## 4 Numerical Model  
### Finite Difference Model  
FLAC3D was used to study the grouting reinforcement effect of surrounding rock and its influence on surrounding rock deformation under different grouting parameters (thickness of reinforced layer and water-cement ratio of grout). The construction method, excavation sequence, and support of the form of the tunnel in the numerical simulation were the same as those in the actual project. The influence of surrounding rock deformation in horizontal and vertical directions on tunnel safety was obviously greater than that in the tunnel axis direction, and the influence of tunnel excavation on tunnel axial direction was not considered here. Therefore, a numerical model with a length of 120.00 m and a depth of 60 m was established for analysis. According to the actual situation, the stratum was divided into three parts, 0-4 m was miscellaneous fill, 4-26 m was fully weathered granite, and 26-60 m was fragmented strongly weathered granite, as shown in Figure 10. The supporting structure of the three heading tunnels in the model was built with shell elements, the lining of the main tunnel was established by solid elements, and the middle partition wall and two side partition walls were built with solid elements. The underground pipeline adopted a shell element, the diameter of the circular pipe was 0.4 m, 0.5 m, and 0.6 m respectively, and the size of the square pipe was 0.45 \(\times\) 0.4 m and 1.2 \(\times\) 0.8 m. According to the field conditions, the boundary conditions of the model were determined. The horizontal displacement of the left and right sides of the model was constrained, the  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Water–Cement Ratio** & **0.75** & **0.80** & **0.85** & **0.90** & **0.95** & **1.00** \\ \hline \(\varphi\) (\({}^{\circ}\)) & 26.3 & 25.5 & 24.8 & 24.0 & 23.3 & 22.5 \\ \(c\) (MPa) & 0.42 & 0.38 & 0.33 & 0.29 & 0.24 & 0.20 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Shear strength parameters of grouted surrounding rock with different water-cement ratio.  
Figure 9: Relationship between the UCS of rock and soil before grouting and the shear strength parameter increase rate of rock and soil after grouting when water–cement ratio \(\eta\) = 1: (**a**) Relation between the increase rate of \(\varphi\) and the UCS of rock and soil before grouting; (**b**) relation between the increase rate of \(c\) and the UCS of rock and soil before grouting.  
horizontal and vertical displacement of the bottom surface of the model was constrained, and the top surface of the model was set as a free surface. The surrounding rock was set as an elastoplastic constitutive model and the mechanical behavior conformed to the Mohr-Coulomb failure criterion:  
\[\tau\ =\ c+\sigma\tan\varphi, \tag{5}\]  
where \(\tau\) is the shear strength (MPa), \(\varphi\) is the internal friction angle (\({}^{\circ}\)), \(\sigma\) is the normal stress (MPa), and \(c\) is the cohesive force (MPa). It shows that the shear strength of the rock mass is affected by the normal stress acting on the plane. The main reason for material failure is not the maximum shear stress, but the most dangerous combination of \(\tau-\sigma\) on a certain plane.  
Table 3 shows the specific parameters of the surrounding rock. Considering the symmetry of the model, to simplify the calculation, only the left half of the model was analyzed in the numerical calculation.