Durner 1994

WATER RESOURCES RESEARCH, VOL. 30, NO. 2, PAGES 211-223,FEBRUARY 1994
Hydraulicconductivityestimationfor soilswith
heterogeneouspore structure
WolfgangDurner
InstituteofTerrestrialEcology,SoilPhysics,FederalInstituteofTechnology,Z•irich,Switzerland
Abstract.The hydraulicconductivityfunction,whichis requiredto solvetheRichards
equation,isdifficulttomeasure.Thereforepredictionmethodsarefrequentlyused
wheretheshapeof the conductivityfunctionis estimatedfromthe moreeasily
measuredwaterretentioncharacteristic.Errorsin conductivityestimationscanarise
eitherfroman invalidityof the predictionmodelfor a givensoil,or fromanincorrect
descriptionoftheretentiondata.Thisseconderrorsourceisparticularlyimportantfor
soilswithheterogeneousporesystemsthat cannotbe adequatelydescribedby the
usuallyusedretentionfunctions.To describetheretentioncharacteristicsofsuchsoils,
aflexible0(½)functionwasformedby superimposingunimodalretentioncurvesofthe
vanGenuchten(1980)type. By combiningthisretentionmodelwith the conductivity
predictionmodelofMualem(1976),conductivityestimationsforsoilswith
heterogeneousporesystemsareobtained.Estimatedconductivitiesbythismodeland
theclassicalvan Genuchten-Mualemmethodcandiffer by ordersof magnitude.Thus
reporteddisagreementsbetweenmeasuredandestimatedconductivitiesmayinsome
casesbedueto aninadequatedescriptionof theretentiondataratherthandueto a
failureof the predictionmodel.
Introduction
Theincreasingconcernwith groundwaterpollutionand
contaminationof soilsby hazardoussubstanceshasstimu-
latedthedevelopmentof numerousmathematicalmodelsof
pollutanttransportin soils.Today,thenumericalsimulation
ofwaterand pollutanttransportin unsaturatedsoilshas
becomea standardtool for assessingenvironmentalpoilu-
tionproblems.Despitetheexistenceandtheuseof simplis-
ticmodelsfor managementpurposes[e.g., Carsel et al.,
1985;Barraclough,1989](seeoverviewsby Addiscottand
Wagenet[1985]andLoagueandGreen[1991]),aswellasthe
recentintensivedevelopmentof specificmacroporemodels
[e.g.,Jarviset al., 1991;Chen and Wagenet, 1991;Gerke
andvanGenuchten,1993],the mostimportantapproaches
tomodelingtransientwater and solutetransportin the
vadosezoneare basedon the Richardsequation.To solve
thisequation,theknowledgeof the effectivesoilhydraulic
properties,namely,thesoilmoisturecharacteristic0(½)and
theK(O) or K(½) relationshipoverthe wholerangeof
moistureconditionsis required.Here, 0 (cubiccentimeters
percubiccentimeter)is the volumetric water content, ½
(centimeters)is the matricpressurehead,andK (centime-
tersperday)is thehydraulicconductivity(a scalarproperty
fortheone-dimensionalcase).For usein models,thehy-
draulicpropertiesare commonlyexpressedby analytical
functions,oftencalledparametericmodels,as,forexample,
listedby van Genuchtenand Nielsen [1985],Bruceand
Luxmoore[1986],andMualem[1986].The parametersof
thesemodelscanbe obtainedby fittingthe functionsto
experimentalwater retention and conductivity data. Alter-
natively,theycanbeestimatedfrommoreeasilymeasured
Copyright1994bytheAmericanGeophysicalUnion.
Papernumber93WR02676.
0043-1397/94/93WR_02676505.00
soilproperties[Bloemen,1980;Saxtonet al., 1986;WOsten
and vanGenuchten,1988;Vereeckenet al., 1989],or deter-
minedbyparameterestimationtechniqueswhentheinverse
problemis solved[Zachmannet al., 1981,1982;Hornung,
1983;Kool et al., 1987;Kool andParker, 1988].
Of all hydraulicproperties,the unsaturatedhydraulic
conductivityK is mostdifficultto measure.Thereforethe
use of indirect methodswhere K(0) is estimatedfrom more
easilymeasuredsoilpropertieshasbecomemoreandmore
common[van Genuchtenand Leij, 1992].A potentially
powerfulclassofmethodsresultsfrompore-sizedistribution
models,wherethewaterretentioncurveofaporousmedium
is interpretedasstatisticalmeasureof its equivalentpore-
sizedistribution[Corey,1992].In thisapproach,theconduc-
tivityisestimatedbyapplyingtheconceptof viscousfluid
flowthroughcapillariesandbyusingaconceptualmodelto
describethe pore interactionand pore connectivity
[Mualem,1986].Threebasicgroupsofstatisticalprediction
modelsmaybedistinguished:(1)thecapillary-bundlemodel
ofPurcell[1949]withitssubsequenttortuositycorrections
byYuster[1951],Burdine[1953],WyllieandGardner[1958],
andAlexanderand Skaggs[1986];(2) the cut andrandom
rejoinmodelof ChildsandCoilis-George[1950]withits
subsequentmodificationsbyMarshall[1958],Millingtonand
Quirk[1961],Campbell[1974]andothers;and,mostre-
cently,(3)theslabmodelofMualem[1976].Mualemand
Dagan[1978]andMualem[1986,1992]providecomprehen-
sivereviewsfor thesegroupsof conductivityestimation
models.
Closed-formexpressionsfortheretentionfunctionandthe
conductivityfunctionareoftenusedinparameterestimation
techniqueswheretheinverseproblemissolved[Kooland
Parker,1987a,b].Applyingapredictivemodelthatcouples
theconductivityfunctionwiththeretentionfunctionisthen
ofparticularadvantagebecauseitminimizesthenumberof
211

212 DURNER: HYDRAULIC CONDUCTIVITY ESTIMATION FOR SOILS
parameters needed to express the hydraulic relationships.
For the validity of this approachit is essentialthat the
prechosenfunctional representationof both the retention
curve and the conductivity curve as well as the model that
couplesthe two functions are appropriateover the whole
moisture range.
In principle, the statisticalmodelscanbe usedto estimate
absoluteconductivityvalues,but in practicethey are used
only to derive the shapeof the conductivityfunctionfrom
the shapeof the retentioncurve.To obtainabsolutevalues,
theestimatedconductivitycurvemustbe scaledwithatleast
onemeasuredconductivity,the "matchingvalue" [Nielsen
et al., 1960].Most frequently,the saturatedconductivityis
taken as matchingvalue, althoughseveral authors[van
GenuchtenandNielsen, 1985;Mualem, 1986;Luckneretal.,
1989;Nielsen and Luckher, 1992]have warnedagainstthis
practice.
Errors of conductivity estimates may arise from two
sources:(1) errorsdueto aninappropriatedescriptionofthe
soil moisture characteristic and (2) errors stemmingfrom
failureof thepredictionmodel.Beginningwiththeworkof
Nielsen et al. [1960], numerousstudies have compared
estimated with measured conductivities and have discussed
the validity of predictionmethodsfrom an empiricalor
semiempiricalpoint of view [e.g., KabIan et al., 1989;
Michiels et al., 1989;Stephens,1992].As a generalresult
from theseinvestigations,conductivityestimationmethods
appearto berelativelyreliablefor sandysoilswithnarrow
particlesize distributions,whereasfor loamyandclayey
soils they often fail. This failure is mostly attributedto
incorrect values of the tortuosity coefficient (or pore-
interactionfactor) r (dimensionless),whichis an empirical
parameterin thepredictingequations(see(6)inthetheory
section).SchuhandCline [1990]haveshownthatßcanvary
overa widerange.In additionto theseexperimentalinves-
tigations,theoreticaldiscussionsand reviews[Brutsaert,
1968;Gardner,1974;Mualem, 1976;Parkesand Waters,
1980;van Genuchtenand Nielsen,1985;Mualem, 1986;
AlexanderandSkaggs,1986]aswellasmathematicalsensi-
tivity analyses[Sakellariou-Makrar•tonakiet al., 1987;Vo-
gel and Cislerova,1988;Sidiropoulosand Yannopoulos,
1988]havebeenpublished.Despitetheextensivetreatment
ofthepredictionproblemanddueto thelackof a reliable
data base of measurementsin undisturbedsoils,the results
appearambiguousandinconclusive.A statementwhichwas
publisheda decadeagoappearsto be stillvalid:"The
problemofwhich[conductivity]measurementmethodand
whichpredictionmethodaremostvalidisundoubtedlyone
of the most vexingonesin contemporarysoil physics"
[Ragabet al., 1981,p. 384].
