Saturated Hydraulic Conductivity Prediction Frommicroscopic Pore

WATER RESOURCES RESEARCH, VOL. 35, NO. 10, PAGES 3149-3158, OCTOBER 1999

Saturated hydraulic conductivity prediction from microscopic pore geometry measurements and neural network analysis I. Lebron, M. G. Schaap,and D. L. Suarez U.S. SalinityLaboratory, ARS, USDA, Riverside,California

Abstract. Traditional modelsto describehydraulic properties in soilsare constrainedby the assumptionof cylindricalcapillarity to accountfor the geometryof the pore space. This studywas conductedto developa new methodologyto directlymeasure the porosity and its microscopiccharacteristics. The methodologyis basedon the analysisof binary imagescollected with a backscatteredelectron detector from thin sectionsof soils.Pore surfacearea, perimeter, roughness,circularity, and maximum and averagediameter were quantifiedin 36 thin sectionsprepared from undisturbedsoils. Saturated hydraulic conductivityKsat, particle size distribution,particle density,bulk density,and chemical propertieswere determinedon the samecores. We usedthe Kozeny-Carmanequation and neural network and bootstrapanalysis to predict a formation factor from microscopic, macroscopic,and chemicaldata. The predictedKsa t was in excellentagreement with the measuredKsa t (R 2 = 0.91) whena hydraulicradius rr• definedas pore area divided by pore perimeter and the formationfactor were includedin the Kozeny-Carmanequation.

1. Introduction been successfullyapplied to the measurementof porosityin rocks by Chretienand Bisdom [1983], Schoonderbeeket al. Flow and transport of water and solutesin soils are con- [1983],Berryman [1998], Berryman and Blair [1987],Ehrlich et trolled by size,geometry, and characteristicsof the soil poros- al. [1991], and Blair et al. [1996] and in soilsby Bruand et al. ity. Most of the characteristicsof soil pores are microscopic, [1996] and Vogel[1997]. suchas roughnessand circularity.Conventional models of liq- Image-basedtechnology provides unique information about uid distribution,flow, and solutetransport rely solelyon cylin- pore geometryby directmeasurement. While other techniques drical capillarity,ignoring the role of surfacearea, angularity, are usedto obtain indirect or deductedvalues using a number and connectivity.It is known that the geometry of the soil of assumptions,microscopy and image analysissupply actual pores is very irregular; for example, Figure 1 showsa thin valuesof the pore space.The size of the poresresolved in the sectionfrom an undisturbedsample of Gilman silt loam soil. imagedepends on the magnificationand the sizeof the pixels. We identify polygonal and angular pores associatedwith Berrymanand Blair [1987], usingcorrelation functions, devel- feldsparsand quartz while the more elongatedpores are, in oped a methodologyto derive the characteristiclength at general, associatedwith clays.These different shapesare im- which the physicalproperties of the heterogeneoussoil micro- portant becausethe water retention in the pores dependson structureare spatiallycorrelated. More recently,Blair et al. the angularityand roughnessof the pores [Tuller et al., 1999]. [1996], with a two-point correlationfunction, determined the Also, elongatedpores may be more susceptibleto the effect of adequateimage resolution to obtain imageparameters consis- the chemicalcomposition of the soil water sincethey are as- tent with those used in a simple flow model, such as the sociatedwith clays,which have colloidalproperties. Kozeny-Carmanequation, for predictionof the permeability. If we look closelyat one of the featuresin Figure 1, we see Neural networktechniques have been usedto predictwater a montmorillonitedomain (Figure 2). The arrangementof this retention propertiesin soils by Pachepskyet al. [1996] and organized structure modifies with changesin the chemical Schaapand Bouten[1996] usingmacroscopic parameters. We composition,as has been demonstratedin previous studies can find no examplesin the literature of the use of neural with electrophoreticmobility for montmorillonites[Shainberg networksin combinationwith pore microfeaturesto predict and Otoh, 1968]and illires[Lebron et al., 1993].The hypothesis flow in soils. of this studyis that the self-arrangementof the particlesin the The objectivesof this studyare as follows:(1) to establisha soil is not totally random and if we understandthe forces methodologyusing a two-pointcorrelation function and image determiningthis self-arrangement,we will be in a better posi- analysissoftware to analyzesoil pore spaceand pore geometry tion to model pore spaceand, consequently,water flow. from microphotographsand (2) to interpretthese data in the The use of thin sectionsto measuregrain sizeshas been contextof the model proposedby Kozenyand Carman using performed for more than 60 years in petrological studies neural networkand bootstrapanalysis to predictsoil hydraulic [Friedman,1958, 1962]. The advanceof new technologiesal- properties. lowsthe use of more powerfulmicroscopes and sophisticated softwarepackages to quicklyprocess thousands of measure- 2. Materials and Methods ments.Scanning electron microscopy and image analysishave The soil used in this studywas Gilman silt loam soil, from This paper is not subjectto U.S. copyright.Published in 1999 by the CoachellaValley, California. The field, 9.7 ha of saline sodic American GeophysicalUnion. soil,was monitoredand mappedin a previoussurvey for elec- Paper number 1999WR900195. trical conductivity(EC) and sodiumadsorption ratio (SAR)

3149 3150 LEBRON ET AL.: SATURATED HYDRAULIC CONDUCTIVITY PREDICTION

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Figure 1. Scanningelectron micrograph from a thin sectionof an undisturbedsample from Gilman silt loam soil, CoachellaValley, California.

