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Page 1: Predicting Bimodal Soil–Water Characteristic Curves

oils areturated soilr gap-gradedpresents apore-size

series inoil–water

ercentages.with dual

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Predicting Bimodal Soil–Water Characteristic CurvesLimin Zhang, M.ASCE,1 and Qun Chen2

Abstract: Failure of colluvial soil deposits induced by rainfall is a common geohazard in the natural terrain. Many colluvial swidely and gap graded with a minimal sand fraction. In order to study the pore water pressures in such gap-graded, unsadeposits through a seepage analysis and to evaluate the stability of these soil deposits, the soil–water characteristic curves fosoils must be known. Usually, gap-graded soils exhibit bimodal grain-size and pore-size distributions. This technical notetheoretical continuum method for the determination of soil–water characteristic curves for soils with a bimodal or multimodaldistribution. Based on the capillary law, the water content in a multimodal soil is equal to the sum of water stored in each porethe soil. Therefore, the bimodal or multimodal soil–water characteristic curves can be obtained by combining the unimodal scharacteristic curves for all components of the soil corresponding to the pore series weighted by the respective volumetric pThe proposed method is verified using experimental soil–water characteristics data of sand–diatomaceous earth mixturesporosity.

DOI: 10.1061/~ASCE!1090-0241~2005!131:5~666!

CE Database subject headings: Colluvium; Soil suction; Soil water; Infiltration; Grain size; Rainfall.

ajorigh-any

andsoilesegap-is, th

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e finearse

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Introduction

Failure of colluvial soil deposits in the natural terrain is a mconcern in the construction and maintenance of railways, hways, waterways, and many other infrastructure projects. Mcolluvial soils are widely and gap graded with a minimal sfraction. The knowledge of the pore-water pressures in thedeposits is of primary interest to evaluate the stability of thdeposits. In order to study the pore water pressures in suchgraded, unsaturated soil deposits through a seepage analyssoil–water characteristic curves~SWCC! for gap-graded soimust be known.

A well-graded soil and a gap-graded soil may have diffetypes of grain structures and pore-size distributions. Fig.~a!shows the possible soil structure, the cumulative pore-size dbution and the pore-size density curve for a well graded soil.the cumulative pore-size distribution and the pore-size decurve of the soil are unimodal and there is only one pore serthe soil. When a certain range of soil grains is missing in athe soil is gap-graded. As shown in Fig. 1~b!, the cumulativepore-size distribution and the pore-size density curve of thegraded soil are both bimodal. If the particle sizes of the cograins are far larger than the sizes of the fine grains and thgrains do not completely fill the pores formed by the co

1Assistant Professor, Dept. of Civil Engineering, Hong Kong UnivScience and Technology, Clear Water Bay, Hong Kong. [email protected]

2Associate Professor, School of Hydraulic and HydroeleEngineering, Sichuan Univ., Chengdu, Sichuan 610065, China.

Note. Discussion open until October 1, 2005. Separate discusmust be submitted for individual papers. To extend the closing daone month, a written request must be filed with the ASCE ManaEditor. The manuscript for this technical note was submitted for reand possible publication on December 30, 2003; approved on Sept28, 2004. This technical note is part of theJournal of Geotechnical andGeoenvironmental Engineering, Vol. 131, No. 5, May 1, 2005. ©ASCE

ISSN 1090-0241/2005/5-666–670/$25.00.

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J. Geotech. Geoenviron. Eng

e

grains, the soil can be considered as being poorly mixed andwill be two pore series in the soil governed by the coarse g~large pores! and the fine grains~small pores!, respectively.

When experimental data are not available, estimation mecan be used to obtain the SWCCs of soils. Many mathemmodels have been proposed in the literature to represent uniSWCCs~e.g., Gardner 1958; Brooks and Corey 1964; Mua1976; van Genuchten 1980; Fredlund and Xing 1994; LeongRahardjo 1997!. These models are usually used to best-fit exmental soil–water characteristics data. Several models for pring the SWCC from the particle-size distribution and volumass properties have also been proposed~e.g., Gupta and Larso1979; Fredlund et al. 1997!.

