Cross-property relations for transport in porous media. Rigorous link between fluid permeability, conductivity and relaxation times

Rigorous relations are derived which relate the fluid permeability for flow through porous media to other measurable properties of the media. One expression relates exactly the static fluid permeability k to the electrical formation factor F (inverse of the dimensionless effective conductivity) and an effective length parameter L, i.e., k = L²/8F. This length parameter involves a certain average of the eigenvalues of the Stokes operator and reflects information about electrical and momentum transport. From the exact relation for k, a rigorous upper bound follows in terms of the principal viscous relaxation time Θ₁ (proportional to the inverse of the smallest eigenvalue): k ≤ νΘ₁/F, where ν is the kinematic viscosity. We also demonstrate that νΘ₁ ≤ DT₁, where T₁ is the diffusion relaxation time for the analogous scalar diffusion problem and D is the diffusion coefficient. Therefore, one also has the alternative bound k ≤ DT₁/F. The latter expression relates the fluid permeability on the one hand to purely diffusional parameters on the other. Finally, we derive an exact expression that relates the dynamic permeability to another diffusion parameter, namely, the dynamic mean survival time of a Brownian particle.