## 3 Materials and Methods
### Simulation Results  
The evolution of vertical displacements and vertical stresses in the model during step-by-step simulation is shown in Figure 7 and Figure 8, respectively. The vertical displacement (subsidence) scale is 1:10. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(E_{1}\), & \(E_{2}\), & \(E_{3}\), & \(v_{12}\) & \(v_{23}\) & \(v_{31}\) & \(\begin{array}{c}G_{12}\), \\ (GPa) \\ \end{tabular} & \(\begin{array}{c}G_{23}\), \\ (GPa) \\ \end{tabular} & \(\begin{array}{c}G_{31}\), \\ (GPa) \\ \end{tabular} & \(\begin{array}{c}Density\), \\ (GPa) \\ \end{tabular} & \(\begin{array}{c}Density\), \\ (GPa) \\ \end{tabular} \\ \hline \multicolumn{8}{c}{Quaternary sediments} \\ \hline
1.4 & 6.3 & 1.4 & 0.27 & 0.1 & 0.25 & 0.55 & 2.86 & 0.56 & 2000 \\ \hline \multicolumn{8}{c}{Rock strata group \#1} \\ \hline
2.1 & 19.2 & 2.1 & 0.23 & 0.12 & 0.22 & 0.85 & 8.57 & 0.86 & 2350 \\ \hline \multicolumn{8}{c}{Rock strata group \#2} \\ \hline
2.9 & 24 & 2.9 & 0.23 & 0.12 & 0.22 & 1.18 & 10.71 & 1.19 & 2550 \\ \hline \multicolumn{8}{c}{Rock strata group \#3} \\ \hline
2.45 & 19.2 & 2.45 & 0.23 & 0.12 & 0.22 & 1.00 & 8.57 & 1.00 & 2450 \\ \hline \multicolumn{8}{c}{Rock strata group \#4} \\ \hline
2.6 & 22.8 & 2.6 & 0.23 & 0.12 & 0.22 & 1.06 & 10.18 & 1.07 & 2500 \\ \hline \multicolumn{8}{c}{Rock strata group \#5} \\ \hline
2.1 & 19.2 & 2.1 & 0.23 & 0.12 & 0.22 & 0.85 & 8.57 & 0.86 & 2350 \\ \hline \multicolumn{8}{c}{Rock strata group \#6} \\ \hline
2.45 & 19.2 & 2.45 & 0.23 & 0.12 & 0.22 & 1.00 & 8.57 & 1.00 & 2450 \\ \hline \end{tabular}
\end{table} 
Table 2: Rock mass properties for numerical simulation. Step 1: Loading the model with gravity, followed by writing the values of stresses and displacements at all nodes of the model into a file using the \"Instate\" command in the ADPL programming language (Figure 7a). Step 2: Reading the created file with stored values and recalculating the numerical model to reset displacements. This step establishes the initial stress-strain state of the rock mass (Figures 7b and 8a). Step 3: Simulating coal extraction with the formation of the fractured zone and caved zone (Figures 7c and 8b). Before coal seam excavation, the stress field in the model (as well as in situ) corresponds to the gravitational stress component and can be calculated by multiplying the rock mass density by its depth. Thus, in Figure 8a, the uniform increase in stress with depth can be observed. This stress pattern characterizes the natural stress-strain state of the rock mass before mining. The corresponding vertical displacement distribution is shown in Figure 7b, where displacements are zero, indicating the undisturbed state of the rock mass. Figure 8: Evolution of vertical stresses (Pa) during step-by-step simulation: (**a**) loading with gravity (Step 1), and resetting displacements (Step 2), (**b**) coal extraction in 815 and 817 longwall panels (Step 3). Figure 7: Evolution of vertical displacements (m) during step-by-step simulation: (**a**) loading with gravity (Step 1), (**b**) resetting displacements (Step 2), and (**c**) coal extraction in 815 and 817 longwall panels (Step 3). As a result of coal seam extraction in the number 815 and 817 longwalls (left part of the model), a fractured zone and a caved zone formed in the goat. The roof strata sag, failure, and fall filled the void that was left after coal extraction. During this process, the rock mass expanded and provided support to the overlying strata. This led to the rock mass movement and surface subsidence. According to the simulation results, the surface subsidence above the goat of the 815 and 817 longwalls is equal to 1.81 m. Increased stresses were formed over the edge of the number 817 longwall. The vertical stress concentration factor is as follows:  
\[\text{Stress concentration}=\frac{\sigma_{max}}{\sigma_{0}}=\frac{63.7}{16.8}=3.79, \tag{1}\]  
where \(\sigma_{max}\)--maximum vertical stress, MPa; \(\sigma_{0}\)--lithological stress, MPa.