On the relationships between the pore size distribution index and characteristics of the soil hydraulic functions

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1 WATER RESOURCES RESEARCH, VOL. 41, W07019, doi: /2004wr003511, 2005 On the relationships between the pore size distribution index and characteristics of the soil hydraulic functions S. Assouline Institute of Soil, Water and Environmental Sciences, Agricultural Research Organization, Volcani Center, Bet Dagan, Israel Received 21 July 2004; revised 6 March 2005; accepted 4 April 2005; published 19 July [1] Continuous mathematical expressions of the soil hydraulic functions, namely, the water retention curve (WRC) and the relative hydraulic conductivity function (RHC), are indispensable for the solution of the equations describing flow processes in soils. The WRC model of Brooks and Corey is a widely used empirical, easy to derive but discontinuous expression that introduces the pore size distribution index, l, and Mualem s approach is widely used to predict the RHC in terms of the WRC. When this approach is applied to the Brooks and Corey expression, the prediction of the RHC depends solely on l. The present study used measured hydraulic functions of a wide range of soil types to derive parameter equivalence between the parameters y c and l of the Brooks and Corey expression and the parameters x and m of the continuous expression that results from the WRC model of Assouline et al. Strong relationships were found between m and l and between x and (y c l ). Also, it was found that l is inversely proportional to the coefficient of variation, e, of the WRC. On the basis of a previous relationship between the parameter h in the RHC model of Assouline and e a relationship between l and h is derived. This leads to a new simple expression for the prediction of the RHC. Citation: Assouline, S. (2005), On the relationships between the pore size distribution index and characteristics of the soil hydraulic functions, Water Resour. Res., 41, W07019, doi: /2004wr Introduction [2] The soil hydraulic functions, namely, the water retention curve (WRC) and the hydraulic conductivity function, are indispensable for the solution of the equations describing flow processes in soils. The WRC describes the relationship between the soil capillary head, y, and the volumetric water content, q, or the effective saturation degree, S e, defined as S e =(q q r )/(q s q r ) where q s and q r are the saturated and residual volumetric water contents, respectively. The hydraulic conductivity function describes the relationship between the unsaturated hydraulic conductivity, K, and q, S e,ory. This function can be expressed relative to the saturated hydraulic conductivity, K sat,ofthe soil of interest, and is then designated the relative hydraulic conductivity function (RHC), K r. [3] The numerical solution of unsaturated flow problems requires continuous functions of the WRC, and intensive efforts are invested in developing appropriate mathematical expressions. One approach involves finding appropriate empirical functions that lead to a good description of the data [Brooks and Corey, 1964; van Genuchten, 1980; Russo, 1988]. Another approach involves the development of expressions for the WRC based on the similarity in shape between the WRC and the particle size distribution of the soil [Arya and Paris, 1981; Haverkamp and Parlange, 1986; Kosugi, 1994; Assouline et al., 1998; Or and Tuller, 1999]. Fractal representation of soils [Tyler and Wheatcraft, 1989; Rieu and Sposito, 1991; Pachepsky et al., Copyright 2005 by the American Geophysical Union /05/2004WR W ] and conceptualization of the soil random structure [Chan and Govindaraju, 2003] were also used to derive expressions for the WRC. However, for some analytical solutions of flow problems, easy to derive expressions are preferred, even though they are not continuous functions. The expression suggested by Brooks and Corey [1964] is a widely used empirical model of this class. This model introduces the pore size distribution index, which relates information on pore size distribution to soil water retention in a power expression that is convenient for analytical derivations. Another widely used empirical model that represents the class of the continuous functions and is applied in most of the codes for numerical solution of the unsaturated flow equation was suggested by van Genuchten [1980]. To allow continuity between analytical and numerical studies, where these two expressions are used, van Genuchten [1980] and Morel-Seytoux et al. [1996] have provided different ways to convert Brooks and Corey parameters to van