Multiscale Simulations for Coupled Flow and Transport Using the Generalized Multiscale Finite Element Method

Eric Chung,Jun Ren,Wing Leung,Yalchin Efendiev

doi:10.3390/computation3040670

Abstract

In this paper, we develop a mass conservative multiscale method for coupled flow and transport in heterogeneous porous media. We consider a coupled system consisting of a convection-dominated transport equation and a flow equation. We construct a coarse grid solver based on the Generalized Multiscale Finite Element Method (GMsFEM) for a coupled system. In particular, multiscale basis functions are constructed based on some snapshot spaces for the pressure and the concentration equations and some local spectral decompositions in the snapshot spaces. The resulting approach uses a few multiscale basis functions in each coarse block (for both the pressure and the concentration) to solve the coupled system. We use the mixed framework, which allows mass conservation. Our main contributions are: (1) the development of a mass conservative GMsFEM for the coupled flow and transport; (2) the development of a robust multiscale method for convection-dominated transport problems by choosing appropriate test and trial spaces within Petrov-Galerkin mixed formulation. We present numerical results and consider several heterogeneous permeability fields. Our numerical results show that with only a few basis functions per coarse block, we can achieve a good approximation.

Highlights

Many porous media problems occur over multiple scales
We consider a multiscale method for a coupled flow and transport system, where the flow equation for the pressure field is described by a steady state elliptic equation and the transport equation for the concentration field is described by a convection-dominated parabolic equation
We present a representative set of numerical experiments that demonstrate the performance of the mixed Generalized Multiscale Finite Element Method (GMsFEM) for approximating the coupled flow and transport Equation (2)

Summary

Introduction

Some type of model reduction or averaging are often employed. Compute the permeabilities on a coarse (computational) grid and solve the flow and transport equations together on a coarse grid. These approaches are easy to implement with a minimal code modification; these approaches can result in large errors for complex heterogeneities because limited information is used in upscaling. Multiscale basis functions are computed and used to solve flow equations. The main contributions of this paper are (1) to use multiscale basis functions for both flow and transport equations and (2) design of a novel mixed multiscale methods for convection-dominated transport equations within the context of Generalized Multiscale Finite Element Method (GMsFEM)

Objectives

Results

Conclusion