Online Generalized Multiscale Finite Element Method for Thermoporoelastic Problems with Phase Transitions and Fractional Derivatives

Abstract

In this paper, we study a coupled thermoporoelasticity problem with phase transitions in heterogeneous porous media. The mathematical model is formulated as a system of equations for temperature, fluid pressure, and solid displacement, where the heat transfer equation includes phase transition effects, the flow equation is described within the poroelastic framework, and the mechanical response is governed by elasticity equations coupled with thermal and hydraulic fields. In addition, fractional derivatives are introduced to describe memory effects and nonlocal behavior in time. The considered problem is highly multiscale due to the heterogeneity of the medium, the presence of phase interfaces, and the complexity of the coupled physical processes. For the numerical approximation, we develop an online Generalized Multiscale Finite Element Method (GMsFEM). The offline stage is based on the construction of local snapshot spaces and spectral decomposition for the main variables, which enables the reduction of the original fine-scale system to a coarse-scale model. To improve the accuracy of the multiscale approximation, we use online basis functions constructed from local residual information. This enrichment procedure allows one to efficiently capture unresolved fine-scale features and significantly reduce the approximation error. The presented method provides an accurate and computationally efficient tool for the simulation of thermoporoelastic processes with phase transitions and fractional dynamics in heterogeneous media.