(SAR = Na+/(Ca2+ + Mg2+)ø'5, where the cationconcentra- chemicalcomposition (using inductively coupled plasma emis- tionis expressed in mmolL-•). A samplingscheme was estab- sionspectroscopy). Because of the scarcityof sample,EC, pH, lished to cover a maximum range of EC and SAR which re- and SAR were measured in a combination of the three nearest sulted in 36 undisturbedsoil samples.The undisturbedsoil samples;the samevalue appearsfor the three combinedsoils. cores were 12 cm in diameter by 12 cm in height and were The resultsof the analysesare shownin Table 1. Thin sections collected from the top 25 cm of the soil by hand, to avoid were prepared by impregnation of the sampleswith epoxy excessivemechanical compression of the natural structure.Sat- EPO-TEK 301 (Epoxy TechnologyInc., Billerica, Massachu- urated hydraulic conductivityKsa t was measured on all soil setts).After hardening,a thin section3.5 x 2.5 cm was cut at cores using Colorado River water as eluent (water used for the plane perpendicularto the water flow, mountedon a glass irrigation on those fields); the chemicalcomposition of the slide, and polished. The polishing processwas done with a ColoradoRiver water is as follows: EC, 1.15dS m-•; pH, 8.15; seriesof diamond polishersto avoid the introductionof con- SAR,2.8; HCO•-, 2.8;CI-, 3.1;SO42-, 5.8; Na+, 5.0;K +, 0.12; taminants and in the absenceof water to preserve soluble Ca2+, 4.0; and Mg2+, 2.5; whereion concentrationsare in minerals. mmolc L- •. Thin sectionswere observedin a scanningelectron micro- Sampleswere saturatedby firstwetting by capillaryrise from scope(SEM) (AMRAY 3200,AMRAY Inc., Bedford,Massa- below, then gradually raising the water level until water chusetts)with the backscatterelectron detector. The intensity ponded on the surface;Ksa t was measuredwith the constant of the backscatteredelectrons (BE) is a functionof the atomic head method. After the Ksat was measured,bulk densityPB weight of the element, with heavier elements having higher wasdetermined, and the air-driedcores were cut verticallywith backscatteringproperties. The result is that elements with a knife (in the directionof the water flow) into two equalparts. higher atomicweight give imagesthat are brighter than lighter One part was usedto prepare thin sections,and the other half elementsin the sample.Charging effects are counteractedby was usedto analyzefor particle densityp,• [Blakeand Hartge, usinglow vacuum in the specimenchamber and by the intro- 1986], particle size distribution[Gee and Bauder, 1986], and duction of air moleculesto dissipatethe charge accumulated LEBRON ET AL.: SATURATED HYDRAULIC CONDUCTIVITY PREDICTION 3151

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Figure 2. Scanningelectron micrographof a montmorillonite domain. on the surface of the sample. No coating of the sample is and contrastof the imageare adequate,the histogrampresents needed. The selectionof the magnificationis made with a a bimodal distribution,since the carbon-basedepoxy displays two-pointcorrelation function (see section3.2) and the quan- as a black color and the soil minerals are bright in a wide tiffcation and classificationof pore spaceswith commercially spectraof grey colors.In general,the quality of the histogram available software (Princeton Gamma-Tech Inc., Princeton, for soil thin sectionsis dependent on the clay content; the New Jersey).The image collectedby the scanningelectron higherthe clay content,the poorer the definitionof the bimo- microscopein the backscatterelectron detector (1024 x 800 dality. The reasonfor this is that the averagepore size in clay pixels) is transformedinto a binary image, and maximum, soils is smaller than that in sandy soils, and consequently, minimum,and averagediameter (Dmax, Dmin, and D•vg, re- individual pixels are more likely to contain both pore and spectively),surface area A, and pore perimeterP are quanti- particle features;therefore, when the pore size approachesthe fied by directly measuringthe number of pixels that conform detectionlimit of our SEM (see section3.3), the definitionof eachfeature in the binaryimage. Roughness R is calculatedby the imageis poor,as is the resu!tanthistogram. R = P(•Davg), and the circularityC is calculatedby C = The micrographis convertedinto a binary image; this con- rrO2max/ ( 4A ) . version is based on the histogram where a threshold level To increasethe size of the samplingarea, we collected 10 needsto be establishedin order to resolvepore featuresfrom picturesfrom each thin sectiondistributed in a regular grid; the totalarea sampled is equivalentto 2.5 mm2. Eachof the particle features.Figure 3 showsa typical histogramfor a BE image in which the particle and pore spacesare representedby pictureswas processed;average parameters are presentedin Table 1. individual peaks. The valley between the peaks containsam- biguouspixels that need to be resolvedinto particlesor pores. We evaluated three methods on the basis of the observation 3. Development of the Microscopic Method that the histogramis a superpositionof two overlappingGauss- 3.1. Determination of the Threshold Grey Level ian curves.However, for finer-texturedsoils the representation The images collectedwith the BE detector contain a grey of the histogramsis complexand is not normally distributedin scalethat can be representedby a histogram.If the brightness all cases.Since in the present study we have sampleswith a 3152 LEBRON ET AL.: SATURATED HYDRAULIC CONDUCTIVITY PREDICTION

