## 3 Methods
### Analytical Methods
#### 3.1.2 Analytical Methods Considering Water Table Drawdown  
Considering a tunnel in drained conditions with lowered water table, as shown in Figure 7:  
[PERSON] and [PERSON] (2018) adopted a lowered water level to replace the initial water level in the [PERSON] solution (2018), so an analytical equation for the calculation of groundwater inflow into tunnels in the drained condition could be used. However, the lowered water level in the equation was only obtained by numerical calculation, which may greatly hinder the application of the formula. Subsequently, [PERSON] et al. (2018) proposed a semi-analytical formula for the virtual water level, which is used to replace the lowered water level in the adjusted mirror image method, so the analytical equation was simplified and described as follows:  
\[Q=2\pi K\frac{0.3(r/h)^{-0.014r-0.22}h}{\ln(2h/r)} \tag{5}\]  
\[s=\left\{\begin{array}{ll}h-e^{(1.83r-27.31)r/h}h,\ r/h<0.1\\ h,r/h\geq 0.1\end{array}\right. \tag{6}\]  
[PERSON] et al. (2018) considered a drained tunnel in an open aquifer with a free water table and obtained two new equations for the water inflow and the water table drawdown, based on the integral method and the appropriate Green function. The equations are described as follows:  
\[Q=2\pi K\frac{h}{\ln\left[\frac{2H}{\pi r}sh\frac{\pi d}{H}\sqrt{1+\frac{sh^{2 }\left(\pi d/H\right)}{\sin^{2}\left(\pi h/H\right)}}\right]} \tag{7}\]  
\[s=\frac{Q}{2\pi K}\ln\frac{\sin^{2}(\pi h/2H)+sh^{2}[\pi(x+d)/2H]}{\sin^{2}( \pi h/2H)+sh^{2}[\pi(x-d)/2H]} \tag{8}\]  
[PERSON] et al. (2018) proposed semi-analytical formulas for the calculation of groundwater inflow rate and water table drawdown of shallow and deep tunnels in drained conditions, based on the trench model and the image method. The formulas are shown as follows:  
Figure 7: Circular tunnel with lowered water table in drained conditions.  
\[\begin{cases}Q=\frac{K}{L-r}\big{(}2 hd+h^{2}\big{)}&\text{(shallow tunnel)}\\ Q=2\pi K\frac{b^{\prime}}{\ln(2h^{\prime}/r)}&\text{(deep tunnel)}\end{cases} \tag{9}\]  
\[s/(h-r)=\begin{cases}-1.3468e^{-40.404r/h}+1.1106,&r/h<0.062\\ 0.449e^{3.0583r/h}+0.4573,&r/h\geq 0.062\end{cases} \tag{10}\]  
where \(d=(ar/h+b)\cdot D\) and \(h^{\prime}=(cr/h+e)\cdot h\). Furthermore, \(a=0.0193r^{2}-0.3223r+0.3654\), \(b=-0.0017r^{2}+0.0311r+0.3538\), \(c=-0.1371r^{2}+2.5061r-18.2290\), and \(e=0.0060r^{2}-0.1124r+1.3456\), referred to by [PERSON] et al. [3].  
Actually, the underground powerhouse of JPSPS is relatively regular, which can be equivalent to a well with a large radius and can be solved using the well theory. The big well method can be used to calculate the water inflow based on [PERSON]'s equation [12; 35], so the equations can be expressed for the unconfined aquifer and confined-unconfined aquifer, respectively:  
\[\begin{cases}Q=1.366K\frac{(2h-s)s}{\ln(R/r)}&\text{(Unconfined well)}\\ Q=1.366K\frac{\big{[}2 hH-H^{2}-(h-s)^{2}\big{]}}{\ln(R/r)}&\text{(Confined- unconfined well)}\end{cases} \tag{11}\]  
where \(R\) is the influence radius and can be obtained using [PERSON]'s formula [35]:  
\[R=2.45\sqrt{hKt/\mu} \tag{12}\]  
where \(\mu\) is the specific yield, \(t\) is the time required for the groundwater level to stabilize.