Influence of pore roughness and pore-size dispersion in estimating the permeability of a porous medium from electrical measurements

The influence of pore-size dispersion and pore roughness on the dynamic and the dc permeability of porous media is analyzed through simple models and computer simulations. In particular, attention is focused on the systematic errors that arise from these geometric features when using the empirical fit for the dynamic permeability k(ω)≈(Λ²/F)f ₁(Λ²ω/v) [Johnson et al., J. Fluid Mech. 176, 379 (1986); Zhou and Sheng, Phys. Rev. B 39, 12027 (1989)] and the estimate for the dc permeability k_dc≈Λ²/8F [Johnson et al., Phys. Rev. Lett. 57 2565 (1986)]. Here, Λ and F are, respectively, the electrically weighted volume-to-surface ratio and the formation factor. It is found that, for in-parallel models, A underestimates the effective hydrodynamic radius L, while for in-series models, A overestimates L. Departures from the aforementioned fits are substantial even if moderate-sized dispersen exists for the in-parallel geometry. For the in-series model, the agreement with the empirical fits is very good, unless a broadband pore-radius distribution f(R), R≪1, exists. The quality of the empirical estimates in pores with rough surfaces is investigated in microgeometries consisting of "corrugated" tubes. It is shown that the electric field can produce large Joule heating on the pore surface, and lead, in some cases, to an underestimate of the effective hydrodynamic radius by Λ, which is inversely proportional to the surface dissipation. The accuracy of the estimates based on the Λ parameter depends on whether an "effective tube" containing the electric field and the velocity field at all frequencies can be identified. Careful numerical calculations of Stokes flows on various "corrugated" capillaries are made over a wide frequency range, and the ac and dc permeabilities are compared with the aforementioned empirical theories.