Machine Learning for Online Multiscale Model Reduction for Poroelasticity Problem in Heterogeneous Media

In this study, we address the poroelasticity problem in heterogeneous media, which involves a coupled system of equations for fluid pressures and displacements. This problem is crucial in geomechanics for modeling the interaction between fluid flow and deformation in porous media, with applications spanning oil and gas reservoirs, groundwater systems, and geothermal energy production. We introduce an innovative approach by integrating machine learning techniques to train online multiscale basis functions, enhancing the Generalized Multiscale Finite Element Method (GMsFEM). This methodology allows for an adaptive and efficient representation of both macroscopic and local heterogeneities in the system, significantly reducing computational costs. The offline multiscale basis functions are precomputed using local spectral problems, while the online basis functions are dynamically updated using machine learning models trained on local residual data. This approach ensures rapid error reduction and robust convergence, leveraging the computational efficiency of machine learning. We demonstrate the effectiveness of this method through numerical experiments, showcasing its potential in advancing the simulation and modeling of poroelasticity problems in heterogeneous media.