## 4 Numerical Model and Construction Plan  
There is a pedestrian passage that was manually excavated in the 1980s in the planned tunnel site. The bedrock is bare, with a sectional size of 1.5 m (width) \(\times\) 2.2 m (height). A new horseshoe tunnel with a length of 100 m and a sectional size of 5.88 m (width) \(\times\) 6.35 m (height) is planned in the area. Figure 4 illustrates its sectional diagram. The exploration-results-based preliminary construction plan shows that the entrance location of the new tunnel will coincide with the existing tunnel location, and the axes of two will separate gradually. Moreover, the new tunnel section is adjacent to the slope and overlying road, so excavation expansion based on existing tunnels will certainly be detrimental to the slope and the road. These outcomes, combined with the results of the surrounding rock grade classification, clearly shows that the tunnel trunk section is distributed primarily with grade III rocks that have a rather high strength, quite intact rock masses, and are medium-thick layered with a good stability. Although the arch and side wall are fundamentally stable without support, excessive blasting vibration easily leads to collapse. To prevent the collapse of the existing tunnel caused by the blast working of new tunnel, backfilling of the existing tunnel is recommended prior to the excavation of the new tunnel. Meanwhile, smooth blasting and pre-split blasting technologies are adopted during the construction to achieve small-charge blasting with a vibration velocity not exceeding 1.5 cm/s to minimize the disturbance to surrounding rocks and to ensure the safety and stability of the tunnel wall. Since the new tunnel is close to the surfaces of the slope and ground, conducting a systematic study on the spatial positions of the new and existing tunnels is necessary on the premise of considering the in situ construction conditions comprehensively. In principle, the shallower the overburden of the new tunnel, the smaller its distance from the free face of the slope, and accordingly, the lower the construction cost. However, huge potential risks will be faced regarding engineering safety. Conversely, if the engineering safety is guaranteed, the construction cost will increase substantially. Furthermore, the surface load is also a crucial factor since the overlying road cannot be closed during construction. Meanwhile, whether concrete backfilling for the existing tunnel is implemented or not will also affect the displacement field distribution of the surrounding rocks after the tunnel excavation. Thus, whether backfilling of the existing tunnel is implemented is also included in the numerical study, and the backfill material is the same as the lining concrete material. Hence, this study needed to consider nine relative spatial positions of the new tunnel, three magnitudes of the surface load, and two backfilling conditions of the existing tunnel, which constitute a total of 54 models. Figure 5 illustrates the tunnel relationships under nine different spatial positions. Figure 4: Sectional size drawing of the tunnel. As shown in Figure 5, the overburden, height, and width of the new tunnel was labeled C, H, and W, respectively, while the distance from the tunnel to the vertical line of slope crest was labeled D. Accordingly, this study considered a total of nine spatial position combinations when C/H = 0.764, C/H = 0.606, and C/H = 0.449, as well as when D/W = 0.378, D/W = 0.251, and D/W = 0.124. It is noteworthy that the overburden of the existing tunnel was 8.95 m, and its distance from the adjacent slope was 4.23 m.  
The numerical model shown in Figure 6a had a 3D size of 100.0 m \(\times\) 54.2 m \(\times\) 52.7 m (L \(\times\) W \(\times\) H), where the width of the road overlying the new tunnel was 10 m, the vertical height of adjacent slope was 38.1 m, and the slope angle was approximately 82\({}^{\circ}\). An excavation simulation is performed on the new tunnel in the negative direction along the x-axis.