Table 1. Clay, Silt, and Sand Percentages,Saturated Hydraulic ConductivityKsat, Porosity Measured With Image Analysis 4>sE•t,Average Pore Diameter Davg, Circularity C, RoughnessR, ElectricalConductivity (EC), pH, SodiumAdsorption Ratio (SAR), Pore Area A, Pore PerimeterP, ParticleDensity Pa, and Bulk DensityPB of Gilman Silt Loam Soil, CoachellaValley, California Sample Clay, Silt, Sand, Ksat, (•SEM, D avg, EC, A, P, Pa, Number % % % mmh- • % tam C R dSm- • pH SAR tam2 tam g cm3 g cm3

1 9 35 56 3.18 31.1 12.9 2.97 1.06 25.4 7.25 71 444 118 2.57 1.38 2 10 39 51 2.84 35.5 12.2 3.21 1.05 25.4 7.25 71 505 116 2.55 ... 3 10 50 37 2.12 29.2 10.9 3.15 1.04 25.4 7.25 71 144 58 2.71 1.38 4 16 50 34 3.24 32.4 11.4 3.23 1.03 62.7 7.36 217 498 120 2.67 1.37 5 34 61 5 0.03 26.0 10.6 3.13 1.05 62.7 7.36 217 97 57 2.71 1.35 6 26 67 7 0.05 27.3 10.6 3.01 1.05 62.7 7.36 217 81 51 2.65 1.36 7 7 21 73 19.9 34.7 11.8 3.16 1.02 1.50 7.44 3.8 887 145 2.55 1.39 8 8 16 76 17.6 32.7 11.4 3.21 1.05 1.50 7.44 3.8 385 100 2.57 ... 9 4 7 89 50.3 37.6 11.8 3.39 1.02 1.50 7.44 3.8 2204 195 2.54 ... 10 11 25 64 2.77 36.7 11.0 3.34 1.03 1.16 7.63 4.7 530 112 2.57 1.49 11 8 23 69 6.80 39.4 11.1 3.13 1.03 1.16 7.63 4.7 677 103 2.58 ..- 12 9 28 63 2.04 28.7 10.7 3.34 1.03 1.16 7.63 4.7 188 69 2.59 1.44 13 10 40 50 2.09 35.5 12.8 3.52 1.04 1.20 7.50 3.0 361 114 2.55 1.31 14 12 34 54 2.31 29.8 11.9 3.33 1.04 1.20 7.50 3.0 230 85 2.54 ..- 15 10 41 49 4.38 31.0 11.4 3.21 1.05 1.20 7.50 3.0 191 79 2.66 1.37 16 18 48 34 0.14 25.6 11.5 3.27 1.06 6.41 7.90 12 121 64 2.69 1.42 17 16 49 36 0.54 21.4 10.4 3.21 1.03 6.41 7.90 12 125 57 2.77 1.50 18 15 54 31 0.37 34.0 11.3 3.48 1.04 6.41 7.90 12 283 94 2.78 1.43 19 17 57 26 0.36 32.5 11.4 3.21 1.07 3.20 7.65 2.0 161 82 2.69 1.35 20 23 56 21 0.64 21.9 10.4 3.21 1.03 3.20 7.65 2.0 102 56 2.78 ... 21 24 60 16 0.19 30.8 10.4 3.17 1.03 3.20 7.65 2.0 185 67 2.74 1.28 22 15 50 35 0.79 33.0 12.1 3.21 1.06 3.22 7.25 3.5 273 93 2.62 1.40 23 12 44 49 2.25 30.8 11.0 3.21 1.03 3.22 7.25 3.5 252 80 2.64 ... 24 15 47 37 1.29 37.2 11.8 3.21 1.06 3.22 7.25 3.5 321 93 2.62 1.29 25 10 33 56 1.74 28.7 13.0 3.52 1.05 3.31 7.90 8.0 276 104 2.63 1.40 26 9 32 59 2.76 29.5 11.5 3.32 1.04 3.31 7.90 8.0 228 84 2.51 ... 27 11 33 56 1.71 25.4 11.2 3.20 1.04 3.31 7.90 8.0 159 72 2.57 1.38 28 13 32 55 0.36 26.1 11.8 3.24 1.03 4.17 7.50 5.2 275 98 2.72 1.51 29 11 29 60 0.73 35.3 12.1 3.40 1.06 4.17 7.50 5.2 219 93 2.61 1.47 30 9 28 63 0.38 19.4 10.7 3.18 1.03 4.17 7.50 5.2 153 60 2.68 1.48 31 15 30 55 0.55 23.0 10.8 3.17 1.04 5.40 8.10 7.2 155 69 2.52 1.53 32 16 44 40 0.44 22.3 11.0 3.16 1.03 5.40 8.10 7.2 202 72 2.71 1.31 33 17 45 38 0.19 30.0 11.2 3.20 1.05 5.40 8.10 7.2 198 74 2.79 1.53 34 20 57 30 0.06 35.4 11.5 3.21 1.07 65.8 7.95 260 192 81 2.77 1.41 35 21 54 25 0.09 30.9 11.2 3.21 1.05 65.8 7.95 260 135 68 2.52 1.31 36 20 54 25 0.09 28.8 11.8 3.21 1.08 65.8 7.95 260 129 73 2.81 1.28

