## 2 Materials and Methods
### Finite Element Method Simulation
The initial values of elastic modulus \(\text{E}_{\text{m}}\) of transversally isotropic model for the rock mass layers using [PERSON]'s rock mass classifications [65] and the strength values of the laboratory parameters of the rock samples were estimated based on formulas (Equations (3) and (4)) as follows [66; 67]: \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{\(\text{R}_{c}\) [MPa]} & \multicolumn{2}{c}{\(\text{R}_{e}\) [MPa]} & \multicolumn{2}{c}{**Roof and Floor Class**} \\  & **Average** & **Range** & & \\ \hline Roof & 24.2 & 14.6–37.5 & 2.4 & III 1  \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geomechanical properties of the seam 215 and surrounding rocks. Figure 2: The diagram of the transversally isotropic model. \[\bullet\] For \(\text{R}_{\text{c}}\leq 100\) MPa: \[\text{E}_{\text{m}}=\left(1-\frac{\text{D}}{2}\right)\cdot\sqrt{\frac{\text{R}_{ \text{c}}}{100}}\;10^{\frac{\text{GS}-10}{40}}\] (3) \[\bullet\] For \(\text{R}_{\text{c}}>100\) MPa: \[\text{E}_{\text{m}}=\left(1-\frac{\text{D}}{2}\right)\cdot 10^{\frac{\text{GS}-10}{40}}\] (4) where: \(\text{R}_{\text{c}}\)--the uniaxial compressive strength of intact rock material, which provides a system for estimating the reduction in rock mass strength for different geological conditions (Table 2). \(GSI\)--value of the geological strength index for the rock mass [65], \(D\)--the degree of disturbance, to which the rock mass has been subjected by blast damage and stress relaxation [67]. Based on geological data from a borehole (Table 3) 3D numerical model of AOI was prepared (Figure 3). The dimension of the 3D model was 3420 \(\times\) 2780 \(\times\) 600 m. The model was constructed using hexahedral (hex) elements, with a total number of elements close to 3,000,000. In the area of the longwall panel, the element size was set to 1 m. From the edges of the longwall panel to the model boundaries, a bias was used, resulting in element sizes ranging from 0.5 m to 10 m. In the vertical direction (z-direction), the element size was consistently set to 1 m.  
These sizes were chosen to balance computational efficiency with the accuracy required to capture the deformation behavior accurately. The geometric conditions for each longwall were as follows:  
* LW501--200 m width, 570 m long, 1.90 m high and with a 8 m caving zone;  
Figure 3: Numerical model of AOI. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Stratum** & **Layer’s Floor Depth [m]** & **Thickness [m]** \\ \hline
1 & Quaternary & 10 & 10 \\
2 & Sandstone & 185 & 175 \\
3 & Silstone & 200 & 15 \\
4 & Sandstone & 240 & 40 \\
5 & Silstone & 275 & 35 \\
6 & Sandstone & 300 & 25 \\
7 & Silstone & 320 & 20 \\
8 & Sandstone & 418 & 98 \\
9 & Coal & 420 & 2 \\
10 & Sandstone & 600 & 180 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Geological characterization of the mining area used in numerical simulations. * LW502--200 m width, 520 m long, 1.90 m high and with a 8 m caving zone. Numerical calculations were carried out for three-dimensional models in two steps (Figure 4), which were simulation of: 1. Initial state of stress and strain, 2. Coal extraction with roof collapse--for this purpose, so-called equivalent elements with reduced parameter values in the highly disturbed mining zone (cave-in zone) were implemented. Results of initial parameters for the analyzed region of KWK Piasat--Ziemowit mine were compiled in Table 4.  
Changes in the values of rock layer parameters were made with the assumption of a constant change coefficient for all rock layers.