## 4 Numerical Simulation Methods
### Calibration of the Double-Yield Model for Goaf Area  
#### 4.5.1 Mechanical Characteristics of the Goaf Area  
During the mining process, the roof of the goat area undergoes dynamic changes from loose to compacted conditions, leading to variations in its mechanical properties. The model proposed by [PERSON] accurately depicts this stress-strain relationship and has been widely utilized in engineering practice [37; 38].  
\[\sigma=\frac{E_{0}\epsilon_{g}}{1-\epsilon_{g}/\epsilon_{g~{}max}} \tag{10}\]  
In the equation: \(\sigma\) is the vertical stress acting on the goat area rock mass, MPa; \(\epsilon_{g}\) is the strain experienced by the goaf area rock mass under vertical stress; \(E_{0}\) is the initial modulus of elasticity of the goat area rock mass, GPa.  
Through laboratory testing, the compressive strength of the roof strata is determined to be 45.26 MPa. Based on these data, the dilatancy coefficient of the goaf area rock mass is calculated to be 1.31, the maximum strain is 0.23, and the initial elastic modulus is 68.99 GPa. The specific stress-strain correspondence is detailed in Table 3.  
After mining, the rock layers from the upper part of the goaf area to the caving zone will undergo compaction. Therefore, the height of the caving zone is calculated as follows:  
\[h_{c}=\frac{100h_{m}}{c_{1}h_{m}+c_{2}} \tag{11}\]  
where: \(h_{c}\) is the height of the caving zone, m. \(h_{m}\) is the mining height, m. \(c_{1}\) and \(c_{2}\) are correction factors.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Lithology** & **Thickness** & **Density** & **Bulk** & **Shear** & **Cohesion** & **Friction** & **Tensile** \\  & **/m** & **/(kg/m3)** & **Modulus** & **Modulus** & **/MPa** & **/(\(\circ\))** & **Angle** & **Strength** \\ \hline Overlying strata & 30.0 & 2540 & 9.92 & 7.35 & 3.81 & 37 & 3.28 \\ Coarse sandstone & 12.5 & 2660 & 10.27 & 8.42 & 5.75 & 39 & 3.50 \\ Middle sandstone & 4.1 & 2660 & 11.27 & 9.42 & 6.75 & 42 & 4.16 \\ Coarse sandstone & 11.2 & 2660 & 10.27 & 8.42 & 5.75 & 39 & 3.5 \\ Siltstone & 3.6 & 2540 & 9.92 & 7.35 & 3.81 & 37 & 3.28 \\ \(4^{-2}\) coal & 7.4 & 1320 & 1.87 & 1.06 & 1.55 & 31 & 0.45 \\ Siltstone & 2.6 & 2540 & 9.92 & 7.35 & 3.81 & 37 & 3.28 \\ Mudstone & 14.6 & 2150 & 3.02 & 2.56 & 1.72 & 31 & 1.07 \\ Underlying strata & 30.0 & 2540 & 9.92 & 7.35 & 3.81 & 37 & 3.28 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Model mechanical parameters.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Strain** & **Stress/MPa** & **Strain** & **Stress/MPa** & **Strain** & **Stress/MPa** \\ \hline
0.01 & 0.72 & 0.09 & 10.01 & 0.17 & 41.49 \\
0.02 & 1.51 & 0.1 & 11.93 & 0.18 & 51.63 \\
0.03 & 2.37 & 0.11 & 14.16 & 0.19 & 66.10 \\
0.04 & 3.32 & 0.12 & 16.77 & 0.2 & 88.38 \\
0.05 & 4.37 & 0.13 & 19.87 & 0.21 & 127.17 \\
0.06 & 5.54 & 0.14 & 23.60 & 0.22 & 211.60 \\
0.07 & 6.85 & 0.15 & 28.19 & 0.23 & 537.23 \\
0.08 & 8.33 & 0.16 & 33.98 & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Stress–strain relationship of goaf materials in Double-Yield model.  
With a mining height of 7.4 m and an average strength of the roof strata of 45.26 MPa, these parameters are substituted into the formula for calculation, yielding a caving zone height of 23.5 m.