Predicting the coefficient of permeability of soils using the Kozeny-Carman equation

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Authors: Robert P. Chapuis and Michel Aubertin

Date: 2003

Type: Rapport / Report

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Citation:

Chapuis, Robert P. et Aubertin, Michel (2003). Predicting the Coefficient of Permeability of Soils Using the Kozenny-Carman Equation. Rapport technique. EPM-RT-2003-03.

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PREDICTING THE COEFFICIENT OF PERMEABILITY OF SOILS USING THE KOZENY-CARMAN EQUATION

Robert P. Chapuis and Michel Aubertin

Département des génies civil, géologique et des mines École Polytechnique de Montréal

EPM-RT-2003-03

Pred

ict

ing

the

coeff

ic

ient

of

permeab

i

l

ity

of

so

i

ls

us

ing

the

Kozeny-Carman

equat

ion

Robert

P

.

Chapu

is

and

M

iche

l

Aubert

in

Department CGM, École Polytechnique de Montréal P.O. Box 6079, Sta. CV, Montreal, QC, Canada, H3C 3A7

Tél.:(514) 340 4711, ext. 4427 – Fax:(514) 340 4477 E-mail: robert.chapuis@polymtl.ca

2003

Robert P. Chapuis, Michel Aubertin

Tous droits réservés

Dépôt légal :

Bibliothèque nationale du Québec, 2003

Bibliothèque nationale du Canada, 2003

EPM-RT-2003-03

Predicting the Coefficient of Permeability of Soils Using the Kozeny-Carman Equation

par : Robert P. Chapuis et Michel Aubertin

Département des génies civil, géologique et des mines.

École Polytechnique de Montréal

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Abstract

Thesaturated hydraulicconductivity ofasoilcan be predicted usingempiricalrelationships, capillary models,statistical modelsand hydraulicradiustheories. A well-knownrelationship between permeabilityand properties of pores was proposed by Kozenyandlater modified by Carman. The resulting equation is largely known under the name of Kozeny-Carman, although these authors never published together. In the geotechnical literature, there is a large consensus thatthe Kozeny-Carman (KC) equation appliesto sands but notto clays. Such opinion, however, is supported only by partial demonstration. This report evaluatesthe background andthe validity of the KC equation with laboratory permeability tests. Considered test results were taken from publicationsthat provided allinformation neededto make a prediction: void ratio, and eitherthe measured specific surface for cohesive soils, orthe gradation curve for non-cohesive soils. This report shows howto estimatethe specific surface of a non-cohesive soil fromits gradation curve. The results presented here show that, as a general rule, the KC equation predicts fairly well the saturated hydraulic conductivity of mostsoils. Many ofthe observed discrepancies can be relatedto either practical reasons (e.g.inaccurate specific surface value, steady flow not reached, unsaturated specimens, etc.) ortheoretical reasons (some wateris motionless, andthe predictive equationisisotropic whereas hydraulicconductivityisananisotropic property). Thesesissues are discussedinrelationtothe predictive capabilities ofthe KC equation.

Key words: permeability, prediction, gradation curve, specific surface Résumé

Laconductivité hydrauliquesaturée d'unsol peutêtre prédite par desrelationsempiriques, des modèles capillaires, des modèles statistiques et des théories de rayon hydraulique. Une relation bien connue entre perméabilité et propriétés des pores fut proposée par Kozeny et modifiée par Carman. L'équationrésultanteestlargementconnuesousle nom Kozeny-Carman(KC), bien que ces auteurs n'aientjamais publié ensemble. Danslalittérature géotechnique,il existe unlarge consensus àl'effet quel'équation de Kozeny-Carman s'applique aux sables mais pas aux argiles. Cependant, cette opinion n'est appuyée que par une démonstration partielle. Cet article examine les fondements etla validité del'équation KC àl'aide d'essais de perméabilité enlaboratoire. Les résultats d'essais proviennent de diverses publications qui ont fourni toute l'information requise pourfaire une prédiction:indice des videsetsoitlasurfacespécifique mesurée pourlessols cohérents, soit la courbe granulométrique pour les sols pulvérulents. L'article montre comment calculerlasurfacespécifique d'unsol pulvérulentà partir desacourbe granulométrique. Les résultats présentésiciindiquent qu'en général,l'équation de Kozeny-Carman prédit assez bienla conductivité hydrauliquesaturée dela plupart dessols. Plusieurs des divergencesconstatées peuventêtrereliéessoità desraisons pratiques(e.g. valeurimprécise delasurfacespécifique, régime permanent pasétabli,échantillons nonsaturés,etc.)soità desraisonsthéoriques(une partie del'eauestimmobile,etl'équation de prédictionestisotropealors quelaconductivité hydraulique est une propriété anisotrope). Ces aspects sont discutés dansl'article enrelation avec la capacité de prédiction del'équation de Kozeny-Carman.

Introduct

ion

Since Seelheim (1880) wrote that the permeability should be related to the squared value of somecharacteristic pore diameter, manyequations have been proposedto predictthe saturated hydraulicconductivity, k, of porous materials. Accordingtostate-of-the-art publications(e.g. Scheidegger 1953, 1954, 1974; Bear 1972; Houpeurt 1974),the k-valueforasingle fluid flow can be predicted using empirical relationships, capillary models, statistical models and hydraulic radiustheories. The best modelsinclude atleastthree parametersto account forthe relationships betweenthe flowrate andthe porous space, for examplethe size ofthe pores,theirtortuosity and their connectivity.

A frequently quoted relation was proposed by Kozeny (1927) and later modified by Carman (1937, 1956). Theresultingequationislargely knownasthe Kozeny-Carman(KC)equation, althoughthetwoauthors have never publishedtogether. Thisequation was developedafter considering a porous material as an assembly of capillarytubes for whichthe equation of Navier -Stokescan be used.It yieldedthe hydraulicconductivity k asafunction ofthe porosity n (or void ratio e),the specific surface S (m2/kg of solids), and afactorC totakeinto accountthe shape andtortuosity of channels. Sinceits first appearance (Carman 1937)tothe present,this equation hastaken severalforms,includingthefollowing onethatis commonly used:

( )

where k is the hydraulic conductivity or coefficient of permeability, C a constant, g the gravitational constant, µw the dynamic viscosity of water,ρw the density of water,ρsthe density

of solids, D_Rthe specific weight (D_R=ρs/ρ_w) of solids, S the specific surface and,e the void ratio.

This equation predicts that, for a given soil, there should be a linear relationship between k and e3/(1+e).It can also be usedto predicttheintrinsic permeability,K (unit m2), knowingthat:

w w w w w w K g K K k= γ/µ = ρ /µ = ρ /ν [2]

where γwisthe unit weight of water(γw = gρw)and νwthe kinematic viscosity of water(µw =

gνw).

Accordingtoclassicalsoil mechanicstextbooks(e.g. Taylor 1948, Lambeand Whitman 1969), the Kozeny-Carman equation is approximately valid for sands, andis not valid for clays. Thesame opinion appears alsoin classical hydrogeologytextbooks(e.g. Freezeand Cherry 1979; Domenico and Schwartz 1990).

