unsaturated swcc

A model to predict the water retention curve frombasic geotechnical properties

M. Aubertin, M. Mbonimpa, B. Bussire, and R.P. Chapuis

Abstract: The water retention curve (WRC) has become a key material function to define the unsaturated behaviorof soils and other particulate media. In many instances, it can be useful to have an estimate of the WRC early in aproject, when little or no test results are available. Predictive models, based on easy to obtain geotechnical properties,can also be employed to evaluate how changing parameters (e.g., porosity or grain size) affect the WRC. In this paper,the authors present a general set of equations developed for predicting the relationship between volumetric water con-tent, , (or the corresponding degree of saturation, Sr) and suction, . The proposed model assumes that water retentionresults from the combined effect of capillary and adhesion forces. The complete set of equations is given together withcomplementary relationships developed for specific applications on granular materials and on fine-grained soils. It isshown that the model provides a simple and practical means to estimate the water retention curve from basic geo-technical properties. A discussion follows on the capabilities and limitations of the model, and on additional tools de-veloped to complement its use.

Key words: water retention curve, unsaturated soils, prediction, porosity, grain size, liquid limit.

Rsum : La courbe de rtention deau (CRE) est devenue une fonction cl pour dfinir le comportement non saturdes sols et autres matriaux meubles. Dans beaucoup de cas, il peut tre utile davoir une valuation de la CRE durantles premires phases dun projet, lorsque peu ou pas de rsultats dessais sont disponibles. Des modles prdictifs,bass sur les proprits gotechniques de base, peuvent aussi tre utiliss pour valuer comment le changement desparamtres (en termes de porosit ou de granulomtrie) affecte la CRE. Dans cet article, les auteurs prsentent un en-semble dquations dveloppes pour prdire la relation entre la teneur en eau volumique (ou le degr de saturationSr correspondant) et la succion . Le modle propos est bas sur lhypothse que la rtention deau rsulte de lactioncombine de forces capillaires et dadhsion. Le systme dquations est donn avec des relations complmentairesdveloppes pour des applications spcifiques sur des matriaux granulaires et sur des sols fins. On montre ainsi quele modle constitue un moyen simple et pratique pour estimer la courbe de rtention deau partir des proprits go-techniques de base. Une discussion suit sur les capacits et les limitations du modle, ainsi que sur des outils addition-nels dvelopps pour complter son usage.

Mots cls : courbe de rtention deau, sols non saturs, prdiction, porosit, granulomtrie, limite de liquidit.

Aubertin et al. 1122

Introduction

In geotechnical engineering, subsurface water is often di-vided into free water in the saturated zone and moisture re-tained in the unsaturated (vadose) zone (e.g., Bowles 1984;

Smith 1990). More attention is now being paid to better de-fine the response of water in the vadose zone. Geotechniquefor unsaturated media has become a rapidly expanding field,which is related to a wide variety of applications including:estimation of field capacity (Meyer and Gee 1999); effi-ciency of covers with capillary barrier effects (Bussire et al.2003); bearing capacity of foundation materials (Rassam andWilliams 1999); seepage through dams (Chapuis andAubertin 2001); compressibility and swelling soil response(Rampino et al. 2000); shear strength and constitutive be-havior (e.g., Alonso et al. 1990; Vanapalli et al. 1996); con-taminant transport (Yong 2001); land subsidence (Thu andFredlund 2000); and freezethaw of road structures (Konradand Roy 2000). Fredlund (2000) and Looney and Falta(2000) present recent overviews of various applications ofunsaturated soil mechanics and physics.

The distribution and motion of water in the unsaturatedzone are closely related to forces of molecular attractionwhich are responsible for water adhering to solid surfaces

Can. Geotech. J. 40: 11041122 (2003) doi: 10.1139/T03-054 2003 NRC Canada

1104

Received 7 January 2002. Accepted July 4 2003. Published onthe NRC Research Press Web site at http://cgj.nrc.ca on15 October 2003.

M. Aubertin,1,2 M. Mbonimpa,2 and R.P. Chapuis.Department of Civil, Geological and Mining Engineering,cole Polytechnique de Montral, P.O. Box 6079, StnCentre-Ville, Montreal, QC H3C 3A7, Canada.B. Buissire.2 Department of Applied Sciences, Universit duQubec en Abitibi-Tmiscamingue (UQAT), 445 boul.de lUniversit, Rouyn-Noranda, QC J9X 5E4, Canada.1Corresponding author (e-mail: [email protected]).2Industrial NSERC PolytechniqueUQAT Chair onEnvironment and Mine Wastes Management.

I:\cgj\CGJ40-06\T03-054.vpOctober 8, 2003 9:38:39 AM

Color profile: DisabledComposite Default screen

(i.e., hygroscopic or adsorbed water on soil particles) and forsurface tension at the interface with air causing capillaryretention (Bear 1972; Marshall et al. 1996). In saturated me-dia, the adsorption forces tend to reduce the pore spaceavailable for water flow, hence reducing the area for freewater flow and hydraulic conductivity, while capillary forcesdisappear in the absence of a waterair interface. In unsatu-rated conditions, these two distinct but complementary typesof force affect the behavior of soil. The corresponding prop-erties, including hydraulic conductivity and shear strength,are no longer material constants but rather depend on therelative amount of water and air in the pore space.

In a porous system, increasing the value of suction, ,(defined by ua uw, where ua is the air pressure and uw is thewater pressure) tends to reduce the volumetric water content,. The quantity of water retained in a soil by suction dependson many factors, namely: shape, size, and distribution ofpore space; mineralogy and surface activity of solid grainparticles; and the chemical composition of interstitial water.The desaturation is typically more pronounced in coarse-grained materials (such as sand and gravel) than in fine-grained materials (such as silt and clay). The value of at agiven also depends on the path, whether it occurs during awetting or drying phase. Different paths may induce some-what different curves (e.g., Maqsoud et al. 2002), but suchhysteresis phenomena will not be addressed directly herein,as only the drainage path is considered here (to simplify thepresentation).

For a given porous media, parameters and are relatedto each other, and form a fundamental material propertyknown as either the water retention curve (WRC) (Marshallet al. 1996; Aubertin et al. 1998; Delleur 1999), the soilwater characteristic curve (Fredlund and Rahardjo 1993;Barbour 1998), the soil suction curve (Yong 2001), the soilmoisture-retention curve (Kovcs 1981), and various othernames (e.g., Klute 1986; Carter 1993; Hillel 1998; Looneyand Falta 2000). The WRC is used in many applications torepresent the relationship between and . The former pa-rameter is related to porosity, n, and degree of saturation, Sr( = nSr); can also be expressed as a function of the morecommon gravimetric water content, w, (i.e., = w(1 n)Dr,where Dr is the relative density of the solid particles). The value, on the other hand, is typically expressed with pressureunits (e.g., kPa). Suction can also be expressed as a pressurehead, as is done here for the proposed model because it isbased on the use of an equivalent capillary rise, hco (seeexplanations in the section below). The total is the sum ofthe matric suction, m, and of the osmotic suction, ; how-ever, the latter is not considered explicitly in this presenta-tion. For engineering applications, typically takes a po-sitive value under negative pore-water pressure, uw, abovethe phreatic water surface, where uw = ua or = ua uw = 0(with ua taken as the reference air pressure).

A large number of experimental investigations have beendevoted to developing and applying techniques to obtain the relationship of soils. Accordingly, various direct and in-direct measurement methods have been proposed, and manyof these have been reviewed in textbooks and monographs(e.g., Klute 1986; Carter 1993; Fredlund and Rahardjo 1993;Marshall et al. 1996; Looney and Falta 2000). The evolution

and understanding of the WRC and of the correspondingmeasurement techniques have been recently presented byBarbour (1998), who provided a historical perspective.

When the volumetric water content diminishes followinga suction increase, the flow of water becomes more difficultbecause of the reduced area and increased tortuosity of thewater phase. The unsaturated hydraulic conductivity, ku,thus, depends on (or ). At a sufficiently low water con-tent, the water phase becomes discontinuous and ku is re-duced to a very small (near zero) value.

To perform the groundwater flow analyses required in anumber of applications in geotechnique and soil physics, en-gineers and scientists use Richards (1931) equation, whichis a combination of Darcys law and the mass conservationequation. In this equation, ku is a function of the state vari-ables, which are typically expressed in terms of or of (e.g., Leong and Rahardjo 1997a; Leij et al. 1997). As directmeasurement of ku can be difficult, time consuming, andcostly, it is customary to use, as a starting point, some rela-tionship between and to estimate the ku function (e.g.,Mualem 1976, 1986; Fredlund et al. 1994), because theWRC can be evaluated more easily than ku in the laboratoryor in the field. The resulting versus data, obtained fromdirect measurements, are plotted and used to derive specificmathematical functions with curves that run through the datapoints. Such equations have been proposed by a large num-ber of authors including Brooks and Corey (1964), vanGenuchten (1980), Bumb et al. (1992), Fredlund and Xing(1994), to name only a few. Many of these equations havebeen compared to each other over the years (e.g., Khire et al.1995; Leij et al. 1997; Leong and Rahardjo 1997b; Burgerand Shackelford 2001), showing that each has some advan-tages and limitations, depending on the technique and databank used for comparison.

