Prediction of the saturated hydraulic conductivity from Brooks and Corey s water retention parameters

Transcription

1 WATER RESOURCES RESEARCH, VOL. 49, , doi:1.12/wrcr.2269, 213 Prediction of te saturated ydraulic conductivity from Brooks and Corey s water retention parameters Paolo Nasta, 1,2 Jasper A. Vrugt, 2 and Nunzio Romano 1 Received 2 November 212; revised 2 April 213; accepted 18 April 213; publised 28 May 213. [1] Prediction of flow troug variably saturated porous media requires accurate knowledge of te soil ydraulic properties, namely te water retention function (WRF) and te ydraulic conductivity function (HCF). Unfortunately, direct measurement of te HCF is time consuming and expensive. In tis study, we derive a simple closed-form equation tat predicts te saturated ydraulic conductivity, K s from te WRF parameters of Brooks and Corey (1964). Tis pysically based analytical expression uses an empirical tortuosity parameter () and exploits te information embedded in te measured pore-size distribution. Our proposed model is compared against te current state of te art using more tan 25 soil samples from te Grenoble soil catalog (GRIZZLY) and ydraulic properties of European soils (HYPRES) databases. Results demonstrate tat te proposed model provides better predictions of te saturated ydraulic conductivity values wit reduced size of te 9% confidence intervals of about 3 orders of magnitude. Citation: Nasta, P., J. A. Vrugt, and N. Romano (213), Prediction of te saturated ydraulic conductivity from Brooks and Corey s water retention parameters, Water Resour. Res., 49, , doi:1.12/wrcr Introduction [2] Large-scale application of ydrological models requires explicit knowledge of te soil ydraulic properties, namely te soil water retention function (WRF) and te ydraulic conductivity function (HCF). Te former is a relationsip between te volumetric soil water content, (L 3 L 3 ), and te soil matric pressure ead, (L), wereas te latter relates te soil ydraulic conductivity, K (L T ), to te soil water content or alternatively to te matric pressure ead. Under full saturation, te values of and K are referred to as saturated water content ( s ) and saturated ydraulic conductivity (K s ), respectively. Under very dry conditions, owever, te amount of water tat is immobilized in small aggregates or narrow pores and does not contribute to water flow is conventionally defined as te residual water content ( r ). Te pysical interpretation of tis parameter still remains rater ambiguous [Haverkamp et al., 25; Leij et al., 25]. [3] Even toug conventional laboratory measurements of te WRF are generally practicable [Dane and Hopmans, 22], direct determination of saturated ydraulic conductivity, K s, and (unsaturated) HCF is rater time consuming, tedious, and expensive [Hopmans et al., 22; Scelle et al., 21; Nasta et al., 211]. An alternative solution, 1 Department of Agriculture, Division of Agricultural, Forest and Biosystems Engineering, University of Napoli Federico II, Napoli, Italy. 2 Department of Civil and Environmental Engineering, University of California, Irvine, California, USA. Corresponding autor: P. Nasta, Department of Agriculture, Division of Agricultural, Forest and Biosystems Engineering, University of Napoli Federico II, Napoli, I-855, Italy. (paolo.nasta@unina.it) 213. American Geopysical Union. All Rigts Reserved /13/1.12/wrcr.2269 wic is especially useful for large-scale ydrological modeling, is to indirectly estimate te HCF using measurements of te pore-size distribution [Peters and Durner, 26; Dane et al., 211]. [4] As far as we are concerned, te work of Cilds and Collis-George [195] constitutes te first publised contribution tat predicts te unsaturated HCF from measurements of te soil water retention relationsip, wic in turn provides knowledge of te corresponding pore-size distribution. Te soil is assumed to be made up of a bundle of capillary tubes of different radii and water flow is computed by combining Poiseuille s equation wit Darcy s law [Burdine, 1953; Brutsaert, 1967]. Te model proposed by Mualem [1976] represents te most refined approac of tis kind and as been combined wit popular analytical relationsips to describe te WRF [Brooks and Corey, 1964; van Genucten, 198; Kosugi, 1996]. Bimodal