**Shixiong Hu, Xiao Liu and Xianzhong Li**


Received: 10 January 2020; Accepted: 19 February 2020; Published: 24 February 2020

**Abstract:** The optimum design of gas drainage boreholes is crucial for energy security and sustainability in coal mining. Therefore, the construction of fluid–solid coupling models and numerical simulation analyses are key problems for gas drainage boreholes. This work is based on the basic theory of fluid–solid coupling, the correlation definition between coal porosity and permeability, and previous studies on the influence of adsorption expansion, change in pore free gas pressure, and the Klinkenberg effect on gas flow in coal. A mathematical model of the dynamic evolution of coal permeability and porosity is derived. A fluid–solid coupling model of gas-bearing coal and the related partial differential equation for gas migration in coal are established. Combined with an example of the measurement of the drilling radius of the bedding layer in a coal mine, a coupled numerical solution under negative pressure extraction conditions is derived by using COMSOL Multiphysics simulation software. Numerical simulation results show that the solution can effectively guide gas extraction and discharge during mining. This study provides theoretical and methodological guidance for energy security and coal mining sustainability.

**Keywords:** fluid–solid coupling; coal containing gas; permeability; energy safety; sustainability

### **1. Introduction**

Since the 21st century, global resource shortage and environmental pollution have become difficult problems in human sustainability development [1,2]. Policy makers in various countries have focused on sustainable energy and low-carbon development by proposing sustainable strategies and methods [3]. With the advent of low-carbon economy, coalbed methane, as a clean, efficient, and safe energy source, has been eliciting considerable research attention. This unconventional natural gas is mainly present in coal seams in free and adsorption states [4]. Coal is a typical dual-porosity/permeability system containing a porous matrix surrounded by fractures. The coal matrix is separated by a natural fracture network composed of butt cleats and face cleats. The cleat system provides an effective flow channel for gas. The developed pore structure is the main space for coalbed methane. During gas percolation of porous media, the effective stress of the porous medium skeleton changes because of the change in pore pressure; the porosity and permeability of porous media also change to a certain extent [5]. These changes can affect the gas flow in pores and the redistribution of gas pressure within a certain range. Therefore, in studying the migration rule and deformation characteristics of gas in a porous medium, such as coal, the interaction between the gas flow in a porous medium and the deformation of the porous medium body should be considered. The mutual coupling between the gas seepage flow field and the stress field in the porous medium should also be considered.

Through relevant laboratory tests and field practice, people have gradually realized that the seepage property of gas in coal is related to the mechanical properties of coal, gas pressure, in situ stress, temperature, and other factors. Sommerton et al. studied the effect of stress on the gas seepage property of coal [6]. Brace investigated the law of permeability change of rock mass under stress [7]. McKee et al. analyzed the relationship between stress and coal porosity and permeability and studied the phenomenon in which the depth of coal seams and effective stress increase, the width of cleats in coal seam decreases, and permeability decreases exponentially [8]. Enever et al. obtained the influence rule between the effective stress and permeability of coal [9]. In accordance with the generalized form of the power law, Sun established a mathematical model for the flow of compressible gas in coal seams. This model, which is called the nonlinear gas flow model, was based on the measured gas parameters in the Zhongmacun Coal Mine of Jiaozuo Mining Bureau. A numerical simulation of the pressure distribution of the homogeneous gas seepage flow field was carried out using various models [10–12]. Li et al. studied the relationship among coal adsorption expansion, deformation, porosity, and permeability in consideration of the adsorption deformation characteristics of the coal skeleton and obtained the relationship among porosity, permeability, and swelling deformation [13,14]. Tao et al. analyzed the problems existing in the theory of nonlinear gas flow and theory of fluid–solid coupling of coal-bed gas. The research achievements in the fields of nonlinear gas flow theory, gas flow theory of the geothermal field effect, and coal-bed gas fluid–solid coupling theory have been scrutinized extensively [15,16]. Zhu et al. considered the Klinkenberg effect and proposed a coupled mathematical model of solid deformation and gas flow [17–20].

