## 3 Middle Rock Pillar Classification Method
### Numerical Model Establishment  
The influence range of the excavation stress is normally 3-5 times of the tunnel radius. A numerical model with a length of 200 m in the X direction, a width of 110 m in the Z direction, and a height of 160 m in the Y direction was established in Abaqus software. The tunnels were located in the position with a depth of 100 m and distance of 60 m to  
Figure 3: Research section diagram. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**State** & **Dry** & **Slightly Damp** & **Occasional Water** & **Frequent Water** & **Massive Water** \\  & & & **Seepage** & **Seepage** & **Seepage** \\ \hline Water yield (L/min \(\times\) 10 m) & 0 & \(\leq\)10 & 10–25 & 25–125 & \(>\)125 \\ Hydraulic pressure (MPa) & 0 & 0 & \(\leq\)0.1 & 0.1–0.25 & \(>\)0.25 \\ Score \(T_{3}\) & 0 & \(-\)2 & \(-\)4\(\sim\)\(-\)2 & \(-\)8\(\sim\)\(-\)4 & \(-\)16 \\ \hline \hline \end{tabular} Note: The value range “-” includes the value on the right, but excludes the value on the left, that is, “\(<\)\(T_{3}\)\(\leq\)”.  
\end{table} 
Table 8: Score of middle rock pillar permeability. the bottom. The main line tunnel section was set parallel to the axis Z, and the ramp tunnel section was extended horizontally with an angle of 8.28\({}^{\circ}\) to Z axis. The two tunnels transitioned gradually from small clear distances to separate tunnels. The length of the research section was 110 m, and the clear distance between the two tunnels ranged from 1.6 to 17.6 m. In the numerical model, the eight-node reduced integration solid element (C3 DBR) was used to simulate the strata, tunnel, initial lining, secondary lining, and temporary support. The thickness of the initial lining solid unit was 20 cm, the temporary support was mainly I-steel, the thickness of the solid elements was 18 cm, and the thickness of the secondary lining solid elements was 40 cm. A hexahedral mesh was used to divide the model, and local mesh encryption was performed for the tunnel excavation section. The numerical model was subdivided into 117,355 elements (Figure 4a). The numerical models of primary lining, secondary lining, and temporary support are shown in Figure 4b. The front, back, two sides, and bottom surfaces of the model were constrained by normal displacement, and the front excavation face and top surface were not constrained. According to a survey report, the rock mass of the small clear-distance tunnel section was Grade III. The physical and mechanical parameters of the rock mass and lining in the numerical simulation were determined by referring to the Specifications for Design of Highway Tunnels [34], as listed in Table 9. The mechanical behavior of the rock mass conformed to the Mohr-Coulomb failure criterion, and the lining structure was an elastomer. To comprehensively analyze the quantification of the width-span ratio index of the middle rock pillar, four excavation schemes, namely, the full section, step, center diaphragm (CD) method, and double wall heading method were simulated in the numerical simulation [35; 36; 37; 38]. In each excavation scheme, the main line tunnel was advance excavated with a footage of 5 m, and the staggering distance between the main line and the ramp tunnels was 15 m. In the simulation process, the geostress was balanced first, and then the rock mass excavation and lining were simulated step by step. Static analysis was adopted throughout the whole numerical simulation process. Figure 4: Numerical calculation model: (**a**) overall numerical model; (**b**) tunnel lining. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Materials** & **Density (kg/m\({}^{3}\))** & **Elastic Modulus (GPa)** & **Poisson’s Ratio** & **Cohesion (MPa)** & **Friction Angle (\({}^{\circ}\))** \\ \hline Surrounding rock & 2200–2300 & 6–10 & 0.25–0.3 & 2.0 & 39–50 \\ Primary lining support & 2500 & 30 & 0.20 & / & / \\ Secondary lining support & 2500 & 32.5 & 0.15 & / & / \\ Temporary support & 7900 & 200 & 0.30 & / & / \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Material parameters.