Genuchten parameters. [4] The mathematical expression that arises from the WRC model of Assouline et al. [1998] is also a continuous function that represents the WRC over the whole range of y. This model links the soil particle volume distribution to the corresponding pore volume distribution, by assuming a series of sequential fragmentations caused by cycles of wetting and drying, physical, chemical and biological effects, and cultivation practices. It has been found to be flexible enough to represent a wide range of WRC shapes. The first objective of the present study was to consider the parameter equivalence and the relationships between the pore size distribution index of Brooks and Corey [1964] 1of8

2 W07019 ASSOULINE: PORE SIZE DISTRIBUTION INDEX AND HYDRAULIC FUNCTIONS W07019 and the parameters of the WRC expression suggested by Assouline et al. [1998]. [5] The introduction of the pore size distribution index provided the basis for models of the RHC [Corey, 1992]. Since the experimental determination of the RHC is tedious and time consuming, considerable efforts are invested in developing models that can predict the RHC from WRC characteristics [Childs and Collis-George, 1950; Burdine, 1953; Mualem, 1976; Kosugi, 1999; Schaap and Leij, 2000; Tuller and Or, 2001; Assouline, 2001; Assouline and Tartakovsky, 2001; Chan and Govindaraju, 2003]. The approach of Mualem [1976] leads to the expression of the K r (S e ) relationship in terms of a power function. When the Brooks and Corey expression for the WRC is applied, the power value of K r (S e ) depends solely on the pore size distribution index, l. In the model of Assouline [2001] the power value was found to be strongly related to the coefficient of variation, a statistical characteristic of the WRC that involves its first two moments. This model was found to improve the prediction of the K r (S e ) function [Assouline, 2001; Assouline and Tartakovsky, 2001]. The second objective of the present study was to determine the relationships between the pore size distribution index of the Brooks and Corey expression and the power value in the RHC expression suggested by Assouline [2001]. 2. Expressions for the WRC and the RHC Functions [6] The Brooks and Corey [1964] model for the WRC represents the effective saturation degree, S e, as a power function of the capillary head, y: S e ðyþ ¼ ðy=y c Þ l y < y c S e ðyþ ¼ 1 y y c ð1þ where l is the pore size distribution index, and y c is generally taken as the air entry pressure although it may be slightly smaller [Corey, 1992]. [7] The premise of the WRC model of Assouline et al. [1998] is that the particle volume distribution in soils results from a series of sequential fragmentations. An expression is proposed to represent S e ({y}), assuming that the fragmentation process is uniform and random, and that the probability for a particle to fragment is proportional to its volume: m S e ðyþ ¼ 1 exp x jyj 1 jy L j 1 ; 0 jyj jy L j where x and m are two fitting parameters, and y L is the capillary head corresponding to a very low water content, q L, which represents the limit of the domain of interest of the WRC. The WRC expression in equation (2) is mainly characterized by the first and second moments, namely, the mean, r G, and the variance, s 2 : ð2þ r G ¼ x 1=m Gð1 þ 1=mÞþ1= jy L j ð3þ s 2 ¼ x 2=m Gð1 þ 2=mÞ G ð1 þ 1=mÞ ð4þ where G is the gamma function. The statistical parameter that addresses both of these characteristics is the coefficient of variation, e, defined as the ratio between the square root of the variance and the mean: e ¼ s=r G [8] The model of Mualem [1976] for the RHC is K r ðs e 2 Þ ¼ Se n 6 4 Z Se 0 Z ds y 7 ds 5 y where s is a dummy variable of integration, and n a power for which an optimal value of n = 0.5 was suggested [Mualem, 1976]. The model of Assouline [2001] for the RHC is K r ðs e Þ ¼ Z Se 0 Z 1 0 3h ds y 7 ds 5 y where the power h is a parameter that depends on soil structure and texture. For the soil data set used to calibrate this RHC model, h was found to be related to the coefficient of variation, e, of the WRC (equation (5)) according to h ¼ 1:18 e 0:61 ¼ 0:83 Therefore, with the relationship in equation (8), it is possible to predict the RHC of soils by using equation (7), given their WRC. [9] When the Brooks and Corey function (equation (1)) is adopted to describe the WRC, the RHC expression resulting from the application of Mualem s model (equation (6)) is K r ðs e Þ ¼ S ð2þ2:5l The application of Assouline s model (equation (7)) in that case leads to the expression [Assouline, 2001] K r ðs e e Þ ¼ S ðhþhl e Þ=l Þ=l ð5þ ð6þ ð7þ ð8þ ð9þ ð10þ 3. Soil WRC and RHC Data [10] Sets of measured data representing the WRCs and RHC functions of a wide range of soil textures were used to characterize and validate the parameters equivalence of the WRC and the RHC expressions. The sources for the data sets for the WRCs were the catalog of Mualem [1974], the laboratory subset of the UNSODA database [Nemes et al., 1999], and the work of Laliberte et al. [1966]. In all the cases, data corresponding to the main drying WRC were used. Data for the RHCs are scarcer and the data used were taken from the catalog of Mualem [1974] and from Laliberte et al. [1966]. Intentionally, data corresponding to clay soils were not considered to avoid possible shrinkswell effects. [11] The calibration data set for the WRCs and the RHC functions is given in Table 1. The data were taken from Mualem [1974] and represent soil types from sand to silty 2of8

3 W07019 ASSOULINE: PORE SIZE DISTRIBUTION INDEX AND HYDRAULIC FUNCTIONS W07019 Table 1. Best Fit Values of the Parameters in Equations (1) and (2) With Their Respective Root Mean Square Deviations, h in Equation (7), and the Computed Coefficient of Variation, e (Equation (5)) for the Calibration Data Set a Soil Type Index q s x m Equation (2) l y c Equation (1) h e Sable de riviere Pouder River sand Rehovot sand Gilat sandy loam Rubicon sandy loam Pachapa sandy clay loam Pachapa loam Weld silty clay loam Amarillo silty clay loam Silt Mont Cenis a The values correspond to WRC data with volumetric water content, q, incm 3 /cm 3 and capillary head, y, in bars. RMSD, root mean square deviations. clay loam soils. The value of q s was set equal to the reported volumetric water content at y = 0 and the value of y L was taken as y L = 15 bars. Equations (1), (2) and (7) were fitted to each soil data set, and the best fit values of the parameters along with the computed values of the coefficient of variation, e (equation (5)) are presented in Table 1. The root mean square deviation (RMSD) between computed and measured (y, q) values represent the respective level of agreement between equations (1) and (2) and the WRC data. In 7 cases out of the 10, equation (2) performed better than equation (1). Equation (2) is flexible enough to reproduce step-function-like shapes as well as sigmoidal ones, and its advantage over equation (1) is that it does not present the discontinuity in the derivative of the WRC at the air entry value y c. [12] Details on the validation data set for the WRCs and the RHC functions are given in Table 2. The methodology applied for the calibration and the validation of the different relationships in Figures 1 4 was as follows: Curve fitting procedure was applied to the data from the calibration set to determine the suggested relationships; the fitted curve (solid line) and the 95% confidence limits resulting from the regression (dashed lines) were plotted with the calibration data (solid circles). The data from the validation set were also plotted (open circles) to check the applicability of the fitted relationship within the 95% confidence limits. The curve fitting procedure applied is an iterative nonlinear regression using the Levenberg-Marquardt method to minimize the sum of the squares of the differences between observed and computed values of the dependent variable [Glantz and Slinker, 1990]. 4. Results and Discussion 4.1. WRC Parameters and Characteristics [13] The parameter equivalence between the expressions of Brooks and Corey [1964] and of Assouline et al. [1998] for the WRC is depicted in Figures 1 (top) and 1 (bottom). Table 2. Validation Data Set for the WRC and the RHC Function Soil Type Source Index WRC RHC Silt loam UNSODA 1280 Y Silt loam UNSODA 1281 Y Sand UNSODA 1290 Y Sandy loam UNSODA 1390 Y Silt loam UNSODA 1490 Y Sand UNSODA 2220 Y Sand UNSODA 2540 Y Sandy loam UNSODA 2541 Y Sandy clay loam UNSODA 2542 Y Sand UNSODA 2550 Y Sandy loam UNSODA 2551 Y Sandy clay loam UNSODA 2552 Y Sand UNSODA 3070 Y Sand Mualem [1974] 4124 Y Y Sinai sand Mualem [1974] 4122 Y Y Adelaide dune sand Mualem [1974] 4125 Y Y Bet Dagan sandy loam Mualem [1974] 3508 Y Y Adelanto loam Mualem [1974] 3404 Y Y Guelph loam Mualem [1974] 3407 Y Y Caribou silt loam Mualem [1974] 3301 Y Y Ida silt loam Mualem [1974] 3305 Y Y Adelanto clay loam Mualem [1974] 3106 Y Columbia sandy loam Laliberte et al. [1966] Y Y Touchet silt loam Laliberte et al. [1966] Y Y 3of8

