*2.2. Gas Migration Equation in Coal*

Based on surface physical chemistry, considering the influence of effective stress, gas pressure, coal matrix adsorption, and desorption gas, the porosity equation is [25]:

$$\phi\_{\rm i} = \phi\_{\rm i0} + \frac{\mathbf{K}\_{\rm s}}{\mathbf{K}\_{\rm s} - a\_{\rm m}(\mathbf{P}\_{\rm m0} - \mathbf{P}\_{\rm m})} \frac{\mathbf{p}\_{\rm L} \varepsilon\_{\rm L} (\mathbf{P}\_{\rm m0} - \mathbf{P}\_{\rm m})}{(\mathbf{P}\_{\rm m0} + \mathbf{P}\_{\rm L})(\mathbf{P}\_{\rm m} + \mathbf{P}\_{\rm L})} - \frac{[a\_{\rm m}(\mathbf{P}\_{\rm m0} - \mathbf{P}\_{\rm m}) + a\_{\rm l}(\mathbf{P}\_{\rm l0} - \mathbf{P}\_{\rm l})](1 - \phi\_{\rm l0})}{\mathbf{K}\_{\rm s} - a\_{\rm m}(\mathbf{P}\_{\rm m0} - \mathbf{P}\_{\rm m}) - a\_{\rm l}(\mathbf{P}\_{\rm l0} - \mathbf{P}\_{\rm l})} \tag{6}$$

In the formula, φf0 is the initial fracture porosity; ε<sup>L</sup> is the adsorption deformation of coal; and PL is Langmuir pressure constant, MPa.

There is a cubic relationship between permeability and porosity [26]:

$$\mathbf{k} = \mathbf{k}\_0 \left(\frac{\Phi\_\mathbf{f}}{\Phi\_\mathbf{f0}}\right)^3 \tag{7}$$

Considering the influence of the Klinkenberg effect in gas-bearing double porous media, the permeability calculation formula is obtained:

$$\begin{split} \mathbf{k}\_{\mathbf{t}} = \mathbf{k} \left( 1 + \frac{\mathbf{c}}{\mathbf{P}\_{\mathbf{f}}} \right) = \mathbf{k}\_{0} \left( \frac{\Phi\_{\mathbf{f}}}{\Phi\_{\mathbf{f}0}} \right)^{3} \left( 1 + \frac{\mathbf{c}}{\mathbf{P}\_{\mathbf{f}}} \right) = \mathbf{k}\_{0} \left[ 1 + \frac{\mathbf{K}\_{\mathbf{g}}}{\mathbf{K}\_{\mathbf{s}} - \mathbf{a}\_{\mathbf{m}} (\mathbf{P}\_{\mathbf{m}0} - \mathbf{P}\_{\mathbf{m}})} \right] \frac{\mathbf{P}\_{\mathbf{f}} \mathbf{L}\_{\mathbf{L}} (\mathbf{P}\_{\mathbf{m}0} - \mathbf{P}\_{\mathbf{m}})}{(\mathbf{P}\_{\mathbf{m}0} + \mathbf{P}\_{\mathbf{L}}) (\mathbf{P}\_{\mathbf{m}0} - \mathbf{P}\_{\mathbf{L}})} \\ - \frac{[a\_{\mathbf{m}} (\mathbf{P}\_{\mathbf{m}0} - \mathbf{P}\_{\mathbf{m}}) + a\_{\mathbf{f}} (\mathbf{P}\_{\mathbf{f}0} - \mathbf{P}\_{\mathbf{f}})] \left( \frac{1}{\Phi\_{\mathbf{f}0}} - 1 \right)}{\mathbf{K}\_{\mathbf{s}} - a\_{\mathbf{m}} (\mathbf{P}\_{\mathbf{m}0} - \mathbf{P}\_{\mathbf{m}}) - \mathbf{K}\_{\mathbf{s}} - a\_{\mathbf{f}} (\mathbf{P}\_{\mathbf{f}0} - \mathbf{P}\_{\mathbf{f}})} \end{split} \tag{8}$$

In the formula, k0 is the initial permeability of coal, mD; and c is the Kirsch coefficient. Coal is regarded as a porous medium composed of pores and fracture. The gas content in coal includes the gas content in coal matrix pores and coal fractures. The mass of gas in the unit volume coal matrix is composed of adsorbed gas and free gas in coal matrix pores and and free gas in coal fractures.

