## 3 Case Study
### Project Specification
#### 3.2.2 Distribution Types of the Duncan-Chang Model Parameters
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Parameter**} & \(\varphi_{0}/^{\circ}\) & \(\Delta\varphi_{f}/^{\circ}\) & \(R_{f}\) & \(K\) & \(K_{b}\) & \(m\) & \(n\) \\ \hline \multirow{2}{*}{Mean} & Lower limit & 38.38 & 0.78 & 699.54 & 332.01 & 0.25 & 0.41 \\  & Upper limit & 41.21 & 0.81 & 858.70 & 405.55 & 0.30 & 0.44 \\ \hline \multirow{2}{*}{SD} & Lower limit & 5.27 & 0.06 & 302.52 & 137.58 & 0.08 & 0.05 \\  & Upper limit & 6.99 & 0.08 & 381.92 & 186.23 & 0.12 & 0.08 \\ \hline \multirow{2}{*}{VC} & Lower limit & 0.13 & 0.07 & 0.37 & 0.36 & 0.31 & 0.13 \\  & Upper limit & 0.17 & 0.10 & 0.51 & 0.51 & 0.42 & 0.19 \\ \hline \end{tabular} Note: SD, standard deviation; VC, variation coefficient.  
\end{table} 
Table 3: Interval estimation of the Duncan–Chang E-B model parameters in the overburden. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{3}{*}{**Parameters**} & \multicolumn{6}{c}{**K-S Test**} & \multicolumn{6}{c}{**A-D Test**} \\ \cline{2-9}  & \multicolumn{1}{c}{\begin{tabular}{c} **Truncated** \\ **Normal** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **Lognormal** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **External** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **Weibull** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **Truncated** \\ **Normal** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **Lognormal** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{\
\begin{tabular}{c} **Extremal** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{
\begin{tabular}{c} **Weibull** \\ **Distribution** \\ \end{tabular} } \\ \hline \(\varphi/^{\circ}\) & ✓ & ✓ & ✗ & ✓ & ✓ & ✗ & ✗ \\ \(R_{f}\) & ✓ & ✓ & ✗ & ✓ & ✓ & ✗ & ✗ \\ \(K\) & ✓ & ✗ & ✓ & ✗ & ✗ & ✗ & ✗ & ✗ \\ \(K_{b}\) & ✓ & ✗ & ✓ & ✓ & ✓ & ✗ & ✓ & ✓ \\ \(m\) & ✗ & ✓ & ✗ & ✗ & ✗ & ✓ & ✗ & ✗ \\ \(n\) & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The marginal distribution of the Duncan–Chang model in the overburden. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{3}{*}{**Parameters**} & \multicolumn{6}{c}{**AIC**} \\ \cline{2-6}  & \multicolumn{1}{c}{\begin{tabular}{c} **Truncated Normal** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **Lognormal** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **Extremal** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{\
\begin{tabular}{c} **Weibull** \\ **Distribution** \\ \end{tabular} } & \multicolumn{1}{c}{
\begin{tabular}{c} **Optimal Marginal** \\ **Distribution** \\ \end{tabular} } \\ \hline \(c/kPa\) & / & 731.14 & / & 709.16 & Weibull distribution \\ \(\varphi/^{\circ}\) & 376.34 & / & 528.79 & 359.73 & Weibull distribution \\ \(R_{f}\) & / & / & \(-133.27\) & \(-184.43\) & Weibull distribution \\ \(K\) & 744.71 & / & / & 730.80 & Weibull distribution \\ \(K_{b}\) & / & / & / & / & \(\mathrm{Lognormal}\) distribution \\ \(m\) & / & / & / & / & Weibull distribution \\ \(n\) & \(-85.36\) & / & / & \(-100.70\) & Weibull distribution \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The optimal marginal distribution of the Duncan–Chang model in the core wall.