## 3 Simulation of Water Intrusion into Subway Station Operating Conditions
### Trend of Mass Flow Rate Changes  
In this paper, the mass flow rate at the station entrance and the tunnel exit, and their difference (D-value), are statistically plotted over time when floodwater intrudes into the station, as illustrated in Figure 9a-c. A comparison of the three graphs in Figure 9 reveals that the three working conditions exhibit comparable trends. From the mass flow rate at the tunnel exit, it can be observed that the floodwater commences its flow into the tunnel following an interval of time during which it has been in the station. The initial mass flow rate of the floodwater is relatively low. The mass flow rate into the tunnel demonstrates an increase over time, with a initially rapid growth rate that subsequently decelerates. Concurrently, the central portion of the curve exhibits a linear correlation. The D-value curve persists in its decline with the passage of time, and its overall trajectory is antithetical to that of the mass flow rate curve at the tunnel exit. A comparison of the three operating conditions reveals that, with a constant water inflow, the mass flow rate from the subway station to the tunnel exhibits a steady increase over time, ultimately reaching a fixed value equivalent to the inflow at the station entrance. Therefore, the total accumulated water within the station remains relatively constant. Moreover, it was determined that the lower the flood elevation of the intruding subway station, the longer it takes for the flood volume to stabilise within the station and the more gradual the mass flow rate-time curve. Figure 8: Velocity vector diagrams for 30 s under different working conditions. Figure 9: _Cont._The results of the simulation were found to be linear. In order to guarantee the accuracy of the aforementioned numerical results, a theoretical derivation was conducted for verification purposes. As previously stated, the relationship between the inflow of water in the subsurface space and the depth of water in front of the surface inlet can be expressed as follows:  
\[Q=ah_{w}^{b} \tag{4}\]  
where, \(Q\) is the single width flow rate at the entrance of the underground space (m\({}^{2}\)/s); \(h_{w}\) is the depth of water on the ground at the entrance of the underground space (m); and \(a\), \(b\) are obtained from Equation (3). Owing to  
\[h_{w}=v_{t}t-h_{e} \tag{5}\]  
where \(v_{t}\) is the road surface water rising speed (m/s); \(t\) is the length of time water has been standing outside the station(s); and \(h_{e}\) is the step height in front of the entrance to the underground space (m), we find that  
\[Q=a(v_{t}t-h_{e})^{b} \tag{6}\]  
Figure 9: Changes in mass flow rate of subway station invaded by floods under different operating conditions. Integrating the flow rate in Equation (6) over the length of time, the total amount of floodwater inflow to the subsurface space at a given \(T\)-second moment is calculated and can be expressed as  
\[V_{T}=\frac{aB}{v_{t}(b+1)}(v_{t}T-h_{e})^{b+1} \tag{7}\]  
where \(B\) is the width of the entrance staircase to the underground space (m). Then the depth of water \(h_{T}\) (m) in the underground space at \(T\) seconds is  
\[h_{T}=\frac{V_{T}-h_{T}S}{A_{s1}} \tag{8}\]  
where \(A_{s1}\) is the flooded floor area in the underground space (m\({}^{2}\)), due to the faster rate of flood water intrusion as well as more subway entrances. Therefore, it can be assumed that the flooded area in the underground space \(A_{s1}\) is equal to the area of the underground space \(A_{s}\), and \(S\) is the area that the water flowed through per unit of time at the tunnel exit (m), so  
\[h_{T}=\frac{aB(v_{t}T-h_{e})^{b+1}}{v_{t}(b+1)(A_{s1}+S)} \tag{9}\]  
Therefore, the rate of rise of the groundwater level \(u_{t}\) (m/s) is  
\[u_{t}=\frac{aB(v_{t}T-h_{e})^{b}}{(A_{s}+S)} \tag{10}\]  
In the numerical simulation setup, the depth of the water in front of the surface portal is maintained at a constant value.