Assouline or HydPropReview VZJ 2013

7/23/2019 Assouline or HydPropReview VZJ 2013

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Conceptual and ParametricRepresentaon of Soil HydraulicProperes: A ReviewThe water retenon curve (WRC) and the hydraulic conducvity funcon (HCF) are key ingre-

dients in most analycal and numerical models for ow and transport in unsaturated porous

media. Despite their formal derivaon for a representave elementary volume (REV) of soil

complex pore spaces, these two hydraulic funcons are rooted in pore-scale capillarity and vis-

cous ows that, in turn, are invoked to provide interpretaon of measurements and processes,

such as linking WRC with the more dicult to measure HCF. Numerous conceptual and para-

metric models were proposed for the representaon of processes within soil pore spaces and

inferences concerning the two hydraulic funcons (WRC and HCF) from surrogate variables. We

review some of the primary models and highlight their physical basis, assumpons, advantages,

and limitaons. The rst part focuses on the representaon and modeling of WRC, including

recent advances such as capillarity in angular pores and lm adsorpon and present empirical

models based on easy to measure surrogate properes (pedotransfer funcons). In the second

part, we review the HCF and focus on widely used models that use WRC informaon to pre-

dict the saturated and unsaturated hydraulic conducvity. In the third part, we briey review

issues related to parameter equivalence between models, hysteresis in WRC, and eects of

structural changes on hydraulic funcons. Recent technological advances and monitoring net-

works oer opportunies for extensive hydrological informaon of high quality. The increase

in measurement capabilies highlights the urgent need for building a hierarchy of parametersand model structures suitable for dierent modeling objecves and predicons across spaal

scales. Addionally, the commonly assumed links between WRC and HCF must be reevaluated

and involve more direct measurements of HCF. The modeling of ow and transport through

structured and special porous media may require special funcons and reecng modica-

ons in the governing equaons. Finally, the impact of dynamics and transient processes at

uid interfaces on ow regimes and hydraulic properes necessitate dierent modeling and

representaon strategies beyond the present REV-based framework.

Abbreviaons: BCC, bundle of cylindrical capillaries; HCF, hydraulic conducvity funcon; PSD, parcle sizedistribuon; PTF, pedotransfer funcon; PVD, pore volume distribuon; REV, representave elementaryvolume; WRC, water retenon curve.

Most operaonal models or flow and transport in porous media rely on theconcept o representative elementary volume (REV) (Bear, 1972; Scheidegger, 1974) toacilitate continuum description o water retention and flow in complex pore spaces. Suchrepresentation invokes various simpliying assumptions concerning capillarity and viscousflow within soil pores. Te outcome o such volume averaged pore-scale processes yields theamiliar macroscopic hydraulic unctions such as the soil water retention curve (WRC)and the hydraulic conductivity unction (HCF). Te WRC describes the relationshipbetween soil matric potential (or capillary head),y, and the soil water content, q(expressed

volumetrically or gravimetrically), under equilibrium conditions. Te pioneering worko Buckingham (1907) established the basis or expressing the hydraulic conductivity asa unction o the hydrologic state variables qand y. Te HCF takes on a orm similar to

Darcys saturated hydraulic conductivityKsparameter (a coefficient o proportionality

between flux and hydraulic gradient; Darcy, 1856). Te main difference between Darcysconstant,Ks,and Buckinghams (1907) HCF stems rom unsaturated conditions givingrise to a highly nonlinear HCF that varies with qor y. Te HCF is ofen scaled by Ks,and expressed in a dimensionless orm as Kr(q) = K(q)/KsorKr(y) = K(y)/Ks. Te soil

water content may also be scaled and represented as dimensionless effective saturation, Se,

defined by Se(q) = [(q qr)/(qs qr)], with qsand qrbeing the soil saturated and residual

volumetric water contents, respectively.

Experimental determination o WRC by means o traditional methods (e.g., hangingwater column, pressure membrane apparatus) is laborious and time-consuming, typically

The water retention curve (WRC)

and the hydraulic conducvity func-

on (HCF) are key ingredients in most

analycal and numerical models for

flow and transport in unsaturated

porous media. We review some of

the primary models and highlight

their physical basis, assumptions,

advantages and limitaons.

S. Assouline, The Dep. of EnvironmentalPhysics, Instute of Soil, Water and Envi-ronmental Sciences, A.R.O.- Volcani Center,Bet Dagan 50250, Israel; D. Or, Dep. of Envi -ronmental Sciences (D-UWIS), Instute ofTerrestrial Ecosystems (ITES), Soil and Ter-restrial Environmental Physics (STEP), SwissFederal Instute of Technology (ETH), Zurich,

Switzerland. Contribuon of the Agricul-

tural Research Organizaon, Instute ofSoil, Water and Environmental Sciences, BetDagan, Israel, no. 606/12. *Correspondingauthor ([email protected]).

Vadose Zone J.doi:10.2136/vzj2013.07.0121Received 8 July 2013.

Special Section: VZJ

Anniversary Issue

Shmuel Assouline*Dani Or

Soil Science Society of America5585 Guilford Rd., Madison, WI 53711 USA.All rights reserved. No part of this periodical maybe reproduced or transmied in any form or by anymeans, electronic or mechanical, including pho-tocopying, recording, or any informaon storageand retrieval system, without permission in wringfrom the publisher.


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yielding only a ew pairs o (y, q) values or a soil sample. Teselimitations motivated eorts to relate WRC to other simplerand easy-to-measure surrogate soil properties. Recent measuringdevices based on the evaporation method (Gardner and Miklich,1962; Wind, 1968) give water retention data under laboratory con-ditions almost as a continuous unction (Schindler and Mller,2006; Peters and Durner, 2008a; Schelle et al., 2010). However,the need or continuous representation o the WRC or numericalmodeling provided the motivation or the development o numer-ous mathematical expressions fitting the measured (y, q) data.

Te experimental determination oKris considerably more compli-cated than that o WRC. Hence, or many practical applications,

the HCF is deduced rom WRC inormation (soil pore sizes), ofensupplemented by direct measurements o soil Ks. Most modelsthat link inormation deduced rom WRC to soil HCF estima-tion invariably assume simple pore geometry to enable capillaryand viscous considerations or obtaining mathematically tractableHCF expressions. Among the conceptual models or porous media

proposed over the years, the simplest and most widely used consid-ers soil pores as an assembly o parallel cylindrical capillaries (Fattand Dykstra, 1951). An increment o complexity was added by con-sidering a statistical distribution o capillary radii or the assumedbundle o capillaries (Purcell, 1949; Burdine, 1953). Additionally,

Childs and Collis-George (1950) considered interconnectionsalong the bundle o capillaries that were conceptually cut andrandomly rejoined. More recently, authors have reormulatedthe concepts o Childs and Collis-George (1950) by consideringbundles o capillaries with random pore size distributions that may

vary along each capillar y (Dullien et al ., 1977).

In the ollowing we review conceptual approaches or model-ing and representing soil WRC and HCF starting with modelsor soil WRC in the irst part, ollowed by conceptual andempirical models or soil KsandKr(HCF) in the second part.

We wi ll address herein only the hydrau lic properties at equilib-rium. Dynamic conditions could affect the expression o thesehydraulic unctions (Diamantopoulos and Durner, 2012). Inthe third part o the review, we consider topics related to model

parameter equivalence, representation o WRC hysteresis, effectso structural changes on soil hydraulic unctions, and proper-ties o coarse porous media. Finally, we close with a look at keychallenges and opportunities or improved representation o soil

hydraulic unctions.

6 Funcons for the Soil WaterRetenon Curve

Te most widely used approaches or representing soil WRC couldbe grouped into: (i) empirical or parametric unctions that fit a

wide range o experimental data , (ii) expressions based on soil par-

ticle size distribution, (iii) expressions on ractal representation o

soil pore spaces, and (iv) pedotranser unctions based on simple

soil surrogate variables. A related class o WRC expressions wasderived rom pore-scale water retention in different geometries(uller et al., 1999; Lebeau and Konrad, 2010; Likos and Jaaar,2013) that were upscaled to represent sample scale WRC. However,these geometry-explicit models have not been widely used in prac-

tice and were implemented primarily or the estimation o the soilhydraulic conductivity unction.