When discussingvalidity problems,mostinvestigators
have concentratedon the failure of the predictivemodel,
moreor lessdisregardingtheproblemofinterferenceswith
thefirst errorsource.No investigationhasbeenpublishedso
far on the systematicerrorsof conductivityestimation
methodsfor soilswith heterogeneouspore systemswhich
arisefromtheuseof nonoptimalwaterretentionfunctions.
Theobjectiveofthispaperistoanalyzeconductivityesti-
mationsfor suchsoils.After reviewingthecharacteristicsof
heterogeneouspore systems,conductivityestimatesare
obtainedby combininga flexibleretentionfunctionwith
Mualem's[1976]model.The resultsof thismethodare
comparedwithresultsobtainedbyclassicalunimodalmod-
els.Thedifferencesareanalyzedandthegenerallimitations
ofstatisticalconductivityestimationmethods,particularly
for soilswith a nonnegligiblefraction of largepores,are
discussed.
Theory
Water Retention Curves and Equivalent
Pore Size Distributions
Theporesystemofa rigidsoilcanbe characterizedbyits
equivalentpore-sizedistribution,which is commonlycalcu.
latedfrom the specificmoisturecapacityC* = dO/de.
Accordingto Laplace's law, the matric pressureat whicha
water-filled pore startsto drain is inverselyproportionalto
theequivalentradiusr of theporenecks.If C* asameasure
of pore densityis plottedversus• on a linear scale,pores
withlargeradiiofdifferentordersof magnitudearesqueezed
together,whichmaskstheshapeofthepore-sizedistribution
at the hydrologically important range close to saturation.
Sincethe matricpressureheadvariesover ordersofmagni.
tude,theretentioncharacteristicsarefrequentlyplotted0na
logarithmicpressureheadscale,whichis convenientlyex-
pressedinpF units,definedfor theunsaturatedrangebypœ
-- log•0(-½) [Schofield,1935],where ½ is expressedin
centimeters.The pF scale is equivalent to a logarithmic
scalefor the pore radius,and the derivativeof thewater
contentversuspF thereforeappropriatefor visualizingthe
pore-sizedistributionover differentordersof magnitude.
The so-definedpore-sizedensityis relatedto C* by
dO(r) dO(pF) dO(½t) de dO()
d logxor dpF d logo I,l d 1ogo
= [loge (10)]I½1C*.
Thepore-sizedistributionsthatare depictedinFigures
1-11 are calculatedby (1). Accordingly,the shadedarea
betweentwopF valuesindicatesthefractionofporespace
thatdrainswhenthepressureheadischangedbyoneorder
of magnitude.
Numerousmathematicalfunctionshavebeenusedinthe
literatureto express0(½)in analyticalform[Brooksand
Corey,1964;King,1965;Brutsaert,1966;Visser,1968:
Laliberte,1969;FarrelandLarson,1972;Rogowski,1972:
Su andBrooks,1975;Haverkampet al., 1977;Gillham,
1976;Simmonsetal., 1979;D'Hollander,1979;vanGenu.
chten,1980].Mostpopularamongthesefunctionsarethe
equationsofBrooksandCorey[1964](hereinanotationthat
isslightlymodifiedfromtheoriginalversion)'
= >
s, = I 1
and,morerecently,theequationofvanGenuchten[1980]:
Se= [1+ -m,
whereSeistheeffectivesaturation(dimensionless),defined
bySe= (0- 0•)/(0•- 0•),withOsand0•indicatingthe
saturatedandresidualvolumetricwatercontents,respec-
tively,a> 0(centimeters)isascalingfactorthatdetermines
thepositionoftheporesizemaximum,andX,n,andmare
dimensionlesscurve-shapeparameters••bjecttoA>0,m
>0, andn > 1.

DURNER: HYDRAULIC CONDUCTIVITY ESTIMATION FOR SOILS 213
-103
• 8
0.25 ......................................................................................1 •
_10-2 _104 _10ø _10• _102
reducedpressureheadW*
Figure1. Retentioncurve(solidline), first derivative
dSeldlog•* (shadedarea),andsecondderivatived2Se/d
(log•,)2 (dashedline)ofthevanGenuchtenfunctionwith
n = 1.5 and rn = 1 - 1/n. The reducedpressurehead
(dimensionless)is definedby •/a. The shadedarearepre-
sentsthepore-sizedistribution.The positionof maximum
poredensityisat½*= -(I/m)
Equation(2)and(3)caneasilybecombinedwithconduc-
tivitypredictionmodelsto obtainclosed~formexpressions
forK(O) andK(•) [Mualem, 1992].By this, both 0(•) and
K(½)are determinedby the parametersof the retention
curveplus one additionalmatchingfactor to scale the
predictedhydraulicconductivityfunction.BrooksandCorey
[1964]combinedtheirfunctionwith thepredictivemodelof
Burdine[1953]; Campbell [1974] combinedthe samereten-
tionfunctionwith a simplifiedversion of the Childs and
Coilis-George[1950]predictionmodel.The retentionmodel
ofvanGenuchten[1980]is mostfrequentlycoupledwith the
predictivemodelof Mualem [1976].The resultingset of
hydraulicfunctionshas been thoroughlyreviewed by van
Genuchtenand Nielsen [1985], Luckner et al. [1989], and
Nielsenand Luckner [1992]. Kool and Parker [1987a] ex-
tendedit to include hysteresisand air entrapment.
All the above-listed retention functions reflect unimodal
and,if continuouslydifferentiable,smoothnormalto lognor-
malshapedpore-sizedistributions.This is illustratedby
Figure1,wheretheretentionmodelof van Genuchten[1980]
isdepictedtogetherwithitsunderlyingpore-sizedistribution
(shadedarea). As comparedto the Brooks and Corey
thnction,the van Genuchten function has the favorable
propertythat its derivative dO/dpF is continuousand be-
comesasymptoticallyzero toward the fine and largepores.
Theextensionof the pore-sizedistributiontowardthefine
poresisdeterminedbytheproductmn(whichfor a½>> I is
equivalentto • in the Brooks and Corey function), whereas
theextensiontowardthe largeporesis determinedby the
ratiom/n.Thelargerthisratiois,thesteeperisthedecrease
inporedensityfromitsmaximumtowardthelargepores.As
•ill beshownlater, the width of the pore-sizedistribution
towardthelargeporesis of particularimportancefor esti-
matingthe conductivityfunction. Summarizing,the van
Genuchtenretentionmodelinitsgeneralform(3)represents
acontinuous,smooth,unimodal,and bell-shapedpore-size
distributionwiththeparametera primarilydeterminingthe
positionoftheporedensitymaximumandtheparametersm
andndeterminingthewidthtowardthefineandlargepore
sizes.VanGenuchtenandNielsen[1985]comparedvarious
•ater retention models and concluded that (3) fits the water
retentiondataof mostsoilsalmostperfectly.