Measurement of the WRC in the laboratory and in thefield, although less demanding than that of ku, can also berelatively time consuming and expensive. In some situations,it may be useful to have estimates of the WRC beforehand,especially during the preliminary stages of a project whenthe available information is limited. Predictive models forthe WRC, which are typically based on simple geotechnical(pedologic) properties such as grain size and porosity, canalso be useful when analyzing and validating test results,and also for evaluating the variations that can be expected inthe field within a nonhomogeneous deposit.

Quite a few predictive models, sometimes referred to aspedotransfer functions (e.g., Bouma 1989; Vereecken et al.1992; Schaap and Leij 1998), have been proposed over thelast two decades or so (see Elsenbeer 2001, and the corre-sponding Monograph Issue on the topic). These include anumber of functional regression methods using empiricalWRC equations, where the fitting parameters are related toparticular textural parameters (e.g., Cosby et al. 1984;Schaap et al. 1998). Other discrete regression methods makeno a priori assumption regarding the shape of the WRC, andconstruct empirical models from the regression equationslinking the values and basic properties (e.g., Gupta andLarson 1979; Rawls and Brackensiek 1982; Rawls et al.1991; Vereecken et al. 1992; Tietje and Tapkenhinrichs1993). It appears preferable, however, to use predictive mod-

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els for which the WRC equation is based on physical charac-teristics of the media. This latter type of model includesthose of Arya and Paris (1981), Haverkamp and Parlange(1986), Tyler and Wheatcraft (1989, 1990), Haverkamp etal. (1999), and Arya et al. (1999).

A model of this type was also proposed by Kovcs (1981).Still relatively unknown and seldom used in the geotechnicalfield, the Kovcs (1981) model assumes that a distinction ex-ists between capillary and adhesive forces, which are bothacting simultaneously to induce suction. This approach po-tentially provides a conceptual view that can be related tothe actual processes involved (e.g., Celia et al. 1995; Nitaoand Bear 1996). However, the Kovcs (1981) model, in itsoriginal version, did not easily lend itself to practical engi-neering applications because some of the key parametersneeded for its use were not completely defined. A few yearsago, the authors applied this model after some modifications(hence, the name Modified Kovcs or MK model) to theWRC of tailings and silts (Aubertin et al. 1998). In this pa-per, the MK model is extended for general applications tovarious types of materials, from coarse sand to some fine-grained soils. The basic assumptions and theoretical consid-erations behind these extensions are briefly presented. Thedescriptive and predictive capabilities of the MK model areshown using test results taken from the literature and ob-tained in the authors facilities. It should be mentioned at theonset that the model has been developed only for isotropicand homogeneous materials, under a drainage path; influ-ence factors such as internal microstructures, anisotropy, andvolume changes are neglected in this presentation.

The MK model

The proposed MK model presented in this paper retainsthe same physical concepts from which the original Kovcs(1981) model was constructed. The modifications introducedserve to generalize the statistical function used to describethe pore-size distribution of the media appearing in the capi-llary component. Some constitutive parameters are alsoexpressed more specifically as a function of basic soil prop-erties, as will be described in the following sections.

The equivalent capillary riseThe MK model makes use of a parameter defined as the

equivalent capillary rise, hco [L], in the porous medium. Dueto the central role played by hco, which is expressed as an el-evation, the MK model equations presented here rely onpressure head units to define (given in cm of water). Therole of the parameter hco is the same as the average capillaryrise in the original Kovcs (1981) model. This parameter isderived from the well-known expression used for the rise hc[L] of water in a capillary tube having a diameter d [L]. Thevalue of hc is given by (Smith 1990; Chin 2000):

[1] hdc

w w

w

=4

cos

where w [MT2] is the surface tension of water (w =0.073 N/m at 20 C), w [] is the contact angle between wa-ter and the tube surface (w = 0 for quartz and glass), andw [ML2T2] is the unit weight of water (w = 9.8 kN/m3 at20 C).

Equation [1] indicates that capillary rise is inversely pro-portional to the diameter of the tube. When applied to porespace in soils above the phreatic surface, this equation helpsto understand why there can be a much greater water rise infine-grained soils where the voids (somewhat similar to cap-illary tubes) are small, than in coarse-grained soils where thevoids are typically larger. In soils, however, the pore size isnot uniform so hc is not easily defined with eq. [1]. Thispore system can be substituted by a system of regular chan-nels with a diameter expressed as the equivalent hydraulicpore diameter, deq [L], defined as (Bear 1972; Kovcs 1981):

[2] d VAeq

v

v

= 4

where Vv [L3] and Av [L2] are the volume and surface of thevoids, respectively. In practice, Av approximately corre-sponds to the surface area, AG [L2], of the solid grains. Byrelating AG to the massic specific surface area Sm [L2M1],eq. [2] can be transformed as (Scheidegger 1974):

[3] d eSeq s m

= 4

In this equation, e [] is the void ratio and s [ML3] is thesolid grain density.

The hco in a soil is obtained by replacing d (in eq. [1]) bydeq, and can, therefore, be expressed as:

[4] h Se

cow w

w

s m=

cos

This is one of the fundamental equations from which theMK model is built. As it will be indicated below, hco issomewhat equivalent (at least for granular soils) to theheight of the capillary fringe above the still water table in ahomogeneous deposit, as defined in many geotechnique text-books (e.g., Lambe and Whitman 1979; Bowles 1984).

Although Sm can be directly measured by various tech-niques (e.g., Lowell and Shields 1984; Igwe 1991), in mostpractical cases, the value of Sm is not readily available to ap-ply in eq. [4]. For coarse-grained soils, the specific surfacearea can nevertheless be estimated from the grain-size distri-bution using the following expression (Kovcs 1981):

[5] SDm s H

=

where [] is a shape factor (6 18; = 10 is used hereas in the Kovcs original model), and DH [L] is an equiva-lent particle diameter for a heterogeneous mixture. The DHfor a heterogeneous mix of particles theoretically representsthe diameter of a homogeneous mix (with a single size) thathas the same specific surface area as the heterogeneous one.

Equation [4] can thus be rewritten to calculate the equiva-lent capillary rise in a soil above the water table:

[6] heDco,G

w w

w H=

cos

where subscript G stands for granular (low-plasticity, low-cohesion) materials, as opposed to fine-grained materials(which will be discussed below). In this equation, w will betaken as zero (e.g., Marshall et al. 1996).

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1106 Can. Geotech. J. Vol. 40, 2003



In granular soils, Sm (and DH) can be evaluated by subdi-viding the grain-size curve based on standard mesh sizes(Chapuis and Lgar 1992). For practical geotechnical appli-cations, the value of DH can also be approximated using thefollowing function (Aubertin et al. 1998; Mbonimpa et al.2000, 2002):[7] DH = [1 + 1.17 log (CU)]D10where D10 [L] is the diameter corresponding to 10% passingon the cumulative grain-size distribution curve, and CU [] isthe coefficient of uniformity (CU = D60/D10).

For the equivalent capillary rise in granular soils, eq. [6]is then expressed as follows:

[8] h beDco,G

=

10

with

[9a] bC

=

+

w w

U w[1.17 log (cos

) ]1For the values of , w, w and w given above, with hco

and D10 expressed in cm, eq. [9a] becomes:

[9b] bC

=

+

0 751

.

)1.17 log ( UAccording to eq. [9], for a sand with D10 = 0.01 cm, CU =

5, and e = 0.5, then b = 0.412 cm2 and the equivalent capil-lary rise in a granular material, hco,G, is about 83 cm; for asilt with D10 = 0.0005 cm, CU = 20 and e = 0.5, and b =0.297 cm2, hco,G would be approximately 1190 cm. Thesevalues of hco,G are in the same range of magnitude as theheight of the capillary fringe obtained from the simplifiedexpression proposed by Bowles (1984), which only consid-ers the influence of D10.

For some plasticcohesive soils, the above equations donot provide reliable estimates of Sm and hco, particularlywhen the liquid limit, wL (%), is above about 30% to 40%.For such fine-grained soils, other factors such as surface ac-tivity influence their water retention capacity (lets recallhere that the effect of internal microstructure due to loadinghistory is not explicitly taken into account in this presenta-tion). For these soils, Sm (in eq. [4]) is better estimated usingthe relationship that exists between the specific surface areaand wL. The following empirical expression, recently pro-posed by Mbonimpa et al. (2002), is used here:[10] S wm L=

where [L2M1] and (unitless) are material parameters.Using a relatively large number (78) of test results from var-ious sources, it has been established that 0.2 m2/g and 1.45 for materials with 22 m2/g Sm 433 m2/g and18% wL 127% (see details in Mbonimpa et al. 2002).

