## 4 Numerical Modeling  
The results from seven boreholes (BDR-08, 10, 21, 22, 24, 25, and 26) along the diversion tunnels of Project 5 show that 75% of the \(RQD/J_{n}\) values were less than 16.5, and only 7.6% of the \(RQD/J_{n}\) values were greater than 25. Similarly, for Project 6, of 13 boreholes (BH-01, 02, 03, 04, 05, 08, 09, 10, 11, 12, 13, 15, 26), in most of them, the degree of jointing was high to very high, resulting in \(RQD\) < 50%. Only BH-09 and BH-26 in Project 6 had predominantly moderate jointing, which corresponds to a range of \(RQD\) from 50% to 75% [37]. For simplicity and to reduce the number of cases that were numerically evaluated, \(J_{r}\) = 3, \(J_{a}\) = 1, and \(J_{w}\) = 1 were used for empirical support design. The average properties of the intact rock for different rock types of projects 5 and 6 are given in Table 9.  
For the numerical validation of Equations (22) and (23), different overburdens were used for different scenarios of \(\sigma_{c}/\sigma_{1}\), within the limitations of this study (5 \(>\sigma_{c}/\sigma_{1}>\) 2). For different \(RQD/J_{n}\) and \(\sigma_{c}/\sigma_{1}\) values, \(SRF\) values were determined using Equations (22) and (23). Based on the calculated tunneling quality index (\(Q\) or \(Qc\)) from Equations (1) and (2), preliminary supports were determined from the support chart of the \(Q\) system; the details are shown in Table 10. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case No.** & \(a\) & \(b\) & \(c\) \\ \hline
1 & 2.44 & 0.804 & 3 \\
2 & 2.123 & 1.023 & 1.5 \\
3 & 2.058 & 1.173 & 0.75 \\
4 & 2.08 & 1.275 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Constants \(a\), \(b\), and \(c\) for Equations (24) and (25) based on Table 4. \begin{table}
\begin{tabular}{c c} \hline \hline
**Joint Orientation** & **Rating of \(C_{0}\)** \\ \hline Favorable & 1 \\ Fair & 1.5 \\ Unfavorable & 2 \\ Very unfavorable & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Adjustment ratings for joint orientation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Project** & \begin{tabular}{c} **Rock** \\ **Type** \\ \end{tabular} & \begin{tabular}{c} **Case** \\ **Number** \\ \end{tabular} & \begin{tabular}{c} **Unit Weight** \\ **(KN/m3)** \\ \end{tabular} & \begin{tabular}{c} **UCS** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Young’s Modulus** \\ **E (GPa)** \\ \end{tabular} & \
\begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} &
\begin{tabular}{c} **mi** \\ \end{tabular} \\ \hline
5 & GN & 1 & 28.6 & 100 & 42 & 0.2 & 23 \\  & UMA & 2 & 31.7 & 80 & 24 & 0.152 & 25 \\ \hline
6 & SS-1 & 3 & 27 & 80 & 40 & 0.2 & 17 \\  & SS-2 & 4 & 27 & 50 & 30 & 0.15 & 17 \\ \hline \hline \end{tabular}  
mi = material constant in the generalized Hoek–Brown criterion [54]. UCS: uniaxial compressive strength.  
\end{table} 
Table 9: Intact rock properties for different rock types of projects 5 and 6. Table 11 shows the rock mass properties that were extrapolated from the intact rock strength properties of Table 9, and the geological strength index (GSI) values through the generalized Hoek-Brown failure criterion [54]. The GSI values that are shown in Table 11 are for the particular cases of Table 10, and are based on the rock mass fabric index approach, which was calculated from \(RQD/J_{n}\) and \(J_{r}/J_{a}\) values [45]. The empirically determined support system was used in numerical modeling to analyze the results before and after the support in terms of the plastic zones in the vicinity of the tunnels and displacement in rock mass. The total displacement at different points along the tunnel perimeter (Figure 10) are shown in Table 12. For the analysis, FLAC (Fast Lagrangian Analysis of Continua) version 7.0 is used. FLAC is an explicit two-dimensional (2D) finite difference program that is suited for sequential excavation modeling.