## 2 Materials and Methods
### Experimental Procedure
#### 2.2.3 Gas Seepage Tests of Stressed Coal Samples  
It is well established that coal is a porous medium with a high adsorption capacity, and its intricate pore structure offers abundant adsorption sites and pathways for gas migration. Upon coal and gas outbursts, gas migration is promptly actuated by a significant and steep gas pressure gradient that emerges within the coal body. The migrating gas flow plays a crucial role in the outburst-induced coal fragmentation process, serving as an energetic medium that not only facilitates coal fragmentation but also conveys the resultant crushed coal particles. Specifically, gas migration occurs primarily through diffusion in mesopores and micropores with pore widths less than 1 \(\upmu\)m, driven by the random thermal motion of gas molecules, or through seepage predominantly in macropores and fissures with pore widths greater than 1 \(\upmu\)m, directed by a gas pressure gradient. Generally, seepage gas is recognized to move as a laminar flow, obeying the [PERSON]'s law, a formula empirically derived by French engineer [PERSON] in 1856 [57], which describes the linear relationship between the discharge rate of a fluid through a porous medium and the hydraulic gradient driving the flow. \[Q=cA\frac{\Delta h}{L} \tag{3}\]  
where \(Q\) is the seepage flow, \(A\) is the cross-sectional area through fluid, \(\Delta h\) represents the seepage head, and \(I=\Delta h/L\) means the hydraulic gradient, \(c\) is the seepage coefficient. When substituting the hydraulic gradient with the gas pressure gradient, the [PERSON]'s law describing gas seepage in the horizontal direction will be obtained as  
\[V=\frac{k}{\mu}\frac{\partial p}{\partial x} \tag{4}\]  
where \(V\) and \(\mu\) denote the seepage velocity and dynamic viscosity of gas, respectively, \(\partial p/\partial x\) indicates the gas pressure gradient, and \(k\) is the permeability of the medium matrix. Equation (4) is the general expression of [PERSON]'s law frequently used to solve the problem of gas seepage within coal seams. Based on the above equation, the permeability \(k\) can be defined as the volume of unit-viscosity fluid traversing through a unit cross-sectional area of a medium in unit time under a unit of pressure gradient [57]. Notably, the permeability \(k\) is an inherent constant solely determined by the structural characteristic of the seepage medium, and completely independent of the properties of the fluid. Therefore, we can obtain  
\[k=\frac{\mu V}{\partial p/\partial x} \tag{5}\]  
from Equation (4), i.e.,:  
\[k=\frac{\mu QL}{A(p_{1}-p_{2})} \tag{6}\]  
where \(p_{1}\) and \(p_{2}\) represent the pressure at the inlet and outlet of the seepage medium, respectively. In Equation (5), for steady-state seepage of liquid, the parameters of seepage velocity \(V\) and pressure gradient \(\partial p/\partial x\) both remain constant due to the incompressibility of liquid, thus ensuring the computed permeability \(k\) is a constant as well. However, this issue becomes more intricate for gas seepage due to its significant compressibility. Unlike in liquid seepage, the gas velocity does not remain constant along the seepage direction but accelerates near the outlet as a result of gas expansion. In the same way, the gas pressure gradient also increases near the outlet, so that the computed permeability \(k\) remains constant only when both the gas seepage velocity \(V\) and gas pressure gradient \(\partial p/\partial x\) simultaneously increase at an unchanged ratio \(V/(\partial p/\partial x)\). Hence, it is imperative to revise Equation (6) for its application in gas seepage for permeability calculations, given its inherent unreasonableness under such circumstances. Supposing gas expands isothermally as an ideal gas, with the exception of extremely high flow velocities, the value of \(p(\partial p/\partial x)\) remains consistent throughout the expansion process.