In practice, eq.[1)is notfrequently used. Thereasonseemstolieinthe difficultyto determinethe soil specific surfacethat can be either measured or estimated. Several methods are availablefor measuringthespecificsurface(e.g. Dallavale 1948, Dullien 1979, Lowelland Shields 1991) butthey are not commonly usedin soil mechanics and hydrogeology. In addition, such methodsseemaccurate onlyfor granularsoils withfew non-plasticfine particles. These practical difficulties may explain whythe KC predictive equationis not commonly used.

Chapuisand Légaré(1992) proposeda methodforestimatingthespecificsurface ofa non -cohesivesoilfromitscomplete grainsizecurve. This methodis used hereintoevaluatethe capability ofthe KCequationto predictthesoil k-value. Manylaboratorytestresults were gathered forthe evaluation. They weretaken from publicationsthat provided alltheinformation needed for this evaluation: void ratio, and either a measured specific surface for a cohesive soil orthecomplete gradationcurvefora non-cohesivesoil. Thereport presentssuccessively(1)

some background onthe Kozeny-Carman equation, (2)the resultsthat various authors presented to validate or invalidate this equation, and an analysis oftheir argumentation, (3)thetest results that are usedinthe present evaluation,(4)the methodtoestimatethespecificsurfacefromthe gradation curve, and(5)the comparison of measured and predicted k-values.

Background

Kozeny (1927) developed atheory for a series of capillarytubes of equallength and obtained thefollowing equation(quotation withthe original notations):

) / ( ) / ( 2 1 3 _σ µ γ ν= I cp _[3]

where v wasthe Darcy velocity, γ the unit weight ofthefluid, Ithe hydraulic gradient, µ its viscosity, c a geometric constant, p the porosity ofthe material and σ1itsspecificsurface

expressed in squared meters per unit bulk volume of the porous material. Kozeny (1927) gave the values offactor c for differenttube cross-sections: 0.50(circle), 0.562(square), 0.597 (equilateraltriangle) and 0.66(thin slot).

Carman(1937, 1938aand b, 1939) verifiedthe Kozenyequation(eq.[3]),introducedthe notion of hydraulic radius and expressed the specific surface per unit mass of solid (it does not vary withthe porosity asin eq. [3]. Furthermore, Carman (1939) consideredthat water does not movein straight channels but aroundirregularly shaped solid particles. Hetriedtotakethisinto account byintroducing angular deviations of 45° fromthe mean straighttrajectory. He proposed anequationsimilartoeq.[1], with C = 0.2and n3/(1-n)2where n isthe usual notationfor porosity. Notethat presently,thereisa preferenceto use e3/(1+e) = n3/(1-n)2asineq.[1]. According to Carman (1939) a factor C = 0.20 gavethe best fit with experimental results. This value of 0.20includedsimultaneouslythe notions of equivalent capillary channel cross-section andtortuosity. Later,these notions were consideredindependently by other authors (e.g. Sullivan and Hertel 1942; Rose and Bruce 1949; Wyllie and Rose 1950, etc.).

The Kozeny-Carmanequation wasalso usedasastarting pointto develop diphasicflow equations (Rose and Bruce 1949; Thornton 1949; Rapoport and Lea 1951; Wyllie and Spangler 1952; Wyllie and Gardner 1958a and 1958b, etc.). Some authors, following Sullivan and Hertel (1942), have replacedthe specific surfaceterm, S2, by atermd wherem2 d_misthe pore diameter of the equivalent capillary. Sometextbooks (e.g. Freeze and Cherry 1979; Domenico and Schwartz 1997) presentthe KC equation withd m2 instead of S2in eq.[1], sometimes calling d_m a representative grain size, without anyindication of howto calculatethis equivalent diameter.

To concludethis brief history ofthe development ofthe Kozeny-Carman equation,itis worth mentioningthat Kozeny(1927) proposed hisequationin German without knowledge ofthe previous and somewhat similar works by Blake (1922), whereas a few yearslater, Fair and Hatch (1933) proposed in English a similar equation. It is also interesting to note that if many recent textbooksreferto Carmen, Carman himself cited D'Arcyinstead of Darcy(1856).

Opinion No.1:the KC equationis validfor non-plastic soils

This opinionis widespread. Thetest results of Carman (1937, 1938a and b, 1939), and others he reported, clearly establishedthe validity ofthe equation for materials havingthe size of gravel and sand,including variousindustrial materials. In soil mechanics, Taylor (1948)illustratedthe relationship between k and e3/(1+e) withthe results reproduced in Fig. 1. He registered also a goodcorrelation between k and e2,arelationship previously proposed by Terzaghi(1925)for

clays. However, Taylor (1948) did not usethe complete formulation ofthe KC equation withthe specific surface and did not evaluatethe value ofthe constant, C.

Figure 1: Relationship between k ande3/(1+e)for a sand, accordingto Taylor(1948). Itshould berememberedthat Taylor, Kozenyand Carman were notinterestedin hydraulic conductivity for the same reasons. For Terzaghi (1925, 1943) and Taylor (1948), a relationship between k ande enables a passage from a valuek1(e₁) – measured at a void ratio e₁ or at a given

dry density –toanother value k₂(e₂) of that same soil densified at e₂. For Kozeny(1927)and Carman (1937),the air or water permeabilitytest was usedto determinethe specific surface, S, of industrial powders. At atime whenthe determination ofS by other methods wastoo slow (over 24 h) andinaccurate, a 30-min air permeabilitytest provided afast methodto controlthe "quality" of anindustrial powder.

Opinion No.2:the KC equationisinadequatefor clays

This opinion is also widespread. Using test results from Terzaghi (1925) and Zunker (1932) for natural clays undergoing consolidation, Carman (1939) foundthatthe experimentalratio k (1 -n)2/n3 was not a constant but rather a decreasing function of porosityn. Thus, he concludedthat clays do not obeyeq.[1]. Carman(1939)ascribedthe divergencetoathin waterlayerthat would be immobilized at the surface of clayey particles. He calculated the water thickness that would be requiredto explainthe divergence fromthe equation. He obtained athickness of 72 Å for the clay tested by Zunker (1932), and of 103, 110 and 99 Å respectively for the three clays tested by Terzaghi(1925).

In soil mechanics, Taylor (1948, section 6.13) adopted Carman’s opinion (1937) but he also wrotethatthere was no methodtocalculatethe motionless waterfilmthickness.Inaddition,

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0 2 4 6

measured k-value (10-4 _m/s)

e

3 /

(1

+e

Taylor (1948) gave experimental resultsto show alinear correlation between logk ande forfine -grainedsoils andsuggestedthatthe dispersion of experimentalresults probably camefrom differencesinthe degree of saturation andinternal structure of compacted clays.

