Computational homogenization of unsteady flows in a periodic porous medium

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.