A Generalized Multiscale Finite Element Method for poroelasticity problems I: Linear problems

Abstract

In this paper, we consider the numerical solution of poroelasticity problems that are of Biot type and develop a general algorithm for solving coupled systems. We discuss the challenges associated with mechanics and flow problems in heterogeneous media. The two primary issues being the multiscale nature of the media and the solutions of the fluid and mechanics variables traditionally developed with separate grids and methods. For the numerical solution we develop and implement a Generalized Multiscale Finite Element Method (GMsFEM) that solves problem on a coarse grid by constructing local multiscale basis functions. The procedure begins with construction of multiscale bases for both displacement and pressure in each coarse block. Using a snapshot space and local spectral problems, we construct a basis of reduced dimension. Finally, after multiplying by a multiscale partitions of unity, the multiscale basis is constructed in the offline phase and the coarse grid problem then can be solved for arbitrary forcing and boundary conditions. We implement this algorithm on two heterogeneous media and compute error between the multiscale solution with the fine-scale solutions. Randomized oversampling and forcing strategies are also tested.