Michaels and Lin (1954) showedthat most soils give alinear correlation between logk ande. They presented test results for a kaolinite powder and different fluids used to form the clay that waslatertested withthesamefluid. Parts oftheirresultsare givenin Lambeand Whitman (1969). Michaels and Lin(1954)triedto verify whetherthe KC equation was validfor clays, and more specifically whethertheintrinsic permeability, K (eq. [2]), was really a geometric property of pores as assumedinthe equation. Their results, for water and ethanol only, are reproducedin Fig. 2. Theseindicated(1)thatthelinearrelationship between K and e3/(1+e) was not well verified,and(2)thattheintrinsic permeability depended onthetype offluid. Consequently, other properties of the fluid, such as its polarity, and characteristics of the solid-fluid interface, should beconsideredto obtaina more general predictiveequationforclays(e.g. Bardonand Jacquin 1968; Goldmanetal. 1990). Later, Al-Tabbaaand Wood(1987) used morerecent testing equipment and methodsto get permeability values for kaolinite percolated by waterinthe vertical and horizontal directions. Accordingtotheir results (Fig. 3),the directional k-values are notlinearly correlatedtothe ratio e3/(1+e), howeverthe firstinvariant defined as I_1k = (2k_h +k_v) /3 ofthe k-matrix(where k_h and k_v arethe horizontal and vertical hydraulic conductivities respectively)islinearly correlatedto e3/(1+e) as predicted bythe KC equation. The variations of this first invariant and the anisotropy ratio k_h / k_vwere examined by Chapuis et al. (1989b) for sand, and by Chapuis and Gill (1989) for sand, clay and sandstone, as afunction ofthe compaction mode(or stress history) and voidratio.

Figure 2: Intrinsic permeability of kaoliniteto water and methanol (from Michaels and Lin 1954).

0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 2.5 e3/ (1+e) me as ur ed K x 10 11 c m 2

Note: The powerlaw

(dottedlines) seems to

give a better fit than the

straightlines.

methanol

Figure 3: Hydraulic conductivity of kaolin(from results of Al-Tabbaa and Wood 1987). Inaddition, Lambeand Whitman(1969) demonstratedtheinfluence of micro-and macro -structures on the hydraulic conductivity of fine grained soils, as already mentioned by Terzaghi (1922), after testing specimens compacted either dry or wet of the optimum Proctor. Since the frequently quoted paper of Mitchell et al. (1965), this issue has been widely studied in relation with claylinersfor environmental projects(e.g. Chapuis 2002).

Comments onthe usual opinions

The previously mentionedcommon opinionsaboutthe KCequation have been based on partial verifications, usually without anyindependent measurement ofthe specific surface. Accordingto publishedresults,itappearsthatthe KCequation has been onlyapproximately verified, despite having a soundtheoretical basis. The problem arises mainly with clayey particles because solid-fluid interactions are not considered in the equation. Furthermore, the k value predicted by this equation is isotropic because it involves only scalar parameters, whereas permeabilityis often anisotropic (Chapuis et al. 1989b). This may be sufficientto seriouslylimit the predictive capacities ofthe equation, as well asthose of other similar equations.

Test

resu

lts

and

ana

lys

is

About 300laboratorytestresults(many ofthemtakenfromtheliterature) were usedto evaluatethe capacity ofthe Kozeny-Carman equationto predictthe k-value. Theselected references that reported the results usually gave all the required information, i.e. void ratio and eitherthespecificsurfaceas measuredforcohesivesoils, orthecomplete grainsizecurvefor non-cohesive soils.

Thetestresults used here(see Table 1) weretakenfrom Mavisand Wilsey(1937)forfive

0.E+00 1.E-09 2.E-09 3.E-09 4.E-09 5.E-09 6.E-09 7.E-09 8.E-09 0.0 1.0 2.0 3.0 4.0 ratio e3/ (1+e) k ( m/ s) kh kv I1k=(2kh+kv)/3

sands(Ottawa,Iowa, pit-run Iowa, uniform Iowa, and non-uniform Iowa); Morris and Johnson (1967)for overtwentysoils; Loiselleand Hurtubise(1976)for various non-plastictills; École Polytechnique and Terratech forthe James Bay Corporation before 1983 on varioustills; Chapuis et al. (1989b) for a sand (kh andkv,two compaction modes); Mesri and Olson (1971) for

threeclays(smectite,illiteand kaolinite); Olsen(1960)forthreeclays(kaolinite,illite, Boston blue clay); Navfac DM7 (1974) for clean sands and gravels; Tavenas et al. (1983b) for Champlain sea clays of St-Zotique, St-Thuribe and St-Alban. In addition,the authors have used several oftheir owntestresults on homogenized minetailings(e.g. Aubertin et al. 1993, Bussière 1993), other unpublishedresultsforsands,siltsandtillsfrom Quebec,andalsosand-smectite mixes with high percentages of smectite(Chapuis 1990, 2002).

Table 1: Data examinedinthis report

────────────────────────────────────────────────────── Soil Reference methodfor S

────────────────────────────────────────────────────── five sands Mavis & Wilsey(1937) Chapuis & Légaré sand and gravel Navfac DM7(1974) Chapuis & Légaré non cohesive soils Morris & Johnson(1967) Chapuis & Légaré non-plastictills Loiselle and Hurtubise(1976) __ or Fig.4 or BET sand Chapuis et al.(1989b) Chapuis & Légaré three clays Mesri and Olson(1971) provided by authors three clays Olsen(1960) provided by authors Champlain clays Tavenas et al.(1983b) Locat et al.

sand-smectite Chapuis(1990, 2002) Olsen

minetailings Bussière(1993) Chapuis & Légaré sands, silts &tills authors’data(unpublished) Chapuis & Légaré ────────────────────────────────────────────────────── Estimates of specific surfacefor non-plastic soils

The specific surface S of a soil is seldom evaluated (and used) in soil mechanics and hydro -geology. However,itis an essential parameter for bituminous mixes,to verify whetherthe solid particles are adequately coated with bitumen. In such mixes,the filleristhe major contributorto thespecificsurface. Usual methodstoevaluate S areapproximateand often based onlocal experience. Varioussimple predictiveequationsareavailable(e.g. Hveem 1974; Duriezand Arrambide 1962;standard Can/Bnq-2300-900). CrausandIshai(1977) proposedarelatively complex analytical method. Thislengthy methodintroduces ashapefactorthatis visually evaluated under a microscope. It depends onthe operator and can be seen as a "fudge factor"to obtain a betterfit between predicted and measured S values.

Chapuisand Légaré(1992) proposedan operator-independent methodthat wascompared withfour other methods.It assumesthat simple geometric considerations can be usedto estimatethe specific surface of a non-plastic soil. If d isthe diameter of a sphere orthe side of a cube,the specific surface S of a group of spheres or cubesis given by:

S (d) = 6/d ρs in m2/kg [4]

where ρs is the density (kg/m3) of the spheres or cubes. Starting with eq. [4], many theoretical

shapefactors,roughnessfactors, or projectionfactors(e.g. Dallavale 1948; Orrand Dallavale 1959; Gregg and Sing 1967). Inthe case of fine-grained non-plastic soils, such as fillers usedin bituminous mixes, Chapuis and Légaré(1992) have proposedto apply eq.[4] asfollows:

S =(6/ρs)Σ [(PNo D- PNo d)/d] in m2/kg [5]

where (PNo D - PNo d)isthe percentage by weight smallerthan size D (PNo D) andlargerthan next

size d (PNo d). Equation [5] was applied to the five fillers used as references by Craus and Ishai

(1977)for whichthespecificsurface had been measured accordingtothestandard method (ASTM C 204 2002) based onthe work of Blaine(1941)and of Oberand Frederick(1959). Table 2illustrates howto usethe complete grain size curve ofthelimestone fillerto calculateits specific surface, S

The grain size curves always have a minimum measurable particle size, Dmin, e.g. 5µmforthe

filler of CrausandIshai(1977)in Table 2.Inthe method of Chapuisand Légaré(1992),an equivalentsize, d_eq., must be definedforall particlessmallerthanthe minimumsizeforthe curve. This equivalent size correspondstothe mean size with respect ofthe specific surface.Itis given by:

∫

Whenthe minimum size, Dmin,is 5µm,the equivalent diameter, deq.,is 2.9µm(see Table 2).

