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Methane flow in coal is associated with the content of both organic matter (OM) and inorganic matter (IOM). Coal matrix contains nanopores ranging in size from a few to hundreds of nanometers, which leads to a non-Darcy effect where the measured permeability of a gas (apparent permeability) is higher than that of a liquid (intrinsic permeability). In this study, a generalized Lattice Boltzmann model (GLBM) is employed for gas flow through the reconstructed coal matrix consisting of OM, IOM, and fractures. The apparent permeability model is proposed to calculate the total flow flux accounting for multiple transport mechanisms including viscous flow, slip flow, transitional flow, and the Knudsen diffusion. The impact of effective pore radius and gas surface diffusion on permeability in the gas adsorption-desorption process is also considered in the model. What’s more, the weighting factors are adopted to adjust the contribution of the viscous flow and the Knudsen flow. The effect of total organic/inorganic content and the development of fractures on the apparent permeability of the reconstructed coal matrix is also studied. It is found that the apparent permeability is extremely low when a fracture is nonexistent, and varies almost linearly with the total organic/inorganic content. A fracture plays a significant role in determining apparent permeability and the velocity distribution of the coal matrix.

Permeability is a key factor in the exploitation of coalbed methane, which directly determines whether the target coal reservoir is valuable for mining. For a long time, people evaluated the permeability of the coal reservoir based on Darcy’s law. However, recent extensive studies [

Coal is a typical tight porous medium, with a wide distribution of pore size, which is distributed in the range of 1-100 nm [

The coal seam is a kind of low permeability reservoir. According to well test data [

The maceral composition of coal consists of organic matter (OM) and inorganic matter (IOM) [

In recent years, the Lattice Boltzmann Method (LBM) on fluid flow in porous media attracted wide attention [

In this paper, based on previous studies, we proposed a GLBM model for predicting coalbed methane permeability. This model included the following sections: (1) The generalized Lattice Boltzmann method was employed to solve the generalized Navier-Stokes equations, which was applied to the calculation and prediction of local permeability at representative elementary volume (REV) scale. (2) The fully coupled viscous flow, slip flow, transition flow, and the Knudsen diffusion were considered in the simulation, and the contribution of these flow mechanisms to the total flow flux was automatically adjusted by the weight factor. (3) The impact of adsorbed gas molecules on the pore space and the contribution of surface diffusion to the total flow flux were taken into consideration. (4) The separate evolution of the local permeability of OM and IOM in the coal matrix was taken into consideration.

For isothermal flows of incompressible fluids in porous media at the REV scale, the generalized Navier-Stokes equations which were proposed by Nithiarasu [

The governing equations of mass and momentum for the generalized Navier-Stokes equations can be given by

Note that equation (

Guo and Zhao [

As in the standard LB model, the density and velocity of flow are defined as

Because the force

By using the Chapman-Enskog technique with the pressure

The pores of OM in coal are usually less than 100 nm in diameter and have a large amount of gas adsorbed on their surfaces [

According to Poiseuille’s law, for capillary tubes with a radius

The volume flux of gas viscous flow is^{3}/s).

As gas molecules adsorbed on the inner surface of a capillary, the loss of the cross-sectional area for free gas transmission may be large [

The gas adsorption amount is a function of pressure. When the reservoir pressure is

Then, the relationship between effective porosity and initial porosity is

After considering the reduction effect of gas adsorption to the pore space, the volume flow flux can be rewritten as

Consequently, the permeability involving gas adsorption becomes a function of pressure:

The Navier-Stokes equations with appropriate slip boundary conditions are sufficient for modeling gas flow in the slip flow regime [

Klinkenberg [

The Knudsen number is defined as the ratio of the mean free path of molecules to the characteristic pore size of porous media:

The mean free path is defined as

Considering the impact of the effective pore size and gas slippage on apparent permeability, the effective radius

The mass flow flux with the gas slippage taken into consideration is

As the Knudsen number is in the range of

The corresponding volume flow flux is

Then, the mass flow flux contributed by the Knudsen diffusion is

Due to the different concentrations of adsorbed gas in each site on the inner surface of a nanopore, surface diffusion makes a significate role in gas transport. The adsorbed gas and the bulk flow gas can be connected by the relationship of the Langmuir isotherm adsorption equation:

The mass flow flux contributed by surface diffusion is

In the viscous flow regime, the probability of intermolecular collisions dominates the total collisions, which occur under high-pressure conditions or in systems without boundaries. For the Knudsen diffusion, only the collision between the gas molecules and the solid walls is considered, which occurs in systems with near-vacuum pressure or extreme restriction [

Considering the influence of porosity and tortuosity of porous media, the total flux can be expressed as

The total mass flow of gas transport in the coal matrix can be considered as a combination of viscous flow (includes the slip flow and transition flow), the Knudsen diffusion, and surface diffusion. In this paper, based on the combined effects of viscous flow, the Knudsen diffusion, and surface diffusion, the apparent permeability of OM is obtained:

Nanopores are also distributed in the IOM of coal, which may have a certain adsorption effect on gas molecules, but they can be ignored compared to the strong adsorption effect of OM. Therefore, it is assumed that the pores of IOM only serve as channels for fluid percolation. According to equation (

As in Appendix

Roy et al.’s experimental condition and fluid parameters.

