Characterization of flow through random media via Karhunen–Loève expansion: an information theory perspective

We leverage on information theory to assess the fidelity of approximated numerical stochastic groundwater flow simulations. We consider flow in saturated heterogeneous porous media, where the Karhunen–Loève (KL) expansion is used to express the hydraulic conductivity as a spatially correlated random field. We quantify the impact of the KL expansion truncation on the uncertainty associated with punctual values of hydraulic conductivity and flow velocity. In particular, we compare the statistical dependence between variables by considering (a) linear correlation metrics (Pearson coefficient of correlation) and (b) metrics capable of accounting for nonlinear dependence (coefficient of uncertainty based on mutual information). We test the selected metrics by analyzing the relationship between hydraulic conductivity fields generated via Monte Carlo sampling with different levels of truncation of the KL expansion and the corresponding fluid velocity fields, obtained through the numerical solution of Darcy’s flow. Our analysis shows that employing linear correlation metrics leads to a general overestimation of the correlation level and information theory based indicators are valuable tools to assess the impact of the KL truncation on the output velocity values. We then analyze the impact of the number of retained modes on the spatial organization of the velocity field. Results indicates that (i) as the number of modes decrease the spatial correlations of the velocity field increases; (ii) linear indicators of spatial correlation are again larger than their nonlinear counterparts.