Combining eqs. [4] and [10] gives:

[11] he

wco,P L1.45

=

where subscript P stands for plasticcohesive materials.From the previous developments, parameter [L] can beexpressed as:

[12a]

= w ww

scos

For values of w given in N/m, w [], w in kN/m3, in m2/g, and s in kg/m3, eq. [12a] becomes:[12b] 0.15sFrom eqs. [11] and [12], one can calculate that for a fine-grained soil with wL = 40%, s = 2700 kg/m3, and e = 0.8,then = 405 cm and hco,P = 106 500 cm; for wL = 80%, s =2700 kg/m3, and e = 0.8, one obtains = 405 cm and hco,P =290 968 cm.

The WRC equationsThe MK model uses hco as a reference parameter to define

the relationship between the degree of saturation, Sr (or vol-umetric water content, ), and matric suction, (expressedhere as a pressure head). As mentioned above, both the orig-inal Kovcs (1981) model and the MK model assumes thatwater is held by capillary forces, responsible for capillarysaturation, Sc, and by adhesive forces, causing saturation byadhesion, Sa. In these models, both components act simulta-neously, and are thus, included in measurements made todetermine the relationship.

The Sc component equation is obtained from a cumulativepore-size distribution function, while the equation of Sa isgiven by an interaction law with van der Waals type attrac-tion between grain surface and water dipoles. The Sc compo-nent is more important at relatively low suction values,while the Sa component becomes dominant at higher suctionwhen most capillary water has been withdrawn.

The proposed set of equations for the MK model is writ-ten as follows for the degree of saturation:

[13] Sn

S S Sr c a c= = + *( )1

In this equation, a truncated value of the adhesion compo-nent, Sa*, is introduced in place of Sa used in the originalmodel, to make sure that the adhesion component does notexceed unity at low suction (0 Sa* 1); it is expressed as:[14] S Sa a* = 1 1where represents the Macauley brackets ( y = 0.5(y +y)); for Sa 1, Sa* = 1, and for Sa < 1, Sa* = Sa (defined be-low).

The contributions of the capillary and adhesion compo-nents to the total degree of saturation are defined as func-tions of hco and using eqs. [15][17].[15] S h m hmc co co= + 1 12 2[( / ) ] exp[ ( / ) ]

[16] S a C he

a cco n

n

=

/( / )

( )/

/ /

2 3

1 3 1 6

with

[17] C = +

+1 1

1 0ln( / )ln( / )

r

r

Equation [15], providing the expression to evaluate Sc (0 Sc 1) is a generalization of the one developed by Kovcs(1981), in which the statistical exponential function has been

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expanded to better reflect the influence of pore-size distribu-tion through the distribution parameter m. The statisticaldistribution expression used for Sc is one that can also be ap-plied for grain-size curves, as there is a well-known similar-ity between the latter and the WRC (e.g., Aubertin et al.1998). On the WRC, parameter m influences the air entryvalue a (or AEV), which theoretically corresponds to thesuction when the largest pores start to drain, and also therate of decline beyond a in the capillary range (Aubertin etal. 2003). For practical applications, the value of m will beexpressed as a function of basic geotechnical properties.

Equation [16] is based on Kovcs (1981) developmentsmade from approximations used in theoretical physics, inwhich a sixth-order hyperbola is used to relate the adhesionsaturation (due to the film of water adsorbed on grain sur-faces) to suction. In this equation, ac is the adhesion coeffi-cient (dimensionless) and n is a normalization parameterintroduced for unit consistencies (n = 1 cm when is givenin cm, corresponding to n 103 atmosphere). ParameterC (eq. [17]), taken from Fredlund and Xing (1994), forcesthe water content to zero when y reaches a limit imposed bythermodynamic equilibrium ( = 0 at = 0 = 107 cm of wa-ter, corresponding approximately to complete dryness). Ineq. [17], r represents the suction at residual water content;r is also equal to the water entry value, WEV, when no dis-tinction is made between drainage and wetting path. As willbe shown below, r can be defined from basic soil properties(as is the case with a and hco).

Figures 1a and 1b show typical curves drawn from theMK model in a semi-log Sr plane, illustrating the contri-butions of Sc and Sa* for granular (Fig. 1a) and plasticcohesive (Fig 1b) materials. Hypothetical (but representa-tive) values for D10, CU, e, wL, s, hco, m, ac, and r havebeen used for these sample plots. The parameters r and aare also shown on Figs. 1a and 1b, together with 90 and 95to define suctions for preset degrees of saturation (of 90%and 95%, respectively). Suctions 90 and 95 are comparedto a in the discussion. The effect of varying parameters m,ac, and r on the WRC is presented graphically in a compan-ion technical report (Aubertin et al. 2003).

Parameter determination and modelapplications

The MK model presented above includes a set of equa-tions that provides an estimate of the WRC from full satura-tion (Sr = 1, = n) to complete dryness (Sr = 0 = ). To applythe model however, three parameters in the constitutiveequations have to be defined explicitly: parameter m ineq. [15], ac in eq. [16], and r in eq. [17]. Based on investi-gations carried out by the authors on a diversity of soils andparticulate media (identified in Tables 1 and 2), it has beenfound that the values of m, ac and r can be predeterminedfrom basic geotechnical properties.

The experimental data used here for granular materialshave been taken from various investigations performed onsands, low plasticity silts, and tailings. The authors resultshave been obtained with either plate extractors or Tempecells according to procedures described in Aubertin et al.(1995, 1998), while some others have been taken from the

literature (see Table 1). All the experimental results on fine-grained materials have been taken from the literature (seeTable 2). However, to limit the possibility of significantshrinkage during testing (see Discussion), results on fine-grained soils cover a relatively small range of liquid limitand porosity values.

Residual suctionThe residual suction r introduced in the expression for

C was evaluated first. For granular materials, values of rwere determined using the tangent method (Fig. 1a), as de-scribed by Fredlund and Xing (1994). The WRCs were ob-tained by fitting the experimental data with a descriptiveequation (i.e., the van Genuchten (1980) model implementedin the code RETC; van Genuchten et al. 1991); the tangentmethod was then applied to the corresponding WRC best-fit.The subsequent analysis indicated that the following rela-tionship provides an adequate estimate of the residual suc-tion (see Fig. 2):

[18] rH

=0 42

1 26.

( ) .eDA simple relationship was also established between r and

the equivalent capillary rise, hco,G (eq. [8]), for granular ma-terials:

[19] r co,G= 0 86 1 2. .hThis equation is illustrated in Fig. 3, with the correspond-

ing data. As can be seen, hco,G is typically somewhat lowerthan the value of r obtained from the above describedmethod. In the following, the value of r for granular soils isdetermined from either eqs. [18] or [19], which give almostidentical residual suction values (see Aubertin et al. 2003 formore details).

Equation [18] is frequently impractical (or inapplicable)for fine-grained soils because D10 and CU are often un-known. Furthermore, the WRC of plasticcohesive materialsis rarely bilinear around the residual water content in a semi-log plot. Considering also that the experimental results arenot customarily available at high suctions, it can be arguedthat the change in slope on the WRC corresponding to resid-ual suction is generally not well defined (see also the discus-sion by Corey (1994)). A direct determination of r, thus,becomes difficult for these soils.

In the following applications of the MK model, it is as-sumed that the r values for plasticcohesive soils can be re-lated to hco,P (eq. [11]) using the same dependency as forgranular soils (eq. [19]). Hence, the expression used for fine-grained soils becomes:

[20] r L= 0 86

1 21 74

.

.

.

ew

Parameters ac and mOnce r was determined for each material in the database

(see Tables 1 and 2), the remaining parameters (ac and m)were evaluated from a fitting procedure, so that calculatedWRCs match experimental data as closely as possible (withthe MK model). Figures 410 compare typical fitted curves

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Fig. 1. (a) Illustration of the capillary and adhesion saturation contributions to the total degree of saturation for a noncohesive(low plasticity) soil, showing the WRC obtained with the MK model (for D10 = 0.006 cm, CU = 10, e = 0.6; r = 190 cm, m = 0.1and ac = 0.01); a is the pressure (suction) corresponding to the air entry value (AEV), r is the pressure corresponding to the resid-ual water content (also called water entry value, WEV), and 90 and 95 are suctions corresponding to a degree of saturation, Sr, of90% and 95%, respectively. (b) Illustration of the capillary and adhesion saturation contributions to the total degree of saturation for acohesive (plastic) soil, showing the WRC obtained with the MK model (for wL = 30%, e = 0.6, s = 2700 kg/m3, r = 9.7 105 cm,m = 3 105, and ac = 7 104); a is the pressure (suction) corresponding to the air entry value (AEV), 90 and 95 are suctionscorresponding to a degree of saturation, Sr, of 90% and 95%, respectively.



(identified as MK model best-fit) using experimentaldata on different granular and fine-grained materials; basicproperties (D10, CU, and e for granular materials; wL, e, andrs for plasticcohesive soils) are also given in each figure.The values of ac and m that lead to the best-fit curves pre-sented in Figs. 410 are given in Table 3.

Further investigations have been conducted to relate theparameter values m and ac to basic geotechnical properties.For granular soils, where hco,G is given by eq. [8], the valueof the pore-size distribution parameter m can be closely ap-proximated by the inverse of the uniformity coefficient: m =1/CU. When CU = 1, m = 1 and Kovcs (1981) originalequation for Sc is then recovered from the MK model. Forthese same granular materials, analysis shows that ac can beconsidered approximately constant, with ac = 0.01.