water retention relationsips ave been introduced for structured soils to improve te prediction of te unsaturated ydraulic conductivity [Durner, 1994; Tuller and Or, 22; Dexter and Ricard, 29; Romano et al., 211]. [5] In tis study we focus our attention to predict te saturated ydraulic conductivity, K s, using parameters of te WRF. Tis study extends previous work on tis topic [e.g., Messing, 1989; Rawls et al., 1998; Timlin et al., 1999; Poulsen et al., 1999; Kawamoto et al., 26; Han et al., 28]. Examples include te models proposed by Misra and Parker [199] and Guarracino [27], and more recently Mattews et al. [21]. All tese models use te WRF of van Genucten [198]. Alternative approaces make use of Kozeny-Carman equation and derive te saturated ydraulic conductivity from some properties of te soil pores [Carman, 1939; Auja et al., 1984; Auja et al., 1989; Capuis and Aubertain, 23; Green et al., 23; Aminrun et al., 24]. 2918

2 [6] Te goal of te present study is to introduce a simple, yet robust closed-form equation tat predicts te saturated ydraulic conductivity from Brooks and Corey s WRF parameters s, b, and [Brooks and Corey, 1964]. For simplicity, te residual water content is assumed to be zero. Te proposed model draws inspiration from te seminal work of Cilds and Collis-George [195] and infers te soil water flux from te pore-size distribution using R max as te maximum effective pore radius (corresponding to te bubbling matric ead, b, of Brooks-Corey s retention relation) [Mattews et al., 21]. An empirical macroscopic tortuosity-connectivity calibration parameter, ereafter referred to as, is introduced to fit te measured K s values. [7] We compare te predictions of our model against tose derived wit te parametric relationsips of Misra and Parker [199] and Guarracino [27] using soil water retention and saturated ydraulic conductivity data from te Grenoble soil catalog (GRIZZLY) [Haverkamp et al., 1997; Haverkamp et al., 25] and ydraulic properties of European soils (HYPRES) [Wösten et al., 1999; Lilly et al., 28] databases. 2. Teory 2.1. Parametric Relations for Describing te Soil Water Retention Function [8] We adopt a statistical approac and model te soil as a porous medium made up by a bundle of capillary tubes wit radii, r (L) of different sizes tat are randomly distributed wit a probability function (r) (L ). Te term (r)dr (L L) can be viewed as te fractional volume of pores wit respect to te soil bulk volume. If tese pores are completely filled wit water, te volumetric water content, (r) is defined as Z r ðþ¼ r ðþdr; r were te volume fraction of water is expressed as d ¼ (r)dr. By using te Young-Laplace equation, we can relate te pore radius, to te matric pressure ead, (L), and rewrite equation (1) as follows ðþ¼ Z dr ðþ r d ¼ d Z f ðþd; were f () ¼ d/d (L ) denotes te first derivative of te WRF, also known as te soil water capacity function. Once all pores are filled wit water, te soil is saturated and te volumetric water content is defined as s (L 3 L 3 ). If te pore-size distribution is truncated up to te largest pore radius, ereafter referred to as R max (L) and we assume tat (R max ) ¼ s, ten s ¼ Z Rmax ðþdr: r ð1þ ð2þ ð3þ [9] To describe te soil WRF we use te parametric relation of Brooks and Corey [1964], ereafter referred to as BC-WRF ðþ¼ ð s r Þ b þ r for < b ; ð4aþ ðþ¼ s for b ; ð4bþ were r (L 3 L 3 ) denotes te residual water content, b (L) is te bubbling matric ead, and (-) caracterizes te sape of te WRF. In practice, b is associated wit te largest pore radius (R max ) of te pore-size distribution since it defines te tresold at wic te saturated soil starts draining and air replaces water into te void spaces. Te parameter b is terefore also referred to as te air-entry value. For values of b, te soil pores are completely filled wit water, and te soil ydraulic conductivity can be considered approximately constant and equal to K s. If < b te ydraulic conductivity decreases nonlinearly wit soil water content. Note tat te residual water content is difficult to measure in practice, and is often estimated from extrapolation of te WRF to te (very) dry range [Scelle et al., 21]. For matematical convenience, we assume tat r ¼. Tis leaves us wit tree parameters, s, b, and tat define te WRF