The Biot–Darcy–Brinkman model of flow in deformable double porous media; homogenization and numerical modelling

Abstract

In this paper we present the two-level homogenization of the flow in a deformable doubleporous structure described at two characteristic scales. The higher level porosity associated with the mesoscopic structure is constituted by channels in a matrix made of a microporous material consisting of elastic skeleton and pores saturated by a viscous fluid. The macroscopic model is derived by the homogenization of the flow in the heterogeneous structure characterized by two small parameters involved in the two-level asymptotic analysis, whereby a scaling ansatz is adopted to respect the pore size differences. The first level upscaling of the fluid-structure interaction problem yields a Biot continuum describing the mesoscopic matrix coupled with the Stokes flow in the channels. The second step of the homogenization leads to a macroscopic model involving three equations for displacements, the mesoscopic flow velocity and the micropore pressure. Due to interactions between the two porosities, the macroscopic flow is governed by a Darcy-Brinkman model comprising two equations which are coupled with the overall equilibrium equation respecting the hierarchical structure of the two-phase medium. Expressions of the effective macroscopic parameters of the homogenized double-porosity continuum are derived, depending on the characteristic responses of the mesoscopic structure. Some symmetry and reciprocity relationships are shown and issues of boundary conditions are discussed. The model has been implemented in the finite element code SfePy which is well-suited for computational homogenization. A numerical example of solving a nonstationary problem using mixed finite element method is included.

Geometrical model of lobular structure and its importance for the liver perfusion analysis

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Robust finite element methods and solvers for the Biot–Brinkman equations in vorticity form

“…We address the analysis of the Biot–Brinkman equations, which serve as a model for filtration of viscous flow in deformable porous media [3, 22, 53, 54]. The system has been recently analysed in [35] for the case of multiple network poroelasticity, using $$$ \mathbf{H}\left(\operatorname{div},\Omega \right) $$$‐conforming displacements and filtration fluxes (or seepage velocities) for each compartment, also designing robust preconditioners.…”