Apparent permeability study of rarefied gas transport properties through ultra-tight VORONOI porous media by Discrete Velocity Method
Introduction
As a type of relatively clean fuel with lower conventional pollutants and emissions of greenhouse gases compared with other fossil fuels, natural gas accounts for a growing share of global energy consumption (Wang et al., 2017a; Paltsev et al., 2011; Mehmani and Prodanović, 2014). Benefited from newly advanced technologies for exploitation including gas injection displacement (Entov et al., 2002; Huo et al., 2017) and hydraulic fracturing techniques (Savins, 1978; Rahm, 2011; Lecampion et al., 2018; Li et al., 2019), unconventional resources, such as tight gas or shale gas have already become a significant source of natural gas production in North America and is gradually triggering another energy revolution in Asia and Europe (Gao and Li, 2016; Al-Douri et al., 2017; Wang et al., 2017b). Compared to conventional gas reservoirs, unconventional natural gases are stored in a special type of geological condition with dense and low permeability shale rock dominated by multi-scale pores, which greatly increases costs and difficulties of extraction (Liu and Zhang, 2019; Loucks et al., 2009; Bilgen and Sarıkaya, 2016). To further understand the exploitation of shale gas, investigating the main factors on permeability in complex porous shale rock could have practical benefits in long-term estimation of gas production.
Advanced computational fluid dynamics could be directly employed to reveal the mechanism of transport process. However, the majority of pores which store a large amount of shale gas within kerogen matters are nano-pores with widths ranging from 2 to 50 nm (Loucks et al., 2009). When the characteristic flow length of shale rock H is of the same order as or even smaller than the mean free path of methane molecule λ, the ratio of these two lengths, which is defined as the Knudsen number (Kn = λ/H), may become relatively large (Gad-el-Hak, 2001). With different Knudsen numbers, the flow regime falls into the following four flow regimes: continuum flow (Kn < 0.001), slip flow (0.001 < Kn < 0.1), transitional flow (0.1 < Kn < 10), and free molecular flow (Kn > 10) (Cercignani, 1969). With the decrease of mean pressure in the nanopores of conventional gas reservoirs, the shale gas flow could transit from the slip flow regime to the transition regime where the conventional Navier-Stokes equations derived from the continuum assumption fail since the heat flux and shear stress in the gas dynamic models cannot be simply expressed in terms of the lower-order macroscopic quantities. Under such circumstances, a number of rarefaction effects have been revealed, including the velocity slip (Tang et al., 2005), temperature jump (Sone et al., 1989), Knudsen paradox (Steckelmacher, 1986), and thermal transpiration (Reynolds, 1879). For shale gas in unconventional reservoirs, a direct consequence of the rarefaction effects is the famous Klinkenberg effect, which describes that the apparent gas permeability ka could be significantly larger than the intrinsic permeability kin and the traditional Darcy's law fails with the decrease of gas pressure, or the increase of Knudsen number (Klinkenberg, 1941). Therefore, accurately capturing the rarefaction effects is the basis for studying the apparent permeability in microscale.
From a mesoscopic and statistical perspective, the Boltzmann equation for the velocity distribution function (VDF) of gas molecules, which is derived based on the gas kinetic theory, is a fundamental way to describe such rarefied gas dynamics (Wu et al., 2013). In practice, for the low-velocity shale gas flow which is confined in the extremely low permeability porous media, a deterministic solution is commonly sought for the gas kinetic models that linearizes the velocity distribution function by global equilibrium state in the Boltzmann equation to the linearized Bhatnager-Gross-Krook (LBGK) kinetic equation with a simpler collision relaxation term (Bhatnagar et al., 1954). The VDFs are discrete in the molecule velocity space by discrete velocity method (DVM) (Goldstein et al., 1989). This treatment not only retains the capability to solve the non-equilibrium high-order terms beyond the Navier-Stokes level, but also more suitable for incompressible flows compared with the Direct Simulation Monte Carlo method (DSMC), avoiding the statistical noise at low-speed conditions (Morris et al., 2011). Recently, the DVM solution to the LBGK has shown the great potential for the study on the transport mechanism of rarefied gas. Wu et al. investigated the apparent gas permeability of simple porous media and QSGS media in detail and their results reveal the importance of tortuosity and anisotropy in gas transport (Su et al., 2017; Wu et al., 2017; Germanou et al., 2018). Liu et al. extended the classical of Knudsen paradox from straight channel to bent boundary conditions. A new phenomenon called rarefaction throttling effect was observed, which shows that the mass flow rate of rarefied gas at the large Knudsen number regime could be influenced by the non-straight boundary more significantly (Liu et al., 2018). The above works demonstrate the promising prospect of this high-fidelity numerical method for capturing the non-equilibrium flow in porous media.