Derivation of Predictive Model for K s from Brooks and Corey s WRF [1] If we mimic te soil as a bundle of N parallel nonintersecting capillary tubes wit average (effective) radius r, ten te total discarge Q (i.e., te volumetric water flow rate) (L 3 T ) flowing troug tis ideal porous medium can be computed from Poiseuille s law [Hillel, 1971] Q ¼ N r4 w g J; ð5þ were J (L L ) denotes te ydraulic gradient between te edges of te tubes, g (L T 2 ) represents te gravitational acceleration, and w (ML 3 ) and w (ML T ) signify te density and dynamic viscosity of te fluid (water), respectively. [11] Te specific discarge, q (L T ) troug our ypotetical soil is defined as te total discarge per unit bulk cross-sectional area, A (L 2 ), and is matematically equivalent to q ¼ Q A ¼ Nr2 A r 2 w g r 2 J ¼ " w g A J; ð6þ were " A ¼ Nr 2 /A (L 2 L 2 ) denotes te areal porosity. Te assumption of a single effective pore radius for all N tubes is rater unrealistic. We terefore relax tis assumption and assume tat our ypotetical porous medium is made up of a bundle of nonintersecting straigt cylindrical pores of lengt L s, but wit varying pore radii. If " A,i represents te relative areal porosity of eac pore-size class i, we can estimate te specific discarge as follows 2919

3 q ¼ wg J XNps i¼1 " A;i ri 2 ; ð7þ were N ps signifies te total number of pore-size classes between and R max. If all pores are filled wit water and contribute to flow, ten tis will lead to te specific discarge at full saturation. [12] Te fraction of water-filled pores, PN ps " A;i can also be i¼1 written in a more general continuous form by using equation (3). Tis leads to te following expression for te specific discarge wic is equivalent to q ¼ wg J q ¼ Z wg b J Z Rmax ðþr r 2 dr; ð8aþ f ðþ :149 2 d; ð8bþ wit r.149/jj at 2 C if expressing r in units of cm. [13] A closer inspection of equation (8b) reveals tat te specific discarge is quadratically proportional to te pore radius. Indeed, q will be influenced mostly by te largest pores tat are predominantly generated by te aggregation and arrangement of te primary soil particles [Durner, 1994]. If we assume saturated conditions, ten we can also use Darcy s Law to calculate te average water flux emanating from te soil sample q ¼ K s J ; were J denotes te actual ydraulic gradient between te top and bottom of te porous medium. Tis macroscopic flux depends not only on te distribution of pore sizes (textural effect) and on te arrangement of te solid particles (structural effect), but also on te pore geometry and caracteristics (tortuosity and connectivity effect) and on te pysical properties of te fluid (viscosity and density effect) [Corey, 1994; Dullien, 1975; Walley et al., 212]. Te macroscopic water flux, q in equation (9) will terefore deviate from te specific discarge, q computed in equation (8b). To reconcile tese differences, Carman [1956] proposed to pragmatically adjust te modeled specific discarge using q ¼ q L a ; ð1þ L s were L a denotes te lengt of te tortuous pat followed by te water particles under te actual ydraulic gradient, J. If we now insert equation (1) into equation (8a), solve for q and subsequently combine tis equation wit equation (9), we yield te following expression for te saturated ydraulic conductivity, K s K s J ¼ wg J L Z Rmax s L a ðþr r 2 dr; ð9þ ð11þ [14] Friction losses are generally larger in and among te tortuous tubes. It is terefore reasonable to pose wic results in te following relation K s J L a L s J ¼ J L a ; ð12þ L s ¼ wg Z L Rmax s L a ðþr r 2 dr; ð13þ in wic we ave assumed tat K s K(R max ). Tis equation is equivalent to (see equations (8a) and (8b)) K s ¼ Z wg b f ðþ :149 2 d; ð14þ were ¼ (L s /L a ) 2 represents an empirical parameter ( <<1), commonly referred to as te macroscopic tortuosity-connectivity factor [Carman, 1956; Bear, 1972; Vervoort and Cattle, 23]. [15] If we assume water at a temperature of 2 C, and tus w ¼.998 g cm 3, in units of cm, w ¼.12 g cm s, and g ¼ cm s 2, equation (14) can be written as follows K s ¼ 4: Z b Z b f ðþ 1 2 d; f ðþð:149=þ 2 d¼ 9:579 ð15þ wit K s in cm. [16] Te first derivative, f () of te BC-WRF is computed analytically from equation (4a) for < b f ðþ¼ d d ¼ s b ð Þ ¼ s b ðþ1þ b : ð16þ [17] By combining equations (15) and (16), te saturated ydraulic conductivity, K s (cm ) can now be calculated from te BC-WRF parameters K s ¼ 9: s b wic after integration ¼ 9: s b Z b Z b ð þ1 ð þ3 Þ 1 2 d Þ d b 2 K s ¼ 9: s b þ 2 ¼ 9: þ 2 s 1 b þ2 b ; ð17þ ; ð18þ 292

