Quantitative visualization and characteristics of gas flow in 3D pore-fracture system of tight rock based on Lattice Boltzmann simulation
Introduction
The tight rock matrix in unconventional gas reservoirs is generally characterized by the small pore/throat and poor pore connectivity due to the complexity of sedimentation and diagenesis (Zou et al., 2012; Adebayo et al., 2019), which largely limits the large-scale and efficient development of reservoirs. Currently, there are many effective stimulation methods of reservoirs to improve the permeability and production of unconventional gas reservoirs, such as hydraulic fracturing, liquid N2 fracturing, and CO2 fracturing (Guo et al., 2015; Hou et al., 2016; Huang et al., 2019). By using the effective fracturing methods, the induced fractures can connect pores or natural fractures in the tight rock matrix, and then a complex pore-fracture system from micro-scale to macro-scale (Hou et al., 2018) is formed as shown in Fig. 1. The complex pore-fracture system provides high conductive channels for gas flow, thus the geometry of the pore-fracture system after the external stimulation is a primary factor to control future gas production (Warpinski and Teufel, 1987; Chen et al., 2015; Benamram et al., 2016). The gas flow simulations of the fractured reservoirs at the macro-scale have been achieved by using the constitutive equations and some macroscopic properties of the fractured reservoirs such as porosity and permeability are usually used (Middleton et al., 2015; Wang et al., 2016). The properties of the pore structure and micro-fracture morphology have an obvious impact on the formation's performance (Wang et al., 2016). Understanding of the local flow information for internal micro-structure is useful to reveal the gas flow physics and to investigate the macroscopic relations. However, experimental measurements of internal fluid migration in such contexts are often laborious since tight rock is like a “black box”.
Compared with the physical experiments, the numerical simulation has advantages of the “faster, better and more economical”. So far, significant efforts have been made to numerically model gas flow in the complex fracture networks based on the continuum hypothesis. The dual continuum model (Liu et al., 2019) and discrete fracture models (DFM) (Zidane and Firoozabadi, 2014) are the two most common approaches to study gas flow in fractured porous media. However, in these numerical methods, a volume-averaging method is employed and the complex microscopic structure of porous media which is directly related to flow regimes is ignored. For the tight rock, the pore size is mainly distributed in the range of nanometer to micrometer, where various gas flow regimes can be induced. Furthermore, the application of the common methods is clearly restricted due to complicated gridding issues and expensive computational costs, especially for the 3D models (Fuentes-Cruz et al., 2014). Lattice Boltzmann method (LBM), as a mesoscopic method based on kinetic theory, have attracted more and more attention because that it is unlimited by the continuum hypothesis and does not need to track the trajectory of the fluid molecules (Benzi et al., 1992; Chen and Doolen, 2012). Compared with the traditional computational fluid dynamics methods, LBM has three obvious advantages (Xu et al., 2019; Hou et al., 2020): (1) Good numerical stability; (2) Natural parallelism; (3) Simple fluid-solid boundary.
Fluid flow in porous media is most popularly modeled at pore scale by combining 3D digital rock and LBM. Faÿ-Gomord (2017), Degruyter (2010) and Ju et al. (2017a) presented the fluid flow distribution in the 3D digital core of tight chalk, sandstone and shale by the LBM simulation with non-slip boundary, respectively. In their studies, the permeability of the sample is estimated based on the Darcy's law. However, in such small pores of the tight rock, the mean free path (MFP) of gas and the size of pores are very close (Zhao et al., 2016a), which will cause a so-called Knudsen layer near the pore/micro-fracture wall and have a significant deviation from the classic Navier–Stokes predictions (Landry et al., 2016). To determinate the deviation, effective relaxation time modified by Knudsen number (Kn) in lattice Boltzmann (LB) model is employed (Wang et al., 2017a, 2017b). Kn, as a characteristic parameter of gas properties and pore space geometry, can indicate gas flow regimes in the microscale pores. Due to the complex pore geometries of 3D real rock, the vast majority of related studies mainly focus on some simple geometries (Ansumali and Karlin, 2002; Guo et al., 2006, 2008; Verhaeghe et al., 2009; Suga et al., 2010; Chai et al., 2010; Zheng et al., 2012; Liou and Lin, 2014), such as tubes and slits. Suga and Ito (2011), and Chai et al. (2010) also used a constant Kn to simulate fluid flow in the complex geometries. In complex porous rock, a locally varying Kn will be induced by locally varying pore sizes (Landry et al., 2016). Recently, Zhao et al. (2016a) proposed to calculate the effective relaxation time in LB model by using the local Kn. It should be noted that the MFP in the unbounded system is used to obtain the value of local Kn in this model. Nevertheless, the MFP of gas molecules in the bounded system is smaller than that in the unbounded system due to the wall effects (Wang et al., 2017a, Wang et al., 2017b). Landry et al. (2016) presented a multi-relaxation-time lattice Boltzmann model with the local effective viscosity to simulate slip flow regimes in arbitrary porous media, but the accuracy of simulation results has not been verified. Wang et al. (2017b) defined an averaged pore wall function to calculate the local MFP of 2D micro-porous media based on the power-law (PL) (Dongari et al., 2011) and the effect of geometric morphologies of 2D pore structure on gas flow is studied by applying a slip-based LB model with the local Kn.