The pBvalues not shownare missingvalues; for thesesamples we usedthe averagep• - 1.43 g cm3. wide range in textural composition,we will use the mathemat- where n and m are the vertical and horizontallags and Nmax= ical minimum of the histogramto establishthe thresholdlevel N- n and Mmax = M- m; f(i, j) = 1 for pixelsin pores to convert the picture into a binary image. and 0 for pixelsin particles.The calculationof S2(m, n) with

3.2. Two-Point Correlation Functions (1) is very inefficientfor an image of significantsize. A more A two-point correlation function provides the probability that two points at a specificdistance are locatedin the region of spaceoccupied by one constituentof a two-phasematerial. These functionshave been usedextensively to calculatediffer- ent propertiesof heterogeneouscomposites and provideinfor- mation about the representativenessof the image. In our case the function is derived from the binary BE image and provides the probabilitythat two pixels separatedby a distancek are simultaneouslyin either a pore or a particle.When considering pores, if k = 0 (zero lag), the probabilityis equal to the porosity 4•. At large distancesand isotropic conditionsthe probabilityis equalto 4)2 [Blairet al., 1996]. Two-pointcorrelation functions of the binarypicture f(i, j) J with N vertical and M horizontalpixels can be calculatedas the 0 50 100 150 200 250 two-dimensionalcorrelation matrix S2(m, n) expressedas Grey level 1 MmaxNmax S2(m,rt): mmaxNmaxZ Z f(i,j) f(i + m, j +n) (1) Figure 3. Histogramof the greylevel of a backscatteredelec- i=1 j=l tron image of a soil thin section. LEBRON ET AL.: SATURATED HYDRAULIC CONDUCTIVITY PREDICTION 3153

