A Three-Dimensional Homogenization Approach for Effective Heat Transport in Thin Porous Media

Heat transport through a porous medium depends on the local pore geometry and on the heat conductivities of the solid and the saturating fluid. Through upscaling using formal homogenization, the local pore geometry can be accounted for to derive effective heat conductivities to be used at the Darcy scale. We here consider thin porous media, where not only the local pore geometry plays a role for determining the effective heat conductivity, but also the boundary conditions applied at the top and the bottom of the porous medium. Assuming scale separation and using two-scale asymptotic expansions, we derive cell problems determining the effective heat conductivity, which incorporates also the effect of the boundary conditions. Through solving the cell problems, we show how the local grain shape, and in particular its surface area at the top and bottom boundary, affects the effective heat conductivity through the thin porous medium.