In this paper, in order to study the effect of gas rarefaction and micro-fracture configuration on gas flow in the 3D pore-fracture system at the pore scale, we extend the PL approach to three dimension to calculate the local MFP of 3D digital rock and a 3D regularized lattice Boltzmann model considered the gas slippage effect is employed. The rest of the paper is organized as follows. The reconstruction of the 3D pore-fracture system for tight rock is introduced in Section 2. Section 3 presents the extended mathematical model for simulating gas flow in the 3D complex porous media and the model is validated with the test results. In Section 4, gas flow in the reconstructed pore-fracture system is visualized and analyzed. Section 5 discusses the effect of morphology and numbers of micro-fracture on rarefaction effects. The main conclusions are shown in Section 6.
Section snippets
3D pore structure of tight rock and its representative elementary volume
The tight sandstone obtained from a deep mine, Xuzhou, Jiangsu, China is selected as the experimental material. Before the 3D pore structure reconstruction of the tight sandstone, the X-ray diffraction experiments and mercury intrusion tests on the tight sandstone samples are performed. The main minerals of the tight sandstone are quartz (87.3%), feldspar (6.1%), and calcite (3.9%), and its pores are mainly composed of transition pores, mesopores, and macropores based on the definition of pores
Evolution equation
In the LB model, the single relaxation lattice Bhatnagar–Gross–Krook (LBGK) model can greatly simplify the calculation process and has been widely used. This model is also employed in the paper, and its basic equation can be expressed as follows (Bhatnagar, 1954):where is the distribution function at the lattice site x and time t; is the equilibrium distribution function of i direction; is the discrete distribution function of
Fracture roughness
In this section, gas seepage simulations in the pore-fracture systems with different fracture roughness are carried out by using the established LB model. In the simulation, the upper and down sides of the digital core are set as the inlet and outlet, respectively. The pressure boundary is applied to the inlet and outlet, where the outlet boundary is set as a fixed pressure value of 5 MPa, and the gas pressure gradient between the inlet and outlet is 0.1 MPa/m. The fluid-solid boundary is
Discussion
Base on the results of Section 4.3, the gas rarefaction effect induced by gas pressure is serious in micro-porous media because that the MFP of the gas molecular is close to the characteristic length of the pore-fracture system. And from the study results of Wang et al. (2017b), the gas rarefaction effect is also highly dependent on the morphology of micro-porous media. To quantitatively evaluate gas rarefaction effect as affected by the pore-fracture structure, two key fracture parameters: the
Conclusions
In this study, a regularized LB model with the slippage boundary condition is applied to simulate the gas flow in the 3D pore-fracture structure at the micro-scale. The 3D pore-fracture systems are reconstructed based on the REV of the intact sandstone and the fractal fracture function. The PL wall function is modified and extended to 3D porous media, to calculate the local relaxation time for the 3D LB model. The LB model is verified with the experiment data. Effects of the fracture morphology
Credit author statement
Peng Hou: Methodology, Software, Formal analysis, Investigation, Data curation, Writing – original draft. Liang Xin: Conceptualization, Methodology, Validation, Writing – review & editing, Supervision. Feng Gao: Writing – review & editing, Supervision. Jiabin Deng: Software, Formal analysis. Jian He: Validation, Resources, Writing – review & editing. Yi Xue: Writing – review & editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work had been financially supported by the Open Fund (No. PLC2020034) of the State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Chengdu University of Technology), the Open Fund of the State Key Laboratory for GeoMechanics and Deep Underground Engineering, China University of Mining & Technology (No. SKLGDUEK1905), and the Research Fund of the State Key Laboratory of Coal Resource and Safe Mining, CUMT (No. SKLCRSM19KF001).
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