes desiccation,cracking may occur. This will limit the negative porepressure that can develop to a value less than 1 atm,and also the electrical resistance will increase, leadingto a loss in efficiency.

Hydrogen gas is generated at the cathode

� �4H O � 4e → 2H ↑ � 4OH (9.150)2 2

Cations in solution are drawn to the cathode wherethey combine with (OH)� that is left behind to formhydroxides. The pH may rise to values as high as 12at the cathode. Some alumina and silica may go intosolution in the high pH environment.

More detailed information about electrochemical re-actions during electroosmosis can be found in Titkovet al. (1965), Esrig and Gemeinhardt (1967), Chilingarand Rieke (1967), Gray and Schlocker (1969), Gray(1970), Acar et al. (1990), and Hamed et al. (1991).

Soil strength increases resulting from consolidationby electroosmosis and the concurrent electrochemicalhardening have application for support of foundationson and in fine-grained soil. Pile capacity for a bridge

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SELF-POTENTIALS 305

Figure 9.41 Natural electroosmosis due to self-potential dif-ferences between oxidizing and reducing soil layers. The ox-idizing soil layer is positive relative to the reducing layer(redrawn from Hilbert, in Veder, 1981).

foundation in varved clay at a site in Canada was wellbelow the design value and inadequate for support ofthe structure (Soderman and Milligan, 1961; Milligan,1994). Electrokinetic treatment using the piles as an-odes resulted in sufficient strength increase to providethe needed support. Recently reported model tests byMicic et al. (2003) on the use of electrokinetics in softmarine clay to increase the load capacity of skirt foun-dations for offshore structures resulted in increases insoil strength and supporting capacity of up to a factorof 3.

9.19 ELECTROKINETIC REMEDIATION

The transport of dissolved and suspended constituentsinto and out of the ground by electroosmosis and elec-trophoresis, as well as electrochemical, reactions havebecome of increasing interest because of their potentialapplications in waste containment and removal of con-taminants from fine-grained soils. The electrolysis re-actions at the electrodes described in the precedingsection, wherein acid is produced at the anode and baseat the cathode, are of particular relevance. After a fewdays of treatment the pH in the vicinity of the anodemay drop to less than 2, and that at the cathode in-crease to more than 10 (Acar and Alshewabkeh, 1993).

Toxic heavy metals are preferentially adsorbed byclay minerals and they precipitate except at low pH.Iron or aluminum cations from decomposing anodescan replace heavy-metal ions from exchange sites, theacid generated at the anode can redissolve precipitatedmaterial, and the acid front that moves across the soilcan keep the metals in solution until removed at thecathode. Geochemical reactions in the soil pores im-pact the efficiency of the process. Among them arecomplexation effects that reverse ion charge andreverse flow directions, precipitation/dissolution,sorption, desorption and dissolution, redox, and im-mobilization or precipitation of metal hydroxides in thehigh pH zone near the cathode.

Some success has been reported in the removal oforganic pollutants from soils, at least in the laboratory,as summarized by Alshewabkeh (2001). However, it isunlikely that large quantities of non-aqueous-phase liq-uids can be effectively transported by electrokineticprocesses, except as the NAPL may be present in theform of small bubbles that move with the suspendingwater.

An in-depth treatment of the fundamentals of elec-trokinetic remediation and the practical aspects of itsimplementation are given by Alshewabkeh (2001) andthe references cited therein.

9.20 SELF-POTENTIALS

Natural DC electrical potential differences of up toseveral tens of millivolts exist in the earth. These self-potentials are generated by differing chemical condi-tions in adjacent soil layers, fluid flow, subsurfacechemical reactions, and temperature differences. Theself-potential (SP) method is one of the oldest geo-physical methods for characterization of the subsurface(National Research Council, 2000). Self-potentialsmay be the source of phenomena of importance in geo-technical problems as well.

The magnitude of self-potential between differentsoil layers depends on the contents of oxidizing andreducing substances in the layers (F. Hilbert, in Veder,1981). These potentials can cause a natural electroos-mosis in which water flows in the direction from thehigher to the lower potential, that is, toward the cath-ode. The process is shown schematically in Fig. 9.41.An oxidizing soil layer is positive relative to a reducinglayer, thus inducing an electroosmotic water flow to-ward the interface. If water accumulates at the inter-face, there can be swelling and loss of strength, leadingultimately to formation of a slip surface.

Generation of Self-Potentials in Soil Layers

Soils in an oxidizing environment are usually yellowor tan to reddish brown and are characterized by oxidesand hydrates of trivalent iron and a low pH relative toreducing soils, which are usually dark gray to blue-gray in color and contain sulfides and oxides and hy-droxides of divalent iron. The local electrical potentialof the soil � depends on the iron concentrations andcan be calculated from Nernst’s equation:

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Figure 9.42 Electrical potentials measured in a trench cut into a slide (from Veder, 1981).Reprinted with permission of Springer-Verlag.

3�RT cFe� � 0.771 � ln (9.151)� �2�F cFe

in which the concentrations are of Fe in solution inmoles/liter pore water. The difference in potentials be-tween two layers gives the driving potential for elec-troosmosis. Values calculated using the Nernstequation are too high for actual soil systems becauseit applies for conditions of no current flow, and theflowing current also generates a diffusion potential act-ing in the opposite direction. Hilbert, in Veder (1981),gives the electrical potential as a function of the in situpH, that is,

� � 0.186 � 0.059 pH (9.152)

Reasonable agreement has been obtained betweenmeasured and calculated values of � for different soillayers. The end result is that potential differences ofup to 50 mV or so are developed between differentlayers. Potentials measured in a trench excavated in aslide zone are shown in Fig. 9.42.

Excess Pore Pressure Generation by Self-Potentials

The pore pressure that may develop at an interface be-tween two different soil layers is given by Eq. (9.142)in which V is the difference in self-potentials betweenthe layers. For a given value of V, the magnitude ofpore pressure depends directly on ke /kh. For example,if ke � 5 � 10�9 m2/s V and kh � 1 � 10�10 m/s,

then ke /hh � 50 m/V. If the self-potential difference is50 mV, then from Eq. (9.142) a pore pressure value of

u � 50 � 9.81 � 0.05 � 25 kPa

is generated, which is not an insignificant value. If wa-ter that is driven toward the interface cannot escape orbe absorbed by the soil, then the effective stress willbe reduced by this amount. If the water is absorbedinto the clay layer, then softening will result. Eitherway, the resistance to sliding along the interface willbe reduced.

