Weakly periodic boundary conditions for the homogenization of flow in porous media

BackgroundSeepage in porous media is modeled as a Stokes flow in an open pore system contained in a rigid, impermeable and spatially periodic matrix. By homogenization, the problem is turned into a two-scale problem consisting of a Darcy type problem on the macroscale and a Stokes flow on the subscale.MethodsThe pertinent equations are derived by minimization of a potential and in order to satisfy the Variationally Consistent Macrohomogeneity Condition, Lagrange multipliers are used to impose periodicity on the subscale RVE. Special attention is given to the bounds produced by confining the solutions spaces of the subscale problem.ResultsIn the numerical section, we choose to discretize the Lagrange multipliers as global polynomials along the boundary of the computational domain and investigate how the order of the polynomial influence the permeability of the RVE. Furthermore, we investigate how the size of the RVE affect its permeability for two types of domains.ConclusionsThe permeability of the RVE depends highly on the discretization of the Lagrange multipliers. However, the flow quickly converges towards strong periodicity as the multipliers are refined