However, to obtain the conductivity estimates as closed-
form functions,the parametersof the van Genuchtenequa-
tion mustbe subjectedto the additionalconstraintm + l/n
- i, where i is an integer value, for the van Genuchten-
Mualem model generally taken as unity [van Genuchten,
1980].This constraint eliminates someof the flexibility of the
function, since a pore-size distribution that extends far
toward fine pores is then always coupledwith a relatively
broad distribution towards the large pores, and vice versa.
Thisis in fairly goodaccordancewith experimentaldatafor
many soils.There are casesof fine-textured soils, however,
where this constraint attributes a part of the pore spaceto
unrealistically large pores.
There is some discussion whether the van Genuchten
functionis a physicallyrealisticdescriptionof 0(•) overthe
whole moisturerangewhen its parametern lies in the range
1 < n < 2 [Nielsen and Lucknet, 1992]. In this case, the
secondderivative,d20/d•2, becomesinfinitetowardsatu-
0
ration,andtheintegral.f0,• d0forlim•0 isunbounded.
This means that an infinite amount of work were necessary
to extract all water from the soil [Mualern, 1978;Sakellario-
Makrantonaki et al., 1987]. According to our view, for
practicalpurposesthisdoesnotrestricttheusefulnessofthe
van Genuchten function as an empirical model for the
retentioncurve. Nevertheless,it appearsimportantto real-
izethatit expressestheretentioncharacteristicaswellasthe
derivedpropertyC* andtherelatedconductivityestimation
in a purelydescriptivefashion.The coefficientsshouldnot
be interpretedasparametersin a physicalsense.
Water Retention Characteristics of Soils
With HeterogeneousPore Systems
Undisturbed soilsfrequently have pore systemsthat are
differentfrom the unimodal,approximatelynormaldistrib-
utedtypewhichis representedby Figure1. Consequently,
attemptsto fit theirretentiondatawitha simplesigmoidal
curveleadto unsatisfyingresults.A schemeof typicalfitting
errorsisdepictedinFigure2. It appearsthatsuchdeviations
aretacitlyassumedto bewithintheconfidenceintervalof
thecorrespondingmeasuredvalues.If theyare not, such
deviationswould be relevant as will be shown later. This
could be used to define the term "heterogeneous" with
respectto hydrologicalproperties.Wecandesignatea pore
systemasheterogeneousif thepore-sizedistributionof a
representativeelementaryarea[BearandBachmat,1991]
cannotbe correctlydescribedby the van Genuchtenfunc-
tion(oranyotheroftheabovementionedunimodalretention
functions).
Poresystemsthatarenonconformwithasimplesigmoidal
retentioncharacteristicmay be a resultof specificparticle-
size distributionsor be due to the formation of secondary
poresystemsbyvarioussoilgeneticprocesses.Aggregation
processesinloamysoilsmayleadtopore-sizedistributions
thatcausetypicalfittingproblemsin the midporerange,as
illustratedin Figure 2 I,bottomleft) [e.g., Sharma and
Uehara, 1968;Smettemand Kirkby, 1990].Besidesaggre-
gation,biologicalsoil-formingprocessesmayleadtosecond-
aryporesystemsin thelarge-porerangein soilsof any
texture{Figure2 (topleft))[e.g.,Othmeret al., 1991;De
Jongetal., 1992].Morainicsoilsandsolifluctionsoilstend
tohavea significantfractionofporesovera widerangeof
tensions,whichleadsto fittingproblemsas illustratedin
Figure2 (topright)[e.g.,ParkesandWaters,1980;Jacob-

pressure head
Figure 2. Schematictypes of deviationsbetween retention
data of structured soils and fitted unimodal retention curves.
The circles indicate typical trends, which help to identify
structured pore systems even in cases where only few
supporting data are available. (Top left) Increasing devia-
tions between data and fitted curve when approachingsatu-
ration, observed in undisturbed soils of any texture [e.g.,
Othrner et al., 1991;De Jong et al., 1992]. (Top fight) More
or less straight retention characteristic in the wet rangeup to
pF3-4, where the main pore system is located. Typical for
morainic and solifiuction soils [e.g., Schj•nning, 1985; Ja-
cobsen, 1989]. (Bottom left) Different slopes of fitted curve
and data in the midpore range. Typical for aggregatedloams
[e.g., Sharma and Uehara, 1968; Smettem and Kirkby,
1990]. (Bottom right) Considerable change in water content
very close to saturation,experimentally observedin uncon-
solidatedsands(see,for example, King [1965](liquid: soltrol
C), Ragab et al. [1981],and Stephensand Rehfeldt[1985]).
mi) aresubjectto theconditionso•i > 0, mi > O,ni > 1.
Weexplicitlydonotimposetheadditionalconstraintmi +
1/ni= 1. Figure3 showsthesuperpositionoftwohypothet.
ical pore-sizedistributionswhich definea bimodalpore
systemaccordingto (4). The dark-shadedarea canbe
interpretedasfine-texturedpore system,with a maximum
pore-sizedensityaroundpF4 (& r = 0.2/am), whereasthe
light-shadedareacan be seenas a secondaryporesystem
with its maximum density in the range of r = 0.1 mm.
The superpositionof two unimodalporesystemshasbeen
previouslyusedby PetersandKlavetter[1986]to describe
the hydraulicpropertiesof fracturedrock. Othmeretal.
[1991]andRossand Smettem [1993]usedthis approachto
describe the retention characteristics of soils with distinct
secondarypore systems.Pruesset al. [1990]modeledthe
coupledtransportof water, heat, and air in partiallysatu-
ratedporousrockby explicitlyconsideringfractureeffects.
They comparedthe resultswith those obtainedby treating
theporousmediumasa singlecontinuumandderivedsimple
criteriafor the applicabilityof the effective continuumap-
proximation.WangandNarasimhan[1985,1990]addressed
theproblemof findingeffectiveparametervaluesandcom-
paredfracturesin tuff with macroporesin soil.
The multimodal retention model, (4), keeps the functional
propertiesof thebasicvan Genuchtenmodel,but is,dueto
the increased number of coefficients, able to account forthe
deviations discussedabove. It is continuously differentiable,
asymptoticto a zero slopetowardsthe fineandlargepores,
andstronglymonotonicover the whole moisturerange.By
keepingk small,the functionis well-behavedfor interpola-
tion purposes,reducingthe noisein measureddatawhile
followingtheshapeof the measuredcurve.Hampton[1990]
showedthat the useof unconstrainedsplinesfor describing
sen, 1989;Lafolie et al., 1989].With unconsolidatedsands,
retention characteristics of the type in Figure 2 (bottom
right)havebeenobserved,evenin caseswherethiscannot
bedueto boundaryeffectsat thebeginningofdrainage[e.g.,
King, 1965;Ragab et al., 1981;Stephensand Rehfeldt,
1985].In naturalsoils,the schematictypesof Figure2 will
always appearto some degreein combination(compare
examplesof Boelset al. [1978],CurrieandRose[1985],
Dixon [1972],Hantschelet al. [1987],Jeppsonet al. [1975],
Parker and van Genuchten[1985],Puckett et al. [1985],and
Tsuji et al. [1975]).Note that theseheterogeneouspore
systemscanbefoundforfieldaveragesaswellasforsingle
soilsamples[e.g.,AndersonandCassel,1986].
Multimodal Retention Function
Toproperlydescribetheretentioncharacteristicsofsoils
withheterogeneousporesystems,we introducea multimo-
dal retention function which is constructedby a linear
superpositionof subcurvesof the van Genuchtentype
[Durner, 1991, 1992]:
k
Se--Z wi
i=!
1+ (ai!½!)n" '
(4)
wherek is the numberof "subsystems"thatformthetotal
pore-sizedistribution,andwi areweightingfactorsforthe
subcurves,subjectto0 < wi < 1and5'.wi = 1.Asforthe
unimodalcurve,the parametersof the subcurves(ori, ni,
._o 0.8
'• 0.6
ß.= 0.4
• 0.2
0 2 4
pF
0.2 ...........