effectiveanalysis is to perform a two-dimensionalfast Fourier Table 2. SEM Measurementsfor Sample 25 at Different transform(FFT) of the matrixf(i, j) as Magnifications:Pixel Size h, Mean Pore Diameter Din, Area A, Perimeter P, and Porosity•SEM P(a, b) = FFT[f(i,/)] (2) Magnification, h, Din, A, P, where a and b are indicesin the complexfrequency matrix P. x /xm Pixels (I)sEM (I)TcF p,m2 For eachfrequency a, b the correlationis performedas [Press 35 2.691 12.87 26.7 29.2 442 131 et al., 1988] 50 1.884 8.25 30.3 29.8 415 141 100 0.942 5.70 30.8 33.4 171 79.5 C(a, = P(a, (3) 200 0.471 4.32 32.7 34.0 64.5 36.0 whereP*(a, b) is the complexconjugate of P(a, b). Through 500 0.188 2.63 32.4 34.9 26.3 21.3 1000 0.094 2.05 38.4 50.8 13.8 16.9 the inverseFFT of C(a, b) we obtain the two-dimensional correlationmatrix S2(m, n): Also given is porositycalculated by the two-point correlationfunction CI)TcF. S2(m,n) = FFT-•[C(a, b)] (4) The column$2(m = 1, n) and row $2(m, n = 1) are the vertical and horizontal correlationfunctions and provide in- formationon the isotropyof the sample.The two-dimensional where h is the pixel size and N is the number of pixelsalong correlation matrix is summarized into a one-dimensional 'two- each side of a squareimage. Berrymanand Blair emphasize point correlationfunction by usingthe procedureoutlined by that there is no advantageto increasingmagnification if the Berryman[1998]. averagepore diameteris larger than a quarter of the width of the digitizedimage and recommendhaving ---100 grains in the 3.3. Determination of Optimal Magnification picture for measurementof porosity. Magnification is the most important factor affecting the Table 2 showsthe r, pixel size, magnification,and porosity quantificationin the analysisof an image. If we measurethe valuescalculated with (4) for sample25 in Table 1. Calculating same parameter, for example,the hydraulicradius (rn = the right-handside of (5) for an imageof 1000pixels, we obtain the value h -> 0.1. When this value is combined with the A/P), at the samelocation of the same sampleat different magnifications,we observethat the valueofrn variesby >50% requirementof havingat least 100 featuresin the picture,we (Figure4). The definitionof the SEM imagedetermines the determinethat magnificationswithin the acceptablerange are qualityof the measurements,which dependson the pixel size, 200-35 x. A magnificationof 50x waschosen as a compromise which, in turn, is inverselyproportional to the magnification. between a bigger samplingsize and a satisfactorydetection Berrymanand Blair [1987]also found that imagemagnification limit. The detectionlimit is 7 •m at 50x magnificationand the was important in the determinationof a surfaceto volume resolutionchosen (1024 x 800 pixels). ratio for usein a Kozeny-Carmanmodel. They providea methodology,based on a two-pointcorrelation function (4), for 4. Prediction of Saturated findingthe appropriatepixel sizeto usewhen preparing images Hydraulic Conductivity of crosssections to be usedto studywater flow. The two-point correlationfunction that we use in this studywas calculated 4.1. Kozeny-Carman Equation using(2), (3), and (4). From (4) we obtain a characteristic There are a number of methods available to relate SEM data length which is an estimate of the mean pore radius r and to saturated hydraulic conductivity.Empirical relationships define the range in which this r mustbe in order to determine suchas fractal analysisor neural network havebeen usedin the reasonableporosity measurements. Berryman and Blair [1987] literature, however,it is more informativeto include as many recommendthat the pixel sizebe ---1% of the size of an aver- physicalrelationships as possible. The model traditionallyused age pore radiusand that to predictKsa t is the Kozeny-Carmanequation, but different 2r approachescan be taken; for example,Ahuja et al. [1989], h-> N(1 - •b) (5) usingmacroscopic measurements, proposed Ksat= Oq•e• (6) where q• is the effectiveporosity, defined as the difference between the soil water retention at saturation and the water y= 5.82-1.70 log x R2 =0.98 retention at 33 kPa. The parametersQ and B are fitting parameters, and they depend on the calibrationdata set [cf. Gimenezet al., 1997]. A different approachis that followedby Blair et al. [1996], who used the Kozeny-Carmanrelation to interpret data de- rivedfrom SEM imagesof sandstones.They usedthe expression: Ksat---Cqbr• (7)

