A two-scale finite element formulation of Stokes flow in porous media

• A nonlinear multiscale technique for permeability of porous materials is presented.
• A Stokes flow is present on the subscale and a Darcy flow on the macroscale.
• The tangent of the subscale problem is computed with respect to macroscale quantities.
• On the subscale, strong periodicity is imposed on both the velocity and pressure.
• Numerical analysis shows the accuracy of the homogenized results.

Seepage through saturated porous material with an open pore system is modeled as a non-linear Stokes flow through a rigid matrix. Based on variationally consistent homogenization, the resulting macroscale problem becomes a Darcy-type flow. The prolongation of the Darcy flow fulfills a macrohomogeneity condition, which in a Galerkin context implies a symmetric macroscale problem. The homogenization is of 1st order and periodic boundary conditions are adopted on a Representative Volume Element. A nonlinear nested multiscale technique, in which the subscale problem is used as a constitutive model, is devised. In the presented numerical investigation, the effects of varying physical parameters as well as of the discretization are considered. In particular, it is shown that the two-scale results agree well with those of the fully resolved fine-scale problem.