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Modeling the effects of gas slippage, cleat network topology and scale dependence of gas transport in coal seam gas reservoirs
Fuel 264 (2020) 116715
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Full Length Article
Modeling the effects of gas slippage, cleat network topology and scale dependence of gas transport in coal seam gas reservoirs
T
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Xu Yua, , Lincheng Xua, Klaus Regenauer-Lieba, Yu Jinga, Fang-Bao Tianb a b
School of Minerals and Energy Resources Engineering, UNSW Sydney, Australia School of Engineering and Information Technology, UNSW Canberra, Australia
A R T I C LE I N FO
A B S T R A C T
Keywords: Coal seam gas flow Lattice Boltzmann method X-ray micro-computed tomography Gas slippage CSG recovery
Gas flow in coal cleats plays a significant role for the permeability of coal seams. Investigations of gas transport in coal are carried out in this work using a numerical scheme with a coal sample scanned by the micro-computed tomography. We analyse a coal sample from the Bowen Basin (Australia) to characterize the natural discrete fracture network defined by connected cleats and matrix blocks. Due to the heterogeneity of coal micro-structures, the numerical simulator is combined with a grid-adaptive scheme. The accuracy of the simulator is validated for gas slip regime (0.01 < Kn < 1.0 ) in the micro-straight channel. Predictions of permeability and porosity are conducted through the method. The results show that the porosity and permeability of the digitalized coal model change along with the length and stabilizes at >400 voxels and >700 voxels, respectively. The Knudsen number (Kn) has an important effect on the gas slippage which can enhance the coal seam gas transport for narrow cleat networks. The increase of Kn ranging from 0.0005 to 0.01 increases the mass flux as well as the permeability of the coal for a given cleat aperture. The investigations of the gas transport in cleats prove the capability of the numerical simulator for modeling gas slip flow. It also provides a potential tool for solving gas transport in the complex geometry of a natural coal seam gas (CSG) reservoir.
1. Introduction Coal seam gas (CSG) production has made recent breakthrough propelling Australia into the top spot for LNG export for the first time in November 2018 due to an increase of 23% from CSG production in Queenslandhttps://www.energy.gov.au. Yet, elsewhere in the world CSG has not made similar progress and similar to shale gas recovery the success of harvesting these unconventional energy sources is not universally repeatable. We propose that this is due to the lack of quantitative tools to investigate gas flow through tight formations such as coal or shale gas reservoirs. In coal the main gas flow is enabled by narrow natural channels, known as cleats [24,19]. In this contribution, we present a robust mesoscopic, numerical framework that models the gas flow at the pore scale of coal cleats considering the phenomenon of gas slippage. Coal seam gas (CSG) production stems from the desorbed gas of the microporous structures of coal. Coal features a dual-porosity consisting of small matrix pores and cleats. The desorbed gas diffuses through the matrix to the cleat system and flows along the cleat network. The cleats, therefore, provide the primary channels for transferring the CSG to the wells. It is well documented that the permeability of coal seams is
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predominantly affected by the intensity and aperture of the cleats of coal seams [4,31]. Most CSG simulations and laboratory experiments consider a macroscale formulation where the permeability is considered to increase with the intensity and aperture of cleats existing in coal seams [24,19,39]. Conversely, permeability is assumed to decrease because of cleat closure or decreasing intensity of coal cleats. Many laboratory studies of coal fractures have been done but little data on the in situ properties of coal cleats has been obtained. For the investigations of the CSG transport in porous structures of coal, experimental and theoretical contributions [8,36,10,26,13,64,65] have been made in the previous references. These macroscopic models are constructed for the investigation of gas permeability and gas diffusion models for coal [48,47,43]. The Darcy’s law and Fick’s law are used for modelling gas flow in coal fractures [39] and gas diffusion in coal is regarded to be homogeneous [48]. Many permeability models have been built for averaging the fluid flow in coal by theoretical and experimental methods. Such coal permeability models usually are functions of the porosity, fluid pressure, and ad/desorption. For example, the power-law dependencies between the permeability and porosity built on simple
Corresponding author. E-mail addresses: [email protected], [email protected] (X. Yu).
https://doi.org/10.1016/j.fuel.2019.116715 Received 21 September 2019; Received in revised form 11 November 2019; Accepted 20 November 2019 0016-2361/ © 2019 Elsevier Ltd. All rights reserved.
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porous structures, such as packed beeds or cubes[30], have been extended for the applications in the investigations of the CSG production [36]. The effects of fluid pressure and volumetric deformation have also been considered in models built by Gray and Harpalani [14,16]. Due to the internal dual-porosity of the coal matrix [12,48,47], a bidispersepore model is correspondingly introduced and applied to model the gas transport in macro- and micro-pores [44–46,34]. However, the bidisperse model is built analytically based on packed pellets of various sizes and then transferred for applications in gas diffusion in coal. Among these models, the Palmer-Mansoori(P-M) model (1998) [35] and the Shi-Durucan (S-D) model [45,46] are currently two popular choices used in reservoir simulations of the CSG production. The above-mentioned models are based on the state of the macroscopic scale. The current models therefore are not built to consider the mesoscopic structure of the coal and its cleat system. Similarly, direct investigations of gas flow in coal cleat networks, have been poorly investigated. Recently, by the combination of X-ray micro-CT imaging technique porescale models for investigations and predictions of fluid transport properties, such as porosity, permeability, reaction diffusion, have attracted a lot of attention. Some pioneer research works have been constructed for the characterization and analysis of coal micro-structure and numerical simulations based on the computed-tomography images [32,40,27,61]. The tomography technique provides a complementary method for building an accurate pore-scale model for the numerical studies. It has been also applied as a useful tool for the studies of the the effects of the sorption deformation and effective stress on the permeability of coal [63,62]. Limitations of experimental studies, complementary numerical studies are needed for better interpretations of the gas transport mechanism. An important effect that has not yet been considered is the Klinkenberg gas slippage effect which has a significant impact on the permeability of a porous medium for liquid and gas flow [23]. It is reported that gas slippage occurs when the pore aperture approaches
Fig. 2. A schematic diagram of 2-D micro-channel with the channel height H and length L. The ratio of the height and length of the channel is 1.0.
the mean free path of the gas seen in Fig. 2 which is indexed by a ratio named Knudsen number. Experimental and theoretical studies have shown that the gas slippage effect increases the permeability of coal for high Knudsen numbers [33,15,51]. Niu et al. concluded that the slippage effect can not be neglected by experiments showing three Australian coal samples where the Knudsen number turned out to be larger than 0.1. Therefore, it is of importance to consider the gas slippage for the numerical investigations of the CSG transport in coal. In this contribution a numerical simulator is developed based on the mesoscopic lattice Boltzmann method for modeling the fluid flow in the coal cleat network combined with the immersed boundary for adjusting the heterogeneous structure of coal. The simulations allow full assessment of the effect of permeability of coal and the Klinkenberg gas slippage effects on the gas flow. The Lattice Boltzmann method allows a mesoscopic assesment of the fluid flow process without Darcy Fig. 1. 3D visualisation of segmentation and reconstruction of coal-cleat system by using X-ray computed tomography and the figure is referred to the Ref.[21] including (a) A coal sample scanned by the micro-CT imaging, (b) Reconstructed 3-D model with discrete fracture network (DFN) method and (c) 2-D cleat network parallel to y-z surface extracted from 3-D model of (b).
