On the apparent permeability of porous media in rarefied gas flows

Abstract

The apparent gas permeability of a 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 30; 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 modelling/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 medium; 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 types 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 1.5. † Email address for correspondence: lei.wu.100@strath.ac.uk

Investigation of the Klinkenberg effect in a micro/nanoporous medium by direct simulation Monte Carlo method

“…(8) is a characteristic length of the flow. In previous studies concerning flow through a porous medium, various characteristic lengths have been used [21][22][23][24]. In this study, we use both the size of the elements (D), as many studies in literatures did, and also the pore size (D p ) to define the Knudsen numbers:…”

“…In a recent study of Wu et al [24], it was pointed out that all the currently widely used empirical solutions, derived from straight cylindrical tube, should be reformulated for a porous medium. However, the results in Fig.…”

“…7 prove that these correlations are still applicative for flows in more complex geometries, provided that an appropriate Knudsen number is used. Furthermore, some data for the porosity of ε = 0.84 by Wu et al [24], which were originally plotted with another definition of Knudsen number, are converted to the present Kn p . It can be seen in Fig.…”

On the unintentional rarefaction effect in LBM modeling of intrinsic permeability

“…Considering the same resolution of N x * N y = 800 * 400, significant deviation of the LBM data (green curve) from the DVM data (black curve) can be seen when Kn ≥ 5 × 10 −3 and grows up to 20% at Kn = 5×10 −2 . It is noted that both LBM and DVM use fully diffuse boundary condition on the solid wall to facilitate the comparison (Ansumali and Karlin, 2002;Wu et al, 2017).…”

Numerical investigation on performance of three solution reconstructions at cell interface in DVM simulation of flows in all Knudsen number regimes

“…The first test case is the force-driven Poiseuille flow with different Knudsen numbers. 23,45,46 This problem can be considered as a viscous fluid flow between two infinite parallel plates with temperature of T 0 = 273 K separated by a distance of L in the y-direction. An external force is exerted in the x-direction, so that Equation (1) can be rewritten as…”