4 and rearranging leads to te following closed-form relation K s ¼ 9: þ 2 s 2 b ; ð19þ wit b in cm. Tis equation will be referred to as Model- Nasta, Vrugt, Romano (Model-NVR) in te remainder of tis paper and te calculated K s values will be denoted wit K s,nvr Description of Two Oter Existing Models to Predict K s [18] To demonstrate te advantages of te proposed model we compare te predictions of K s,nvr derived wit Model-NVR against tose derived wit te closed-form equations of Guarracino [27] and Misra and Parker [199]. [19] Tese two alternative models bot rely on te WRF proposed by van Genucten [198], ðþ¼ r þ ð s r Þ½1 þ ðjj Š m ; ð2þ were (cm ), m (-), and n (-) are tree sape parameters wose values need to be derived from calibration against te measured retention data. Te value of m is typically derived troug te common relationsip, m ¼ 1/n [Mualem, 1976]. [2] Guarracino [27] estimates te saturated ydraulic conductivity, K s,g (cm ) from te pore-size distribution using a fractal law distribution wic is truncated between a minimum and a maximum pore radius. Tis simple model, ereafter referred to as Model-G, uses te following closed form equation K s;g ¼ 1: D 4 D s 2 ; ð21þ wit fractal dimension, D (-) originally set to 1.996, but considered a calibration parameter in te present analysis. Tis model uses only two parameters of te van Genucten-water retention function (VG-WRF), and s and assumes te inverse of te air entry value, to identify te bubbling matric ead, b. [21] Misra and Parker [199] proposed te following parametric relationsip, ereafter referred to as Model-MP, to estimate K s,mp (cm ) from te VG-WRF parameters and s 3:89 15 K s;mp ¼ 5=2 s 2 ; ð22þ MP wit tortuosity factor, MP (-) originally set at 2.5 [Corey, 1979] and considered constant. [22] Table 1 summarizes te WRF and calibration parameters used in eac of te tree predictive models. In all our calculations, we assume tat r ¼. 3. Soil Databases [23] To evaluate te tree predictive K s models, we use 62 soil samples from te Grenoble soil catalog GRIZZLY [Haverkamp et al., 1997; Haverkamp et al., 25] and 197 Þ n Table 1. Summary of te Tree Different Models Used Herein Wit te WRF-Type (BC or VG), Number of Parameters (p), WRF Parameters and Empirical Parameters soil samples from te HYPRES database [Wösten et al., 1999; Lilly et al., 28]. Data consist of soil bulk density, b (g cm 3 ), texture classes (% clay, silt and sand), saturated water content, s (cm 3 cm 3 ), water retention data, () and saturated ydraulic conductivity K s,obs (cm ). [24] Te U.S. department of agriculture texture distribution for te different soil samples is presented in Figure 1. All, but two texture classes (silty and sandy-clay) are represented. Quite conveniently, te GRIZZLY database includes estimates of te BC-WRF parameters, b (cm) and (-) and VG-WRF parameters, (cm ), n (-), and m (-). For HYPRES te BC-WRF (equation (4)) and VG-WRF (equation (2)) parameters were derived by nonlinear optimization in MATLAB (te MatWorks, Inc.) [Coleman and Li, 1996]. Summary statistics of te soil properties and ydraulic parameters of bot databases are reported in Table 2. Te soil caracteristics of bot databases are quite similar, wit te exception of K s tat differs considerably. 4. Results 4.1. Case Study I: Te GRIZZLY Database [25] Eac of te tree models considered erein contains a single empirical parameter, {, D, MP } tat requires calibration against measured K s data (K s,obs ). We use te MATLAB Optimization Toolbox (Te MatWorks, Inc.) [Coleman and Li, 1996], and calibrate eac model by minimizing te sum of squared error (SSE) between (log 1 - transformed) observed, K s,obs and predicted, K s,pred saturated ydraulic conductivity values SSEðÞ ¼ XNs i¼1 Model-NVR Model-G Model-MP WRF BC VG VG p WRF parameters s,, b s, s, Empirical parameter D MP i i 2; log 1 log 1 Ks;pred i ðþ ð23þ K i s;obs using te N s ¼ 62 samples of te GRIZZLY database. [26] Table 3 lists te optimized values of te empirical parameters of Model-NVR, G, and MP, respectively, and teir