2
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∫Γ FIB (xb ) δ (xb − x ) dS,
assumption and enables therefore to tackle exactly the range of scales encountered in gas flow through coal seams[52,53]. The method also allows integration of the topology of cleats obtained from micro-CT analyses. In previous work, we have presented simulations of fluid flow using the immersed boundary lattice Boltzmann method[50,49,55,57]. Based on the numerical method proposed in Ref. [58] investigations of gas flow in the coal cleats have been conducted in this study.
where f IB is the body force acting on a fluid point x , FIB is the Lagrangian force density acting on the fluid by the IB, xb is the Lagrangian point on the immersed interface, δ () is Dirac delta function proposed in Ref.[38], and Γ represents the involved interfaces. By using the feedback IB scheme, the Lagrangian force density is determined by
2. Methodology
FIB (xb) = k ⎡Ub (X b) − ⎣
f
f In recent years, X-ray micro-computed tomography (CT) has allowed unprecedented insight into the microstructures of coal and statistical methods have been developed to adequately incorporate this information for mesoscopic simulations [2,41]. The obtained data is provided at high resolution making it an ideal methodology for the model construction of numerical simulations. In this paper we use the methodology to investigate gas flow in the fractured coal using a reconstructed micro-CT model of a coal sample from the Bowen Basin of Australia. For measurements and characterizations of coal, the micro-CT has inherent advantages over other destructive methods of characterizing coal microstructure. The digital coal structure used in the simulations is built based on a discrete fracture network model. The segmentation and analysis results of the current coal sample have bee published by Jing et al. 21. For details of the segmentation and reconstruction method, the reader is referred to Refs. [21,20].
(5)
IB (x )
=
∑ FIB,i δ (xb,i − x )ΔS, xb,i ∈ Γ,
(6)
2.3. Boundary conditions and adaptive Cartesian mesh scheme For the numerical simulations of gas flow in a straight fracture in Section 3.1, the periodic boundary condition is applied in the y-axis and the fluid flow in the x-axis is driven by pressure gradient. The pressuredriven boundary condition is functioned by body force acting on the fluid. In the sequential section, investigations of gas flow in the microcleat system are conducted in the reconstructed coal-cleat network. Symmetrical boundary conditions are used in both of x- and y-axis directions and gas flow is also driven by pressure differences. A adaptive meshing scheme with Cartesian grid system is used in this work for simulations of gas flow in complex coal-cleat systems of coal based on a previous work [55]. The core part of the scheme is to add fine meshes around the fluid–solid interface and coarse meshes in the far field which can significantly improve the computational stability and efficiency. The mesh refinement ratio of two adjacent blocks of different refinements is two in this work.
A simulator is developed based on the immersed boundary-lattice Boltzmann (IB-LB) method presented in the Ref.[55] which is applied in this work for the micro-gaseous flow in coal cleat networks. The main governing equations used in the simulator are briefly introduced and listed below. The lattice Boltzmann equation (LBE) with the multiplerelaxation-time model is expressed by
mi (x + ei δt , t + δt ) − mi (x , t )
3. Results
(1)
where m = Mg is the velocity moment composed of the momentum matrix M and distribution function g , meq = Mg eq is the equilibrium moment with the equilibrium function g eq , ei is the discrete velocity, si is a series of relaxation rates, the force component G is expressed as
3.1. Gas flow in a micro-channel To study the problems of gas flow in micro-scale channel, the Knudsen number (Kn) is an important influence factor affecting fluid flow character. When the Knudsen number is high by lowering the height of a channel, the phenomenon of slippage occurs which plays an significant role for computations of permeability. Fig. 2 shows the simple structure of a 2-D fracture channel. This section presents validations of micro-scale gas flow within a range of Knudsen number between 0.01 and 1.0. Cercignani derivated an analytical solution of second-order accuracy from the linearized Boltzmann equation relating slip velocity to Knudsen number[5] which is give by.
(2)
where f is an external body force. The macroscopic variables, such as density, velocity, pressure, are calculated by ρ = ∑ gi , u = (∑ gi ej + 0.5fδt )/ ρ , p = ρcs2 respectively, and cs is the lattice sound speed. The equilibrium distribution function g eq is determined by
(u·ej )2 u · ej u2 g eq = ωj ρ ⎡1 + 2 + − 2⎤ j 4 ⎢ c 2 c 2 cs ⎥ s s ⎦ ⎣
∫Ω u (y) δ (y − xb) dy⎤⎦,
(4)
where kis the feedback coefficient analytically calculated by [55] as a sum of polynomials of the local normal vector to the immersed boundary, Ω is the whole fluid domain, Ub is the target boundary velocity, ui is the fluid velocity at x i , FIB, i is the Lagrangian force density at xb, i , ΔS is the local area associated with xb, i , and ΔVi is the local lattice volume. The approximation scheme of the feedback coefficient can reduce the velocity error at the immersed boundaries and improve the numerical stability of simulations.
2.2. Development of the IB-LB simulator for fluid flow
G = ωρδt [f·(ej − u )/cs2 + (f·ej )(u·ej )/cs4]
=
and
2.1. Segmentation of micro-CT data
s = si (mieq (x , t ) − mi (x , t )) + ⎛1 − i ⎞ (MG )j , 2⎠ ⎝
IB (x )
(3)
us / U = 1.146Kn
where ωj is the jth weight. The D2Q9 model is used for the two-dimensional simulation. The fluid kinematic viscosity ν is related to several specific relaxation rates as detailed in Refs. Optimized or suboptimized choice of the relaxation rates, si , for these three models can be found in the references provided, and thus is not discussed here. For enhancement of the performance of the numerical computation, an improved immersed boundary (IB) method is used in which the boundary conditions are applied by spreading the Lagrangian force density into the bulk fluid and given by
∂u ∂u2 + 0.907Kn2 2 ∂y ∂y
(7)
where us and U is the slip velocity and reference velocity respectively. In this case, three non-dimensional parameters which are Knudsen UL λ number (Kn = H ), Reynolds number (Re = ν ), and Mach number U
(Ma = c ), are considered for the micro-channel flow. The parameters s considered in these cases are listed in Table 1. Studies of grid convergence analysis and validation comparing the present results with Cercignani’s analytical solution, have been conducted for 0.01 < Kn < 1 and presented in the Fig. 3(a) and 3(b). The results show that the 3
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Table 1 Parameters of Knudsen number (Kn), Reynolds number (Re) and Mach number (Ma) for micro-channel flow. No.