TABLE 2-- Specific surface(m2/kg) of alimestonefiller (ρρρρs = 2880 kg/m3); gradation curvefrom Craus and Ishai(1977).

───────────────────────────────────────────── Size Cumulative Difference X S = 6/dρs X S

(mm) passing(%) (PNo D- PNo d) m2/kg m2/kg ───────────────────────────────────────────── 0.074 100 ---- --- --- -0.060 94 0.06 34.72 2.08 0.050 89 0.06 41.67 2.08 0.040 83 0.06 52.08 3.13 0.030 76 0.07 69.44 4.86 0.020 65 0.11 104.17 11.46 0.010 45 0.20 208.33 41.67 0.005 24 0.21 416.67 87.50 deq.= 0.0029 0.24 718.39 172.41 Specific surfaceS (m2/kg) = 325.2 ───────────────────────────────────────────── Notes: 1. The value of S is obtained asΣ(XS) = 325.2 m2/kg.

2. The equation giving "d equivalent"is providedinthetext.

This method can be appliedtofillers becausethey have no plasticity(Langlois et al. 1991) and their finest particles (< 5 µm) are mainly inactive rock flour. Table 3 gives the values of S as calculated with four different methods. It appearsthatthe proposed method (eqs. [4-6]), without using any visually estimated shape factor, correctly evaluates the specific surface of non-plastic fine powders,exceptfor hydratedlimefor whichthe grainsizecurveis noteasyto obtain by

sedimentation. The method of Eqs. [4-6] was usedto estimatethe specific surface of non-plastic soilsto be usedinthe KC equation.

TABLE 3-- Estimated specific surfaces(m2/kg)forfillers (from Chapuis and Légaré 1992)

─────────────────────────────────────────── Predicted values of S (m2/kg)

Filler S measured P-1 P-2 P-3 P-4 type ASTM C 204 1977 1992 1962 1962 ─────────────────────────────────────────── Limestone 263 258 325 600 346 Hydratedlime 869 750 615 1104 553 Glass beads 86 78 98 120 72 Dolomite 202 183 206 324 195 Basalt 247 217 247 420 240 ─────────────────────────────────────────── Notes: P-1 = Craus andIshai(1977)

P-2 = Chapuis and Légaré(1992)

P-3 and P-4 = Duriez and Arrambide(1962,tome 1, p.288) forfillers classified as veryfine orfine Estimation of specific surfacefortills and Champlain clays

In the case of cohesive soils, authors have usually provided the specific surface, except for a fewtest results on Champlain clays (Tavenas et al. 1983b). Similarly, severaltest results fortills (13reports), provided bytheJames Bay Corporation, did notinclude data onspecificsurface. For applyingthe KC equationtothese soils,the specific surfaces of Champlain clays and Quebec tills were determined asfollows. Locat et al.(1984) estimatedspecificsurfaces ofseveral Quebecclays usingthe methylene blue method(Tran 1977). Figure 4 plotstheestimated S versusthe percentage of particlessmallerthan 2 µmas obtained bysedimentation.Itappears thattestedclays withalow plasticity(8 < IP < 15, where IPisthe plasticityindex) havea specific surface S between 23 and 30 x 103 m2/kg, independently of the percentage of particles smallerthan 2 µm (seethetwo horizontallinesin Fig. 4). Thisisthe case for clays ofthe Great Whale River, Shawinigan, Chicoutimiand Outardes. Thisfinding was usedtoestimatethe specificsurface ofseveraltestedtills havingalow plasticity or no plasticity:forthecoarse fraction down to 2 µm, S was calculated by the method of Chapuis and Légaré (1992) and, aS value of 27 x 103 m2/kg was attributed to the fraction smaller than 2 µm. It can be seen alsoin Fig. 4 that the S values provided by Locat et al. (1984) for Champlain and North-West Quebec claysfall withinslopinglinesin Fig. 4. Similarzones may be definedin Fig. 5 where S is plotted versusthe sum [IP + (% < 2µm)]. The S-values of Champlain claystested by Tavenas et al. (1983b) were given by Locat et al(1984),for examplefor St-Alban, or evaluated using Figs. 4 and 5, for example for St-Zotique. Inthis case,theinitial void ratio, e, was closeto 2.5 andthe natural water content, w, was closeto 91% whereasIP = 36% and(% < 2µm) = 80. The specific surface S wasthen estimatedfrom Figs. 4-5 asS = 61± 5 x103 m2/kg.

Figure 4: Correlation between specific surface S and percentage smallerthan 2µµmµµ(results of Locat et al. 1984).

Figure 5: Correlation between measured specific surface andthe sum of plasticityindex plusthe percentage offines smallerthan 2 microns.

0 10 20 30 40 50 60 70 80 90 0 20 40 60 80 100

% smaller than 2 microns

S (1 0 3 m 2 /k g) IPlower than 15 IP from 15 to 35 IP higher than 35 0 10 20 30 40 50 60 70 80 90 0 20 40 60 80 100 120 140 160 IP + % smaller than 2 microns

S (1 0 3 m 2 /k g) IPlower than 15 IP from 15 to 35

More generally, the specific surface of any clay, S, may be assessed from its liquid limit,LL (e.g. Muhunthan 1991). The results of De Bruyn et al. (1957), Farrar and Coleman (1967), Locat etal.(1984),and Sridharanetal.(1984, 1988), have been gatheredin Fig. 6toillustratethat thereis an approximatelylinear correlation between 1/S and 1/LL. The best fit straightline (R2 = 0.88) of Fig.6 correspondstothe equation:

1/S (m2/g) = 1.3513(1/LL)- 0.0089 [7] when the liquid limit, LL, is lower than 110. A power law function of LL for S could also be used(Mbonimpa et al. 2002) but provides basicallythesame estimate of S for LL values. Equation [7] usually predicts an S value within ± 25% ofthe measured value when 1/LL > 0.167 (LL < 60%) as shownin Fig. 6. Poorer predictions are achieved using eq. [7] for soils withLL > 60%, especially clayey soils containing some bentonite. Taking againthe example of St-Zotique that had anLL = 61%, eq. [7] predictsS = 75±19 x103 m2/kg whichis closeto but slightly higher and moreinaccuratethanthe S values predictedfrom Figs 4 and 5.

Figure 6: Correlation betweentheinverse ofthe specific surface, S, andthe inverse oftheliquidlimit,LL, of clays.

App

l

icat

ion

and

eva

luat

ion

It was mentionedearlierthat previousevaluations of Kozeny-Carmanequation were partial and usuallylimitedtothe k-e component ofthe relationship. A relativelythorough evaluationis presented hereafter. Forthat purpose,the next figures present log [k / 1m/s] versuslog [e3/DR2

S2(1+e)]. Accordingto Eq.[1],the experimental data should verifythefollowingrelationship: log[k /(1m/s)] =A +log[e3/DR2S2(1+e)] [8]

where A = 0.29to 0.51fora C value(seeeq.1) between 0.2and 0.5assuggested by Carman (1939). Thefigures presented belowillustratethe correlationsthat have been obtainedfor

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.00 0.01 0.02 0.03 0.04 0.05 1/ LL(%) 1 / S ( m 2 /g ) Locat et al. (1984) De Bruyn et al. (1957)

Farrar & Coleman(1967)

Sridharan et al. (1984)

Sridharan et al. (1988)

best fit, R2=0.844

125% of bestfit

several soiltypes using afactor A = 0.5,resultingin:

log[kpredicted/(1m/s)] =0.5 +log[e3/DR2S2(1+e)] [9]

where kpredictedisin m/s,DR ande have no dimension andS isin m2/kg.

Sand and gravel

Resultsforsandand gravelare presentedin Figs. 7to 9. Their S-values were determined usingthe method of Chapuis and Légaré (1992). The predicted k-values were obtained using Eq. 9. Measured k-valuesare in the range 10-1 to 10-5 m/s. Most data for the sands of Mavis and Wilsey(1937) are alignedin a narrow band along Eq.[9](Fig. 7).

Figure 7: Predicted versus measured k-valuesforthe sands of Mavis and Wilsey(1937) The position ofany pointin Fig. 7 depends mainly ontwofactors:the uncertaintyinthe values ofspecificsurface, S,and degree ofsaturation, Sr, ofthetestedspecimen. Forcoarse

(non-plastic)soils,the uncertainty dueto S should notexceed 10%,thus 20%for S2, which represents an uncertainty ∆y= ± 0.08 cyclein Fig. 7. The uncertainty dueto Sris deemed higher,

becausetheexperimental k-values of Fig. 7 were obtainedinrigid-wall permeameters, where usually Sris unknown. Thecurrentstandardforthistestis ASTM D2434(2002),in which

"saturation"issupposedto have been obtained after using a vacuum pump. Thisstandard procedure, however, does not provideany meanstocheck whetherthesoilspecimenisfully saturated (degree of saturation Sr = 100%) or not. Such a method was proposed by Chapuis et al.

(1989a), who haveestablisheditsaccuracyandshownthatifa rigid-wall permeameter is used (withouttherecently defined precautions), Sr usuallylies between 75 and 85%. Thenthe

measured k value represents only about 15 to 30% of k (Sr=100%), which results in an

underestimate of x, ∆x = - 0.5to –0.8in Fig. 7. Such a condition could well applytothe results of Mavisand Wilsey(1937). The measured unsaturated hydraulicconductivity k(Sr) may be

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 measured k (m/s) pr ed ic te d k ( m/ s) Sr = 100% Sr = 75% equality

evaluated byseveral equations. Here asimplified equation proposed by Mualem(1976)is retained:

k(Sr)/k(sat) =(Sr-S0)3/(1-S0)3 [10]

in whichS0isthe degree of saturation correspondingto a residual water contenttaken as 0.2 for

sandand gravel.Ifa degree ofsaturation Sr = 75%isconsideredforthetests of Mavisand

Wilsey (1937), and Eq. [9]is usedto predict k(sat) with Eq. [10]to predict k(Sr),it appears (Fig.

7)thatthe measured k-values are well-predicted bythe KC equation.

To use the chart of Navfac DM7 (1974) the sand must have a coefficient of uniformity, CU,

between 2 and 12, a void ratio e between 0.3 and 0.7, a diameterD10 between 0.1 and 3 mm, and

a ratio D10/D5 lower than 1.4. The last condition meansthatthe grain-size distribution curve of

the sand cannot end with aflat portion. Aflatfinal portion mayindicate arisk of segregation and particle movement withinthe soil, or risk of suffossion. Such risks can be evaluated by usingthe criteria of Kezdi(1969), Sherard(1979) or Kenneyand Lau(1985, 1986). Thesesuffossion criteria have beenshownto be mathematicallysimilarandtheycan bereplaced by minimum values forthe secant slope ofthe grain-size distribution curve(Chapuis 1992, 1995). Eight sands were defined by straight-line grain size curves using eight values of D₁₀ (0.2-0.3-0.4-0.5-0.6-0.8 -1.0-1.5 mm) and a coefficient of uniformity of 7 which representsthe means of 2 and 12that are thelimits ofthe chart. TheS-values ofthe eight sands were estimated usingthe method of Eqs. [4-6]. The KC equation predicts saturated k-values (for Sr = 100%) that are usually higher than

the measured k-values (Fig.8). A better agreement between predicted and measured k-values is obtained using the Mualem equation (Eq. [10]) and assuming a Sr-value of about 85% for these

tests.

Figure 8: Predicted versus measured k-valuesforthe sands of Navfac DM7(1974) 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00

1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 measured k (m/s) pr ed ic te d k ( m/ s) Sr = 100% Sr = 85% equality

Inthetests of Figs 7 and 8,the soil specimens were most probably not fully saturated (Sr = 75

to 85%). The more recenttests of Chapuis et al. (1989a), however, were designedto ensure full saturation of the tested sand as checked by a mass and volume method. Here the KC equation predicts saturated k-valuesthat are closetothe measuredk-values atSr=100%(Fig. 9).

Figure 9: Predicted versus measured k-valuesforthe sandtested by Chapuis et al.(1989a) inthe vertical direction after static compaction.

The resultsin Figs 7to 9 confirmthatthe KC equation(Eq.[9]) provides afair estimate ofthe vertical hydraulicconductivity, kv, ofsaturatedsandand gravel. Whenthespecimensare not

fully saturated (75% < S_r <100%), Eq. [9] can be usedjointly with Eq. [10]to estimatethek(Sr)

-value.

Other non cohesive soils

Other data provided by MorrisandJohnson(1967)forsandsandsiltysandsareexamined hereafter. The specific surface S ofthese non cohesive soils was established usingthe method of Chapuisand Légaré(1992). Whenthecomplete grain-size distributionsare usedtoassess S, thenthe predictedk-values (Eq. [9]) are relatively dispersed aroundthe equalityline(y = x)for Sr

between 85 and 100% (Fig. 10). This dispersionin Fig. 10 may be duetothe factthatthe grain -size distribution of several soils specimens had a slopelowerthan 20%inthe fine size zone,thus indicatingarisk ofsegregationand particle movement orsuffossion(Chapuis 1992, 1995).It alsoindicatesthat such specimens were probably formed artificially by mixing several soillayers that may have been naturallyadjacent but not mixed. In such cases, the fine particles that can move with water do not really belongtothe solid skeletonrestrainingthe water seepage, andthus should not be considered in the S-value that contributes to the k-value of such soils. Consequently,the grain-size distribution ofthese soils was modifiedto follow a minimum slope of 20%asshownin Fig. 11. Newspecificsurfacesforthesesoils were obtainedfromtheir modified gradation curves and usedto predicttheir new k-values(Eq.[9])that are compared with

1.E-05 1.E-04 1.E-03

1.E-05 1.E-04 1.E-03

measured kv (m/s) pr ed ic te d kv ( m/ s) Sr = 100% equality

the measured k-valuesin Fig. 12. Now the comparison is better, although for several tests the difference between predicted and measured k-values exceeds one order of magnitude. Again,the discrepancy may beattributedtoincompletesaturationinrigid-wall permeametertestsandto gradation curves withrisks of suffossion.

