Prediction of unsaturated soil hydraulic conductivity

with electrical conductivity

Claude Doussan1 and Stephane Ruy1

Received 24 July 2008; revised 23 June 2009; accepted 7 July 2009; published 7 October 2009.

[1] Soil hydraulic conductivity (K) varies greatly with matric potential (h) and exhibits ahigh variability at the field scale. However, this key property for estimating water flux insoils is difficult to measure. In contrast, soil electrical conductivity (s) is easier to measureand is influenced by the same parameters affecting K. We derive a simple relationshipbetween s and K(h) and test it against laboratory and literature data. Importantly, we showthat parameters of this s-K(h) relationship can be completely determined with accessiblemeasurements of saturated hydraulic conductivity, electrical conductivity of the soilsolution, and clay content. This results in K(h) estimation with a RMSE ranging between0.4 and 0.5 for log K, i.e., of the order of most experimental determinations of K. A furthertest of the s-K(h) relationship on the large UNSODA hydraulic database shows goodagreement and the robustness of the relationship. Such a relation could be useful in thespatial monitoring of water fluxes at the field scale using electrical resistivity tomographyif the s(h) relationship can be obtained.

Citation: Doussan, C., and S. Ruy (2009), Prediction of unsaturated soil hydraulic conductivity with electrical conductivity, Water

Resour. Res., 45, W10408, doi:10.1029/2008WR007309.

1. Introduction

[2] Unsaturated hydraulic conductivity (K), known tovary over orders of magnitude with water content (q) ormatric potential (h) variations, is a key property for estimat-ing water flux in the vadose zone. However, this is also one ofthe most difficult to predict [Revil and Cathles, 1999]. At thefield scale, the high horizontal and vertical variability ofhydraulic conductivity [e.g., Reynolds and Elrick, 1985;Mohanty et al., 1994] requires numerous measurements toget confident predictions of soil water transfer and itsvariability [Warrick et al., 1977].[3] Measurements of unsaturated hydraulic conductivity,

either in the laboratory on core samples or in the field,are time consuming, difficult to set up and error prone,making these measurements impractical to study field vari-ability [Doussan et al., 2002; Slater and Lesmes, 2002]. Tocope with these difficulties, soil hydraulic properties arealso often obtained from pedotransfer functions (PTFs)which are based on more readily available soil propertiessuch as soil granulometry or texture [Wosten et al., 2001].However, the reliability of prediction of PTFs dependsheavily on the data set from which they were derived[Wagner et al., 2001].[4] The use of a proxy variable, easily measurable in the

field or in the laboratory, and tightly linked to the hydraulicconductivity would definitively help in the determination ofthe latter. Electrical conductivity (s) is such a candidatebecause electric conduction (via electrolytes) and fluid flowtake place in the same network of pores and cracks and are

influenced by the same parameters of the porous medium(pore diameters, connectivity, tortuosity). In soil science,soil electrical conductivity has been traditionally used todetermine soil salinity [Rhoades et al., 1990; Corwin andLesch, 2005], to delineate soil units, clay content or hardpanlayer [Samouelian et al., 2005; Weller et al., 2007] and,more recently, to identify soil structure [Besson et al., 2004;Samouelian et al., 2004], to map soil water content [Michotet al., 2003; Zhou et al., 2001]. Strikingly, little or nostudies have been dedicated to the estimation of soilunsaturated hydraulic conductivity with the help of electri-cal resistivity in soil physics [Mualem and Friedman, 1991].This is, however, an old but continuing and challenging areaof research in consolidated, saturated porous medium,mainly in the field of petroleum engineering or hydro-geology [e.g., Katz and Thompson, 1986; Huntley, 1986;Borner et al., 1996; Revil and Cathles, 1999; Purvance andAndricevic, 2000; Binley et al., 2005; Slater, 2007].[5] The objective of this study is to show the usefulness

of electrical conductivity for determining unsaturated soilhydraulic conductivity variations with water potential orwater content using concepts derived from theoreticalstudies on saturated porous materials. With this aim inview, the derived hydraulic-electrical conductivity (s-K)relationship is tested with experimental data that wereobtained by measuring K, h, q, s, on undisturbed soil coresof contrasting textures and equivalent data issued fromliterature. As this relationship (basic approach) may requirenumerous laboratory measurements, we also test two otherderived, simplified approaches (variant approach and sim-plified approach) which need less data and might be appliedeither in the laboratory or in the field. This simplified1EMMAH, UMR 1114, INRA, UAPV, Avignon, France.

Copyright 2009 by the American Geophysical Union.0043-1397/09/2008WR007309$09.00

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approach is validated on a large hydraulic database toexemplify its potential use and robustness.

