A Hybrid HDMR for Mixed Multiscale Finite Element Methods with Application to Flows in Random Porous Media

Abstract

Stochastic modeling has become a popular approach to quantifying uncertainty in flows through heterogeneous porous media. In this approach the uncertainty in the heterogeneous structure of material properties is often parametrized by a high-dimensional random variable, leading to a family of deterministic models. The numerical treatment of this stochastic model becomes very challenging as the dimension of the parameter space increases. To efficiently tackle the high-dimensionality, we propose a hybrid high-dimensional model representation (HDMR) technique, through which the high-dimensional stochastic model is decomposed into a moderate-dimensional stochastic model, in the most active random subspace, and a few one-dimensional stochastic models. The derived low-dimensional stochastic models are solved by incorporating the sparse-grid stochastic collocation method with the proposed hybrid HDMR. In addition, the properties of porous media, such as permeability, often display heterogeneous structure across multiple spatial scales. To treat this heterogeneity we use a mixed multiscale finite element method (MMsFEM). To capture the nonlocal spatial features (i.e., channelized structures) of the porous media and the important effects of random variables, we can hierarchically incorporate the global information individually from each of the random parameters. This significantly enhances the accuracy of the multiscale simulation. Thus, the synergy of the hybrid HDMR and the MMsFEM reduces the dimension of the flow model in both the stochastic and physical spaces, and hence significantly decreases the computational complexity. We analyze the proposed hybrid HDMR technique and the derived stochastic MMsFEM. Numerical experiments are carried out for two-phase flows in random porous media to demonstrate the efficiency and accuracy of the proposed hybrid HDMR with MMsFEM.