However, the fluid–solid coupling model that scholars have established still has certain limitations. For example, the Klinkenberg and adsorption expansion effects on gas migration in coal were not considered simultaneously. Existing research shows that strain on adsorption expansion occurs after the coal body adsorbs gas. When the strain is limited by certain factors, adsorption expansion stress follows, which causes a certain degree of primary deformation of the coal skeleton and affects the development of coal pores. Thus, the effect of adsorption expansion on gas migration should not be ignored [21]. On the basis of previous studies, the author comprehensively considers the influence of the two factors of migration of coalbed methane in coal and establishes a mathematical model of fluid–solid coupling in a low-permeability coal seam. Thereafter, relevant partial differential equations are derived. The establishment of the model expands the theory of fluid–solid coupling of gas-bearing coals under multi-field conditions and clarifies the law of gas occurrence and seepage in coal. The establishment of the fluid–solid coupling equation can characterize gas flow in coal seams from the perspective of time and improve the research method of gas dynamic flow law. The rationality of the established mathematical model is verified by using a specific example of the effective extraction radius of coal mine gas. This model provides a theoretical basis for the design and layout of gas drainage boreholes in coal mining and a reasonable reference for decision makers to control coal mine gas effectively [22,23].

#### **2. Mathematical Model**

#### *2.1. Basic Assumptions*


#### *2.2. Physical Property Parameter Model of Coal*

The compression and adsorption expansion of deformation result in different degrees of deformation of the coal skeleton because of the changes in conditions of the coal seam, such as crustal stress and gas pressure. Owing to the different depths of the coal seam, the gas pressure and crustal stress also change to varying degrees, resulting in changes in coal seam porosity and permeability.

2.2.1. Deformation Mechanism of Coal Containing Gas

According to previous research results [24], two kinds of deformation mechanisms exist under the joint action of internal and external stresses of coal containing gas.


**Figure 1.** Structural deformation.

This study assumes that the coal seam is thermostatic. Thus, the deformation of the particles under the influence of temperature changes is ignored.

#### 2.2.2. Porosity Mathematical Model

According to the relevant definition of porosity, the porosity change of coal can be expressed as follows:

$$\varphi = \frac{V\_p}{V\_b} = \frac{V\_{l0} + \Delta V\_p}{V\_{l0} + \Delta V\_b} = 1 - \frac{V\_{s0} + \Delta V\_s}{V\_{l0} + \Delta V\_b} = 1 - \frac{1 - q\rho\_0}{1 + c} \left( 1 + \frac{\Delta V\_s}{V\_{s0}} \right) \tag{1}$$

where ϕ is coal porosity, ϕ<sup>0</sup> is the initial porosity of coal, *Vp* is the pore volume of coal, *Vp*<sup>0</sup> is the initial pore volume of coal, *Vb* is the total apparent volume of coal, *Vb*<sup>0</sup> is the initial total apparent volume of coal, Δ*Vp* is the variation in the pore volume of coal, Δ*Vb* is the total apparent volume change in coal, *Vs* is the volume of the coal skeleton, Δ*Vs* is the volume variation of the coal skeleton, and *e* is the volumetric strain of coal.