2. Theoretical Background

2.1. Electrical Conductivity of Soil

[6] The electrical conductivity of soil can be formulatedas a complex quantity s*:

s* ¼ s0 þ is00 ð1Þ

where i =ffiffiffiffiffiffiffi�1p

. Both the real (s0) and imaginary part (s00) ofelectrical conductivity may vary with frequency of thecurrent in natural geological media. The classical inter-pretation of this variation of conductivity with frequency isthat current conduction occurs in two parallel pathways:through an electrolytic (water) pathway and through aninterfacial (surface of particles) pathway [Slater, 2007;Ghorbani et al., 2008]. The in-phase (real part, s0)conductivity is attributed to ionic conduction in the poresand ionic migration in the electrical double layer (EDL) thatdevelops at the fluid-grain interface. The quadrature(imaginary part, s00) conductivity results from polarization(local accumulation of charges) that occurs with more orless rapid redistribution of charge within the EDL or fromdifferences in conductivity-polarizability of the soil grainmixture producing charge accumulation at the interfaces.Determination of frequency dependence of s00 (and s0) inrelation with porous medium composition or properties,such as hydraulic conductivity, is an active present field ofresearch [Cosenza et al., 2007; Binley et al., 2005; Davis etal., 2006; Binley and Kemna, 2005; Slater, 2007].[7] At saturation, in-phase conduction (s0) can be decom-

posed into within-pore (spor) and surface (sint) parallelconduction pathways. This can be written as [Waxman andSmits, 1968; Rhoades et al., 1990; Mualem and Friedman,1991; Slater, 2007]

s0 ¼ spor þ sint ¼ 1=Fsatð Þsw þ sint ð2Þ

where sw is the electrical conductivity of the soil solutionand Fsat is the electrical formation factor at saturation (Fsat =sw/s

0 when sint = 0). Fsat is interpreted as a dimensionlessscale invariant parameter characterizing pore space topol-ogy (function of porosity and pore connectivity) and 1/Fsat

is a measure of the effective interconnected porosity [Reviland Cathles, 1999].[8] Variation of s0 with water content has been repre-

sented with more or less semiempirical models [Archie,1942; Waxman and Smits, 1968; Bussian, 1983; Rhoades etal., 1990; Mualem and Friedman, 1991; Ewing and Hunt,2006]. The Waxman and Smits [1968] (WS) model isclassically used in geological media containing clay andexhibiting significant surface conduction (sint) and has beenverified recently with synthetic pore-scale modeling ofshaly sand [Devarajan et al., 2006]. It can be summarizedwith

s0 Sð Þ ¼ Sn

Fsat

sw þss

S

� �ð3Þ

where S is the water saturation ( = q/qsat), qsat being thewater content at saturation, ss is the surface conductivity

associated with counter ions of the EDL and sint = ss/(SFsat)in equation (2). In the following, we will use the electrical

resistivity ratio F =sw

s0or F =

sw

spor

, for the ratio between soil

water and bulk soil (or within-pore) electrical conductivity,function of soil water content or soil matric potential (seesection 2.2).

2.2. Hydraulic-Electrical Conductivity Relationships

[9] Thorough reviews of hydraulic-electrical conductivityrelationships, mainly for saturated porous media, can befound in work by Lesmes and Friedman [2005] and Slater[2007]. Briefly, most of these relationships have beenderived on the basis of two main kinds of approaches.[10] The first is the use of the capillary tube analogy of

porous medium and the derived Kozeny-Carman (KC) typeequation for permeability. Electrical parameters are used inthis analogy on the premise that they represent effectiveparameters of the pore space, controlling flow, rather thantotal volume parameters (for example, F is analogous to theinterconnected effective porosity/tortuosity). An example isgiven by the equation derived by Revil and Cathles [1999],which gives accurate results in the case of sand or sedi-mentary rocks of low clay content:

k ¼ d2f3m

24ð4Þ

where d is the mean grain diameter, f is the porosity, m isthe cementation factor and k is the intrinsic permeabilityrelated to saturated hydraulic conductivity Ksat by

Ksat ¼ krwgm

ð5Þ

with rw and m respectively the fluid density and dynamicviscosity, g the gravitational acceleration. More recently, anexperimental power law correlation has been shown ondifferent consolidated and unconsolidated sedimentarymaterials between the surface area per unit pore volumeSpor, commonly used as a measure of the hydraulic radius inthe KC equation, and the quadrature conductivity s00 at lowfrequency. This leads to expression of Ksat in terms ofelectrical parameters only:

Ksat ¼a

Fsat � s00cð6Þ

where a and c are constants depending on the porousmedium. However, some recent results tend to show that thepower law relationship between Spor and s00 could be not sostrong [Slater and Lesmes, 2002; Slater, 2007].[11] The second approach, in the prediction of K with

electrical parameters comes from application of percolationtheory to porous medium for fluid and current flow [Katzand Thompson, 1986; Thompson et al., 1987], stating thattransport in a disordered system with a broad distribution ofconductance is related to conductances exceeding a thresh-old value. This threshold value is the largest conductancesuch that the set of connecting conductances forms aninfinite percolating cluster, which is the same for fluidand current flow. This results in the equation

k ¼ 1

226

l2cFsat

ð7Þ

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where lc is the characteristic length scale (associated to apore diameter) of the threshold conductance, Fsat is theformation factor. lc was estimated with mercury porosimetryby the equivalent pore diameter of the pressure correspond-ing to the inflection point in the pressure–injected mercuryvolume curve. This equation proved to be accurate for alarge set of sandstone samples. Thompson et al. [1987]reported agreement with experimentally determined valuesof permeability to be within a factor of two for mostsamples and a factor of three for all samples. Moreover,based on dimensional analysis arguments [Guyon et al.,2001], statistical and percolation concepts [Katz andThompson, 1986; Gueguen and Dienes, 1989], semiempi-rical description of pore space with capillary bundles [Walshand Brace, 1984] or physical arguments, such as equalitybetween Joule dissipation at the macroscopic level and inthe pore space [Johnson et al., 1986], numerous differenttheoretical formulations have shown that hydraulic con-ductivity is proportional to the square of pore diameter andinversely related to the formation factor (i.e., (7) can be