Kn
Re
U / cs
δx / H
1 2 3 4 5
1.0 1.0 1.0 1.0 1.0
0.02171 0.02171 0.02171 0.02171 0.02171
0.0173 0.0173 0.0173 0.0173 0.0173
0.015625 0.0078125 0.00390625 0.001953125 0.0009765625
6 7 8 9 10 11
0.01 0.05 0.1 0.2 0.5 1.0
2.171 0.434 0.2171 0.1085 0.0434 0.02171
0.0173 0.0173 0.0173 0.0173 0.0173 0.0173
0.001953125 0.001953125 0.001953125 0.001953125 0.001953125 0.001953125
Fig. 4. A diagram of geometry-adaptive mesh refinement in a cleat joint including (a) segmented cleat joint before meshing and (b) geometry-adaptive grids.
lattice nodes. Procedures of geometry-adaptive mesh generation can be divided into the following steps. Step 1: X-ray micro-CT scans tomographic slices are obtained as raw data. The reconstruction processes and segmentation are carried out sequentially and the results exported for simulations. Step 2: For the immersed boundary method the internal interfaces/ fracture walls need to be identified. This process is performed in a separate numerical program. Step 3: In this step, the geometry-adaptive mesh is generated according to the shape of coal-fracture walls which have been extracted in step 1. The mesh grids grow in four directions of x, -x, y, -y (2-D). Grids are mainly developed within the cleat channels. No grids exist in the solid block except those necessary for the boundary conditions. Fig. 4(a) and 4(b) present a micro-CT image of a cleat joint before and after geometry-adaptive grids generation respectively, according to the above procedures. It is shown that fine meshes are generated along the cleat wall and coarse meshes are created in the middle of the channel in Fig. 4(b). The Neumann boundary condition is imposed at inlet and outlet as we consider a pressure-driven flow in the coal-cleat network. For the simulations of gas flow in 2D cleat network, the minimum mesh size used in the 2D cleat study is 1/32H (H the cleat aperture). This mesh generation scheme is explained in Lincheng et al.’s work [55]. Readers who are interested in the scheme can find it in the literature. (See Fig. 5).
Fig. 3. Results for gas slip flow in a micro-channel including (a) grid convergence analysis at Kn = 1.0 and (b) slip velocity for 0.01 < Kn < 1.0 .
improved IB-LB method is capable to simulate micro-scale fluid flow 3.2. Fluid flow in 2-D coal cleat network 3.2.1. Geometry-adaptive mesh refinement The IB-LB method is applied for the fluid flow in coal-cleat networks with a combination of the geometry-adaptive Cartesian grids in which no grids are generated in the solid matrix except those necessary for the boundary conditions.In this work, the microstructure of coal is created by the segmentation and reconstruction schemes for cleat-matrix network based on the CT imaging of the coal sample. A special boundary method (referred as ”non-ghost cell method”) has been established for the suspending grids in the non-fluid region. In the implementation process, each non-fluid block is represented by a virtual fluid lattice node. Mesh grids grow in the normal direction of cleat boundaries and are divided into several blocks with various grid sizes. Computations of fluid flow through the porous structure of coal are carried out on the 4
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Fig. 5. A schematic diagram of dividing the 2-D coal cleat structure (overall 960 voxels) into multiscale-length cells.
3.3. Scale-dependence analysis of coal-cleat porosity and permeability The scale dependence of the macroscopic rock properties, such as porosity, permeability is well known in the literature [3,60,17,1]. Coal has a heterogeneous porous structure that changes the porosity and permeability by increasing the length of the coal sample. From the micro-CT data shown in the Fig. 1(c), the connected porosity ∊ is esti∑N mated by counting the number of pixels of connected cleats ∊ = ∑ Nc where N c and N are number of pixels in cleats and solid matrix respectively. Nine cleat structures of various lengths are generated from the two-dimensional cleat network (Fig. 1(c)), including 200, 300, 400, 500, 600, 700, 800, 900, and 960 (voxel). The approximated porosity is calculated correspondingly and presented in Fig. 6(a). Simulations of gas flow are carried out based on the nine coal-cleat structures. The result of estimated porosity is presented in Fig. 6(a) showing that the microscale porosity of connected cleat network changes and gradually stays at around 4% when increasing the model size. In this case, the minimum length of representative element volume is 700voxel/2.91mm defined by a stabilization of the value of average porosity around 4%. The corresponding permeabilities are calculated by the current IB-LBM method and shown in Fig. 6(b). It is found that the permeability of the coal-cleat system increases along with the sample length until it also stabilizes for a Length > 700voxel .
Fig. 6. Length dependence analysis includes (a) porosity of coal calculated based on the digitalized data and (b) Permeability computations for various length scale.
3.4. Effects of Knudsen number The Klinkenberg effect plays a very significant role in the microgaseous flow in porous media [23] and the impact becomes bigger for Knudsen number (Kn) larger than 0.001(slip regime). Due to the interest in other research areas where flow through nanotubes in e.g. MEMS and shale gas is considered a significant research effort has been dedicated to investigate gas flow with coupled Klinkenberg effects [66,6,7,59]. The main channels for coal seam gas are defined by a natural fracture network known as cleats which occur in two sets of orthogonal fractures named ”face cleat” and ”butt cleat” [24,19]. Face cleat is usually parallel to coal seam floor and its connectivity plays a dominant role in the contribution to the cleat parallel permeability. Klinkenberg derived the effective gas permeability at a finite pressure by considering gas slippage which is represented by the Knudsen number Kn [23]. When Knudsen number is larger than 0.001, the Klinkenberg effect has a higher effect on gas flow behavior, especially
Fig. 7. Evolution curves of gas flow in 2-D cleat model for various Knudsen numbers containing the black line for dimensionless mass flux and red line for the permeability.
in low permeability media [42,37,54,18]. In the present study, the numerical code considers 0 < Kn < 0.01 and the result is illustrated in Fig. 7. It is shown that the permeability increases along with the increase of Knudsen number. 4. Discussion Gas transport in cleats plays a key role in the CSG production from coal seams. The goal of this paper is to present a numerical simulator for direct modeling mesoscopic effect on gas flow via characterizing the 5
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(1) A numerical simulator composed of immersed boundary and lattice Boltzmann method is introduced. Gas flow in a straight microchannel is investigated for the gas slip flow. The slippage effect is determined by the Knudsen number. The grid convergence result shows that the slip velocity profiles converge with decreasing grid size. The accuracy of the simulator is validated by the comparison with the Cercignani’s second-order solution. (2) The porosity and permeability of the coal sample feature scale dependence below a critical length. Simulations of gas flow at length from 200 to 960 voxels show that the porosity and permeability grow with length and stabilize at 400 voxels (porosity) and 700 voxels (permeability), respectively. (3) Investigations of the Klinkenberg effects on gas transport in coal are carried out. It is shown that the Klinkenberg effect plays a significant role in coal seam gas production. The outlet mass flux and permeability show a substantial increase with increasing Knudsen number for the same channel width.
