Computational homogenization of unsteady flows in a periodic porous medium

TL;DR

This paper develops a computational homogenization method for modeling unsteady viscous flow in periodic porous media, incorporating memory effects through an integro-differential Darcy law.

Contribution

It introduces a novel approach to homogenize unsteady flows with memory effects by solving auxiliary problems and approximating the memory kernel as a sum of exponentials.

Findings

Successfully applied to a 2D test problem of unsteady filtration.

Transformes nonlocal problem into local differential equations.

Provides stable numerical schemes for time integration.

Abstract

The work is devoted to the development and computational implementation of the homogenization method for modeling unsteady flows of a viscous incompressible fluid in periodic porous media taking into account memory effects. At the macrolevel, the flow is described by an integro-differential Darcy law with a tensor memory kernel determined by solving unsteady problems on the periodicity cell. The developed approach to computational homogenization is based on finding the steady-state and unsteady components of the conductivity tensor from solving auxiliary boundary value and spectral problems on the periodicity cell. The nonlocal macroscopic problem is transformed into a local system of differential equations by approximating the memory kernel as a sum of exponentials. Issues of spatial finite element approximation are discussed, and stable two-level schemes in time are constructed. The results of applying the developed computational homogenization technology for unsteady filtration problems in porous media to a two-dimensional test problem are presented.