First-principles calculations of dynamic permeability in porous media

By starting from the linearized Navier-Stokes equation for the fluid and the elastic wave equation for the solid frame of a porous medium, the first-principles definition of the frequency-dependent permeability κ(ω) and the recipe for its calculation are derived through the application of the homogenization procedure. It is shown systematically that, in the limit of wavelength much larger than the typical pore size, the fluid may be regarded as incompressible in the permeability calculation, and the solid-frame displacement acts as an additional pressure source term for the fluid flow. The physics underlying the generic asymptotic frequency dependence of κ(ω) is introduced through its analytic solution in the case of a cylindrical tube. To calculate κ(ω) for periodic porous media models, we have formulated a finite-element approach for the numerical solution of the incompressible fluid equations at low and intermediate frequencies and of the Laplace equation at high frequencies. The numerical results for the sinusoidally modulated tube, the fused-spherical-bead lattice, and the fused-diamond lattice indicate a large range of values for the static permeability ${\kappa }_0$ as well as the other asymptotic parameters such as the tortuosity α and the surface length parameter Λ as defined by Johnson et al. However, despite their variability, almost all the numerical data on periodic models are shown to satisfy the approximate scaling relation κ(ω)≃${\kappa }_0$f(ω/${\omega }_0$), where ${\omega }_0$ is a characteristic frequency particular to the medium, and f is a universal function independent of microstructures. We advance arguments that delineate the physical reason for this scaling behavior as well as the condition for its validity. The scaling prediction is then generalized to the case of random porous media through both numerical simulations and the critical-path argument. Our theory gives a simple explanation to the observed correlations in sedimentary rocks, and the scaling prediction is supported by experimental κ(ω) measurements on fused glass beads and crushed-glass samples.