## 3 Three-Dimensional Numerical Simulation
### Numerical Model Construction and Parameter Selection  
To obtain a more comprehensive understanding of the overlying rock failure and development law of coal seam mining in Menkeqing mining area, based on the actual parameters of the working face, the model is set with a 400 m strike length, a 300 m tilt width, and 240 m height and is 400 m \(\times\) 300 m \(\times\) 240 m size. The model includes 2-2 and 3-1 coal seams. Since the actual dip angle of the coal seam is 2\({}^{\circ}\) on average and the dip angle is small, the average dip angle of the coal seam is 0\({}^{\circ}\) in the simulation, the simulated excavation width of the working face is 200 m, the mining length is 200 m, and the mining thickness of the coal seam is 5 m. The 3D model is divided into 1,200,000 3D units with a  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Rock Straa**} & **Elastic** & **Bulk** & **Shear** & **Tensile** & \multirow{2}{*}{**Poisson’s**} & **Friction** & \multirow{2}{*}{**Cohesion/MPa**} & **Bulk** \\  & **Modulus** & **Modulus** & **Modulus** & & & & **Ratio** & **Angle\({}^{C}\)** & **Density/kg/m\({}^{3}\)** \\  & **/GPa** & **/GPa** & **/GPa** & **/MPa** & & & & & \\ \hline Sliststone & 18.00 & 15.00 & 6.92 & 3.00 & 0.30 & 22.50 & 1.24 & 2430 \\ Gritsione & 18.20 & 18.96 & 6.79 & 2.85 & 0.34 & 32.05 & 1.78 & 2420 \\ Coal seam 2-2 & 12.00 & 10.00 & 4.62 & 2.00 & 0.30 & 28.00 & 1.10 & 2300 \\ Medium & 20.10 & 17.01 & 7.71 & 3.41 & 0.30 & 31.78 & 2.11 & 2540 \\ sandstone & 21.60 & 20.00 & 8.18 & 0.77 & 0.32 & 38.47 & 2.86 & 2550 \\ Sandy mudstone & 15.30 & 15.00 & 5.75 & 2.28 & 0.33 & 32.78 & 1.47 & 2410 \\ Mudstone & 14.00 & 15.56 & 5.19 & 2.20 & 0.35 & 28.03 & 1.17 & 2340 \\ Coal seam 3-1 & 12.00 & 10.00 & 4.62 & 2.00 & 0.30 & 28.00 & 1.10 & 2300 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Physical and mechanical parameters of the working face rock strata.  
Figure 2: Schematic diagram of the rock mechanical loading test (compressive strength test).  
total of 1,241,391 nodes. The numerical model adopts the Moore-Coulomb rock strength criterion. Figure 3 depicts the 3D numerical model diagram.  
To eliminate the influence of boundary effects on the accuracy of numerical simulation results, certain parameters need to be set for the model. Based on the geometric scale of the model, mining depth, and actual on-site conditions, a 100 m protective coal pillar was left at each end of the working face direction, and a 50 m protective coal pillar was left at each end of the inclination direction. Usually, to limit the range of mechanical motion in the model, it is necessary to constrain the boundaries of the X, Y, and Z axes separately. Once the boundaries are constrained and fixed, there will be no displacement or velocity in the fixed direction. The horizontal displacement of all nodes on the fixed boundary surfaces is as follows: X = 0 m and X = 400 m. The displacement in the Y direction of all nodes on the fixed boundary surfaces is as follows: Y = 0 m and Y = 300 m. There is fixed displacement in the Z direction of the bottom boundary (Z = 0 m). To generate gravity, a gravity acceleration of 9.8 m/s\({}^{2}\) is usually loaded in the negative Z-axis direction and the density of the model is set. The entire model is divided into 18 layers of rock, with an average density of 2500 kg/m\({}^{3}\) for the overlying rock model. To simulate the real stress environment, a certain load needs to be applied to the model to accurately reflect its stress situation, with a compensation load of 12.5 MPa. The single excavation step is 20 m.