Scaling of constitutive relationships in unsaturated heterogeneous media: A numerical investigation

Abstract

Numerical models of groundwater flow in the unsaturated zone require relative conductivity and moisture retention curves at a discretization scale much larger than that at which these constitutive relationships are measured. This study seeks to identify simple scaling laws linking parameters of block‐scale constitutive relationships with the parameters of their measurement‐scale counterparts represented here using Gardner‐Russo models. The two parameters of these models, saturated conductivity and pore‐size distribution parameter, are viewed as cross‐correlated regionalized variables with isotropic or anisotropic spatial correlation structures and are simulated using the sequential Gaussian method. Upscaled constitutive curves for steady gravity drainage through three‐dimensional heterogeneous domains are then constructed by repeating numerical flow experiments for a series of applied infiltration rates and plotting the resulting upscaled relative conductivities and average water saturations against average matric potentials. The experiments reveal clear log‐linear relationships between upscaled relative conductivities and average matric potentials, the slopes of which are used to define an upscaled pore‐size distribution parameter. The reciprocal of this quantity, which represents an upscaled characteristic capillary length, is very well approximated by a spatial power average of its measurement‐scale values. Furthermore, the averaging exponent is found to be the same as that associated with the upscaling of saturated conductivities over the same flow domains. When average matric potentials are normalized using this upscaled characteristic capillary length, relative conductivity and moisture saturation values for widely different flow field realizations collapse along well‐defined curves. However, for wet conditions the form of these curves differs from that of the models assumed at the measurement scale.