Macroscopic non-linear filtration law for porous media containing cylindrical and spherical inhomogeneities

This paper provides the macroscopic non-linear filtration law of a two-phase porous medium with cylindrical or spherical inhomogeneities. At the local scale, the fluid flow in both phases of the composite porous material obeys the Forchheimer law. The macroscopic law is obtained in the framework of the non-linear variational homogenization method, considering unit cells with concentric cylinders or spheres subjected to homogeneous boundary conditions. In order to derive a closed-form expression of the macroscopic law, we employ the kinematic approach with trial velocity fields inspired by a linear solution. The resulting analytical model is then compared with numerical upper and lower bounds, demonstrating its high accuracy. Finally, we provide comparisons with numerical results for unit cells containing a population of polydisperse inclusions.