Higher-Order Approximations for Saturated Flow in Randomly Heterogeneous Media via Karhunen-Loéve Decomposition

摘要

In this study, we consider transient saturated flow in randomly heterogeneous porous media and try to obtain higher-order solutions of mean head and mean flux, as well as their associated uncertainties based on the combination of Karhunen-Loeve decomposition and perturbation methods. We first decompose the log hydraulic conductivity Y = lnKs as an infinite series on the basis of a set of orthogonal Gaussian standard random variables. The coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of the log hydraulic conductivity. We then write head as an infinite series whose terms h(n) represent the hydraulic head of nth order in terms of σY, 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 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 (and mean flux) up to fourth order in σY and the head (and flux) variances up to third order in σY 2. We conduct Monte Carlo simulations (MC) and compare MC results against approximations of various orders from our moment-equation approach on the basis of Karhunen-Loeve decomposition (KLME). We also compare our results with those from first-order conventional moment-equation approach (CME). It is evident that the KLME approach with higher-order corrections is superior to firstorder approximations and is computationally more efficient than Monte Carlo simulations and the conventional moment-equation approach.