Quantification of the information content of Darcy fluxes associated with hydraulic conductivity fields evaluated at diverse scales

Abstract

We rest on an Information Theory perspective and assess (i) the average information content and (i) the information shared between Darcy flux fields associated with fields of hydraulic conductivity (K) characterized by differing support (or measurement) scale. We treat hydraulic conductivity as a spatial random field, characterized by a given distribution and correlation structure. The latter is modeled through a truncated power law variogram (TPV), which explicitly takes into account a characteristic length scale of the support volume of K through a lower cutoff scale. We then frame our study in a numerical Monte Carlo context where groundwater flow is evaluated across a collection of realizations of hydraulic conductivity characterized by different values of the TPV parameters and subject to uniform in the mean flow. We quantify information through the Shannon entropy of the probability mass functions of the Darcy flux components as well as the mutual information and the multivariate mutual information respectively shared by pairs and triplets of Darcy flux components related to hydraulic conductivity fields evaluated at diverse scales and associated with various levels of heterogeneity. Partitioning of multivariate mutual information according to unique, redundant and synergetic contributions is also quantified. We found consistent trends (i) in the variation of the average information content with respect to the size of lower cutoff scale and (ii) in the way information is shared between pairs and triplets of Darcy flux components associated with diverse support scales of the underlying conductivities.