When two or more pore series exist in a soil, the corresping SWCC can be bimodal or multimodal. The aforementioprediction methods, however, are intended for unimodal SWSeveral dual-porosity or multiporosity models for the soil–wcharacteristics were proposed for structured porous media.and van Genuchten~1993! developed a one-dimensional duporosity model for simulating preferential movements of wand solutes in structured soils. The water content of the bulkwas expressed as the sum of the water contents of the frand the matrix pore system weighted by respective volumweighting factors. Durner~1994! extended the unimodal vGenuchten-Mualem model~van Genuchten 1980! to fit bimodaland multimodal water retention functions by introducing weiing factors for combining individual functions. The weightfactors were determined by best fitting the measured watertion functions. Burger and Shackelford~2001a,b! presentepiecewise-continuous forms of the Brooks–Corey~1964!, van Genuchten~1980!, and Fredlund–Xing~1994! SWCC functions toaccount for the bimodal patterns of experimental SWCCs forletized diatomaceous earth and sand–diatomaceous earth mwith dual porosity. Each SWCC was separated into two segmjoined at a matching point. The data in each segment werefit independently using a unimodal SWCC function. Hence,

sets of parameters and one additional parameter related to the

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Page 2: Predicting Bimodal Soil–Water Characteristic Curves

ccor-

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common suction at the matching point must be assigned in adance with the experimental data. Zhang and Fredlund~2003!presented a method for predicting bimodal SWCCs from psize distributions for unsaturated fractured rocks. The poredistributions of the rock matrix and the fracture must be knowuse that method.

Indeed, existing methods can be used to fit and represeSWCCs for structured soils and fractured rocks when suffiexperimental data are available. However, they may not bedirectly for predicting bimodal or multimodal SWCCs for sowith large grains that prohibit a routine measurement of SWCThe objective of this technical note is to present a methodpredicting bimodal or multimodal SWCCs for bimodal or mumodal soils using the unimodal SWCCs for the characteristiccomponents that correspond to the respective pore seriesexperimental SWCC data of mixtures of sand and diatomacearth are used to verify the proposed method.

Soil–Water Characteristic Curves for Bimodal Soils

The theoretical basis for the estimation of SWCCs from the psize distribution of soil established by Fredlund and Xing~1994!can be introduced into this technical note. The pores in theare considered as a collection of different sized cylindrical tuThe saturated volumetric water content of the soilus can be expressed as

us =ERmin

Rmax

fsrddr s1d

whereRmax and Rmin5maximum and minimum pore radii in thsoil; r5pore radius; andfsrd5actual pore-size density functioWhen the soil is not fully saturated, the pore water pressurthe soil will be negative. The matric suction follows the capill

Fig. 1. Structures, pore-size distributions, and pore-size decurves for unimodal and bimodal soils

law

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J. Geotech. Geoenviron. Eng

c =C

rs2d

where c5matric suction in the soil; C=2T cosa, whereT5surface tension of water; anda5contact angle between waand soil particle. Eq.~2! implies that water tends to fill smapores first. When all the pores with radii less than or equalRare filled with water, the volumetric water contentusRd is

usRd =ERmin

R

fsrddr s3d

Two extreme suction conditions can be defined as followcording to Eq.~2!

cmax=C

Rmins4d

cmin =C

Rmaxs5d

wherecmax and cmin5maximum and minimum suctions, resptively. Substituting the capillary law into Eq.~3! and assuming thmaximum suction,cmax, being infinity, a general form of thSWCC is~Fredlund and Xing 1994!

uscd = usEc

`

gssdds s6d

where gssd5scaled pore-size density as a function ofc; ands5dummy variable for suction. When the soil is fully saturatecapproaches zero and the value of the cumulative fune0

`gssdds is equal to unity.As shown in Fig. 1~b!, there exist two pore series in a bimo

soil and the combined pore-size density curve for the soil imodal. The combined pore-size density functionfsrd for the soilmass is equal to the sum of the pore-size density functions folarge-pore seriesf lsrd and the small-pore seriesfssrd

fsrd = f lsrd + fssrd s7d

The two pore series in the soil are assumed to be conneTherefore, the suctions in the two characteristic pore series asame. According to above assumptions and the capillary lawvolumetric water content of the soil can be expressed as

uscd =ERmin

R

flsrddr +ERmin

R

fssrddr = uslEc

`

glssdds+ ussEc

`

gsssdds

s8d

where usl and uss5volumetric water contents of the large-pseries and the small-pore series, respectively, when they aresaturated.