For fine-grained (plasticcohesive) soils, for which hco,P isgiven by eq. [11], both m and ac values can be taken as con-stants (with m = 3 105 and ac = 7 104) in the predictiveapplications. In this case, the influence of grain-size distri-bution appears to be somewhat superseded by the dominant

effect of the surface activity (defined here through the wLdependency).

Sample resultsThe relationships and parameter values defined above are

used to evaluate the WRC of various materials. Tables A1and A2, in the Appendix, give detailed calculation results toshow explicitly how the presented set of equations are usedto obtain the WRC with the MK model. Figures 410 com-pare representative fitted curves (with best-fit ac and m val-ues) and predicted curves (with preset ac and m values). Ascan be seen on these figures, there is generally a good agree-ment between the predicted and the measured WRC, despitethe differences sometimes observed between the best-fit andpredicted parameter values (especially with m for loose orfine-grained materials). Such good agreement is obtainedwith the majority of results identified in Tables 1 and 2.

The proposed set of equations has also been successfullyused by the authors on actual field projects (e.g., Aubertin etal. 1999; Nastev and Aubertin 2000) where in situ data and

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Source Material D10 (cm) CU () e ()Sydor (1992) Coarse sand 0.05800 1.3 0.750

Borden sand 0.00910 1.7 0.590Modified Borden sand 0.00800 1.8 0.640

Bruch (1993) Beaver Creek sand consolidated at 5 kPa 0.00930 2.6 0.269Beaver Creek sand consolidated at 10 kPa 0.00930 2.6 0.267

Ricard (1994) Tailings Bevcon 0.00035 11.4 0.6740.00035 11.4 0.7100.00035 11.4 0.795

Tailings Senator 0.00031 11.9 0.7980.00031 11.9 0.929

Tailings Sigma 0.00033 14.6 0.6950.00033 14.6 0.7460.00033 14.6 0.802

Tailings Sigma + 10% bentonite 0.00010 35.0 0.6980.00010 35.0 0.944

Kissiova (1996) Tailings Bevcon 0.00038 11.1 0.5700.00038 11.1 0.6800.00038 11.1 0.920

Tailings Sigma 0.00034 14.7 0.6600.00034 14.7 0.720

Sacrete sand 0.01450 3.5 0.5700.01450 3.5 0.630

MacKay (1997) London Silt 0.00060 5.5 0.634Ottawa sand 0.00937 1.7 0.587

Lim et al. (1998) Beaver Creek sand consolidated at 5 kPa 0.00930 2.6 0.618Rassam and Williams (1999) Tailings at 50 m 0.00600 5.0 0.637

Tailings at 150 m 0.00184 9.5 0.637Authors results Tailings Sigma (fine) 0.00040 9.8 0.720

0.00040 9.8 0.740Tailings Sigma (coarse) 0.00040 8.6 0.640

0.00040 8.6 0.7100.00040 8.6 0.780

Till 0.00035 4.5 0.6200.00035 4.5 0.7000.00035 4.5 0.7200.00035 4.5 0.790

Table 1. Nature, origin, and basic geotechnical properties of the granular materials.



laboratory independent measurements have confirmed thatthe predicted WRCs correspond well to actual values. It hasalso been validated recently during additional investigations(including ongoing graduate student projects on other soils,using column tests); these additional results will be the ob-ject of future publications.

The MK model can equally be used in a more classical(descriptive) manner to derive the WRC from a few relevant

testing results. In this instance, the model can then beapplied to evaluate the expected influence of changing prop-erties on the WRC, such as varying grain size, porosity, orliquid limit.

Despite the encouraging results obtained so far, there arenevertheless some limitations and expected discrepancies be-tween the predicted curves and actual data; some factors arediscussed below.

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Fig. 2. Relationship between the residual suction, r,exp, determined from the experimental data, the void ratio, e, and the equivalentgrain-size diameter, DH, for granular materials identified in Table 1 (DH is defined by eq. [7]).

Fig. 3. Relationship between the residual suction, r,exp, determined from the measured data, and the equivalent capillary rise, hco,G(eq. [8]), for granular materials.



Discussion

Recent updates to the MK modelThe proposed model presented above provides a simple

means to estimate the WRC for granular and fine-grained

soils, as well as for other particulate media. The MK modelequations have been developed as an extension of theKovcs (1981) model, which was selected initially becauseof its physical bases regarding water retention phenomena(Ricard 1994; Aubertin et al. 1995, 1998). The model makes

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Fig. 4. Application of the MK model to a coarse, uniform, and relatively loose sand (data from Sydor (1992)).

Source Material wL (%) Dr () e ()Alimi-Ichola and Bentoumi (1995) Gault clay 40 2.650 0.942Vanapalli et al. (1996) Sandy clay till (consolidated at 200 kPa) 35.5 2.730 0.430

Sandy clay till (consolidated at 25 kPa) 35.5 2.730 0.540Sandy clay till (consolidated at 25 kPa) 35.5 2.730 0.545

Huang et al. (1998) Silty sand PPCT11 22.2 2.680 0.536Silty sand PPCT21 22.2 2.680 0.502Silty sand PPCT16 22.2 2.680 0.463Silty sand PPCT26 22.2 2.680 0.425

OKane et al. (1998) Till cover 40 2.770 0.493Vanapalli et al. (1998) Guadalix Red silty clay 33 2.660 0.480Fredlund (1999) Record 3713 35.5 2.65 0.545

Record 3714 35.5 2.65 0.545Record 3715 35.5 2.65 0.449Record 3716 35.5 2.65 0.474Record 3717 35.5 2.65 0.474Record 3718 35.5 2.65 0.518Record 3720 35.5 2.65 0.546Record 3728 35.5 2.65 0.438Record 55 35.5 2.65 0.475Record 65 35.5 2.73 0.546Record 66 35.5 2.73 0.438Record 70 35.5 2.73 0.444Record 71 35.5 2.73 0.518Record 72 35.5 2.73 0.472Record 73 35.5 2.73 0.545Record 75 35.5 2.73 0.430Record 76 35.5 2.73 0.372

Table 2. Nature, origin, and basic geotechnical properties of the plasticcohesive materials.



a distinction between capillary and adhesive forces responsi-ble for moisture retention; this helps to understand the some-what different approaches taken here to define the materialparameters for granular (Sc dominated; see Fig. 1a) andplasticcohesive (Sa dominated; see Fig. 1b) materials. Re-cent efforts have been aimed at developing the necessary re-lationships so that the evaluation of these two componentsfor predictive purposes could be accomplished using onlybasic geotechnical properties, such as the grain size (through

D10, CU), porosity, n (or void ratio e), solid grain density(s), and liquid limit (wL).

The main modifications included in this generalized ver-sion of the model, compared to the previous (Aubertin et al.1998) version, can be summarized as follows:(1) The equivalent capillary rise, hco, is defined explicitly

using eqs. [8] and [11], and its physical meaning isbetter understood in terms of capillary fringe and resid-ual suction.

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Fig. 5. Application of the MK model to a fine, uniform, and dense sand (data from Bruch (1993)).

Fig. 6. Application of the MK model to tailings Sigma (silty material, well-graded, and loose) (data from authors).



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(2) The specific surface, Sm, is evaluated with more explicitequations for granular (eq. [5]) and fine-grained(eq. [10]) soils. The former equation was already in-cluded in the previous version, while the latter expres-sion stems from the authors recent work on hydraulicconductivity functions (Mbonimpa et al. 2002). Equa-tion [10] extends the use of the MK model to some

fine-grained (plasticcohesive) soils for which the WRCdoes not seem to be significantly influenced by com-pressibility and internal structure.

(3) Instead of a fixed value for the residual suction, r(15 000 cm), the updated model uses a variable r in theC equation (eq. [17]), which depends on material prop-erties.


Fig. 7. Application of the MK model to tailings Sigma mixed with 10% bentonite (data from authors).

Fig. 8. Application of the MK model to Guadalix red silty clay (data from Vanapalli et al. (1998)).



(4) The pore-size distribution parameter m, appearing in thecapillary component, Sc (eq. [15]), has been explicitlyrelated to the uniformity coefficient (for granular mate-rials), hence, reflecting its grain-size dependency.

(5) The adhesion component is expressed with a truncatedterm (Sa*) to eliminate the possibility of exceeding themaximum degree of saturation. This term is particularly

important for plasticcohesive soils (when wL > 30%40%); it may also be important for other relatively fine-grained materials without cohesion such as hard rocktailings. The value of coefficient ac is now defined moreprecisely.

Also, the available results indicate that the predictionsmade with the MK model appear to compare well with those

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Fig. 9. Application of the MK model to a till (data from OKane et al. (1998)).

Fig. 10. Application of the MK model to Indian Head Till (data from Fredlund (1999)).