1500 15 .15
poreradius(I,Lm)
.0015
Figure3. Constructionofamultimodalretentionfunction,
(4).(Top)Bimodalretentioncurve(solidline),unimodal
subcurveforthetexturalporesystem(dottedline,wi =0.6,
al =0.005cm-l nl = 14 mi= 1- 1/ni)andunimodal
subcurvefor a secondaryporesystem(dashedline.w2=
0.4,a2= 0.1cm-1 n2= 2 2 m2= 1- 1/n,).(Bottom)
Pore-sizedistributionsof bimodalfunction(s•lidline),of
texturalporesystem(darkshaded),andof secondarypore
system (light shaded).

pF pF
retentiondatacanleadtoconsiderableproblemsinnumer- 0
icalsimulationsofwatertransportduetotheroughshapeof 1.2 10ø
thewatercapacityfunction.Incontrasttoconstrainedspline
interpolations,whichappearstobethebestalternative•,, 10scheme,(4)requireslesscoefficients.Finallyitscoefficients• 0.8
can,toacertaindegree,beinterpretedinthesamemannerasthebasicvanGenuchtencoefficients,namely,•i=l/oti, 0.4
indicatingthepositionsofporedensitymaxima,andhimi ß •ø•
andni/mi, determiningthewidthoftheunderlyingpore-size 0.2
distributions.However,if theporesystemisnotdistinctly 0
bimodalor multimodal,thecoefficientsof the multimodal
retentionfunctiontendtobehighlycorrelated.It istherefore
importantthattheynotbeconsideredas"parameters"with
aphysicalmeaning;rathertheymustbeseenascurveshape
coefficientslikethoseof anyalternativeinterpolationfunc-
tion.
Themultimodalretentionfunctionis fitted to data by
minimizingtheobjectivefunctionZ:
N
g(P)= • •oill0i- • (½i,P)II (5)
i=1
whereP= {0s,Or,ai,hi,mi,wi}Tistheparametervector,
Nisthenumberof measureddata, ooiaretheweightsfor the
leastsquaresminimization,p.determinestheminimization
norm,andOiandb(½i,P) arethemeasuredandcalculated
watercontentsat pressure½i, respectively.If p. = 2, (5)
resultsin a weightedleast'squaresscheme,which corre-
spondstoamaximumlikelihoodfitundertheassumptionof
independentandnormallydistributeddataerrors.If outliers
aremorefrequent,p.canbe setto unity.In thiscase,(5)
minimizesthe sumof absolutedeviationsbetween measured
watercontentsandthe retention curve. The fitting procedure
weuseis basedon a multidimensionalbracketing method.
Thenumericalprocedureisrobustandunconditionallycon-
vergent,butcomputationallynot very efficient.For each
coefficientPi, thefollowingpropertiesmaybetailoredfor
specificneeds'(1) initial guessof Pi, (2) Pi constantor
optimized,and(3)allowedminimumandmaximumvalueof
Pi.Theoptimizationisterminatedwhentheimprovementof
theobjectivefunctionorthechangesofallparametervalues
arebelowtheir tolerance values between two iteration steps,
(•Z< Sz,Api< epi),orwhenthemaximumnumberof
iterationsisexceeded.(The programis availablefromthe
authoronrequest.)
ConductivityEstimation
Therelativehydraulicconductivityfunctioniscomputed
by numericalevaluation of Mualem's [1976] predictive
modelonbaseof the unimodalor multimodalrepresentation
ofq•(Se):
g--SeLf()]'
(6)
x•heref isgivenby
Se1,is;.
f(Se)-- ½(5;)
(7)
In[6),SJisanempiricalcorrectionfunction,whichallows
foraprioriunknowneffectsofporeconnectivityandtortu-
1õ 1.5 0.01õ 0 0.2 0.4 0.6 0.8
poreradius(I.U'n) effectivesaturation
Figure4. Conductivityestimationfor a structuredpore
system.(Left) Bimodalretentioncurve (solid line) and
underlyingpore-sizedistribution(shadedarea).(Right)Pre-
dictedrelativeconductivityfunctionsK(½) (fromtopleftto
bottomright,relatedtotopaxis)andK(Se) (fromtopfight
to bottomleft, relatedto bottomaxis).The coefficientsare
}t,•= 0.1, a• = 0.2cm-1, - ,
= 0.1, a2 = 0.001cm-l n2 = 22 andm2 = 1 - 1/n,.
osityin Mualem'smodel.Mualem[1976]foundanaverage
valueofr = 0.5tobeoptimalfora setof45samplesoils.The
absoluteconductivityKabs(Se)canbecalculatedbymatch-
ingthepredictedfunctionK(S•) to a matchingvalueKref,
measuredat somereferencesaturationSref, accordingto
Kabs(Se)= (Se/Sref)•'KrefK(Se)' (8)
Results and Discussion
ConductivityEstimationfor HeterogeneousPoreSystems
Forthepurposeofcomparingconductivityestimatesfor
bimodalandmultimodalsoils,we assumeat this pointthat
boththe retentionmeasurementsof the examplesoilsand
thepredictivemodelitselfarefreeoferror.Ofcourse,these
assumptionsare notgenerallyjustifiedandhaveto be
discussedlater in their own right.
InFigure4,thehydraulicpropertiesofahypotheticalsoil
with a narrowtexturalpore-sizedistributionand distinct
secondaryporesystemareshown.Twofindings,whichare
ofgeneralnature,areclearlyillustrated:(1)theshapeofthe
retentioncurveisdirectlyreflectedintheshapeof K(½)and
(2)thesecondaryporesystemincreasestheconductivityby
someordersofmagnitudewhiletakingonlyfewpercentof
porespace.TheparticularshapeofK(0)cannotbereflected
byunimodalmodels.The"similarity"betweentheshapes
of0(½)andK(½)isphysicallymeaningful:atpressureranges
wherethe specificwatercapacityis high,indicatedby a
steepslopeoftheretentionfunction,theconductivitymust
dropsteeplyaswell,sinceduringadrainageprocessalarge
portionoftheporespaceemptiesin thatrange.A steep
decreaseoftheconductivityclosetosaturationistypicalfor
soilswith a secondaryporesystemin the large-porerange.
Compositeconductivityfunctionswithsimilarcharacteris-
ticshavebeenderivedfromexplicitconsiderationof two
poroussystems,e.g.,for fracturedrock[Wangand
Narasimhan,1985;PetersandKlavetter,1988]orforsoils
withmacropores[Othmeretal., 1991;ChenandWagenet,
1991;Chenet al., 1993].
If oneattemptstodescribeabimodalsoilwithsimplified
unimodalhydraulicfunctions,theerrorinvolvedismuch

100
......"' 0 2 pF 4 6...... 0 2 pF4 6
..-..........,. c .... 11.1.:::::::...%I,,,"" //' 0.4 ............ ),
10'••or ?? function
,•o-•/ ,/ A'tr•' • o.•t-........ "•-_I •
.......:....,¾,.....i ,::..,i...............h• 10•
• o '150 1.5 0.015 0.1 0.2 0.3 0.4 0.5
10'4 ix)reradius(!.u'n) watercontent
Figure7. Hydraulicpropertiesof RideauClay Loam[De
Jonget al., 1992].See Figure6 for explanations.