Figure 10: Predicted versus measured k-valuesforthe non-cohesive soilstested by Morris and Johnson(1967). The complete particle gradation curve was used

to calculatethe specific surface, S.

Figure 11: Example of a slopeflatterthan 20%(per cycle) and modification ofthe gradation curvetotakeinto accountthe degree offreedom offine particles.

1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1. E-08 1. E-07 1. E-06 1. E-05 1. E-04 1. E-03 1. E-02 1. E-01 1. E+ 00 1. E+ 01 measured k (m/s) pr ed ic te d k ( m/ s) Sr = 100% Sr = 85% equality 0 10 20 30 40 50 60 70 80 90 100 0.001 0.01 0.1 1 10 100 grain size (mm) pe rc en ta ge p as si ng ( %) test slope 20%

the corrected curve follows the 20%line

Figure 12: Predicted versus measured k-valuesforthe non-cohesive soilstested by Morris and Johnson(1967). The modified particle gradation curve(Fig.11) was usedto calculate S. Tills and silty sands of Quebec

When the specific surface S of such soils can be adequately evaluated as discussed before, Eq. [9] gives a good prediction of the k-value as shown in Fig. 13. When the S-value is not accurately evaluated,the predictionis not so good as shown below.

For the Quebec tills tested for Hydro-Quebec by Loiselle and Hurtubise (1976), the specific surface S was assessed bythree methods. The first method wasto estimateS usingthe complete gradationcurveandthe method of Chapuisand Légaré(1992)for non-plasticsoils.Inthe second method,the 1st method was used onlyforthefractioncoarserthan 2 µm, and then the fraction smallerthan 2 µm was assumedto have a specific surface of 27x103 m2/kg. Thusit was assumedthatthe fine fraction oftills from James Bay and Outardes was somewhat similartothe low plasticityclays oftheseregions,for which Locatetal.(1984) measuredaspecificsurface between 23 and 30x103 m2/kg (see Figs. 4 and 5). Inthethird method,the 1st method was used onlyforthefractioncoarserthan 0.63 mm,andthenthefractionsmallerthan 0.63 mm was assumedto haveaspecificsurface of 1.7x103 m2/kg. This value was obtained usingthe BET (Brunauer-Emmett-Teller) methodforafewtillspecimens ofthe Laurentidesarea(North of Montreal), having a plasticityindex, IP,lowerthan 5 and grain size curves similartothosetested by Loiselle and Hurtubise (1976). Only the fraction smaller than 0.63 mm was tested using the BET method (currently considered asthe best methodto evaluate S forfine-grained soils), andits S-value was always closeto 1.7x103 m2/kg.

Predictedand measured k-valuesforthetillstested by Loiselleand Hurtubise(1976)are gatheredin Fig. 14. The 1st method, based only on gradation(downto approximately 1.3 microns), predicts a k-valuethatis usually 3to 10times higherthanthe measured k-value. The

1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1. E-08 1. E-07 1. E-06 1. E-05 1. E-04 1. E-03 1. E-02 1. E-01 1. E+ 00 1. E+ 01 measured k(m/s) pr ed ic te d k ( m/ s) Sr = 100% Sr = 85% equality

2 method, based onaspecificsurface of 27x10 m/kgforthefractionsmallerthan 2 µm, predicts a k-valuethatis usually 3to 10timeslowerthanthe measured k-value. The 3rd method, based onthe BETspecificsurface of 1.7x103 m2/kgforthefractionsmallerthan 0.63 mm, predicts more correctly the k-value than the two previous methods. Thus the BET method (real measurements) gave better predictionsforthe k-valuethanthe 1stand 2nd methodsthatare estimates based on assumptions.

Figure 13: Predicted versus measured k-valuesfor non-plastictills and silty sand specimens(authors results).

Figure 14: Predicted versus measured k-valuesforthetillstested by Loiselle and Hurtubise(1976).

1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03

1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03

measured K-value, m/s

pr ed ic te d K-va lu e, m/ s silty sands till - Val d'or till -Rouyn till - St-Sauveur non-plastic silt equality 1.E-11 1.E-10 1.E-09 1.E-08 1.E-07 1.E-06

1.E-11 1.E-10 1.E-09 1.E-08 1.E-07 1.E-06

measured k (m/s) pr ed ic te d k ( m/ s) SS-gradation SS-2microns SS-BET equality

Several other northerntillstested for SEBJ (James Bay Corporation) had fines with a plasticityindex IP higherthan 5, butthe IP values ofthetestedspecimens were notreported. Thus it was impossible to estimate the S value and then to correlate predicted and measured k-values. Whenthe IPis higherthan 5,thesetills have anS valuethatis much higherthanthat of tillstested by Loiselle and Hurtubise (1976). As a resultthe measuredk-values forthese northern tills arelowerthanthose of Fig. 14, and may reach 10-10to 10-11 m/s, a rangethat may be found withthe plastictills of Southernand Eastern Quebec(Appalachianareas). Toillustratethe influence offinesintheevaluation of S and k, the ratio of measured k-value over predicted k-value(1st method based on gradation only)is plotted versusthe D10in Fig. 15. Thisfigure

indicates that the prediction worsens when the D10 decreases, as anticipated, given the

importance of the fine particles forthe value ofS. For such plastic soils, a good prediction ofk requires either anindependent determination of S, orthe complete gradation curve andthe Atterberglimits ofthetestedspecimen.Inthelattercase, Figs 5 or 6 wouldthen be usedto assess S andthen k.

Figure 15: Ratio kmeasured/kpredicted versustheD10forthetills of Fig.14.

Minetailings

Many mine tailings have been tested for permeability at Polytechnique (L'Écuyer et al. 1992; Aubertin et al. 1993, 1996; Bussière 1993; Monzon 1998). Thesetailings arefinely crushed hard rock particles, with gradations ofsilts,and usually no orlittle plasticity. Here,theirspecific surface has been estimated fromtheir complete gradation curve usingthe method of Chapuis and Légaré(1992). Afewresults(Bussière 1993)are presented herefor homogenizedsamples. Measured k-values have been obtained for fully saturated specimenstestedin either rigid-wall or flexible wall permeameters.It was checkedthat both permeameters gave similarresults.

The predicted k-values do not matchthe measured k-valuesinthecase of minetailings,as

0.0001 0.001 0.01 0.1 1 10 100 0.0001 0.001 0.01 0.1 1 D10 of the till, mm ra ti o k me as . / k p re d.

shownin Fig. 16. The difference may be explained by several factors. First,the fine particles of tailings are angular, sometimes acicular. The voidratio,e, oftailingsis usually much higherthan the void ratio of silts having similar grain-size curves. As a result, the void space between the solidsis not similartothe void space of a natural soil. Accordingto numerous experiments,the measured k-value oftailings depends on aratioe3+a/(1+e) where ais positive, and ontheirliquid limit whenitis higherthan 40 (Aubertin et al. 1996; Mbonimpa et al. 2002). In addition,tailings are proneto several phenomena (Bussière 1993) such as creation of newfines during compaction (particle breakage) and chemical reactions during permeability testing. Consequently, the predicted k-value musttakethese phenomenainto account.

