On the apparent permeability of porous media in rarefied gas flows

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. Specifically, we point out that the Navier-Stokes equations with the first-order velocity-slip boundary condition are often misused to predict the apparent gas permeability of the porous media; that is, any nonlinear dependence of the apparent gas permeability with the Knudsen number, predicted from the Navier-Stokes equations, is not reliable. Worse still, for some type of gas-surface interactions, even the ``filtered'' linear dependence of the apparent gas permeability with the Knudsen number is of no practical use since, compared to the numerical solution of the Bhatnagar-Gross-Krook equation, it is only accurate when the ratio between the apparent and intrinsic permeabilities is $\lesssim1.5$.