0 0.1 0.2 0.3 0.4 0.5
water content
Figure 5. Schematic representation of the conductivity
calibration problem for bimodal pore system. Line A, con-
ductivity curve of a bimodal pore system; line B, unimodal
predicted curve matched to unsaturated conductivity; line C,
unimodal predicted curve matched to saturated conductiv-
ity; and line D, unimodal predicted curve with optimized
pore interaction factor r.
large[for theconductivityestimationsthanfortheretention
function. Matching the unimodal conductivity function with
the measuredsaturatedconductivity value Ks, asis common
practice,will leadto an overestimationof the unsaturat6d
conductivity in the whole unsaturatedmoisturerange(Fig-
ure 5, curve C). If, on the other hand, the unimodalpredicted
function is matched to an unsaturated conductivity value,
the increaseof hydraulicconductivitynear saturationis not
reflected, thus leading to a gross underestimationof the
soil's conductivity near saturation (Figure 5, curve B).
However, the useof this functionin simulationsof unsatur-
ated water transportleads to a valid descriptionof the
hydraulicsystemas longas saturationis notapproached
[WangandNarashimhan,1985].The double-porositychar-
0 1000.6
0.4
0.2
pF pF
2 4 6 0 2 4 6....
i i 1
'"•,'....-.'[•-unimodal-",, • trimodal
:•-'.,-.,.c:-'x
15 0.15 0.0015 0 0.2 0.4 0.6
poreradius(I.u'n) watercontent
Figure6. Hydraulicpropertiesof SolarVillageClay
[Hampton,1989].(Left) Measuredretentiondata(solid
circles),fittedbimodal(solidline),andunimodal(dashed
line)retentioncurve,andcorrespondingbimodal(black
shaded)andunimodal(lightshaded)pore-sizedistributions.
(Right)Predictedbimodal(solidlines)andunimodal(dashed
lines)conductivitycurves.TheK(½)curves,fromtopleftto
bottomright,arerelatedto thetopaxis.TheK(0) curves,
fromtopfighttobottomleft,arerelatedtothebottomaxis.
The coefficientsof thefunctionsarelistedinTable1.
acteristic of the conductivity function cannot be reflectedby
the unimodal estimation, even if the tortuosity factor• is
manipulated(Figure 5, curve D).
In Figures6, 7, and 8, measuredretention data of three
soils are depicted, representing structured pore systems
which can be frequently found. The chosen examplesare
similar to the schematic soil types described in Figures2
(bottomleft), 2 (top left), and 2 (top right), respectively.In
additionto the experimentaldata, Figures6-8 showtheleast
squaresfittedunimodalandmultimodalretentioncurves,the
underlying pore-size distributions, and the estimatedcon-
ductivity curves. Some physical propertiesof the soilsand
the coefficients of the fitted functions are listed in Table 1.
Unlike the unimodal function, the multimodal functionde-
scribesthemeasureddataof eachsoilaccurately,anditis
obvious that in each case the two functions representfun-
damentally different pore systems.
Figure6 (left)depictsthehydraulicpropertiesofacolumn
packedwith smallclay aggregates[Hampton,1989].The
pore-sizedistributionisdistinctlybimodal,witha maximum
poredensityoftheintra-aggregateporesatr ---0.01/xm,and
a secondarymaximumfortheinteraggregateporesatr --20
/xm.Whereastheprimaryporesystemisdeterminedbythe
particle-sizedistributionofthatsoil,theporespacebetween
theaggregatesdependsonlyonthepackingandistherefore
essentiallyindependentof the soiltexture.Notethatthe
existenceandextentof secondaryporesystemsinundb
turbedsoilsposesa limitationto thesuccesspotential0f
regressionmethodsthatseekto predicthydraulicrelation-
shipssolelyfromparticle-sizedistributions.Figure7(left}
showsdataof anundisturbedcoreof Rideauclayloam[D}
pF
0 2 4
0.5 i i i i
0.4
0.:3
0
15 1.5 0.015
poreradius(llm)
pF
6 0 2 4 610
10•li10•
0 0.1 0.2 0.3 0.4
water content
Figure8. HydraulicpropertiesofROnhaveSandyL0an•
[SchjOnning,1985].SeeFigure6 for explanations.

Table 1. Parametersof Example Soils
Classification
Rideau R0nhave
Solar Vii!age Clay Sandy
Clay* Loam? Loams AggregatedLoamõ
Texture
Type -3
Density,gcm
Porosity,%
Samplevolume,cm3
log(Ks),ms-•
Unimodalvan Genuchten
Os,cm3cm-3
Or,cm3cm-3-1
a, cm
m
clay loam heavy clay coarsesandy
aggregates loam
repacked undisturbed undisturbed
1.3 1.55
52.7 41.3
12.3 345 100/250
-5.7 -4.6 -5.3
0.556 0.439 0.403
0.000 0.000 0.000
0.0328 0.0027 0.0212
1.160 1.134 1.181
0.138 0.!18 0.153
loam aggregates
repacked
0.98
0.63
750-1500
-4.4
Bimodal Bimodal
Multimodal van Genuchten
Trimodal Bimodalll Bimodalô Bimodal**
Os,cm3cm-3 0.556 0.473
Or' cm3cm-3 0.044 0.132
wl 0.54 0.14
-• 0.0252 0.6634
al, cm
nl 2.482 2.060
rnl 0.243 0.515
w2 0.46 0.86
- • O.00006 O.O011Ot2, cm
n2 2.272 1.286
rn2 0.825 0.223
W3
-1
Ot3, ½m
rt3
m2
0.403 0.633 0.633 0.633
0.000 0.160 0.160 0.154
0.21 0.68 0.65 0.62
0.715 0.4422 0.3320 0.2799
1.349 4.675 3.123 2.849
0.210 0.304 0.680 1.0130
0.16 0.32 0.35 0.38
0.0093 0.0119 0.0112 0.0084
1.550 3.471 2.527 1.741
0.355 0.422 0.604 1.000
0.62
0.00017
1.818
0.450
*Source:D. R. Hampton(personalcommunication,1989).
?Source:DeJonget al. [1992].
$Source:SchjOnning[1985].
õSource:Smettemand Kirby [1990].
IlSoconstraint.
ôHere,rni = 1 -- 1/ni.
**Here mi = 1.
Jonget al., 1992].Thepore-sizedistributionis againbi-
modal,but the secondarypore systemis now lesspro-
nouncedandshiftedtowardlargerporesizes,ascompared
tothepreviousqxample.Forthissoilwecandeducethe
existence,but notthe actualshapeof the secondarypore
system,becausethe measurementsare insufficientin the
importantrangenearsaturation.Forthepurposeofanalyz-
ingbimodalconductivityestimationsof suchporesystems
weassumeherethat the fittedbimodalfunctionis a valid
modelofthesoil'strueretentionpropertiesevenintherange
whereno data are available. As a third example, the reten-
tionpropertiesofanundisturbedsampleofR0nhavesandy
10am[SchjCnning,1985]aredepictedinFigure8(left).Ina
strictsense,thisporesystemisunimodal.Nevertheless,it
cannotbeadequatelydescribedbytheclassicalsigmoidal
retentioncurves.Thecontinuous,almostlineardecreaseof
thewatercontentfrompFOtopF3indicatesaconsiderable
poredensityovera widerangeofporesizes,whereasthe
mainporesystemislocatedin therangeof finepores.
Acomparisonoftheconductivityestimatesfromtheuni-
andthemultimodalretentionfunctions(Figures6(right)to8
(right))showsthefollowing.
1. Conductivityestimatesfromuni-andmultimodalde-
scriptionsoftheretentioncurvecandifferbyordersof
magnitude.ThesedifferencesarevisibleinboththeK(½)
and the K(0) plane.
2. Thelargestdifferencesoccurin thewet moisture
range,whichis veryimportantfor solutetransportpro-
cesses.
3. ThedifferencesintheestimatedK(½)curvesarisein
mostcasesfromdifferentslopesnearsaturation.Asthesoil
dries,thecurvesremainapproximatelyparallel(Figure7)or
approacheachotheragain(Figures6and8),providedthe
sameresidualwatercontentisapproachedbytheretention
functions.