Parametric and Empirical Expressions for WRCWe reer to a definitive review o parametric models by Kosugi etal. (2002) and present in the ollowing only the most widely used

expressions representing WRC. One o the simplest parametricmodels or WRC was proposed by ani (1982), it is an exponentialunction with a single fitting parameter:

oe

o

( ) 1S e

y-

y y y = + y

[1.1]

where yo is the capillary head at the WRCs inflection point.

Equation [1.1] was extended by Russo (1988):

o

2

2

eo

( ) 1k

S e

y +-y

y y = + y

[1.2]

wherekis an empirical constant (related to the power lin Eq. [2.17]below). Interestingly, Russo (1988) derived Eq. [1.2] starting roman exponential HCF as discussed in the next section.

Brooks and Corey (1964) proposed a widely used power unction

o yto represent Se(y):

e cc

e c

( ) ;

( ) 1 ;

S

S

-l y y = y < y y

y = y y

[1.3]

Te Brooks and Corey (1964) model contains a discontinuityat the pressure value o ycthat represents the air entry va lue, or

the capillary pressure at which the largest connected pores inthe soil sample are invaded by air. Te dimensionless parameter

lreflects the pore size distribution index (large lvalues corre-spond to sandy soils). Campbell (1974) proposed an expression

similar to Eq. [1.3].

With the advent on numerical solutions or unsaturated flow, the

discontinuity in the WRC derivative at ycintroduced numerical

difficulties. Additionally, evidence suggests that the discontinuityis not present in field-measured WRC data (Milly, 1987). o over-

come this l imitation, Clapp and Hornberger (1978) and Hutsonand Cass (1987) suggested replacing the sharp discontinuity atyc

with a parabolic curve joining the two-part unction.


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Brutsaert (1966) proposed an expression that provides or acontinuous description o WRC or entire range rom Se= 0 toSe= 1:

1

e( ) 1 ( )nS a

- y = + y [1.4]

where aand nare empirical parameters resulting rom best-fitprocedures to data. Brutsaerts (1966) expression did not receivemuch attention until van Genuchten (1980) introduced a similar

expression with a third parameter, m:

e( ) 1 ( )mnS

- y = + a y [1.5]

Compared to the Brooks and Coreys model (Eq. [1.3]), the expres-sions in Eq. [1.4] and [1.5] do not account directly or an air entry

matric potential, but the parameter ais inversely proportional to

the air entry matric potential. Te expression by van Genuchten

(1980) provided good fit to WRC data rom many soils, particu-larly or data near saturation (van Genuchten and Nielsen, 1985),and thus became very popular among modelers. Nimmo (1991)and Ross et al. (1991) ound that the van Genuchten model is suc-

cessul at high and medium water contents, but ofen give poorresults at low water contents. Campbell and Shiozawa (1992) pro-

posed a modification o the van Genuchten model or improving

model fit to dry range data.

Kosugi (1994, 1996) proposed a WRC model that results romrepresenting pore radii distribution based on a three-parameterlognormal distribution and applying the capillary rise equation to

deduce the corresponding capillary head distribution. Te result-ing expression or the WRC is:

2c

c oe c

e c

ln1

( ) erfc ;2 2

( ) 1 ;

S

S

y -y -s y -y y = y < y s

y = y y

[1.6]

his model presents our itting parameters: qr, yc, yo, and adimensionless parameter, s, which is related to the width o the

pore radius distribution. Te model was tested or fify soils andproduced consistent and reasonable results.

Linking Soil Parcle Size Distribuonto Pore Sizes and WRCTe WRC is ofen interpreted as representative o distribution o

pore sizes o a prescribed simple geometry (e.g., cyl inder, sphere).Tese textural pores are the voids resulting rom packing o solidparticles. It is thus attractive to seek quantitative links betweeneasy-to-measure particle size distribution (PSD) and the desired

WRC representing the related pore size distribution. Te proposedmodels have used similarity in the distributions o particle to poresizes, or invoked packing arguments giving rise to the WRC.

Arya and Paris (1981) developed a model to predict the WRC o a

soil rom its PSD, bulk density, and particle density, based on the

assumption that the mean pore radius is proportional to the meanparticle radius. Te pore radii are then converted in volumetricwater content and in equivalent soil water capillary head using theequation o capillarity (y= 14.9 r1or rand yin mm). Model

predictions or seven soil materials show close agreement with theexperimental data. Tis model was urther developed in Arya etal. (1999).

Haverkamp and Parlange (1986) also assumed a linear relation-ship between soil particle diameter and equivalent pore radius to

derive an analytical expression or the WRC. Te application othe model to sandy soils (with no organic matter) yielded reason-able agreement.

Nimmo et al. (2007) relied on the Arya and Paris model to developa property transer model that translates PSD into WRC. Fredlundet al. (2002) suggested a model o the WRC based on the PSD,assuming a constant packing actor independent o grain size toconvert PSD into pore volume distribution (PVD). Mohammadi

and Vanclooster (2011) proposed a model that related PSD toWRC assuming a packing state parameter that characterized eachsize raction in the PSD.

Detailed experimental results (Crisp and Williams, 1971; Glover

and Walker, 2009) paint a different picture than the constant pro-

portionality between PSD and PV D. Resu lts suggest that suchproportionality holds only or mean va lues o particle and poresizes, but not or the entire distribution. Hence, it is not surprisingthat the coefficient o proportionality between particle and poresizes is expected to vary with soil type, mean particle size, and

packing. Even simple geometrical considerations or the packing omonodisperse spheres show that the ratio o inscribed pore diam-

eter, d, to particle diameter,D, varies with packing angle:

(1 cos )

cos

d

D

- J=

J [1.7]

where J is hal-packing angle between centers o neighboringspheres (with d/D= 0.41 or J= 45 corresponding to cubic

packing; and d/D= 0.15 or J= 30 corresponding to tetrahe-dral packing). Analyses o Glover and Walker (2009) show a valueo d/D= 0.29 or packs o spherical particles but a wide rangeo proportionality between particles and pore sizes or differentnatural media. Rouault and Assouline (1998) have shown that or

polydisperse dense packs, linear relationship between particle andpore sizes is inadequate, and, instead, a power unction describes

better that relationship. In short, while particle and pore size


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distributions are linked, the relationships are not straightorward,and ofen the resulting pore size distribution is broader than the

particle size distribution even or monodisperse packs (see Fig. 1reproduced rom Crisp and Williams, 1971; and results o Zhang

et al., 1997; Assouline and Rouault, 1997).

Chan and Govindaraju (2003, 2004) conceptualized a porousmedium composed o polydisperse and lognormally distributedspheres (similar to the assumptions made by Shirazi and Boersma,1984; Buchan, 1989; Shiozawa and Campbell, 1991). Tey appliedconcepts rom orquato (2002) to statistically link PSD or poly-

disperse spherical particles and the resulting WRC with reasonableresults or sand and loamy sand soils.

An interesting aspect concerns the statistical distribution o par-ticles or aggregate sizes in relation to the underlying ragmentation

process in their ormation (Bittelli et al., 1999). Te lognormal dis-tribution o particle sizes results rom breakup process where theragmentation probability is independent o particle size (enchov

and Yanev, 1986). Evidence suggests that in soil, the ragmentation

probability o soil particles or aggregates is proportional to theirsizes (Hadas and Wol, 1984). For a uniorm and random rag-mentation process, the probability distribution unction o particlesizes is an exponential unction (enchov and Yanev, 1986):

a a mina a a

a max a min

( ) 1 exp( ) ; v v

F v v vv v

- = - -e =-

[1.8]

where vais the soil particle/aggregate size constrained between aminimal, va min, and a maximal value, va max. Invoking a powerunction to link between particles and pores size (Assouline andRouault, 1997; Rouault and Assouline, 1998), such as vp= gva

u,the resulting probability distribution unction o the pore sizes is

a Weibull distribution:

p pminp a p

pmax pmin

( ) 1 exp[ ( ) ] ;u v v

F v v vv v

- = - -w =

- [1.9]

where vpis the soil pore size varying between a minimal value,

vp min , and a maximal one, vp max. Assouline et a l. (1998, 2000)applied such approach to model the related WRC according to:

Se(y) = 1 exp[x(|y|-1 |yL|

-1)m]; 0 |y| |yL| [1.10]

where xand mare two fitting parameters, and yLis the capillaryhead corresponding to a very low water content, qL, whichrepresents the limit o interest or a particular WRC application.

