## 4 Conclusions  
The phase field models are used in COMSOL to systematically study the flow pattern of CH\({}_{4}\) in coal seams after hydraulic fracturing at the pore scale. The Navier-Stokes equations and Cahn-Hilliard equations are used to simulate the two-phase flow of gas and water in porous media. The flow patterns and fluid saturation are analyzed from the perspectives of viscous and capillary instability, wettability, and permeability. The following conclusions are drawn:  
1. It is found that during the CH\({}_{4}\)-water displacement process, fingering phenomena are very pronounced when \(lgCa<-4.37\). The boundary between capillary fingering and viscous fingering is approximately determined to be at \(lgM=-1.22\), \(lgCa=-4.77\)  
Figure 14: Snapshots of the dual-permeability model after stabilization by CH\({}_{4}\) breakthrough: (**a**) is the case of \(lgM=0\) for different \(Ca\) values; (**b**) is the case of \(lgCa=-3.15\) for different \(M\) values. Figure 15: Effect of different \(Ca\) and \(M\) on CH\({}_{4}\) saturation before and after breakthrough. (**a**) Changes of CH\({}_{4}\) concentration before and after breakthrough at different \(lgCa\); (**b**) Changes of CH\({}_{4}\) concentration before and after breakthrough at different \(lgM\). while the boundary between viscous fingering and stable displacement is approximately at \(lgM=-0.52\), \(lgCa=-4.07\). 2. The CH\({}_{4}\)-water two-phase flow patterns and invading phase (CH\({}_{4}\)) saturation are quantitatively analyzed under different capillary forces (\(Ca\)) and viscous forces (\(M\)). As \(Ca\) and \(M\) increased, the invading CH\({}_{4}\) gradually transitioned from capillary fingering to viscous fingering and eventually to stable displacement, with higher displacement efficiency. The saturation of invading CH\({}_{4}\) at breakthrough and stability is compared. At \(lgM=-0.52\) and \(lgCa=-4.37\), displacement efficiency decreases due to the appearance of preferential paths. 3. By changing the wettability angle of the porous media, the CH\({}_{4}\)-water two-phase flow in hydrophilic, neutral, and hydrophobic coal bodies is simulated. The backflow occurs only under hydrophilic conditions and that drainage efficiency increases significantly with increasing wettability angle. 4. A dual permeability model is constructed under different \(Ca\) and \(M\) values to simulate the effect of varying local permeability on two-phase displacement in real coal seams. CH\({}_{4}\) preferentially invaded high-permeability regions, and displacement efficiency in low-permeability regions increases with unfavorable viscosity ratios (\(lgM<0\), \(lgCa=-3.15\)) and increasing \(lgCa\). As can be seen from the above conclusions, the phase field method has high accuracy, stability and wide applicability. It can accurately describe the distribution of different phases of two-phase fluids and interface evolution by introducing continuous phase field variables. And it can accurately simulate the dynamic behavior of complex interfaces. Therefore, this method can be used to predict the amount of gas emission in water-bearing coal seam after the hydraulic permeability enhancement of coal seam. Moreover, the method can be used for simulation and validation when considering two-phase flow interface problems in other fields such as petroleum engineering and CO\({}_{2}\) sequestration. The simulations in this paper were conducted under the condition of constant temperature and pressure. Future studies will consider more situations such as the flow of high-temperature and high-pressure two-phase fluids. Z.W.--writing original draft, data curation and data analysis. C.G. and N.L.--review and editing, conception, methodology and software. K.F.--software. X.Z. and T.L.--data curation, investigation. All authors have read and agreed to the published version of the manuscript. This work was funded by the National Natural Science Foundation of China (52204247 and 12302448) and the Natural Science Foundation of Zhejiang Province (LQ21E040005 and LQ23E040009).