In the actual state, the volume strain increment of coal particles caused by the bulk deformation of coal particles is mainly composed of three parts. The first part is the strain increment caused by pore gas pressure compressing the coal particles. The second part is the increment in expansion strain caused by coal particles adsorbing the coalbed methane. The third part is the strain increment caused by thermoelastic expansion. The strain increment caused by thermoelastic expansion is zero because we assume that the temperature of the coal seam is constant. In reference to Figure 2, the relationship among the three parts can be expressed as follows:

$$\frac{\Delta V\_s}{V\_{s0}} = \frac{\Delta V\_{sp}}{V\_{s0}} + \frac{\Delta V\_{sf}}{V\_{s0}} + \frac{\Delta V\_{st}}{V\_{s0}},\tag{2}$$

where <sup>Δ</sup>*V*sp *Vs*<sup>0</sup> is the strain increment caused by pore gas pressure compressing the coal particles, <sup>Δ</sup>*V*sf *Vs*<sup>0</sup> is the swelling strain increment caused by coal particles adsorbing the coal bed gas, and <sup>Δ</sup>*V*st *Vs*<sup>0</sup> is the strain increment caused by thermal elastic expansion. <sup>Δ</sup>*V*st *Vs*<sup>0</sup> is zero because the temperature of the coal seam is assumed to be constant. The total volume strain of coal particle deformation can be expressed as follows:

$$\frac{\Delta V\_s}{V\_{s0}} = \frac{\Delta V\_{sp}}{V\_{s0}} + \frac{\Delta V\_{sf}}{V\_{s0}} = \frac{\varepsilon\_p}{1 - \varphi\_0} - \frac{\Delta P}{K\_s} \tag{3}$$

where ε*<sup>p</sup>* is the expansion strain generated by the adsorption of gas per unit volume.

$$\varepsilon\_{\mathcal{P}} = \frac{2a\rho RT}{3V\_mK\_s} \ln(1 + bP),\tag{4}$$

where *T* is the thermodynamic temperature of the coal seam (K), *a* is the limit adsorption capacity per unit mass of combustibles under a reference pressure (m3/Mg), *b* is the adsorption constant (MPa<sup>−</sup>1), *R* is the universal gas constant (*R* = 8.3143 J/(mol·K)), <sup>ρ</sup> is the coal density (kg/m3), *Vm* is the molar volume of gas (22.4 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>m</sup>3/mol), and *Ks* is the volume modulus of the coal skeleton (Pa).

**Figure 2.** Deformation relationship of coal particles.

Equations (3) and (4) are substituted into Equation (1) to obtain a mathematical model of the dynamic evolution of porosity.

$$\varphi = 1 - \frac{1 - \varphi\_0}{1 + e} \left( 1 + \frac{\varepsilon\_p}{1 - \varphi\_0} - \frac{\Delta P}{K\_\circ} \right) = 1 - \frac{1 - \varphi\_0}{1 + e} \left( 1 + \frac{2a\rho RT}{3V\_m K\_\circ (1 - \varphi\_0)} \ln(1 + bP) - \frac{P - P\_0}{K\_\circ} \right) \tag{5}$$

#### 2.2.3. Permeability Evolution Mathematical Model

Permeability is an important indicator that describes the difficulty of gas migration in gas-bearing coal seams, the permeability of coal seams, and gas drainage difficulty. Therefore, a correct mathematical model of permeability evolution must be established for gas control in coal mines.

The permeability of gas-containing coal is closely related to the stress state of coal. Different stress states cause changes in coal-rock skeleton deformation and pore volume. When porosity changes, permeability also changes. To establish the relationship between coal permeability and porosity, we can refer to the Kozeny–Carman equation [25–27].

$$k = \frac{q}{k\_z S\_p^2} = \frac{qV\_p^2}{k\_z A\_S^{2'}}\tag{6}$$

where *k* is permeability (md), *kz* is a dimensionless constant (*kz* = 5), *Sp* is the pore surface area per unit pore volume of coal (cm2), and *As* is the total surface area of coal pore (cm2).

The permeability in the initial state is assumed to be

$$k\_0 = \frac{qV\_p^2}{k\_z A\_S^{2'}}\tag{7}$$

where *k*<sup>0</sup> is the initial permeability (md), *As*<sup>0</sup> is the total surface area of the coal pore in the initial state (cm2).

