Global-Local Nonlinear Model Reduction for Flows in Heterogeneous Porous Media

TL;DR

This paper introduces a combined global-local nonlinear model reduction technique that leverages multiscale and empirical interpolation methods to efficiently simulate complex nonlinear flows in heterogeneous porous media.

Contribution

It develops a novel integrated approach using GMsFEM and empirical interpolation to reduce computational costs in nonlinear multiscale flow simulations.

Findings

Significant reduction in computational complexity.

Accurate approximation of fully-resolved solutions.

Effective handling of high-contrast heterogeneous media.

Abstract

In this paper, we combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational complexity associated with nonlinear flows in highly-heterogeneous porous media. To solve the nonlinear governing equations, we employ the GMsFEM to represent the solution on a coarse grid with multiscale basis functions and apply proper orthogonal decomposition on a coarse grid. Computing the GMsFEM solution involves calculating the residual and the Jacobian on the fine grid. As such, we use local and global empirical interpolation concepts to circumvent performing these computations on the fine grid. The resulting reduced-order approach enables a significant reduction in the flow problem size while accurately capturing the behavior of fully-resolved solutions. We consider several numerical examples of nonlinear multiscale partial differential equations that are numerically integrated using fully-implicit time marching schemes to demonstrate the capability of the proposed model reduction approach to speed up simulations of nonlinear flows in high-contrast porous media.