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.