Figure 16: Predicted versus measured k-valuesfor minetailings. Accordingtotheresults of Fig. 16,the best-fitlinear equation can be expressed as:

logkmeasured = 1.46logkpredicted + 1.99 [11]

Consequently,itis proposed hereto predict the k-value of mine tailings as follows. The S-valueisfirst determined usingthecomplete gradationcurveandthe method of Chapuisand Légaré(1992). Thenthe k-valueis predicted using Eq.[9] modified by Eq.[11]to give Eq.[12]: log[k / 1 m/s] = 1.46(0.5 +log[e3/DR2S2(1+e)]) + 1.99 [12]

The predicted k-values (using Eq. [12]) versus measured k-values for tested tailings now are fairly close as shownin Fig. 17.

Here a note of caution must be made:the k-valuethat can be predicted using either Eq.[12] or otherequations(Aubertinetal. 1996; Mbonimpaetal. 2002)isthat of homogenizedtailings testedinlaboratorysaturatedconditions. The k-value ofintactsaturatedsamples oftailings cannot be predicted bythe KC equation(L'Écuyer et al. 1992), becauseintacttailings are typicallyfinelystratified(cm or mm scale) and have a high anisotropy ink. The KC equation

1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 predicted k (m/s) me as ur ed k ( m/ s) tailings equality linear best fit

assumesthatthe specimenis homogeneous, andit cannot predictthe k-value of a heterogeneous stratified specimen.

Figure 17: Predicted versus measured k-valuesfor minetailings using Eq.12 with theS-value estimatedfromthe gradation curve.

Fine-grained plastic soils

The measured k-values were provided by Mesriand Olson(1971)forsmectite,illiteand kaolinite, by Olsen(1960)for kaolinite,illiteand Boston blueclay, by Al-Tabbaaand Wood (1987)for kaolinite,and by Tavenasetal.(1983b)for Champlain(Quebec)intactclays. The specific surface of these clays was either provided inthe publications or evaluated using Figs. 4 or 5. Other measured k-valuesareforseveralsoil-bentonite mixescontaininga highcontent (over 20%) of bentonitethat completely fillsthe pore space (Chapuis 1990, 2002), for whichthe specific surface of bentonite wastaken as 6x105 m2/kg,the average value provided by Mesri and Olson(1971). The predicted versus measured k-valuesfor allthese clays are shownin Fig. 18.

In Fig. 18,the few pointsthat are far fromthe equalityline represent older results for smectite andillite,for whichthe k(e) proposedcurvesfalltothe 10-12to 10-13 m/srange. Theauthors obtainedthese k values not directlyfrom permeabilitytests butindirectlyfromconsolidation curves using Terzaghi’stheory. Such anindirect methodis presently knownto provide unreliable k-values (Tavenas et al. 1983a). Further developmentsintestingtechniques, better understanding of phenomena andimproved accuracy(e.g. Haugetal. 1994, Hossain 1995, Tavenasetal. 1983a)as wellas durationconsiderationsforclayssuchassmectite(e.g. Chapuis 1990) have produced k(e)curvesthat do notfall below 10-11 m/s. In Fig. 18 the more recent test data for smectite are closetothe equalityline. They were obtained usingtriaxial equipment and specimens 2-3 cm highthat have given values downto 1to 5x10-11 m/s, after verylongtimes (2 to 4 weeks)toinsurefull hydration(100%) and complete consolidation orswelling ofthis special clay(Chapuis 1990).

1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 measured k (m/s) pr ed ic te d k ( m/ s) wit h eq .1 1 tailings equality

Figure 18: Predicted versus measured k-valuesfor clays.

The uncertainty for data shownin Fig. 18 depends mainly on four factors:the uncertaintiesin estimated specific surface, S, and degree of saturation,Sr, duringthetest, whether or not enough

time wasallowedtoreachsteady-stateconditionsafterconsolidationand/orswelling,andthe thinrigidified waterlayeratthesurface ofclay particles. Several oftheseissues havealready been discussedinthe case of sand and gravel. Fortested clays,the specific surface was provided bythe authors andis relatively well known:it should not produce an error greaterthan 20%,thus 40% for S2, which represents∆x= ± 0.15in Fig. 18, even for Quebecintact clays. Thereislittle uncertainty related to the degree of saturation of tested clay specimens. It is probably close to 100% because most specimens weretestedintriaxial cells with a high back-pressuretoincrease the saturation (Lowe and Johnson 1960; Black and Lee 1973; Daniel et al. 1984; Rad and Clough 1984, 1986; Camapum de Carvalho et al. 1986; Donaghe et al. 1986).

Consideringthesmall uncertaintyrelatedtorecent dataforlongtesting duration,the KC equation (Eq. [9]) provides a fair estimate ofthe k-value ofintact specimens of natural clays and oflaboratory-madesamplesthatare prepared withaclay, hydrated,saturatedandconsolidated before permeabilitytesting.

However, the k-value of compacted clays (clay liners and covers) cannot be predicted by the KC equation (Eq. [9]). The k-value of compacted clay does not depend only onits void ratio and specific surface, but also on the preparation and compaction modes (e.g. Terzaghi 1922; Lambe 1954, 1958; Bjerrumand Huder 1957; Peirceetal. 1987; Wrightetal. 1997). Mitchelletal. (1965) foundthat clay specimens compacted wet of optimum may have k-values 2 or 3 orders of magnitudelessthanspecimenscompacted dry ofthe optimum. Also,theyidentifieda dual porosityincompactedclay. The porosity oftheclay mass(equivalentto primary porosityin hydrogeology) correspondstothefinestructure atthe micronscale ofsolid particles. The porosity between clay clods (equivalent to secondary porosity) corresponds to a macrostructure resulting from excavation, transport, handling and remolding by field equipment. The resulting

1.E-13 1.E-12 1.E-11 1.E-10 1.E-09 1.E-08 1.E-07 1.E-13 1.E-12 1.E-11 1.E-10 1.E-09 1.E-08 1.E-07 measured k (m/s) pr ed ic te d k ( m/ s)

Al-Tabbaa & Wood

Champlain

bentonite-authors

Mesri & Olson

Olsen

k-value can be predicted by an equationtakinginto accountthe primary and secondary porosities. The observed k-valueis mostlyrelatedtothesecondary porosityandcan beexpressed bya power law that is close to a cubic law, the theoretical law of flow in narrow apertures (Chapuis 2002).

Similarly,the k-value ofsoil-bentonite mixesin whichthe bentonite powder does notfill completelythe voidspacecannot be predicted using Eq.[9]. It can, however, be predicted by other methods(Chapuis 1990, 2002) consideringthe dual porosity of such mixes.

