## 2 Materials and Methods
### Finite Element Method Simulation  
Based on many years of authors experience as well as on the basis of literature analysis, the transversally isotropic model is the most reflecting constitutive model of the rock mass. Research conducted over many years has proven that rocks have anisotropic properties [26; 49; 50; 51; 52]. In 1970, [PERSON] and [PERSON] [53] conducted laboratory studies to determine the value and nature of shale anisotropy. They cut samples from the rock block in six different directions and subjected them to a uniaxial compression test. As a result of the research carried out in this way, they found that the anisotropy of rocks increases with the increase in the acting load and that it is possible to adopt a simple anisotropy model with nine constants of the type: E\({}_{1}\), E\({}_{2}\), E\({}_{3}\), G\({}_{12}\), G\({}_{23}\), G\({}_{31}\), v\({}_{12}\), v\({}_{13}\), v\({}_{23}\) called the orthotropic model. Similar studies on the orthotropy of rocks were carried out by [PERSON] [54] examining the behavior of the Mesozoic shale and [55]. As a result of these studies, they concluded that it is possible to adopt an anisotropic model with five elastic constants called the anisotropic axisymmetric model (transversally isotropic model in Figure 2). Anisotropic properties can be caused by both geological processes during rock formation and subsequent tectonic processes, as well as processes resulting from human activity (anisotropy of rocks resulting from cracks resulting from exploitation). Research on the anisotropy of mechanical properties of rocks was carried out, among others, by [PERSON] [56], [PERSON] [57], [PERSON] and [PERSON] [58], [PERSON] [59], and [PERSON] [60]. The most important conclusions from the research on rock anisotropy indicate:  
* Sedimentary rocks have anisotropic properties,
* In most cases:
* The elastic modulus perpendicular to the lamination is greater than the modulus of elasticity parallel to the lamination,
* The strength of the rock in the direction perpendicular to the interleaving plane is greater than in the direction parallel, \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Date of Measurements dd/mm/yyyy**} \\ \hline
**Leveling** & **GNSS** \\ \hline
7 June 2021 & \\
28 September 2021 & \\
25 November 2021 & \\
11 March 2022 & \\
12 May 2022 & 4 August 2022 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dates of geodetic measurements: levelling and GNSS. In transversally isotropic model, assuming the isotropy plane in directions 1-2, the modulus of elasticity is defined as follows: \(\text{E}_{1}=\text{E}_{2}\), \(\text{E}_{3}\), \(\
u=\
u_{12}=\
u_{13}=\
u_{23}\), \(\text{G}_{13}\). Additionally, the modulus of shear elasticity in the isotropy plane \(\text{G}_{12}\) can be distinguished, calculated from the formula:  
\[\text{G}_{12}=\frac{\text{E}_{1}}{2\cdot(1+\ \
u_{12})} \tag{1}\]  
When there is no laboratory data on the value of the \(\text{G}_{13}\) form elasticity parameter, the formula developed by [PERSON] [61] can be used:  
\[\text{G}_{13}=\frac{\text{E}_{1}\cdot\text{E}_{3}}{\text{E}_{1}\cdot(1+2\cdot \
u)+\text{E}_{3}} \tag{2}\]  
Numerical modelling of the impact of underground mining on the land surface using a transversely isotropic model was presented, among others, by [PERSON] [62], [PERSON], [PERSON] and [PERSON] [63], [PERSON] and [PERSON] [64] and [PERSON] [19; 39]. Based on the abovementioned literature analysis and the experience concerning numerical modelling of underground hard coal mining, it was decided to use the transversally isotropic model in numerical analyses. The initial parameters for iterative procedure were estimated on the basis of the results of strength tests of rock samples (Table 2) and the methodology presented in the work by [PERSON] [39].