generalized as k = c � l2 � 1

Fsat

, where l is a characteristic pore

diameter and c a constant).[12] Considering water relations, unsaturated soils are

characterized by the water content (extensive variable)and the matric potential (intensive variable). The latterresults from the interactive capillary and adsorptive forcesbetween the water and the soil matrix [Hillel, 1998],analogous to capillary pressure. It represents the pressure(or energy per unit volume) which is needed to be applied tothe soil to extract the soil water at state of free water. It isthus negative relative to atmospheric pressure and decreasesmore or less abruptly with decrease of saturation (waterretention curve: Figure 1).[13] If the matric potential of the soil is h (expressed as a

matric head m < 0), then the maximum correspondingequivalent water filled pore diameter d is (capillary equation)

d ¼ 2 g cosarw g hj j ð8Þ

where g is the surface tension of water (7.2 � 10�2 Nm�1 at 25�C), a is the contact angle between water and

soil (�0 for soils), d is the diameter of the largest filledpore (in contact with air) at the matric potential h. At 20�C,

(8) results in d � 29

hj j, where h is in m and d is in mm.

[14] In this analysis, we hypothesize that equation (7), orits more general form, holds for unsaturated porous media;that is, the unsaturated hydraulic conductivity still dependson a critical pore diameter, defining the largest filled poreenabling a connected percolating cluster and that this largestfilled pore can be derived from the matric potential by (8).Thus, we test in the following if lc = d or lc = a.db (i.e.,lc scales as a power law of d with a and b constants) inconjunction with (7) can describe the unsaturated hydrau-lic conductivity curve of soils. This leads to considerationof the following.

For lc = d

K hð Þ ¼ A

226� B � 1

F hð Þ � 29:10�6� �2� hj j�2 ð9Þ

For lc = a.db

K hð Þ ¼ A

226� B0 � 1

F hð Þ � 29:10�6� �2b� hj j�2b ð10Þ

where A =rwgm

is the fluid scale factor between intrinsic

permeability (k�m2) and hydraulic conductivity (K�m/s),A � 9.77 106 for water (at 20�C), and F(h) is the ratiobetween soil water and bulk soil electrical conductivity forunsaturated condition as a function of matric potential h (orof soil saturation S). B and B0(= B. a2) are scale parametersrelating K(h) to Ksat and h is expressed in hydraulic headunits (m).[15] Equations (9) and (10) diverge when h tends to zero,

i.e., when the soil becomes saturated. Indeed, betweensaturation and the air entry point of the soil, h decreasesbut the soil stays saturated, and thus K is equal to Ksat. Toaccount for this physical process, the following condition isadded:

K hð Þ ¼ Ksat if K hð Þcalculated > Ksat ð11Þ

where K(h)calculated is obtained from either (9) or (10).[16] The electrical resistivity ratio F to be used here (in

equation (9) or equation (10)) should theoretically be F =sw/spor, i.e., the within-pore electrical conductivity becausewater flow takes place in the interconnected pore network.spor can be estimated from the fit of WS equation (3) bymeasurement of s0(S) at different saturation and salinitiesand setting ss = 0 in equation (3). Approximation of F inthis way will be used in what we call below the ‘‘basic’’and ‘‘variant’’ approaches. However, as it will be shownbelow, in a more simple way, measured bulk soil realelectrical conductivity can also be used to approximate Fand F = sw/s

0 in this case. This will be used in what wecall below the ‘‘simplified’’ approach (Figure 2).

3. Material and Methods

3.1. Soil Samples and Sampling

[17] Laboratory experiments were done with three con-trasted soils: a silty clay loam (Avignon, France), a loam

Figure 1. Water retention curve of the three studied soils.

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(Collias, France) and a sand (Fontainebleau, France). Table 1shows some properties of these soils. Triplicate undisturbedcores of the silty clay loam and the loam soils werecollected in the topsoil of untilled lysimeters with PVCrings 3 cm high, 10 cm inner diameter, at INRA experi-mental station (Avignon, France). These cores were used forelectrical measurements. An additional core 7 cm high,15 cm inner diameter was taken for determination ofunsaturated hydraulic conductivity in the laboratory. Thesand is a pure silica sand and was packed/vibrated underwater in the cylinders.[18] The few data from literature that we could find,

containing hydraulic conductivity-electrical measurements,were also used in this analysis: a sandy loam (Columbia)and a fine sand (Oso Flaco) from Tuli and Hopmans [2004]and a dune sand (Tottori sand) from Inoue et al. [2000].

3.2. Hydraulic Measurements

[19] Unsaturated hydraulic conductivity was determinedin the laboratory with Wind’s evaporation method [Tamariet al., 1993] and steady state drip infiltration at low suction.Saturated hydraulic conductivity (Ksat) was measured with apermeameter by the constant head method. The retentioncurve was determined with suction tables and pressureplates in the course of electrical measurements. Matricpotential heads lower than �150 m were reached by lettingthe samples dry. In this case, water content was convertedinto matric potential by using data from Chanzy [1991] forthe same soils (vapor equilibrium of soil samples with salinesolutions). The retention curve of the 3 studied soils ispresented Figure 1.