Landslide Stabilization Using Short-CircuitConductors

If slope instability is caused by a slip surface betweenreducing and oxidizing soil layers, then a simple meansfor stabilization can be used (Veder, 1981). Short-circuiting conductors, such as steel rods, are driveninto the soil so that they extend across the slip surfaceand about 1 to 2 m into the soil below. The mechanismthat is then established is shown in Fig. 9.43.

Electric current generated by reduction reactions inthe oxidizing soil layer and oxidizing reactions in thereducing layer flows through the conductors. Becauseof the presence of oxidizing agents such as ferric iron,oxygen, and manganese compounds, in the upper ox-idizing layer that take up electrons, electrons pass fromthe metal conductor to the soil. That is, the introduc-tion of electrons initiates reducing reactions. In the re-ducing layer, on the other hand, there is already a

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THERMALLY DRIVEN MOISTURE FLOW 307

Figure 9.43 Mechanism for slide stabilization using short-circuiting conductors (adapted from Veder, 1981).

surplus of electrons. If these pass into the conductor,then the environment becomes favorable for oxidationreactions. Thus, positive charges are generated in thereducing soil layer as the conductor carries electronsaway. The oxidizing soil layer then takes up these elec-trons.

Completion of the electrical circuit requires currentflow through the soil pore water in the manner shownin Fig. 9.43, where adsorbed cations, shown as Na�,plus the associated water, flow away from the soil layerinterface. This electroosmotic transport of water re-duces the water content in the slip zone. Thus, short-circuit conductors have three main effects (Veder,1981):

1. Natural electroosmosis is prevented because theshort-circuiting conductors eliminate the poten-tial difference between the two soil layers.

2. Electrochemical reactions produce electroos-motic flow in the opposite direction, thus helpingto drain the shear zone.

3. Corrosion of the conductors produces high va-lence cations that exchange for lower valence ad-sorbed cations, for example, iron for sodium,which leads to soil strengthening.

Several successful cases of landslide stabilizationusing short-circuiting conductors have been describedby Veder (1981) and the references cited therein. Typ-ically, steel rods about 25 mm in diameter are used,spaced a maximum of 3 to 4 m apart in grid patternscovering the area to be stabilized. Conditions favorable

for use of short-circuiting conductors are (1) intact co-hesive soils with a low hydraulic conductivity, (2)shear between oxidizing and reducing clay layers, and(3) a relatively thin, well-defined shear zone.

9.21 THERMALLY DRIVEN MOISTURE FLOW

Thermally driven flows in saturated soils are rathersmall. Gray (1969) measured thermoelectric currentson the order of 1 to 10 �A/�C cm, with the warm sidepositive relative to the cold side. Thermoosmotic pres-sures of only a few tenths of a centimeter water headper degree Celsius were measured in saturated soil. Netflows in different directions have been measured in dif-ferent investigations, evidently because of differenttemperature dependencies of chemical activity coeffi-cients. These small thermoelectric and thermoosmoticeffects in saturated soils may be of little practical sig-nificance in geotechnical problems.

On the other hand, thermally driven moisture flowsin partly saturated soils can be large, and that theseflows can be very important in subgrade stability,swelling soils, and heat transfer and storage problemsof various types. Theoretical representations of mois-ture flow through partly saturated soils based solely onthe application of irreversible thermodynamics, such asdeveloped by Taylor and Cary (1964), have not beencompletely successful. They underestimate the flowssubstantially, perhaps because of the inability to ade-quately represent all the processes and interactions.

A widely used theory for coupled heat and moistureflow through soils was developed by Philip and DeVries (1957). It accounts for both liquid- and vapor-phase flows. Vapor-phase flow depends on the thermaland isothermal vapor diffusivities and is driven by tem-perature and moisture content gradients. The liquid-phase flow depends on the thermal and isothermalliquid diffusivities and is driven by the temperaturegradient, the moisture content gradient, and gravity.The two governing equations are:

For vapor-phase flow:

qvap � �D �T � D �� (9.153)TV �V�w

and for liquid-phase flow:

qliq � �D �T � D �� k i (9.154)TL �L ��w

where qvap � vapor flux density (M /L2 /T)�w � density of water (M /L3)

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T � temperature (K)� � volumetric water content (L3 /L3)

DTV � thermal vapor diffusivity (L2 /T /K)D�V � isothermal vapor diffusivity (L2 /T)qliq � liquid flux density (M /L2 /T)

DTL � thermal liquid diffusivity (L2 /T /K)D�L � isothermal liquid diffusivity (L2 /T)

k� � unsaturated hydraulic conductivity (L /T)i � unit vector in vertical direction

The thermal vapor diffusivity is given by

D d�0 0D � v�[a � ƒ(a) � �]h� (9.155)� � � �TV � dTw

The isothermal vapor diffusivity is given by

D � hg d�0 0D � v�a (9.156)� � � �� V � RT d�w

where D0 � molecular diffusivity of water vapor in air(L2 /T)

v � mass flow factor � P / (P � p)P � total gas pressure in pore spacep � partial pressure of water vapor in pore

space� � tortuosity factora � volumetric air content (L3 /L3)h � relative humidity of air in pores� � ratio of average temperature gradient in

the air-filled pores to the overall temper-ature gradient

g � acceleration of gravity (L /T2)R � gas constant (FL /M /K)�0 � density of saturated water vapor (M /L3)� � suction head of water in the soil (negative

head) (L)ƒ(a) � a /ak for 0 � a � ak

� 1 for a � ak

ak � a at which liquid conductivity is lost orat which the hydraulic conductivity fallsbelow some arbitrary fraction of the sat-urated value

The thermal liquid diffusivity is given by

� d�D � k (9.157)� �� TL � � dT

The isothermal diffusivity is given by

d�D � k (9.158)� ��L � d�

in which � is the surface tension of water (F /L).Use of the above equations requires knowledge of

four relationships to describe the properties of the soilsin the system:

1. Hydraulic conductivity as a function of watercontent

2. Thermal conductivity as a function of water con-tent

3. Volumetric heat capacity (see Table 9.2)4. Suction head as a function of water content

The hydraulic conductivity and suction relationshipsare hysteretic; that is, they depend on whether the soilis wetting or drying. Examples of the variations of thedifferent properties needed for the analysis are shownin Fig. 9.44 as a function of degree of saturation andvolumetric water content. The data are for a crushedlimestone that is used for a trench backfill around bur-ied electrical transmission cables. This material is usedbecause of its low thermal resistivity, which makes itsuitable for effective dissipation of heat from the bur-ied cable, provided the saturation does not fall belowabout 40 percent.

