A Combination Method of Mixed Multiscale Finite-Element and Laplace Transform for Flow in a Dual-Permeability System

Tang-Wei Liu,Xue-Lin Qiu,He-Hua Xu

doi:10.5402/2012/202893

Abstract

An efficient combination method of Laplace transform and mixed multiscale finite-element method for coupling partial differential equations of flow in a dual-permeability system is present. First, the time terms of parabolic equation with unknown pressure term are removed by the Laplace transform. Then the transformed equations are solved by mixed FEMs which can provide the numerical approximation formulas for pressure and velocity at the same time. With some assumptions, the multiscale basis functions are constructed by utilizing the effects of fine-scale heterogeneities through basis functions formulation computed from local flow problems. Without time step in discrete process, the present method is efficient when solving spatial discrete problems. At last, the associated pressure transform is inverted by the method of numerical inversion of the Laplace transform.

Highlights

There have been some important developments in numerical methods of flow in fractured porous media
Developing multiscale methods allow us to overcome above difficulty while retaining a satisfactory accuracy
Multiscale finite-element methods are regarded as numerical methods and strategies in which basis functions are ISRN Applied Mathematics computed by solving local homogeneous PDEs subject to special boundary conditions 1, 3– 11

Summary

Introduction

There have been some important developments in numerical methods of flow in fractured porous media. Multiscale FEMs are efficiency and convenience for elliptic equation of steady flow model. Despite their advantage, they still have some practical limitations in solving parabolic equations for nonsteady flow model 12. They still have some practical limitations in solving parabolic equations for nonsteady flow model 12 Their major drawback is that it is necessary to take small time steps. We can see that numerical methods for parabolic equation with general FEMs require the solution of some simultaneous algebraic equations at each time step. The present method, the combined use of the Laplace transform and the multiscale finite-element method, is used to solve some problems of flow in fractured porous media.

Mathematical Model and Laplace Transform

Mixed Finite-Element Method

Multiscale Basis Functions

Conclusion