The gas adsorption in the coal matrix follows the Langmuir equation:

$$\mathbf{m}\_1 = \rho\_{\mathbf{g}0} \rho\_{\mathbf{s}} \frac{\mathbf{a} \mathbf{b} \mathbf{P}\_{\mathbf{m}}}{\mathbf{b} \mathbf{P}\_{\mathbf{m}} + 1} \frac{1 - \mathbf{A} - \mathbf{B}}{1 + 0.31 \mathbf{B}} \tag{9}$$

The free gas in coal matrix pores and coal fractures can be calculated by gas state equation. The free gas quality in coal matrix pores:

$$
\Delta \mathbf{m}\_{\mathbf{2}} = \phi\_{\mathbf{m}} \rho\_{\mathbf{m}} = \phi\_{\mathbf{m}} \mathbf{P}\_{\mathbf{m}} \frac{\mathbf{M}\_{\mathbf{c}}}{\mathbf{RT}} \tag{10}
$$

The free gas quality in coal fracture:

$$\mathbf{m}\_{\mathbf{f}} = \phi\_{\mathbf{f}} \rho\_{\mathbf{f}} = \phi\_{\mathbf{f}} \mathbf{P}\_{\mathbf{f}} \frac{\mathbf{M}\_{\mathbf{c}}}{\mathbf{RT}} \tag{11}$$

Gas quantity in unit volume coal pore:

$$\mathbf{m\_m} = \mathbf{m\_1} + \mathbf{m\_2} = \rho\_{\mathbf{g}0} \rho\_s \frac{\mathbf{a} \mathbf{b} \mathbf{P\_m}}{\mathbf{b} \mathbf{P\_m} + 1} \frac{1 - \mathbf{A} - \mathbf{B}}{1 + 0.31 \mathbf{B}} + \phi\_\mathbf{m} \mathbf{P\_m} \frac{\mathbf{M\_c}}{\mathbf{R} \mathbf{T}} \tag{12}$$

In the formula, ρg0 is the density of gas under standard conditions, kg/m<sup>3</sup> ; ρ<sup>s</sup> is coal density, kg/m3 ; a is the maximum adsorption capacity per unit volume of coal, m3/t; b is the adsorption constant of coal, MPa<sup>−</sup>1; A is coal ash; B is coal moisture; φ<sup>m</sup> is the porosity of the coal matrix, %; Mc is the molar mass of gas, kg/mol; R is the ideal gas constant, J/(mol·K); T is the temperature in coal, K; m1 is the amount of gas adsorbed per unit volume of coal matrix, kg/m<sup>3</sup> ; m2 is the amount of free gas per unit volume of coal matrix, kg/m3 ; mf is the amount of free gas per unit volume of coal fracture, kg/m<sup>3</sup> ; and mm is the gas content per unit volume of coal matrix, kg/m3 .

In the original state, without mining disturbance, coal pore pressure is equal to coal fracture pressure. When the gas is extracted by a negative pressure borehole, the borehole breaks the dynamic balance of coal structure and gas pressure in the original state. As a result, the pressure difference between coal pore and gas pressure in coal fracture is produced. The migration of gas in coal pores conforms to Fick's diffusion law, and the migration in fractures conforms to Darcy's linear seepage law. The change in gas in coal matrix pores of unit coal body per unit of time follows the law of conservation of mass. The amount of gas change is equal to the amount of desorption gas of coal body outside the absorption unit minus the amount of diffusion gas of unit body:

$$\begin{aligned} \frac{\partial \mathbf{m\_m}}{\partial t} &= \mathbf{Q\_s} - \mathbf{q\_s} = \mathbf{D} \nabla \left( m\_1 \right) - \frac{\mathbf{M\_c}}{\tau \mathbf{RT}} \left( \mathbf{P\_m} - \mathbf{P\_f} \right) \\ \mathbf{Q\_s} &= \mathbf{D} \nabla \left( \rho\_{\mathbf{g}0} \rho\_s \frac{\mathbf{abM\_c(\mathbf{P\_{m0}} - \mathbf{P\_m})}{(\mathbf{bP\_m} + 1)\mathbf{RT}} \frac{1 - \mathbf{A} - \mathbf{B}}{1 + 0.31 \mathbf{B}} \right) \\ \mathbf{q\_s} &= \frac{\mathbf{M\_c}}{\tau \mathbf{RT}} \left( \mathbf{P\_m} - \mathbf{P\_f} \right) \end{aligned} \tag{13}$$

In the formula, Qs is the external mass source of the unit, kg/(m<sup>3</sup> ·s); qs is the unit mass source, kg/(m<sup>3</sup> ·s); D is the gas diffusion coefficient, <sup>m</sup>2/s; <sup>τ</sup> is desorption time of adsorbed gas, d; and ∇ is a Hamiltonian operator.