Figure 2. Flowchart for estimating unsaturated hydraulic conductivity (K) from electrical conductivityrequiring more or less experimental data. The basic approach is the most demanding, while the simplifiedapproach relies only on bulk electrical s0 conductivity measurements, without Waxman and Smits [1968]model fitting.

Table 1. Physical Properties of the Three Investigated Soils

Soil

Texture

qsat(m3 m�3)

Ksat

(m s�1)D10

(mm)D50

(mm)Percent Sand,50–2000 mm

Percent Silt,2–50 mm

Percent Clay<2 mm

Avignon siltyclay loam

15.7 51 33.3 0.38 4.07 � 10�5 0.9 6.8

Collias loam 37.7 48.7 13.6 0.44 1.86 � 10�5 2 31.8Fontainebleausand

99.5 0.3 0.2 0.36 3.09 � 10�4 181.9 256.3

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3.3. Electrical Measurements

[20] Measurement of electrical conductance of soil wasdone with a pair of electrodes made of stainless steelperforated discs of the same diameter as the soil samples[Jouniaux et al., 2006; Revil et al., 2002]. One electrodewas glued at the bottom of the sample and also served as asupport for the soil sample. The second electrode wasadjusted, gently pressed, at the top the sample. Fine soilwas added on grids to maximize the electrical soil-electrodecontact. Electrodes and fine soil were placed on the moistsamples ensuring maximum adherence between soil andelectrodes. Electrodes were left in place and adhered to soilduring the whole measurement sequence. This allowed forthe measurement of the mean electrical conductivity of thewhole sample. Prior to any measurements, the soil sampleswere saturated and immersed in CaCl2 (2H2O) solution andleft to equilibriate with soil solution for one week. Thesolution was renewed and checked for conductivity untilequilibrium was attained (about one week more). Imposedelectrical conductivity of the soil solution for silty clay loamand loam was 7.14 10�2 S m�1 and 6.5 10�2 S m�1 for thesand. Other conductivities of the soil solution were alsotested to verify the consistency of the Waxman and Smitsmodel to data (5 10�3 to 3.3 S m�1). Electrical measure-ments were done with soil samples installed in the suctiontable or the pressure chamber in order to reach the desiredmatric potentials, from saturation to �150 m matric poten-tial head [Rhoades et al., 1976]. Additional measurementswere performed for matric potential lower than �150 m byair drying the samples.[21] Electrical conductance of the bulk soil was deter-

mined with a Hioki 3522 LCR meter piloted by a PC. Thissetup can record magnitude and phase variation of complexconductance with frequency (from 0.03 to 105 Hz here). TheLCR meter was compensated for open and short circuit.Accuracy of the LCR meter is better than 10�1 W onresistance and 1 mrad on phase measurements. Measuredelectrical conductance [S, S] was converted to conductivity[s, S m�1] with s = gS, where g is a geometrical factor. In

this simple electrode setting, g =l

Ssurf; where l is the height

and Ssurf is the surface area of the sample.[22] In this two electrodes measurement setup, electrode

polarization can affect conductivity measurements at lowfrequency. Electrical measurements were done in the 0.1 Hzto 100 kHz frequency range and impedance response of thesamples showed a quasi purely resistive behavior, negligiblecapacitive effects, in the 1–10 kHz frequency range (phaseangle < 1.2�). The in-phase conductivity measurements (s0)were selected when the quadrature conductivity (s00) wasminimal in this frequency range (Figure 3). Complementarymeasurements with resistor components showed no capac-itive coupling of the measuring device and setting in thisfrequency range (phase angle <0.04�). Resistivity measuredin the kHz range may induce about 4% variation comparedto resistivity measured in the 1–10 Hz range [Binley et al.,2005]. Impact of this variation can be neglected in theresults presented below. Finally, an experimental check ofthis two electrodes setup was also done with a four electrodearray (Wenner array, 1.5 cm spacing) on sand samples withdifferent water saturations. Relative differences in measuredconductivities between the two setups ranged between 2 and12% at most, confirming that the two electrodes measure-ments are representative of the sample conductivity. Theelectrical conductivity was temperature corrected and isexpressed at 25�C.

3.4. Mathematical Treatments

[23] Linear, nonlinear fitting of equations and statisticaltreatments were done with the R software [R Develop-ment Core Team, 2006]. Model accuracy is estimated with

the root-mean-square error (RMSE)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNi¼1

logKi � log Kið Þ2

N

s,

where K and K are respectively the measured and calculatedvalues of the hydraulic conductivity and N is the number of(K, K) pairs. Data from graphs in the literature were digitizedfrom documents for further treatments with Engauge Digi-tizer free software (http://digitizer.sourceforge.net/).