The vapor flow is made up of a flow away from thehigh-temperature side that is driven by a vapor densitygradient and a return flow caused by variation in thepore vapor humidity as reflected by variations in soilsuction. At moderate soil suction values, for example,a few meters for sand and several tens of meters forclay, the thermal vapor diffusivity predominates, andmoisture is driven away from the heat source (McMil-lan, 1985).

The isothermal diffusivity term only becomes im-portant at very high suction levels. The liquid flow con-sists of a capillarity-driven flow toward the heat sourceand an outward liquid flow due to variations in watersurface tension with temperature. McMillan’s analysisshowed that for both sand and clay the isothermal liq-uid diffusivity term was 4 to 5 orders of magnitudegreater than the thermal liquid diffusivity term. Thuscapillarity-driven flow predominates for any significantgradient in the volumetric moisture content. The verysmall thermal liquid diffusivity is consistent with theobservations noted earlier for saturated soils in whichmeasured water flows under thermal gradients aresmall.

The total water flow q in an unsaturated soil underthe action of a temperature gradient and its resultingwater content gradient equals the sum of the vapor-phase and liquid-phase movements. Thus, from Eqs.(9.153) to (9.158),

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THERMALLY DRIVEN MOISTURE FLOW 309

Figure 9.44 Examples of properties used for analysis of thermally driven moisture flow ina partially saturated, compacted, crushed limestone: (a) particle size distribution, (b) suctionhead as a function of volumetric water content, (c) hydraulic conductivity as a function ofdegree of saturation and volumetric water content, (d) isothermal liquid diffusivity as afunction of degree of saturation and volumetric water content, (e) isothermal vapor diffusivityas a function of degree of saturation and volumetric water content, and (f) Thermal waterdiffusivity as a function of degree of saturation and volumetric water content. Thermal re-sistivity as a function of water content for this soil is shown in Fig. 9.14.

q� �(D � D )�T � (D � D )�� k iTV TL �V �L ��w

� �D �T � D � � k iT � � � (9.159)

in which

D � D � D � thermal water diffusivityTV TL

(9.160)

and

D � D � D � isothermal water diffusivity� �V �L

(9.161)

Equation (9.159) is the governing equation for mois-ture movement under a thermal gradient in unsaturatedsoils as proposed by Philip and De Vries (1957). Dif-ferentiation of this equation and application of thecontinuity requirement gives the general differentialequation for moisture flow:

�� k�� (D �T) � �(D ��) � (9.162)T ��t �z

The heat conduction equation for the soil is

�T kt� � �T (9.163)� ��t C

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Figure 9.44 (Continued )

where kt � thermal conductivityC � volumetric heat capacity

The ratio of thermal conductivity to the volumetricheat capacity is the thermal diffusivity A.

Both transient and steady-state temperature distri-butions computed using the Philip and De Vries theoryincorporated into numerical models have agreed wellwith measured values in a number of cases. The actualmoisture movements and distributions have not agreedas well, for example, Abdel-Hadi and Mitchell (1981)and Cameron (1986). The numerical simulations havebeen done using transform methods, finite differencemethods, the finite element method, and the integratedfinite difference method. Cameron (1986) reformulatedthe equations in terms of suction head rather thanmoisture content and incorporated them into the finiteelement model of Walker et al. (1981) for solution oftwo-dimensional problems.

9.22 GROUND FREEZING

Heat conduction in soils and rocks is discussed in Sec-tion 9.5, and values for thermal properties are given in

Table 9.2. Three topics are considered in this section:(1) the depth of frost penetration, which illustrates theapplication of transient heat flow analysis, (2) frost ac-tion in soils, a phenomenon of great practical impor-tance that can be understood through consideration ofinteractions of the physical and physicochemical prop-erties of the soil, and (3) some effects of freezing onthe behavior and properties of the soil after thawing.These topics are also covered in some detail by Konrad(2001) and the references therein.

Depth of Frost Penetration

Accurate estimation of the depth of ground freezingduring the winter, the depth of thawing in permafrostareas during the summer, and the refrigeration and timerequirements for artificial ground freezing for tempo-rary ground stabilization are all problems involvingtransient heat flow analysis. They differ from the con-duction analyses in the preceding sections in that thephase change of water to ice must be taken into ac-count. Prediction of the maximum depth of frost pen-etration illustrates this type of problem. Theoreticalsolutions of this problem are based on a mathematical

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GROUND FREEZING 311

Figure 9.45 Thermal energy as a function of temperaturefor a wet soil. Figure 9.46 Assumed conditions for the Stefan equation.

analysis developed by Neumann in about 1860 (Berg-gren, 1943; Aldrich, 1956; Brown, 1964; Konrad,2001).

The relationship between thermal energy u and tem-perature T for a soil mass at constant water content isshown in Fig. 9.45. In the absence of freezing or thaw-ing

�u� C (9.164)

�T

The Fourier equation for heat flow is

�Tq � �k (9.165)t t �z

In the absence of freezing or thawing, thermal conti-nuity and conservation of thermal energy require thatthe rate of change of thermal energy of an element plusthe rate of heat transfer into the element equal zero,that is, for the one-dimensional case

�u �q� � 0 (9.166)

�t �z

Using Eqs. (9.164) and (9.165), Eq. (9.166) may bewritten

2�T � TC � k (9.167)t 2�t �z

or

2�T � T� a (9.168)2�t �z

where a � kt /C is the thermal diffusivity (L2 /T). Equa-tion (9.168) is the one-dimensional, transient heat flowequation.

At the interface between frozen and unfrozen soil,z � Z, and the equation of heat continuity is

dZL � q � q (9.169)s ƒ udt

where Ls is the latent heat of fusion of water and qƒ �qu is the net rate of heat flow away from the interface.Equation (9.169) can be written

dZ �T �Tƒ uL � k � k (9.170)s ƒ udt �z �z

where the subscripts u and f pertain to unfrozen andfrozen soil, respectively. Simultaneous solution of Eqs.(9.168) and (9.170) gives the depth of frost penetra-tion.