Te expression in Eq. [1.10] can improve in some cases the repre-sentation o the WRC at the transition between saturation and

unsaturated conditions. o illustrate the main trends o the expres-sions presented above, the two-parameter models o Brooks andCorey (1964) (Eq. [1.3]) and Assouline et al. (1998) (Eq. [1.10])are fitted to the experimental data o the WRC o a sand (sablede riviere) and a loam (Pachappa loam) as they were reported inMualem (1974a). Te results are depicted in Fig. 2. For the sand,the power unction in Eq. [1.3] seems to fit best the WRC, while,

in the case o the loam, an expression that present an inflectionpoint like Eq. [1.4-1.6] and Eq. [1.10] is in better agreement withthe experimental data.

Hwang (2004) compared the perormances o nine PSD models

and concluded that the model o Fredlund et al. (2000), the modelproposed by Skaggs et al. (2001), and the Weibull distributionbased on the model o Assouline et al. (1998) provided the bestfits to data corresponding to 1385 Korean soils. Cornelis et al.(2005) compared 10 WRC models and concluded that the modelso van Genuchten (1980), Kosugi (1994, 1996), and Assouline etal. (1998) resulted in the best overal l fit or data rom 48 horizons

o 24 soils rom Belgium.

Fig. 1. Links between particle and pore size distributions o sand(modified rom Crisp and Williams, 1971).


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In all the expressions above, one has to determine theWRC residual water content, qr. Te definition o thisvariable remains ambiguous; some consider qras a fitting

parameter without ascribing much physical significance toits value; others have considered this point on the WRC

as the marker or loss o phase continuity that, in turn,affects a wide range o processes rom termination o stage1 evaporation (Lehmann et al., 2008; Or et al., 2013) tothe onset o film flow or HCF (uller and Or, 2001).Regardless o the interpretation o this WRC parameter,

evidence suggests that the value o qr is closely linkedwith soil specific surace area as demonstrated by ullerand Or (2005). In their analyses, uller and Or (2005)

presented evidence or the near-universal behavior o the

WRC dry-end due to dominance o film adsorption (byvan der Walls orces) and thus predictable dependencyo qron soil surace area tested or a wide range o soils.

Wang et al. (2008) extended this relationship and deriveda WRC model based on soil specific surace area and on

the physical and chemical behavior o the water and airphases in the unsaturated porous medium. Te modelperormance was good or the soils tested and improved the rep-resentation o the dry end o the WRC.

Mul-Modal Water Retenon CurvePorous media containing large contrasts in pore sizesor example,pores orming between and within soil aggregates, soils with sig-nificant ractions o large biopores within a finer matrix (roots,earthworms), or in ractured porous media with significant rac-ture void ractioncould be characterized by a bimodal WRC.Zhang and van Genuchten (1994) proposed a model o the WRC

that represents a sigmoidal curve when its our fitting parametersversion is used and describes bimodal curves when the five fitting

parameters version is used.

However, the practical representation o the bimodal WRC couldbe achieved by superposition o two WRC fitted to each o the poredomains. Te challenge o fitting several WRC to such dual porositymedia or multi-modal pore size distributions is relatively minor andofen one ends up with an expression such as (Durner, 1994):

e1

( ) 1 ( ) i

i

k mni i

i

S w-

=

y = + a y [1.11]

wherekis the number o the domains (each with an index i), andwirepresents the weighting actor or the ith sub-WRC correspond-ing to each domain. Tis procedure is illustrated in Fig. 3 or thebimodal case (k= 2). Note, however, that even when the value othe weighting parameter (wi) or a pore subdomain is relativelysmall with minor effect on the apparent multimodal WRC (e.g.,or ractured rock), the impact on the HCF could be significant,and the treatment o the resulting HCF should be consideredcareully. For certain flow conditions, large voids (however small

Fig. 2. Te water retention curve or a sand and a loam soil. Te experimental data(circles) were reported by Mualem (1974a). Te curves correspond to Eq. [1.3](dashed line) and [1.10] (solid line).

Fig. 3. A porous medium characterized by a bimodal particle sizedistribution (upper plot) and the corresponding water retention curve(lower plot) resulting rom Eq. [1.11] with w1= 0.6 and w2= 0.4.


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their raction might be) may dominate the flow regime entirelyas in the case o ractured rock or macropores in soil near satura-tion. Schaap and van Genuchten (2006) and Jarvis (2008) havesuggested models that improve the description o the hydraulicunctions near saturation to account or macroporosity.

Pore Scale Mechanisc WRCAngular Poresand Surface AreaIn addition to general dependency on pore size, liquid retentionand flow dynamics in porous media are greatly influenced by poreshape and angularity. Even cursory inspection shows that soil porespaces do not resemble cylindrical capillaries (assumed in ideal-ized representation o capillar y phenomenon), and most pores are

angular with rough suraces (Li and Wardlaw, 1986; Mason andMorrow, 1991; uller et al., 1999). Angular pore shapes are notonly a more realistic representation o natural pore spaces, but theycapture a richer range o capillary liquid behavior than possiblein the classical bundle o cylindrical capillaries (BCC) model. Akey aspect in angular pores is the possibility o dual occupancy

o liquid and gaseous phases within the same invaded pores thatprovide a r icher depiction o water retention and hydraulic con-tinuity in unsaturated soils (Dullien et a l., 1986; Or and uller,2000; uller and Or, 2002). Tese elements were incorporated in

several pore scale models, such as proposed by uller et al. (1999),

that provided description o water retention within angular porousmedium that accommodate both capillary and adsorptive orcesaffecting water retention. Te basic unit pore is comprised o a

polygon-shaped pore (e.g. , triangle, square) l inke d with inter-nal surace area or adsorption (Fig. 4). Te unit pore is definedby three parameters, the polygonal pore size, L(or capillary

processes), and two dimensionless values, band d(that scale slit-

spacing bL, and slit-length dL), or film adsorption surace area.Te simple geometry allows accounting or different soil texturaland structural classes by adjusting pore width (L) and the propor-

tions o exposed suraces. Te relative saturation o a pore, Sw, withareaAporeor a chemical potential (P) (or capillary pressure,y) is:

2 2

w 2pore pore

( ) ( )( )

r FS F

A A

P g s= = g

P [1.12]

where sis the surace tension o the liquid, and ris the radius ocurvature o the liquidvapor interace that is dependent on the

chemical potentialP=s/rr, or the capillary pressurey=s/r, andF(g) a shape or angularity actor dependent on angularity o thepore cross section only.

Te pore-scale model enabled derivation o hydraulic unctionsrom measurable soil properties (e.g., specific surace area). Tegeometry o the proposed unit pore is simple and tractable andenables upscaling to sample scale using a statistical rameworksimilar to that used or representation o pore radii distributionin the BCC (Laroussi and de Backer, 1979; Kosugi, 1994). Details

o the method are presented in Or and uller (1999). More recently,Likos and Jaaar (2013) extended the treatment to pores orm-ing between sand grains demonstrating reasonable representationor WRC or coarse-textured media based on their geometrically

explicit pore scale model.