microstructure (cleat network) with the micro-CT technique. This study of the CSG migration in coal cleats is conducted on a reconstructed coal structure considering the gas slippage effects. The current simulator is built using LBM which has a limitation for modeling micro-gaseous flow for large Kn. Therefore, modified immersed boundary method (IBM) is used in this work as an extension to be able to deal with the gas–solid interactions for modeling gas slippage effects. The method is particularly suited for this analysis due to its capability of dealing with complex boundaries [38]. We have shown in this work that the new simulator is capable of dealing with the gas flow in the straight channel for the slip flow regime 0.01 < Kn < 1.0 . However, it is found that it is of low accuracy for solving curved-wall problems [56]. The Kn in the present formulation is defined as the ratio of the molecular mean free path length to the mean cleat aperture. Unfortunately, direct measurements of Kn under in situ conditions in CSG reservoirs are not available, however, extrapolations have reported cleat widths ranging from 0.001 to 20 mm based on microscopic measurements of Bowen Basin coal samples [11]. Law et al. (1993) and Laubach et al. (1998) observed coal samples in the laboratory and measured the mean cleat aperture ranging from 0.01 to 0.2 mm [24,25]. Similar results can be found in other references [9,29,22]. The value of Knudsen number of coal cleats is, therefore, estimated to be less than 0.05 based on the above data which is the range adopted for our simulations of gas flow in two-dimensional cleat network. According to the current research, it is found that the effects of gas slippage grow with the increase of the Kn which can improve the outlet mass flux and the permeability of coal seams. Fig. 7 shows that for Kn > 0.001 the Klinkenberg effect is of importance for the CSG production and cannot be neglected. Temperature and confining pressure have a big influence on gas flow in coal and the Klinkenberg effect. The main effect of temperature and confining pressure is to change the aperture of the cleats with a minor change on mean free path of the gas. The effect of cleat aperture change as a function of thermal and mechanical strains is subject of ongoing work requiring analysis of X-ray microtomography experiments in environmental cells replicating pressure and temperature conditions in the reservoir. Therefore, in this paper the Knudsen number is used as a free parameter to allow incorporation of the temperature and pressure effects on gas flow through coal cleats upon completion of the experiments. While the simulator has shown to be a robust and numerically efficient multiscale formulation the main limitation of the current scheme is the lack of incorporation of adsorption induced volumetric strain which can adversely affect the cleat aperture leading to an increase in Knudsen number. Future work will, therefore, consider the pressure propagation in fluid and solid at the same time with an extended version of the simulator. Little work has been available on this important topic except for a pioneering paper by Ref. [28] who consider the stress propagation from a solid body to the fluid with the lattice Boltzmann method. This implementation would provide a potential to simulate the gas migration through the coal cleat and matrix as well as the important stress–strain evolution. Due the novelty of the mesoscopic approach few laboratory measurements are available at the micro-scale which is another area of future work.
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 research/project was undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government. Xu Yu would like to thank the China Scholarship Council (CSC) and UNSW for providing scholarships for his study. This work was supported by the Australian Research Council (ARC Discovery grants No. DP140103015, DP170104550, DP170104557, ARC Discovery Early Career Researcher Award Grant No. DE160101098). References [1] Aramideh S, Vlachos PP, Ardekani AM. Pore-scale statistics of flow and transport through porous media. Phys Rev E 2018;98:013104. [2] Blunt MJ, Bijeljic B, Dong H, Gharbi O, Iglauer S, Mostaghimi P, Paluszny A, Pentland C. Pore-scale imaging and modelling. Adv Water Resour 2013;51:197–216. [3] Bosl WJ, Dvorkin J, Nur A. A study of porosity and permeability using a lattice boltzmann simulation. Geophys Res Lett 1998;25:1475–8. [4] Busch A, Gensterblum Y. Cbm and co2-ecbm related sorption processes in coal: a review. Int J Coal Geol 2011;87:49–71. [5] Cercignani C. Theory and application of the Boltzmann equation/ [by] Carlo Cercignani. London: Scottish Academic Press; Distributed by Chatto and Windus Edinburgh; 1975. [6] Chai Z, Shi B, Guo Z, Lu J. Gas flow through square arrays of circular cylinders with klinkenberg effect: A lattice boltzmann study. Commun Comput Phys 2010. https:// doi.org/10.4208/cicp.010809.081209a. [7] Chen L, Fang W, Kang Q, De’Haven Hyman J, Viswanathan HS, Tao WQ. Generalized lattice boltzmann model for flow through tight porous media with klinkenberg’s effect. Phys Rev E Stat Nonlin Soft Matter Phys 2015;91:033004. [8] Clarkson C, Bustin R. The effect of pore structure and gas pressure upon the transport properties of coal: a laboratory and modeling study. 2. adsorption rate modeling. Fuel 1999;78:1345–62. [9] Close JC, Mavor MJ. Influence of coal composition and rank on fracture development in fruitland coal gas reservoirs of san juan basin; 1991. [10] Danesh NN, Chen Z, Aminossadati SM, Kizil MS, Pan Z, Connell LD. Impact of creep on the evolution of coal permeability and gas drainage performance. J Natural Gas Sci Eng 2016;33:469–82. [11] Gamson PD, Beamish BB, Johnson DP. Coal microstructure and micropermeability and their effects on natural gas recovery. Fuel 1993;72:87–99. [12] Gan H, Nandi S, Walker Jr P. Nature of the porosity in american coals. Fuel 1972;51:272–7. [13] Gerami A, Mostaghimi P, Armstrong RT, Zamani A, Warkiani ME. A microfluidic framework for studying relative permeability in coal. Int J Coal Geol 2016;159:183–93. [14] Gray I, et al. Reservoir engineering in coal seams: Part 1-the physical process of gas storage and movement in coal seams. SPE Reservoir Eng 1987;2:28–34. [15] Harpalani S, Chen G. Gas slippage and matrix shrinkage effects on coal permeability. Proceedings of the 1993 International Coal bed Methane Symposium.
5. Conclusion In this paper, the numerical investigations of gas transport using an IBM-LBM simulator are carried out enabling the assessment of the effect of microstructure of a coal sample on gas flow. The microstructure (cleat-network) has been obtained from a micro-computer tomography analysis. Another novelty of the method is the introduction of an adaptive mesh-refinement scheme allowing multi-scale simulations at much reduced cost. The main innovation of this paper is the assessment of the Klinkenberg effect on gas transport in natural coal cleat. The following conclusions are drawn from this research. 6
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Tuscaloosa, AL, USA: University of Alabama; 1993. p. 285–94. [16] Harpalani S, Zhao X. An investigation of the effect of gas desorption on coal permeability formation. Proceedings of the 1989 Coalbed Methane Symposium Tuscaloosa, Alabama. 1989. p. 57–64. [17] Heap MJ, Kennedy BM. Exploring the scale-dependent permeability of fractured andesite. Earth Planet Sci Lett 2016;447:139–50. [18] Hu G, Wang H, Fan X, Yuan Z, Hong S. Mathematical model of coalbed gas flow with klinkenberg effects in multi-physical fields and its analytic solution. Transp Porous Media 2009;76:407. [19] Jeran, CMMDP, Cleats in bituminous coalbeds. Report. UNITED STATES DEPARTMENT OF THE INTERIOR Rogers C.B. Morton, Secretary BUREAU OF MINES; 1974. [20] Jing Y, Armstrong RT, Mostaghimi P. Impact of mineralization on digital coal properties. Energy Fuels 2017;31:11558–68. [21] Jing Y, Armstrong RT, Ramandi HL, Mostaghimi P. Coal cleat reconstruction using micro-computed tomography imaging. Fuel 2016;181:286–99. [22] Karacan CÖ, Okandan E. Fracture/cleat analysis of coals from zonguldak basin (northwestern turkey) relative to the potential of coalbed methane production. Int J Coal Geol 2000;44:109–25. [23] Klinkenberg L, et al. The permeability of porous media to liquids and gases. Drilling and production practice American Petroleum Institute; 1941. p. 200–13. [24] Laubach SE, Marrett RA, Olson J, Scott A. Characteristics and origins of coal cleat: a review. Int J Coal Geol 1998;35:175–207. [25] Law BE, Rice DD, et al. Hydrocarbons from coal. USA: Association of Petroleum Geologists; 1993. [26] Li H, Shi S, Lu J, Ye Q, Lu Y, Zhu X. Pore structure and multifractal analysis of coal subjected to microwave heating. Powder Technol. 2019;346:97–108. [27] Liu J, Sarout J, Zhang M, Dautriat J, Veveakis E, Regenauer-Lieb K. Computational upscaling of drucker-prager plasticity from micro-ct images of synthetic porous rock. Geophys J Int 2017;212:151–63. [28] Marconi S, Chopard B. A lattice boltzmann model for a solid body. Int J Mod Phys B 2003;17:153–6. https://doi.org/10.1142/S0217979203017254. [29] Mazumder S, Wolf KH, Elewaut K, Ephraim R. Application of x-ray computed tomography for analyzing cleat spacing and cleat aperture in coal samples. Int J Coal Geol 2006;68:205–22. [30] McCauley J. Models of permeability and conductivity of porous media. Physica A 1992;187:18–54. [31] Moore TA. Coalbed methane: a review. Int J Coal Geol 2012;101:36–81. [32] Mostaghimi P, Armstrong RT, Gerami A, Warkaini ME, Ramandi HL, Pinczewski V. Micro-ct imaging and microfluidics for understanding flow in coal seam reservoirs. International Symposium of the Society of Core Analysts, Newfoundland, Canada. 2015. p. 16–21. [33] Niu Y, Mostaghimi P, Shikhov I, Chen Z, Armstrong RT. Coal permeability: gas slippage linked to permeability rebound. Fuel 2018;215:844–52. [34] Palmer I. Permeability changes in coal: analytical modeling. Int J Coal Geol 2009;77:119–26. [35] Palmer I, Mansoori J, et al. How permeability depends on stress and pore pressure in coalbeds: a new model. SPE annual technical conference and exhibition Society of Petroleum Engineers; 1996. p. 557–64. [36] Pan Z, Connell LD. Modelling permeability for coal reservoirs: a review of analytical models and testing data. Int J Coal Geol 2012;92:1–44. [37] Persoff P, Hulen JB. Hydrologic characterization of four cores from the Geysers coring project Technical Report CA (United States): Lawrence Berkeley National Lab; 1996. [38] Peskin CS. Flow patterns around heart valves: a numerical method. J Comput Phys 1972;10:252–71. [39] Purl R, Evanoff J, Brugler M, et al. Measurement of coal cleat porosity and relative permeability characteristics. SPE Gas Technology Symposium Society of Petroleum Engineers; 1991. p. 93–104. [40] Ramandi HL, Liu M, Tadbiri S, Mostaghimi P. Impact of dissolution of syngenetic and epigenetic minerals on coal permeability. Chem Geol 2018;486:31–9. [41] Ramandi HL, Mostaghimi P, Armstrong RT, Saadatfar M, Pinczewski WV. Porosity and permeability characterization of coal: a micro-computed tomography study. Int J Coal Geol 2016;154:57–68.
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Full Length Article
Modeling the effects of gas slippage, cleat network topology and scale dependence of gas transport in coal seam gas reservoirs
T
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Xu Yua, , Lincheng Xua, Klaus Regenauer-Lieba, Yu Jinga, Fang-Bao Tianb a b
School of Minerals and Energy Resources Engineering, UNSW Sydney, Australia School of Engineering and Information Technology, UNSW Canberra, Australia
A R T I C LE I N FO
A B S T R A C T
Keywords: Coal seam gas flow Lattice Boltzmann method X-ray micro-computed tomography Gas slippage CSG recovery
Gas flow in coal cleats plays a significant role for the permeability of coal seams. Investigations of gas transport in coal are carried out in this work using a numerical scheme with a coal sample scanned by the micro-computed tomography. We analyse a coal sample from the Bowen Basin (Australia) to characterize the natural discrete fracture network defined by connected cleats and matrix blocks. Due to the heterogeneity of coal micro-structures, the numerical simulator is combined with a grid-adaptive scheme. The accuracy of the simulator is validated for gas slip regime (0.01 < Kn < 1.0 ) in the micro-straight channel. Predictions of permeability and porosity are conducted through the method. The results show that the porosity and permeability of the digitalized coal model change along with the length and stabilizes at >400 voxels and >700 voxels, respectively. The Knudsen number (Kn) has an important effect on the gas slippage which can enhance the coal seam gas transport for narrow cleat networks. The increase of Kn ranging from 0.0005 to 0.01 increases the mass flux as well as the permeability of the coal for a given cleat aperture. The investigations of the gas transport in cleats prove the capability of the numerical simulator for modeling gas slip flow. It also provides a potential tool for solving gas transport in the complex geometry of a natural coal seam gas (CSG) reservoir.
1. Introduction Coal seam gas (CSG) production has made recent breakthrough propelling Australia into the top spot for LNG export for the first time in November 2018 due to an increase of 23% from CSG production in Queenslandhttps://www.energy.gov.au. Yet, elsewhere in the world CSG has not made similar progress and similar to shale gas recovery the success of harvesting these unconventional energy sources is not universally repeatable. We propose that this is due to the lack of quantitative tools to investigate gas flow through tight formations such as coal or shale gas reservoirs. In coal the main gas flow is enabled by narrow natural channels, known as cleats [24,19]. In this contribution, we present a robust mesoscopic, numerical framework that models the gas flow at the pore scale of coal cleats considering the phenomenon of gas slippage. Coal seam gas (CSG) production stems from the desorbed gas of the microporous structures of coal. Coal features a dual-porosity consisting of small matrix pores and cleats. The desorbed gas diffuses through the matrix to the cleat system and flows along the cleat network. The cleats, therefore, provide the primary channels for transferring the CSG to the wells. It is well documented that the permeability of coal seams is
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predominantly affected by the intensity and aperture of the cleats of coal seams [4,31]. Most CSG simulations and laboratory experiments consider a macroscale formulation where the permeability is considered to increase with the intensity and aperture of cleats existing in coal seams [24,19,39]. Conversely, permeability is assumed to decrease because of cleat closure or decreasing intensity of coal cleats. Many laboratory studies of coal fractures have been done but little data on the in situ properties of coal cleats has been obtained. For the investigations of the CSG transport in porous structures of coal, experimental and theoretical contributions [8,36,10,26,13,64,65] have been made in the previous references. These macroscopic models are constructed for the investigation of gas permeability and gas diffusion models for coal [48,47,43]. The Darcy’s law and Fick’s law are used for modelling gas flow in coal fractures [39] and gas diffusion in coal is regarded to be homogeneous [48]. Many permeability models have been built for averaging the fluid flow in coal by theoretical and experimental methods. Such coal permeability models usually are functions of the porosity, fluid pressure, and ad/desorption. For example, the power-law dependencies between the permeability and porosity built on simple
Corresponding author. E-mail addresses: [email protected], [email protected] (X. Yu).
https://doi.org/10.1016/j.fuel.2019.116715 Received 21 September 2019; Received in revised form 11 November 2019; Accepted 20 November 2019 0016-2361/ © 2019 Elsevier Ltd. All rights reserved.
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porous structures, such as packed beeds or cubes[30], have been extended for the applications in the investigations of the CSG production [36]. The effects of fluid pressure and volumetric deformation have also been considered in models built by Gray and Harpalani [14,16]. Due to the internal dual-porosity of the coal matrix [12,48,47], a bidispersepore model is correspondingly introduced and applied to model the gas transport in macro- and micro-pores [44–46,34]. However, the bidisperse model is built analytically based on packed pellets of various sizes and then transferred for applications in gas diffusion in coal. Among these models, the Palmer-Mansoori(P-M) model (1998) [35] and the Shi-Durucan (S-D) model [45,46] are currently two popular choices used in reservoir simulations of the CSG production. The above-mentioned models are based on the state of the macroscopic scale. The current models therefore are not built to consider the mesoscopic structure of the coal and its cleat system. Similarly, direct investigations of gas flow in coal cleat networks, have been poorly investigated. Recently, by the combination of X-ray micro-CT imaging technique porescale models for investigations and predictions of fluid transport properties, such as porosity, permeability, reaction diffusion, have attracted a lot of attention. Some pioneer research works have been constructed for the characterization and analysis of coal micro-structure and numerical simulations based on the computed-tomography images [32,40,27,61]. The tomography technique provides a complementary method for building an accurate pore-scale model for the numerical studies. It has been also applied as a useful tool for the studies of the the effects of the sorption deformation and effective stress on the permeability of coal [63,62]. Limitations of experimental studies, complementary numerical studies are needed for better interpretations of the gas transport mechanism. An important effect that has not yet been considered is the Klinkenberg gas slippage effect which has a significant impact on the permeability of a porous medium for liquid and gas flow [23]. It is reported that gas slippage occurs when the pore aperture approaches
Fig. 2. A schematic diagram of 2-D micro-channel with the channel height H and length L. The ratio of the height and length of the channel is 1.0.
the mean free path of the gas seen in Fig. 2 which is indexed by a ratio named Knudsen number. Experimental and theoretical studies have shown that the gas slippage effect increases the permeability of coal for high Knudsen numbers [33,15,51]. Niu et al. concluded that the slippage effect can not be neglected by experiments showing three Australian coal samples where the Knudsen number turned out to be larger than 0.1. Therefore, it is of importance to consider the gas slippage for the numerical investigations of the CSG transport in coal. In this contribution a numerical simulator is developed based on the mesoscopic lattice Boltzmann method for modeling the fluid flow in the coal cleat network combined with the immersed boundary for adjusting the heterogeneous structure of coal. The simulations allow full assessment of the effect of permeability of coal and the Klinkenberg gas slippage effects on the gas flow. The Lattice Boltzmann method allows a mesoscopic assesment of the fluid flow process without Darcy Fig. 1. 3D visualisation of segmentation and reconstruction of coal-cleat system by using X-ray computed tomography and the figure is referred to the Ref.[21] including (a) A coal sample scanned by the micro-CT imaging, (b) Reconstructed 3-D model with discrete fracture network (DFN) method and (c) 2-D cleat network parallel to y-z surface extracted from 3-D model of (b).
2
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∫Γ FIB (xb ) δ (xb − x ) dS,
assumption and enables therefore to tackle exactly the range of scales encountered in gas flow through coal seams[52,53]. The method also allows integration of the topology of cleats obtained from micro-CT analyses. In previous work, we have presented simulations of fluid flow using the immersed boundary lattice Boltzmann method[50,49,55,57]. Based on the numerical method proposed in Ref. [58] investigations of gas flow in the coal cleats have been conducted in this study.
where f IB is the body force acting on a fluid point x , FIB is the Lagrangian force density acting on the fluid by the IB, xb is the Lagrangian point on the immersed interface, δ () is Dirac delta function proposed in Ref.[38], and Γ represents the involved interfaces. By using the feedback IB scheme, the Lagrangian force density is determined by
2. Methodology
FIB (xb) = k ⎡Ub (X b) − ⎣
f
f In recent years, X-ray micro-computed tomography (CT) has allowed unprecedented insight into the microstructures of coal and statistical methods have been developed to adequately incorporate this information for mesoscopic simulations [2,41]. The obtained data is provided at high resolution making it an ideal methodology for the model construction of numerical simulations. In this paper we use the methodology to investigate gas flow in the fractured coal using a reconstructed micro-CT model of a coal sample from the Bowen Basin of Australia. For measurements and characterizations of coal, the micro-CT has inherent advantages over other destructive methods of characterizing coal microstructure. The digital coal structure used in the simulations is built based on a discrete fracture network model. The segmentation and analysis results of the current coal sample have bee published by Jing et al. 21. For details of the segmentation and reconstruction method, the reader is referred to Refs. [21,20].
(5)
IB (x )
=
∑ FIB,i δ (xb,i − x )ΔS, xb,i ∈ Γ,
(6)
2.3. Boundary conditions and adaptive Cartesian mesh scheme For the numerical simulations of gas flow in a straight fracture in Section 3.1, the periodic boundary condition is applied in the y-axis and the fluid flow in the x-axis is driven by pressure gradient. The pressuredriven boundary condition is functioned by body force acting on the fluid. In the sequential section, investigations of gas flow in the microcleat system are conducted in the reconstructed coal-cleat network. Symmetrical boundary conditions are used in both of x- and y-axis directions and gas flow is also driven by pressure differences. A adaptive meshing scheme with Cartesian grid system is used in this work for simulations of gas flow in complex coal-cleat systems of coal based on a previous work [55]. The core part of the scheme is to add fine meshes around the fluid–solid interface and coarse meshes in the far field which can significantly improve the computational stability and efficiency. The mesh refinement ratio of two adjacent blocks of different refinements is two in this work.
A simulator is developed based on the immersed boundary-lattice Boltzmann (IB-LB) method presented in the Ref.[55] which is applied in this work for the micro-gaseous flow in coal cleat networks. The main governing equations used in the simulator are briefly introduced and listed below. The lattice Boltzmann equation (LBE) with the multiplerelaxation-time model is expressed by
mi (x + ei δt , t + δt ) − mi (x , t )
3. Results
(1)
where m = Mg is the velocity moment composed of the momentum matrix M and distribution function g , meq = Mg eq is the equilibrium moment with the equilibrium function g eq , ei is the discrete velocity, si is a series of relaxation rates, the force component G is expressed as
3.1. Gas flow in a micro-channel To study the problems of gas flow in micro-scale channel, the Knudsen number (Kn) is an important influence factor affecting fluid flow character. When the Knudsen number is high by lowering the height of a channel, the phenomenon of slippage occurs which plays an significant role for computations of permeability. Fig. 2 shows the simple structure of a 2-D fracture channel. This section presents validations of micro-scale gas flow within a range of Knudsen number between 0.01 and 1.0. Cercignani derivated an analytical solution of second-order accuracy from the linearized Boltzmann equation relating slip velocity to Knudsen number[5] which is give by.
(2)
where f is an external body force. The macroscopic variables, such as density, velocity, pressure, are calculated by ρ = ∑ gi , u = (∑ gi ej + 0.5fδt )/ ρ , p = ρcs2 respectively, and cs is the lattice sound speed. The equilibrium distribution function g eq is determined by
(u·ej )2 u · ej u2 g eq = ωj ρ ⎡1 + 2 + − 2⎤ j 4 ⎢ c 2 c 2 cs ⎥ s s ⎦ ⎣
∫Ω u (y) δ (y − xb) dy⎤⎦,
(4)
where kis the feedback coefficient analytically calculated by [55] as a sum of polynomials of the local normal vector to the immersed boundary, Ω is the whole fluid domain, Ub is the target boundary velocity, ui is the fluid velocity at x i , FIB, i is the Lagrangian force density at xb, i , ΔS is the local area associated with xb, i , and ΔVi is the local lattice volume. The approximation scheme of the feedback coefficient can reduce the velocity error at the immersed boundaries and improve the numerical stability of simulations.
