Stochastic Simulations for Flow in Nonstationary Randomly Heterogeneous Porous Media Using a KL-Based Moment-Equation Approach

In this study, we extend the Karhunen–Loève moment equation (KLME) approach, an approach based on KL decomposition, to efficiently and accurately quantify uncertainty for flow in nonstationary heterogeneous porous media that include a number of zones with different statistics of the hydraulic conductivity. We first decompose the log hydraulic conductivity Y = lnKs for each zone by the KL decomposition, which is related to a set of eigenvalues and their corresponding orthogonal deterministic eigenfunctions. Based on the decomposition for all individual zones, we develop an algorithm to find the eigenvalues and eigenfunctions for the entire domain. Following the methodology proposed by Zhang and Lu [J. Comput. Phys., 194 (2004), pp. 773–794], we solve the head variability up to second order in terms of σ2 Y and compare the results with those obtained from Monte Carlo (MC) simulations. It is evident that the results from the KLME approach with higher-order corrections are close to those from the MC simulations, but the computational cost for the KLME method is much lower than that for the MC simulations.