Can Moderately Rarefied Gas Transport Through Round and Flat Tight Channels of Fractured Porous Media be Described Accurately?

Abstract

This paper provides critical insights into the rigorous formulation of moderately rarefied gas transport through narrow channels in naturally and induced fractured porous media, such as gas shale rocks, approximated as round (cylindrical) and flat (slit) types. This formulation considers critical improvements over the previous attempts by proper implementation of the effective equivalent mean hydraulic radius of tight flow channels, the wall-slip effect accommodation of Maxwell (Philos Trans R Soc Lond A 170:231–256, 1879), the variable cross-section hard sphere model of gas molecules and the modified bulk mean free-path of Bird (Phys Fluids 26(11):3222–3223, 1983. https://doi.org/10.1063/1.864095), the apparent viscosity and mean free path for the confined-state gas behavior modification, the flow through narrow capillary tubes represented by a Hagen–Poiseuille-type equation, the Knudsen diffusivity, and an improved relationship between the apparent permeability and the intrinsic permeability. The description of gas transport through extremely tight channels is accomplished by superposition of the Poiseuille bulk flow (convection) and the Knudsen transport (diffusion) mechanisms. This approach is applied to investigate the accuracy of several previous studies on the modeling of gas transport through extremely tight narrow channels of round and flat types under moderately rarefied conditions. Although the simulation results reported by the previous studies of Roy et al. (J Appl Phys 93(8):4870–4879, 2003), Javadpour (J Can Pet Technol 48(8):16–21, 2009), and Veltzke and Thoming (J Gas Mech 698:406–422, 2012) appear to follow the trends observed in the experimental studies of Roy et al. (2003) flowing argon gas through a round channel (tube) and Ewart et al. (J Gas Mech 584:337–356, 2007. https://doi.org/10.1017/S0022112007006374) flowing helium gas through a single straight flat narrow channel, it is concluded that these results are not actually accurate for several reasons. The accuracy of the basic model presented by Javadpour (2009) suffers from some formulation issues and the low-order accuracy of the numerical approximations. The complicated model presented by Veltzke and Thoming (2012) is impractical and difficult because of the two-dimensional solutions of the Navier–Stokes equations with no-slip boundary condition and produces inaccurate solutions because of the improper definition of the effective radius of the straight flat narrow channel. The improved pressure equation of the compressible rarefied gas flow in tight channels developed in the present paper is highly nonlinear. The possibility of numerical calculation errors associated with the solution of the differential pressure equation was eliminated completely by an application of an integral transformation by facilitating a pseudo-transfer or flow potential function, and the solution of this equation was obtained accurately and fully analytically. It is shown that the proper formulation and accurate analytical solution developed in this paper can indeed lead to significantly accurate matches of the same experimental data than those reported by the previous studies. Thus, the deviation of the previous simulation results from the experimental data cannot be attributed simply to possible experimental errors associated with the laboratory tests but also to the limitations in formulation and inaccuracies in numerical solution. The exercises presented in this paper reveal that the previous modeling efforts certainly involve various types of errors and the experimental data cannot be matched by the gas transport models simply by adjusting the values of the unknown model parameters unless the models and their parameters are theoretically meaningful.