# Simplified Calculation Method for Active Anti-Floating of Elliptical Basements by Relief Wells
## 2 Simplified Calculation of Multi-Well System for Elliptical Pit
### Inside-Outside seepage Field Solved in Series  
After obtaining the analytical solution of the inner seepage field, the resistance coefficient method can be used to solve the seepage field of the inner and outer sides of the pit in series using the fsolve module of the scipy library in Python. The resistance coefficient method has three resistance coefficients: \(\xi_{1},\xi_{2},\xi_{3}\), can be obtained through the solution of these three coefficients of the total flow \(Q\), so as to calculate the seepage field distribution of the pit under the system of multiple wells, for the underground structure of the interception of the IDPR anti-floating system to provide a basis for design. Depending on the stratigraphic environment in which the pit is located, the calculations can be divided into the pressure-bearing mode and the submerged mode. As shown in Figure 4, the outer head is \(H_{d}\), the inner head is \(H_{R}\), the thickness of the cut-off wall is \(d\), the permeability coefficient is \(K_{w}\), the head at the edge of the affected area is \(H_{0}\), the permeability coefficient of the weakly permeable layer at the bottom of the waterproof curtain is \(K_{1}\), the thickness is \(D\), the thickness of the outer side is \(T_{1}\), the thickness of the inner side is \(T_{2}\), and the water level of the relief well is \(h_{w}\). In Figure 4a, \(K_{0}\) and \(T_{0}\) are the permeability coefficient and  
Figure 3: seepage field inside elliptical pit cut-off wall. thickness of the pressure-bearing highly permeable layer on the outside of the cut-off wall, and \(K\) and \(T\) are the permeability coefficient and thickness of the pressure-bearing highly permeable layer on the inside of the cut-off wall. In Figure 4b, \(K_{0}\) and \(T_{0}\) are the permeability coefficients and thicknesses of the submerged layer on the outside of the cut-off wall, and \(K\) and \(T\) are the permeability coefficients and thicknesses of the hydrophobic layer on the inside of the cut-off wall. The specific calculation procedure can be found in the literature [18], only the main formulae are given here. In pressure-bearing mode, the main aquifer outside the pit cut-off wall is the pressurized soil layer. Among them, the thickness of the relatively highly permeable layer is the thickness of the pressurized highly permeable layer on the outside of the cut-off wall, i.e., \(T_{q}=T_{0}\), the permeability coefficient is \(K_{0}\), and \(\xi_{1}\) is the external seepage resistance coefficient, at which time there is the following [23]:  
\[\left. \begin{array}{l}H_{0}-H_{d}=\frac{Q}{K_{0}}\xi_{1}\\ \xi_{1}=\frac{1}{2\pi T_{0}}\ln\frac{R_{0}}{R_{d}}\end{array} \right\} \tag{14}\]  
If in submerged mode, then  
\[\left. \begin{array}{l}T_{0}=\left(h_{0}-h_{d}\right)/2\\ h_{0}=H_{0}-z_{0}\\ h_{d}=H_{d}-z_{0}\end{array} \right\} \tag{15}\]  
depending on the main aquifer outside the pit cut-off wall, the total flow rate of the seepage field outside the pit cut-off wall can be obtained from the given site head by selecting the above formulae for calculating the seepage field outside the pit cut-off wall in pressure-bearing or submerged mode. Figure 4: Schematic of pit profiles under the IDPR scheme: (**a**) Pressure-bearing mode; (**b**) submerged mode. “Reproduced with permission from [PERSON] et al. [18], Simplified calculation and design method of multi-well system for anti-uplifting based on intercepting and discharging water; published by Chinese Journal of Geotechnical Engineering, 2021”. The next step is to calculate the seepage field at the cut-off wall. In pressure-bearing mode, the thickness of the relatively highly permeable layer at this point is \(T_{q}=(T_{0}/2+T/2)\) and the permeable part of the wall is calculated as follows:  
\[ \begin{array}{c}Q_{1}=\frac{(H_{d}-H_{R})2\pi RK_{w}}{\xi_{a}}\\ \xi_{a}=\frac{d}{T_{0}/2+T/2}\end{array} \Bigg{\}} \tag{16}\]  
The calculations for the leached part of the wall base are as follows:  
\[