The change in gas in the coal fracture of unit time follows the law of conservation of mass. The amount of gas change is equal to the amount of fracture seepage plus the amount of coal matrix pore diffusion source:

$$\frac{\partial \mathbf{m}\_{\mathbf{f}}}{\partial \mathbf{t}} = \frac{\partial (\rho\_{\mathbf{g}} \boldsymbol{\Phi}\_{\mathbf{f}})}{\partial \mathbf{t}} = \mathbf{q}\_{\mathbf{s}} - \nabla (\rho\_{\mathbf{g}} \mathbf{V}) = \frac{\mathbf{M}\_{\mathbf{c}}}{\tau \mathbf{R} \mathbf{T}} (\mathbf{P}\_{\mathbf{m}} - \mathbf{P}\_{\mathbf{f}}) + \nabla (\rho\_{\mathbf{g}} \frac{\mathbf{k}}{\mu} \nabla \mathbf{P}\_{\mathbf{f}}) \tag{14}$$

In the formula, V is the velocity of gas seepage in coal fracture, m/s; <sup>V</sup> <sup>=</sup> <sup>−</sup> <sup>k</sup> <sup>μ</sup>∇Pf; and μ is the gas dynamic viscosity, Pa·s.

The formula of pore gas pressure of the coal matrix changing with time can be obtained from the Equation (13):

$$\frac{\partial \mathbf{p\_m}}{\partial t} = \frac{[\mathbf{D}\,\nabla(m\_1) - \frac{\mathbf{M\_c}}{\pi \text{RT}}(\mathbf{P\_m} - \mathbf{P\_f})](1 + 0.31\,\text{B})(\mathbf{bP\_m} + 1)^2 \text{RT}}{(1 - \mathbf{A} - \mathbf{B})\mathbf{ab}\rho\_{\text{g0}}\rho\_{\text{s}}\text{RT} + \phi\_{\text{m}}\mathbf{M\_c}(1 + 0.31\,\text{B})(\mathbf{bP\_m} + 1)^2} \tag{15}$$

The formula can be obtained from the Equation (11) and the Equation (14):

$$\rho\_{\rm f} \frac{\partial \mathbf{P\_{f}}}{\partial \mathbf{t}} + \mathbf{P\_{f}} \frac{\partial \phi\_{\rm f}}{\partial \mathbf{t}} = \frac{\mathbf{P\_{m}} - \mathbf{P\_{f}}}{\pi} + \nabla \left(\frac{\mathbf{k}}{\mu} \mathbf{P\_{f}} \nabla \mathbf{P\_{f}}\right) \tag{16}$$

The deformation equation of gas-bearing coal is (1)–(5), (6), (8) is the change formula of porosity and permeability of coal, and (9)–(16) is the equation of gas occurrence and migration in coal pores and fractures. In the process of negative pressure drilling, the mechanical structure characteristics of coal body and the balance of gas in the original state of coal body are broken, and the coal body structure is slightly deformed. The coal deformation leads to changes in porosity and permeability, which affects the migration of gas in coal. The gas-solid coupling model is established by combining the deformation control Equation (5) of a gas-bearing coal body with the mass conservation control Equations (13) and (16) of gas migration in coal pores and fractures.

#### **3. Numerical Simulation**

#### *3.1. Model Assumptions*


#### *3.2. Model Introduction*

In this paper, a coal mine of No. 4 coal seam gas extraction is taken as the research object. No. 4 coal has an average thickness of 10.75 m, and the coal coefficient is 18.99%. The apparent density of coal is 1.24~1.59 t/m3, with an average of 1.36 t/m3. The true density is 1.41~1.59 t/m3, with an average of 1.49 t/m3. The coal seam has an initial gas pressure between 1.01 MPa and 1.29 MPa. The gas content in the coal seam is 3.91~4.58 m3/t.

Based on the gas-solid coupling model constructed above, with the help of the multiphysics coupling simulation software COMSOL Multiphysics, the physical field coupling calculation is carried out by using the custom partial differential equation module (PDE) and the solid mechanics module in structural mechanics. The transient solver is selected. The migration of gas in coal is described by the PDE module, and the deformation of the coal structure is described by the solid mechanics module. The gas-solid coupling formula derived above is embedded in the formula of the module.

A three-dimensional geometric model of 30 m × 40 m × 7 m is established as shown in Figure 1a. The radius of the central extraction borehole is 75 mm. The coal seam has the overlying strata load of 12 MPa, and the direction extends the *Z*-axis vertically downward. The coal body is set not to produce energy exchange with the outside, so the outer boundary of coal is set to zero flux. It is assumed that there is no displacement around the coal seam in its normal direction and the bottom surface is fixed. The borehole extraction gas negative pressure is set to 20 kPa. The initial gas pressure of coal matrix pores and coal fractures is 1.01 MPa. When meshing, as shown in Figure 1b below, it is set to user-controlled meshing. The mesh is divided into a free subdivision tetrahedron with a curvature factor of 0.6. The maximum element size is 2 m and the minimum element size is 0.5 m. The parameters used for numerical simulation are shown in Table 1.

(**a**) geometric model (**b**) mesh subdivision

**Figure 1.** Geometric model and meshing.