101 102 103 where •b is the porosityand r• is the hydraulicradius. The constantC can be definedby usingthe Poiseuilleequation: Magnification Figure 4. Hydraulic radius rn as a function of the magnification. C = vGr2 (8) 3154 LEBRON ET AL.' SATURATED HYDRAULIC CONDUCTIVITY PREDICTION where C incorporateseffects of accelerationof gravity!7, pore relationshipsthrough minimization of the followingobjective geometryG, pore tortuosityr, and the kinematicviscosity of function: the pore water v. C has the largestvalue for parallel, circular 36 pores(G - 8, r = 1). However,for real soilsit is mostlikely that the assumptionof parallel, circular pores will lead to a Ob= • (F- F') 2 (11) considerableoverestimation of the saturatedhydraulic conduc- i=1 tivity. Furthermore, pore connectivityis an additional factor that may play a role in soils. where the numberof samplesis 36 andF andF' are calculated In this studywe will calculatethe hydraulicradius r/_/from with (10) and the neural network-predictedF factors. total pore area A and total pore perimeter P Typically,usage of neural networksleads to black box models in which the flow of information can be difficult to track. rH=A/P (9) Furthermore, neural networks tend to be sensitive to nonlinear instabilityand overfitting [Hsieh and Tang, 1998]. Nonlinear This definitionwill includenoncircularity and roughnessof the instabilitystems from the possibilitythat many local minima poreswith bothA and P measureddirectly with the SEM. For may exist on the surfaceof the objectivefunction. Overfitting equal areasthe hydraulicradius will becomesmaller when the results from the function approximation of the neural net- noncircularityor pore roughnessincreases. The porosity 4>in works,which, without further precaution,may lead to the un- (7) is alsomeasured from the SEM imageanalysis (rkSEM)' The desired inclusion of noise and artifacts into the model. A constantC in (7) is splitinto a constantpart C• = 17/v(equal model that is overfittedwill give very good predictionsfor its to 8.46x 109)and a nonconstantpart F thatlumps effects of calibrationdata set but poor predictionsfor independentdata. pore tortuosity,geometry, and connectivity.This factor F is a Theseproblems can largelybe overcomeby repeatedlyrestart- soil-specificreduction factor and is analogousto the "forma- ing the neural network calibrationwith different initial coeffi- tion factor" that is used in many hydrologicalstudies of rock cients, by limiting the number of iterations [Schaap et al., strata[e.g.,Adler et al., 1992;Berryman and Blair, 1987;Blair et 1999], and by validatingthe neural network model on inde- al., 1996]. To remove a bias toward high hydraulicconductiv- pendent data. ities,we express(7) in a logarithmicform: The bootstrapmethod [Efronand Tibshirani,1993] was used log (Ksat)= log (C,) + log (F) + log [(kS•M(A/P)2] (•0) to generateuncertainty estimates of predictedF and to gen- erate independentcalibration and validation data setsfor the Equation (10) will be used in two different ways. First, we neural network calibration. Bootstrap theory assumesthat assumethat the conceptof straightcylindrical pores appliesto multiple alternativerealizations of the populationcan be sim- our samplesand predict gsa t with(10) usingmeasured rkSEM, ulatedfrom the singledata set that is available.By calibrating A, andP withC = 8.46x 109and F = • (G = 8 andt - the model on these alternativedata sets,different predictions 1). Second,we will use measuredgsa t and solve(10) for F. result,leading to uncertaintyabout the true model. The alter- This approachis necessarybecause we cannotaccount for the native data setshave the samesize as the original data set and individualcontributions of pore tortuosity,geometry, and con- are created by random resamplingwith replacement.There- nectivity.Subsequently, we will investigatethe relationshipof fore, in a data set of n samples,each samplehas a chanceof F with microscopicproperties ((DSEM, •4, P, pore roughness, 1 - [(n - 1)/n] n of being selectedonce or multiple times. and circularity),macroscopic properties (clay and sandper- Becausesome samplesare selectedmore than once, each al- centageand bulk density),and soil chemicalproperties (pH, ternativedata set contains-64% (for n - 36) of the original EC, and SAR). We will usea combinationof a neuralnetwork data. Neural networks were calibrated on each of these data , and the bootstrapmethod to find a predictivemodel for F. sets.The bootstrapmethod was combined with the TRAINLM routine of the neural network toolbox [Demuth and Beale, 4.2. Neural Network and Bootstrap Analysis 1992] of the MATLAB © package(version 4.0, MathWorks A big advantageof neural networksover traditional regres- Inc., Natick, Massachusetts).The neural network code was siontechniques is that they do not need an a priori selectionof modified to avoid local minima in the objectivefunction. We the underlyingmodel expressions(linear, exponential,etc). used 60 alternative data sets, leading to 60 neural network The term "neural network" representsa large collection of models;the averageof the 60 predictionswas the true F, and numericaltechniques that resemblebiological neural systems. the correspondingstandard deviation gave the uncertaintyin Many different types of neural network exist, each with a F. particularrange of applications[Hecht-Nielsen, 1991; Haykin, 1994]. In this studywe used the most commontype of three-layer 5. Results and Discussion feed-forwardnetworks as alsoused by Pachepskyet al. [1996], Schaap and Bouten [1996], and others. This type of neural 5.1. Microscopic Method network can be seen as a "universalfunction approximator" The use of microscopictechniques to describe the pore becauseit can estimateany continuousnonlinear function with spacein a soil has the advantagethat we can considerpore a desired degree of accuracy.Because they need no a priori propertiesthat are otherwiseunavailable. However, special model specifications,neural networks are capable of finding precautionsneed to be taken if the objectiveis to relate those complexrelationships between input data (in our case, for propertieswith macroscopicevents. As a preliminary evalua- example, clay percentage,bulk density,pore roughness,and tion of the consistencyof our microscopictechnique, we com- pH) and output data (the factorF). An iterativecalibration pared the porosityof our samplesobtained by the imageanal- procedurebased on the Levenberg-Marquardtalgorithm [Mar- ysisfrom the micrographs((DSEM) with the porositycalculated quardt, 1963;Demuth and Beale, 1992] was usedto find these using LEBRON ET AL.: SATURATED HYDRAULIC CONDUCTIVITY PREDICTION 3155

iOB 45= 1 (12) pa where PB is the bulk densityand Pa is the soil particle density. Figure 5 showsthat the porositymeasured with SEM system- atically underestimatesthe total calculatedporosity (12) by -0.15 cm3/cm3. Most likely this underestimationis due to unresolvedpores <7/xm. To explain the differencebetween qbSEM and calculatedporosities, we derived the pore volume contained in pores <7 /xm using a pedotransfer function [Schaapet al., 1998]. By usingthe capillarylaw it followsthat a pore of 7/xm in diameteris equivalentto a pressurehead of 43 kPa. After adding the porosity associatedwith the water content at 43 kPa pressurehead to qbSEM we found a much better agreementwith the calculatedvalues (Figure 5). Even thoughthe pedotransferfunction provides predictions with an implicit error, we estimatethat our porositymeasurements are comparableto those obtainedconventionally. The two-point correlationfunction calculationsfor all our A samplesindicated that the magnificationused (50x) provides acceptableresolution. As mentionedearlier, the resolutionof the imagesis related to the clay contentof the sample.Figure 6 showsthe micrographsand the correlation functionsfor samples5 and 9 (Table 2), whoseclay content covers the range of our samples.The correlationfunctions show that the char- aateristiclength (where the function crosses the (/:)•EM) ismuch smallerthan the sizeof the micrograph(1880/•m). The accu- mulationof 10 mici'ographsper sample ensures a goodrepre- sentation of the microfeatures. From these resultswe concludethat a magnificationof 50x leadsto a sufficientresolution of relevant small pores,as well as a goodspatial coverage. Furthermore, the magnificationwill also lead to an exclusion of features that are too small to have an effect on saturated hydraulic conductivity[Ahuja et al., 1989; Berrymanand Blair, 1987].