The total volume of coal and the volume change of a single coal particle are Δ*Vb* and Δ*Vs*, respectively, when the initial state changes to a new one. Based on the definition of porosity, the new porosity is

$$\varphi = \frac{V\_{p0} + (\Delta V\_b - \Delta V\_s)}{V\_{l0} + \Delta V\_b} \tag{8}$$

The new pore surface area can be expressed as follows:

$$S\_p = \frac{A\_{s0}(1+\partial)}{V\_{p0} + (\Delta V\_b - \Delta V\_s)} \, ^\prime \tag{9}$$

where ∂ is the increasing coefficient of the pore surface area of coal (%).

For Equations (6) and (7), the ratio of the new permeability to the original permeability is computed by the following:

$$\frac{k}{k\_0} = \frac{qS\_{p0}^2}{q\alpha S\_p^2} = \frac{1}{1+c} \frac{1}{(1+\partial)^2} \left(\frac{V\_{p0} + \Delta V\_p}{V\_{P0}}\right)^3. \tag{10}$$

The total surface area of the unit volume of coal particles is almost unchanged in the stress and strain process of coal [28]. This occurrence can be ignored. Thus, ∂ is approximately zero. According to [24], Equation (10) can be simplified as follows:

$$\frac{k}{k\_0} = \frac{1}{1+\varepsilon} \left(\frac{V\_{p0} + \Delta V\_p}{V\_{P0}}\right)^3 = \frac{1}{1+\varepsilon} \left(1 + \frac{\varepsilon}{\varphi\_0} - \frac{\Delta V\_s}{V\_{s0}} \Delta \frac{(1-\varphi\_0)}{\varphi\_0}\right)^3. \tag{11}$$

The combination of Equations (3), (4) and (11) can be obtained:

$$k = \frac{k\_0}{1+\varepsilon} \left[ 1 + \frac{\varepsilon}{q\nu\_0} + \frac{\Delta P(1-q\nu\_0)}{q\nu\_0 K\_\delta} - \frac{2a\rho RT\Delta \ln(1+bP)}{3q\nu\_0 V\_m K\_\delta} \right]^3. \tag{12}$$

The above equation is a mathematical model for the evolution of the permeability of coal containing gas.

#### *2.3. E*ff*ective Stress of Gas-Bearing Coal*

Gas-containing coal is a complex deformable pore-fracture dual media. Coal has a strong adsorption capacity for gas and produces a certain adsorption expansion stress, which changes the stress distribution of coal.

Therefore, when studying the problem of fluid–solid coupling of coal containing gas and rock, the relationship between the effective stress of the coal seam and its adsorption-expansion stress should be considered simultaneously. In view of this problem, Wu et al. [29,30] derived the following formula for calculating the expansive stress of coal. The expansion stress formula of coal can be expressed as follows:

$$
\sigma\_p = E \varepsilon\_p = \frac{2a\rho RT(1 - 2v)\ln(1 + bP)}{3V\_m} \tag{13}
$$

where σ*<sup>p</sup>* is the expansion stress (Pa), *v* is the Poisson ratio, and *E* is the elastic modulus of coal (Pa).

In accordance with the effective stress law of Terzaghi and in consideration of the expansion stress absorbed by coal, the effective stress equation of gas-bearing coal can be expressed as follows:

$$
\sigma'\_{i\dot{j}} = \sigma\_{i\dot{j}} - \alpha P \delta\_{i\dot{j}} - \sigma\_p \delta\_{i\dot{j}\dot{\prime}} \tag{14}
$$

where σ *ij* is the effective stress of gas-bearing coal (MPa), σ*ij* is the overall stress of gas-containing coal (MPa), and α is the Biot coefficient.

