# On the apparent permeability of porous media in rarefied gas flows

## TL;DR

This paper investigates the apparent permeability in porous media during rarefied gas flows, revealing key factors like tortuosity and gas-surface interactions that explain experimental observations and improve modeling accuracy.

## Contribution

It identifies two main factors—tortuous flow paths and non-unitary tangential momentum accommodation—that explain Klinkenberg's experimental results, advancing understanding of gas flow in porous media.

## Key findings

- The correction factor decreases with increasing Knudsen number due to tortuosity and surface interactions.
- Klinkenberg's results are valid only when the permeability ratio is below approximately 30.
- Numerical results serve as benchmarks for modeling rarefied gas flows in complex geometries.

## Abstract

The apparent gas permeability of the porous medium is an important parameter in the prediction of unconventional gas production, which was first investigated systematically by Klinkenberg in 1941 and found to increase with the reciprocal mean gas pressure (or equivalently, the Knudsen number). Although the underlying rarefaction effects are well-known, the reason that the correction factor in Klinkenberg's famous equation decreases when the Knudsen number increases has not been fully understood. Most of the studies idealize the porous medium as a bundle of straight cylindrical tubes, however, according to the gas kinetic theory, this only results in an increase of the correction factor with the Knudsen number, which clearly contradicts Klinkenberg's experimental observations. Here, by solving the Bhatnagar-Gross-Krook equation in simplified (but not simple) porous media, we identify, for the first time, two key factors that can explain Klinkenberg's experimental results: the tortuous flow path and the non-unitary tangential momentum accommodation coefficient for the gas-surface interaction. Moreover, we find that Klinkenberg's results can only be observed when the ratio between the apparent and intrinsic permeabilities is $\lesssim30$; at large ratios (or Knudsen numbers) the correction factor increases with the Knudsen number. Our numerical results could also serve as benchmarking cases to assess the accuracy of macroscopic models and/or numerical schemes for the modeling/simulation of rarefied gas flows in complex geometries over a wide range of gas rarefaction.