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 Thöming (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 Thöming (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.

中文翻译：

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能否准确描述通过破裂多孔介质的圆形和扁平致密通道的中等稀薄气体传输？

本文提供了对通过天然和诱导裂缝性多孔介质（如气页岩）中的狭窄通道进行的中等稀薄气体输送的严格公式的重要见解，这些介质近似为圆形（圆柱形）和扁平（狭缝）类型。该公式考虑了通过正确实施紧密流动通道的有效等效平均水力半径、麦克斯韦壁滑效应调节（Philos Trans R Soc Lond A 170:231–256, 1879）、变量气体分子的横截面硬球模型和 Bird 的修正体积平均自由程（Phys Fluids 26(11):3222–3223, 1983. https://doi.org/10.1063/1.864095）、表观粘度和限制状态气体行为修正的平均自由程，通过由 Hagen-Poiseuille 型方程表示的狭窄毛细管的流量、Knudsen 扩散系数以及表观渗透率和固有渗透率之间改进的关系。通过极密通道的气体传输描述是通过 Poiseuille 整体流（对流）和 Knudsen 传输（扩散）机制的叠加来完成的。该方法用于研究先前几项关于在中等稀薄条件下通过圆形和扁平类型的极紧密狭窄通道进行气体传输建模的研究的准确性。尽管 Roy 等人先前的研究报告了模拟结果。(J Appl Phys 93(8):4870–4879, 2003)、Javadpour (J Can Pet Technol 48(8):16–21, 2009) 和 Veltzke 和 Thöming (J Gas Mech 698:406–422, 2012) 似乎遵循 Roy 等人的实验研究中观察到的趋势。(2003) 使氩气流过圆形通道（管）和 Ewart 等人。(J Gas Mech 584:337–356, 2007. https://doi.org/10.1017/S0022112007006374 ) 使氦气流过单个平直狭窄通道，得出的结论是这些结果实际上并不准确，原因有几个。Javadpour (2009) 提出的基本模型的准确性受到一些公式问题和数值近似的低阶准确性的影响。Veltzke 和 Thöming (2012) 提出的复杂模型不切实际且困难，因为 Navier-Stokes 方程的二维解具有无滑移边界条件，并且由于直线的有效半径定义不正确而产生不准确的解平坦狭窄的通道。本文提出的可压缩稀薄气体在紧密通道中流动的改进压力方程是高度非线性的。通过促进伪传递或流动势函数的积分变换的应用，完全消除了与压差方程解相关的数值计算错误的可能性，并且该方程的解是准确且完全解析地获得的。结果表明，与以前的研究报告的数据相比，本文开发的适当公式和准确的分析解决方案确实可以显着准确匹配相同的实验数据。因此，先前模拟结果与实验数据的偏差不能简单地归因于与实验室测试相关的可能的实验误差，也归因于公式的限制和数值解的不准确性。本文中提出的练习表明，以前的建模工作肯定涉及各种类型的错误，除非模型及其参数在理论上有意义，否则仅通过调整未知模型参数的值就无法与气体输运模型匹配实验数据。