4 W07019 ASSOULINE: PORE SIZE DISTRIBUTION INDEX AND HYDRAULIC FUNCTIONS W07019 Figure 1. The relationships between (top) m and l and (bottom) x and (y l c ). The solid and open circles represent the calibration and the validation data, respectively. The dashed lines represent the 95% confidence limits of the fitted curve (solid line). The fitted relationship between the parameter m and the pore size distribution index l to the calibration data in Table 1 is expressed by m ¼ 1:494 l 1:24 ¼ 0:98 ð11þ The 95% confidence limits of the fitted relationship are represented by the dashed lines. The data from the validation set (open circles) fit well within the 95% confidence limits, indicating that equation (11) represents also the validation data. [14] The parameter x was found strongly related to (y l c ), and the resulting relationship based on the data in Table 1 is x ¼ 1:57 y l c 1:208 ¼ 0:99 ð12þ The data from the validation set are also well represented by this relationship. Consequently, by using the relationships in Equations (11) and (12), it is possible to transform soil WRC data expressed in terms of the Brooks and Corey model to enable application of the Assouline et al. [1998] model and to avoid the discontinuity in the derivative of the WRC when necessary, as in the case of numerical solutions of the flow equations. [15] The information on the soil structure and texture, represented by the WRC, is reflected in its first two moments, namely, the mean and the variance. Moreover, it is assumed that this information is condensed in the ratio between them, through the coefficient of variation, e. The pore size distribution index, l in the Brooks and Corey model (equation (1)) also characterizes soil structure and texture. Therefore a relationship between l and e can be expected. On the basis of the data in Table 1 the l(e) relationship that was found, which is also depicted in Figure 2, is l ¼ 0:721 e 0:857 ¼ 0:98 ð13þ It is a strong relationship that fits also very well the data of the validation set (Table 2) as depicted by the spreading of 4of8

5 W07019 ASSOULINE: PORE SIZE DISTRIBUTION INDEX AND HYDRAULIC FUNCTIONS W07019 Figure 2. The relationship between l, the pore size distribution index in equation (1), and e, the coefficient of variation characterizing the WRC (equation (5)). The solid and open circles represent the calibration and the validation data, respectively. The dashed lines represent the 95% confidence limits of the fitted curve (solid line). the open circles around the fitted curve (solid line) and within the 95% confidence limits (dashed lines). The pore size distribution index, l in the Brooks and Corey model (equation (1)) is generally considered as an empirical parameter [Corey, 1992]. The relationship in equation (13) suggests that l is closely related to the pore size distribution characteristics of the soils as they are represented by the WRC first two moments New RHC Expression [16] The RHC model of Assouline [2001] revealed a relationship between the parameter, h, and the coefficient of variation, e, of the WRC (equation (8)). The relationship between h and e that resulted from the data in Table 1 is shown in Figure 3. It is expressed by h ¼ 1:10 e 0:624 ¼ 0:88 ð14þ which is similar to equation (8). Except for two out of the three silt loam soils (Caribou and Ida soils), the data representing the validation set agree relatively well within the 95% confidence limits of this curve. However, validation for the range 1.5 < e < 3.5 was not available. Figure 3. The relationship between h, the power in equation (7), and e, the coefficient of variation characterizing the WRC (equation (5)). The solid and open circles represent the calibration and the validation data, respectively. The dashed lines represent the 95% confidence limits of the fitted curve (solid line). 5of8

6 W07019 ASSOULINE: PORE SIZE DISTRIBUTION INDEX AND HYDRAULIC FUNCTIONS W07019 Figure 4. The relationship between h, the power in equation (7) and l, the pore size distribution index in equation (1). The solid and open circles represent the calibration and the validation data, respectively. The dashed lines represent the 95% confidence limits of the fitted curve (solid line). [17] On the basis of equations (13) and (14), a relationship may be expected between the pore size distribution index, l, of the Brooks and Corey model for the WRC (equation (1)) and h in the RHC model (equation (7)). For the data in Table 1 this relationship, depicted in Figure 4, is expressed by h ¼ 1:40l 0:717 ¼ 0:84 ð15þ It also represents the data from the validation set, soils 3301 and 3305 (Caribou and Ida silt loams) excluded. In the cases where the pore size distribution index, l, of soils is known, equation (15) can be used to estimate h to predict the RHC function according to Assouline s [2001] model. It also allows an alternative expression to the RHC when the Brooks and Corey model for the WRC (equation (1)) is