Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods

Abstract

An efficient method for uncertainty analysis of flow in random porous media is explored in this study, on the basis of combination of Karhunen‐Loeve expansion and probabilistic collocation method (PCM). The random log transformed hydraulic conductivity field is represented by the Karhunen‐Loeve expansion and the hydraulic head is expressed by the polynomial chaos expansion. Probabilistic collocation method is used to determine the coefficients of the polynomial chaos expansion by solving for the hydraulic head fields for different sets of collocation points. The procedure is straightforward and analogous to the Monte Carlo method, but the number of simulations required in PCM is significantly reduced. Steady state flows in saturated random porous media are simulated with the probabilistic collocation method, and comparisons are made with other stochastic methods: Monte Carlo method, the traditional polynomial chaos expansion (PCE) approach based on Galerkin scheme, and the moment‐equation approach based on Karhunen‐Loeve expansion (KLME). This study reveals that PCM and KLME are more efficient than the Galerkin PCE approach. While the computational efforts are greatly reduced compared to the direct sampling Monte Carlo method, the PCM and KLME approaches are able to accurately estimate the statistical moments and probability density function of the hydraulic head.