A multiscale probabilistic collocation method for subsurface flow in heterogeneous media

Abstract

Owing to the spatial variability of the media properties, uncertainty quantification for subsurface flow and solute transport usually requires high‐resolution simulations. In this work, a multiscale probabilistic collocation method (MSPCM) is developed for solving such problems in a computationally efficient manner. The subsurface flow problem is cast in a stochastic framework, and probabilistic collocation strategy is used to represent the original stochastic differential equation. The resulting equations are a set of decoupled deterministic equations with respect to collocation points. A multiscale finite element method is utilized to solve these deterministic problems on a coarse mesh. Coarse‐scale basis functions are constructed on a field in which the conductivity varies spatially at each set of stochastic collocation points. The coarse‐scale solution is then obtained by solving a modified coarse formulation that takes into account the fine‐scale heterogeneity. The fine‐scale solution is reconstructed after the coarse‐scale solution is available. Since the PCM and multiscale finite element method are implemented at different levels, the MSPCM inherits their respective advantages, in which a stochastic problem is decomposed by fewer realizations and is solved on a coarser grid. The performance of the proposed method is demonstrated with numerical examples. The capability of MSPCM in reproducing the probability density functions (PDFs) of head and velocity is investigated. The numerical results show that the MSPCM with proper coarsening level is able to capture small‐scale heterogeneity with a coarse mesh and to generate satisfactory probability density functions of head and velocity.