Prediction of Steady State Flow in Randomly Heterogeneous Formations by Conditional Nonlocal Finite Elements

Abstract

We consider the effect of measuring randomly varying hydraulic conductivity K(x) on one’s ability to predict deterministically, without any upscaling, two-dimensional steady state flow subject to random sources and/or boundary conditions. Such prediction is possible by means of first ensemble moments of heads and fluxes, conditioned on measured values of K(x); the uncertainty associated with such prediction can be quantified by means of the corresponding conditional second moments. As these predictors vary generally more smoothly over space than their random counterparts, they are resolved on coarser grids without upscaling by nonlocal Galerkin finite elements. We compare the head and flux predictions resulted from using two methodologies of inferring conditional ensemble moments of K(x) from available data. The first approach relies on the known statistical distribution of K(x) to generate conditional (thus non-stationary) fields of the natural log-hydraulic conductivity Y(x) = ln K(x) with prescribed mean and variance. In the second approach, the experimental measurements of Y at selected locations are used (by means of Kriging) to estimate Y(x) at points where it is not known, and to evaluate autocovariance of estimation error associated with such a prediction. The results obtained from both approaches are compared with conditional Monte Carlo simulation (MCS). Our nonlocal finite element solution based on the first approach is in excellent agreement with MCS. The nonlocal finite element solution based on the Kriging estimate smoothes spatial variability of the unbiased head and flux predictors, and their covariances.