## 4 Numerical Modeling
The application of such systems is not universal, due to the case histories on which each system is based. However, a few systems, including RMR, \(Q\), and _RMi_, have gotten international attention. Compared to the other classification systems, these systems have still been widely used in tunnel design since their development, because they are continuously upgraded in terms of characterizations or support, or both [3]. The evaluations of the three systems indicate that the \(Q\) and _RMi_ systems incorporate stress-related factors, making their application more suitable for tunnel support design, where stress is the major factor governing the ground behavior. However, these systems do not cover tunnels designed in jointed rock mass and high in situ stress fields. The two systems are limited by the case histories on which they are based. In this study, the application of these two systems are empirically extended from the relevant tunnel case records. In the literature, the _SRF_ factor is either a function of \(RQD/J_{n}\) or \(\sigma_{c}/\sigma_{1}\). In this study, the empirical equation that was developed for the characterization of _SRF_ is a function of both \(RQD/J_{n}\) and \(\sigma_{c}/\sigma_{1}\), along with \(\sigma_{c}\). Figure 6a shows that \(SRF_{Q}\) (the _SRF_ value that was developed for the original \(Q\)-system equation, Equation (1)) significantly increases with relative block size. A maximum change in \(RQD/J_{n}\) of less than seven results in a variation of 5.4 for \(SRF_{Q}\), while for a maximum change in \(RQD/J_{n}\) greater than seven, \(SRF_{Q}\) varies by approximately 15.05, which is larger by a factor of about 2.75. A total \(SRF_{Q}\) change of 20.82 can be observed with \(RQD/J_{n}\) for a given value of \(\sigma_{c}/\sigma_{1}\). With an increase in \(RQD/J_{n}\), the difference of \(SRF_{Q}\) for different values of \(\sigma_{c}/\sigma_{1}\) is comparatively low for a given value of \(RQD/J_{n}\). An opposite trend can be seen in Figure 6b. For \(RQD/J_{n}\), the rate of decrease of \(SRF_{Q}\) with respect to low values of \(\sigma_{c}/\sigma_{1}\) is large compared to high values of \(\sigma_{c}/\sigma_{1}\). For a \(\sigma_{c}/\sigma_{1}\) value below 3.5, the maximum fluctuation in \(\sigma_{c}/\sigma_{1}\) for a given value of \(RQD/J_{n}\) results in a variation of the \(SRF_{Q}\) value of about 2.49. Fluctuation in \(\sigma_{c}/\sigma_{1}\) by an equal amount in its upper limit (above 3.5) can change \(SRF_{Q}\) by 1.35 approximately, which is about 1.84 times smaller. The maximum change in \(SRF_{Q}\) is 4.09 with \(\sigma_{c}/\sigma_{1}\) for any given value of \(RQD/J_{n}\). The available data indicate that the maximum value of the relative block size is 12. Figure 7 shows that the variation of \(SRF_{QC}\) with \(RQD/J_{n}\) also depends on \(\sigma_{c}\) as well as \(\sigma_{c}/\sigma_{1}\). For the same value of \(\sigma_{c}/\sigma_{1}\), high principal stress (\(\sigma_{1}\)) requires rock with a high \(\sigma_{c}\) value. The high level of stress means that more support is required, and hence, a high value of \(SRF_{QC}\). The maximum variation of \(SRF_{QC}\) with \(RQD/J_{n}\) is 18.5 for \(\sigma_{c}=100\) MPa, with \(\sigma_{c}/\sigma_{1}\) ranging from three to four (Figure 7b), and the minimum variation is 7.38 for \(\sigma_{c}=37.5\) MPa (Figure 7c). During the design stage of a tunnel, when inadequate information exists, then, with knowledge of \(\sigma_{c}\) (which is the most basic parameter to be determined) and the location of the tunnel, Figure 7 can be used to approximate values of \(SRF_{QC}\) for different values of \(RQD/J_{n}\). In the construction phase of a tunnel, when most of the information is in hand at the excavation face, Equations (22) and (23) can be used to calculate \(SRF_{QC}\). The maximum rating of \(\sigma_{c}\) is 100, as shown in Table 3; therefore, the calculated \(SRF_{Q}\) and \(SRF_{QC}\) have the same value for tunnel sections where \(\sigma_{c}=100\) MPa.