## 3 Field Test of the Dege Tunnel
The three-dimensional temperature field coupling model for the Dege tunnel is proposed based on the measured ground temperature and the ground temperature gradient. The numerical model is divided into a shallow-buried section and a deep-buried section, which is the same as the actual tunnel. According to field measurements, the ground temperature in the tunnel is 11.5 \({}^{\circ}\)C, and the ground temperature gradient is 6% for the shallow-buried section. For the deep-buried section, the ground temperature and ground temperature gradient are 15 \({}^{\circ}\)C and 2%, respectively. The length, width and height of the proposed model are 2943 m, 70 m and 53 m, respectively, as shown in Figure 7. The left and right boundaries of the model are far enough from the tunnel, and the heat fluxes on the boundaries have only a minor effect on the tunnel temperature fields. Thus, the left and right boundary conditions of the model are assumed as adiabatic [11], while the upper and lower boundaries are assumed as heat flux boundaries. The heat flux is calculated according to the thermal conductivity of surrounding rock and the geothermal gradient. The heat flux of the lower boundary is 0.114 w/m\({}^{2}\) for the shallow-buried section, while 0.038 w/m\({}^{2}\) for the deep-buried section. On the other hand, the heat flux of the upper boundary is \(-\)0.114 w/m\({}^{2}\) for the shallow-buried section while \(-\)0.038 w/m\({}^{2}\) for the deep-buried section. The front and back boundaries are assumed as convective boundaries with the convection coefficient of 15 w/m\({}^{2}\)\(\cdot\)k [23]. In the proposed model, 340,676 hexahedral elements are used to represent the air in the tunnel, the surrounding rock and the tunnel lining. The \(k\)-\(\epsilon\) equations are applied to the turbulence model, where the constants \(C_{\epsilon 1}\), \(C_{\epsilon 2}\), \(C_{\mu}\), \(\alpha_{k}\) and \(\sigma_{\epsilon}\) are taken as 1.44, 1.92, 0.09, 1.0 and 1.3, respectively. The air is assumed to be incompressible, and its density is obtained based on the altitude and the average annual temperature of the tunnel region with a value of 0.86 kg/m\({}^{3}\)[24]. The specific heat capacity, thermal conductivity and the viscosity of the air are taken as 1005 J/kg\({}^{\circ}\)C, 0.02 w/m\({}^{\circ}\)C, and 1.72 \(\times\) 10\({}^{-5}\) kg/m\(\cdot\)s, respectively. According to the experimental measurements and mathematical calculations, the thermal physical parameters of the surrounding rock, the concrete material and the heat-insulating material (FLOLIC) are employed in the numerical study as listed in Table 1.  
Figure 7: Employed three-dimensional model for the Dege tunnel. According to the meteorological data of the Dege tunnel, the daily temperature variation at the tunnel entrance (near Chengdu) can be fitted to:  
\[T=T^{\prime}-11.85\times\sin\biggl{(}\frac{\pi t}{12}+\pi/4\biggr{)}\]  
where \(T^{\prime}\) is the daily mean temperature (\({}^{\circ}\)C) and \(t\) is time (hour). The daily temperature variation at the tunnel exit (near Lasa) can be fitted to:  
\[T=T^{\prime}-10.20\times\sin\biggl{(}\frac{\pi t}{12}+\pi/4\biggr{)}\]  
The daily mean temperature variation at the tunnel entrance (near Chengdu) in one year can be fitted to:  
\[T^{\prime}=-0.85-10.35\times\sin\biggl{(}\frac{\pi t}{4320}+\pi/2\biggr{)}\]  
The daily mean temperature variation at the tunnel exit (near Lasa) in one year can be fitted to:  
Figure 3.0