Conditional simulations of flow in randomly heterogeneous porous media using a KL-based moment-equation approach

Abstract

In this study, we extend the KLME approach, a moment-equation approach based on the Karhunen–Loève decomposition (KL), developed by Zhang and Lu [An efficient, higher-order perturbation approach for flow in randomly heterogeneous porous media via Karhunen–Loève decomposition. J Comput Phys 2004;194(2):773–94] to efficiently incorporate existing direct measurements of the log hydraulic conductivity. We first decompose the conditional log hydraulic conductivity Y = ln K s as an infinite series on the basis of a set of orthogonal Gaussian standard random variables { ξ i }. The coefficients of this series are related to eigenvalues and eigenfunctions of the conditional covariance function of the log hydraulic conductivity. We then write head as an infinite series whose terms h ( n) represent the head contribution at the nth order in terms of σ Y , the standard deviation of Y, and derive a set of recursive equations for h ( n) . We assume that h ( n) can be expressed as infinite series in terms of the products of n Gaussian random variables. The coefficients in these series are determined by substituting decompositions of Y and h ( m) , m < n, into those recursive equations. We solve the conditional mean head up to fourth-order in σ Y and the conditional head covariances up to third-order in σ Y 2 . The higher-order corrections for the conditional mean flux and flux covariance can be determined directly from the higher-order moments of the head, using Darcy’s law. We compare the results from the KLME approach against those from Monte Carlo (MC) simulations and the conventional first-order moment method. It is evident that the KLME approach with higher-order corrections is superior to the conventional first-order approximations and is computationally more efficient than both the Monte Carlo simulations and the conventional first-order moment method.