## 4 Numerical Analysis of a High Rock Temperature Tunnel
### Numerical Model  
In this paper, the Jiwoxiga Tunnel is taken as the calculation model. The longitudinal profile of the tunnel is shown in Figure 15. The governing equation and calculation method in Section 3.1 above are adopted. AB and CD are adiabatic boundaries, BD is a heat flux boundary, and AC is the Robin (third-type) boundary condition.  
The boundary AC is primarily determined by air temperature (\(T_{amb}\)) and the convective heat transfer coefficient (\(h_{c}\)) between the air and the ground surface. In the tunnel area,  
Figure 14: Location of the Jiwoxiga tunnel.  
Figure 15: The longitudinal profile of the Jiwoxiga tunnel.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number** & **Depth** & **Road Shoulder Temperature** & **Bottom Hole Temperature** \\  & **(m)** & **(°C)** & **(°C)** \\ \hline BD3Z-85 & 76.5 & 44.7 & 45 \\ JWZ-3 & 105.2 & 47.9 & 48 \\ BD3Z-88 & 80.3 & 36.1 & 45.6 \\ BD3Z-89 & 72.8 & 36.1 & 37.5 \\ JWZ-1 & 120 & 49.5 & 65.4 \\ BD3Z-90 & 73.6 & 45.3–48 & 57.2 \\ BD3Z-91 & 70.8 & 39.6–41.6 & 47.5 \\ BD3Z-93 & 88.5 & 24.1 & 25 \\ BD3Z-94 & 82 & 21 & 21.6 \\ BD3Z-95 & 88 & 18 & 18.2 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Borehole depth and temperature data.  
the altitude is high, and the terrain is complicated, so it is difficult to obtain long-term temperature data. Therefore, meteorological data from the vicinity of the tunnel site are used. According to the meteorological data of the tunnel site, the mean annual air temperature is approximately 6.9 \({}^{\circ}\)C, and the variation in the monthly mean temperature is shown in Figure 16. The air temperature can be fitted by a sine function, as shown in Equation (4).  
\[T_{amb}=6.93-9 sin\left(\frac{2\pi}{8760}t_{h}+\frac{\pi}{3}\right)+\frac{2.6 \times t_{h}}{365\times 24\times 50} \tag{4}\]  
where \(T_{amb}\) is the air temperature (unit: \({}^{\circ}\)C) and \(t_{h}\) is the time (unit: hours). In Equation (4), the impact of climate warming is considered, with a temperature increase of 2.6 \({}^{\circ}\)C in 50 years [33; 34; 35]. According to meteorological data, the average annual wind speed in the tunnel area is 2.23 m/s. By substituting the wind speed into Equation (3), we can obtain the heat transfer coefficient between the ground surface and ambient air.  
Table 4 lists the calculating parameters of the tunnel. Generally, the temperature of the thermostatic layer is chosen as the initial temperature of the rock mass [36]. The field test data show that the layer of rock with a constant temperature in the Jiwoxiga tunnel is approximately 40 m thick. The temperature in this layer is 40 \({}^{\circ}\)C. Based on the existing borehole temperature data, the heat flux was calculated according to Ref. [37], and then the average value was taken. Therefore, the BD heat flux at the bottom boundary was 0.92 W/m\({}^{2}\). According to Figure 15, a 1:1 numerical model is established. The finite element meshes are shown in Figure 17. The initial temperature distribution of the rock mass surrounding the tunnel was determined by a separate 100-year simulation of natural conditions [33].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Physical** & **Density** & **Thermal Conductivity** & **Isobaric Specific Heat** \\
**Parameters** & **(kg/m\({}^{3}\))** & **(W/(m\({}^{\circ}\)C))** & **Capacity J/(kg\({}^{\circ}\)C)** \\ \hline Values & 2920 & 2.07 & 1156 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The thermal physical parameters of the rock surrounding the tunnel.  
Figure 16: Air temperature outside the tunnel.