Simplifiedsigmoidalretentionfunctionscannotreflectthe
smallbutimportantchangesinwatercontentthattakeplace
inporerangesdistantfromthemainporesystem.The
limitingcaseforthisinabilityisretentionfunctionsofthe
typeof(2),wherebothwatercontentandconductivityare

constant between saturation and the air entry point. Func-
tions of thistype representdiscontinuouspore-sizedistribu-
tions and are by definition not adequatefor conductivity
estimations in loamy and clayey soils.
Whether the conductivity estimation by a nonoptimal
retention function will tend to lead to an overestimation or
an underestimationcan be simply evaluatedby comparing
the slope of the fitted retention curve with the slopeof the
measured data. If the fitted retention curve underestimates
the true water release near saturation (as in Figures 2 (top
left), 2 (top right), 2 (bottomright), 7 (left) and8 (left)), the
predictedchangeofK(½) will aswell be smallerthanthetrue
change.Consequently,if matchedto Ks, thepredictedcurve
results in an overestimation. In studies where conductivity
predictionshave been comparedwith measurements,this
type of predictionerrorhas,indeed,frequentlybeenfound
[e.g., Parkesand Waters, 1980].If, on the otherhand,the
fitted retention curve indicatesa steeperslope than the
measuredwater content data (as in the wet rangeof Figures
2 (bottomleft) and6 (left)),thepredictedchangeofK willbe
larger than the change that were predictedby a more
accurateretentionfunction. Accordingly,thetwo estimated
K(•) curvesof the morainicsoil(Figure8) approacheach
otheragainatdrierconditions,sincetheunimodal0(½)curve
hasa steeperslopethanthemultimodaloneat intermediate
pore sizes.
Usingconductivityestimationsfrommultimodalretention
curvesas an investigationtool, someof the classicalprob-
lems of the predictionmethodswill be discussedin the
followingsections,namely(1) theroleof theparameterOr,
(2)thequestionwhethertheparticularmathematicalformof
the retention function is of relevance, and (3) the sensitivity
of conductivitypredictionson the slopeof the retention
curve near saturation, which will be found to be most
important.
Role of 0r and Os
Throughoutthehistoryof conductivityestimations,the
searchfor a "correct" residualwater content0r hasbeena
continuoussourceof vexation.Statementsonthe roleof Or
for conductivitypredictionsrangefrom"veryimportant"
[Sidiropoulosand Yannopoulos,1988]to "unimportant"
[VogelandCislerova,1988].Partly,thismaybeduetothe
factthatthereis noagreementonthedefinitionand,conse-
quently,thecorrectdeterminationofOr[Sidiroupoulosand
Yannopoulos,1988;Lucknetet al., 1989;Nimmo,1991;
NielsenandLuckner,1992].Mualem[1976],whoconsidered
the correctvalueof the residualwatercontentto be impor-
tantfortheprediction,devoteda considerablepartofhis
classicalpaperto thedeterminationof Or-VanGenuchten
[1980]proposedtosetOrequaltoameasuredwatercontent
at a highsuction.Recentreviews[vanGenuchtenand
Nielsen,1985;NielsenandLuckher,1992]recommendtreat-
ing Orasa purefittingparameter.
From(6)we canseethatin principletheconductivity
predictioninthewetrangeisnotaffectedbythevalueofOr,
sincethehydraulicconductivityofa soilisdeterminedat
anysaturationbyonlyafewpercentofthelargestwater-
filledpores.Durner[1992]confirmedthisindependency
usingwaterretentiondataofa loamfromRoskilde,Den-
mark(similartoFigure8).Contrarytoaunimodalmodel,a
changeof Orfrom0.0to 0.19hadnoinfluenceonthe
conductivitypredictionintherangeoflargepores,sinceit
didnotaltertheshapeofa fittedbimodalretentioncurvein
thatrange.Anyinfluenceof Orinunimodalhydraulicmodels
is thereforecausedby interferencesof the curveshape
parameters.For the vanGenuchtenretentionmodel,asan
example,a decreaseof thevalueof Orcausesa shiftofthe
parametern toward a smallervalue. A similareffectis
causedby an increaseof the parameterOs.Thereforeifthe
interest lies purely in the wet moisture range, we can
disregarddatapointsin thedry rangeif thisleadstoabetter
descriptionin the rangeof interest.
Effectsof parametercorrelationscanbe shownbydataof
Kablan et al. [1989], who comparedconductivitymeasure.
mentsfor a fine sandwith conductivity estimatesbythevan
Genuchten-Mualem model. In the fitting procedureforthe
retentioncurve,the parameterOswas setequalto thelargest
singlemeasuredwater content.Sincefield datawitha
considerablescatter were used, this value was far higher
thanan optimallyfitted value.As a consequence,the0pti.
mizedvaluesof Orand n bothbecametoo small,andthe
predictedconductivityfunctioncorrespondinglyunderesti.
mated the decrease in conductivity at low saturation,thus
leadingto the conclusionthat the predictivemodelworked
well at highsaturations,but failed at low saturations.The
exampleexhibitsa dilemmain fittingproceduresforthe
retention curve. To get a best overall fit, all parameters
shouldbe simultaneouslyoptimized [compareNielsenand
Luckner,1992].An independentdetermination,asisdesir-
ablefor parametersof physicalsignificance,e.g., intrans-
portmodels,ishereinappropriate.Ontheotherhand,ifan
optimizedvalueof Osislowerthatactuallyobservedvalues,
informationrelevantfor K(0) at largeporesmaybemasked
bytheoverallfittingprocedure[Mualem,1992].
Mathematical Formulation of the Retention Function
Asisexpressedby(6),theconductivitypredictionmodels
estimatetherelativeconductivityfunctionpurelyfromthe
shapeoftheretentionfunction.Thuswecanpostulatethat
thespecificmathematicalformofa retentionmodelhasno
influenceon the conductivitypredictionas longasit pro-
videsaproperdescriptionofthemeasureddata.Weverify
thisbydescribingastronglyaggregatedloamwithabimodal
poresystem(databySmettemandKirkby[1990])usingthree
differentretentionmodels,basedonthebimodalformof(4):
(1)theunconstrainedmodelwithvaryingmiandhi,(2)the
constrainedmodelwithmi" 1- l/hi, and(3)asimplified
formwithmi = 1.Forcomparison,wealso•interpolatethe
retentiondatalinearly,withextrapolationsin thedryrange
to watercontentzeroat pF7. To definethepore-size
distributiontowardthelargepores,themeasuredvalueof0
atpFOistakenasOs.Figure9showstheretentioncharac-
teristic,theunderlyingpore-sizedistribution,andthepre-
dictedK(½)curve,asrepresentedbythislinearinterpola-
tion.Thepore-sizedistributionhasaveryruggedshapeand
isnoncontinuous.Suchadescriptionofthehydraulicprop-
ertieswouldcauseproblemsin numericalsimulations
[Hampton, 1990].
Acomparisonofallfourretentionmodelsandtherelated
estimatedconductivityfunctionsisshowninFigure10.The
agreementamongthefourmodelsaswellastheagreement
betweentheestimatedandmeasuredconductivitiesisal-
mostperfect(Figure10(right)).We concludethatany
smoothfunctionwhich describesthe retentiondataaccu-
ratelyappearstobesuitableinthiscase,regardlessofits

DURNER:HYDRAULICCONDUCTIVITYESTIMATIONFORSOILS 219
0
o.4
0.2
pF
,,
i i i i
15oo 150 15 3
poreradius
!.5 0 I 2
pF
1
1oø •_
10'2'• >
I0"•• 0,2
Figure9. Hydraulic properties of AggregatedLoam
[SmettemandKirkby, 1990].(Left) Experimentalretention
data(solid circles), linear interpolation (solid line), and
pore-sizedistribution(shadedarea),asrepresentedby the
linearinterpolated retention curve. (Right) Estimated con-
ductivityfunction.