Stefan Formula The simplest solution is to assumethat the latent heat is the only heat to be removed dur-ing freezing and neglect the heat that must be removedto cool the soil water to the freezing point, that is, thethermal energy stored as volumetric heat is neglected.This condition is shown by Fig. 9.46. For this case Eq.(9.168) does not exist, and Eq. (9.170) becomes

dZ TsL � k (9.171)s ƒdt Z

where Ts is the surface temperature. The solution ofthis equation is

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Figure 9.47 Freezing index in relation to the annual temperature cycle.

1 / 22k � T dtƒ s� �Z � (9.172)

Ls

The integral of Ts dt is a measure of freezing intensity.It can be expressed by the freezing index F, which hasunits of degrees � time. Index F is usually given indegree-days. It is shown in relation to the annual tem-perature cycle in Fig. 9.47. Freezing index values arederived from meteorological data. Methods for deter-mination of freezing index values are given by Linellet al. (1963), Straub and Wegmann (1965), McCormick(1971), and others. Maps showing mean freezing indexvalues are available for some areas. It is importantwhen using such data sources to be sure that there arenot local deviations from the average values that aregiven. Different types of ground cover, local topogra-phy and vegetation, and solar radiation all influencethe net heat flux at the ground surface.

The Stefan equation can also be used to estimate thesummer thaw depth in permafrost; that is, the thicknessof the active layer. In this case the ground thawingindex, also in degree-days and derived from meteoro-logical data, is used in Eq. (9.172) in place of thefreezing index (Konrad, 2001).

Modified Berggren Formula The Stefan formulaoverpredicts the depth of freezing because it neglectsthe removal of the volumetric heats of frozen and un-frozen soil. Simultaneous solution of Eqs. (9.168) and

(9.170) has been made for the conditions shown in Fig.9.48, assuming that the soil has a uniform initial tem-perature that is T0 degrees above freezing and that thesurface temperature drops suddenly to Ts below freez-ing (Aldrich, 1956). The solution is

1 / 22kT tsZ � � (9.173)� �Ls

where k is taken as an average thermal conductivityfor frozen and unfrozen soil. The dimensionless cor-rection coefficient � depends on the two parametersshown in Fig. 9.49. The thermal ratio � is given by

T0� � (9.174)Ts

and the fusion parameter � is

C� � T (9.175)sLs

An averaged value for the volumetric heats of frozenand unfrozen soil can be used for C in Eq. (9.175).

In application, the quantity Tst in Eq. (9.173) is re-placed by the freezing index, and Ts in (9.175) is givenby F / t, where t is the duration of the freezing period.

The coefficient � corrects the Stefan formula for ne-glect of volumetric heat. For soils with high water con-tent C is small relative to Ls; therefore, � is small and

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GROUND FREEZING 313

Figure 9.48 Thermal conditions assumed in the derivation of the modified Berggren for-mula.

the Stefan formula is reasonable. For arctic climates,where T0 is not much above the freezing point, � issmall, � is greater than 0.9, and the Stefan formula issatisfactory. However, in more temperate climates andin relatively dry or well-drained soils, the correctionbecomes important.

A comparison between theoretical freezing depthsand a design curve proposed by the Corps of Engineersis shown in Fig. 9.50 for several soil types. The the-oretical curves were developed by Brown (1964) usingthe modified Berggren equation and the thermal prop-erties given in Fig. 9.13.

Consideration should be given to the effect of dif-ferent types of surface cover on the ground surfacetemperature because air temperature and ground tem-perature are not likely to be the same, and the effectsof thermal radiation may be important. Observed

depths of frost penetration may be misleading if esti-mates for a proposed pavement or other structure areneeded because of differences in ground surface char-acteristics and because the pavement or foundationbase will be at different water content and density thanthe surrounding soil.

The solutions do not account for flow of water intoor out of the soil or the formation of ice lenses duringthe freezing period. This may be particularly importantwhen dealing with frost heave susceptible soils orwhen developing frozen soil barriers for the cutoff ofgroundwater flow. Methods for prediction of frostdepth in soils susceptible to ice lens formation and therate of heave are given by Konrad (2001). The initia-tion of freezing of flowing groundwater requires thatthe rate of volumetric and latent heat removal be highenough so that ice can form during the residence time

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Figure 9.49 Correction coefficients for use in the modified Berggren formula (from Aldrich,1956).

of an element of water moving between the boundariesof the specified zone of solidification.

Frost Heaving

Freezing of some soils is accompanied by the forma-tion of ice layers or ‘‘lenses’’ that can range from amillimeter to several centimeters in thickness. Theselenses are essentially pure ice and are free from largenumbers of contained soil particles. The ground sur-face may ‘‘heave’’ by as much as several tens of cen-timeters, and the overall volume increase can be manytimes the 9 percent expansion that occurs when waterfreezes. Heave pressures of many atmospheres arecommon. The freezing of frost-susceptible soils be-neath pavements and foundations can cause major dis-tress or failure as a result of uneven uplift duringfreezing and loss of support on thawing, owing to thepresence of large water-filled voids. Ordinarily, icelenses are oriented normal to the direction of cold-frontmovement and become thicker and more widely sep-arated with depth.

The rate of heaving may be as high as several mil-limeters per day. It depends on the rate of freezing in

a complex manner. If the cooling rate is too high, thenthe soil freezes before water can migrate to an ice lens,so the heave becomes only that due to the expansionof water on freezing.

Three conditions are necessary for ice lens forma-tion and frost heave:

1. Frost-susceptible soil2. Freezing temperature3. Availability of water

Frost heaving can occur only where there is a watertable, perched water table, or pocket of water reason-ably close to the freezing front.

Frost-Susceptible Soils Almost any soil may bemade to heave if the freezing rate and water supplyare controlled. In nature, however, the usual rates offreezing are such that only certain soil types are frostsusceptible. Clean sands, gravels, and highly plasticintact clays generally do not heave. Although the onlycompletely reliable way to evaluate frost susceptibilityis by some type of performance test during freezing,soils that contain more than 3 percent of their particlesfiner than 0.02 mm are potentially frost susceptible.

