## 3 Results
### Effect of Axial Pressure on Seepage Velocity  
Different seepage velocity and pressure gradients were generated by changing the axial pressure step by step. The seepage discharge (\(Q\)) under various loads and confining pressures was recorded, and the seepage velocity (\(v\)) of the broken coal sample was calculated according to Equation (3) [32]:  
\[v=\frac{Q}{A} \tag{3}\]  
where \(Q\) is the seepage discharge under standard conditions (Room temperature: 20 \({}^{\circ}\)C, atmospheric pressure: 101 kPa), m\({}^{3}\)/s; \(A\) is the cross-sectional area of the specimen, m\({}^{2}\);  
The pressure gradient is calculated according to the osmotic pressure difference between both ends of the coal sample [36]:  
\[G_{P}=-\frac{P_{1}-P_{2}}{H} \tag{4}\]  
Figure 5: Axial pressure-permeability fitting curve. where \(P_{1}\) is the pore pressure when the fluid flows into the coal sample, MPa; \(P_{2}\) is the pore pressure when the fluid flows out of the coal sample, and the outlet of the permeability is connected with the air. \(H\) is the height of the coal sample, m.  
The Forchheimer-type permeability equation was employed to characterize the permeability law of broken coal. Equation (5) accurately describes the nonlinear permeability state [31]:  
\[-\frac{dp}{dx}=\frac{\mu}{k}v+\beta\rho v^{2} \tag{5}\]  
where \(k\) denotes permeability, m\({}^{2}\); \(\beta\) is a non-Darcy factor; \(\rho\) is fluid density, kg/m\({}^{3}\). The permeability curve of a typical coal sample with \(n=1.0\) was obtained, as shown in Figure 6.  
The relationship between the seepage velocity and the pressure gradient of the coal sample during the axial pressure loading is shown in Figure 6. As shown by the curve, linear flow occurs below the boundary line, resembling a quadrilateral region. Within this quadrilateral region, the flow retains linear characteristics, while beyond these boundaries the flow exhibits non-linear behavior. Reduced axial pressure promotes linear flow behavior, whereas increased confinement pressure not only shifts the boundary downward but also amplifies non-linear flow characteristics. Under constant confining pressure, the non-linear flow characteristics show a progressive increase as the axial load increases from 500 N to 700 N. This evolution is primarily due to the increase in confining stress, which induces particle rearrangement and densification of broken coal. The resulting compaction leads to a reduction in both the porosity and the number of interconnected permeability channels within the coal sample. Consequently, the permeability process intensifies, amplifying the deviation from the linear region. Comparison of Figure 5(a,b) shows that the extent of the linear region increases as the confining pressure increases. The reason for this may be that the increase in confining pressure increases the strength of the coal sample. As a result, the osmotic pressure has greater difficulty in altering the internal structure of the coal sample, thereby weakening the effect of the pressure gradient at both ends. Consequently, permeability has a wider linear range. Based on the linear region shown in Figure 6, the corresponding linear regions for different Talbot power exponents are shown in Figure 7. As the Talbot power exponent increases, C1 and C2 increase progressively and their difference increases, indicating an upward shift and expansion of the linear region. Similarly, D1 and D2 also increase  
Figure 6: Seepage velocity-pressure gradient relationship curve. progressively and their difference increases, indicating a rightward shift and expansion of the linear region. This phenomenon suggests that the Talbot power exponent improves linear permeability as shown in Figure 7. The upward and rightward shift of the linear region indicates that the seepage velocity of the coal sample increases under identical stress conditions, indicating improved permeability capacity and efficiency.