Online conservative generalized multiscale finite element method for flow models

TL;DR

This paper introduces an online enrichment approach using GMsFEM for two-phase flow in heterogeneous porous media, enhancing pressure, velocity, and saturation accuracy through systematic basis function updates and flux postprocessing.

Contribution

It develops an online GMsFEM framework with a postprocessing technique for conservative fluxes, improving accuracy in flow models with heterogeneous coefficients without repeated basis updates.

Findings

Enhanced pressure and velocity accuracy demonstrated

Conservative fluxes constructed via postprocessing

Effective in various heterogeneous permeability scenarios

Abstract

In this paper, we consider an online enrichment procedure using the Generalized Multiscale Finite Element Method (GMsFEM) in the context of a two-phase flow model in heterogeneous porous media. The coefficient of the elliptic equation is referred to as the permeability and is the main source of heterogeneity within the model. The elliptic pressure equation is solved using online GMsFEM, and is coupled with a hyperbolic transport equation where local conservation of mass is necessary. To satisfy the conservation property, we aim at constructing conservative fluxes within the space of multiscale basis functions through the use of a postprocessing technique. In order to improve the accuracy of the pressure and velocity solutions in the online GMsFEM we apply a systematic online enrichment procedure. The increase in pressure accuracy due to the online construction is inherited by the conservative flux fields and the desired saturation solutions from the coupled transport equation. Despite the fact that the coefficient of the pressure equation is dependent on the saturation which may vary in time, we may construct an approximation space using the initial coefficient where no further basis updates follow. Numerical results corresponding to four different types of heterogeneous permeability coefficients are exhibited to test the proposed methodology.