Application of generalized boundary conditions for homogenization of thermal and filtration properties of soils

In the paper, generalized boundary conditions were used for the homogenization of coefficients of the Laplace partial differential equation in the context of Darcy flow and heat diffusion phenomena. The mesoscopic boundary value problem was defined and analyzed from the variational perspective and the finite element formulation of the homogenization problem was provided. The matrix equation for the apparent macroscopic properties, resulting from FEM discretization, was derived and utilized in two illustrative examples: homogenization of the filtration coefficient of clay amended with expanded shale and thermal conductivity of the soil with multiple fractions. It is shown, that generalized boundary conditions can provide very good homogenization results without the assumption of the periodicity of the material. For best results, the microscopic length parameter has to be properly estimated.