Based on Eq.~8!, the volumetric water content of a fully sarated soil is

us = usl + uss s9d

In fact, us is equal to the total porosity of the soil and the srated volumetric water content of each pore series can bpressed as

usl = plnpl s10d

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Page 3: Predicting Bimodal Soil–Water Characteristic Curves

theries inensityents;se-

ividu-

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-eries

riesll-

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uss= psnps s11d

wherepl and ps are, respectively, volumetric percentages ofcomponents with the large-pore series and the small-pore sethe soil mass that can be calculated easily based on the dvalues and the percentages by dry weight of the soil componnpl andnps5porosities of the components with the large-poreries and the small-pore series when they are considered indally. Accordingly, Eq.~8! can be rewritten as

uscd = plnplEc

`

glssdds+ psnpsEc

`

gsssdds= plulscd + psusscd s12d

whereulscd andusscd5SWCC functions of the components wthe large-pore series and the small-pore series, respectivel~12! shows that the SWCC of the soil is the combination ofSWCCs for the two characteristic components weighted byindividual volumetric percentages.

Using the widely used van Genuchten~1980! function to represent the SWCC for the two components with large-pore sand small-pore series, Eq.~12! can be rewritten as follows:

uscd = plnplF 1

1 + salcdnlGml

+ psnpsF 1

1 + sascdnsGms

s13d

whereal , nl, andml5fitting parameters for the large-pore secomponent, andas, ns, andms5fitting parameters for the smapore series component.

The bimodal SWCC can also be described using theestablished Fredlund and Xing~1994! function as follows:

uscd = plnpl31 −

lnS1 +c

crlD

lnS1 +106

crlD45 1

lnFe+ Sc

alDnlG6

ml

+ psnps31 −

lnS1 +c

crsD

lnS1 +106

crsD45 1

lnFe+ S c

asDnsG6

ms

s14d

where e5base of natural logarithm;cr5soil suction in the residual condition;a, n, and m5three parameters of the SWCfunction; subscriptsl and s represent the large-pore series coponent and the small-pore series component, respectively.

Multimodal Soil–Water Characteristic Curves

A soil with a multimodal grain-size distribution may also havmultimodal pore-size distribution. Let us assume that the soibe characterized by several overlapping porous continua apore series can be superposed to obtain the overall pordistribution of the soil. As an extension of Eq.~12!, the SWCCfunction can be expressed as

uscd = oi=1

N

piuiscd s15d

whereN5number of pore series;pi5polumetric percentage of thsoil component with theith pore series; anduiscd5SWCC associated with theith pore series. At full saturation, the volume

water content of the soil can be represented as the sum of the

668 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINE

J. Geotech. Geoenviron. Eng

.porosities of the soil components weighted by their respevolumetric percentages

us = oi=1

N

usi = oi=1

N

pinpi s16d

whereusi andnpi5saturated volumetric water content and poity of the soil component with theith pore series.

Verification of Multimodal Soil–Water CharacteristicCurve Functions

Experimental Data

The experimental data obtained by Burger and Shacke~2001a,b! are cited to verify the proposed method for predictinmultimodal SWCC. The constituent materials used in the stuBurger and Shackelford were two sizes of processed diatceous earth~DE! pellets and two sands with different grain-sdistributions. The DE pellets were given the names CG1CG2, corresponding to average pellet diameters of approxim1 and 2 mm, respectively. The two sands were referred to as20–60 and Sand 10–20 because 90% by weight of themretained between the No. 20 and No. 60 sieves, and the Nand No. 20 sieves, respectively. The measured specific gand the maximum density for each material are shown in Ta~also in Table 1 of Burger and Shackelford 2001b!. In their studySWCCs were measured for 12 different mixtures of CG1 pewith Sand 20–60, CG2 pellets with Sand 20–60, and CG2 pwith Sand 10–20 at four different percentages of DE ranging4 to 30% by dry weight. The measured SWCCs with three dent percentages of DE are later used in the present studypercentages of DE by volume in the mixtures,pn, can be calculated according to the density and the percentage by dry weigeach soil constituent as follows:

pn =1

1 +s1 − pwdrDE

pwrSA

s17d

wherepw5percentage of DE by dry weight; andrSA andrDE5drydensities of the sand and the DE, respectively. The percentaDE in the mixtures by dry weight and by volume are listedTable 2.

