Theory of dynamic permeability and tortuosity in fluid-saturated porous media

TLDR

The study examines the response of a Newtonian fluid in a rigid isotropic porous medium to an infinitesimal oscillatory pressure gradient. The authors aim to demonstrate the relevance of their findings to the acoustic properties of non‑rigid porous media and to show how dynamic permeability/tortuosity can be measured with superfluid 4He. They construct a simple model based on exact high‑ and low‑frequency parameters, validate it against numerical simulations on large lattices with randomly varying tube radii, and propose measuring dynamic permeability/tortuosity using superfluid 4He. The authors derive analytic properties of the linear response function, introduce a new parameter Λ that characterizes high‑frequency behavior, and show that their model accurately predicts responses—including for fractal pore walls—matching numerical simulations.

Abstract

We consider the response of a Newtonian fluid, saturating the pore space of a rigid isotropic porous medium, subjected to an infinitesimal oscillatory pressure gradient across the sample. We derive the analytic properties of the linear response function as well as the high- and low-frequency limits. In so doing we present a new and well-defined parameter Λ, which enters the high-frequency limit, characteristic of dynamically connected pore sizes. Using these results we construct a simple model for the response in terms of the exact high- and low-frequency parameters; the model is very successful when compared with direct numerical simulations on large lattices with randomly varying tube radii. We demonstrate the relevance of these results to the acoustic properties of non-rigid porous media, and we show how the dynamic permeability/tortuosity can be measured using superfluid 4He as the pore fluid. We derive the expected response in the case that the internal walls of the pore space are fractal in character.