Scaling Water Retention Curves for Soils with Lognormal Pore-Size Distribution

The scaling theory approach has been widely used as an effective method to describe the variation of soil hydraulic properties. In conventional scaling, reference retention curves and scaling factors are determined from minimization of residuals. Most previous studies have shown that scaling factors are lognormally distributed. In this study, we derived physically based scaling factors, assuming that soils are characterized by a lognormal pore-size distribution function. The theory was tested for three sets of retention data. Two data sets included samples of a sandy loam soil, and one set included samples of a loamy sand soil. Individual soil water retention data were fitted to the retention model proposed by Kosugi (1996). The parameters of the model are the mean and variance of the log-transformed poreradius distribution. Scaling factors and parameters of the reference curve were computed directly from the parameters of individual soil water retention functions. Assuming that (i) the soil pore radius of a study area is lognormally distributed and (ii) soil samples are obtained from random sampling of effective soil pore volume from the study area, we have proposed a theoretical interpretation of the lognormal scaling factor distribution. Scaling results for all three data sets compared well with those obtained using the conventional scaling method. D WATER FLOW in soils requires knowledge of the soil hydraulic properties. The hydraulic properties of unsaturated soil are represented by the water retention characteristic (the relationship between the volumetric soil water content, 6, and the capillary pressure head, h, and the unsaturated hydraulic conductivity, K, function. Both properties are variable in hetK. Kosugi, Graduate School of Agriculture, Kyoto Univ., Kyoto 6068502, Japan; J.W. Hopmans, Dep. of Land, Air, and Water Resources, Hydrology Program, Univ. of California, Davis, CA 95616. Received 15 Aug. 1997. *Corresponding author ([email protected]). Published in Soil Sci. Soc. Am. J. 62:1496-1505 (1998). erogeneous soils, or in field plots that are apparently homogeneous. The concept of similar media was introduced by Miller and Miller (1956) to develop scaling theory for the analysis of such variations in field soils. Scaling provides a means to relate hydraulic properties of different soils to those of a reference soil using scaling factors. In recent years, the scaling theory approach has been widely used as an effective method to describe the variation of soil hydraulic properties. Previous studies proposed different methods to determine scaling factors for soil hydraulic properties. While Russo and Bresler (1980) suggested the scaling method to compute reference hydraulic properties directly from observed hydraulic data, most previous studies employed functional models for the reference retention and conductivity curves. Warrick et al. (1977) adopted polynomial functions to express reference h(6) and K(Q) curves. Simmons et al. (1979) used a logarithmic function for the scaling of the water retention characteristic. The combined soil water retention-hydraulic conductivity model proposed by Brooks and Corey (1964) was used by Ahuja and Williams (1991) in the scaling of 6(/z) and K(h) relationships. Clausnitzer et al. (1992) proposed a method of simultaneous scaling of h(Q) and K(Q) curves by employing the combined soil water retention-hydraulic conductivity model proposed by vanGenuchten (1980). Moreover, Zhang et al. (1993) showed scaling results for four different soil water retention models. Most studies showed that scaling factors are lognormally distributed (e.g., Warrick et al., 1977; Hopmans, 1987; Clausnitzer et al., 1992; Zhang et al., 1993). Abbreviations: C, conventional; PB, physically based; PDF, probability density function; REV, representative elementary volume; RSS, residual sum of squares. KOSUGI AND HOPMANS: SCALING WATER RETENTION CURVES FOR SOILS 1497 On the basis of the lognormal distribution of scaling factors, stochastic models have been proposed to analyze the effects of variable soil hydraulic properties on saturated and unsaturated soil water flow. By employing Monte Carlo simulation and using the scaling factor as a lognormally distributed random variable, Clapp et al. (1983) concluded that heterogeneity in soil hydraulic properties may account for approximately 75% of the observed standard deviation of field water content. Ahuja et al. (1984) examined infiltration phenomena by using lognormally distributed scaling factors for the saturated hydraulic conductivity. Assuming that the scaling factor distribution function is lognormal, the statistical properties of soil water regime (Hopmans and Stricker, 1989; Kim et al., 1997) and solute transport (van Ommen et al., 1989; Bresler and Dagan, 1979) for large soil domains (agricultural field, watershed) were analyzed. Most recently, Nielsen et al. (1998) concluded that scaling opportunities to describe field soil water behavior continue to appear both promising and provocative. Thus, many studies have demonstrated the potential of scaling to describe variability of soil hydraulic properties. However, these studies have not emphasized the statistical significance of the scaling factor distribution. No theoretical interpretation has been proposed for the apparent lognormal distribution of scaling factors. This is most likely so because most previous studies used empirical curve-fitting equations for soil hydraulic properties, which do not address the physical significance of their parameters. The objectives of this study were to present a physically based method of scaling soil water retention curves using the physically based retention model introduced by Kosugi (1996) and to propose a theoretical interpretation for scaling factor distributions. The Kosugi (1996) model assumes the soil pore radii to be lognormally distributed. Consequently, the parameters of the retention model have physical significance and are directly related to the statistical properties of the soil pore-size distribution. The lognormal soil pore-size distribution has been assumed in some previous studies. Based on the fact that many soils show a lognormal particle-size distribution, Brutsaert (1966) proposed the lognormal distribution to describe poresize distribution. Gardner (1956) introduced the possibility of characterizing soil structure using a lognormal pore-size distribution, assuming a relationship between aggregate size and pore size. Most recently, Nimmo (1997) proposed such a relationship and subsequently derived a model to describe the soil structural influence of soil water retention using a lognormal aggregate-size distribution model. Pachepsky et al. (1995) derived the fractal dimension of soil pores assuming a lognormal pore-size distribution. THEORY After establishing the functional form of the soil water retention model for soils with a lognormal pore-size distribution, we show how statistical theorems provide physically based parameters for the soil water retention function of a study field. Based on the similar media concept, the distribution of physically based scaling factors is derived that, combined with the reference retention curve, characterizes the spatial variability of soil water retention data. Lognormal Distribution Model for Soil Pore-Size Distribution and Water Retention Curve The probability density function (PDF) of soil pore radius r, p(r), is defined as (Brutsaert, 1966) p(r) = dSJdr [1] where 5e is the effective saturation Se = (6 er)/(6, 8r) [2] described by 6S and 6r, denoting the saturated and residual volumetric water content (L L~), respectively, and the dimension of p(r) is LT. In Eq. [I], p(r)dr = dSe, represents the volume of pores of radius r —> r + dr per unit effective pore volume of soil. The effective pore volume is defined as the product of the total soil volume and the effective porosity, (6S 6 r) , of the soil. Integrating Eq. [1] yields the cumulative pore-radius distribution function: