## 3 Theoretical Analysis of Tunnel Portal Stability
### Model Establishment  
In this section, the numerical simulation method is used to study the influences of upper and lower side dimensions of trapezoidal freezing soil mass on portal deformation.  
According to the engineering design data, the outer and inner diameters of the tunnel are 14.5 m and 13.3 m, respectively. The distance between the two tunnels is 21 m, and the buried depth of the tunnel center is 17.68 m. The length of the numerical model is 151.5 m, the width is 90 m, and the height is 69.93 m. The model is divided into five strata: silty clay, muddy silty clay, silt, fine sand, and medium-weathered siltstone, from top to bottom. The thickness for each stratum is 1 m, 7.5 m, 18 m, 20.5 m, and 22.93 m, respectively. The  
Figure 8: Optimized reinforcement scheme diagram of the Wuhu River Crossing end.  
numerical calculation model is shown in Figure 9, and was established using FLAC3D numerical simulation software.  
The boundary condition was set as followed. The Y direction displacement was fixed on the front and rear boundary of the model; the X direction displacement was fixed on the left and right boundary of the model. The upper boundary of the model was free, and the lower boundary was fixed in the X, Y, and Z directions. The Mohr-Coulomb constitutive model was adopted for every soil layer.  
The physical and mechanical parameters of the in situ soil, cement-soil, and freezing cement-soil are shown in Tables 1-3, respectively. These parameters were obtained as follows. Firstly, a series of indoor experiments were conducted, including an unconfined compressive test, a triaxial compression test, and a physical parameters test. Thus, the physical and mechanical parameters of soil, cement-soil, and freezing cement-soil were obtained. However, the indoor test conditions were not exactly the same as those of the field. Therefore, the obtained parameters cannot be used in the numerical model. Thus, the parameters should be corrected. In order to guarantee the model to be in reasonable agreement with the field observations, the parameters were corrected according to the field observations. The corrected parameters are shown in Tables 1-3, and were used in the numerical model.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Soil Horizons** & **Elastic Modulus** & **Poisson’s** & **Volume Weight** & **Cohesion** & **Internal Friction** & **Soil** \\  & _E_/MPa** & **Ratio**\(\
u\) & _\(\gamma\)/(**kN/m3**) & _c_/kPa** & **Angle**\(\varphi\)/(\({}^{\circ}\)) & **Thickness/m** \\ \hline Silty clay & 6.2 & 0.32 & 17.1 & 12 & 6 & 1 \\ Muddy silty clay & 10.0 & 0.30 & 18.4 & 15 & 22 & 7.5 \\ Silt & 20.6 & 0.28 & 19.5 & 2 & 27 & 18 \\ Fine sand & 25.0 & 0.28 & 20.0 & 3 & 30 & 20.5 \\ Medium-weathered & \multirow{2}{*}{\(21\times 10^{3}\)} & \multirow{2}{*}{0.23} & \multirow{2}{*}{21.9} & \multirow{2}{*}{\(15\times 10^{3}\)} & \multirow{2}{*}{38} & \multirow{2}{*}{22.93} \\ siltstone & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physical and mechanical parameters of the in situ soil.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Cement-Reinforced** & **Elastic Modulus** & **Poisson’s** & **Internal Friction** & **Cohesion** & **Tensile Strength** & **Volume Weight** \\
**Soil Layer** & **E/MPa** & **Ratio**\(\
u\) & **Angle**\(\varphi\)/(\({}^{\circ}\)) & _c_/kPa** & _\(\varphi\)/MPa_ & _\(\gamma\)/(**d**kN/m3**) \\ \hline Silty clay & 50.4 & 0.3 & 26.2 & 120 & 0.028 & 18.0 \\ Muddy silty clay & 83.16 & 0.28 & 40.1 & 150 & 0.074 & 19.0 \\ Sand & 165.2 & 0.25 & 50.1 & 300 & 0.085 & 20.0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Physical and mechanical parameters of the cement-soil.  
Figure 9: Three-dimensional numerical calculation model.