Fractals and Percolaon TheorySince the early 1980s with the studies o Burrough (1981), and lateron with the work o yler and Wheatcraf (1989) and subsequentlyRieu and Sposito (1991), many researchers have attempted to har-ness the utility and generality o ractal theory (Mandelbrot, 1967)to represent complex sizes and roughness properties o soil particles(Borkovec et al., 1993) in a compact manner (typically as powerlaw). Such representation was not only motivated by experimentalobservations o particle and aggregate size distributions (reflecting

weathering and ragmentation by soil orming processes), but ofenserving as a stepping stone or compact and general description o

soil pore spaces (yler and Wheatcraf, 1989; Young and Craword,1991; Bartoli et a l., 1991; Perrier et al., 1996; Gimnez et al ., 1997;

Xu and P. Dong , 2004) and scaling in soi ls and heterogeneousporous media (Guadagnini et al., 2013).

Te criticism o Baveye and Boast (1998) notwithstanding, a keystep remains as to how to establish similarity between assumed

Fig. 4. Calculated and measured liquid saturation as a unctiono chemical potential (matric potential) or Hygiene sandstoneemploying a single unit cell with triangular pore cross section andnegligibly small surace area (see uller and Or, 2001).


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pore space ractal dimension,D, and a practical exponent or the

power law representing the pore size distribution and the WRC.

yler and Wheatcraf (1990) obtained a power law expressionor the WRC similar to the Brooks and Corey (1964) (and theCampbell, 1974) model, where the power (2D) is equivalent tolinEq. [1.3] or a porous medium similar to a Sierpinski carpet. Rieu

and Sposito (1991) have shown that starting rom a ractal repre-sentation o aggregation o structured soils yields ractal voids andthus (with appropriate scaling o solid mass and pore spaces; Perrieret al., 1996) recovers the amiliar power unctions or the WRC.Note, however, that a ractal model o soil aggregate sizes maynot necessarily lead to a power unction or the WRC. Moreover,

studies have shown that a power unction o soil WRC does notguarantee a ractal geometry or soil pore spaces (Craword et al.,1995; Bird et al., 1996; Assouline and Rouault, 1997; Rouault andAssouline, 1998).

Pachepsky et al. (1995a) showed that within the ractal ramework

or soil WRC, deviation rom power law representation could beattributed to a multiractal structure o soil pores. Tis implies adependence o the ractal dimension on pore radii. Pachepsky etal. (1995b) have developed a ractal-based WRC or sel-similar

pore sizes with a correction actor,f(r) that accounts or depen-dence o the ractal dimension on pore radii. Representingf(r) by

a lognormal probability distribution unction o the pore radii, theresulting q(y) unction resembles the expression in Eq. [1.6].

More recently, Hunt and coworkers (Hunt and Gee, 2002; Hunt,

2005) expanded the application and interpretation o such ractalpore space models in the context o percolation theor y to capi-

talize on universal traits in the theory or inerences o criticalvolumes or connectivity and critical path to provide insights onkey transport properties (HCF, Diffusion, etc.) as affected by waterretention in such spaces.

Pedotransfer FunconsExperimental challenges in determining WRC limit itsavailability in soil databases. On the other hand, such databases

generally provide inormation on other (easier-to-measure)soil physical and mechanical properties. Hence, empirical andstatistical unctions, c alled pedotranser unctions (PFs), aterm coined by Bouma (1989), were proposed to relate simple

and handy soil properties to WRC (see reviews in Rawls et al.,1991; imlin et al., 1996; Pachepsky and Rawls, 2004; Saxtonand Rawls, 2006; Vereecken et al. , 2011). Most o the variablesound in PFs are (i) oven-dry bulk density or porosity, (ii)organic carbon content; (iii) and soil textu resand, silt, a ndclay content (Gupta and Larson, 1979; Rawls, 1983; Saxtonet al., 1986; Ritchie et al., 1987; Vereecken et al., 1989; Rawlsand Brakensiek, 1989; Pachepsky et al., 1999; Wsten et al.,1999; Weynants et a l., 2009). Pachepsky et al . (1998) ound that

inserting a penetration resistance parameter into PFs improvedthe estimate o WRC based on soil texture and bulk density.

he empiricalcorrelative approach at the basis o the PFoffers reasonable initial estimates or certain large-scale analyses(Romano, 2004). However, the limited physical basis or the esti-mates o WRC and applicability within the range o values used orthe regression analysis, necessitate extra caution or their generalapplication (Chirico et al., 2007). Ofen, data used or the calibra-tion o the PFs are rom specific locations (regions or countries)

and thereore, direct transer to elsewhere may lead to significant

errors. Gijsman et al. (2002) compared the perormances o eight

different PFs and observed significant discrepancy among theresults due to the local nature o the data basis. For the combina-tion between the compared PFs and the database used in thatstudy, the PF proposed in Saxton et al. (1986) appeared to per-orm the best. However, considering different PFs and usingdifferent databases, the conclusions might differ. In the studies o

ietje and apkenhinrichs (1993) and Kern (1995), best peror-

mances were achieved by the PFs in Vereecken et al. (1989) andRawls and Brakensiek (1989), respectively.

6 Modeling the Soil HydraulicConducvity Funcon

The Saturated Hydraulic ConducvityFor viscous and relatively slow flows characteristic o porous media(ofen with Reynolds numberRe< 1), the water flux, q, according

to Darcys law is proportional to the gradient o total hydraulichead, DF/L, expressed as:

sq KL

DF=- [2.1]

whereF=P/rg+zrepresents the total hydraulic head driving theflow across distanceL, andKs[L

1] is a proportionality constantknown as the saturated hydraulic conductivity, also related to the

permeabilityk[L2] o the porous medium (a property o pore spacegeometry) and m[M L11], the dynamic viscosity o the fluid.Te relationship between the saturated hydraulic conductivityKs

and the permeability,k, is given by:

s

gK k

r = m

[2.2]

where r[M L3] is the fluid density, andg[L 2] is accelerationdue to gravity. In groundwater applications, the permeability isofen expressed in units o Darcy (D), leading to the ollowingunit equivalence (or water): 1 D = 1 1012m2?105m s1.

Ofen, the permeability is expressed as a unction o a pore spacecharacteristic length, l, (Scheidegger, 1960):


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2

( )

lk c

f n= [2.3]

wheref(n) is related to porosity, pore shape actor, and tortuosity

o the porous medium o interest, and ca dimensionless constant.

Several characteristic lengths were deduced rom different porerelated attributes (Bernab and Bruderer, 1998), including thehydraulic radius (Kozeny, 1927; Scheidegger, 1960; Johnson andSchwartz, 1989), critical pore radius (Katz and Tompson, 1986;

Arns et al., 20 05), air entry matric potential (Assouline and Or,2008), and mean grain diameter, d(Revil and Cathles, 1999).

An interesting expression was proposed by Revil and Cathles(1999) in which a cementation exponent, mthat varies with pore

space characteristics was introduced:

2 3

24

md nk = [2.4]

Tis and similar expressions establish links between electrical con-ductivity measurements and permeability. Based on the pioneering

work o Archie (1942), the va lues o the cementation exponentwere near 2.0 or consolidated sandstone and 1.3 or loose sand.Lesmes and Friedman (2005) presented a table or coarse mediareporting a similar range (m= 1.3 to 2.0).

Considering the simplest capillary model, that is, a bundle o par-

allel cylindrical capillaries o constant radius r, the permeabilityaccording to Eq. [2.3] is expressed as a unction o porosity, n,according to:

2

2 2

n rk

= [2.5]

A widely used generalization to porous media is known as theKozeny equation (Scheidegger, 1960):

3

2

c nk

S=

t [2.6]

where tis the soil tortuosity, Sis the surace exposed to the fluid

per unit volume, and cis a constant related to the pore shape that

varies between 1/2 or a circle to 2/3 or a strip. A modificationo Eq. [2.6] leads to the well-known KozenyCarman expression

(Carman, 1937):

3

2 2o

1

5 (1 )

nk

S n=

- [2.7]

where Sois surace exposed to luid per unit volume o solidmaterial.