Physical parameters | Value |
---|---|

Mean pore radius (r/nm) | 100 |

Porosity | 0.2 |

Tortuosity | 1 |

Temperature (K) | 300 |

Molar mass (kg/mol) | 39.948 |

Gas dynamic viscosity (Pa ⋅ s) | |

Length of porous media ( | 60 |

Constant pressure at outlet (kPa) | 4.8 |

Comparison of LB simulation results and experimental data.

Therefore, the model proposed in this paper can accurately describe the coalbed methane in the nanopores in the real state.

Figure

Schematic diagram of coal and the 2D reconstructed image.

Zhao et al. [

To explore the influence of the volume fraction of OM and IOM on permeability, we reconstructed 8 groups of porous media with the random distribution of OM and IOM, as shown in Figure

Random reconstruction images of coal matrix with different contents of OM and IOM. The volume fractions of OM and IOM are (a)

The physical parameters for simulation in three-component porous media.

Groups | Inorganic matter (IOM) | Organic matter (OM) |
---|---|---|

Physical parameters | ||

Simulation region size ( | ||

Temperature (K) | 303.15 | |

Mean pressure (MPa) | 6 | |

Gas density (kg/m^{3}) | 43.6089 | |

Gas dynamic viscosity (Pa ∙ s) | ||

Tortuosity | 4 | |

Porosity (%) | 20 | 5 |

Adsorption capacity (m^{3}/t) | — | 20 |

Langmuir pressure (Mpa) | — | 2 |

Median (nm) | 10 | 3 |

0.8 | 0.35 |

As shown in Figure

Changes of global permeability with OM content.

In fact, in the process of coalbed methane mining, we are more concerned about the data of the whole production cycle. However, due to the nature of LBM and the limited range of pressure (density) fluctuation in the simulation, the results obtained directly from the real parameters of the coal reservoir may not be correct. In this part, the density and viscosity coefficient of coalbed methane under different pressures are used to calculate the global permeability, and the results are summarized to indirectly obtain the data of the whole pressure drop process. It should be noted that the density and viscosity coefficients of methane are discrete under different conditions. We employed an online calculation software (

Methane density, viscosity coefficient, and the corresponding fitting function under different pressure conditions ((a) density, (b) dynamic viscosity, and (c) kinematic viscosity).

There is a kind of double-pore structure composed of pores and fractures, in which fractures are common in coal. As can be seen from Figure

To further explore the impact of fractures on permeability, four groups of three-component porous media are reconstructed in this part, as shown in Figure

The reconstructed coal matrix with different fracture development.

Velocity distribution in different fracture development.

The impact of fracture development on permeability.

Figure

Figure ^{-4} mD in the pressure drop range, which is extremely low permeability; the results were verified by the Finite Element Method as can be seen in Appendix

In this paper, we established a GLBM model for predicting coalbed methane permeability in the REV scale, which was fully coupled with the viscous flow, slip flow, transition flow, and the Knudsen diffusion. The model also considered the impact of the effective pore radius and surface adsorption and adopted the weighting factors to automatically adjust the contribution of viscous flow and the Knudsen diffusion to the total gas flow. Besides, based on the morphology and distribution characteristics of OM, IOM, and fractures in the coal matrix, a random reconstruction algorithm of three-component porous media is developed. For the accuracy of the simulation, we adopted a series of pressure-matched parameters.

The results are as follows: (1) Due to the wide range of pore size distribution, the distribution of mesopores and macropores is relatively large, which makes the contribution of IOM to the gas transport capacity also relatively large. (2) The permeability of the coal matrix itself is extremely low, and the contribution of a fracture to gas permeability is very large. (3) The greater the content of the fracture (or the fractal dimension), the larger the permeability; it is necessary to employ reservoir stimulation techniques such as hydraulic fracturing in the development process of coalbed methane.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This work was supported by a project of the Education Department of Hunan (Grant No. 19C0370) and the Youth Project of Natural Science Foundation of Hunan Province (Grant No. 2020JJ5023).

In Appendix A, we simulated the fluid flow between two parallel plates filled with a porous medium to verify the GLBM with the proposed apparent permeability model. In Appendix B, the results of methane flow in the coal matrix containing OM and IOM were verified by the finite element software COMSOL Multiphysics.