A computational macroscale model for the time fractional poroelasticity problem in fractured and heterogeneous media

TL;DR

This paper develops a multiscale computational model for time fractional poroelasticity in fractured heterogeneous media, combining fractional calculus, fracture modeling, and multiscale finite element methods to efficiently simulate complex subsurface processes.

Contribution

It introduces a novel multiscale approach incorporating fractional derivatives and fracture modeling, enabling accurate coarse-grid simulations of complex poroelastic media.

Findings

The method achieves good accuracy with fewer multiscale basis functions.

Numerical results validate the effectiveness of the proposed approach.

Error analysis demonstrates the method's robustness across different fractional powers.

Abstract

In this work, we introduce a time memory formalism in poroelasticity model that couples the pressure and displacement. We assume this multiphysics process occurs in multicontinuum media. The mathematical model contains a coupled system of equations for pressures in each continuum and elasticity equations for displacements of the medium. We assume that the temporal dynamics is governed by fractional derivatives following some works in the literature. We derive an implicit finite difference approximation for time discretization based on the Caputo time fractional derivative. A Discrete Fracture Model (DFM) is used to model fluid flow through fractures and treat the complex network of fractures. We assume different fractional powers in fractures and matrix due to slow and fast dynamics. We develop a coarse grid approximation based on the Generalized Multiscale Finite Element Method (GMsFEM), where we solve local spectral problems for construction of the multiscale basis functions. We present numerical results for the two-dimensional model problems in fractured heterogeneous porous media. We investigate error analysis between reference (fine-scale) solution and multiscale solution with different numbers of multiscale basis functions. The results show that the proposed method can provide good accuracy on a coarse grid.