## 3 Results and Discussion
### Effect of Permeability and Inhomogeneity on Flow in Pore Media  
In real coal seams, the pores are generally of varying sizes, and local permeability also differs. A local porous media often contains many pore networks of varying scales, encompassing multiple permeability ranges. This section constructs a porous media model with dimensions of 0.02 \(\times\) 0.00514 m\({}^{2}\), as shown in Figure 11. The wettability of the porous media is set to neutral, i.e., \(\theta_{c}=\pi/2\), with no-slip boundary conditions, and the medium is pre-saturated with water. The temperature of the entire model is constant. The effects of temperature and gravity on the fluid could be ignored. During the injection, displacement is carried out at a constant flow rate of 0.01 m/s from the inlet boundary (left to right), with the outlet pressure set to zero. The top and bottom boundaries are closed, and different permeabilities are simulated by varying the particle size to change the pore throat radius (Rt). As shown in Figure 12a, the permeability of the porous medium significantly increases with the pore throat radius (Rt). While at Rt = 50 \(\upmu\)m and Rt = 150 \(\upmu\)m, CH\({}_{4}\) partially break through the outlet of the porous media. At smaller pore sizes (Rt = 50 \(\upmu\)m), many water regions in the upper part of the porous media are trapped. This is because a smaller pore throat radius means narrower flow channels, which increases fluid flow resistance. Higher flow resistance requires CH\({}_{4}\) to exert more pressure to move the water. If the pressure is insufficient to overcome this resistance, CH\({}_{4}\) could not smoothly break through the porous media's outlet. Additionally, the interfacial tension between water and methane causes water to adhere more easily to the pore walls. At smaller pore throat radii, water is more likely to form liquid bridges, further hindering gas flow. As shown in Figure 12b, the volume fraction (Vn) of CH\({}_{4}\) in the porous media rapidly increases with the increase in Rt, indicating that the quality of permeability greatly affects the efficiency of fluid transport. To simulate a realistic permeability scenario within the coal seam, the particle size in the upper part of the porous media in Figure 13 was reduced by 36% to simulate a high permeability region. A dual permeability region was constructed to investigate the effect of heterogeneity on CH\({}_{4}\)-water two-phase flow, as shown in Figure 13. The labeled blue circles are particles in the highly permeable layer, whose diameter is 36% smaller than the particles in the low-permeable part of the layer. The arrows mark the inlet and outlet. Figure 11: Model of porous media with Rt = 50 \(\upmu\)m.  
Figure 14 shows the simulation results of the dual permeability model under different \(Ca\) and \(M\) values. Figure 14a simulates different capillary numbers with \(lgM=0\), while Figure 14b simulates different viscous forces with \(lgCa=-3.15\). As expected, CH\({}_{4}\) preferentially breaks through in the high permeability region but could not completely expel the water within the medium. As shown in Figure 14a, when \(lgCa=-5.15\), CH\({}_{4}\) mainly invades the high permeability region but could not break through the porous media outlet over time. When the \(lgCa\) value increases to \(-4.15\), CH\({}_{4}\) completely displaces the water in the high permeability region, while the water in the low permeability region is hardly displaced. As the \(lgCa\) value continues to increase to \(-3.15\), the CH\({}_{4}\) interface in the low permeability region gradually advances to the halfway point of the porous media before stagnating. However, the breakthrough channel width in the high permeability region decreases from 3 pore widths to 2. When the \(lgCa\) value reaches \(-2.15\), the water in the low permeability region is almost completely displaced by CH\({}_{4}\).