corresponding SSE values. Note tat te optimal values of Model-G and Model-MP are comparable to teir original (default) values of D ¼ and MP ¼ 2.5, respectively. [27] We now evaluate te predictive capabilities of eac individual model. Te results are displayed in te left column of Figure 2 wic sows a comparison between te observed and estimated K s values for Model-NVR (Figures 2a and 2b), Model-G (Figures 2c and 2d) and Model-MP (Figures 2e and 2f). For convenience, te ydraulic conductivity values are plotted on a log-log scale (base-1) and te 1:1 line is grapically summarized (dased line). Te rigt column of Figure 2 plots istograms of te log 1 transformed residuals. 2921

5 Figure 1. Soil texture distribution (Sa¼sand; Lo¼loam; Si¼silt; Cl¼clay) of te 62 soil samples of te GRIZZLY database (open squares) and of te 197 soil samples of te HYPRES database (solid circles). [28] Te left subplots in Figure 2 illustrate tat te predictions of te proposed Model-NVR cluster most closely around te 1:1 line wit 9% confidence intervals derived from a Student s t distribution (wit t ¼ for 62 ¼ 61 degrees of freedom) tat appear most tigt. Model-MP exibits te poorest performance, wit te largest scatter and spread from all te tree considered models. For all te tree models te residuals are approximately Gaussian (according to te 2 test) and tus log-normally distributed. Model-G and Model-MP exibit somewat larger residuals tan Model-NVR. Altogeter, te results presented in Figure 2 tus favor te proposed Model-NVR for predicting te saturated ydraulic conductivity of te samples in te GRIZZLY database. [29] Te performance of te tree models as also been quantified using simple summary statistics of te data fit. Table 3 lists te root mean square error (RMSE), te coefficient of determination (R 2 ), Akaike information criterion (AIC) and Bayes information criterion (BIC). All of te tree models demonstrate a relatively good agreement between te predicted and observed K s values wit RMSE values tat are smaller tan 1 order of magnitude. Model- NVR exibits te best performance wit te lowest RMSE values, te igest R 2 and te smallest average spread of te 9% confidence limits. Te AIC and BIC can be used to decide wic of te tree models is most favored given te available data. Tese two measures are defined troug te following information criterion I i ¼ 2ln L max;i þ ð pi Þ; ð24þ were L max (L 2 T) is te (maximum) likeliood of model i and (p i ) represents a penalty term tat penalizes for te number of parameters, p [Diks and Vrugt, 21]. Indeed, te AIC and BIC diagnostics trade off quality of fit against model complexity. If te residuals are Gaussian distributed (see Figure 2), te value of L max can be computed from te SSE using SSE 2ln L max ¼ N s ln þ N s : N s 1 ð25þ [3] Te penalty term for AIC is (p) ¼ 2p and for BIC tis term is given by (p) ¼ pln(n s ). Te model wit te lowest values for AIC and/or BIC is most supported by te available data. Note tat AIC and BIC typically use te number of calibration parameters as measure of model complexity (penalty term). Eac model used erein contains one empirical calibration parameter, and tus we can resort to metrics suc as te RMSE and SSE to decide wic model is statistically preferred. Yet we purposely Table 2. Descriptive Statistics of te 62 Soil Samples of te GRIZZLY Database and 197 Samples of te HYPRES Database GRIZZLY HYPRES Variables Units Mean SD Min Max Mean SD Min Max b gcm Clay % Silt % Sand % s cm 3 cm log 1 K s cm log 1 b cm log 1 cm n m

6 Table 3. Calibrated Values of te Empirical Parameters (, D, and MP ) for Eac of te Tree Different Models (NVR, G, MP) Using te Samples From te GRIZZLY Data Set a Model-NVR Model-G Model-MP Empirical parameter ¼.138 D ¼ MP ¼ 39.9 SSE RMSE R AIC BIC a Model fit is expressed by te sum of squared error (SSE), te root mean square error (RMSE), te coefficient of determination (R 2 ), Akaike s information criterion (AIC) and Bayes information criterion (BIC). use te AIC and BIC to explicitly recognize differences in te number of input (WRF) parameters. [31] Bot te AIC and BIC listed in Table 3 indicate tat Model-NVR is preferred over Model-G and Model-MP. In oter words, te prediction of K s is significantly improved by adding information about te pore-size