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obtained with many other predictive models, which ofteninclude more input parameters (and thus, require more infor-mation for their application). For instance, predicted volu-metric water content values with MK, at a given suction,generally approach the measured values within a range of0.020.05, which is comparable to the best available pedo-transfer functions (e.g., Wsten et al. 2001; Schaap et al.2001; Rawls et al. 2001; Zhuang et al. 2001), even whenthese rely on particular measured points on the WRC (e.g.,water content at suctions of 10, 33, or 1500 kPa). It must besaid, however, that these are preliminary and qualitative re-marks, as the authors have mainly concentrated their recentefforts on the practical development of the model equationsfor geotechnical applications. More work on this aspect willthus, have to be performed in the future, to obtain a formalcomparison with other models.

The potential user should also be aware of some remain-ing limitations for the MK model, as discussed below. Someadditional relationships for a and r, using the same mate-rial properties as above, are proposed to complement thetool kit made available to estimate WRC and related keyparameters.

Prediction of the residual suction and air entry valueThe residual suction, r and the air entry value, a (or

AEV), are familiar points on the WRC, which are often usedfor practical applications. The residual suction has alreadybeen defined above with eqs. [18] and [19]. However, the ac-tual value of r (and of the corresponding volumetric watercontent) can often be difficult to determine precisely; evenits physical meaning has been questioned (e.g., Corey 1994).For the applications shown here, more than one approachcan be used to define r. It can be measured, for example, onthe fitted curve adjusted to the available experimental data,with either the MK model or with another model (such asthe van Genuchten (1980) equation); r then corresponds tothe intersection point between two tangents (see Fig. 1a). Itcan also be estimated from the predicted WRC obtained withthe MK model (using eqs. [13][17]). Here, the calculated rvalue (from eq. [18] or [20]) may be somewhat differentthan the one deduced with the tangent method with the com-plete WRC obtained from the MK model, because the for-mer gives a point-wise estimate while the latter represents apredicted curve ranging over several orders of magnitude.Alternatively, r could also be estimated with MK by usingthe suction at which Sc becomes negligible (Sc 0.010.05for instance). Unfortunately, at this point, not enough isknown about the significance and measurement of residual

suction to be more specific about the best way to predict itsvalue (particularly for fine-grained soils).

More than one method also exists for the determination ofthe AEV (e.g., Corey 1994; Aubertin et al. 1998). As theMK model aims at obtaining the entire WRC, it is not par-ticularly well suited for defining precisely specific points onthe curve, such as a (or r). As a general indication, it hasbeen observed that the MK model typically overestimatesthe AEV by about 17%, on average, for granular materials(Aubertin et al. 2003). A more precise predictive estimate ofthe AEV can nevertheless be obtained directly from basicgeotechnical properties (for granular materials), using thefollowing expression (Aubertin et al. 2003):

[21] a,estH

=

beD x

11( )

where b1 and x1 are fitting parameters, and DH is given byeq. [7]. As can be seen in Fig. 11, similar predictions can beobtained with b1 = 0.43, x1 = 0.84, and with b1 = 0.15, x1 =1.0, in eq. [21] (for the range of data considered here). Equa-tion [21] can also be used to estimate the AEVdeduced fromthe MK model WRC (i.e., a,MK) by using b1 = 0.6, x1 = 0.8.Additional methods to define and predict the AEV arepresented in a companion technical report (Aubertin et al.2003), which also shows that the material AEV obtainedfrom experimental measurements on granular materials istypically close to 75% of 95 and 60% of 90, where 95 and90 are, respectively, the suction for a degree of saturation of95% and 90% (defined on the WRC obtained from the MKmodel; see Fig. 1a).

For plasticcohesive materials, the authors have not yetbeen successful at developing an acceptable correlationbetween a and basic geotechnical properties. The approachrelating the AEV to 95 and 90 in the MK model appears tobe applicable in this case, but too few data have been ana-lyzed at this stage to fully support this preliminary assump-tion.

The equivalent capillary rise, hcoThe equivalent capillary rise, hco, constitutes a central pa-

rameter in the MK model. This parameter value is obtainedby eqs. [8] and [9] for granular materials and by eqs. [11]and [12] for plasticcohesive soils. Here, this parameter the-oretically represents the capillary head in a pore with a dia-meter equal to the equivalent hydraulic diameter, deq,defined according to eq. [3]. As shown in Fig. 3, hco is rela-tively close, but typically somewhat lower than r. Hence, it


Fitted WRC Predicted WRCMaterial m ac m acCoarse, uniform and relatively dense sand (data from Sydor (1992); see Fig. 4) 0.827 0.007 0.769 0.01Fine and dense sand (data from Bruch (1993); see Fig. 5) 0.091 0.013 0.388 0.01Tailings Sigma (silty material, coarse and loose) (Authors data, see Fig. 6) 0.161 0.010 0.102 0.01Tailings Sigma mixed with 10% bentonite (Ricard (1994); see Fig. 7) 0.019 0.009 0.029 0.01Guadalix Red silty clay (data from Vanapalli et al. (1998); see Fig. 8) 3.6106 7.6104 3.0105 7.0104Till (data from OKane et al. (1998); see Fig. 9) 8.1106 6.5104 3.0105 7.0104Indian Head Till (Record 728; data from Fredlund (1999); see Fig. 10) 1.0109 7.0104 3.0105 7.0104

Table 3. Parameters m and ac leading to the best-fit, for WRC shown in Figs. 4-10, and the selected values for predictive purposes.



can be seen as representative of the height of the capillaryfringe in homogeneous soils.

A recent investigation furthermore indicates that for gran-ular materials, one can relate hco,G to the AEV using the fol-lowing relationship (Aubertin et al. 2003):[22] h b xco,G a= 2 2

Figure 12 illustrates this equation (data identified in Ta-ble 1) with b2 5.3 and x2 1 when a is determined fromthe experimental curve (a,exp); eq. [22] is also shown on thesame figure with b2 1.7 and x2 1.2, applicable when a isgiven by the predictive MK model (a,MK; see Fig. 1a).

For plasticcohesive soils, the physical meaning of hco,Premains unclear, as indicated by the high values calculatedwith eq. [12] given above. Again, the fact that the residualsuction is not well defined in fine-grained soils does not fa-vor a clear understanding of the conditions represented bysuch high suction values (for r and hco,P). Hence, the equiv-alent capillary rise, hco,P , should only be viewed, at thispoint, as a required input for the MK model application.

Parameters with preset valuesAs shown above, the MK model can be applied in a de-

scriptive manner by adjusting directly parameters ac and mto the experimental results (see Figs. 410). Its main use atthis point, however, is aimed at making predictions based oneasy to obtain basic properties.

In the predictive application of the MK model, values arepreassigned to the adhesion coefficient. A value of ac equalto 0.01 is suggested in the case of granular materials (forwL below 30% 40%), and of 7 104 in the case ofplasticcohesive materials.

A fixed value is also assigned to the pore-size distributionparameter m (3 105) in the case of plasticcohesive mate-rials, while m has been related to the uniformity coefficient(m = 1/CU) for granular materials.

The user could possibly improve predictions by definingmore precisely the underlying properties on which those pa-rameters depend, including the specific surface area, Sm, andthe particle shape factor, a (Aubertin et al. 2003); this aspectneeds to be studied further to evaluate the real influence ofsuch additional information.

When considering the quality of predictions made withthe MK model and the discrepancies sometimes observed,the variety of the WRC measurement techniques and associ-ated uncertainties have to be kept in mind. These may have asignificant effect on the curves and on the estimates obtainedfor parameters r, m, and ac in the MK model, and it can beexpected that the quality of the predictions is influenced bythe data themselves. Nevertheless, the available data appearto be representative of what can be expected under usual lab-oratory conditions, and it is encouraging to see that theagreement between predicted and measured WRC is gener-ally good.

A final note of caution is required as many influence fac-tors have not been taken into account in the present state ofthe MK model. For instance, the influence of compactionconditions that may affect the microstructure (fabric) andpore-size distribution, is neglected; this should be kept inmind by users who may wish to apply the model to com-pacted clay liners, for example. Also neglected by the modelare phenomena such as hysteresis of the WRC (e.g., Mualem1984; Maqsoud et al. 2002), volume change of the sampleduring testing (e.g., Al-Mukhtar et al. 1999; Ng and Pang2000), stressstrain history (Vanapalli et al. 1999), existenceof a heterogeneous or multimodal pore-size distribution(Burger and Shackelford 2001), and presence of very coarseparticles (Yazdani et al. 2000). Further work is under way tointroduce some of these factors into the MK model.

Additional work being conducted also aims at using theMK model to evaluate the relative hydraulic conductivityfunction kr() by integrating the equations presented above.

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Fig. 11. Proposed relationships to estimate the air entry value, a,exp (obtained from the experimental data), for granular materials.



This function, kr(), is combined with a predictive model forthe saturated hydraulic conductivity, ks (Mbonimpa et al.2000, 2002), to obtain the complete unsaturated ku function.

Conclusion

This paper presents a set of equations developed to predictthe water retention curve (WRC). These proposed expres-sions are derived from the Kovcs (1981) model, which hasbeen modified to define explicitly all input parameters, andgeneralized to extend its application to a variety of porousmedia, including granular and fine-grained soils.