#### *2.4. Establishment of a Fluid–Solid Coupling Model of Gas-Bearing Coal*

#### 2.4.1. Gas Content Equation

The existing mine gas in the coal bed occurs mostly in free, adsorption, and dissolved states. The dissolved gas content is not considered in this study because the amount is notably small. The gas content is the sum of the free-state and adsorbed-state gas contents. The adsorption gas content accounts for more than 90% of the total, and the content of adsorption gas is related to moisture, coal ash, and gas pressure. The free gas content mainly depends on coal porosity and the magnitude of gas pressure. According to previous research and the modified Langmuir adsorption equilibrium equation, the gas content of a coal seam can be obtained as follows [31]:

$$Q = \left(\frac{abcP}{1+bP} + \varphi \frac{P}{P\_n}\right) \rho\_{n\prime} \tag{15}$$

where *Pn* is the gas pressure under standard conditions (*Pn* = 0.10325 MPa), and ρ*<sup>n</sup>* is the coalbed methane density under standard conditions (kg/m3).

In the above equation,

$$c = \rho \frac{1 - A - M}{1 + 0.31M'} $$

where *Q* is the gas content (kg/m3), *c* is the coal quality correction parameter (kg/m3), *A* is the ash content of coal (%), and *M* is the coal moisture (%).

#### 2.4.2. Stress Field Equation of Gassy Coal

Assuming that the gas-bearing coal is an isotropic linear elastic medium, the stress field changes obey the following equation.

(1) Balance equation

$$
\sigma\_{\dot{i}\dot{j},\dot{j}} + F\_{\dot{i}} = 0 \langle \dot{i}, \dot{j} = 1, 2, 3 \rangle,\tag{16}
$$

where *Fi* is the bulk stress (N/m3).

According to the effective stress Equation (14) of gas-bearing coal, the equilibrium differential equation expressed by effective stress is obtained by the introduction of Equation (16). Then the equilibrium differential equation can be expressed as follows:

$$\left(\sigma\_{ij,j}' + \left(aP\delta\_{i\bar{j}}\right)\_{,\dot{j}} + \left(\sigma\_p \delta\_{i\bar{j}}\right)\_{,\dot{j}} + F\_{\bar{i}} = 0\tag{17}$$

(2) Geometric equation

In the spatial distribution of gas-bearing coals, let *u* (x, y, z), *v* (x, y, z), and *w* (x, y, z) be the displacement components in directions x, y, and z, respectively, and continuous single-valued functions of coordinates. Then, the strain and displacement components satisfy the following geometric equations, which can be expressed as tensor symbols.

$$\varepsilon\_{i\bar{j}} = \frac{1}{2} (u\_{i,\bar{j}} + u\_{j,\bar{i}}) \left( i, \bar{j} = 1, 2, 3 \right) \tag{18}$$

(3) Constitutive equation of gas-bearing coal

The constitutive equation of gas-bearing coal describes the relationship between the stress and strain of coal. The constitutive relationship in this study is based on the strains caused by the adsorption expansion of gas-bearing coal, compression of the coal particle body, and rock stress.

The linear strain caused by gas adsorption by coal particles is as follows:

$$
\varepsilon\_{\rm PX} = \frac{2a\rho RT}{9V\_mK\_s} \ln(1 + bP) \tag{19}
$$

The linear compression strain of coal particles caused by the change in pore gas pressure is as follows:

$$
\varepsilon\_{PY} = -\frac{\Delta P}{3\mathcal{K}\_{\\$}}.\tag{20}
$$

According to Hooke's law, the strain due to crustal stress is computed using the following:

$$
\varepsilon\_D = \frac{1}{2G} \left( \sigma' - \frac{v}{1+v} \Theta' \right). \tag{21}
$$

According to the above expressions, the total strain of gas-bearing coal is as follows

$$\varepsilon = \varepsilon\_{PX} + \varepsilon\_{PY} + \varepsilon\_D \ = \frac{2a\rho RT}{9V\_{\text{m}}K\_{\text{s}}} \ln(1 + bP) - \frac{\Delta P}{3K\_{\text{s}}} + \frac{1}{2G} \left(\sigma' - \frac{\upsilon}{1 + \upsilon} \Theta'\right) . \tag{22}$$

By using the above formula as a reference, the following equation can be derived.

$$\sigma'=2G\varepsilon+\frac{\upsilon}{1+\upsilon}\Theta'-2G\left(\frac{2a\rho RT\ln(1+bP)}{9V\_{\text{m}}K\_{\text{s}}}-\frac{\Delta P}{3K\_{\text{s}}}\right) \tag{23}$$

where *G* is the shear modulus (MPa), and Θ is the effective volume stress.