distribution troug te BC-WRF parameter Case Study II: Te HYPRES Database [32] To test te predictive performance of eac of te tree models we now use te HYPRES data set using te calibrated empirical parameters listed in Table 3. We grapically summarize te results in Figure 3 using a similar format as used previously in Figure 2. Te left column compares measured and predicted ydraulic conductivities for eac of te tree considered parametric models, wereas te rigt column plots istograms of te associated error residuals. [33] Te scatter plots at te left and side demonstrate tat te observed K s values of te HYPRES samples are substantially iger tan tose measured previously for te GRIZZLY samples (Table 2). Tis difference does not seem to affect te performance of te proposed Model- NVR. Te RMSE of tis model (.78) is almost identical to its counterpart of.76 derived for te GRIZZLY database. Tis performance is significantly better tan tat of Model- G and Model-MP wic exibit RMSE values of 1.21 and 1.24, respectively. [34] Te predictions of Model-NVR group closely around te 1:1 line wit 9% confidence intervals derived from Student s t distribution (wit t ¼ for 197 ¼ 196 Figure 2. GRIZZLY database: Comparison between observed and predicted K s values. Left column Twodimensional scatter plots of observed and predicted saturated ydraulic conductivities for models, (a) NVR, (c) G, and (e) MP. For convenience, we also include te 1:1 line (dased line) and 9% confidence limits (dotted lines). Rigt column: Histograms of te corresponding residuals of te (b) NVR, (d) G, and (f) MP models. Figure 3. HYPRES database: Comparison between observed and predicted K s values. Left column Twodimensional scatter plots of observed and predicted saturated ydraulic conductivities for models, (a) NVR, (c) G, and (e) MP. Te 1:1 line (dased line) and 9% confidence limits (dotted lines) are conveniently included. Rigt column: Histograms of te corresponding residuals of te (b) NVR, (d) G, and (f) MP models. 2923

7 Table 4. Performance of Eac of te Tree Prediction Models (NVR, G, MP) Using te 197 Soil Samples From te HYPRES Data Set a degrees of freedom) wic are substantially smaller tan tose derived wit te oter two models. Moreover, te istograms of te (log)-residuals computed wit Model-NVR center around zero, wereas a significant bias is observed for te oter two models. Altogeter tese results demonstrate an improved ability of Model NVR to predict te measured K s values. [35] Tis conclusion is furter supported by te summary diagnostics of te model fits listed in Table 4. Note, owever tat all of te tree models exibit a relatively low coefficient of determination (R 2 ). Tis igligts tat single-average calibrated values of te empirical parameters cannot fully capture te observed variation. Te AIC and BIC criteria once again indicate tat te exploitation of all WRF parameters ( s, b, and ) in te proposed Model- NVR as desirable statistical advantages. Te fit to te observed K s data is significantly enanced and te prediction uncertainty reduced. 4. Conclusion [36] Wen direct measurement of te saturated ydraulic conductivity is not affordable, it is necessary to use an alternative approac tat predicts te K s values from available soil ydraulic properties. In tis paper we introduced a simple but robust closed-form parametric relation tat predicts te saturated ydraulic conductivity from te BC water retention parameters. Te proposed model not only significantly improves te prediction of te saturated ydraulic conductivity, but te associated confidence intervals are also considerably smaller. We posit tat furter improvements are possible, if te procedure for te calibration of is furter refined. Notation A Bulk cross-sectional area of te soil, L 2 D Fractal dimension in Model-G f () Water capacity function, L g Gravitational acceleration, LT 2 Matric suction ead, L b Bubbling matric ead of te BC-WRF, L I Information criterion J Hydraulic gradient, LL K Hydraulic conductivity, LT Saturated ydraulic conductivity, LT K s Model-NVR Model-G Model-MP SSE RMSE R AIC BIC a Te empirical parameters (, D, and MP ) ave been set to teir calibrated values in Table 3. Te model fit is expressed wit te sum of squared error (SSE), te root mean square error (RMSE), te