4. Results

4.1. Water Potential– and Water Content–ElectricalConductivity Relationships

[24] Variation of electrical conductivity (s0) with waterpotential is shown in Figure 4. Almost log linear decreasesof s0 are observed over ranges of matric water potentialdepending on the soil type. For matric water potential lowerthan �150 to �160 m, electrical conductivity of the siltyclay loam and of the loam soils converge to the same value,with a decrease in the slope of the log linear relationship.This can be linked to the fact that for matric potential lowerthan �150 m, a great part of the water may occur in theform of film over the solid surface [Hillel, 1998]. In the caseof sand, very low values of s0 are found for water potentiallower than 1 m.[25] Variation of s0 with water content, together with fit to

the WS equation (equation (3)) are shown in Figure 5. WSequation adequately fit the data (see parameters in Table 2),except for the silty clay loam soil, when saturation is lowerthan � 0.56, i.e., lower than ��170 m matric head. In thiscase, the WS equation underestimates s0. The contrary

Figure 3. Example of the variation of soil complexelectrical conductivity, real and imaginary parts, withcurrent frequency for the experimental setting used. Caseof the sand submitted to a matric potential h = �0.21 m.Polarization of electrodes occurs in the Hz range, while inthe kHz range, the sample behavior is nearly purelyresistive.

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results for the loam when saturation is lower than �0.05.These deviations of the WS model seem related to thechange of slope in the s0–water potential relationshipobserved in Figure 4, and possibly to the change of waterconformation (from filled pores to film over solid surface) atlow matric potential (<�150 m) as denoted above. However,the WS model represents the experimental data quitesatisfactorily in the range of experimental measurementsof hydraulic conductivity (from saturation to �10, �50 mmatric potential) or in the range of matric potential classi-cally used in soil physics (0 to �150 m). Table 2 empha-sizes the influence of surface electrical conductance fornatural soils, particularly for the loam and silty clay loamsoils, where ss is 0.11 and 0.35 S m�1, respectively. Thesevalues are in the same range as values determined for shalysandstone, i.e., 0.1–0.5 S m�1 [Taylor and Barker, 2006] orin soils [Nadler and Frenkel, 1980; Shainberg et al., 1980].

4.2. Surface Electrical Conductivity–TextureRelationship

[26] Except for sand, the possible high value of surfaceconductivity for soils highly impacts on the measured

values of the bulk soil conductivity (s0). As the value ofthe within-pore conductance (spor in equation (2)) would beof importance in the relation with hydraulic conductivity, afirst estimation of the relative influence of ss is needed.Surface conductance is related to cation exchange capacity[Waxman and Smits, 1968] and, thus, the clay content of thesoil [Rhoades et al., 1989]. We examined such a relation-ship with published data in the literature for experimentsshowing variations of soil electrical conductivity with watercontent for different fluid conductivities. These data werefitted to the WS equation (equation (3)) to obtain ss for awide range of soils and textures (13 soils, clay content from0 to 36%, silty clay loam to sand textures). Figure 6 showsthat, except one soil, ss does correlate with the ratio Clay/(Sand + Silt), resulting in

ss ¼ 0:654Clay

Sand þ Siltþ 0:018; r2 ¼ 0:968 ð12Þ

[27] Figure 7 shows the relationship between surfaceconductance obtained from WS fit and those estimated fromRhoades et al. [1989] regression (ss = 0.023%Clay �0.0209). ss from the Rhoades equation is almost twice theWS value. However, Rhoades regression was based on aslightly different s (S) relationship than WS. Clearly fromFigure 7, introduction of the Clay/(Sand + Silt) ratio enablesa better estimation of surface conductance with clay contentlower than 6% than Rhoades regression.

Figure 5. Variation of the electrical conductivity (real part) of the three studied soils with watersaturation. Symbols are measured data, and lines are the best fit to the Waxman-Smits model [Waxmanand Smits, 1968] (equation (3) in the text). Arrows denote water saturation at which matric potential hreaches �150 m.

Figure 4. Variation of the electrical conductivity (realpart) of the three studied soils with matric water potential ofthe soil sample. Bars represent ±1standard deviation oftriplicate samples.

Table 2. Fitted Parameters of the Waxman and Smits Equation

Relating Soil Electrical Conductivity and Water Saturation for the

Three Soils Investigated in This Studya

Fsat n ss (S/m)

Avignon silty clay loam 5.48 5.96 0.359Collias loam 4.54 1.88 0.109Fontainebleau sand 4.62 2.58 0.036

aSee equation (3).

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4.3. Hydraulic-Electrical Conductivity Relationship

[28] The electrical resistivity ratio F was estimated eitherwith the calculated within-pore electrical conductivity (spor)from the WS fit of experimental data (basic and variantapproaches) by setting ss = 0, or with the measured bulkreal component of electrical conductivity s0 (simplifiedapproach). Hydraulic conductivity variation with waterpotential was then fitted to equations (9) and (10) withthese F values. Figures 8 (soils from this study) and 9 (datafrom literature) correspond to the fit of equation (9) with thebasic approach. Figures 8 and 9 show that these equationsdo fit the soils we experimentally studied and those from

literature. Using the basic approach and the equation (9),with only one free parameter (parameter B), gives an overallRMSE = 0.44 (n = 94 K-h pairs) for the estimated log K ofthe six soils (see Table 3 for individual soils). When usingequation (9) with measured s0 in place of spor, we found asame goodness of fit: overall RMSE = 0.44 (Figure 10). Ahigher accuracy is found, particularly for the Collias loam,when using equation (10), whatever the approach, resultingin an overall RMSE = 0.33 for log K. However, this increasein accuracy is done at the expense of adding another freeparameter, namely, the b exponent in equation (10). Thisfitted exponent varies between 0.6 and 3 (Table 3), with amean value equal to 1.34 (±0.86) for the basic approach or1.5 (±0.86) for the simplified approach, as compared to 1 inequation (9). We couldn’t find any significant relationshipbetween b and the other soil parameters investigated. Toavoid unnecessary free parameter for K prediction, we willfocus in the following on K prediction with equation (9)(i.e., b = 1). In this case, all parameters can be estimated ormeasured. Indeed, in contrast to the b exponent ofequation (10), the B scaling parameter of equation (9) provedto be highly correlated with Ksat and ss, with different

