Calculation of the permeability and apparent permeability of three-dimensional porous media

In this study, creeping and inertial incompressible fluid flows through three-dimensional porous media are considered, and an analytical–numerical approach is employed to calculate the associated permeability and apparent permeability. The multiscale homogenization method for periodic structures is applied to the Stokes and Navier–Stokes equations (aided by a control-volume type argument in the latter case), to derive the appropriate cell problems and effective properties. Numerical solutions are then obtained through Galerkin finite-element formulations. The implementations are validated, and results are presented for flows through cubic lattices of cylinders, and through the dendritic zone found at the solid–liquid interface during solidification of metals. For the interdendritic flow problem, a geometric configuration for the periodic cell is built by the approximate matching of experimental and numerical results for the creeping-flow problem; inertial effects are then quantified upon solution of the inertial-flow problem. Finally, the functional behavior of the apparent permeability results is analyzed in the light of existing macroscopic seepage laws. The findings contribute to the (numerical) verification of the validity of such laws.