![]() Present, “ On gaseous self-diffusion in long capillary tubes,” Phys. Daneri, “ Flow of a rarefied gas between two parallel plates,” J. Gibelli, “ A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization,” J. Pareschi, “ Numerical methods for kinetic equations,” Acta Numer. Stefanov, “ A dusty gas model-direct simulation Monte Carlo algorithm to simulate flow in micro-porous media,” Phys. Ejtehadi, “ A review and perspective on a convergence analysis of the direct simulation Monte Carlo and solution verification,” Phys. Li, “ Efficient prediction of gas permeability by hybrid DSBGK-LBM simulations,” Fuel 250, 154– 159 (2019). Li, Multiscale and Multiphysics Flow Simulations of Using the Boltzmann Equation ( Springer, 2020). Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows ( Clarendon Press, Oxford, 1994). Torrilhon, “ Modeling nonequilibrium gas flow based on moment equations,” Annu. Struchtrup, “ Macroscopic transport equations for rarefied gas flows,” in Macroscopic Transport Equations for Rarefied Gas Flows ( Springer, 2005), pp. Zhaoli, “ Review of micro seepage mechanisms in shale gas reservoirs,” Int. Wei, “ A review on slip models for gas microflows,” Microfluid. Kremer, An Introduction to the Boltzmann Equation and Transport Processes in Gases ( Springer Science & Business Media, 2010). Zoback, “ Gas transport and storage capacity in shale gas reservoirs-A review. Bakajin, “ Fast mass transport through sub-2-nanometer carbon nanotubes,” Science 312, 1034– 1037 (2006). The key advantages of promoting the Enskog equation for upscaling flows in porous media lie in its ability to capture the non-equilibrium physics of tightly confined fluids while being computationally more efficient than fundamental simulation approaches, such as molecular dynamics and derivative solvers. While we observe slight deviations in the Enskog density and velocity profiles from the MD when the reduced density is greater than 0.2, this limit is well above practical engineering applications, such as in shale gas. Our results showed (a) very good agreement between EDMD, PHS-MD, and Enskog solutions across density, velocity, and temperature profiles for all the simulation conditions and (b) numerical evidence that deviations exist in the normalized mass flow rate vs Knudsen number curve compared to the standard curve without confinement. Our in-house Event-Driven Molecular Dynamics (EDMD) code and a pseudo-hard-sphere Molecular Dynamics (PHS-MD) solver are used to study force-driven Poiseuille flows in the limit of high gas densities and high confinements. In this paper, we perform the first verification study of the Enskog equation by using particle simulation methods based on the same hard-sphere collisions dynamics. Modeling dense gas flows inside channels with sections comparable to the diameter of gas molecules is essential in porous medium applications, such as in non-conventional shale reservoir management and nanofluidic separation membranes.
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