Title: On the apparent permeability of porous media in rarefied gas flows,
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.