Verification Procedure and Results

The fitted unimodal SWCCs for the two sands obtained by Bu

Table 1. Index Properties of Constituent Materials~after Burger anShackelford 2001b!

Diatomaceousearth Sand

Property CG1 CG2 Sand 20–60 Sand 10–

Specific gravityof solids

2.28 2.24 2.65 2.65

Maximum dry densitysg/m3d

0.54 0.62 1.39 1.66

Minimum totalporosity ~%!

76.4 72.5 47.5 37.2

and Shackelford~2001a,b! using the original forms of the van

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Page 4: Predicting Bimodal Soil–Water Characteristic Curves

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eters

allyeriesdal

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CG200%

.3

.7

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Genuchten~1980! model and the Fredlund and Xing~1994!model, as well as the fitted SWCCs for the two DEs using~13! and ~14! are shown in Figs. 2 and 3, respectively. TSWCCs for the two sands are unimodal. The fitting paramfor the SWCCs are listed in Table 3, in which theus values aretaken from Table 1. The SWCCs for CG1 and CG2 are typicbimodal because of the intrapellet and interpellet pore sexisting in the DEs. The fitting parameters for the two bimoDEs are shown in Table 4. According to Eq.~9!, the sum ofusl

anduss for each of the DEs is equal to the total porosity showTable 1.

Table 2. Percentages of Diatomaceous Earth by Dry Weight and b

CG1 and Sand 20–60

pw s%da 4.1 14.1 27.6 4.4

pn s%db 10.0 29.8 49.7 9.4apw5percentage of diatomaceous earth by dry weight.bpn5percentage of diatomaceous earth by volume.

Fig. 2. Measured and predicted soil–water characteristic curvesvan Genuchten function for sand–diatomaceous earth mixtures

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Using Eq.~15!, the SWCCs for the mixtures can be obtaiby combining the respective SWCC of each constituent inmixtures weighted by their volumetric percentages. The meaSWCCs ~Burger and Shackelford 2001b! and the predicteSWCCs using Eq.~15! for the three different mixtures are ashown in Figs. 2 and 3. The percentage of DE shown infigures is by dry weight. As shown in Figs. 2 and 3, there isdiscrepancy between the measured and predicted SWCCspatterns of the SWCCs for the mixtures are all bimodal ormodal. Some mixtures, such as the mixtures of CG1 andwith Sand 20–60, are trimodal because the SWCCs for the 1

me in Mixtures

2 and Sand 20–60 CG2 and Sand 10–20

15.2 29.4 3.6 12.7 25

28.8 48.4 9.1 28.1 47

Fig. 3. Measured and predicted soil–water characteristic curvesFredlund–Xing function for sand–diatomaceous earth mixtures

y Volu

CG

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Page 5: Predicting Bimodal Soil–Water Characteristic Curves

o thes of

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DE are bimodal and the particle size of the sand is not close tparticle sizes of CG1 and CG2. The SWCCs for the mixtureCG2 with Sand 10–20@Figs. 2~c! and 3~c!# appear to be bimodbecause the particle size of Sand 10–20 is similar to that ofand therefore there are only two distinct pore series in thetures. The results illustrate that the multimodal equation forSWCC in the present study can capture the multimodal charaistics of water storage and closely fit the measured data. Intion, both the van Genuchten model and the Fredlund andmodel can represent the bimodal or trimodal SWCCs eqwell.

Conclusions

A method is proposed for describing SWCCs for gap-gradedthat exhibit bimodal or multimodal pore-size distributions. Tsoil can be considered as a composite continuum with twmore pore series governed by the characteristic componentssoil. Based on the capillary law, the volumetric water contenthe soil can be obtained by summing the volumetric contewater stored in each pore series in the soil. Thus, the SWCthe bimodal or multimodal soil can be obtained.

The proposed method is verified by experimental SWCCof mixtures of sand and DE. The SWCCs of the mixtures obtaby combining the SWCCs of the DE and the sand usingmethod presented in this technical note agree well with thesured curves by Burger and Shackelford~2001a,b!. This demonstrates that the proposed multimodal soil-water charactecurve equation can commendably depict the multiporosity cacteristics of the soil mixtures. Both the van Genuchten~1980!model and the Fredlund and Xing~1994! model can represent tmultimodal SWCCs equally well.