In the proposed MK model, the degree of saturation, Sr,includes two components acting jointly: one created by cap-illary forces (Sc) whose contribution is more important atrelatively low suction, and one associated with adhesiveforces (Sa) which mainly contributes at higher suction. Bothcomponents can be evaluated from basic (and generallyavailable) material properties, including the effective diame-ter, D10, uniformity coefficient, CU, liquid limit, wL, voidratio, e, and solid grain density, s. These properties are firstused to define the equivalent capillary rise, hco, which con-stitutes the central parameter in the MK model.

The set of equations used to predict the WRC containsthree parameters required for model application: the residualsuction r, the pore-size distribution parameter, m, and theadhesion coefficient, ac. In the case of granular materials, arelationship has been developed between r and basic geo-technical properties; a simple relationship between r andhco is also presented. The function, r(hco), established forgranular materials is also used for fine-grained soils wherethe residual suction is difficult to define directly. The twoother parameters, m and ac, have been obtained first by acurve-fit procedure so that calculated WRC match experi-mental data as closely as possible with the MK model.

These values are then expressed from basic geotechnicalproperties.

In all the cases considered here, the MK model allowed agood representation of the experimental WRC, for granularand fine-grained materials. It can, therefore, be presented asa useful tool for predicting the WRC during the preliminaryphases of a project, and for estimating how such WRC mayvary with changing material properties.

The model is not meant to replace the required tests to ob-tain representative data for particular conditions, but ratherprovides potential users with a simple and practical meansof foreseeing the possible characteristics of the WRC. Thesame can be said about the complementary relationshipsproposed to estimate specific points on the curve, includingthe air entry value, from material basic properties. Whenexperimental data become available, these can be used toimprove upon the predictive capabilities of the proposedequations, by making the necessary adjustments for specificmaterials; the MK model can then be used in a more classi-cal descriptive manner. The discussion presented at the endof the paper finally draws attention to the inherent limita-tions of the proposed model.

Acknowledgments

The postdoctoral grant provided to M. Mbonimpa by theInstitut de Recherche en Sant et Scurit du Travail duQubec (IRSST) is thankfully acknowledged. Special ac-knowledgements are also given to Antonio Gatien and to thegraduate students who performed the tests over the years,and to Dr. John Molson for helping improve the quality ofthe manuscript. The authors also received financial supportfrom the Natural Sciences and Engineering Research Coun-cil of Canada (NSERC) and from a number of industrial par-ticipants of the PolytechniqueUQAT Chair in Environmentand Mine Wastes Management.

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Fig. 12. Relationships between the equivalent capillary rise, hco,G, and the air entry value for granular materials; a,exp and a,MK areobtained on the experimental and on the MK-model-predicted WRC, respectively.



References

Alimi-Ichola, I., and Bentoumi, O. 1995. Hydraulic conductivityand diffusivity in vertical and horizontal inflow. In Unsaturatedsoils / Sols non saturs. Vol. 1. Edited by E.E. Alonso and P.Delage. A.A. Balkema, Rotterdam. pp. 335341.

Al-Mukhtar, M., Qi, Y., Alcover, J.-F., and Bergaya, F. 1999.Oedometric and water retention behavior of highly compactedunsaturated smectites. Canadian Geotechnical Journal, 36(4):675684.

Alonso, E., Gens, A., and Josa, A. 1990. A constitutive model forpartially saturated soils. Gotechnique, 40(3): 405430.

Arya, L.M., and Paris, J.F. 1981. A physico-empirical model topredict the soil moisture characteristic from particle size distri-bution and bulk density data. Soil Science Society of AmericaJournal, 45: 10231030.

Arya, L.M., Leij, F.J., van Genuchten, M.T., and Shouse, P.J. 1999.Scaling parameter to predict the soil water characteristic fromparticle-size distribution data. Soil Science Society of AmericaJournal, 63: 510519.

Aubertin, M., Ricard, J.F., and Chapuis, R.P. 1995. A study of cap-illary properties of mine tailings: Measurements and modeling.In Proceedings of the 48th Canadian Geotechnical Conference,Vancouver, B.C., 2527 September. pp. 1724.

Aubertin, M., Ricard, J.-F., and Chapuis, R.P. 1998. A predictivemodel for the water retention curve: application to tailings fromhard-rock mines. Canadian Geotechnical Journal, 35: 5569.[Erratum, 36: 401 (1999)]

Aubertin, M., Bussire, B. Monzon, M., Joanes, A.-M., Gagnon,D., Barbera, J.-M., Aachib, M.,Bdard, C. Chapuis, R.P., andBernier, L. 1999. tude sur les barrires sches construites partir des rsidus miniers. Phase II : Essais en place. Rapportde recherche Projet CDT P1899. Rapport NEDEM/MEND2.22.2c, 331 pp.

Aubertin, M., Mbonimpa, M., Bussire, M., and Chapuis, R.P.2003. Development of a model to predict the water retentioncurve using basic geotechnical properties. Technical ReportEPM-RT-2003-01. cole Polytechnique de Montral, Montral.51 pp.

Barbour, S.L. 1998. Nineteenth Canadian Geotechnical Collo-qium : The soilwater characteristic curve: a historical perspec-tive. Canadian Geotechnical Journal, 35(5): 873894.

Bear, J. 1972. Dynamics of fluids in porous media. Dover Publica-tions Inc., New York.

Bouma, J. 1989. Using soil survey data for quantitative land evalu-ation. Advances in Soil Science, 9: 177213.

Bowles, J.E. 1984. Physical and geotechnical properties of soils.2nd ed. McGraw-Hill Book Company, New York.

Brooks, R.H., and Corey, A.T. 1964. Hydraulic properties ofporous media. Colorado State University, Fort Collins, Colo.Hydrology Paper No. 3.

Bruch, P.G. 1993. A laboratory study of evaporative fluxes inhomogeneous and layered soils. M.Sc. thesis, University of Sas-katchewan, Saskatoon, Saskatchewan.

Bumb, A.C., Murphy, C.L., and Everett, L.G. 1992. A comparisonof three functional forms to representing soil moisture character-istics. Ground Water, 3: 177185.

Burger, C.A., and Shackelford, C.D. 2001. Evaluating dual poros-ity of pelletized diatomaceous earth using bimodal soil watercharacteristic curve functions. Canadian Geotechnical Journal,38(1): 5366.

Bussire, B. Aubertin, M., and Chapuis, R.P. 2003. The behaviorof inclined covers used as oxygen barrier. Canadian Geo-technical Journal, 40: 512535.

Carter, M.R. 1993. Soil sampling and methods of analysis. Lewis,Boca Raton, Fla.

Celia, M.A., Reeves, P.C., and Ferrand, L.A. 1995. Pore scalemodels for multiphase flow in porous media. U.S. NationalReport to the International Union of Geodesy and Geophysics19911994. Reviews of Geophysics, 33: 10491057.

Chapuis, R.P., and Aubertin, M. 2001. A simplified method to esti-mate saturated and unsaturated seepage through dikes understeady state conditions. Canadian Geotechnical Journal, 38(6):13211328.

Chapuis, R.P., and Lgar, P.P. 1992. A simple method for deter-mining the surface area of fine aggregates and fillers in bitumi-nous mixtures. In Effects of aggregates and mineral filler onasphalt mixture performance. American Society for Testing andMaterials (ASTM), Special Technical Publication (STP), 1147:177186.

Chin, D.A. 2000. Water-resources engineering. Prentice Hill, Up-per Saddle River, N.J.

Corey, A.T. 1994. Mechanics of immiscible fluids in porous media.Water Resources Publications, Highlands Ranch, Colo.

Cosby, B.J., Hornberger,G.M., Clapp, R.B., and Ginn, T.R. 1984.A statistical exploration of the relationships of soil moisturecharacteristics to the physical properties of soils. Water Re-sources Research, 20: 682690.

Delleur, J.W. 1999. The handbook of groundwater engineering.CRC Press, New York.

Elsenbeer, H. 2001. Editorial: Pedotransfer fuctions in hydrology.Journal of Hydrology (Special Issue), 251(34): 121122.

Fredlund, M.D. 1999. Soilvision 2.0. Version 2.0 [computer pro-gram]. A knowledge-based database system for unsaturated-saturated soil properties. Soilvision Systems Ltd., Saskatoon,Sask.

Fredlund, D.G. 2000. The 1999 R.M. Hardy Lecture: The imple-mentation of unsaturated soil mechanics into geotechnical engi-neering. Canadian Geotechnical Journal, 37: 963986.

Fredlund, D.G., and Rahardjo, H. 1993. Soil mechanics for unsatu-rated soils. John Wiley and Sons, Inc., New York.

Fredlund, D.G., and Xing, A. 1994. Equations for the soilwatercharacteristic curve. Canadian Geotechnical Journal, 31(4):521532.

Fredlund, D.G., Xing, A., and Huang, S. 1994. Predicting thepermeability function for unsaturated soils using the soilwatercharacteristic curve. Canadian Geotechnical Journal, 31: 533546.

