An efficient, high-order perturbation approach for flow in random porous media via Karhunen–Loève and polynomial expansions

Abstract

In this study, we attempt to obtain higher-order solutions of the means and (co)variances of hydraulic head for saturated flow in randomly heterogeneous porous media on the basis of the combination of Karhunen–Loève decomposition, polynomial expansion, and perturbation methods. We first decompose the 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 the series are related to eigenvalues and eigenfunctions of the covariance function of log hydraulic conductivity. We then write head as an infinite series whose terms h ( n) represent head of nth order in terms of σ Y , the standard deviation of Y, and derive a set of recursive equations for h ( n) . We then decompose h ( n) with polynomial expansions in terms of the products of n Gaussian random variables. The coefficients in these series are determined by substituting decompositions of Y and h ( n) into those recursive equations. We solve the mean head up to fourth-order in σ Y and the head variances up to third-order in σ Y 2. We conduct Monte Carlo (MC) simulation and compare MC results against approximations of different orders from the moment-equation approach based on Karhunen–Loève decomposition (KLME). We also explore the validity of the KLME approach for different degrees of medium variability and various correlation scales. 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 the Monte Carlo simulation.