# Simplified Calculation Method for Active Anti-Floating of Elliptical Basements by Relief Wells
## 2 Simplified Calculation of Multi-Well System for Elliptical Pit
### Ellipse to Circle Mapping  
Let the length of the semi-long axis of the ellipse be \({}_{a}\), the length of the semi-short axis be \(b\), and \((z=x+yi)\) be the coordinates of the points in the plane inside the ellipse. The conformal mapping transformation from ellipse to unit circle is then as follows [34]:  
\[\Xi=\sqrt{k}sn(\frac{2K}{\pi}\text{arcsin}\frac{z}{\sqrt{a^{2}-b^{2}}}) \tag{7}\]  
The transformation of an ellipse into a unit circle by the above conformal mapping is shown in Figure 2:  
where the parameter \(k\) in the elliptic function and the elliptic integral \(K\) of the first type are calculated as follows [35]:  
\[k=\left(\frac{2q^{\frac{1}{4}}+2q^{\frac{5}{4}}+2q^{\frac{58}{4}}+\ldots}{1+2q +2q^{4}+2q^{9}+\cdots}\right)^{2} \tag{8}\]  
\[K=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^{2}\sin^{2}t}}=F(k,\frac{\pi}{2}) \tag{9}\]  
\[q=\left(\frac{(a-b)}{(a+b)}\right)^{2} \tag{10}\]  
For elliptic functions, the calculation of each function is shown below:  
\[sn\ u=\frac{m^{\frac{1}{4}}\theta_{1}(z_{0},q)}{\theta_{4}(z_{0},q)} \tag{11}\]  
where \(u=2 Kz_{0}/\pi\); \(m=k^{2}\), and  
\[\theta_{1}(z_{0},q)=2\sum_{n=0}^{\infty}{(-1)^{n}q^{(n+\frac{1}{2})^{2}}\sin \biggl{[}\biggl{(}n+\frac{1}{2}\biggr{)}2z_{0}\biggr{]}} \tag{12}\]  
Figure 2: Schematic diagram of conformal mapping of ellipsoids.  
\[\theta_{4}(z_{0},q)=1+2\sum_{n=1}^{\infty}\left(-1\right)^{n}q^{n^{2}}\cos(2 nz_{0}) \tag{13}\]  
First, the coordinates of the elliptical pit \((z=x+yi)\) are conformally mapped by the above equation. Second, the equivalent coordinates \(\Xi\) after the transformation are substituted into Equation (6) to calculate the potential function of the seepage field inside the circular pit cut-off wall. If there are \(n\) wells, there are \(n\) equations, and the \(n\) flow rates \(q_{i}\) are taken as a \(1\times n\) vector \(x\) to be solved, and the \(n\) equations on the right side of the equations are taken as the \(n\times n\) matrix \(A\). The \(n\) water heads \(\varphi\) on the left side of the equation are used as a \(n\times n\) vector \(b\). Finally, the matrix equation \(Ax=b\) is solved using the linalg.solve command in the numpy library in Python to obtain the vector \(x\). The water discharge from each individual well under the water head inside the target pit cut-off wall, and then the seepage field inside the cut-off wall can be obtained by superimposing the seepage field of each individual well distribution.  
Figure 3 shows the seepage field flow network inside the pit waterproof curtain of an elliptical pit under a non-uniformly arranged relief well, where the blue line is the seepage field flow line and the black line is the seepage field equipotential line.  
After obtaining the seepage field distribution inside the cut-off wall of the target pit, the equivalent radius of the elliptical pit is found according to the area equivalent to the circular pit by \(R_{d}=\sqrt{\left(a\times b\times\pi\right)/\pi}=\sqrt{ab}\).Then the seepage field inside and outside the pit cut-off wall is solved in series by the resistance coefficient method. Finally the total flow and head of the target pit is obtained.