Even with advanced numerical methods, a proper understanding of the mechanism of the transport process in unconventional reservoirs still strongly requires knowledge of the microstructures of porous media. Similar to the instruments using X-rays for medical examinations, the images of a few millimeters across rock samples, normally constraining the resolution to a few microns and even sub-micron resolution, are constructed by micro-CT scanners around 1,0003 to 2,0003 voxels (Yeong and Torquato, 1998). Direct application of computational fluid dynamics on scanned images can yield more precise and realistic flow process. Ho et al. (2019) investigated the rarefied gas flows in one shale sample by the LBGK to test Klinkenberg-type models. However, a deep understanding of the gas transport mechanism and the corresponding extraction of core factors affecting the permeability depend on comparing differences under diverse parameters. The high costs of experimental test as well as the computational resource make the direct simulation results more suitable as a follow-up verification rather than summarizing the rules.
Rather than indulging in infinitely real but unnecessary details, many researchers attempt to review the representations of porous media from a statistical prospect, to capture key features of the porous media, e.g. porosity, pore/throat distribution and corresponding connectivity (Dong and Blunt, 2009; Vogel and Roth, 2001). The basic frame in a statistical sense strongly relies on the network containing the topological relationship between the different pores connected by the throats. This concept could be easily and precisely defined by a VORONOI tessellation of the pore space. A VORONOI polyhedron contains all the solid area closer to itself than any other and yields the topological relationship between vertices and edges directly, which are considered to be well suited as a representative of pores and throats in rock respectively in previous works (Rostron, 2018). Although many studies about gas flow in porous media have been reported, the theoretical geometries, e.g. spherical particles (Gallis et al., 2001), QSGS (Wang et al., 2017c) and fractal structures (Yu and Cheng, 2002), differ significantly from real shale shock in porosity and pore distribution. Moreover, for traditional generation methods of porous media which insert solids to confine flow channels, the inserting process could become more difficult as the porosity decreases. To overcome these drawbacks, we firstly introduce VORONOI tessellation to construct topological relationship for pores and throats in porous media. Therefore, the position of pores and throats could be directly obtained and the width of pores and throats can be accurately controlled to produce the porous media with extra-low porosity. Meanwhile, the clear topological relationship significantly improves the controllability of key parameters of porous medium, such as pore size and corresponding distribution, throats width and length, coordination number of pore and throats, etc., which greatly facilitate the study on how these key parameters influencing the apparent permeability.
The remainder of this paper is structured as follows. We propose a new method for the generation of controllable low-porosity porous media based on the VORONOI tessellation in Section 2. The intrinsic permeability of continuum gas flow in VORONOI porous media is studied in Section 3. The numerical results of apparent permeability in VORONOI porous media and high-order apparent permeability correction model are presented in Section 4. Finally, the conclusions are drawn in Section 5.
Section snippets
Model description
The present study focuses on the mechanisms of transport process in porous media and aims to apply advanced computational gas kinetic methods to simulate gas flow in porous structure reconstructed by controlling the essential morphological parameters including porosity, tortuosity and pore throat ratio. To this end, pressure-driven flows are investigated by adopting deterministic solutions of the Multiple-Relaxation-Time Lattice Boltzmann Method (MRT-LBM) and the Linearized BGK equation (LBGK)
Physical model
To explore how the geometry affects flow properties in VORONOI porous media, the MRT-LBM method is employed to study intrinsic permeability in continuum flow regime. The cube-shaped VORONOI porous media are constructed with side length H. Surfaces of and are applied by inlet and outlet pressure boundary with pin and pout, respectively. Surfaces of , , and are the upper, lower, left and right boundaries, respectively, which are subjected to symmetric
Gas flow in VORONOI porous media in non-equilibrium flow regime
The Linearized BGK equations solved by the Discrete Velocity Method could capture the non-equilibrium effect in gas flow accurately under complex boundaries. However, a large number of discrete velocities are required to capture non-equilibrium effects precisely in highly rarefied gas flow. Specifically, to guarantee the reliability of the results, at least 9 and 24 discrete velocity numbers are required in each velocity direction for the Knudsen numbers of 0.1 and 10.0, respectively. That is,
Conclusions
In this work, we have proposed a new method for the generation of controllable low porosity porous media based on the VORONOI tessellation and its topological relationship. Two numerical simulation methods were employed to study gas flow in VORONOI porous media, including the multiple-relaxation-time lattice Boltzmann method for continuum gas flow and the linearized BGK equations via the discrete velocity method (DVM) for non-equilibrium gas flow. The main findings could be summarized as
Declaration of competing interest
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work. There is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant No. 51825604 and No. 11802231, the Natural Science Basic Research Program of Shaanxi Province under Grant No. 2019JQ-494, and the China Postdoctoral Science Foundation under Grant No. 2019T120928.
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