applied: Considering the RHC function in equation (10) along with the h(l) relationship in equation (15), the following general expression for K r (S e ) can be written where K r ðs e Þ ¼ Se a ð16þ h i a ¼ a l b þ l ðb 1Þ ð17þ The values for the constants a and b in equation (17) can be the coefficient and the power of l, respectively, in equation (15) or slightly different h(l) expressions that can result from the use of different calibration data sets. Consequently, equations (16) and (17) can be used to predict the RHC of soils. The values of a and b can also be considered as empirical values determined by fitting equations (16) and (17) to measured RHC data of soils of interest. [18] The predictive performance of the new K r (S e ) expression in Equations (16) and (17) with a = 1.40 and b = (equation (15)), was evaluated comparatively to that of equation (7) when h is determined by best fit procedure, and to that of the predictive models in equation (9) and in equation (7) when h is determined by applying equation (14). The resulting K r (S e ) values for the soils of the calibration and the validation sets (Tables 1 and 2) were compared with the measured data, and the corresponding RMSD was computed for each soil and presented in Table 3. The new K r (S e ) expression in Equations (16) and (17) provided one of the two best results in 10 out of the 18 cases, when compared to those of equation (9) or Equations (7) and (14). The mean all the 18 cases is also shown in Table 3. When h was determined by best fit procedure, equation (7) presented the best overall performance. However, the three predictive models presented a similar general performance. Therefore the new expression in Equations (16), (17) and (15) can be considered as an alternative RHC model that, along with additional available ones, can contribute to determine a predicted range for K r (S e ) rather than a single, not always, accurate function. This can be valuable to estimate the sensitivity of the solution of flow and transport equations to K r (S e ), or to develop pedotransfer functions. 5. Summary and Conclusion [19] The Brooks and Corey [1964] model is a widely used two-parameter expression for the WRC, that is easy to derive but is discontinuous. Corey [1992] considered one of the parameters, y c, to be a physical one, as it represents the air entry pressure, assumed to correspond to the largest pore size, whereas the second parameter, l, the pore size distribution index, is an empirical constant. The model of Assouline et al. [1998] relates the WRC to the pore size distribution, as it assumes that soil structure evolves by means of a uniform random fragmentation process. The 6of8

7 W07019 ASSOULINE: PORE SIZE DISTRIBUTION INDEX AND HYDRAULIC FUNCTIONS W07019 Table 3. Root Mean Square Deviations Between the Computed RHC According to the Four Models and the Corresponding Measured RHC Data a Soil type Equation (7) Equation (7) Using Equation (14) Equation (9) Equations (16) and (17) Using Equation (15) Sable de riviere Pouder River sand Rehovot sand Gilat sandy loam Rubicon sandy loam Pachapa sandy clay loam Pachapa loam Weld silty clay loam Amarillo silty clay loam Silt Mont Cenis Sand Sinai sand Adelaide dune sand Bet Dagan sandy loam Adelanto loam Guelph loam Columbia sandy loam Touchet silt loam Mean RMSD a The four models are equation (7) with h from best fit; equation (7) with h from equation (14); equation (9); and equations (16), (17), and (15). resulting mathematical expression for the WRC is a continuous function with two parameters, x and m. Clear parameter equivalence was established between x and (y c l ), and m and l. Consequently, WRC data described in terms of the discontinuous function of Brooks and Corey can be easily converted to the expression of Assouline et al. [1998] when continuity is required, as they can be converted to the expression of van Genuchten [1980] by using the parameter equivalence suggested by van Genuchten [1980] or Morel- Seytoux et al. [1996]. [20] The information on the soil pore size distribution, contained within the WRC, can be expressed by the two first moments of the WRC, or by their ratio, the coefficient of variation, e, which can be estimated and expressed in terms of x and m. On the basis of the data of hydraulic properties of a wide range of soil types, a relationship between l and e has been established, showing that l is inversely proportional to the coefficient of variation, e, of the WRC. This reduces the empirical nature of this parameter and relates it to the statistical characteristics of the pore size distribution itself, as they are expressed through e. [21] Assouline