The following equation can be obtained by arranging Equation (23) after introducing the Lame constant.

$$
\sigma' = 2G\varepsilon + \lambda e + \frac{2G\Delta P}{3K\_s} - \frac{4Ga\rho RT\ln(1+bP)}{9V\_mK\_s},
\tag{24}
$$

where λ is the Lame constant.

Assuming that the coal is a linear elastic medium, the constitutive equation of gas-bearing coal-rock deformation conforms to Hooke's law, as follows:

$$
\sigma\_{ij}' = \lambda \epsilon \delta\_{ij} + 2G\varepsilon\_{ij}. \tag{25}
$$

According to the stress–strain relationship of coal in Equation (32) and combined with the above formula, the effective stress constitutive equation of gas-bearing coal expressed in tensor form is derived as follows:

$$
\sigma\_{ij}' = \lambda e \delta\_{ij} + 2G\epsilon\_{ij} + \frac{2G\Delta P}{3K\_s}\delta\_{ij} - \frac{4Ga\rho RT\ln(1+bP)}{9V\_mK\_s}\delta\_{ij}.\tag{26}
$$

(4) Stress field equation of gas-bearing coal

Substituting Equation (26) into Equation (17) yields the following:

$$G\sum\_{j=1}^{3} \frac{\partial^2 u\_i}{\partial \mathbf{x}\_i^2} + \frac{G}{1 - 2v} \sum\_{j=1}^{3} \frac{\partial^2 u\_i}{\partial \mathbf{x}\_i \mathbf{x}\_j} + \left[ a + \frac{2G}{3K\_s} + \left( \frac{1 - 2v}{V\_m} - \frac{2G}{3V\_m K\_s} \right) \frac{2ab\rho RT}{3(1 + bP)} \right] \frac{\partial P}{\partial \mathbf{x}\_i} + F\_i = 0. \tag{27}$$

This formula is the fluid–solid coupling stress field equation of gas-containing coal, and the deformation field of gas-bearing coal is represented by displacement. The variation of the strain field of gas-containing coal caused by crustal stress, gas adsorption by coal particles, and gas pressure change are considered in the equation.

#### 2.4.3. Fluid–Solid Coupling Gas Seepage Equation of Gas-Bearing Coal

#### (1) Gas flow equation

The previous experimental results reveal that when gas migrates in low-permeability gas-bearing coal seams, the gas molecules near the surface of the coal wall show the phenomenon of non-zero velocity, which does not conform to Darcy's law [32]. This occurrence is called the slippage effect or Klinkenberg effect in seepage mechanics. Its permeability can be expressed as follows:

$$k = k\_{\mathcal{S}} \left( 1 + \frac{4\alpha\lambda\_1}{r} \right) = k\_{\mathcal{S}} \left( 1 + \frac{m}{P} \right) \tag{28}$$

where *kg* is the Klinkenberg permeability, ω is the scale factor, λ<sup>1</sup> is mean free path of gas molecules, *r* is the average pore radius, *m* is the Klinkenberg coefficient (MPa), and ∇P is the gas pressure gradient in the coal seam (Pa/m).

