## 3 Numerical Model
### Numerical Simulation Results  
Figure 4 illustrates the progressive nature of fracture expansion. During the initial stage, fractures predominantly propagated within the sandy mudstone layer of the coal seam's roof strata. At this point, the fracture extended upwards along the vertical height to cover the roof sandstone and the underlying coal seam simultaneously, with a relatively fast expansion rate. When the fracture height extended to the upper interface, the pressure inside the fracture made it difficult to open the roof of fine sandstone. Under the induction effect of directional perforation, the fracture propagated downwards, encountering significant resistance at the direct roof-pseudo roof interface (sandstone-mudstone interface). When the stress reached a certain level, the fracture penetrated through the direct roof-pseudo roof interface (sandstone-mudstone interface); when the fracture reached the pseudo roof-coal seam interface, the resistance generated was smaller, ultimately penetrating the coal seam interface to fully open up the coal seam. Through the analysis of statistical data from monitoring the width of points of fracture, as shown in Figure 5, it can be observed that due to the higher plasticity of the coal seam compared to the overlying mudstone, wider fractures can be formed than those in the overlying strata. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multicolumn{1}{c}{**Rock**} &
\begin{tabular}{c} **Fine-Grained** \\ **Sandstone** \\ \end{tabular} & **Siltstone** & **Mudstone** & **Coal Seam** & **Mudstone** \\ \hline Elastic modulus/GPa & 20.00 & 15.50 & 11.80 & 5.10 & 11.80 \\ Poisson’s ratio & 0.23 & 0.25 & 0.28 & 0.33 & 0.28 \\ Tensile strength/MPa & 4.50 & 2.42 & 1.75 & 0.50 & 1.75 \\ Density/(kg\(\cdot\)m\({}^{-3}\)) & 2.54 & 2.05 & 1.85 & 1.35 & 1.85 \\ Permeability/\(10^{-3}\) um\({}^{2}\) & \(2\times 10^{-3}\) & \(10^{-2}\) & \(10^{-3}\) & \(10^{-2}\) & \(10^{-3}\) \\ Vertical stress/MPa & 17.74 & 17.74 & 17.74 & 17.74 & 17.74 \\ Stress of Maximum horizontal principal/MPa & 16.30 & 16.30 & 16.30 & 16.30 & 16.30 \\ Stress of Minimum horizontal principal/MPa & 14.00 & 13.50 & 13.00 & 10.00 & 14.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Rock mechanics parameter settings of the roof fracturing model. Figure 4: Expansion diagram of fracturing fractures in the coal seam roof. The influence of different mudstone thicknesses on the penetration of fractures was analyzed, with mudstone pseudo roof thicknesses set at 1 m, 2 m, 3 m, and 4 m. By applying numerical simulation methods, the development rules of the width of immediate coal seam roof fractures were compared under different conditions of the pseudo roof thickness in mudstone. It can be seen that as the thickness of the mudstone pseudo roof increased, the width of fractures in the overlying strata and within the coal seam gradually decreased. Specifically, the width of fractures in the coal seam roof initially grew rapidly, but slowed down in the later stage; the final fracture width was negatively correlated with the thickness of the mudstone pseudo roof, as shown in Figure 6.  
From Figure 7, it can be seen that with the increase in mudstone pseudo roof thickness, the width of fractures within the coal seam gradually decreased, and the time taken for fractures to develop to the center of the coal seam (point B) increased. Figure 5: Comparison chart of fractures in overlying strata and the coal seam. Figure 6: Statistical chart of the fracture width of the coal seam roof. From Figure 8, it can be seen that with the increase in mudstone pseudo roof thickness, the width of fractures within the coal seam gradually decreased. From Figure 9, it can be observed that with the increase in mudstone pseudo roof thickness, under constant fracturing displacement, the pore pressure in the formation gradually decreased. Figure 8: Fracture width under different pseudo roof thickness conditions. Figure 7: Statistical chart of fracture width of coal seam.