## 3 Numerical Simulation
### Results of Analyses  
Numerical analyses were performed using FLAC 3D, and the LDP curves were obtained according to the rock mass properties (RMR) and initial stress (Po) conditions (Figure 7). In Figure 7, the x-coordinate represents the distance from the tunnel excavation face, while the y-coordinate denotes the normalized deformation for the final deformation caused by tunnel excavation. As the tunnel excavation advances, the preceding deformation occurs ahead of the face. The ratio of the preceding displacement is defined as the normalized displacement \((u_{r}/u_{r}^{m})\) at the face. The LDP in this study shows that as the RMR decreases, the ratio of the preceding displacement decreases, and the displacement converges gradually behind the excavation face. This LDP trend is similar to that reported by previous studies [3]. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Input Data** & **RMR** & **V** & **IV** & **III** & **II** & **I** \\ \hline Rock mass property & **5** & **10** & **15** & **30** & **50** & **70** & **90** \\ \hline Rock mass property & & & & & & & \\ Elastic modulus, E (MPa) & 50 & 100 & 500 & 1500 & 5000 & 14,000 & 20,000 \\ Cohesion, c (kPa) & 80 & 100 & 150 & 500 & 1000 & 3000 & 5000 \\ Internal friction angle \(\Phi\) (\({}^{\circ}\)) & 30 & 30 & 30 & 33 & 35 & 40 & 45 \\ Initial stress, \(P_{0}\) (MPa) & & & & & & & \\ \(P_{0}\) = 3 MPa & A-1 & B-1 & C-1 & D-1 & E-1 & F-1 & G-1 \\ \(P_{0}\) = 5 MPa & A-2 & B-2 & C-2 & D-2 & E-2 & F-2 & G-2 \\ \(P_{0}\) = 10 MPa & A-3 & B-3 & C-3 & D-3 & E-3 & F-3 & G-3 \\ \(P_{0}\) = 15 MPa & A-4 & B-4 & C-4 & D-4 & E-4 & F-4 & G-4 \\ \(P_{0}\) = 20 MPa & A-5 & B-5 & C-5 & D-5 & E-5 & F-5 & G-5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical modeling data for evaluating the LDP under different rock mass qualities and initial stress conditions. Figure 6: Construction site referenced for the properties of material: (**a**) location map; and (**b**) geological structure of the study area. Figure 7: _Cont._  
Notably, the LDP presented herein was found to differ from that reported by [PERSON] [8], which is indicated by a dotted line in Figure 7. The rock mass deformation ahead of the excavation face was found to be less than that determined by [PERSON] [8]. Furthermore, according to the RMR (expressed as groups A to F), different shapes of the LDP are presented. The LDP proposed by [PERSON] [8] showed a trend similar to that of the proposed LDP only at \(x/R_{T}=5\) behind the excavation face, when the RMR was grade V (groups A, B, and C). This indicates that the LDP formula of [PERSON] [8] represents the measured values for a specific site, and thus, it cannot reflect various rock conditions. It should be noted that according to the LDP analysis results obtained in this study, the ratio of the preceding displacement varies from 0.1 to 0.35 according to the change in rock conditions. The LDP of [PERSON] [8], which showed the best fit for the measured values of tunnel internal displacement reported by [PERSON] et al. [9], simply expresses the ratio of the preceding displacement as approximately 0.3 (Figure 3). This can lead to an error when predicting the ratio of the preceding displacement while neglecting the rock conditions; consequently, a complete tunnel behavior analysis cannot be realized. When the initial stress increased, the LDP tended to move toward the right, and this tendency became more dominant under the condition of poor rock quality (Figure 8). This is attributed to the effect of increasing the plastic area around the excavation surface of the tunnel as the initial stress increases. It means that under high-stress conditions, the rock mass around the tunnel  
Figure 7: LDP results from 3D numerical analysis: (**a**) \(P_{0}=3\) MPa; (**b**) \(P_{0}=5\) MPa; (**c**) \(P_{0}=10\) MPa; (**d**) \(P_{0}=15\) MPa; and (**e**) \(P_{0}=20\) MPa. deforms more as compared with low-stress conditions. Furthermore, poor-quality rocks have low load-carrying capacity as compared to strong rocks.