Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Three-Dimensional Media

Abstract

In this paper, we consider the poroelasticity problem in three-dimensional heterogeneous media. The mathematical model contains a coupled system of equations for fluid pressures and displacements. The poroelasticity problem in heterogeneous media is an important topic in the field of geomechanics, involving the modeling of the interaction between fluid flow and deformation in porous media. This topic is of great relevance in many engineering applications, such as oil and gas reservoirs, groundwater systems, and geothermal energy production. In recent years, the development of Online Generalized Multiscale Finite Element Methods (GMsFEM) has offered a promising solution to efficiently tackle this challenging problem. GMsFEM involves the use of a coarse-scale model that captures the macroscopic behavior of the system, combined with a fine-scale model that captures the local heterogeneities. This approach enables accurate and efficient simulations of the poroelasticity problem in heterogeneous media, with reduced computational costs compared to traditional methods. We utilize online basis functions to enhance the offline multiscale space and quickly reduce errors using residual data. Fast convergence of online methods can only be ensured with suitable offline coarse-grid spaces. This paper highlights the potential of online GMsFEM in advancing the modeling and simulation of poroelasticity problems in three-dimensional heterogeneous media. We investigate how the number of online and offline basis functions affects the relative errors between the multiscale solution and the reference (fine-scale) solution.