Acknowledgments

The work described in this paper was substantially supportedgrant from the NSFC/RGC Joint Research Scheme betweeNational Natural Science Foundation of China and the HKong Research Grants Council~Project No. N_HKUST611/03!.

Table 3. Soil–Water Characteristic Curve Fitting Parameters for Sa

Model Material us

van Genuchten Sand 20–60 47.5

function Sand 10–20 37.2

Fredlund–Xing Sand 20–60 47.5

function Sand 10–20 37.2

Table 4. Bimodal Curve Fitting Parameters for Diatomaceous Earth

Model Material usla crl al nl

Eq. ~13! CG1 42.0 — 2.67 149.4

CG2 37.7 — 3.69 137.9

Eq. ~14! CG1 42.0 1.6 0.47 9.4

CG2 37.7 0.5 0.30 8.9ausl=plnpl.buss=psnps.

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The writers would also like to thank Professor D. G. Fredlundhis valuable advice on this work.

References

Brooks, R., and Corey, A.~1964!. “Hydraulic properties of porous mdium.” Hydrology Paper No. 3, Colorado State Univ., Fort CollinColo.

Burger, C. A., and Shackelford, C. D.~2001a!. “Evaluating dual porositof pelletized diatomaceous earth using bimodal soil–water charistic curve functions.”Can. Geotech. J., 38~1!, 53–66.

Burger, C. A., and Shackelford, C. D.~2001b!. “Soil–water characteristcurves and dual porosity of sand-diatomaceous earth mixtureJ.Geotech. Geoenviron. Eng., 127~9!, 790–800.

Durner, W.~1994!. “Hydraulic conductivity estimation for soils with heerogeneous pore structure.”Water Resour. Res., 30~2!, 211–223.

Fredlund, M. D., Fredlund, D. G., and Wilson, G. W.~1997!. “Predictionof the soil–water characteristic curve from grain-size distributionvolume-mass properties.”Proc., 3rd Brazilian Symp. on UnsaturatSoils, Rio de Janeiro, Vol. 1, 13–23.

Fredlund, D. G., and Xing, A.~1994!. “Equations for the soil-water chaacteristic curve.”Can. Geotech. J., 31~4!, 521–532.

Gardner, W. R.~1958!. “Some steady state solutions of the unsaturmoisture flow equation with application to evaporation from a wtable.” Soil Sci., 85~4!, 228–232.

Gerke, H. H., and van Genuchten, M. T.~1993!. “A dual-porosity modefor simulating the preferential movement of water and solutestructured porous media.”Water Resour. Res., 29~4!, 305–319.

Gupta, S. C., and Larson, W. E.~1979!. “Estimating soil water retentiocharacteristics from particle size distribution, organic matter perand bulk density.”Water Resour. Res., 15~6!, 1633–1635.

Leong, E. C., and Rahardjo, H.~1997!. “A review of soil–water characteristic curve equations.”J. Geotech. Geoenviron. Eng., 123~12!,1106–1117.

Mualem, Y. ~1976!. “A new model for predicting the hydraulic condutivity of unsaturated porous media.”Water Resour. Res., 12, 593–622

van Genuchten, M. T.~1980!. “A closed-form equation for predicting thhydraulic conductivity of unsaturated soils.”Soil Sci. Soc. Am. J,44~5!, 892–898.

Zhang, L. M., and Fredlund, D. G.~2003!. “Characteristics of wateretention curves for unsaturated fractured rocks.”Proc., 2nd AsianUnsaturated Soil Conf., D. Karube, A. Iizuka, S. Kato, K. Kawai, anK. Tateyama, eds., Osaka, Japan, 425–428.

cr a n m

— 1.20 23.11 0

— 2.13 178.20 0.0

1.10 1.01 12.84 0.

0.51 0.50 31.65 0.7

ml ussb crs as ns ms

0.014 34.4 — 0.002 1.85 1

0.028 34.8 — 0.012 6.29 0

1.167 34.4 51060 586.2 1.69 4

1.528 34.8 519 123.9 4.31 0

nd

8

1

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