Te concept o permeability o a porous medium can be consideredalso in terms o a flow around submerged bodies, thereore definingrelated drag or riction actors. Te riction actor around a sub-merged sphere is inversely proportional to the Reynolds numbero the flow,Re(Ergun, 1952). Considering steady flow through

porous packs o granular media characterized by Re= {(dVsr)/[m(1 n)]}, with Vsbeing the superficial velocity, and the riction

actor or flow in tubes, a KozenyCarman type o relationshipis obtained:

3s

2 2180

(1 )

V n

L d n

mDF=

- [2.8]

Ofen Kozeny-type expressions tend to overestimatekat low poros-ities (Bernab et a l., 1982); hence, Bourbi et al. (1987) and Zhu etal. (2007) suggested variable power o nin Eq. [2.6] and [2.7], witha value o 3 or large porosity, and a value o 7 to 9 or low porosities.

Te corrections above are linked with the concept o critical or

connected porosity (nc) that was introduced to improve the peror-mances o Kozeny-type equations (Mavko and Nur, 1997; Quispeet al., 2005):

3c

2

( )n nck

S

-=

t [2.9]

Marshall (1958) derived an expression or soil permeability thatrelies on HagenPoiseuilles equation and accounts or the dis-tribution o pore radii in an isotropic material. Tis equationresembles the Kozeny equation in the particular case o a porousmedium with an effective pore radius.

Expressions or soil saturated hydraulic conductivity, Ks, weredeveloped based on the KozenyCarman model, involving a powerrelationship betweenKsand effective soil porosity (Green et al.,2003; Ahuja et al., 1984, 1989a,b):

Ks=aneb [2.10]

where aand bare empirical coefficients and ne is the effectiveporosity, ca lculated as the saturated water content qsminus thewater content at a matric tension o 33 kPa, a va lue usual ly used

to characterize field capacity. Rawls et al. (1998) ound that thepower value, b, can vary between 1.59 and 3.98, and that it could

be related to the pore size distribution index,l , o the Brooks andCorey (1964) model or the WRC (Eq. [1.3]), according to therelationship b = 3 l . Consequently, Eq. [2.10] becomes:

Ks=ane(3l) [2.11]


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Assouline and Or (2008) proposed to consider the air entry matricpotentia l, yc, as a natural characteristic length or the porousmedium, which integrates aspects o pore size and connectivityin a simple ashion (Katz and Tompson, 1986; Arns et a l., 2005).Tey ollowed the analysis o Sisavath et al. (2000) by capitalizing

on the Aissen ormula (Aissen, 1951) that represents complex poreshapes in a relatively simple way (see Fig. 5) to derive the ollowinggeneral expression ork:

2 2c

1 1

( 1)

ck

n=

t + y [2.12]

o account or effects o a broad pore size distribution on estimatesok, an expression was proposed by Wyllie and Sprangler (1952):

r2

c

( )

2

nck

-q l = t l +y [2.13]

where q r is the residual water content and l is the pore size

distribution index characterizing the WRC in the Brooks andCorey model (Eq. [1.3]).

The Unsaturated HydraulicConducvity FunconTree major approaches can be identified in models developed topredict the HCF, namely, the empirical, the macroscopic and the

statistical approaches. A comprehensive review o these differentapproaches can be ound in Mualem (1986) and Brutsaert (2005).

Te empirical approach consists o best-fitting simple mathematicalormulas to available hydraulic conductivity data and unctions

that are integrable and could y ield analytical solutions or prob-lems involving Buckingham-Darcy fluxes. Tese unctions can be

represented by Gardners type HCFs (Gardner, 1958):

r

1( )

1p

K

c

y = y +

[2.14]

G( )r ( )K e

-a yy = [2.15]

where aG, c, andpare fitting parameters. A power unction type

was suggested by Wind (1955) and adapted to the air entry valueycby Brooks and Corey (1964), leading to an equation o Kr(y)similar to the power type orm o Eq. [1.3]. Te major limitation othis approach is the specificity o equations to a range o potentials(Or et al., 2000b) or to specific soil type. Te lack o systematic linkwith WRC limits incorporation o hysteresis into these unctions(especially in theirK(y) orm). Note that the WRC expressionin Eq. [1.2] was derived by Russo (1988) starting rom Gardners(1958) Eq. [2.15] to establish links between parameters o this

part icular HCF (the parameters are relatively easy to measure)and a compatible WRC.

Approaches similar to the statistical representation o soil PSD andthe WRC (as discussed above) were also applied to estimation o

HCF (Kosugi, 1999; Arya et al., 1999; Kosugi et al., 2002; Blanket al., 2008; Arya and Heitman, 2010; Hwang and Hong, 2006;Nasta et al., 2013a). Some o the simplest models or HCF yieldexpressions orKr(Se) relationship in the ollowing general orm:

Kr(Se) = Sem [2.16]

Averjanov (1950) suggested that m= 3.5, very close to the theo-retical value o m= 3.0 derived by Irmay (1954). Tis approachconsiders basically uniorm and parallel capillaries and thusneglects the eect o pore-size distribution on hydraulic con-

ductivity (Childs and Collis-George, 1950). Bresler et al. (1978)ound that m= 7.2 represent relatively well measured data or12 soils reported by Mualem (1974a). Using experimental WRCdata o 50 soils, Mualem (1978) ound that mvaries between 2.5and 24.5. A good correlation was ound between the mvalues othe 50 soils and the corresponding energy per unit volume o soil,

w, required to drain the saturated soil down to the wilting point.Consequently, the derived empirical linear ormula relating mtowincorporates to some extent the effect o pore size distributionin the macroscopic approach. Brooks and Corey (1964) suggestedaccounting or the effect o pore size distribution in the power

value o Eq. [2.16] usinglo Eq. [1.3] by means o the relationship

m= (3 + 2/l). Considering the data o Mualem (1978), Brutsaert(2000) ound that m= (2.18 + 2.51/l) produced a good agreement.Assouline (2005) has proposed that:

m=a[lb+ l(b1)] [2.17]

where the constants aand bcan be considered as empirical con-stants. For the data set used in Assouline (2005),a= 1.40 and b= 0.717.

Fig. 5. A vertical cross section through a pack o modeling clay spheres(5 mm in diameter) arranged in octahedral packing with an inscribedand circumscribed circles used in the Aissen ormulation (Eq. [2.12])are depicted or a typical pore in the cross section (see Assouline andOr, 2008).


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Te oundations or the statistical approach that uses WRC inor-

mation to predict the HCF were apparently established by Purcell(1949), and developed through significant contributions by Childsand Collis-George (1950), Fatt and Dykstra (1951), Burdine(1953), Wyllie and Gardner (1958), Brutsaert (1967), Mualem(1976), and Mualem and Dagan (1978). Te outcome o the sta-tistical approach depends on the conceptual-geometrical modelo the porous medium. For example, the power unction in Eq.[2.16] stems rom considering the porous medium as an assemblyo parallel capillaries tubes having a circular cross section and a

prescribed distribution o radii.

Te statistical approach assumes that the HagenPoiseuille equa-tion is valid at the single pore level. A general orm o the relative

hydraulic conductivity,Kr(Se), can be derived as a unction o the

effective saturation, Se(Mualem and Dagan, 1978; Mualem, 1986;Kosugi et al., 2002):

( ) ( )

( )

d

d

e

0 er e e 1

e

0 e

1

1

eS

l

S

SK S S

S

S

V

n

n

y = y

[2.18]

Te actor Selin Eq. [2.18] is a correction that accounts or the

tortuosity o the flow path (departure rom straight capillaries). Forn = 2, z = 1, and l= 2, one can recognize the model o Burdine(1953), while n= 1, z = 2 and l= 0.5 correspond to the model o

Mualem (1976). In act, the value o the power ldepends on thespecific soil-fluid properties and varies considerably or different

soils. Te value o l= 0.5 suggested by Mualem (1976) was theoptimal value with regards to measured RHC data o the 45 soils

that were considered in that study.

