## 3 Variation of Mechanical Parameters in the Coal Mass  
Coal mass exhibits inherent non-homogeneity, which can be characterized using a normal distribution function with a coefficient of variation to relate it to mechanical properties [24]. In this study, a coefficient of variation of 0.15 is applied to the mechanical parameters. The standard deviation for each selected parameter--elastic modulus, internal friction angle, and cohesion--is calculated based on the original coal seam parameters, as detailed in Table 3.  
Assuming a normal distribution, parameters are selected within the range of (\(\mu-3\sigma\), \(\mu+3\sigma\)) for ten distinct sets of parameter groups (a-j), as detailed in Table 4[25]. By employing the stochastic response surface methodology, calculations are performed to ascertain the necessary number of required matching points for the ten parameter sets. Adjustments are implemented to align with the \(3\sigma\) criterion, with a particular focus on the tenth parameter group.  
Finally, a functional relationship is established between variations in mechanical parameters of coal mass and the results obtained from numerical modeling of coal tunneling processes. This methodology offers valuable insights into the manner in which variability in the mechanical properties of coal mass affects roadway stability.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Height/m** & \begin{tabular}{c} **The Name of the** \\ **Rock Formation** \\ \end{tabular} & \begin{tabular}{c} **Density/** \\ **(kg/m3°)** \\ \end{tabular} & \begin{tabular}{c} **Modulus of Elasticity/** \\ **GPa** \\ \end{tabular} & \begin{tabular}{c} **Shear Modulus/** \\ **GPa** \\ \end{tabular} & \
\begin{tabular}{c} **Bulk Modulus/** \\ **GPa** \\ \end{tabular} &
\begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} \\ \hline
10 & upper roof & 2560 & 27.5 & 11.08 & 18.41 & 0.26 \\
6.4 & immediate roof & 2500 & 9.6 & 3.87 & 8.2 & 0.28 \\
9.6 & coal seam & 1400 & 3.1 & 1.25 & 3.02 & 0.31 \\
4 & immediate bottom & 2500 & 9.6 & 3.87 & 8.2 & 0.28 \\
10 & basic bottom & 2560 & 27.5 & 11.08 & 18.41 & 0.26 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mechanical parameters of rock and coal mass.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**The Name of the** & **Normal Stiffness/** & **Tangential Stiffness/** & **Internal Friction** & **Adhesion/** & **Tensile Strength/** \\
**Rock Formation** & **GPa** & **GPa** & **Angle/°** & **MPa** & **MPa** \\ \hline upper roof & 5100 & 2100 & 25 & 10.2 & 5.6 \\ immediate roof & 3200 & 1400 & 20 & 8.1 & 3.2 \\ coal seam & 2000 & 800 & 18 & 3.0 & 1.1 \\ immediate bottom & 3200 & 1400 & 20 & 8.1 & 3.2 \\ basic bottom & 5100 & 2100 & 25 & 10.2 & 5.6 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mechanical parameters of perimeter rock joints.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mechanical Parameter** & **Mean \(\mu\)** & **Standard Deviation \(\sigma\)** & **Coefficient \(\
u\)** \\ \hline Elastic modulus \(c\)/GPa & 3.1 & 0.465 & 0.15 \\ Internal friction angle \(\varphi/(\lx@math@degree)\) & 18 & 2.7 & 0.15 \\ Cohesion \(E\)/kPa & 3.0 & 0.45 & 0.15 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Statistical characteristics of three mechanical parameters.