Finite volume multiscale finite element method for solving the groundwater flow problems in heterogeneous porous media

[1] In this paper we present a numerical method, the finite volume multiscale finite element method (FVMSFEM), for solving the groundwater flow problems in heterogeneous porous media spanning over many scales. This method is based on an efficient coupling between the finite volume discretization and the multiscale finite element base functions. It can efficiently capture the large-scale structure of the solution on a coarse grid without resolving all the small-scale features and is locally conservative. The underlying idea is to estimate the macroscopic fluxes across the finite control volume interface segments, which bring the local small-scale information of the medium property to the large scales, by employing the multiscale finite element base functions. We describe the strategy for constructing such a method on the basis of the transient flow problems in porous media. Numerical experiments are carried out for groundwater flow in porous media with a random lognormal conductivity field to demonstrate the efficiency and accuracy of the developed method.

[1]  A. Mantoglou,et al.  The Turning Bands Method for simulation of random fields using line generation by a spectral method , 1982 .

[2]  T. Arbogast.  Implementation of a Locally Conservative Numerical Subgrid Upscaling Scheme for Two-Phase Darcy Flow , 2002 .

[3]  Shujun Ye,et al.  Application of the multiscale finite element method to flow in heterogeneous porous media , 2004 .

[4]  E Weinan,et al.  Finite difference heterogeneous multi-scale method for homogenization problems , 2003 .

[5]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[6]  Lynn W. Gelhar,et al.  Stochastic subsurface hydrology from theory to applications , 1986 .

[7]  Louis J. Durlofsky,et al.  Upscaling of Channel Systems in Two Dimensions Using Flow-Based Grids , 2003 .

[8]  H. Tchelepi,et al.  Multi-scale finite-volume method for elliptic problems in subsurface flow simulation , 2003 .

[9]  P. M. De Zeeuw,et al.  Matrix-dependent prolongations and restrictions in a blackbox multigrid solver , 1990 .

[10]  Louis J. Durlofsky,et al.  Representation of grid block permeability in coarse scale models of randomly heterogeneous porous media , 1992 .

[11]  Peter K. Kitanidis,et al.  Effective hydraulic conductivity for gradually varying flow , 1990 .

[12]  Anthony T. Patera,et al.  A parallel Monte-Carlo finite-element procedure for the analysis of multicomponent random media , 1995 .

[13]  E Weinan,et al.  The Heterognous Multiscale Methods , 2003 .

[14]  E Weinan,et al.  Multi-scale Modeling and Computation , 2003 .

[15]  Zhiming Chen,et al.  A mixed multiscale finite element method for elliptic problems with oscillating coefficients , 2003, Math. Comput..

[16]  Zhiming Chen,et al.  Numerical Homogenization of Well Singularities in the Flow Transport through Heterogeneous Porous Media , 2003, Multiscale Model. Simul..

[17]  Todd Arbogast,et al.  A two-scale numerical subgrid technique for waterflood simulations , 2002 .

[18]  Gedeon Dagan,et al.  Analysis of flow through heterogeneous random aquifers: 2. Unsteady flow in confined formations , 1982 .

[19]  L. Durlofsky.  Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media , 1991 .

[20]  Peter K. Kitanidis,et al.  Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach: 1. Method , 1992 .

[21]  J. F. McCarthy,et al.  Comparison of fast algorithms for estimating large-scale permeabilities of heterogeneous media , 1995 .

[22]  Thomas Y. Hou,et al.  Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients , 1999, Math. Comput..

[23]  J. Banavar,et al.  The Heterogeneous Multi-Scale Method , 1992 .