0 ! 2 8 4 5 0 5
pF
I 2 3 4
pF
10ø
10•-•
10.48
10.• '•
Figure11. Dependenceof conductivityestimationsonthe
asymptotic slope of the retention curve near saturation.
(Left) Retentioncurves-dottedline, Brooksand Corey
function(2) (a-t = 10 cm, A = 0.15);solidline, van
Genuchtenfunction(3),withconstraintrn= 1- 1/n(a- t =
10.5cm, n = 1.15, nm = 0.15); dashedline, vanGenuchten
functionwithoutconstraint(a-i = 10.5cm,n = 1.0, nrn=
0.15). (Right)Predictedconductivitycurves.
mathematicaltype. As will be shownnext, thisis, however,
onlytrueaslongasthe functionsdo really correspond,and
donotdifferin their asymptoticbehaviortoward saturation.
Slopeof the RetentionCurve Near Saturation
Theasymptotic slope of the retention function near satu-
rationis the most sensitive determinant for conductivity
estimationsif smoothretention functions (i.e., functions that
representcontinuouspore-size distributions) are used. This
originatesfrom the use of the Hagen-Poiseuillelaw in the
predictionmodels,which statesthatthe flux densityin pores
ofacertainsizeis inverselyproportionalto thesquareof the
poreradius.Consequently,all statisticalconductivitypre-
dictionmethodsintegrateover the inverseof • [Mualern,
1992].The smallerthe suctioniswhere changesin the water
contenttake place, the greater is its contribution to the
conductivityintegral(e.g., (7)). If retentionfunctionswith a
distinctair entrypointare usedfor the conductivityestima-
tion[Brooksand Corey, 1964;Campbell, 1974],the inte-
pF pF
o 3 o 1 2ß
0.6'
1 2
i i' i i
,
1500 150 15
porerad:us(!ran)
ß5 0 0.2 0.4 0,6
water content
10ø
10'2
10'4
Figure10. Conductivityestimationfor AggregatedLoam
[SmettemandKirkby,1990]fromfourmathematicallydif-
ferentretentionfunctions.(Left)Fittedretentionmodels:
linearinterpolated(solidline),bimodalretentionfunction
(equation(4))withrni allowedtovary(dottedline),mi = 1
-' 1/ni(dashedline),andmi = 1(dashed-dottedline).See
Table1 forcoefficientsof themodels.(Right)Estimated
conductivityfunctionsandmeasuredconductivitydata.The
fourretentionmodelsyieldalmostidenticalestimates.The
agreementbetween estimates and measurements is excel-
lent.
grandin (7), 1/½dS•, is bounded,sincedS[, = 0 between
saturationandtheair entrypressure.Thisisdifferentfor any
retentionfunctionwhich approachessaturationasymptoti-
cally. Now the valueof the integranddependson how fast
dS•, approacheszero as 1/0 approachesinfinity, i.e., on the
slopeof theretentionfunction.The absolutesensitivityof K
to Se, definedby Sabs= dK/dSe, becomesinfinitenear
saturation.For the closed-formconductivityfunction of the
van Genuchten-Mualem model,
g = - - s/m)m]2 (9)
this absolutesensitivityis givenby
2Se
'
Sabs-'rSe ~q- l/m)1-m' (10)(1 --Se
whereg(Se) = [1 - (1 - S•I/")m].Whentheeffective
saturationapproachesunity, the first right-handterm andthe
numeratorof the secondterm in (10) approachfinite values,
whereas the denominator of the second term becomes zero,
becausethe value of the parameter m is boundedby 0 < m
< 1. This causes[$abs[to approachinfinity.Due to this
sensitivity characteristic,even small changesin water con-
tent near saturation(indicatingthe existence of large pores)
are very importantfor the shapeof the estimatedconductiv-
ity function. Measurementsclose to Osmust therefore be
taken at small pressureintervals and as accurately as possi-
ble.
Reliable conductivity estimations become impossible if
measurements near saturation are missing, regardless
whether the pore systemis heterogeneousor not. To illus-
trate this, we have fitted three different unimodal functions
to hypotheticalretentiondata(Figure 11 (left)). The reten-
tion models are the Brooks and Corey function, the van
Genuchten function with m = 1 - l/n, and the van
Genuchten function with n = 1.0. Since there are no
measureddata between saturationand Se = 0.8 at pF2,
eachfunctiondescribesthe dataequally well. However, the
correspondingpore-sizedistributions,whichareidenticalin
the fine-porerange, are very differentin their asymptotic
behaviorin the large-porerange.As was discussedabove,
the Brooks and Corey function(dashedline) restrictsthe

220 DURNER:HYDRAULICCONDUCTIVITYESTIMATIONFORSOILS
sizeof thelargestporesconsideredby theconductivity
predictionmodelto theequivalentradiusoftheairentry
pressure(r --- 0.15 mm in the givenexample).The two
smoothfunctionsrepresentacertainamountofporesinthe
large-porerangeandcanevenconsiderporesizesthatare
physicallyabsurd.Theconductivityestimatesbasedonthe
threefunctionsareshiftedagainsteachotherby ordersof
magnitude, and without additional measurementsbetween
saturationandpF2 thereis noway to decidewhichmodel
represents the soil best.
The accuracyof watercontentmeasurementsisgenerally
limited by the error of the measurementmethod. Unfortu-
nately,it becomesexperimentallymoreandmoredifficultto
measuretheretentioncharacteristicof a soilaccuratelyand
reproduciblyat very low suctions.It isfurtherquestionable
whetherretentioncurves,indeed,representthe pore-size
distributionof a porousmediumat high saturation,where
the nonwettingphaseis notcontinuous[Corey,1992].This
meansthat the conductivityestimationbecomesby defini-
tion inaccuratewhen approachingsaturation,even if we
assumethat the predictive model itself is free of error and its
applicability does not depend on the pore sizes. As a
consequence, conductivity estimationsfrom retention func-
tionsthat representa certainporespaceintherangeof very
smallsuctions(say, belowpFO) are unreliable,regardlessof
whether unimodal or multimodal functions are used. This is,
for example, generally the case if the parametern in the
constrainedvan Genuchten equation(m = 1 - l/n) is close
to unity.
ComparisonsBetweenMeasured and
Estimated Conductivities
To assess whether conductivity estimations from more
accurate retention functions are of advantage in practice,
comparisonsof unimodal and multimodal predictions with
experimental data are necessary.To be of statisticalsignif-
icance, suchcomparisonsmustrest on a broaddatabaseof
well-documented measurements of high quality. Existing
data have frequently been measuredon homogenizedand
packedsoilsamples,anda varietyof measurementmethods
have been used. To date, most of the publishedempirical
validation studies on conductivity estimations used rela-
tively small databasesof the authors' own measurements.
This mightbe onereasonwhy manyof thesestudiesresulted
in a new proposed"optimal" value of the tortuosityor
pore-interactionfactorr [e.g.,Kunzeetal., 1968;Greenand
Corey, 1971;Jackson, 1972; Ghosh, 1977;Ragab et al.,
1981;Talsma, 1985;Alexanderand Skaggs, 1986].
In caseswhere measureddata appear basically reliable,
often the retention measurements near saturation are insuf-
ficient. Then, the hydrologicallyimportantsecondarypore
systemis not strictlydetermined,aswasdiscussedabove.
Therefore estimationsby a flexible retention function are
uncertain, and a better or worse agreementwith measure-
mentsmightbe achievedat random.Durner[1992]showed
for Beit Netofa clay (whichservedvan Genuchten[1980]as
an examplefor possiblefailuresof the van Genuchten-
Mualem model) that a conductivityestimationbased on a
bimodal retentionfunction led to a very goodagreementwith
measureddata. However, since for this soil no retention
meaœurementsare available in the pressurerangebetween
saturationandpF2, the saturatedwatercontentwasarbi-
trarilyset2%largerthanthelargestmeasuredwatercontent.
