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Abstract A high-performance gas kinetic solver using multi-level parallelization is developed to enable pore-scale simulations of rarefied flows in porous media. The Bhatnagar–Gross–Krook model equation is solved by the discrete velocity method with an iterative scheme. The multi-level MPI/OpenMP parallelization is implemented with the aim to efficiently utilize the computational resources to allow direct simulation of rarefied gas flows in porous media based on digital rock images for the first time. The multi-level parallel approach is analyzed in detail confirming its better performance than the commonly-used MPI processing alone for an iterative scheme. With high communication efficiency and appropriate load balancing among CPU processes, parallel efficiency of 94% is achieved for 1536 cores in the 2D simulations, and 81% for 12288 cores in the 3D simulations. While decomposition in the spatial space does not affect the simulation results, one additional benefit of this approach is that the number of subdomains can be kept minimal to avoid deterioration of the convergence rate of the iteration process. This multi-level parallel approach can be readily extended to solve other Boltzmann model equations. [ABSTRACT FROM AUTHOR]

• Conduct pore-scale simulations of the shale gas permeability. • Present a comprehensive grid convergence study in a standard 3D Sierpinski carpet. • Compare the computational performance of the two promising DSBGK and DVM methods. • Provide the comparison data for other simulation methods and empirical permeability models. Shale gas flow at the pore scale is very challenging to simulate due to high Knudsen numbers (K n), low speeds and complicated pore geometries. The direct simulation BGK (DSBGK) method and the discrete velocity method (DVM) are promising methods to simulate three-dimensional (3D) gas flows in the shale rock. As the grid-convergence accuracy and computational cost highly depend on the spatial and molecular velocity grids, here we present a rigorous study of the computational performance of the two methods by simulating gas flows through the standard 3D Sierpinski carpet. In this study, we found reasonable agreement between the DSBGK and DVM simulations when comparing the velocity profiles and permeability curves obtained by using fine spatial and velocity grids. However, the grid convergence of the DSBGK simulation is faster than that of the DVM, which requires a very large number of spatial and velocity grids for convergence. Consequently, the DSBGK simulation is orders of magnitude cheaper than the DVM simulation in terms of the CPU time and memory usage. While the DVM requires a high-performance computing facility to meet the memory demand of several hundreds of gigabyte for simulating this 3D flow problem, the DSBGK simulation can run on an ordinary laptop. This work therefore demonstrates the DSBGK method as a practical simulation tool for 3D pore-scale gas flows in the shale rock. Converged results generated in this study can be used as comparison data by other simulation methods. [ABSTRACT FROM AUTHOR]

• GSIS provides accurate and efficient solution for multiscale rarefied gas flow. • New definition for the characteristic flow length of porous media. • Universal slope in the Klinkenberg correlation for permeability of 2D porous media. • MRT-LBM with wall scalling is not able to predict gas permeability of porous media. The apparent gas permeability (AGP) of a porous medium is an important parameter to predict production of unconventional gas. The Klinkenberg correlation, which states that the ratio of the AGP to the intrinsic permeability is approximately a linear function of reciprocal mean gas pressure, is one of the most popular estimations to quantify AGP. However, due to the difficulty in defining the characteristic flow length in complex porous media where the rarefied gas flow is multiscale, the slope in the Klinkenberg correlation varies significantly for different geometries such that a universal expression seems impossible. In this paper, by solving the gas kinetic equation using the general synthetic iterative scheme (GSIS), we compute the AGP in porous media that are represented by Sierpinski fractals and pore body/throat systems. With the abilities of fast convergence to steady-state solution and asymptotic preserving of Navier-Stokes limit, it is shown that GSIS is a promising tool to simulate low-speed rarefied gas flow through complex multiscale geometries. A new definition of the characteristic flow length is proposed as a function of porosity, tortuosity and intrinsic permeability of porous media, which enables to find a unique slope in the Klinkenberg correlation for all the considered geometries. This research also shows that the lattice Boltzmann method using simple wall scaling for the effective shear viscosity is not able to predict the AGP of porous media. [ABSTRACT FROM AUTHOR]