coefficient of determination (R 2 ), Akaike s information criterion (AIC) and Bayes information criterion (BIC). L s Straigt lengt of te bundle of cylindrical pores, L L a Actual lengt of te bundle of cylindrical pores, L L max Maximum likeliood, L 2 T m Sape parameter of te VG-WRF N Number of capillary pores N s Number of samples N ps Number of pore-size classes n Sape parameter of te VG-WRF p Number of parameters (r) Probability distribution function of pores, L Q Volumetric flow rate, L 3 T q Specific discarge, LT r Pore radius, L R max Maximum pore radius, L Sape parameter of te VG-WRF, L " A Soil areal porosity, L 2 L 2 w Dynamic viscosity of water, ML T Pore-size distribution index of te BC-WRF w Water density, ML 3 Macroscopic tortuosity-connectivity in Model- NVR MP Tortuosity in te Model-MP Penalty term in te information criterion Soil water content, L 3 L 3 r Residual water content, L 3 L 3 Saturated water content, L 3 L 3 s [37] Acknowledgments. Te autors wis to tank R. Haverkamp and A. Lilly for graciously providing te GRIZZLY and HYPRES data sets. We also acknowledge te comments and suggestions of W. Durner and two anonymous reviewers tat ave enanced te quality of te current version of tis paper. References Auja, L. R., J. W. Naney, P. E. Green, and D. R. Nielsen (1984), Macroporosity to caracterize spatial variability of ydraulic conductivity and effects of land management, Soil Sci. Soc. Am. J., 48, Auja, L. R., D. K. Cassel, R. R. Bruce, and B. B. Barns (1989), Evaluation of spatial distribution of ydraulic conductivity using effective porosity data, Soil Sci., 148, Aminrun, W., M. M. Amin, and S. M. Eltaib (24), Effective porosity of paddy soils as an estimate of its saturated ydraulic conductivity, Geoderma, 121, Bear, J. (1972), Dynamics of Fluids in Porous Media, Elsevier, New York. Brooks, R. H., and A. T. Corey (1964), Hydraulic properties of porous media, in Hydrology Paper No. 3, 27 pp., Colorado State Univ., Fort Collins, Colo. Brutsaert, W. (1967), Some metods of calculating unsaturated permeability, Trans. ASAE, 1, Burdine, N. T. (1953), Relative permeability calculation size distribution data, Trans. Am. Inst. Min. Metall. Pet. Eng., 198, Carman, P. C. (1939), Permeability of sands soils and clays, J. Agric. Sci., 29, Carman, P. C. (1956), Flow of Gases troug Porous Media, Academic, New York. Capuis, R. P., and M. Aubertin (23), On te use of Kozeny-Carman equation to predict te ydraulic conductivity of soils, Can. Geotec. J., 4, Cilds, E. C., and N. Collis-George (195), Te permeability of porous materials, Proc. R. Soc. A, 21, Coleman, T. F., and Y. Li (1996), An interior, trust region approac for nonlinear minimization subject to bounds, SIAM J. Optim., 6, Corey, A. T. (1979), Mecanics of Heterogeneous Fluids in Porous Media (Reprinted 23), Water Resour. Publ., Littleton, Colo. Corey, A. T. (1994), Mecanics of Immiscible Fluids in Porous Media (Reprinted 23), Water Resour. Publ., Higlands Ranc, Colo. 2924

8 Dane, J. H., and J. W. Hopmans (22), Water retention and storage, in Metods of Soil Analysis, Part 4, SSSA Book Ser. 5, edited by J. H. Dane and G. C. Topp, pp , SSSA, Madison, Wis. Dane, J. H., J. A. Vrugt, and E. Unsal (211), Soil ydraulic functions determined from measurements of air permeability, capillary modeling and ig-dimensional parameter estimation, Vadose Zone J., 1, 1 7. Dexter, A. R., and G. Ricard (29), Te saturated ydraulic conductivity of soils wit n-modal pore-size distributions, Geoderma, 154, Diks, C. G. H., and J. A. Vrugt (21), Comparison of point forecast accuracy of model averaging metods in ydrologic applications, Stocastic Environ. Res. Risk Assess., 24, Dullien, F. A. L. (1975), Single pase flow troug porous media and pore structure, Cem. Eng. J., 1, l 34. Durner, W. (1994), Hydraulic conductivity estimation for soils wit eterogeneous pore structure, Water Resour. Res., 3, Green, T. R., L. R. Auja, J. G. Benjamin (23), Advances in predicting agricultural management effects on soil ydraulic properties, Geoderma, 116, Guarracino, L. (27), Estimation of saturated ydraulic conductivity Ks from te van Genucten sape parameter, Water Resour. Res., 43, W1152, doi:1.129/26wr5766. Han, H., D. Gimenez, and A. Lilly (28), Textural averages of saturated soil