s [2001] model relates the RHC to primary soil properties, such as pore geometry and soil structure, and this enabled a relationship between h, the power value in the RHC model, and e to be found previously. In light of the established l(e) relationship, in the present study h was found to be related to l, and the first consequence of this result is that it enables the RHC of soils characterized by their pore size distribution index, l, to be predicted by means of Assouline s [2001] model. The second consequence is that it offers an alternative expression for estimating the RHC when the Brooks and Corey model for the WRC (equation (1)) is used. Consequently, the new expression (equations (16) and (17)), along with other available K r (S e ) models, can contribute to determine a predicted range for K r (S e ) rather than a single function. [22] Acknowledgments. Thanks are owed to S. Friedman for his insightful comments and to John Selker and three anonymous reviewers for their constructive remarks. This research was supported by research grant US R from BARD, the United States Israel Binational Agricultural Research and Development Fund, which is gratefully acknowledged. Contribution of the Agricultural Research Organization, Institute of Soil, Water and Environmental Sciences, Bet Dagan, Israel, 601/05. References Arya, L. M., and J. F. Paris (1981), A physicoempirical model to predict the soil moisture characteristic from particle-size distribution and bulk density data, Soil Sci. Soc. Am. J., 45, Assouline, S. 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Collis-George (1950), The permeability of porous materials, Proc. R. Soc. London, Ser. A, 201, Corey, A. T. (1992), Pore-size distribution, in Proceedings of the International Workshop, Indirect Methods for Estimating the Hydraulic Properties of Unsaturated Soils, edited by M. T. van Genuchten, F. J. Leij, and L. J. Lund, pp , Univ. of Calif., Riverside. Glantz, S. A., and B. K. Slinker (1990), Primer of Applied Regression and Analysis of Variance, McGraw-Hill, New York. Haverkamp, R., and J. Y. Parlange (1986), Predicting the water retention curve from particle size distribution: I. Sandy soils without organic matter, Soil Sci., 142, Kosugi, K. (1994), Three-parameter lognormal distribution model for soil water retention, Water Resour. Res., 30, Kosugi, K. (1999), General model for unsaturated hydraulic conductivity for soils with lognormal pore-size distribution, Soil Sci. Soc. Am. J., 63, of8

8 W07019 ASSOULINE: PORE SIZE DISTRIBUTION INDEX AND HYDRAULIC FUNCTIONS W07019 Laliberte, G. E., A. T. Corey, and R. H. Brooks (1966), Properties of unsaturated porous media, Hydrol. Pap. 17, Colo. State Univ., Fort Collins. Morel-Seytoux, H. J., P. D. Meyer, M. Nachabe, J. Touma, M. T. van Genuchten, and R. J. Lenhard (1996), Parameter equivalence for the Brooks-Corey and van Genuchten soil characteristics: Preserving the effective capillary drive, Water Resour. Res., 32, Mualem, Y. (1974), A Catalogue of the Hydraulic Properties of Unsaturated Soils, Technion Isr. Inst. of Technol., Haifa. Mualem, Y. (1976), A new model of predicting the hydraulic conductivity of unsaturated porous media, Water Resour. Res., 12, Nemes, A., M. Schaap, and F. J. Leij (1999), The UNSODA unsaturated soil hydraulic database, version 2.0, report, U.S. Salinity Lab., Riverside, Calif. Or, D., and M. Tuller (1999), Liquid retention and interfacial area in variably saturated porous media: Upscaling from single-pore to sample-scale model, Water Resour. Res., 35, Pachepsky, Y. A., R. A. Shcherbakov, and L. P. Korsunskaya (1995), Scaling of soil water retention using a fractal model, Soil Sci., 159, Rieu, M., and G. Sposito (1991), Fractal fragmentation, soil porosity and soil water properties: I. Theory, Soil Sci. Soc. Am. J, 55, Russo, D. (1988), Determining soil hydraulic properties by parameter estimation: On the selection of a model for the hydraulic properties, Water Resour. Res., 24, Schaap, M., and F. J. Leij (2000), Improved prediction of unsaturated hydraulic conductivity with the Mualem van Genuchten model, Soil Sci. Soc. Am. J., 64, Tuller, M., and D. Or (2001), Hydraulic conductivity of variably saturated porous media: Film and corner flow in angular pore space, Water Resour. Res., 37, Tyler, S. W., and S. W. Wheatcraft (1989), Application of fractal mathematics to soil water retention estimation, Soil Sci. Soc. Am. J., 53, van Genuchten, M. T. (1980), A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J., 44, S. Assouline, Institute of Soil, Water and Environmental Sciences, ARO, Volcani Center, P.O.B. 6, Bet Dagan 50250, Israel. (vwshmuel@ agri.gov.il) 8of8