Fluid permeability in porous media: Comparison of electrical estimates with hydrodynamical calculations

The principal dimensionless quantities used to characterize the geometry of porous media are the porosity φ and the electrical formation factor F. However, many properties of interest (e.g., nuclear magnetic relaxation, mercury porosimetry, and viscous fluid flow) depend on the absolute dimensions of the pore space. Among the most important pore scale lengths are the pore volume to surface area ratio ${\mathit{V}}_{\mathit{p}}$/S, the Λ parameter, and the diffusion-limited surface trapping length ${\mathit{w}}_{\mathit{s}}$. We have calculated these lengths for two- and three-dimensional geometrical models of porous media and have used each of them to estimate the permeability k to viscous fluid flow as determined by direct numerical solution of the Stokes equations. Our analysis is based on three families of geometrical models: (1) three-dimensional ordered sphere packs (including the consolidation regime), (2) two-dimensional tortuous-path models, and (3) two-dimensional Koch-curve models. In all cases we find that the rigorous bound recently formulated in terms of ${\mathit{w}}_{\mathit{s}}$ provides rather a weak constraint when compared to the actual value of k. In the sphere-pack models, permeability estimates based on ${\mathit{V}}_{\mathit{p}}$/S are reasonably accurate, but such estimates are much less valuable in the more interesting two-dimensional geometries. Our most important finding is that in all the cases examined the Λ-parameter estimate of permeability is quite reliable. Nevertheless, in the Koch-curve models, as the effective channel cross-sectional area narrows, we are able to see evidence for the breakdown of this estimate. This breakdown is associated with differences in the singularities of the Stokes and Laplace solutions in the vicinity of jagged constrictions in the flow paths.