5.2. Prediction of Saturated Hydraulic Conductivity B

As we reviewed in section4.1, there are different possible 0.45 expressionsof the Kozeny-Carmanequation. We tested the 0.4 relationshipproposed by Ahuja et al. [1989] to predictgsa t in Oursoils samples. When we substitutedqbSE M in (6) andused 0.35 linear regressionbased on log (gsat) values,we obtainedQ = 0.3 304 and B = 4.07 for our data set. However,the prediction ._o 0.25 of Ksat versusmeasured Ksa t had a very low correlationcoef- • 0.2 ficient (0.17, Table 3). The lack of correlationbetween mea- o O 0.15 •SEM2 0.1 0.70 Sample 5 0.05 • 0.60 0 1 O0 200 300 400 500 ._• 0.50 Correlation distance (urn) c o • 0.40 o Corrected o ßo o oOO porosity Figure 6. Binarymicrographs for (a) sample5, (b) sample9, b 0.30 6>ø Oo: øoooo o o o and (c) two-pointcorrelation function for samples5 and 9. o o o o Measured

o porosity '•ß 0.20 o

• 0.10 i . i ' l 0.35 0.45 0.55 0.65 sured and predicted Ksat indicates that the version of the Kozeny-Carmanequation by Ahuja et al. [1989] cannot be Calculatedporosity (cm3/cm 3) reliablyused for our data. The use of a higher effectiveporos- Figure 5. Porosityof the 36 samplesmeasured with the scan- ity (the cutoff pore size in our case correspondsto 43 kPa ning electron microscopebefore and after the correctionfor insteadof the 33 kPa proposedby Ahuja et al. [1989]) doesnot the detectionlimit versusthe calculatedporosity. justifythe poor predictionand indicatesthat fitting parameters 3156 LEBRON ET AL.: SATURATED HYDRAULIC CONDUCTIVITY PREDICTION

Table 3. CorrelationCoefficient R 2 of the Regression are not independentvariables and showa stronglinear corre- Betweenthe MeasuredKsa t and the PredictedKsat Using lation.It is knownthat EC and SAR haveopposite effects in Different Models the developmentof the doublelayer and, consequently, in the flocculate/dispersedstate of the clayparticles; while increases Model Ksat Predictions (R 2) of EC favorflocculation, increases of SAR promotedispersion. Ksat = A 0sBEM 0.17 This directrelation of sodicityand salinityis commonin arid Neural network zoneswhere the majority of the saltsare sodic. 0.71 Input SSC The valuesof pH werenot lineallycorrelated with the chem- Input SSC + PB 0.75 Input SSC + pB + SAR + pH + EC 0.77 icalparameters, and the prediction of F wasimproved after the Ksat= FCCbsE•ar•(F obtainedwith neural 0.91 incorporationof pH in the neuralnetwork analysis. The pH networkand input C + po + pH + R) hasbeen widely neglected in transportmodeling efforts, but The firstmodel is the Kozeny-Carmanequation used by Ahuja et al. Suarezet al. [1984]showed that at highpH an increasein 1 unit [1989]and Gimenez et al. [1999],the secondmodel is directregressions of pH cancause a decreasein Ksat of 1 orderof magnitude;this of differentvariables with measuredKsa t usingneural network analysis, decreaseis equivalentto an increaseof 20 SAR units. The and the third model is the Kozeny-Carmanequation used in the importanceof the pH is related to the sign of the variable presentstudy, for which the factorF was predictedusing neural net- charges,generally located at the edgesof the clays,in the iron work analysis.SSC, sand, silt, and clay;p•, bulk density;SAR, sodium absorptionratio; EC, electricalconductivity; and R, roughness. and aluminumoxides, and in the organicmatter. Above the pointof zero charge(PZC) the variablecharges are negative; consequently,the net electrostaticcharges in the soilparticles withouta physicallybased conceptual model is not adequate are negative,and repulsiveforces occur among the particles. for our soil samples. When the pH of the soil is below the PZC, the edgesare The conceptualmodel needsto accountfor the difference positivelycharged, providing the possibilityfor the electro- between an ideal systemof an array of parallel cylindrical staticforces to form bonds.There is a considerablevariability poresand the intricatedistribution of poresin a natural soil. of PZC valuesfor the different componentsof a soil, but in The pore spaceis the resultof the rearrangementof particles general,we canconsider that thereis a surfacecharge reversal in structuralunits called aggregates. These aggregates are the aroundpH 7-9. When the predictedF valueswere calculated resultof a seriesof attractiveand repulsiveforces of electro- in the rangeof 5-25% clayat constantbulk density and rough- staticand physicalorigin. We introducedthe factorF, whichis nessand variable pH (Figure7), the F factordecreased when a combinationof tortuosity,connectivity, and microscopicpa- the pH increasedand decreasedwhen the clay contentin- rameters.F was best predicted when clay percentage, creased,in agreementwith our previousobservations. roughness,and pH were used as predictorsusing neural net- If we keep pH and pB constantand changeroughness, we work analysis.Clay percentage was the singlevariable with the predicta decreasein F valueswhen the roughnessincreases highestcontribution (R 2 = 0.60 alone)to theprediction ofF. (Figure8); this agreeswith the resultsof Berrymanand Blair The predictionof the formationfactor did not requiresa- [1987].Berryman and Blair found that for the Kozeny-Carman linity or sodicityparameters. However, it is well known,from model for fluid flow in porousmedia it is not the absolute dispersionor aggregatestability tests, that salinityand sodicity specificsurface area but ratherthe roughnessof the porewalls have an importantrole in the dispersionand flocculationof relativeto the meanpore size that influencesthe permeability. clayparticles [Goldberg and Forster, 1990; Hesterberg and Page, When roughnessand pH were constant,the F factordecreased 1993;Lebron and Suarez,1992; Kretzschmar et al., 1993],in the whenthe p• increased(Figure 9); thismay due to the fact that hydraulicconductivity [Suarez et al., 1984],and, consequently, increasesin p• are associatedwith an increasein tortuosity, in the rearrangementof the claydomains [Shainberg and Otho, whichis knownto reducethe Ksat. 1968;Quirk andAylmore, 1971; Lebron et al., 1993]. The predictedKsat, taking into accountthe F factor in the However, when EC or SAR were incorporatedinto the Kozeny-Carmanequation, shows a very goodagreement with neuralnetwork analysis to predictF, therewas no significant the measuredKsa t (Figure 10) with a regressioncoefficient of increase in the correlation coefficient. From Table 1 we ob- 0.91. The predictionswere highlyimproved by the use of rH, servethat in this particularsample population, EC and SAR