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GROUND FREEZING 315

Figure 9.50 Predicted frost penetration depths compared with the Corps of Engineers’ de-sign curve (Brown, 1964). Curve a—sandy soil: dry density 140 lb/ ft3, saturated, moisturecontent 7 percent. Curve b—silt, clay: dry density 80 lb/ ft3, unsaturated, moisture content2 percent. Curve c—sandy soil; dry density 140 lb/ ft3, unsaturated, moisture content 2percent. Curve d—silt, clay: dry density 120 lb/ ft3, moisture content 10 to 20 percent (sat-urated). Curve e—silt, clay: dry density 80 lb/ ft3 saturated, moisture content 30 percent.Curve f—Pure ice over still water.

Frost-susceptible soils have been classified by theCorps of Engineers in the following order of increasingfrost susceptibility:

Group(increasing

susceptibility) Soil Types

F1 Gravelly soils with 3 to 20 percentfiner than 0.02 mm

F2 Sands with 3 to 15 percent finerthan 0.02 mm

F3 a. Gravelly soils with more than20 percent finer than 0.02-mm sands, except fine siltsands with more than 15percent finer than 0.02 mm

b. Clays with PI greater than 12percent, except varved clays

F4 a. Silts and sandy siltsb. Fine silty sands with more than

15 percent finer than 0.02mm

c. Lean clays with PI less than 12percent

d. Varved clays

A method for the evaluation of frost susceptibilitythat takes project requirements and acceptable risksand freezing conditions into account as well as the soiltype is described by Konrad and Morgenstern (1983).

Mechanism of Frost Heave The formation of icelenses is a complex process that involves interrelation-ships between the phase change of water to ice, trans-port of water to the lens, and general unsteady heatflow in the freezing soil. The following explanation ofthe physics of frost heave is based largely on the mech-anism proposed by Martin (1959). Although the Martin(1959) model may not be correct in all details in thelight of subsequent research, it provides a logical andinstructive basis for understanding many aspects of thefrost heave process.

The ice lens formation cycle involves four stages:

1. Nucleation of ice2. Growth of the ice lens3. Termination of ice growth4. Heat and water flow between the end of stage 3

and the start of stage 1 again

In reality, heat and water flows continue through allfour stages; however, it is convenient to consider themseparately.

The temperature for nucleation of an ice crystal, Tn,is less than the freezing temperature, T0. In soils, T0 in

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Figure 9.51 Temperature versus depth relationships in afreezing soil.

pore water is less than the normal freezing point ofwater because of dissolved ions, particle surface forceeffects, and negative pore water pressures that exist inthe freezing zone. The freezing point decreases withdecreasing distance to particle surfaces and may beseveral degrees lower in the double layer than in thecenter of a pore. Thus, in a fine-grained soil, there isan unfrozen film on particle surfaces that persists untilthe temperature drops below 0�C.

The face of an ice front has a thin film of adsorbedwater. Freezing advances by incorporation of watermolecules from the film into the ice, while additionalwater molecules enter the film to maintain its thick-ness. It is energetically easier to bring water to the icefrom adjacent pores than to freeze the adsorbed wateron the particle or to propagate the ice through a poreconstriction.

The driving force for water transport to the ice is anequivalent hydrostatic pressure gradient that is gener-ated by freezing point depression, by removal of thewater from the soil at the ice front, which creates ahigher effective stress in the vicinity of the ice thanaway from it, by interfacial tension at the ice–waterinterface, and by osmotic pressure generated by thehigh concentration of ions in the water adjacent to theice front. Ice formation continues until the water ten-sion in the pores supplying water becomes greatenough to cause cavitation, or decreased upward waterflow from below leads to new ice lens formation be-neath the existing lens.

The processes of freezing and ice lens formationproceed in the following way with time according toMartin’s theory. If homogeneous soil, at uniform watercontent and temperature T0 above freezing, is subjectedto a surface temperature Ts below freezing, then thevariation of temperature with depth at some time is asshown in Fig. 9.51. The rate of heat flow at any pointis �kt(dT /dz). If dT /dz at point A is greater than atpoint B, the temperature of the element will drop.When water goes to ice, it gives up its latent heat,which flows both up and down and may slow or stopchanges in the value of dT /dz for some time period,thus halting the rate of advance of the freezing frontinto the soil.

Ground heave results from the formation of a lensat A, with water supplied according to the mechanismsindicated above. The energy needed to lift the over-lying material, which may include not only the soil andice lenses above, but also pavements and structures, isavailable because ice forms under conditions of super-cooling at a temperature TX � TFP, where TFP is thefreezing temperature. The available energy is

XL(T � T )FP F � (9.176)TFP

The quantity L is the latent heat. Supercooling of 1�Cis sufficient to lift 12.5 kg a distance of 10 mm. Al-ternatively, the energy for heave may originate fromthe thin water films at the ice surface (Kaplar, 1970).

As long as water can flow to a growing ice lens fastenough, the volumetric heat and latent heat can pro-duce a temporary steady-state condition so that (dT /dz)A � (dT /dz)B. For example, silt can supply water ata rate sufficient for heave at 1 mm/h. After some timethe ability of the soil to supply water will drop becausethe water supply in the region ahead of the ice frontbecomes depleted, and the hydraulic conductivity ofthe soil drops, owing to increased tension in the porewater. This is illustrated in Fig. 9.52, where hydraulicconductivity data as a function of negative pore waterpressure are shown for a silty sand, a silt, and a clay,all compacted using modified AASHTO effort, at awater content about 3 percent wet of optimum.

A small negative pore water pressure is sufficient tocause water to drain from the pores of the silty sand,and this causes a sharp reduction in hydraulic conduc-tivity. Because the clay can withstand large negativepore pressures without loss of saturation, the hydraulicconductivity is little affected by increasing reductionsin the pore pressure (increasing suction). The smalldecrease that is observed results from the consolidationneeded to carry the increased effective stress requiredto balance the reduction in the pore pressure. For thesilt, water drainage starts when the suction reaches

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GROUND FREEZING 317

Figure 9.52 Hydraulic conductivity as a function of negativepore water pressure (from Martin and Wissa, 1972).

about 40 kPa; however, a significant continuous waterphase remains until substantially greater values of suc-tion are reached.

