## 3 Theoretical Analysis of Tunnel Portal Stability
### Model Establishment
#### 4.2.2. Portal Deformation of the Wuhu River-Crossing Tunnel  
The portal deformation of the Wuhu River-Crossing Tunnel with the real end reinforcement method was simulated. Figure 11 shows the x and z direction displacement contour of the portal with the real reinforcement method.  
From Figure 11, it can be seen that the x-direction displacement is the largest on the left and right sides of the portal, which are \(-1.74\) mm and \(1.34\) mm, respectively. The z-direction displacement is the largest at the top and bottom of the tunnel, which are \(1.49\) mm and \(-1.56\) mm, respectively. Compared with the displacement of the unreinforced portal, the z-direction displacement of the reinforced portal decreases significantly.  
Figure 12 is the y-direction displacement contour on the longitudinal section of the tunnel with the real end reinforcement method. It can be seen from the figure that when the portal is broken, the soil behind the portal shows an outward movement trend. The y-direction displacement at the center of the portal is the largest, at \(-9.91\) mm.  
Figure 13 shows field observation displacement and numerical displacement at the portal monitoring point. From the figure, it can be seen that the numerical results at the portal are consistent with the field observation results, and the error does not exceed \(10\%\). This demonstrates that the simulated model is in reasonable agreement with the field observation.  
Influences of Upper and Lower Side Dimensions of Trapezoidal Freezing Soil Mass on the Portal Deformation  
From the theoretical analysis in Section 2, it can be seen that the trapezoidal freezing soil mass has a better reinforcement effect than the rectangle freezing soil mass. To further evaluate the performance of trapezoidal freezing soil mass, the influences of trapezoidal freezing soil mass dimensions on the tunnel portal deformation were studied in this section. Firstly, the upper side length of the trapezoidal freezing soil mass \(L_{1}\) was \(3\) m, while the  
Figure 10: The z-direction displacement contour of the unreinforced tunnel end (unit: m).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Freezing** & **Elastic Modulus** & **Poisson’s** & **Internal Friction** & **Cohesion** & **Tensile Strength** & **Volume Weight** \\
**Cement-Soil** & **_E/MPa_** & **Ratio \(\
u\)** & **Angle \(\varphi\)/(\({}^{\circ}\))** & _c/k_**Pa** & \(\sigma\)/MPa** & \(\gamma\)/(kN/m\({}^{3}\)) \\ \hline Silty clay & 170 & 0.3 & 40.25 & 300 & 0.096 & 20.0 \\ Muddy silty clay & 299.02 & 0.28 & 53.2 & 400 & 0.265 & 21.0 \\ Sand & 684.46 & 0.25 & 61.8 & 700 & 0.352 & 22.0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Physical and mechanical parameters of the freezing cement-soil.  
lower side length \(L_{2}\) was 3 m, 4 m, and 5 m, respectively. The influences of the lower side length of the trapezoidal freezing soil mass on the tunnel portal deformation were analyzed. Then, the lower side length \(L_{2}\) was 3 m, while the upper side length \(L_{1}\) was 1 m, 2 m, and 3 m, respectively. The influences of the upper side length on portal deformation were studied. Finally, the deformation of the tunnel portal, with a longitudinal length of 3 m of rectangular freezing soil mass, was simulated.  
The triaxial mixing pile and rectangle freezing method were used in the Wuhu River-Crossing Tunnel. The authors have carried out many studies on the triaxial mixing piles reinforcing the tunnel end alone. The results show that if the triaxial mixing pile's longitudinal length is 5 m, the maximum tunnel portal deformation is approximately 25 mm. The deformation value is above the allowable value of 15 mm [24; 25]. To highlight the effect of the freezing soil mass, the triaxial mixing pile's longitudinal length was 5 m in this section.  
Figure 11: The displacement contour of the portal with the real reinforcement method (unit: m).  
Figure 12: The y-direction displacement contour on the longitudinal section of the tunnel with the real reinforcement method (unit: m).