## 4 Numerical Simulation of Strain Energy of Hard Roof  
### Numerical Model  
According to the geological conditions, the numerical model is established via FLAC\({}^{3\text{D}}\), as shown in Figure 3. The length \(\times\) width \(\times\) height of the model is 180 m \(\times\) 360 m \(\times\) 45 m. The bottom and side displacements of the model are fixed. The Mohr Coulomb criterion has significant advantages in simulating the failure of heterogeneous rock formations. It is a  
Figure 2: Asymmetric three-hinged arch structure of initial roof caving.  
universal geotechnical model widely used in engineering to analyze rock failure. It reflects the failure mechanism of rocks and can effectively simulate the stress and failure state of surrounding rocks [36; 37; 38]. After parameter calibration through numerical simulation, it is found that the Mohr Coulomb criterion can better match the strength test results in the laboratory. The parameters of the calibrated rock are shown in Table 1. The gob adopts a Null model, which can effectively reflect the empty state of the goaf before the initial collapse of the roof.  
This simulation analyzes the strain energy of the roof of the goaf at different distances in the 2102 working face, and the simulation is divided into three steps. First: the overall operational balancing of the model under the action of the original rock stress. Second: transport crossheading during tunneling of the 2102 working face and crossheading of the return air. Third: the mining of the 2102 working face every 10 m for a total of 8 times reaching 80 m for the calculation.  
The research on the initial mining collapse of the working face roof is mostly simplified to a two-dimensional beam model to study the fracture distance. The energy model used in this study provides a detailed analysis of the energy generation, accumulation, and evolution process of the hard roof above the goaf in the three-dimensional mining space of the working face. It can intuitively determine the location of rock layers that affect mining safety and provide a basis for determining hydraulic fracturing parameters in the later stage.  
When performing energy analysis, it is necessary to establish an energy numerical model for calculation. From the perspective of energy, when the rock deforms under the action of external forces, it is assumed that the physical process has no heat exchange  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Lithology** & **Density/ kg/m3** & **Bulk** & **Shear** & **Cohesion/ MPa** & **Angle of Internal** & **Tension** \\
**Mudstone** & 2510 & 5.1 & 2.5 & 0.796 & 41.76 & 0.642 \\ Medium-grained sandstone & 2600 & 26.34 & 18.08 & 2.06 & 46.1 & 1.06 \\ Fine sandstone & 2550 & 20.7 & 13.1 & 1.648 & 42.22 & 0.988 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Key rock parameters used in the numerical model.  
Figure 3: Model overview.  
with the outside world, and the rock can release elastic strain energy [39], which can be expressed as  
\[U_{\rm e}=\frac{1}{2E}\Big{[}\sigma_{1}{}^{2}+\sigma_{2}{}^{2}+\sigma_{3}{}^{2}-2 \mu(\sigma_{1}\sigma_{2}+\sigma_{2}\sigma_{3}+\sigma_{3}\sigma_{1})\Big{]} \tag{4}\]  
where \(E\) is the elastic modulus; \(\mu\) is Poisson's ratio; and \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\) are the maximum, intermediate, and minimum principal stresses, respectively.  
The total strain energy \(W_{\rm i}\) of the element is  
\[W_{\rm i}=U_{\rm i}V_{\rm i} \tag{5}\]  
where \(W_{\rm i}\) is the strain energy density of the i-th unit body, and \(V_{\rm i}\) is the volume of the i-th unit.  
Based on the principle of energy analysis, the calculation program of the elastic strain energy distribution density of coal and rock mass is written in fish language and the calculation program is run in the simulation process.