Momentum transfer within a porous medium. I. Theoretical derivation of the momentum balance on the solid skeleton

The momentum balance on the solid skeleton of a porous medium like porous rocks, foam metals, or porous brushes is, here, theoretically derived with the volume averaging method. It is derived for both homogeneous and non-homogeneous porous media and for the latters no length scales constraints are invoked. The momentum balance on the solid skeleton holds in the whole porous medium and contains volume averaged stresses and velocity. For heterogeneous porous media, it is coupled with the fluid momentum balance through a general averaged quantity, while in the homogeneous case, it is coupled with Darcy’s equation, corrected with the first and the second Brinkman’s term, through a geometrically rescaled Darcy’s term. This latter equation coincides with Biot’s equation for poro-elasticity, but it is here derived with a different formalism. This approach gives the opportunity to derive a new stress boundary condition at the interface between a porous medium and a homogeneous fluid.