ydraulic conductivity predicted from water retention data, Geoderma, 146, Haverkamp, R., C. Zammit, F. Boubkraoui, K. Rajkai, J. L. Arrue, and N. Heckmann (1997), GRIZZLY, Grenoble soil catalogue: Soil survey of field data and description of particle-size, soil water retention and ydraulic conductivity functions, Lab. d Etude des Transferts en Hydrol. et en Environ., Grenoble, France. Haverkamp, R., F. J. Leij, C. Fuentes, A. Sciortino, and P. J. Ross (25), Soil water retention: I. Introduction of a sape index, Soil Sci. Soc. Am. J., 69, Hillel, D. (1971), Soil and Water: Pysical Principles and Processes, Academic, New York. Hopmans, J. W., J. Sim?unek, N. Romano, and W. Durner (22), Inverse metods, in Metods of Soil Analysis, Part 4, SSSA Book Ser. 5, edited by J. H. Dane and G. C. Topp, pp , SSSA, Madison, Wis. Kawamoto, K., P. Moldrup, T. P. A. Ferre, M. Tuller, O. H. Jacobsen, and T. Komatsu (26), Linking te Gardner and Campbell models for water retention and ydraulic conductivity in near Saturated soil, Soil Sci., 171, , doi:1.197/1.ss c. Kosugi, K. (1996), Log-normal distribution model for unsaturated soil ydraulic properties, Water Resour. Res., 32, Leij, F. J., R. Haverkamp, C. Fuentes, Z. Zatarain, and P. J. Ross (25), Soil water retention: II. Derivation and application of sape index, Soil Sci. Soc. Am. J., 69, Lilly, A., A. Nemes, W. J. Rawls, and Y. A. Pacepsky (28), Probabilistic approac to te identification of input variables to estimate ydraulic conductivity, Soil Sci. Soc. Am. J., 72, Mattews, G. P., G. M. Laudone, A. S. Gregory, N. R. A. Bird, A. G. D. G. Mattews, and W. R. Walley (21), Measurement and simulation of te effect of compaction on te pore structure and saturated ydraulic conductivity of grassland and arable soil, Water Resour. Res., 46, W551, doi:1.129/29wr772. Messing, I. (1989), Estimation of saturated ydraulic conductivity in clay soils from moisture retention data, Soil Sci. Soc. Am. J., 53, Misra, S., and J. C. Parker (199), On te relation between saturated conductivity and capillary retention caracteristics, Ground Water, 28, Mualem, Y. (1976), A new model for predicting te ydraulic conductivity of unsaturated porous media, Water Resour. Res., 12, Nasta, P., S. Huyn, and J. W. Hopmans (211), Simplified multistep outflow metod to estimate unsaturated ydraulic functions for coarse-textured soils, Soil Sci. Soc. Am. J., 75, Peters, A., and W. Durner (26), Improved estimation of soil water retention caracteristics from ydrostatic column experiments, Water Resour. Res., 42, W1141, doi:1.129/26wr4952. Poulsen, T. G., P. Moldrup, T. Yamaguci, and O. H. Jacobsen (1999), Predicting saturated and unsaturated ydraulic conductivity in undisturbed soils from soil water caracteristics, Soil Sci., 164, Rawls, W. J., D. Gimenez, and R. Grossman (1998), Use of soil texture, bulk density, and slope of te water retention curve to predict saturated ydraulic conductivity, Trans. ASAE, 41, Romano, N., P. Nasta, G. Severino, and J. W. Hopmans (211), Using bimodal log-normal functions to describe soil ydraulic properties, Soil Sci. Soc. Am. J., 75, Scelle, H., S. C. Iden, A. Peters, and W. Durner (21), Analysis of te agreement of soil ydraulic properties obtained from multistep-outflow and evaporation metods, Vadose Zone J., 9, Timlin, D. J., L. R. Auja, Y. Pacepsky, R. D. Williams, D. Gimenez, and W. Rawls (1999), Use of Brooks-Corey parameters to improve estimates of saturated conductivity from effective porosity, Soil Sci. Soc. Am. J., 63, Tuller, M., and D. Or (22), Unsaturated ydraulic conductivity of structured porous media: A review of liquid configuration-based models, Vadose Zone J., 1, van Genucten, M. T. (198), A closed form equation for predicting te ydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J., 144, Vervoort, R. W., and S. R. Cattle (23), Linking ydraulic conductivity and tortuosity parameters to pore space geometry and pore-size distribution, J. Hydrol., 272, Walley, W. R., G. P. Mattews, and S. Ferraris (212), Te effect of compaction and sear deformation of saturated soil on ydraulic conductivity, Soil Tillage Res., 125, Wösten, J. H. M., A. Lilly, A. Nemes, and C. Le Bas (1999), Development and use of a database of ydraulic properties of European soils, Geoderma, 9,