## 2 Numerical Methods
### Empirical Equations and Physical Models  
In tunnel construction, air leakage is inevitable at the connectors and ruptures of the ventilation duct due to construction operations or transportation. The commonly used calculation formulas for air leakage of duct are shown as flows [32; 38]:  
Calculation formula of air leakage per hundred meters:  
\[P_{100}=\frac{Q_{f}-Q_{e}}{Q_{f}\times\frac{L}{100}}\times 100\% \tag{1}\]  
Japanese high wooden formula:  
\[Q_{e}=Q_{f}e^{zL} \tag{2}\]  
Voronin formula:  
\[Q_{\mathrm{f}}=\varnothing Q_{\mathrm{e}} \tag{3}\]  
Japanese Qinghan tunnel formula:  
\[Q_{f}=\frac{Q_{e}}{\left(1-\beta\right)^{1/100}} \tag{4}\]  
The equation for the air leakage rate in the last 100 m of a duct:  
\[Q_{f}=\left(\left(1+\frac{1-\sqrt{1-M_{100}}}{\sqrt{1-M_{100}}}\right)\times \left(\frac{L}{100}\right)^{\frac{2}{3}}\right)^{2}\times Q^{e} \tag{5}\]  
where \(P_{100}\) is the air leakage rate, \(Q_{e}\) is the ventilation of the working face (m\({}^{3}\)/min), \(Q_{f}\) is the ventilation of the duct, and \(L\) is the length of the duct (m). In addition, the following equations were used to calculate tunnel leakage. And \(Z=-\frac{1}{500}ln\frac{x}{10}\)\(\varnothing=\left(1+\frac{1}{3}DnK\sqrt{R}\right)^{2}\) is the air-leakage coefficient. \(\beta\) is the leakage coefficient. \(M_{100}\) is the air-leakage coefficient. The air leakage rate for standard duct manufacturing is \(\leq\)4%; however, when purchasing a duct, an air leakage rate of 2% is assumed. When the control standard is more stringent, the air duct leakage rate is 0.5% [39]. Owing to different management and maintenance conditions, air leakage in tunnels varies widely. According to engineering experience, the air leakage rate of an air duct in a 100 m tunnel is generally taken as 1% [39]. Five tunnel models with different mesh densities were established in Figure 1. A mesh independence study was performed for a tunnel of length 100 m by conducting simulations on grids with 274,303 elements (grid A), 137,154 elements (grid B), 68,623 elements (grid C), 46,646 elements (grid D), and 27,435 elements (grid E). The five models are shown in Figure 1a. Because this study mainly focuses on the distributions of air velocity and pollutant concentration of the tunnel, the average air velocity in the tunnel is analyzed and compared. Fresh airflows from the duct outlet towards the tunnel. Under the drive of fluid, the fluid inside the tunnel flows from the working face towards the tunnel exit direction. The flow velocity distribution from the working face to the tunnel exit inside the tunnel is shown in Figure 1b. The difference in air velocity between grids A and B is 2.5%, that between grids B and C is 3.7%, that between grids C and D is 8.9%, and that between grids D and grid E is 12.1%. For the accuracy and efficiency of calculation, grids B and C were selected. In order to select the appropriate leakage size and leakage direction, multiple ventilation duct models of different sizes were established. The calculation results of the model were compared with the theoretical calculation results of Equation (1), as shown in Table 1. The model with the leakage size 0.08 m \(\times\) 0.10 m is ultimately determined as the model for subsequent simulation. Based on the selected leakage size model, the direction of duct leakage holes was studied. As can be seen from the table, the leakage is the largest when the hole is located at the bottom of the duct. Thus, the case with the largest amount of leakage was taken as the research object. Figure 1: Grid independence analysis: (**a**) five models of different mesh densities; (**b**) the results obtained with the five models. According to the selected air duct models in Table 1, the simulated tunnel ventilation system is established as shown in Figure 2. The length of the tunnel is 500 m. Ventilation ducts are equipped with air leakage holes with a vertical downward direction of 0.08 m \(\times\) 0.08 m every 100 m.  
The finite element model is established by ADINA software.