Te models o Burdine (1953) and Mualem (1976) lead to closed-orm analytical expressions when some models o the WRC

presented in the first section are used to express dSe/yin Eq. [2.18].Te expression resulting rom the application o Mualems modelto the WRC expression o Brooks and Corey (1964) [Eq. 1.3] is:

Kr(Se) = Se(2 + 2.5l)/l [2.19]

When Mualems model is applied to the van Genuchten (1980)expression or Se(y) (Eq. [1.5]) imposing m= (1 1/n) with n>1,Kr(Se) becomes:

{ }2

0.5 (1/ )r e e e( ) 1 1

mm

K S S S = - - [2.20]

Te expression in Eq. [1.5] lacks second-order continuity at satu-ration when 1 < n< 2 (Luckner et al., 1989; Vogel et al., 2000;

among others). As a result,Kr(Se) expressed by means o Eq. [2.20]could exhibit a high sensitivity just below Se= 1.0, which mightcause numerical instabilities in simulations o near-saturated infil-tration, especially or 1 < n< 1.3 (Durner, 1994; Schaap and van

Genuchten, 2006). ForcingKr(Se) to remain constant or a small

range o yvalues (ew centimeters) below y= 0 could be enough

to obtain stable numerical simulations (Vogel et al., 2000; Schaapand van Genuchten, 2006).

Applying Mualems model to Assouline et al . (1998) expression o

the WRC (Eq. [1.10]) leads to:

1/

2

1/

r e

1 1 1 1,

( )1 1 1

Le

L

e

K S S- m

- m -ux x g ux - + m m y y = x G + m m y

[2.21]

where g(b,u) and G(u) are the incomplete and the completeGamma unctions, respectively, and u= (|y|1 |yL|

1)m . A

detailed description o its derivation and perormances can beound in Assouline and artakovsky (2001).

It is interesting to notice that Brutsaert (2000) suggested anexpression that resembles the one in Eq. [2.19]:

Kr(Se) = Se(2.5 + 2l)/l [2.22]

In terms o Mualems model, the expression o Brutsaert (2000) is:

e2.5

e

0

0.5r e e 1e

0

d

( ) d

S S

K S S S-

y = y

[2.23]

meaning that l= (-0.5). Tis is in agreement with the results oLeij et al. (1997) who have obtained a mean value o l= (-0.72)when 401 pairs o water retention and unsaturated hydraulic con-

ductivity data were considered. Te model in Eq. [2.17], with n =

1, z = 2 and l= 0.5, that corresponds to Mualems model (1976),perorms the best when compared to the models o Averjanov(1950), Wyllie and Gardner (1958) and Millington and uirk(1961). However, some remarks can be made: (i) the value z = 2

stems rom the assumption that the pore configuration is concep-tually described by a pair o capillary elements whose lengths are

proportional to their radii, which is a very strong constraint and a

restrictive assumption considering pore configuration in soils; (ii)

the assumed constant value o l= 0.5 is in disagreement with the

act that the tortuosity actor strongly depends on soil structureand texture in addition to its dependence on soil moisture content(Pachepsky, 1990).


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Assouline (2001, 2004a) has suggested a different model orKr(Se),which requires less restrictive assumptions and improves the pre-diction oKr(q) when compared with measured data:

e e

0

r e 1e

0

d

( )d

S S

K SS

h y=

y

[2.24]

Following the notation in Eq. [2.18], this model corresponds to n

= 1, z =hand l= 0. Te quadratures in Eq. [2.24] can be evaluatednumerically, using WRC data. For some analyticaly(q) unctions,closed-orm analytical expressions orKr(Se) in Eq. [2.24] do exist.Te resulting expression o the HCF when Brooks and Corey(1964) model (Eq. [1.3]) is used in Eq. [2.24] is:

( )/r e e( )K S S

h+hl l= [2.25]

Te resulting expression o the HCF when van Genuchten(1980) expression (Eq. [1.5] with m= 1 1/n) is used in Eq.(2.24) is:

{ }(1/ )r e e( ) 1 1 mmK S Sh

= - - [2.26]

Te resulting expression o the HCF when Assouline et al.(1998) model (Eq. [1.10]) is used in Eq. [2.24] is:

1/

1/

r e

1 1 1 1,

( )

1 1 1

L

L

e

K S- m

h

- m -xa x g xa - + m m y y =

x G + m m y

[2.27]

Te power hin Eq. [2.24] depends on the soil structure andtexture, which both shape the WRC. Tereore, the power

hwas ound to be highly correlated with lo Brooks andCorey (1964) (Eq. [1.3]), and to the coeffi cient o variation,

e, o the WRC expressed according to Eq. [1.10] (Assouline,2005):

h= 1.40l0.717(r2= 0.84) [2.28]

h= 1.10e-0.624(r2= 0.88) [2.29]

Based on a restricted data set o 10 soils, rom sand to sandy

clay loam, it can be shown that his correlated with so the

model o Kosugi (1994) (Eq. [1.6]:

h= 0.99s-0.82(r2= 0.92) [2.30]

A dependency between hand swas also observed by Nasta et al.(2013b) or optimized hvalues rom 20 soil types showing a linearcorrelation between hand s.

Te perormance o the statistical approach is il lustrated in Fig.6, depicting the fit o Eq. [2.19] and [2.27] to experimental datareported by Mualem (1974a) or the sand and the loam soils that

were already presented in Fig. 2. For the sand, Eq. [2.27] improvesthe agreement between the estimated Kr(Se) and the measureddata, while it seems that this is the opposite or the loam soil. Itis interesting to note that at the WRC level, Eq. [1.3] perormedbetter than Eq. [1.10] or the sand than or the loam. Tese resultsemphasize the complexity o the relationship between the WRCand the HCF, and the limits o the statistical approach inherentto the related conceptualization o the pore space.

Fig. 6. Te hydraulic conductivity unction o a sand and a loam soil expressed interms o Se(upper plots) or y(lower plots). Te experimental data (black dots)were reported by Mualem (1974a). Te curves correspond to Eq. [2.19] (dashedline) and [2.27] (solid line).


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Te PF approach was also applied to correlate between soil struc-ture and texture to soil hydraulic conductivity, both saturated andunsaturated (Campbell, 1985; Vereecken et al., 1990; Cosby et al.,1984; Brakensiek et al., 1984; Wsten, 1997; Wsten et al., 1999;

Saxton et al., 1986; Rawls et al., 1998). Wagner et al. (2001) com-

pared the perormances o some o these expressions and concludedthat the PF proposed by Wsten (1997) provided the best esti-mate oK(y). However, their conclusion was that the discrepancybetween the estimated soil hydraulic conductivity and experimen-tal data indicated that the application o PFs must be perormed

with great caution.

Hydraulic Conducvity Funcon Models Basedon Corner and Film FlowsTe detailed picture o liquid-vapor interace configuration inangular pores and in liquid films (discussed above) and assuming

these liquidvapor interaces remain relatively stable under slowlaminar flow conditions enable introduction o hydrodynamicconsiderations or flow in unsaturated pore spaces. Studies by

Ransohoff and Radke (1988), Blunt and Scher (1995), and others,have used the equilibrium interaces as boundaries or flowingliquid. Tis is a reasonable assumption considering the large vis-cous orces required or breakup or deormation o liquidvaporinteraces (Romero and Yost, 1996), especially at low matric poten-tials. uller and Or (2001) considered viscous flow in a direction

perpendicular to the unit cell cross section (similar to derivations

or the BCC model invoked by Fatt and Dykstra, 1951; Millingtonand uirk, 1961; and Mualem, 1976), and ignored network effectsto obtain estimates o saturated and unsaturated hydraulic con-ductivity or a unit pore. For different filling stages o a unit pore

(determined by matric potential and geometrical attributes) one

may identiy three primary flow regimes to derive HCF: (i) flowin completely filled pore spaces, (ii) corner flow in partially filled

pores and grooves, and (iii) fi lm flow on solid sur aces. Specificsolutions o the NavierStokes equations or the geometry andinteracial conditions could be averaged to yield average cross-sec-

tional velocities or each o these regimes. wo key assumptions are(i) that equilibrium liquidvapor interaces remain stable and (ii)

that flow pathways are parallel. Liquidvapor configurations ordifferent matric potentials are estimated and statistically upscaledto obtain unsaturated hydraulic conductivity rom velocity expres-sions weighted by the appropriate liquid-occupied cross-sectional

areas. An example o such calculations or Hygiene sandstone is

shown in Fig. 7 or the same unit cell presented in Fig. 4.

