## 4 Numerical Analysis  
### The Excavation Process and Current Stability Analysis  
The excavation process of the numerical model commenced on the right side of the mining room. The excavation procedure required periodic suspensions at every 2 m length interval to allow the stress in the surrounding rock mass to achieve equilibrium. Figure 6 provides an insight into the vertical stress and displacement at the top of the sidewall throughout the excavation, while Figure 7 presents the vertical stress distribution in the surrounding rock mass during various excavation stages. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Density** & \begin{tabular}{c} **Geological** \\ **Strength Index** \\ **(GSI)** \\ \end{tabular} & \begin{tabular}{c} **Hoek–Brown** \\ **Parameter mi** \\ \end{tabular} & \begin{tabular}{c} **Uniaxial** \\ **Compressive** \\ **Strength** \\ \end{tabular} & \
\begin{tabular}{c} **Elastic** \\ **Modulus** \\ \end{tabular} &
\begin{tabular}{c} **Multiplier** \\ **Factor** \\ \end{tabular} \\ \hline  & kg/m\({}^{3}\) & / & / & MPa & GPa & \\ \hline Gypsum ore & 2400 & 75 & 15 & 25 & 8 & 0.1 \\ Moderately & & & & & & \\ weathered muddy & 2475 & 65 & 10 & 50 & 10 & 0.5 \\ siltstone & & & & & & \\ \hline \end{tabular}
\end{table} 
Table 1: Numerical model parameters for goaf. Evidently, when the excavation face maintained a considerable distance from the sidewall, only minor disturbances were observed. The vertical stress at the monitoring point altered from 1.5 MPa to 2.5 MPa as the excavation length reached 6 m. However, as the thickness of the sidewall diminished, the increment in stress escalated rapidly. For instance, when the excavation depth reached 8 m, the vertical stress surged to 5.5 MPa. The analysis of the stress distribution illustrated that the vertical stress at the monitoring point exhibited intermediate levels within the overall stress distribution of the sidewall. Furthermore, the phenomenon of stress concentration stemming from the excavation emerged initially at the corner of the sidewall and subsequently propagated towards the central region. Upon the completion of the excavation, the vertical stress in the central region reached 10 MPa, yet it could peak at 18 MPa in the corner. This occurrence indicates that the stress on the sidewall during the excavation process could closely approach the uniaxial compressive strength of gypsum rock in a laboratory (25 MPa). The stress concentration might lead to localized rock damage at the corners and free surface. Nonetheless, the general stability of the sidewall in the current stage remains assured. Figure 8 illustrates the progression of the plastic state of the surrounding rock mass at various excavation stages. Following the completion of excavation, the rock roof of the abandoned mining area lost its support, leading it to bend downward under the pressure exerted by the overlying rock. Due to the relatively weak tensile strength of the rock, elements within a certain thickness of the roof underwent tension failure as the stress surpassed the tensile strength limit. As the excavation concluded, the failed elements  
Figure 6: Vertical stress–vertical displacement curve at the monitoring point of current stability. Figure 7: Vertical stress evolvement at different excavation states. within the roof connected with each other, forming a rupture surface. Consequently, a roof collapse occurred, resulting in a nearly 1 m thick fallen region at the mining room top. Contrasted with its tensile strength, gypsum's compressive strength is notably stronger. Hence, it was not until the excavation length reached 8 m that the sidewall began to yield (Figure 8c). Upon the completion of the excavation, both sides of the side wall gave rise to a shear zone Figure 8d). The stress within the shear zone of the gypsum rock did not exceed its yield strength, adhering to the principles of elasticity. This region was enveloped and confined by the yield zone, experiencing a three-directional stress state.