In sand, the volume of water in a pore is large, andthe latent heat raises the freezing temperature to thenormal freezing point. Hence, there is no supercoolingand no heave. Negative pore pressure development atthe ice front causes the hydraulic conductivity to drop,so water cannot be supplied to form ice lenses. Thussands freeze homogeneously with depth. In clay, thehydraulic conductivity is so low that water cannot besupplied fast enough to maintain the temporary steady-state condition needed for ice lens growth. Heave inclay only develops if the freezing rate is slowed to wellbelow that in nature. Silts and silty soils have a com-bination of pore size, hydraulic conductivity, and freez-ing point depression that allow for large heave atnormal freezing rates in the field.

The freezing temperature penetrates ahead of a com-pleted ice lens, and a new lens will start to form onlyafter the temperature drops to the nucleation temper-ature. The nucleation temperature for a new lens maybe less than that for the one before because of reducedsaturation and consolidation from the previous flows,

which have now reduced the distance that water canbe from a particle surface. The temperature drop mustreach a depth where there is sufficient water availableafter nucleation to supply a growing lens. The thickerthe overlying lens, the greater the distance, thus ac-counting for the increased spacings between lenseswith depth. The greater the depth, the smaller the ther-mal gradient, as may be seen in Fig. 9.51, where(dT /dz)A � (dT /dz)A� where A� is on the temperaturedistribution curve for a later time t2. Because of this,the rate of heat extraction is slowed, and the temporarysteady-state condition for lens growth can be main-tained for a longer time, thus enabling formation of athicker lens.

More quantitative analyses of the freezing and frostheaving processes in terms of segregation potential,rates, pressures, and heave amounts are available. TheProceedings of the International Symposia on GroundFreezing, for example, Jones and Holden (1988),Nixon (1991), and Konrad (2001) provide excellentsources of information on these issues.

Thaw Consolidation and Weakening

When water in soil freezes, it expands by about 9 per-cent of its original volume. Thus a fully saturated soilincreases in volume by 9 percent of its porosity, evenin the absence of ice segregation and frost heave. Theexpansion associated with freezing disrupts the origi-nal soil structure. When thawed, the water returns toits original volume, the melting of segregated iceleaves voids, and the soil can be considerably moredeformable and weaker that before it was frozen. Un-der drained conditions and constant applied overburdenstress, the soil may consolidate to a denser state thanit had prior to freezing. The lower the density of thesoil, the greater is the amount of thaw consolidation.The total settlement of foundations and pavements as-sociated with thawing is the sum of that due to (1) thephase change, (2) melting of segregated ice, and (3)compression of the weakened soil structure.

Testing of representative samples under appropriateboundary conditions is the most reliable means forevaluating thaw consolidation. Samples of frozen soilare allowed to thaw under specified levels of appliedstress and under defined drainage conditions, and thedecrease in void ratio or thickness is determined. Anexample of the effects of freezing and thawing on thecompression and strength of initially undisturbed Bos-ton blue clay is shown in Fig. 9.53 from Swan andGreene (1998). These tests were done as part of aground freezing project for ground strengthening to en-able jacking of tunnel sections beneath operating raillines during construction of the recently completedCentral Artery/Tunnel Project in Boston. Detailed

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C4-FTe0 = 1.171

C1-UFe0 = 1.064

10 100

100

120

80

60

40

20

0

1000 10000

Effective Stress, σ�c (kPa)

UUC1-UF(σ1–σ3)max = 109.6 kPa

ε1 = 2.3%su/σ3cell = 0.36

e0 = 1.02; w = 37.5%

UUC4-FT(σ1– σ3)max = 42.4 kPa

ε1 = 12.8%su/σ3cell = 0.14

e0 = 1.13; w = 43.2%

Axial Strain. %

0 5 10 15 20 25

Dev

iato

r S

tres

s, σ

1–

σ 3 (

kPa)

Ver

tical

Str

ain,

εv

(%)

0

10

12

14

16

18

20

22

2

4

6

8

(a)

(b)

Figure 9.53 (a) Comparison between the compression behavior of unfrozen (C1-UF) andfrozen then thawed (C4-FT) samples of Boston blue clay. (b) Deviator stress vs. axial strainin unconsolidated–undrained triaxial compression of unfrozen (UUC1-UF) and frozen andthawed (UUC4-FT) Boston blue clay (from Swan and Greene, 1998).

analysis of the thaw consolidation process and its an-alytical representation is given by Nixon and Ladanyi(1978) and Andersland and Anderson (1978).

Ground Strengthening and Flow Barriers byArtificial Ground Freezing

Artificial ground freezing has applications for formationof seepage cutoff barriers in situ, excavation support,and other ground strengthening purposes. These appli-

cations are usually temporary, and they have the ad-vantage that the ground is not permanently altered,except for such property changes as may be caused bythe freeze–thaw processes. Returning the ground to itspristine state may be important for environmental rea-sons where alternative methods for stabilization couldpermanently change the state and composition of thesubsoil.

Freezing is usually accomplished by installation offreeze pipes and circulation of a refrigerant. For emer-

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CONCLUDING COMMENTS 319

Stress effect

Time, t (hr)

tf00

10

40

80

120

160

20 30

TemperatureEffect

T = –2.2 °C, σ = 0.138 MPa

T = –2.2 °C, σ = 0.55 MPa

T = 0 °C, σ = 0.55 M

Pa

FirstStage

SecondStage

ThirdStage

Nat

ural

Str

ain,

ε -

%

Figure 9.54 Creep curves for a frozen organic silty clay(from Sanger and Sayles, 1979).

gency and rapid ground freezing, expendable refrig-erants such as liquid nitrogen or carbon dioxide in anopen pipe can be used. The thermal energy removaland time requirements for freezing the ground can becalculated using the appropriate thermal conductivity,volumetric heat, and latent heat properties for theground and heat conduction theory in conjunction withthe characteristics of the refrigeration system (Sanger,1968; Shuster, 1972; Sanger and Sayles, 1979). Formany applications the energy required to freeze theground in kcal/m3 will be in the range of 2200 to 2800times the water content in percent (Shuster, 1972).However, if the rate of groundwater flow exceeds about1.5 m/day, it may be difficult to freeze the groundwithout a very high refrigeration capacity to ensurethat the necessary temperature decrease and latent heatremoval can be accomplished within the time any el-ement of water is within the zone to be frozen.