A differentvaluefor 0• mighthavechangedtheestimation
considerably,leadingtoa badagreement.Note,however,
thatit wasnotpossibleto cometo a satisfactoryagreement
betweenmeasuredandestimatedconductivitiesusingthe
unimodal van Genuchten-Mualem model.
Sinceparticularlyforheterogeneoussoilswithsecondary
poresystemsthe problemof measurementsnearsaturation
is furthercomplicatedby macropores,air encapsulations,
and hysteresiseffects,we do not attempt to compareinthis
paper unimodaland multimodalconductivitypredictions
with measurements.It appearsdoubtfulwhethera validation
attemptin this classicalway is viable with presentlyavail-
able data.
Conclusions
The existenceof a secondarypore systemgreatly affects
the shapeof the estimatedhydraulic conductivitycurvein
the wet range. Estimation differencesdue to differentreten-
tion modelscan be larger than differencesdue to the useof
alternative prediction models [Childs and Coilis-George,
1950; Burdine, 1953; Mualem, 1976]. Consequently,the
ignoranceof a secondarypore system may lead to large
errors. We have shown for soils with heterogeneouspore
systemsthat the commonlyappliedunimodalretentionfunc-
tions are inadequate for conductivity estimations, because
they lack the necessaryflexibility in the rangeof largepores.
Their use may lead to wrong results in flow and transport
simulationsor in parameterestimationprocedureswherethe
inverseproblemis solved[Durner, 1991].Validation studies
of hydraulic conductivity prediction methods shouldbe
reevaluatedif the appliedretention modeldid not accurately
describe the measured data. Particularly for loamy and
clayey soilsthe reasonsfor the frequent failure of conduc-
tivity prediction methods should be reconsidered.In some
casesit may not be the failure of the predictive model,but
the use of inadequate retention functions which leadstoa
discrepancybetweenestimatesandmeasurements.Forsoils
with very wide pore-sizedistributions,unimodalretention
modelsare sometimesforcedto representa nonnegligible
part of the pore spacein the range of unrealisticallylarge
poresizes.In suchcases(indicated,e.g., by thevanGenu-
chtenparametern beingcloseto unity) the conductivity
predictionnear saturationis by definitionunreliable.
The almostperfectagreementbetweenmeasuredand
estimatedconductivitiesfor theloamysoilof Smettemand
Kirkby[1990,Figure10b]indicatesthatconductivitypredic-
tionmethodsbasingonstatisticalpore-domainmodelshave
a high successpotential not only for soils with narrow
pore-sizedistributions,but as well for soilswith heteroge-
neousporesystems.As comparedto unimodalretention
functions,theuseofthemultimodalretentionfunction(orof
anyotherflexibleretentionfunction)leadsto morereliable
conductivityestimatesifit improvesthefitofthemeasured
retentiondataandif theshapeofthefittedretentionfunction
in the rangeoflargeporesis strictlydeterminedby measure-
ments.Particularlyfor validationsof flow and transport
modelsonlaboratoryscale,thehydrauliccharacteristicof
thesoilsampleshouldbedescribedasaccuratelyaspossi-
ble.
Theslopeofthewaterretentioncurvenearsaturationis
themostimportantpropertyindeterminingtheshapeofthe
estimatedconductivityfunction.Evensmalluncertaintiesin

DURNER: HYDRAULIC CONDUCTIVITY ESTIMATION FORSOILS 221
measuredwatercontentsnear saturationlead to largeesti-
mationuncertainties.Whena retentioncharacteristichasno
distinctairentrypoint,the measuredsaturatedconductivity
isgenerallynot an appropriatematchingvaluefor the
predictedconductivityfunction.Thisconclusion,whichwas
alreadystatedfor unimodalporesystems[e.g., van Genu-
chtenandNielsen, 1985;Lucknet et al., 1989;Nielsenand
Lucknet,1992],is even more importantfor heterogeneous
poresystems.Theuseofmultimodalfunctionscanhelpto
explainthisproblem;however,it cannotsolveit.
Wemayexpectthat the basicconceptsthat underliethe
statisticalconductivitypredictionmodelsgenerallyholdless
forsecondarypore systems as compared to textural ones.
Theshapeandorganizationof voidsin aggregatedsoilscan
beverydifferent,dependingon soilgenesisand aggregation
state[WangandNarasimhan, 1990].As an example,a net of
planarvoidswill behavehydrologicallyquite differently
fromthe pore spacebetween sphericalaggregates,or the
porespaceduetomicrorootchannels.Nevertheless,eachof
theseporesystemsmay have a similarly shapedretention
characteristic.Therefore we shouldregard •-in the correc-
tionfunction SJ as a purely empirical fitting parameter
[Roulieret al., 1972; Russo, 1988]. For use in parameter
estimationprocedures, Nielsen and Lucknet [1992] even
suggestto decouple the retention and the conductivity
function,and to use the closed-form conductivity equation
of the van Genuchten-Mualem model (9) as an empirical
expressionfor K(Se). For soils with heterogeneouspore
systems,we expectthe bestresultsby stillusingthe shapeof
theestimatedconductivity function while treating the two
parametersKs and ß as unknowns. By this strategy the
informationon the pore system containedin the retention
functionwill be usedto fullest advantage.Close to saturation
theconductivitiesof soilscontaininglarge pores shouldbe
directlymeasured.This means, however, that the attractive-
nessofthe predictionmethodsis considerablyreduced.
Propertiesof heterogeneouspore systemscannotbe iden-
tifiedbyparameterestimationtechniqueswheretheinverse
problemis solved,if a priori unimodalhydraulicproperties
areassumed.Due to the ill-posednessof the mathematical
problem,it appearsontheotherhandimpossibleto usemore
flexiblehydraulicfunctionsand to identify the increased
numberof coefficientsby the inversemethod.Thereforethe
retentioncurvesof soilswith heterogeneouspore systems
shouldbedirectlyfittedto measureddata.Preferably,these
measurementsare obtainedfrom a transient flow experiment
bysimultaneouslymeasuring½and0inhighresolution[e.g.,
Sobczuketal., 1992].Theflowcharacteristicsof theexper-
imentcanthenbeusedto optimizethetwoconductivity-
relatedunknowns,Ksandr, bysolvingtheinverseproblem.
Finally,wewanttorecallthatthereliabilityofconductiv-
itypredictionmethodsisgenerallyrestrictedbysomefun-
damentalproblems,e.g.,theassumptionof isotropicorga-
nizationofporesindependentoftheirsize,thevalidityofthe
Hagen-Poiseuillelawforflowinlargepores,theproblemof
airentrapment,andtheproblemofhysteresis.Thespecific
findingsinthispaperforheterogeneousporesystemsare
neverthelessapplicable.Theresultsarefurtherindependent
ofthechoiceoftheparticularstatisticalconductivitypredic-
tionmodel.Theextensionto thefieldscalerequiresthe
considerationofspatialandtemporalvariabilityaswellas
theconsiderationofscaleproblemsanderrorsinducedby
themeasurementmethod.Thefurtherdevelopmentofsuit-
ablemeasurementmethodsto identifyheterogeneouspore
systemsappearsthereforeto be of primary importance.
Acknowledgments.The authorthanksDuaneHampton,Kalama-
zoo,Michigan,OleJacobsen,Tinglev,Denmark,andRichardDe
JongandG. ClarkeTopp,Ottawa,Ontario,Canada,whoprovided
thesoilphysicaldataofsomeoftheexamplesoilsinthispaper.The
authorgratefullyacknowledgesthegrantof a long-termfellowship
oftheEuropeanEnvironmentalResearchOrganization(EERO).
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(ReceivedDecember8, 1992;revisedSeptember3, 1993;
acceptedSeptember16, 1993.)

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