New Scheme of Finite Difference Heterogeneous Multiscale Method to Solve Saturated Flow in Porous Media

Fulai Chen,Li Ren

doi:10.1155/2014/575298

Abstract

A new finite difference scheme, the development of the finite difference heterogeneous multiscale method (FDHMM), is constructed for simulating saturated water flow in random porous media. In the discretization framework of FDHMM, we follow some ideas from the multiscale finite element method and construct basic microscopic elliptic models. Tests on a variety of numerical experiments show that, in the case that only about a half of the information of the whole microstructure is used, the constructed scheme gives better accuracy at a much lower computational time than FDHMM for the problem of aquifer response to sudden change in reservoir level and gives comparable accuracy at a much lower computational time than FDHMM for the weak drawdown problem.

Highlights

Natural porous media exhibit a significant spatial variability in most attributes of hydrogeological interest
The main difference between the two methods is the microscopic scheme, in contrast to finite difference heterogeneous multiscale method (FDHMM) by Abdulle and E, where the numerical fluxes are computed on the fly using localized and more resolved computations which means that FDHMM by Abdulle and E needs the macroscopic and microscopic evolution at every time step and the new scheme adopts the idea of multiscale finite element method (MsFEM) of Hou et al, by numerically precomputing a finite difference analogue of a multiscale shape function, which provides a fixed expression for the numerical basic flux in terms of the coarse variables
8.8 x (m) solving basic microscopic elliptic problems and estimating basic macroscopic fluxes, it is subtly brought to the large scale for microscale information of the medium property and useful information about the gradients of the solutions of basic microscopic elliptic models

Summary

Introduction

Natural porous media exhibit a significant spatial variability in most attributes of hydrogeological interest. The main difference between the two methods is the microscopic scheme, in contrast to FDHMM by Abdulle and E, where the numerical fluxes are computed on the fly using localized and more resolved computations which means that FDHMM by Abdulle and E needs the macroscopic and microscopic evolution at every time step and the new scheme adopts the idea of MsFEM of Hou et al, by numerically precomputing a finite difference analogue of a multiscale shape function, which provides a fixed expression for the numerical basic flux in terms of the coarse variables.

New Scheme of FDHMM

Evaluation of Numerical Accuracy

Conclusion