Gupta, S.C., and Larson, W.E. 1979. Estimating soil-water reten-tion characteristics from particle size distribution, organic matterpercent, and bulk density. Water Resources Research, 15(6):16331635.

Haverkamp, R., and Parlange, J.-Y. 1986. Predicting the waterretention curve from particle size distribution: 1. Sandy soilswithout organic matter. Soil Science, 142: 325339.

Haverkamp, R., Bouraoui, F., Zammit, C., and Angulo-Jaramillo,R. 1999. Soil properties and moisture movement in the unsatu-rated zone. In The handbook of groundwater engineering. Editedby J.W. Delleur, CRC Press, New York. pp. 5.15.47.

Hillel, D. 1998. Environmental soil physics. Academic Press, NewYork.

Huang, S., Barbour, S.L., and Fredlund, D.G. 1998. Developmentand verification of a coefficient of permeability function fora deformable unsaturated soil. Canadian Geotechnical Journal,35(3): 411425.

Igwe, G.J.I. 1991. Powder technology and multiphase systems: Gaspermeametry and surface area measurement. Ellis Horwood,New York.

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Khire, M., Meerdink, J., Benson, C., and Bosscher, P. 1995. Unsat-urated hydraulic conductivity and water balance predictions forearthen landfill final covers. In Soil suction applications ingeotechnical engineering practice. Edited by W. Wray and S.Houston. ASCE, New York. pp. 3857.

Kissiova, M. 1996. tude des modles de prdiction de la conduc-tivit hydraulique des matriaux meubles non-saturs. M. Ing.,cole Polytechnique de Montral, Montral, Qc.

Klute, A. 1986. Water retention : laboratory methods. In Methodsof soil analysis, part I. Physical and mineralogical methods. 2nded. Edited by A. Klute. Agronomy Monograph No. 9, AmericanSociety of Agronomy. Soil Science Society of America, Madi-son Wis. pp. 635662.

Konrad, J.-M., and Roy, M. 2000. Flexible pavements in cold re-gions: a geotechnical perspective. Canadian Geotechnical Jour-nal, 37(3): 689699.

Kovcs, G. 1981. Seepage hydraulics. Elsevier Science Publishers,Amsterdam.

Lambe, T.W., and Whitman, R.V. 1979. Soil mechanics, SI version.John Wiley and Sons, New York.

Leij, F.J., Russel, W.B., and Lesch, S.M. 1997. Closed-formexpressions for water retention and conductivity data. GroundWater, 35(5): 848858.

Leong, E.C., and Rahardjo, H. 1997a. Permeability functions forunsaturated soils. Journal of Geotechnical and Geoenviron-mental Engineering, ASCE 123(12): 11181126.

Leong, E.C., and Rahardjo, H. 1997b. Review of soil-water charac-teristic curve functions, Journal of Geotechnical and Geoenvi-ronmental Engineering, ASCE 123(12): 11061117.

Lim, P.C., Barbour, S.L., and Fredlund, D.G. 1998. The influenceof degree of saturation on the coefficient of aqueous diffusion.Canadian Geotechnical Journal, 35(5): 811827.

Looney, B.B., and Falta, R.W. 2000. Vadose zone science and tech-nology solutions. Vols. I and II. Battelle Press, Columbus, Ohio.

Lowell, S., and Shields, J.E. 1984. Powder surface area and poros-ity. 2nd ed. Chapman and Hall, London.

MacKay, P.C. 1997. Evaluation of oxygen diffusion in unsaturatedsoils. M.Sc. Eng., University of Western Ontario, London, Ont.

Maqsoud, A., Bussire, B., and Aubertin, M. 2002. Lhystrsisdes sols non saturs utilizs dans les couvertures avec effets debarrire capillaire. In Proceedings of the 55th Canadian Geo-technical Conference and 3rd Joint IAH-CNC and CGS Ground-water Specialty Conference, Niagara Falls, Ontario, Canada.Edited by D. Stolle, A. Piggot, and J.J. Crowd. pp. 181188.

Marshall, T.J., Holmes, J.W., and Rose, C.W. 1996. Soil physics.3rd ed. Cambridge University Press, Cambridge, U.K.

Mbonimpa, M., Aubertin, M., Chapuis, R.P., and Bussire, B.2000. Dveloppement de fonctions hydriques utilizant les pro-prits gotechniques de base. In Proceedings of the 1st JointIAH-CNC and CGS Groundwater Specialty Conference and the53rd Canadian Geotechnical Conference, Montreal, Quebec, Ca-nada, 1518 October. pp. 343350.

Mbonimpa, M., Aubertin, M., Chapuis, R.P., and Bussire, B. 2002Practical pedotransfer functions for estimating the saturated hy-draulic conductivity. Geotechnical and Geological Engineering,20: 235259.

Meyer, P.D., and Gee, G.W. 1999. Advective-diffusion contaminantmigration in unsaturated sand and gravel. Journal of Geotechni-cal and Geoenvironmental Engineering, ASCE 122(12): 965975.

Mualem, Y. 1976. A new model for predicting the hydraulic con-ductivity of unsaturated porous media. Water ResourcesResearch, 12: 513522.

Mualem, Y. 1984. A modified dependent-domain theory of hyster-esis. Soil Science, 137: 283291.

Mualem, Y. 1986. Hydraulic conductivity of unsaturated soil: Pre-diction and formulas. In Methods of soil analysis, Part I: Physi-cal and mineralogical methods. Edited by A Klute. AmericanSociety of Agronomy (ASA) and Soil Science Society of Amer-ica (SSSA), Madison, Wis. Agronomy Monograph No. 9.pp. 799823.

Nastev, M., and Aubertin M. 2000. Hydrogeological modelling forthe reclamation work at the Lorraine Mine Site, Qubec. In Pro-ceedings of the 1st Joint IAH-CNC and CGS Groundwater Spe-cialty Conference, Montreal, Quebec, Canada, 1518 October.pp. 311318.

Ng, C.W.W., and Pang, Y.W. 2000. Experimental investigationsof the soilwater characteristics of a volcanic soil. CanadianGeotechnical Journal, 37: 12521264.

Nitao, J.J., and Bear, J. 1996. Potentials and their role in transportin porous media. Water Resources Research, 32(2): 225250.

OKane, M., Wilson, G.W., and Barbour S.L. 1998. Instrumenta-tion and monitoring of an engineered soil cover system for minewaste rock. Canadian Geotechnical Journal, 35(5): 828846.

Rampino, C., Mancuso, C., and Vinale, F. 2000. Experimental be-haviour and modeling of an unsaturated compacted soil. Cana-dian Geotechnical Journal, 37(4): 748763.

Rassam, D.W., and Williams, D.J. 1999. A relationship describingthe shear strength of unsaturated soils. Canadian GeotechnicalJournal, 36(2): 363368

Rawls, W.J., and Brackensiek, D.L. 1982. Estimating soil water re-tention from soil properties. Journal of Irrigation and DrainageDivision, ASCE 108: 166171.

Rawls, W.J., Gish, T.J., and Brakensiek, D.L. 1991. Estimating soilwater retention from soil physical properties and characteristics.Advances in Soil Science, 16: 213234.

Rawls, W.J., Pachepsky Y.A., and Shen, M.H. 2001. Testing soilwater retention estimation with the MUUF pedotransfer modelusing data from the southern United States. Journal of Hydrol-ogy, 251: 177185.

Ricard, J.F. 1994. tude en laboratoire de la relation capillaire etde la conductivit hydraulique des rsidus miniers. Mmoire deMatrise, cole Polytechnique de Montral, Montral, Qc.

Richards, L.A. 1931. Capillary conduction of liquids in porous me-diums. Physics, 1: 318333.

Schaap, M.G., and Leij, J.F. 1998. Database related accuracy anduncertainty of pedotransfer functions. Soil Science, 163: 765779.

Schaap, M.G., Leij, F.J., van Genuchten, M.Th. 1998. Neural net-work analysis for hierarchical prediction of soil water retentionand saturated hydraulic conductivity. Soil Science Society ofAmerica Journal, 64: 843855.

Schaap, M.G., Leij, F.J., van Genuchten, M.Th. 2001. ROSETTA:a computer program for estimating soil hydraulic parameterswith hierarchical pedotransfer functions. Journal of Hydrology,251: 163176.

Scheidegger, A.E. 1974. The physics of flow through porous me-dia. 3rd ed. University of Toronto Press, Toronto.

Smith, G.N. 1990. Elements of soil mechanics. 6th ed. BSP Profes-sional Books, Oxford.

Sydor, R.C. 1992. Engineered mine tailings cover: verification ofdrainage behaviour and investigations of design. M.Sc. thesis,University of Waterloo, Waterloo, Ont.

Thu, T.M., and Fredlund, D.G. 2000. Modelling subsidence in Ha-noi City area, Vietnam. Canadian Geotechnical Journal, 37(7):621637.

Tietje, O., and Tapkenhinrichs, M. 1993. Evaluation of pedo-



transfer functions. Soil Science Society of America Journal, 57:10881095.

