## 4 Numerical Analysis
### Medium- and Long-Term Stability Analysis  
Based on the analyses in the last section, the numerical model of the abandoned mining area appears to remain stable during the excavation. However, some degree of damage did occur in the sidewall, leading to the formation of a relaxation region. This region may allow groundwater from upper layers to infiltrate the abandoned area through cracks and channels formed by the roof deformation. Over time, the relaxation region of the sidewalls will undergo softening and spalling, exposing the elastic core. This exposure will decrease the effective thickness and horizontal confining stress of the sidewall, leading to further damage and inevitably causing instability in the wall. In light of this, a numerical model was established for further analysis, as depicted in Figure 11. Several assumptions were made for this model. The collapsed roof area shown in Figure 8 and the damaged sidewall area shown in Figure 9 have been completely destroyed, forming a 1 m thick collapsed mass at the bottom of the abandoned mining area. Due to sidewall damage and spalling, its thickness has decreased by 0.1 m compared with the initial model. The constitutive law of the collapsed mass follows the Mohr-Columb model, with detailed parameters listed in Table 2.  
Figure 10: Vertical displacement evolvement at different excavation states. Figure 12 presents the vertical stress and displacement curves at the top of the sidewall, while Figure 13 illustrates the distribution of the vertical stress in the surrounding rock mas under varying spalling thicknesses of the sidewall. With increasing spalling collapse, the vertical stress at the monitoring point gradually rises. The stress curve peaks at a spalling thickness of 0.4 m, after which it starts to decline in conjunction with a continuous increase in vertical displacement. This trend suggests that the sidewall has become unstable. Regarding Figure 13, it indicates that as spalling intensifies, the elastic core experiences a significant increase in vertical stress. When the spalling thickness reaches 0.3 m, the elastic core is subjected to a vertical stress exceeding 20 MPa, which is close to the uniaxial compressive strength of gypsum rock in a laboratory (25 MPa). When the thickness reaches 0.4 m, the stress within the elastic core suddenly becomes disordered and then decreases, signaling a prelude to instability. \begin{table}
\begin{tabular}{c c c c c} \hline
**Density** & **Elastic Modulus** & **Poisson’s Ratio** & **Cohesion** & **Friction Angle** \\ \hline kg/m\({}^{3}\) & GPa & / & kPa & \({}^{\circ}\) \\ \hline Collapsed mass & 2000 & 0.5 & 0.35 & 0 & 35 \\ \hline \end{tabular}
\end{table} 
Table 2: Numerical model parameters for collapsed mass. Figure 11: Numerical model setup for long-term stability. The evolution of the plastic elements (Figure 14) reveals that the tensile damage area at the top of the abandoned mining area remains relatively unchanged following the collapse of the roof. This lack of significant change can be attributed to the loss of tensile strength in the collapsed area during the previous excavation and the subsequent rebalancing of stress in the rock mass outside the collapsed roof region. Consequently, removing this region of rock mass is unlikely to substantially impact the plastic state of the model. In essence, the critical factor for the long-term stability of the abandoned mining area is the condition of the sidewall. Figure 14d illustrates that the shear plastic regions have become interconnected and cannot be constrained. Moreover, gypsum rock within the sustained plastic-yielding state exhibits a non-linear strain-softening behavior. After reaching the peak strength, it rapidly falls to the low residual strength. Figure 12: Vertical stress–vertical displacement curve at the monitoring point of long-term stability. Figure 13: Vertical stress evolvement at different wall damage states.