Estimation of auto-covariance of log hydraulic conductivity from Generalized Sub-Gaussian porosity and particle size random fields

Abstract

We derive analytical formulations relating the spatial covariance ( C Y ) of (log-transformed) hydraulic conductivities to auto- and cross-covariances of porosity ( ϕ ) and representative soil particle sizes within the framework of the classical Terzaghi model. The latter provides an empirical relationship which is widely used to obtain conductivity estimates. We frame the study within recent stochastic approaches and conceptualize appropriate transformations of ϕ and representative soil particle size as Generalized Sub-Gaussian (GSG) spatially cross-correlated random processes. Consistency of the theoretical framework against sample distributions of ϕ and particle size is assessed through the analysis of field data. A perturbation-based approach yields workable expressions of C Y upon truncating the otherwise exact analytical solution at given orders of approximations. Our analytical (truncated) log-conductivity covariance is in agreement with its Monte Carlo-based counterpart. A Global Sensitivity Analysis relying on classical Sobol indices quantifies the relative importance of all parameters embedded in the formulation of C Y . We show that parameters driving the GSG nature of the distribution of (transformed) porosity are key to the main features of C Y . We also document the relevance of properly capturing emergences of possible cross-correlations between ϕ and representative particle size to reconstruct conductivity fields.