The long-term strength and stress–strain character-istics of frozen ground depend on the ice content, tem-perature, and duration of loading. The short-termstrength under rapid loading, which can be up to 20MPa at low temperature, may be 5 to 10 times greaterthan that under sustained stresses. That is, frozen soilsare susceptible to creep strength losses (Chapter 12).The deformation behavior of frozen soil is viscoplastic,and the stress and temperature have significant influ-ence on the deformation at any time. The creep curvesin Fig. 9.54 illustrate these effects. The onset of thethird stage of creep indicates the beginning of failure.The evaluation of stability of frozen soil masses, theprediction of creep deformation, and the possibility ofcreep rupture are complex problems because of het-erogeneous ground conditions, irregular geometries,and temperature and stress variations throughout thefrozen soil mass. Design and implementation consid-erations for use of ground freezing in construction aregiven by Donohoe et al. (1998).

9.23 CONCLUDING COMMENTS

Conductivity properties are one of the four key dimen-sions of soil behavior that must be understood andquantified for success in geoengineering. The otherthree dimensions are volume change, deformation andstrength, and the influences of time. They form thesubjects of the following three chapters of this book.

Water flows through soils and rocks under fully sat-urated conditions have been the most studied, and hy-draulic conductivity properties, their determination andapplication for seepage studies of various types, con-struction dewatering, and the like are central to geo-technical engineering. One objective of this chapter hasbeen to elucidate the fundamental factors that control

the permeability of soils to water and how this propertydepends on soil type, especially gradation, and issensitive to testing conditions, soil fabric, and en-vironmental factors. The understanding of these fun-damentals is important, not only because of theinsights provided but also because many of the sameconsiderations apply to the several other types of flowsthat are known to be important—chemical, electrical,and thermal. Knowledge of one is helpful in the un-derstanding and quantification of the other because themathematical descriptions of the flows follow similarforce-flux relationships.

At the same time it is necessary to take into accountthat the flows of fluids of different composition and theapplication of hydraulic, chemical, electrical, and ther-mal driving forces to soils can cause changes in com-positions and properties, with differing consequences,depending on the situation. Furthermore, as examinedin considerable detail in this chapter, flow coupling canbe important, especially advective and diffusive chem-ical transport, electroosmotic water and chemical flow,and thermally driven moisture flow. Considerable im-petus for research on these processes has been gener-ated by geoenvironmental needs, including enhanced

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and more economical waste containment and site re-mediation strategies.

Ground freezing, in addition to its importance in en-gineering and construction in cold regions, is seeingnew applications for temporary ground stabilizationneeded for underground construction in sensitive urbanareas.

QUESTIONS AND PROBLEMS

1. A uniform sand with rounded particles has a voidratio of 0.63 and a hydraulic conductivity, k, of 2.7� 10�4 m/s. Estimate the value of k for the samesand at a void ratio of 0.75.

2. The soil profile at a site that must be dewateredconsists of three homogeneous horizontal layers ofequal thickness. The value of k for the upper andlower layers is 1 � 10�6 m/s and that of the mid-dle layer is 1 � 10�4 m/s. What is the ratio of theaverage hydraulic conductivity in the horizontaldirection to that in the vertical direction?

3. Consider a zone of undisturbed San Francisco Baymud free of sand and silt lenses. Comment on theprobable effect of disturbance on the hydraulicconductivity, if any. Would this material be ex-pected to be anisotropic with respect to hydraulicconductivity? Why?

4. Assume the specific surface of the San FranciscoBay mud in Question 3 is 50 m2/g and prepare aplot of the hydraulic conductivity in meters/sec-ond as a function of water content over the rangeof 100 percent decreased to 25 percent by consol-idation using the Kozeny–Carman equation.Would you expect the actual variation in hydraulicconductivity as a function of water content to beof this form? Why? Sketch the variation youwould expect and explain why it has this form.

5. At a Superfund site a plastic concrete slurry wallwas proposed as a vertical containment barrieragainst escape of liquid wastes and heavily con-taminated groundwater. The subsurface conditionsconsist of horizontally bedded mudstone and silt-stone above thick, very low permeability clayshale. The cutoff wall was to extend into the slayshale, which has been shown to be able to serveas a very effective bottom barrier. For the finaldesign and construction, however, a 3-ft-widegravel trench was used instead of the slurry wall.Sumps and pumps placed in the bottom of thetrench are used to collect liquids. Explain how thistrench can serve as an effective cutoff and discussthe pros and cons of the two systems.

6. How can the effects of incompatibility betweenchemicals in a waste repository and a compactedclay liner best be minimized?

7. Two parallel channels, one with flowing water andthe other with contaminated water, are 100 ft apart.The surface elevation of the contaminated channelis 99 ft, and the surface elevation of the clean wa-ter channel is at 97 ft. The soil between the twochannels is sand with a hydraulic conductivity of1 � 10�4 m/s, a dry unit weight of 100 pcf, anda specific gravity of solids of 2.65. Estimate thetime it will take for seepage from the contaminatedchannel to begin flowing into the initially cleanchannel. Make the following assumptions and sim-plifications:a. Seepage is one dimensional.b. The only subsurface reaction is adsorption onto

the soil particles.c. The soil–water partitioning coefficient is 0.4

cm3/g.d. Hydrodynamic dispersion can be ignored.

8. For the compacted clay waste containment linershown below and assuming steady-state condi-tions:a. What is the contaminant transport for pure

molecular diffusion?b. What is the contaminant transport rate for pure

advection?c. What is the contaminant transport rate for ad-

vection plus diffusion?d. Why don’t the answers to parts (a) and (b) add

up to (c)?

NOTE: Advection and diffusion are in thesame direction; therefore, J � 0, and the so-lution will be in the form

a x2c � a e � a1 3

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QUESTIONS AND PROBLEMS 321

9. One-dimensional flow is occurring by electroos-mosis between two electrodes spaced at 3.0 m witha potential drop of 100 V (DC) between them.What should the water flow rate be if the coeffi-cient of electroosmotic permeability, ke, is 5 �10�9 m2/s V assuming an open system? If no wa-ter is resupplied at the anode, what maximum con-solidation pressure should develop at a pointmidway between electrodes if the hydraulic con-ductivity of the soil is 1 � 10�8 m/s?