Tyler, S.W., and Wheatcraft, S.W. 1989. Application of fractalmathematics to soil water retention estimation. Soil Science So-ciety of America Journal, 53(4): 987996.

Tyler, S.W., and Wheatcraft, S.W. 1990. Fractal processes in soilwater retention. Water Resources Research, 26(5): 10471054.

van Genuchten, M.Th. 1980. A closed-form equation for predictingthe hydraulic conductivity of unsaturated soils. Soil Science So-ciety of America Journal, 44: 892898.

van Genuchten, M.TH., Leij, F.J., and Yates, S.R. 1991. The RETCcode for quantifying the hydraulic functions of unsaturated soils.EPA/600/2-91/065.

Vanapalli, S.K., Fredlund, D.G., Pufahl, D.E., and Clifton, A.W.1996. Model for the prediction of shear strength with respect tosoil suction. Canadian Geotechnical Journal, 33: 379392.

Vanapalli, S.K., Sillers, W.S., and Fredlund, M.D. 1998. The mean-ing and relevance of residual state to unsaturated soils. InProceedings of the 51st Canadian Geotechnical Conference,Edmonton, Alberta, Canada, 47 October. Preprint Vol. 1.pp. 101108.

Vanapalli, S.K., Fredlund, D.G., and Pufahl, D.E. 1999. The influ-ence of soil structure and stress history on the soil water charac-teristics of a compacted till. Gotechnique, 49: 143159.

Vereecken, H., Diels, J., van Orshoven, J., Feyen, J., and Bouma, J.1992. Functional evaluation of pedotransfer functions for the es-timation of soil hydraulic properties. Soil Science Society ofAmerica Journal, 56: 13711378.

Wsten, J.H.M., Pachepsky, Ya.A., and Rawls, W.J. 2001. Pedo-transfer functions: bridging the gap between available basic soildata and missing soil hydraulic characteristics. Journal of Hy-drology, 251: 123150.

Yazdani, J., Barbour, L., and Wilson, W. 2000. Soil water charac-teristic curve for mine waste rock containing coarse material. InProceedings of the 6th Environmental Speciality CSCE and 2ndSpring Conference of the Geoenvironmental Divison of theCGS, London, Ontario, 710 June. pp. 198202.

Yong, R.N. 2001. Geoenvironmental engineering Contaminatedsoils, pollutant fate, and mitigation. CRC Press, Boca Raton,Fla.

Zhuang, J., Jin, Y., and Miyazaki, T. 2001. Estimating water reten-tion characteristic from soil particle-size distribution using anon-similar media concept. Journal of Hydrology, 251: 308321.

List of symbols

ac adhesion coefficient ()Av surface of the voids (cm2)AG surface area of the solid grains [L2]

b coarse-grained material parameter to calculate hco,G (cm2)b1 coarse-grained material parameter to estimate a (cmx +11 )b2 coarse-grained material parameter to correlate hco,G and

a ()C correction factor ()CU uniformity coefficient () (CU = D60/D10)

d diameter of a tube (cm)D10 diameter corresponding to 10% passing on the cumula-

tive grain-size distribution curve (cm)D60 diameter corresponding to 60% passing on the cumula-

tive grain-size distribution curve (cm)

deq equivalent pore-size diameter (cm)DH equivalent grain-size diameter for a heterogeneous mix-

ture (cm)Dr relative density of the solid particles ()

e void ratio ()hc capillary rise in a tube (cm)

hco equivalent capillary rise in a porous material (cm)hco,G equivalent capillary rise in a granular material (cm)hco,P equivalent capillary rise in a plasticcohesive material

(cm)kr relative hydraulic conductivity ()ks saturated hydraulic conductivity [LT 1]ku unsaturated permeability function [LT 1]m pore-size distribution parameter in the MK model ()n porosity ()

Sa adhesion component of the degree of saturation ()Sa* truncated adhesion component of the degree of satura-

tion ()Sc capillary component of the degree of saturation ()

Sm specific surface area per unit mass of solid (m2/g)Sr degree of saturation ()ua air pressure (cm)uw water pressure (cm)Vv volume of the voids (cm3)w gravimetric water content ()

wL liquid limit (%)x1 coarse-grained material parameter to estimate a ()x2 coarse-grained material parameter to correlate hco,G and

a ()y open variable shape factor ()

w contact angle ()w unit weight of water (kN/m3) volumetric water content () material parameter used to estimate Sm (m2/g) plasticcohesive material parameter required to calcu-

late hco,P (cm) osmotic suctions solid-grain density (kg/m3)

w surface tension of water (N/m) material parameter used to estimate Sm () suction (cm)

0 suction at complete dryness (Sr = 0) (cm)90 suction corresponding to a degree of saturation of 90%

(cm)95 suction corresponding to a degree of saturation of 95%

(cm)a air entry value or AEV (cm)

a,exp air entry value determined from the experimental data(cm)

a,MK air entry value determined from the WRC predictedwith the MK model (cm)

m matric suctionn normalization parameterr residual suction (corresponding to the residual water

content) (cm)r,exp residual suction determined from the experimental data

(cm)

2003 NRC Canada




Appendix A: Sample calculations

2003 NRC Canada


Suction(cm)

Sc (eq. [15])()

C (eq. [17])()

Sa (eq. [16])()

Sa* (eq. [14])()

Sr (eq. [13])()

Predicted (eq. [13]) ()

Measured ()

1 1.000 1.000 1.264 1.000 1.000 0.419 0.41910 1.000 1.000 0.861 0.861 1.000 0.419 0.401

141 0.999 0.996 0.552 0.552 1.000 0.418 0.392281 0.793 0.992 0.490 0.490 0.895 0.374 0.381422 0.455 0.989 0.456 0.456 0.704 0.295 0.306563 0.254 0.985 0.433 0.433 0.577 0.242 0.266703 0.148 0.982 0.416 0.416 0.502 0.210 0.24844 0.090 0.978 0.402 0.402 0.456 0.191 0.213984 0.058 0.975 0.391 0.391 0.426 0.178 0.195

1125 0.038 0.972 0.381 0.381 0.405 0.169 0.1831266 0.026 0.968 0.372 0.372 0.389 0.163 0.1711406 0.019 0.965 0.365 0.365 0.376 0.158 0.1571688 0.010 0.959 0.351 0.351 0.358 0.150 0.1462110 0.004 0.951 0.336 0.336 0.339 0.142 0.138

1.0104 0.000 0.850 0.231 0.231 0.231 0.097 1.0107 0.000 0.000 0.000 0.000 0.000 0.000

Note: D10 is 0.0004 cm; e is 0.72 (); CU is 9.8 (); b (eq. [9]) is 0.347 cm2; hco,G (eq. [8]) is 1206 cm; r (eq. [18]) is 4603 cm; 0(fixed) is 1 107 cm; n (fixed) is 1 cm; ac (fixed) is 0.01 (); m = 1/CU = 0.102 ().

Table A1. Typical calculation results to predict the WRC of granular materials: Case of Sigma tailings (see Fig. 6).

Suction (cm)Sc (eq. [15])()

C (eq. [17])()

Sa (eq. [16])()

Sa* (eq.[14])()

Sr (eq. [13])()

Predicted (eq. [13]) ()

Measured ()

1 1.000 1.000 2.790 1.000 1.000 0.305 0.3059.8 1.000 1.000 1.908 1.000 1.000 0.305 0.305

196 1.000 1.000 1.158 1.000 1.000 0.305 0.305392 0.993 1.000 1.031 1.000 1.000 0.305 0.295784 0.716 1.000 0.919 0.919 0.977 0.298 0.268

1 176 0.428 1.000 0.859 0.859 0.919 0.280 0.2671 568 0.270 0.999 0.818 0.818 0.867 0.264 0.2631 960 0.182 0.999 0.788 0.788 0.827 0.252 0.2542 940 0.085 0.999 0.737 0.737 0.759 0.231 0.2473 920 0.049 0.999 0.702 0.702 0.716 0.218 0.2324 900 0.032 0.998 0.676 0.676 0.686 0.209 0.2275 880 0.022 0.998 0.656 0.656 0.663 0.202 0.2207 840 0.012 0.997 0.624 0.624 0.629 0.192 0.215

43 120 0.000 0.986 0.465 0.465 0.465 0.142 0.175372 400 0.000 0.892 0.293 0.293 0.293 0.089 0.075838 880 0.000 0.783 0.225 0.225 0.225 0.069 0.055

1 486 660 0.000 0.663 0.173 0.173 0.173 0.053 0.0402 916 480 0.000 0.471 0.110 0.110 0.110 0.033 0.024

1 0000 000 0.000 0.000 0.000 0.000 0.000 0.000 0.000Note: wL is 35.5%; e is 0.438; s is 2659 kg/m3; (eq. [12b] is 397.5 cm; hco,P (eq. [11]) is 160 581 cm; r (eq. [20]) is 1 518 208 cm;

0 is 1 107 cm; n (fixed) is 1 cm; ac (fixed) is 7 104 (); m (fixed) is 3 105 ().

Table A2. Typical calculation results to predict the WRC of plasticcohesive soils: Case of Indian Head Till (see Fig. 10).

