4.2.1. Establish Mathematical Model

According to the characteristics of gas migration in the large-space goaf under the condition of high-intensity and rapid mining advance, the Brinkman seepage equation and Fick diffusion equation can be coupled to solve the transient process of gas diffusion under the action of air flow, so as to study the mechanism of the two application modes.

The convection and migration process of gas diffusion in goaf includes two physical processes: gas ventilation convection and concentration diffusion [19,20]. The gas diffusion equation conforms to Fick diffusion law, that is

$$
\delta t \frac{\partial \mathcal{C}}{\partial t} + \nabla \cdot (-D \nabla \mathcal{C}) + v \cdot \nabla \mathcal{C} = R \tag{4}
$$

where, δ*t* is the instantaneous time proportion coefficient, which is equivalent to porosity; *C* is the gas concentration, mol/m3; <sup>∇</sup> is the lalpace operator; *<sup>D</sup>* is the diffusion coefficient, m2/s; *<sup>v</sup>* is the average velocity, m/s; *<sup>R</sup>* is the source term, mol/(m3·s).

Under the condition of fully mechanized coal mining, the goaf caving area is composed of broken caving rock mass, which can be regard as the large porous medium. Its compaction degree is related to the support pressure above the goaf, in which the air flow channel system is relatively complex, and the Brinkman seepage equation is more suitable, that is

$$\frac{d\eta}{k}\mathbf{v} = \nabla \cdot \left(-pI + \eta(\nabla v + (\nabla v)^T)\right) + F \tag{5}$$

where, *v* is the flow velocity of fluid, m/s; *k* is the permeability, m2; η is the dynamic viscosity coefficient, Pa·s; *p* is the fluid pressure, Pa; *I* is the unit vector; *F* is the fluid resistance.

In the calculation model of this paper, the simultaneous equations (Equations (4) and (5)) can solve the transient process of gas diffusion under the action of air flow, which can be calculated by numerical simulation software COMSOL Multiphysics.

COMSOL Multiphysics is a large-scale advanced numerical simulation software developed by the Swedish company COMSOL. Based on the finite element theory, COMSOL Multiphysics is developed to simulate various physical phenomena, which can better simulate the coupling of multiple physical fields and describe physical phenomena by solving various mathematical models [19].

#### 4.2.2. Model Calculation

Referring to the actual size of the comprehensive mechanized coal mining workface, the calculation model of 80 × 195 m was established by COMSOL Multiphysics (in order to simplify the calculation, this paper only studied the goaf area formed by about one week of coal mining without considering the dynamic process). As shown in Figure 9, the width of the roadway is 5.6 m, so the lower 5.6 m part of the left side is treated as the boundary of air inlet, and the upper 5.6 m part of the left side is treated as the boundary of air outlet, and the other boundaries are airtight. Since the air flow in the roadway of workface accounts for the majority, and the air flow penetrating into the goaf only accounts for the minority, the pressure difference between air inlet and outlet is set as 100 Pa. At the same time, the air inlet and outlet are set as the convection boundary, between which is the airtight boundary, and the gas concentration flux was supplied by the other boundaries. There are 1 atm and 3 mol/m3 initial concentration gas in the goaf, and the other numerical simulation parameters are shown in Table 5.

**Figure 9.** Computational mode and its boundary conditions.



When simulating 'gas sealing', the gas separation walls with equal width of roadways were arranged on both sides of the goaf, the parameters of which were set as the foamed concrete in Table 4, and the remaining parts of the model were unchanged. When simulating 'gas replacement', the foamed concrete was used to block the entire cavity of the goaf and replace the gas in the empty hole, which could completely block the gas source. Since the replacement process was dynamic, the parameters of the whole calculation range were constantly changing. In order to simplify the study, the pressure difference between the air inlet and outlet was increased by five times in consideration of the displacement effect of the gas when the filling material was pumped, after which the parameters of the whole goaf were set as the foamed concrete in Table 4 [21–23].

#### *4.3. Analysis of Simulation Results*

#### 4.3.1. No Gas Isolation

We drew the gas concentration distribution map in the goaf when the time was 0, 0.5, 1, 3, and 5 days, as shown in Figure 10a–e. The results show that under the condition of 100 Pa air pressure difference, it will take a certain time to start dispersing the gas because of the existence of 1 atm and 3 mol/m<sup>2</sup> initial concentration in the goaf. With the development of time, from the air inlet to the outlet, a fan-shaped concentration reduction area was gradually formed until it became stable about 5 days later, forming an accumulation area in the upper corner and keeping stable at 5–6 mol/m2. If measures are not taken to isolate gas, the gas accumulation area near the upper corner will make the gas continuously flow out with the air, causing the hidden danger to the workface.

**Figure 10.** Concentration distribution in gas evacuation process: (**a**) *t* = 0 day; (**b**) *t* = 0.5 day; (**c**) *t* = 1 day; (**d**) *t* = 3 days; (**e**) *t* = 5 days.