2.2. Development of the IB-LB simulator for fluid flow
G = ωρδt [f·(ej − u )/cs2 + (f·ej )(u·ej )/cs4]
=
and
2.1. Segmentation of micro-CT data
s = si (mieq (x , t ) − mi (x , t )) + ⎛1 − i ⎞ (MG )j , 2⎠ ⎝
IB (x )
(3)
us / U = 1.146Kn
where ωj is the jth weight. The D2Q9 model is used for the two-dimensional simulation. The fluid kinematic viscosity ν is related to several specific relaxation rates as detailed in Refs. Optimized or suboptimized choice of the relaxation rates, si , for these three models can be found in the references provided, and thus is not discussed here. For enhancement of the performance of the numerical computation, an improved immersed boundary (IB) method is used in which the boundary conditions are applied by spreading the Lagrangian force density into the bulk fluid and given by
∂u ∂u2 + 0.907Kn2 2 ∂y ∂y
(7)
where us and U is the slip velocity and reference velocity respectively. In this case, three non-dimensional parameters which are Knudsen UL λ number (Kn = H ), Reynolds number (Re = ν ), and Mach number U
(Ma = c ), are considered for the micro-channel flow. The parameters s considered in these cases are listed in Table 1. Studies of grid convergence analysis and validation comparing the present results with Cercignani’s analytical solution, have been conducted for 0.01 < Kn < 1 and presented in the Fig. 3(a) and 3(b). The results show that the 3
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Table 1 Parameters of Knudsen number (Kn), Reynolds number (Re) and Mach number (Ma) for micro-channel flow. No.
Kn
Re
U / cs
δx / H
1 2 3 4 5
1.0 1.0 1.0 1.0 1.0
0.02171 0.02171 0.02171 0.02171 0.02171
0.0173 0.0173 0.0173 0.0173 0.0173
0.015625 0.0078125 0.00390625 0.001953125 0.0009765625
6 7 8 9 10 11
0.01 0.05 0.1 0.2 0.5 1.0
2.171 0.434 0.2171 0.1085 0.0434 0.02171
0.0173 0.0173 0.0173 0.0173 0.0173 0.0173
0.001953125 0.001953125 0.001953125 0.001953125 0.001953125 0.001953125
Fig. 4. A diagram of geometry-adaptive mesh refinement in a cleat joint including (a) segmented cleat joint before meshing and (b) geometry-adaptive grids.
lattice nodes. Procedures of geometry-adaptive mesh generation can be divided into the following steps. Step 1: X-ray micro-CT scans tomographic slices are obtained as raw data. The reconstruction processes and segmentation are carried out sequentially and the results exported for simulations. Step 2: For the immersed boundary method the internal interfaces/ fracture walls need to be identified. This process is performed in a separate numerical program. Step 3: In this step, the geometry-adaptive mesh is generated according to the shape of coal-fracture walls which have been extracted in step 1. The mesh grids grow in four directions of x, -x, y, -y (2-D). Grids are mainly developed within the cleat channels. No grids exist in the solid block except those necessary for the boundary conditions. Fig. 4(a) and 4(b) present a micro-CT image of a cleat joint before and after geometry-adaptive grids generation respectively, according to the above procedures. It is shown that fine meshes are generated along the cleat wall and coarse meshes are created in the middle of the channel in Fig. 4(b). The Neumann boundary condition is imposed at inlet and outlet as we consider a pressure-driven flow in the coal-cleat network. For the simulations of gas flow in 2D cleat network, the minimum mesh size used in the 2D cleat study is 1/32H (H the cleat aperture). This mesh generation scheme is explained in Lincheng et al.’s work [55]. Readers who are interested in the scheme can find it in the literature. (See Fig. 5).
Fig. 3. Results for gas slip flow in a micro-channel including (a) grid convergence analysis at Kn = 1.0 and (b) slip velocity for 0.01 < Kn < 1.0 .
improved IB-LB method is capable to simulate micro-scale fluid flow 3.2. Fluid flow in 2-D coal cleat network 3.2.1. Geometry-adaptive mesh refinement The IB-LB method is applied for the fluid flow in coal-cleat networks with a combination of the geometry-adaptive Cartesian grids in which no grids are generated in the solid matrix except those necessary for the boundary conditions.In this work, the microstructure of coal is created by the segmentation and reconstruction schemes for cleat-matrix network based on the CT imaging of the coal sample. A special boundary method (referred as ”non-ghost cell method”) has been established for the suspending grids in the non-fluid region. In the implementation process, each non-fluid block is represented by a virtual fluid lattice node. Mesh grids grow in the normal direction of cleat boundaries and are divided into several blocks with various grid sizes. Computations of fluid flow through the porous structure of coal are carried out on the 4
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Fig. 5. A schematic diagram of dividing the 2-D coal cleat structure (overall 960 voxels) into multiscale-length cells.
3.3. Scale-dependence analysis of coal-cleat porosity and permeability The scale dependence of the macroscopic rock properties, such as porosity, permeability is well known in the literature [3,60,17,1]. Coal has a heterogeneous porous structure that changes the porosity and permeability by increasing the length of the coal sample. From the micro-CT data shown in the Fig. 1(c), the connected porosity ∊ is esti∑N mated by counting the number of pixels of connected cleats ∊ = ∑ Nc where N c and N are number of pixels in cleats and solid matrix respectively. Nine cleat structures of various lengths are generated from the two-dimensional cleat network (Fig. 1(c)), including 200, 300, 400, 500, 600, 700, 800, 900, and 960 (voxel). The approximated porosity is calculated correspondingly and presented in Fig. 6(a). Simulations of gas flow are carried out based on the nine coal-cleat structures. The result of estimated porosity is presented in Fig. 6(a) showing that the microscale porosity of connected cleat network changes and gradually stays at around 4% when increasing the model size. In this case, the minimum length of representative element volume is 700voxel/2.91mm defined by a stabilization of the value of average porosity around 4%. The corresponding permeabilities are calculated by the current IB-LBM method and shown in Fig. 6(b). It is found that the permeability of the coal-cleat system increases along with the sample length until it also stabilizes for a Length > 700voxel .
Fig. 6. Length dependence analysis includes (a) porosity of coal calculated based on the digitalized data and (b) Permeability computations for various length scale.
3.4. Effects of Knudsen number The Klinkenberg effect plays a very significant role in the microgaseous flow in porous media [23] and the impact becomes bigger for Knudsen number (Kn) larger than 0.001(slip regime). Due to the interest in other research areas where flow through nanotubes in e.g. MEMS and shale gas is considered a significant research effort has been dedicated to investigate gas flow with coupled Klinkenberg effects [66,6,7,59]. The main channels for coal seam gas are defined by a natural fracture network known as cleats which occur in two sets of orthogonal fractures named ”face cleat” and ”butt cleat” [24,19]. Face cleat is usually parallel to coal seam floor and its connectivity plays a dominant role in the contribution to the cleat parallel permeability. Klinkenberg derived the effective gas permeability at a finite pressure by considering gas slippage which is represented by the Knudsen number Kn [23]. When Knudsen number is larger than 0.001, the Klinkenberg effect has a higher effect on gas flow behavior, especially
Fig. 7. Evolution curves of gas flow in 2-D cleat model for various Knudsen numbers containing the black line for dimensionless mass flux and red line for the permeability.
