Upscaling Schemes and Relationships for the Gardner and van Genuchten Hydraulic Functions for Heterogeneous Soils

Upscaled soil hydraulic properties are needed for many large-scale hydrologic applications such as regional and global climate studies and investigations of land–atmosphere interactions. Many larger scale subsurface flow and contaminant transport studies also require upscaled hydraulic property estimates. The objectives of this study were to develop a methodology for upscaling hydraulic property functions using a p-norm approach, to examine how p-norm values differ for two commonly used soil hydraulic property models (the Gardner and van Genuchten functions), and to investigate the relative sensitivities of p-norms and the effective hydraulic parameters to the degree of soil heterogeneity (expressed in terms of variances and auto-correlation lengths of the hydraulic parameters) and other environmental conditions. The p-norm approach expresses upscaling schemes such that it reduces their sensitivity to uncertainties (heterogeneities) in site conditions. The upscaling schemes are obtained as the result of two new criteria proposed to upscale soil hydraulic properties in this study— one preserving the ensemble vertical moisture flux across the land– atmosphere boundary, and a second preserving the ensemble soil surface moisture content. The effective soil hydraulic parameters of a heterogeneous soil formation are then derived by conceptualizing the formation as an equivalent homogeneous medium that satisfies the upscaling criteria. Upscaling relationships between the Gardner and van Genuchten models can then also be established for steady-state vertical flow using the statistical structures of the hydraulic parameters of these two models as estimated from field measurements. The upscaling scheme is demonstrated using hydraulic property data collected at 84 locations across a site in the Mojave Desert. Our results show that the p-norms generally vary less in magnitude than the effective parameters when the variances of the hydraulic parameters increase. We also show that, in general, p-norm values are better defined for the van Genuchten model than the Gardner model. Hydraulic parameter auto correlations, as defined by correlation lengths, were found to have little impact in relating the upscaling schemes (p-norm values) for the two hydraulic property models, but correlation between the hydraulic parameters within the hydraulic property models can significantly affect p-norm relationships. UNSATURATED flow in the vadose zone is often simulated using closed-form expressions for the soil hydraulic properties involving the soil water retention and unsaturated hydraulic conductivity functions. Some of the more commonly used hydraulic functions include the Gardner–Russo (Gardner, 1958; Russo, 1988), Brooks– Corey (Brooks and Corey, 1964), and van Genuchten (van Genuchten, 1980) models. Soil hydraulic property functions are generally measured at relatively small (local) scales. Major questions remain about how to best aggregate the spatially variable hydraulic properties within a heterogeneous soil volume to obtain effective hydraulic properties (or hydraulic parameters) at the large (field or watershed) scale. Effective parameters are essential for upscaling point-level measurements and processes to larger scales, and for reducing the complexity of hydrologic systems. During the last several decades, much work has been done to develop and examine different averaging (and hence upscaling) schemes. Sharma and Luxmoore (1979) pointed out that the soil–plant–atmospheric interactions are very complex in terms of evaluating the influence of soil heterogeneity on water budget. They found that results very much depended on the coefficient of variation and the frequency distribution function of the Miller– Miller scaling factor, as well as on the specific soil–plant– atmospheric conditions involved. Kim and Stricker (1996) used Monte Carlo simulations to investigate both the separate and simultaneous effects of horizontal heterogeneity in the soil hydraulic properties and rainfall intensity on the statistical properties for various components of the one-dimensional water balance. Their results showed that heterogeneity has a stronger effect on the annual water balance for loam than for sand. Kim et al. (1997) further investigated the impact of heterogeneity on the spatially averaged water budget of the unsaturated zone using an analytical framework (Kim et al., 1996). Zhu et al. (2004) established optimal averaging schemes and probability distribution functions for parameters of several hydraulic functions, using correspondence among those hydraulic models by preserving the macroscopic capillary length and predicting the same vertical moisture flux. In more recent studies, Zhu and Mohanty (2006) investigated the effective hydraulic parameters for transient infiltration in terms of the optimal averaging schemes for the input hydraulic and environmental parameter fields. Specifically, they explored the effects of microtopography (as reflected in surface ponding depth) and hydraulic parameter correlation on the ensemble-mean behavior, as well as on the optimal effective schemes. Zhu et al. (2006) additionally examined the impact of the skewness (thirdorder moment) of hydraulic parameter distributions on “effective” soil hydraulic parameter averaging schemes for heterogeneous soils in a flat landscape. Numerical or field experimental results showed that the distribution skewness is also important for determining the upscaled effective parameters, in addition to the mean and variance. Several studies have investigated conditions for which different hydraulic property functions produce the same or similar hydrologic responses. For example, Warrick (1995) discussed some of the features that need to be J. Zhu and M.H. Young, Desert Research Institute, Division of Hydrologic Sciences, 755 East Flamingo Rd., Las Vegas, NV 89119; M.Th. van Genuchten, U.S. Salinity Lab., 450 West Big Springs Rd., Riverside, CA 92507-4617. Received 14 Mar. 2006. *Corresponding author ([email protected]). Published in Vadose Zone Journal 6:186–195 (2007). Original Research doi:10.2136/vzj2006.0041 a Soil Science Society of America 677 S. Segoe Rd., Madison, WI 53711 USA R e p ro d u c e d fr o m V a d o s e Z o n e J o u rn a l. P u b lis h e d b y S o il S c ie n c e S o c ie ty o f A m e ri c a . A ll c o p y ri g h ts re s e rv e d . 186 Published online February 27, 2007