Application o the procedure or a wide range o soils by uller andOr (2001) highlighted a key eature o this approach, namely, the

consideration o water flow in thin films that is likely to becomeimportant or dry range o WRC and or fine-textured soils with

considerable internal surace area. An example o the HCF tail isshown or Gilat loam (Fig. 8). It is considered an important trans-

port pathway or arid regions and biological activity in dry soils.

More recently, Peters and Durner (2008b) and Lebeau and Konrad(2010) devised more practical approaches or incorporation o thewater flow in films using simplified representation o the WRCrequired as input or HCF estimation.

Fig. 7. Measurements (Mualem, 1976) and estimated hydraulicconductivity unction or Hygiene sandstone deduced rom a singleunit cell with triangular central pore (see uller and Or, 2001).

Fig. 8. Measured (Mualem, 1976) and calculated hydraulic

conductivity unction or Gilat loam using the van GenuchtenMualem model (Eq. [2.20]) and using the corner-film flows model ouller and Or (2001).


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6Special ConsideraonsParameter Equivalence betweenDierent WRC and HCF ModelsTe most widely used empirical models or WRC are

the Brooks and Corey (1964) (Eq. [1.3]) (avored oranalytical solutions o flow problems) and the vanGenuchten (1980) (Eq. [1.5]) model (ofen avoredor numerical schemes). Ability to compare param-eters and results based on use o these two modelsmotivated several studies that proposed correspon-dence between the model parameters (van Genuchten,1980; Morel-Seytoux et al., 1996; Lenhard et al., 1989;Stankovich and Lockington, 1995; Sommer andStckle, 2010). For example, assuming m= 1 1/nin Eq. [1.5], the comparison between Eq. [1.3] and [1.5] or largeyvalues leads to a= yc

-1and n= l + 1 (van Genuchten, 1980).

Morel-Seytoux et al. (1996) suggested parameter equivalence orEq. [1.3] and [1.5] based on the preservation o the capil lary drive

and the asymptotic behavior oy

(q

) at lowq

values. Under suchconditions, ais proportional to yc-1with the product (ayc

-1)or a given soil being dependent onl . Assouline (2005) suggestedequivalences between the parameters in Eq. [1.10] and those inEq. [1.3] and [1.5]. able 1 provides some o the main parameterequivalences that were proposed.

Similarly, parameters equivalences have been proposed in terms othe HCF models. Rucker et al. (2005) and Ghezzehei et al. (2007)introduced a correspondence between the parameter aGo themodel o Gardner (1958) (Eq. [2.15]) and the parameters aandno the Mualemvan Genuchten model (Eq. [2.20]). A concise

relationship resulted rom the study o Ghezzehei et al. (2007) inthe orm o aG= 1.3an.

Te suggested equivalences enable transer o parameters acrossmodels. It increases also the opportunities or applications, reuse

o previously reported data, and comparison o approaches andresults rom previous studies.

Hysteresis in the Water Retenon CurvePore space morphology and interconnectedness coupled withboundary conditions (rate o drainage or imbibition) may resultin different sequences o pore filling or emptying and affect the

resulting phase distribution (Haines, 1930; Poulovassilis, 1962;Vachaud and Tony, 1971; Smiles et al., 1971). Consequently, or

the same porous medium, different WRCs characterize drying or

wetting processes and even or different rates o wetting and drying,as Davidson et al. (1966) observed: Te size o the pressure incre-

ment or redistribution rate will control not only the number opore sequences which fil l and conduct water, but wil l result in adifferent water content distribution within the pore sequences.

A porous medium can be represented by a bivariate statistical distri-bution unction or the capillary head or water filling or emptying,

f(yw, yd), where ywand ydare the capillary head values o fill-ing or emptying , respectively, o a representative volume o the

porous medium. Te resulting hysteretic WRC is composed o

main drying and wetting curves that define the limits o an infinitenumber o interior scanning curves o several orders. Te main andprimary scanning loops are illustrated in Fig. 9, using data reportedby Huang et al. (2005). Te different approaches that have beenapplied to the modeling o hysteresis in WRC include empiricalexpressions, invasionpercolation theory, and independent anddependent domains theory, as reviewed by Pham and Barbour(2005) and Mualem and Beriozkin (2009).

Several empirical models have been proposed (Hanks et al., 1969;Haverkamp et al ., 1977; Scott et al., 1983; Kool and Parker, 1987;Hogarth et a l., 1988; Nimmo, 1992; Huang et al., 2005). Some

o these models are based on a scaling approach assuming shapesimilarity between main and primary curves. I such similarity

Fig. 9. Hysteresis in the water retention curve: (a) dr ying and wettingevents, departing rom the main wetting curve (black triangles). (b)Wetting and drying events, departing rom the main drying curve(black dots). Data are rom Huang et al. (2005).

able 1. Parameters equivalence or some WRC models based on the studies o vanGenuchten (1980), Morel-Seytoux et al . (1996) and Assoul ine (2005).

ModelsModel

parameters

Brooks and Corey (1964)

(Eq. [1.3])

van Genuchten (1980)

(Eq. [1.5])

yc l a n

van Genuchten (1980)(Eq. [1.5])

a a= yc1

n n= l+1

Assouline et al. (1998)(Eq. [1.10])

m m= 1.49 l1.24 11.21

m x = a

m= 0.51n1.1

x x= 1.57 (ycl) -1.21


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exists between wetting curves, it does not exist between dryingones, thus inducing inconsistencies when applied (Mualem and

Beriozkin, 2009). Tese models ailed to reproduce accuratelythe experimental data, especially in the cases where a large por-tion o the hysteretic domain is within the range o the air entry

matric potential.

he domains-based approach pioneered by Nel (1942, 1943)and Everett (1955) offered a quantitative basis or description ohysteresis. First, the independent domain approach assumes thatdomains in the pore space (pore clusters) are filled or emptied inde-

pendently (Poulovassilis , 1962; Parlange, 1976). opp and Miller

(1966) and Everett (1967) introduced interconnections betweenthe independent elements and opened the way to the dependentdomain approach (opp, 1971; Mualem, 1974b; Mualem andDagan, 1975; Mualem and Miller, 1979; Mualem, 1984).

Comparison between the perormances o the different approachesand models can be ound in Viaene et al. (1994), Pham and Barbour

(2005), and Mualem and Beriozkin (2009). Te test perormed byViaene et al. (1994) using 10 soils indicated that models calibratedby both branches o the main hysteresis loop perormed better thanthose calibrated by a single main branch. Mualem and Beriozkin(2009) demonstrated that primary wetting curves o all orderscould be predicted based on sca ling approaches, while accurateestimates o the primary drying curves o all orders has to rely on

the dependent domain theory. Te hysteresis in the WRC reflectsalso in terms o the HCF (Mualem, 1986). Te stronger expressiono the hysteresis in the HCF is when it is expressed as K(y). It ismuch less noticeable when it is expressed asK(q).

Eects of Soil Structural Changeson the WRC and HCFTe review above emphasizes the crucial role o soil structurein shaping the hydraulic unctions. Soil structure can undergodynamic changes at different time scales both in natural envi-ronments and in agricultural fields due to, among other actors,sedimentation, illuviation, consolidation, cycles o wetting anddrying or reezing and thawing, tillage, swelling, and mechani-cal compaction. Tese changes will induce variations in the soilhydraulic properties, thus affecting many aspects o the soilwater

plantatmosphere system. Appropriate quantitative tools withpredictive ability are thereore essential to account or these effects

on agricultural, hydrological and environmental aspects.