Figure 6. Experimental relationship between surfaceelectrical conductivity of different soils (13 soils) estimatedfrom the fitting of experimental data (water saturation andsoil electrical conductivity) to the model of Waxman andSmits [1968] (equation (3) in the text) and an indicator ofsoil texture (ratio of clay to coarser particles). Experimentaldata are the three soils from this study and data from Tuliand Hopmans [2004], Inoue et al. [2000], Amente et al.[2000], Malicki and Walckzak [1999], Rinaldi and Cuestas[2002], Nadler [1982], Rhoades et al. [1976, 1989], andKalinski and Kelly [1993].

Figure 7. Relationship between surface electrical con-ductivity of different soils estimated from the Waxman andSmits [1968] model (see equation (3) in text) and from theRhoades et al. [1989] model. See Figure 6 for the list ofsoils investigated.

Figure 8. Comparison between measured unsaturatedhydraulic conductivities (log K), estimated from the windmethod and the fitted K–electrical conductivity (s0)relationship (equation (9) in the text). Data are for the threesoils examined in this study.

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regressions if equation (9) is expressed on a spor (basic andvariant approach) or a s0 (simplified approach) basis:

Using spor in equation (9)

log Bð Þ ¼ 0:706 log Ksatð Þ � 4:481 ss þ 1:762 ð13aÞ

with R2 = 0.987, RMSE = 0.096, n = 6, p value = 0.07%

Using s0 in equation (9)

log Bð Þ ¼ 0:645 log Ksatð Þ � 6:322 ss þ 1:274 ð13bÞ

Figure 9. Comparison between measured unsaturatedhydraulic conductivities (log K) and the fitted K–electricalconductivity (s0) relationship (equation (9) in the text).Experimental data are from Inoue et al. [2000] for Tottorisand and Tuli and Hopmans [2004] for Oso Flaco sand andColumbia sandy loam.

Figure 10. Comparison between experimental and fittedunsaturated hydraulic conductivity (log K) values for sixsoils: three from this study (see Figure 8) and three fromliterature (see Figure 9); 94 data points. K is estimated fromelectrical conductivity with equation (9) and F is calculatedfrom bulk soil electrical resitivity (F = sw/s

0).

Table 3. Results of the Fit of the Unsaturated Hydraulic Conductivity of Six Soils With Equations (9) and (10)a

F Variable Soil

K(h) = f(1/F�h�2)b K(h) = f(1/F�h�2b)c

r2 RMSEd (m/s) r2 RMSE (m/s) b

Avignon silty clay loam 0.946 0.44 0.952 0.44 1.04Colias loam 0.858 0.71 0.925 0.34 0.58

spor Fontainebleau sand 0.911 0.52 0.903 0.57 0.78Tottori Dune sand 0.870 0.18 0.939 0.15 1.34Oso Flaco sand 0.970 0.46 0.917 0.22 3.00

Columbia sandy loam 0.881 0.46 0.966 0.31 1.32

Avignon silty clay loam 0.997 0.44 0.947 0.43 1.06Colias loam 0.991 0.69 0.926 0.35 0.65

s0 Fontainebleau sand 0.994 0.55 0.819 0.49 1.28Tottori Dune sand 0.998 0.18 0.926 0.16 1.51Oso Flaco sand 0.995 0.55 0.851 0.28 3.14

Columbia Sandy loam 0.997 0.50 0.967 0.30 1.38

aThe results relate electrical conductivity to hydraulic conductivity, where B and B0, b are fitted constants, h is the matric potential, and

F the electrical resistivity ratio calculated either with the estimated within-pore electrical conductivity (F = sw/spor) or with the measured

bulk real component of electrical conductivity (F = sw/s0). Three of the soils are from this study and three are from literature data.

bEquation (9).cEquation (10).dRMSE, root-mean-square error of log K (K in m/s).

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with R2 = 0.949, RMSE = 0.20, n = 6, p value = 0.54%. Theuse of s0 in place of spor results, as could be anticipated, in aslightly lower correlation for predicting B.[29] Figure 11 shows a comparison between the predic-

tion of the hydraulic-electrical conductivity relationship(equation (9), with F calculated with the measured s0) andthe Mualem–van Genuchten formula [van Genuchten,1980] for hydraulic conductivity using the parameters ofwater retention curve for sand in this case. The electricalformulation (equation (9)) results in an accuracy in theestimation of hydraulic conductivity of the same order orbetter (for 3 soils) than the Mualem–van Genuchten for-mula (Table 4) without the need of estimating parameters ofthe water retention curve.