in low permeability media [42,37,54,18]. In the present study, the numerical code considers 0 < Kn < 0.01 and the result is illustrated in Fig. 7. It is shown that the permeability increases along with the increase of Knudsen number. 4. Discussion Gas transport in cleats plays a key role in the CSG production from coal seams. The goal of this paper is to present a numerical simulator for direct modeling mesoscopic effect on gas flow via characterizing the 5
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(1) A numerical simulator composed of immersed boundary and lattice Boltzmann method is introduced. Gas flow in a straight microchannel is investigated for the gas slip flow. The slippage effect is determined by the Knudsen number. The grid convergence result shows that the slip velocity profiles converge with decreasing grid size. The accuracy of the simulator is validated by the comparison with the Cercignani’s second-order solution. (2) The porosity and permeability of the coal sample feature scale dependence below a critical length. Simulations of gas flow at length from 200 to 960 voxels show that the porosity and permeability grow with length and stabilize at 400 voxels (porosity) and 700 voxels (permeability), respectively. (3) Investigations of the Klinkenberg effects on gas transport in coal are carried out. It is shown that the Klinkenberg effect plays a significant role in coal seam gas production. The outlet mass flux and permeability show a substantial increase with increasing Knudsen number for the same channel width.
microstructure (cleat network) with the micro-CT technique. This study of the CSG migration in coal cleats is conducted on a reconstructed coal structure considering the gas slippage effects. The current simulator is built using LBM which has a limitation for modeling micro-gaseous flow for large Kn. Therefore, modified immersed boundary method (IBM) is used in this work as an extension to be able to deal with the gas–solid interactions for modeling gas slippage effects. The method is particularly suited for this analysis due to its capability of dealing with complex boundaries [38]. We have shown in this work that the new simulator is capable of dealing with the gas flow in the straight channel for the slip flow regime 0.01 < Kn < 1.0 . However, it is found that it is of low accuracy for solving curved-wall problems [56]. The Kn in the present formulation is defined as the ratio of the molecular mean free path length to the mean cleat aperture. Unfortunately, direct measurements of Kn under in situ conditions in CSG reservoirs are not available, however, extrapolations have reported cleat widths ranging from 0.001 to 20 mm based on microscopic measurements of Bowen Basin coal samples [11]. Law et al. (1993) and Laubach et al. (1998) observed coal samples in the laboratory and measured the mean cleat aperture ranging from 0.01 to 0.2 mm [24,25]. Similar results can be found in other references [9,29,22]. The value of Knudsen number of coal cleats is, therefore, estimated to be less than 0.05 based on the above data which is the range adopted for our simulations of gas flow in two-dimensional cleat network. According to the current research, it is found that the effects of gas slippage grow with the increase of the Kn which can improve the outlet mass flux and the permeability of coal seams. Fig. 7 shows that for Kn > 0.001 the Klinkenberg effect is of importance for the CSG production and cannot be neglected. Temperature and confining pressure have a big influence on gas flow in coal and the Klinkenberg effect. The main effect of temperature and confining pressure is to change the aperture of the cleats with a minor change on mean free path of the gas. The effect of cleat aperture change as a function of thermal and mechanical strains is subject of ongoing work requiring analysis of X-ray microtomography experiments in environmental cells replicating pressure and temperature conditions in the reservoir. Therefore, in this paper the Knudsen number is used as a free parameter to allow incorporation of the temperature and pressure effects on gas flow through coal cleats upon completion of the experiments. While the simulator has shown to be a robust and numerically efficient multiscale formulation the main limitation of the current scheme is the lack of incorporation of adsorption induced volumetric strain which can adversely affect the cleat aperture leading to an increase in Knudsen number. Future work will, therefore, consider the pressure propagation in fluid and solid at the same time with an extended version of the simulator. Little work has been available on this important topic except for a pioneering paper by Ref. [28] who consider the stress propagation from a solid body to the fluid with the lattice Boltzmann method. This implementation would provide a potential to simulate the gas migration through the coal cleat and matrix as well as the important stress–strain evolution. Due the novelty of the mesoscopic approach few laboratory measurements are available at the micro-scale which is another area of future work.
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 research/project was undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government. Xu Yu would like to thank the China Scholarship Council (CSC) and UNSW for providing scholarships for his study. This work was supported by the Australian Research Council (ARC Discovery grants No. DP140103015, DP170104550, DP170104557, ARC Discovery Early Career Researcher Award Grant No. DE160101098). References [1] Aramideh S, Vlachos PP, Ardekani AM. Pore-scale statistics of flow and transport through porous media. Phys Rev E 2018;98:013104. [2] Blunt MJ, Bijeljic B, Dong H, Gharbi O, Iglauer S, Mostaghimi P, Paluszny A, Pentland C. Pore-scale imaging and modelling. Adv Water Resour 2013;51:197–216. [3] Bosl WJ, Dvorkin J, Nur A. A study of porosity and permeability using a lattice boltzmann simulation. Geophys Res Lett 1998;25:1475–8. [4] Busch A, Gensterblum Y. Cbm and co2-ecbm related sorption processes in coal: a review. Int J Coal Geol 2011;87:49–71. [5] Cercignani C. Theory and application of the Boltzmann equation/ [by] Carlo Cercignani. London: Scottish Academic Press; Distributed by Chatto and Windus Edinburgh; 1975. [6] Chai Z, Shi B, Guo Z, Lu J. Gas flow through square arrays of circular cylinders with klinkenberg effect: A lattice boltzmann study. Commun Comput Phys 2010. https:// doi.org/10.4208/cicp.010809.081209a. [7] Chen L, Fang W, Kang Q, De’Haven Hyman J, Viswanathan HS, Tao WQ. Generalized lattice boltzmann model for flow through tight porous media with klinkenberg’s effect. Phys Rev E Stat Nonlin Soft Matter Phys 2015;91:033004. [8] Clarkson C, Bustin R. The effect of pore structure and gas pressure upon the transport properties of coal: a laboratory and modeling study. 2. adsorption rate modeling. Fuel 1999;78:1345–62. [9] Close JC, Mavor MJ. Influence of coal composition and rank on fracture development in fruitland coal gas reservoirs of san juan basin; 1991. [10] Danesh NN, Chen Z, Aminossadati SM, Kizil MS, Pan Z, Connell LD. Impact of creep on the evolution of coal permeability and gas drainage performance. J Natural Gas Sci Eng 2016;33:469–82. [11] Gamson PD, Beamish BB, Johnson DP. Coal microstructure and micropermeability and their effects on natural gas recovery. Fuel 1993;72:87–99. [12] Gan H, Nandi S, Walker Jr P. Nature of the porosity in american coals. Fuel 1972;51:272–7. [13] Gerami A, Mostaghimi P, Armstrong RT, Zamani A, Warkiani ME. A microfluidic framework for studying relative permeability in coal. Int J Coal Geol 2016;159:183–93. [14] Gray I, et al. Reservoir engineering in coal seams: Part 1-the physical process of gas storage and movement in coal seams. SPE Reservoir Eng 1987;2:28–34. [15] Harpalani S, Chen G. Gas slippage and matrix shrinkage effects on coal permeability. Proceedings of the 1993 International Coal bed Methane Symposium.
5. Conclusion In this paper, the numerical investigations of gas transport using an IBM-LBM simulator are carried out enabling the assessment of the effect of microstructure of a coal sample on gas flow. The microstructure (cleat-network) has been obtained from a micro-computer tomography analysis. Another novelty of the method is the introduction of an adaptive mesh-refinement scheme allowing multi-scale simulations at much reduced cost. The main innovation of this paper is the assessment of the Klinkenberg effect on gas transport in natural coal cleat. The following conclusions are drawn from this research. 6
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