A hybrid HDMR for mixed multiscale finite element method with application for flows in random porous media

TL;DR

This paper introduces a hybrid high-dimensional model representation (HDMR) combined with a mixed multiscale finite element method (MMSFEM) to efficiently simulate flows in random porous media, reducing computational complexity while maintaining accuracy.

Contribution

It proposes a novel hybrid HDMR technique that decomposes high-dimensional stochastic models into manageable low-dimensional models and integrates it with MMsFEM for improved flow simulation in heterogeneous porous media.

Findings

Hybrid HDMR effectively reduces stochastic model dimensionality.

Integrated approach improves simulation accuracy for flows in random media.

Significantly decreases computational cost compared to traditional methods.

Abstract

Stochastic modeling has become a popular approach to quantify uncertainty in flows through heterogeneous porous media. The uncertainty in heterogeneous structure properties is often parameterized by a high-dimensional random variable. This leads to a deterministic problem in a high-dimensional parameter space and the numerical computation becomes very challengeable 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 a most active random space and a few one-dimensional stochastic models. The derived low-dimensional stochastic models are solved by incorporating sparse grid stochastic collocation method into the proposed hybrid HDMR. The porous media properties such as permeability are often heterogeneous. To treat the heterogeneity, we use a mixed multiscale finite element method (MMsFEM) to simulate each of derived stochastic models. To capture the non-local spatial features of the porous media and the important effects of random variables, we can hierarchically incorporate the global information individually from each of random parameters. This significantly enhances the accuracy of the multiscale simulation. The synergy of the hybrid HDMR and the MMsFEM reduces the stochastic model of flows in both stochastic space and physical space, and significantly decreases the computation complexity. We carefully analyze the proposed HDMR technique and the derived stochastic MMsFEM. A few numerical experiments are carried out for two-phase flows in random porous media and support the efficiency and accuracy of the MMsFEM based on the hybrid HDMR.