Te main approach applied relied on empirical or semi-empiricalmodels considering that changes in soil structure could be wellrepresented by changes in porosity or bulk density (Assouline etal., 1997; Ahuja et al., 1998; Stange and Horn, 2005; Assouline,2006a,b). Tis approach was the premise o a conceptual modelaiming to derive the hydraulic properties o a seal layer develop-ing on the soil surace ollowing the impact o high-kinetic energyraindrops (Mualem and Assouline, 1989; Assouline, 2004b). Or et

al. (2000a) used the FokkerPlanck equation to model soil deor-

mation and coupled it to the probabilistic nature o the PSD todevelop a stochastic model or post-tillage dynamic changes in soilstructure and consequently on soil hydraulic properties.

A large body o studies has investigated the relationships betweenKsand rb(Carman, 1937; Laliberte et al., 1966; Reicovsky et al., 1980;Young and Voorhees, 1982; Mualem and Assouline, 1989; Or et al.,

2000a,b; Or and Ghezzehei, 2002; Green et al., 2003; Assouline,2006b; Berli et al., 2008). A concise review o the dierentapproaches developed to relate the permeability o porous media to

porosity can be ound in Assouline and Or (2008). An interestingresult o that study was that the characteristic length inherent to therelationship between soil structure and permeability could be wellrepresented by the air entry matric potential, yc.

Many soil types, including some o the most ertile soils, containsignificant amounts o active clay minerals that exhibit shrinkswell behavior in response to changes in soil water content and

chemical composition o the soil solution (Warkentin et al ., 1957;Giraldez and Sposito, 1983; Giraldez et al., 1983; uirk, 1986;Revil and Cathles, 1999). Te theory or crystalline and osmoticswelling o clay minerals at the scale o individual clay lamellaeis well established (Derjaguin and Landau, 1941; uirk, 1986;Derjaguin et al., 1987; Israelachvili, 1991; Ohshima, 1995;McBride and Baveye, 2002). However, translation o this behaviorto the prediction o hydraulic properties o swelling soils remains

limited. Changes in pore space attributes induced by the shrinkswell behavior o clay minerals is still presenting a challenge to

predictive modeling o hydraulic properties o clayey soils and oflow and transport processes in soils . uller and Or (2003) capital-

ized on modeling liquid distribution in rigid angular pore spaces(uller and Or, 2001) to provide the ramework or pore-scalegeometrical changes in swelling porous media. Other modelingand measurement rameworks have been proposed or quanti-ying volume changes and soil hydraulic properties (Chertkovand Ravina, 2002; Chertkov, 2012; Garnier et al., 1997; Whiteet al., 2003; among others). A key eature in these models is theneed to combine physicochemical processes with mechanical and

hydrodynamic considerations towards predicting mechanical state,volume changes, and constitutive hydraulic relationships or swell-ing porous media.

The Hydraulic Properes of Coarse Porous MediaEvidence suggests that soil water storage and water flow character-

istics in soils containing appreciable amounts o stones or gravellayers is significantly modified (Fis et al., 2002; okunaga etal., 2002; Unger, 1971; Verbist et al., 2013). For mixed stones ina background o a finer soil matrix, the effect on the hydraulicbehavior may involve hydraulic decoupling o the embedded stonesdue to large pore size contrast, thereby reducing effective wateravailability or plants (Cousin et al., 2003) or reducing averagehydraulic conductivity due to lower-conducting obstacles (Fis et


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al., 2002; Cousin et al., 2003; Coppola et al., 2013). okunaga et

al. (2003) devised methods and gravitational corrections to deter-

mining WRC or gravel layers ound at the Hanord site. Chapuis(2004) discusses methods or estimation o hydraulic conductivityo gravels with emphasis on riction based empirical unctions.Dexter (1993) conducted a series o gravity-driven flows throughbeds o large particles and was able to demonstrate an increase in

flux ocusing with thickness o the bed (and increased in the spatialvariance o outflow). Te explanation was that riv ulets o watertake a random walk and progressively combine with depth (simi-lar to a river network on suraces), and the number o individualrivulets decreases with depth in the gravel bed, pointing to theenhanced potential or preerential flows. Te hydraulic propertieso coarse materials may affect evaporation dynamics rom soils,and a systematic study by Unger (1971) demonstrated significantchanges in evaporative losses and water storage due to gravel layers(similar to effects o artificial mulch).

6

Summary and OutlookAdvances in observation and measurement methods provided newinsights and enhanced understanding o key flow and transport

processes ruled by soil WRC and HCF. Te translation o theseinsights into a broader and more routine parameterization o the

hydraulic unctions characterizing natural soils and porous mediaremains hindered by limited in situ measurements methods andthe omnipresent soil spatial heterogeneity. Yet, mounting pressurerom various modeling communities (hydrology, soil, climate, envi-ronmental regulators) makes the judicious selection o models and

parameters or soil WRC and HCF more critical now than beore.

Researchers are ofen caught in the gap between the growing needor hydrologic parameters and the chronically limited inormationregarding various important quantities. It is not surprising thatsince the seminal paper o van Genuchten (1980), which popular-

ized the link between WRC with HCF, publications reportingdirect measurements o HCF (or surrogate variable such as thesoil water diffusivity) have dropped dramatically in the past ewdecades (only recently, with the renewed interest by the geotech-nical engineering community in unsaturated lows that newmeasurements are being reported). Te reliance on WRC or iner-ence o HCF requires improved conceptual models o capillary and

viscous interactions within realistic pore spaces.

Te parametric models or the WRC reviewed in this study providea range o complexity and reflect various underlying assumptions.

Parameter equivalence enables transer o parameters across models(at least in an approximate ashion) and enriches the opportunitiesor applications and reuse o available data previously reported. Tegrowing popularity o pedotranser models reflects the increasingneed or spatial inormation on hydraulic unctions at resolution

that exceed present measurement capabilities. On the other hand,

with the rapid expansion o ecohydrological observatories (ereno,

CUAHSI) and Earth observing platorms, there is a concurrentincrease in flow o hydrologic inormation that could be used toconstrain WRC and HCF estimates.

At the conceptual level, new and more complex models or soil porespaces have been proposed and tested (Dullien et a l., 1977; uller

et al., 1999). Tese models not only expand the range o geometriesconsidered in representing soil WRC, but also include additional

and underrepresented processes, such as the role o adsorption, dis-joining pressures , complex topolog y, and more, which eed intothe understanding o flow processes, colloidal transport, aquatichabitats or microbial lie, and more.

Te key remaining challenges could be categorized into conceptual,methodological, and scale-related challenges. At the conceptuallevel, there is a growing realization that the bundle o cylindri-cal capillaries must be replaced by more realistic (yet tractable)description o pore spaces and processesadvances in consider-ing retention and flow in films, in angular pores, and expansion

o universal rameworks (percolation theory) offer a promise. Wealso need to expand our efforts judicially using surrogate variables,such as creating better links between particle and pore sizes, andexpand the physical basis or the popular pedotranser unctions.

Te rapid growth in measurement capabilities and deployment o

ecohydrological observatories and networks offer unprecedentedopportunities or extensive data gathering at multiple scales.Nevertheless, without developing a clear plan and methodologies

or quality assurance o derived parameters, the establishment odata sharing protocols, and defining structure and hierarchy orthe inormation derived, the picture may become even more cha-

otic, and the potential o these advances may not be ully realized.Te opportunity or tailoring parameters and data or differentspatial (and temporal) scales offers abilities to describe unsaturatedflow and transport processes at an ever-increasing resolution andquality. Nevertheless, modelers and practitioners must becomemore aware o the inherent limitations o the governing equa-tions and macroscopic representation o WRC and HCF that are

derived rom pore-scale phenomena, yet are applied to very largescales, ofen involving ar more complex flow processes than pos-tulated in their derivation.

AcknowledgmentsTe authors thank Wolgang Durner and an anonymous reviewer or their

constructive comments.

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