4.4. Estimation of Unsaturated Hydraulic ConductivityFrom Soil and Electrical Parameters

[30] Results of section 4.3 suggest that K could beestimated with equation (9) in different ways, with varyingaccuracy, depending on the electrical and soil data at hand(Figure 2). In the first case, the within pore electricalconductivity spor can be estimated. This can be achievedeither by measuring at different saturation/salinities the q-s0

relationship and fitting data to the WS equation (basicapproach), as done with experimental data of this article,

or by estimating ss with equation (12), if clay percentage isknown, and evaluating exponent parameter n and Fsat with afew measurements of q-s0, Fsat can be independentlyestimated with different salinities at saturation (variantapproach). The B parameter is estimated by equation (13a).Using the above way of estimating K results in the sameaccuracy as with the original fitted coefficients with anoverall RMSE = 0.45 for the six soils. The other way ofestimating K is to measure s0 only and to use equations (12)and (13b) to estimate K(h) (simplified approach). This canbe summarized as (SI units h [m], ss [S m�1], K [m s�1])

ss ¼ 0:654Clay

Sand þ Siltþ 0:018

K hð Þ ¼ 6:86 10�4 � K0:645sat

106:322ss� 1

F hð Þ � hj j�2

K hð Þ ¼ Ksat if K hð Þcalculated > Ksat

8>>>>><>>>>>:

ð14Þ

[31] Results of equation (14) are presented Figure 12,which shows that this equation adequately fits the data,knowing relatively accessible parameters in soil physics:bulk electrical conductivity (s0), %Clay and Ksat. Theoverall RMSE between calculated and measured values is0.51 and is only slightly increased in comparison withresults obtained with the basic approach, using WS param-

Figure 11. Comparison between experimental data ofunsaturated hydraulic conductivity (log K) for the sandexamined in this study and K predictions with the Mualem–van Genuchten formula [van Genuchten, 1980] and with theproposed K–electrical conductivity (s0) relationship (equa-tion (9) in the text with F calculated with measured s0).

Table 4. Root-Mean-Square Error of log K Calculated With the Mualem–van Genuchten Formula or the Electrical

Formulation of Hydraulic Conductivity Using s 0

Soils

AvignonSilty ClayLoam

ColiasLoam

FontainebleauSand

TottoriDune Sand

Oso FlacoSand

ColumbiaSandyLoam

RMSE VGa 1.27 0.50 2.0 0.85 0.5 0.36RMSE s 0 0.44 0.69 0.55 0.18 0.55 0.50

aRMSE, root-mean-square error of log K; VG, Mualem–van Genuchten formula.

Figure 12. Comparison between measured and calcu-lated hydraulic conductivity (log K) with the electrical-hydraulic conductivity relationship (equation (14), simpli-fied approach). Measured bulk electrical conductivity s0 isused in equation (14), and other parameters are determinedfrom measured %Clay-Ksat.

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eters fitted from experimental data (see above whereRMSE = 0.44).

4.5. Verification of Electrical-Hydraulic ConductivityRelationship With a Large Soil Database

[32] We tested the predictive ability of equation (14) for alarge, completely independent, data set of soil unsaturatedhydraulic conductivity (UNSODA version 2.0) [Nemes etal., 1999]. We used all the data of the database wheneverh(q), K(h) or K(q), Ksat, %Clay parameters were available,resulting in 190 soils and around 3000 h-K data points. Asno soil electrical conductivity data are available inUNSODA, we estimated the electrical resistivity ratio Fby using an average value for n (saturation exponent) andFsat (formation factor at saturation) of the WS equation (3)derived from our 13 soils literature review (comparesection 4.2). These mean values are n = 2.27 (±1.18) and

Fsat = 4.63 (±0.89). F is calculated by1

F¼ Sn

Fsat

, for each

pair of q-h values, qsat being estimated/retrieved fromUNSODA data for S calculation. In this case, calculated Fcorresponds to the within-pore electrical conductivity. Weused equations (12), (13a), and (9) to estimate K(h) andresults are presented Figure 13. It appears that our estima-tion of K is in good accordance with the experimentalvalues resulting in RMSE = 1.1. It should be stressed herethat electrical conductivity was not measured but estimatedfrom water content values with a unique, average, value ofFsat and n which can vary highly with the soil type.Analysis of the error of estimation according to soil typeshows that soils with more than 40% clay content (silty clayand clay soils) show the higher deviation, with consistentunder estimation of calculated hydraulic conductivity. With-out these soils (19 soils), the RMSE criterion decreases to0.88. This tends to show that clayey soils might presentmarkedly different WS parameters from our estimated mean

values for soils and that the B coefficient is underestimatedwith equation (13a) or equation (13b).[33] These values of the error in the estimation of

hydraulic conductivity can be compared to estimation ofK with pedotransfer functions. Wagner et al. [2001] tested8 common PTFs for predicting unsaturated hydraulic con-ductivity with an independent database (36 soils) fromwhich they were derived. The best PTF, with saturatedhydraulic conductivity as an input, resulted in an RMSE =1.28 that grew up to RMSE = 2.96 in the worst case. Thisshows that our estimation of K behaves better than thosePTFs (RMSE = 1.1 with all soils and 0.88 without clayeysoils) and gives some confidence about the robustness/validity of the derived s-K relationships.