D

iscuss

ion

Many of the previous results have been grouped in Fig. 19 to illustrate that the KC equation can be used to predict fairly well the k-value of different soils when information is available to determine correctlythe specific surface andif adequate precautions aretakenforthe permeability test. These datainclude sands and gravels (Mavis and Wilsey 1937, Navfac 1974), granular soils (Morris and Johnson 1967), mine tailings (authors), tills (authors), Quebec natural (intact) clays (Tavenasetal. 1983b), kaolinite(Mesriand Olsen 1971; Olsen 1960; Al-Tabbaaand Wood 1987), pure bentoniteandsoil-bentonite mixes withatleast 20% bentonite(Chapuis 2002). Usually Eq. [9] predicts a k-valuethatis between 1/3 and 3timesthe measured k-value, whichis withinthe expected margin of variationforlaboratory permeabilitytestresults.

Figure 19: Predicted versus measured k-valuesfor different soils. 1.E-12 1.E-11 1.E-10 1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1. E-12 1. E-11 1. E-10 1. E-09 1. E-08 1. E-07 1. E-06 1. E-05 1. E-04 1. E-03 1. E-02 measured k (m/s) pr ed ic te d k ( m/ s)

Mavis & Wilsey Navfac

tailings (mod. KC) tills - authors Al-Tabbaa & Wood Champlain

bentonite-authors equality

y = 3x y = x/3

Accordingto presentstandardsforlaboratory permeability tests (ASTM D2434, D5084 and D5856, 2002),thereal precision ofthesetesting methodsseems unknownandthereforetheir biascannot be determined. Accordingtotheauthorstestsandliteraturereview,the precision depends on testing procedures and soilintrinsic variability. For example, an excellent precision (k-value within ± 20% for 3 specimens) can be reached with sand and gravel whentwo conditions are met. First, a special procedure (using both vacuum and de-aired water) must be followed withanimproved permeameterforensuringfullsaturation(Chapuisetal. 1989a). Second,thesoil gradation must not be pronetointernalerosion(Chapuis 1992).Itisalso importantto determinethereal gradient usinglateral piezometers(ASTM D2434, 2002). The real degree of saturation must be determined usingthe mass-and-volume method (Chapuis et al. 1989a).

Inthe case of alow k-value specimentestedin a rigid-wall or flexible-wall permeameter,itis knownthatsaturatingthespecimen by back-pressuretakes alongtime(sometimesseveral weeks), whereasinflow and outflow rates are very small. It may be assumedthat whenthe ratio of outflowtoinflowrateis between 0.75and 1.25(e.g. ASTM D5084, 2002),asteady-state condition has beenreached. However,thetest may befarfrom asteady-stateconditionif saturationis not completed. Usuallytheinflow and outflow rates show sometrend,increasing or decreasing withtime. Stopping atesttoo early mayleadto an underestimate ofthe k-value by up totwo orders of magnitude(Chapuis 1990).

Thespecimen preparation method may also influence the test results. This is true for sand: the directionalk-value depends onthe compaction mode (Chapuis et al. 1989b). Thisis equally trueforcompactedclayto be usedinaliner oracover. Thisistruealsoforcompactedtill specimensthat may exhibit a dual porositylike compacted clays(Watabe et al. 2001).In addition, whena wettillspecimenis heavilycompacted byimpact,compaction may generate high pore pressure and produce either local internal erosion or clogging (inhomogeneous material), resultingin either overestimated or underestimated measured k-values. Compaction of drytill, onthe other hand, may produce micro-fissuresandincreasethe k-value. Sucheffects were not documentedforthetests on non-plastictills presented here.

In the case of silty, non-plastic soils, three tests on three specimens of the same sample may give k-values ranging between half and twice the mean value. This seems to be due to at least tworeasons. First, a variation of ±2%inthe 2-micronsfines content mayinduce alarge variationin S-value andin k-value. Second,the soil gradation may be proneto some segregation of fines during placement and/or percolation. Inthe case of clays, when a pasteis prepared at a water content slightly higher than the liquid limit and then consolidated, an excellent precision can bereached(e.g. kaolin,results by Al-Tabbaa and Wood 1987).Inthecase of natural homogeneous clayssampled withthin-wallsamplers,threespecimenstakenatelevationsz, z+1m and z+2m, may give k-values ranging between 75 and 125% ofthe mean value, even when the voidratiosandthe Atterberglimitsare verysimilar. Whentheclay propertiesare more variable,the measured k-values may range between 1/3 and 3timesthe mean value. Here again, special precautions must be taken with natural claysto ensure full saturation. Itis also required to waitlongenoughtocompleteconsolidation(orswelling)andto measureequalinflowand outflow volumes forlong periods oftime, atleast 2-3 days for ordinary clay and 3 weeks or more for a bentonite specimen 2-3 cm-high.

As a result, it is usually admitted that the true k-value of a soil lies between 1/3 and 3 times the value given by a goodlaboratorytest. In a graphlikethat of Fig. 19,the resultinginaccuracy alongthe y-axisis ± 0.5 as shown bythetwo straightlinesthatrun paralleltothe equalityline.

Conc

lus

ion

By using many permeabilitytest results,the authors showthatthe KC equation provides good predictions of the vertical hydraulic conductivity, k, of homogenized soil specimens. Consequently,it may be used with confidenceto estimatethe k-value of a soilintherange of 10-1 to 10-11 m/s. Thisis arough prediction, however, usuallyfallingintherange of 1/3to 3timesthe measured k-value.

Observed differences between predictedand measured k-values may be duetoinaccurate estimates of specific surface, S, to faulty permeability testing procedures (incomplete saturation, etc.) as discussedinthe report, and alsototheoreticallimitations ofthe equation. Thisisotropic equationcannotrepresentcorrectly hydraulicconductivitythatisin mostcasesananisotropic parameter. This is one reason why the predictions are only approximately valid. For example, the kh(e) and kv(e) functions at full saturation, as determined by a series of directional

permeabilitytests(Chapuisetal. 1989b), do not verifyexactlytheequation. Similarly,the results ofthisreportare valid onlyfor hydraulicconductivityandcannot beextrapolatedto anotherliquid. Forsuchanextension, other properties oftheliquid(e.g. polarity),and ofthe solid-liquidinterface should be considered.

A frequent reason for having dispersed dataisinadequate permeabilitytest procedures. Considering difficulty of obtaining excellentlaboratorytest data,it may be concludedthat currentlaboratorytest results are not accurate and precise enoughto givethe best value of factor C inthe Kozeny-Carman equation(Eq.[8]).

Nevertheless,the authors believethatthe KC equationrepresented by Eq.[9]is a good predictivetool for any natural homogeneous soil. Specialistsin geotechnique and hydrogeology should useit more systematically. It can be used for quick estimates ofthe k-value of a series of soil specimens(after determination of S), and as a check ofthe quality of permeabilitytests.

Acknow

ledgments

Thanks are due to Isabelle Montour for calculating specific surfaces of several specimens, to Bruno Bussière and Monica Monzonfor performing manytests ontailings,to Antonio Gatienfor performingtests onsands,silty sands, clays and bentonite mixes, and checking calculations of specific surfaces. Jerry Levay and Bernard Boncompain ofthe James Bay Corporation provided laboratory permeabilitytestreportsfortills. Thisreportis aresult of aresearch program involvingtheoreticalanalysisandfieldwork designedtoimprovethereliability of permeability and aquifertests and sponsored bythe Natural Sciences and Engineering Council of Canada and by MEND(Canmet).

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