Therefore, when the Klinkenberg effect is considered, the equation of gas flow in the coal seam can be expressed by the following:

$$q = -\frac{k\_{\mathcal{S}}}{\mu} \left( 1 + \frac{m}{P} \right) \nabla \mathbf{P}\_{\prime} \tag{29}$$

where *<sup>q</sup>* is the velocity vector of gas flow (m/s), and <sup>μ</sup> is the gas dynamic viscosity (1.08 <sup>×</sup> <sup>10</sup>−<sup>5</sup> Pa·s).

#### (2) Continuity equation

According to the hypothesis, if the model is isolated from the outside and no exchange of substance and energy in any form occurs, the gas flow in the coal seam will conform to the law of conservation of mass, expressed in the form of a differential equation:

$$\frac{\partial \mathbf{Q}}{\partial t} + \nabla \cdot \left(\rho\_{\mathcal{S}} q\right) = I,\tag{30}$$

where Q is the gas content in coal, ρ*<sup>g</sup>* is the gas density when the gas pressure is P (kg/m3), and *I* is the source sink term.

(3) Gas seepage field equation of gas-bearing coal

According to the state equation of coalbed methane in [33], the gas content equation (15), and the gas flow equation (29), the equation of gas seepage flow field can be obtained by combining them with Equation (30). The result is as follows:

$$\frac{M\_{\rm g}}{RT} \bigg[ \wp + \frac{abcP\_{\rm R}}{(1+bP)^2} + \frac{(1-\varrho\_{0})P}{(1+e)K\_{\rm s}} - \frac{2ab\rho RT}{3V\_{\rm m}K\_{\rm s}(1+bP)(1+e)} \bigg] \frac{\partial P}{\partial t} - \frac{M\_{\rm g}P}{RTZ} \bigg[ \frac{k\_{\rm g}}{\mu} \left(1+\frac{m}{P}\right) \nabla P \bigg] = -\frac{M\_{\rm g}P}{RT} \cdot \wp \frac{\partial \varepsilon}{\partial t} \,, \tag{31}$$

where *Mg* is the molar mass of gas (kg/mol). *Z* is the gas compressibility factor, and its value is approximately 1 in the case of a small temperature difference.

In summary, Equations (5), (12), (27), and (30) constitute a fluid–solid coupling model of gas-bearing coal.

#### **3. Numerical Simulation of the Model and Analysis of Its Results**

Deduction and analysis show that the fluid–solid coupling mathematical model of gas-bearing coal is a complex nonlinear equation group. The coupled numerical solution requires the use of a numerical method and COMSOL Multiphysics simulation software.

#### *3.1. Geometric Model Establishment*

The geometric model starts from the 370 m position of the transportation lane on the 14221 working face of Xin'an coal mine. The coal mine is an experimental site for determining the effective influence radius of the drilling borehole down the seam. The main mining coal seam is No. 21 coal, and the average coal seam thickness is 4.22 m. The coal seam has a simple structure and extremely low mechanical strength. The soft coal structure is developed and generally classified as classes III–V. The coal is powdery and easily polluted.

The size of the geometric model is 14 m × 14.22 m. Figure 3 shows the geometric diagram of the model. The model is divided into three layers, in which the coal seam is located in the two rock layers, the thickness is 4.22 m, and the upper part is loaded with a stable load of 11.7 MPa. The borehole is located in the middle of the model and has a diameter of 0.089 m and a negative pressure of 13 kPa. The initial gas pressure in the coal seam is 0.9 MPa, and the gas only migrates in the coal seam. The initial time is t = 0 day, and the simulation time is 100 days.

**Figure 3.** Coal and gas coupling geometric model.

According to the field measurement, the drainage radius of the drilling borehole down the seam is the effective extraction radius of 1.5–1.8 m when the gas is extracted for 30 days. The measured drainage radius of the mine is the effective extraction radius of 3 m when the gas is extracted for 80 days. These data are used to verify the practical application value of the model.

#### *3.2. Model Parameters*

Table 1 shows the relevant parameters of the fluid–solid coupling model of gas-bearing coal.


**Table 1.** Material parameters of gas-bearing coal.