10. a. A soil has a coefficient of electroosmotic per-meability equal to 0.3 � 10�8 m/s per V/mand a hydraulic conductivity of 6 � 10�9 m/s.Starting from the general relationship

J � L Xi ij j

derive an expression for the pore water tensionthat may be developed under ideal conditionsfor consolidation of the clay by electroosmosisand compute the value that should develop at apoint where the voltage is 25 V. Be sure toindicate correct units with your answers.

b. In the absence of electrochemical effects orcavitation, would you consider your answer topart (a) to represent an upper or lower boundestimate of the pore water tension? Why?(HINT: Consider the influence of consolidationon the soil properties that are used to predictthe pore water tension.)

11. In 1892 Saxen established that there is equivalencebetween electroosmosis and streaming potentialsuch that the results of a hydraulic conductivitytest in which streaming potential is measured canbe used to predict the volume flow rate duringelectroosmosis in terms of the electrical current.Starting with the general equations for coupledelectrical and hydraulic flow, derive Saxen’s law.What will be the drainage rate from a soil, inm3/h amp, if the streaming potential is 25 mV/atm? What will be the cost of electrical power percubic meter of water drained if electricity costs$0.10 per kWh and a maximum voltage of 75 Vis used?

12. It might be possible to prevent leakage of hazard-ous and toxic chemicals through waste impound-ment and landfill clay or geosynthetic-clay linersby means of an electroosmosis counterflow barrieragainst hydraulically driven seepage. Consider theimpoundment and liner system shown below.

Assume that the water pressure at the top of the leach-ate collection layer is atmospheric and that the onlyfluxes across the liner are water and electricity. Thecharacteristics of the compacted clay liner are:

Hydraulic conductivity�7k � 1 � 10 m/sh

Electroosmotic coefficients�9 2k � 2 � 10 m /s Ve

�6 3k � 0.2 � 10 m /s ampi

a. Wire mesh is proposed for use as electrodes.Where would you place the anode and cathodemeshes?

b. If the waste pond is to be filled to an averagedepth of 6 m, what voltage drop should bemaintained between the electrodes?

c. What will the power cost be per hectare of im-poundment per year? Power costs $0.09 perkWh.

d. Assume that the leachate collection layer isflushed continuously with freshwater and thatthe liquid waste contains dissolved salts. Writethe complete set of equations that would be re-quired to describe all the flows across the linerduring electroosmosis. Define all terms.

e. Will maintenance of a no hydraulic flow con-dition ensure that no leachate will escapethrough the clay liner? Why?

13. a. Estimate the minimum footing depths for struc-tures in a Midwestern city where the freezingindex is 750 degree-days and the duration ofthe freezing index is 100 days. The mean an-nual air temperature is 50�F. The soil is siltyclay with a water content of 20 percent and adry unit weight of 110 lb/ft3. Assume no icesegregation and compare values according tothe Stefan and modified Berggren formulas.

b. What will be the depth of frost penetration be-low original ground surface level if a surfaceheave of 6 inches develops due to ice lens for-

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mation? Assume a frozen ground temperatureof 32�F.

c. If a pavement is to be placed over the soil, whatthickness of granular base course should beused to prevent freezing of the subgrade? Thebase course will be compacted to a dry densityof 125 lb/ft3 at a water content of 15 percent.If the pavement structure is to contain an 8-inch-thick Portland cement concrete surfacelayer, will your result tend to overestimate orunderestimate the base thickness required?Why?

14. A compacted fine-grained soil is to be used as aliner for a chemical waste storage area. Free liquidleachate and possibly some heavier than waterfree phase nonsoluble, nonpolar organic liquids(DNAPLs) may accumulate in some areas as a re-sult of rupturing and corrosion of the drums inwhich they were stored. Two sources of soil foruse in the liner are available. They have the fol-lowing properties:

Property Soil A Soil B

Unified class (CH) (CL)Liquid limit (%) 90 45Plastic limit (%) 30 25Clay size (%) 50 30Silt size (%) 30 40Sand size (%) 20 30Predominant clay

mineralSmectite Illite

Cation exchangecapacity (meg/100 g)

60 20

a. Which of the two soils would be best suited foruse in the liner? Why?

b. What tests would you use to validate yourchoice? Why?

c. Assume that you have confirmed that it will bepossible to compact the soil to states that willhave hydraulic conductivities in the range of 1� 10�8 to 1 � 10�11 m/s. A liner thickness of0.6 m is proposed. Leachate is likely to accu-mulate to a depth of 1.0 m above the top of theliner. A leachate collection layer will underliethe liner.

d. If the concentration of dissolved salts in theleachate is 1.0 M and the average diffusion co-efficient is 5 � 10�10 m2/s, determine for thesteady state the total amount of dissolvedchemical per unit area per year that will escape

through the liner as a function of the hydraulicconductivity. Show in the same diagram theproportions of the total that are attributable todiffusion and advection.

Assume that the leachate collection layer isfully drained, but for purposes of analysis thefluid level can be considered at the bottom ofthe clay. Determine the leakage rate throughthe liner per unit area as a function of the hy-draulic conductivity and show it on a diagram.

15. The diagram below shows the cross section of atunnel and underlying borehole in which wastecanisters for spent nuclear fuel are located. Suchan arrangement is proposed for deep (e.g., severalhundred meters) burial of nuclear waste in crys-talline rock. The surrounding rock can be assumedfully saturated, and the groundwater table will bewithin a few tens of meters of the ground surface.Thermal studies have shown that the temperatureof the waste canister will rise to as high as 150�Cat its surface. A canister life of about 100 years isanticipated using either stainless steel or copperfor the material. The surrounding environmentmust be safe against leakage of radionuclides fromthe repository for a minimum of 100,000 years.

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QUESTIONS AND PROBLEMS 323

Clay or a mix of clay with other materials such assand and crushed rock is proposed for use as thefill both around the canisters and in the tunnel.a. What are the most important properties that the

backfill should possess to ensure isolation andbuffering of the waste from the outside envi-ronment?

b. What clay material would you propose for thisapplication and under what conditions wouldyou place it?

c. Assess the probable natures and directions ofheat and fluid flows that will develop, if any.

d. What alterations might occur in the materialduring the life of the repository if any? Con-sider the effects of groundwater from the sur-rounding ground, corrosion of the canister, andthe prolonged exposure to high temperature.Would each of these alternations be likely toenhance or impair the effectiveness of the claypack?

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