  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.
