## 3 Semi-Analytical Solution
### General Solution of Single Tunnel Drainage  
The water table and tunnel (or parallel adit) circumferences in the \(Z_{j}\) plane are conformally mapped to two circles with radius 1 and \(\alpha_{j}\) in the \(\zeta_{j}\) plane by mapping functions (where \(j=\) p, t; p represents the parallel adit, and t represents the tunnel. For convenience, follow-up discussions also follow this agreement) (Figure 3) [14]. The mapping function is:  
\[\zeta_{j}=\zeta_{j}+\eta_{j}\mathrm{i}=\frac{z_{j}+A_{j}\mathrm{i}}{z_{j}-A_{ j}\mathrm{i}} \tag{5}\]  
where \(A_{j}=\sqrt{{c_{j}}^{2}-{r_{j}}^{2}}\), \(c_{j}=d_{j}+r_{j}-d_{\mathrm{w}}\), \(z_{j}=x_{j}+y_{j}\mathrm{i}\) is the point of the \(Z_{j}\) plane, \(\zeta_{j}=\zeta_{j}+\eta_{j}\mathrm{i}\) is the point of the mapping plane, and \(\mathrm{i}\) is the imaginary unit.  
From the mapping Function (5), the radius of the parallel adit and tunnel in the \(\zeta_{j}\) plane is: \(\alpha_{j}=(c_{j}-\sqrt{{c_{j}}^{2}-{r_{j}}^{2}})/{r_{j}}\).  
Using conformal mapping, Equation (1) can be expressed in the \(\zeta_{j}\) plane in polar coordinates as:  
\[\frac{\partial^{2}h_{j,k}}{\partial{\rho_{j}}^{2}}+\frac{1}{\rho_{j}}\frac{ \partial h_{j,k}}{\partial{\rho_{j}}}+\frac{1}{\rho_{j}^{2}}\frac{\partial h_{ j,k}}{\partial{\varphi_{j}}^{2}}=0 \tag{6}\]  
where \(h_{j,k}\) is the total head, and \(\rho_{j}\) and \(\varphi_{j}\) are the polar radius and polar angle in \(\zeta_{j}\) plane, respectively. Subscripts \(j\) and \(k\) represent the serial number of the tunnels and the order of iterations, respectively.  
In the \(\zeta_{j}\) plane, Equation (6) (the Laplace equation in polar coordinates) can be solved by using the method of separating variables [25]. Its general solution is shown in Equation (7). When the solution domain is an annulus, the coefficients in the general solution are easily obtained by the boundary conditions.  
\[h_{j,k}=E_{(j,k)}+F_{(j,k)}\ln\rho_{j}+\sum_{m=1}^{\infty}\left[\,(A_{(j,k)m} \rho_{j}^{m}+C_{(j,k)m}\rho_{j}^{-m})\,\cos m\varphi_{j}+\,(B_{(j,k)m}\rho_{j} ^{m}+D_{(j,k)m}\rho_{j}^{-m})\,\sin m\varphi_{j}\right] \tag{7}\]  
where \(E_{(j,k)}\), \(F_{(j,k)}\), \(A_{(j,k)m}\), \(C_{(j,k)m}\), \(B_{(j,k)m}\), \(D_{(j,k)m}\) are coefficients, which can be determined by boundary conditions. The meanings of subscripts \(j\) and \(k\) are as described above, and the subscript \(m\) denotes the order of the Fourier series.  
Figure 3: Plane of conformal mapping.  
According to mapping Function (5), \(\rho_{j}\) and \(\varphi_{j}\) in the \(\zeta_{j}\) plane have the following relationship with \(x\) and \(y\) in the \(Z\) plane:  
\[\rho_{j}=\frac{\sqrt{\left(x_{j}^{2}+y^{2}-A_{j}^{2}\right)^{2}+4A_{j}^{2}x_{j} ^{2}}}{x_{j}^{2}+\left(y-A_{j}\right)^{2}} \tag{8}\]  
\[\varphi_{j}=\arctan\frac{2A_{j}x_{j}}{x_{j}^{2}+y^{2}-A_{j}^{2}} \tag{9}\]  
where \(j=\) p, t; \(x_{\text{p}}=x\); \(x_{\text{t}}=x-l\).  
In addition, the radial hydraulic gradient along the tunnel (or parallel edit) boundary needs to be transformed from the \(Z_{j}\) plane to the \(\zeta_{j}\) plane in subsequent calculations, and the relationship between them is:  
\[\frac{\partial h_{j,k}}{\partial n_{j}}=\frac{\partial h_{j,k}}{\partial\rho_ {j}}\lambda_{j} \tag{10}\]  
where \(\lambda_{j}\)=\(|\frac{\partial\zeta_{j}(z_{j})}{\partial z_{j}}|\) is the modulus of the derivative of the mapping function, and \(n_{j}\) is the radial direction on the circumference of the tunnel (or parallel edit) in the \(Z_{j}\) plane.