-1

-1 p.=1.4o p.=1.40 pH=7.5 R=1.15 pH=7.5 Roug h ness= 1.15

-4 5 10 15 20 25 5 10 15 20 25 Clay % Clay % Figure 8. Predictedvalues of the log of the formationfactor Figure 7. Predictedvalues of the log of the formationfactor F at differentroughness values R in the rangeof 5-25% clay F at differentpH valuesin the rangeof 5-25% claycontent. content. LEBRON ET AL.: SATURATED HYDRAULIC CONDUCTIVITY PREDICTION 3157 which includesarea and perimeter of the pores directlymea- 3.5 suredin the image.This informationcannot be obtainedby any other methodology(that is not imagebased); rH is traditionally inferredthrough indirect methods with a seriesof assumptions suchas the absenceof coulombicforces among particles and 1.5 specificgeometry. As a corroborationof the need for a physicallybased model we appliedneural network analysisdirectly to our data in an attempt to empiricallypredict Ksat without --J 0.5 using the Kozeny-Carmanequation or microscopicparame- o ters. We used different variables with relative success;Table 3 • -o.5 showsthat we were able to predict Ksat with a correlation -1 coefficientrange of 0.71-0.77, but we did not reachthe quality -1.5 of the predictionsmade with (10). -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Despitethe excellentcorrelation between the calculatedand Measured Log(K,•t) (cm/day) measuredKsa t there are many unresolvedproblems that pre- vent the use of the F values predicted in this study for soils Figure 10. PredictedKsa t using the Kozeny-Carmanequa- from nonaridregions. It is for that reasonthat more studiesare tion with (circles)and without (squares)the formationfactor needed to quantify the effects of EC and sodicityon pore versusthe measuredKsa t. geometryand pore distribution.The individualcontribution of eachone of the chemicalparameters will providethe essential informationto developa conceptualmodel for the prediction ceptualmodel of the effect of the chemicalproperties on the of hydraulicproperties. Also, measurementsof tortuosityand pore spaceand pore distributionin the soil. connectivitywith independentmethodologies, such as time domainreflectometry or acoustictechniques, will providethe Acknowledgment. The authorsthank Richard Myers from San Di- necessaryinformation for a more physicallybased model. The ego Petrographics;his carefullyprepared soil thin sectionsare greatly results of such measurements will be addressed in future ex- appreciated. periments. References 6. Conclusions Adler, P.M., C. G. Jacquin,and J. F. Thovert, The formationfactor of reconstructedporous media, Water Resour. Res., 28, 1571-1576, The proposedmethodology provides a quantificationof pa- 1992. rametersof the pores at a scalenot possibleto measurewith Ahuja, L. R., D. K. Cassel,R. R. Bruce, and B. B. Barnes,Evaluation macroscopictechniques. 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