5. Conclusion

[34] We derived a relationship between electrical andhydraulic conductivity for unsaturated soils on the prem-ise that capillary equation could be used to extend theKatz and Thompson [1986] equation to unsaturated po-rous medium. Laboratory experiments showed the consis-tency of this relationship. Importantly, we presentedevidences, using our laboratory data and review ofliterature data, that parameters of this relationship betweenhydraulic and electrical conductivity can be completelydetermined (equation (14)) with different accuracy, depend-ing on the data at hand (Figure 2). In particular, the soilelectrical conductivity that can be used may be the within-pore electrical conductivity (necessitating the estimation ofthe s-q relationship) or, simply, the bulk soil conductivity, atthe expense of a slight decrease in accuracy of prediction.Apart from soil electrical conductivity, additional accessibledata such as Ksat, sw or %Clay are needed. Altogether, thisresults in K(h) estimation with an RMSE ranging between0.4 to 0.5 for log K, i.e., of the order of the experimentalerror for determination of K(h). The range of validity of theproposed s-K relationship was further tested on the largeUNSODA hydraulic database (�190 soils and 3000 datapoints). Within-pore electrical conductivity was estimatedfrom water content and average parameters for soils in theWaxman and Smits equation (equation (3)). This resulted ina relatively good agreement between measured and calcu-lated K (RMSE = 1.1) with the largest deviation observedfor clayey soils (>40% clay; RMSE = 0.88 without thesesoils). These results compare favorably or better than exist-ing PTF and show the robustness of the derived s-Krelationship.[35] Implications of such a s-K relationship range from

laboratory to field-scale applications. In the laboratory,concomitant determination of electrical conductivity andmoisture content, for water retention measurements allowsthe simultaneous estimation of K(h) or K(q) if Ksat ismeasured (Figure 2). This can lead to fastest determinationof hydrodynamic properties of soil samples, without the needfor relying on an a priori functional form between waterretention and hydraulic conductivity (such as Mualem–vanGenuchten), because effective porosity or tortuosity termsare included in the electrical conductivity. In this case,the three derived approaches can be applied. The basicapproach is based on the overall fit of the WS model(determination of parameters n, Fsat, ss) and requiresconcomitant determination of electrical conductivity with

Figure 13. Test of the proposed electrical-hydraulicrelationship (equation (14)) for estimation of the unsaturatedhydraulic conductivity with the soil hydraulic databaseUNSODA [Nemes et al., 1999]. The plot encompasses�190 soils and 3000 K data points. Grey dots representsoils with more than 40% clay content. See the text for thedetermination of K-s0 relationship parameters (section 4.5).

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soil water retention curve for different soil solution salinitiessw (at least 3). The variant approach is based on a partial fitof the WS model (n, Fsat), the parameter ss being estimatedfrom equation (12), Fsat can be estimated from s0 variationat saturation with different sw, and n from electrical mea-surements of the water retention curve. Finally, the simpli-fied approach (equation (14)) can be used, necessitating onlythe concomitant determination of s0 and soil matric potential.[36] At the field scale, electrical resistivity tomography

(ERT), which gives a spatial mapping (2-D vertical or 3-D)of soil electrical resistivity, is becoming a widely used toolfor mapping soil water movement and variations [e.g., Zhouet al., 2001; Michot et al., 2003, Srayeddin and Doussan,2009] and ERT might potentially gives estimation of thedistribution of water content in space. If some measure-ments of h are available or the h(q) relationship is known forthe soil horizons together with estimation of Ksat and%Clay, combined use of ERT measurement, hydraulicmeasurements and the derived s-K relationship, based onthe simplified approach (equation (14)) could lead to adirect estimation of the spatial variability with time of soilwater fluxes. Recent results from induced polarizationmeasurements show a power law relationship betweenCole-Cole relaxation time [Binley et al., 2005] or slope ofthe log-log variation of imaginary conductivity variationwith frequency [Tong and Tao, 2008] with saturated per-meability, and possibly unsaturated hydraulic conductivity,of sandstones. Moreover, relaxation time may also be linkedto van Genuchten parameters of the retention curve [Binleyet al., 2005]. Combined use of induced polarization and DCelectrical measurements could thus possibly enables con-comitant estimation of Ksat and K of soil samples in thelaboratory or in the field.[37] In conclusion, combined use of electrical/hydraulic

measurements could be a step forward in the direct moni-toring of field-scale variability of water fluxes. However,experimental data linking hydrodynamic properties andelectrical properties are still scarce for unsaturated soils.Concomitant experimental determination of these transferproperties, especially for clayey soils deserves furtherinvestigation, both experimentally and theoretically, in orderto improve or test the accuracy of the s-K relationship.

[38] Acknowledgments. We thank A. Tuli and J. W. Hopmans forproviding us with an electronic form of their published data. We alsowarmly thank N. Denchick for assistance in performing some of theexperiments. We acknowledge reviewer A. Hordt, the anonymousreviewers, Associate Editor L. Slater, and Editor S. Tyler for their helpfulsuggestions and comments. This study benefited from an INSU/PNRHgrant (France).

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����������������������������C. Doussan and S. Ruy, EMMAH, UMR 1114, INRA, UAPV, Domaine

Saint Paul-Site Agroparc, F-84914 Avignon CEDEX 9, France. (doussan@avignon.inra.fr)

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