## 3 Semi-Analytical Solution
### Schwartz Iteration Method
#### 3.2.2 Calculation Process  
Boundary conditions for step 1 of the first iteration (in the \(\zeta_{\mathrm{p}}\) plane):  
At the water table:  
\[h_{\mathrm{p,1}}\left|{}_{\mathrm{p}=1}\right.=0 \tag{13}\]  
At the tunnel circumference:  
\[\frac{\partial h_{\mathrm{p,1}}}{\partial\rho_{\mathrm{p}}}\left|{}_{\mathrm{ p}=\mathrm{a}_{\mathrm{p}}}=i_{\mathrm{p}}/\lambda_{\mathrm{p}}=f_{\mathrm{p,1}}( \varphi_{\mathrm{p}})\right. \tag{14}\]  
The boundary condition (14) can be expanded using the Fourier series:  
\[f_{\mathrm{p,1}}(\varphi_{\mathrm{p}})=a_{(\mathrm{p,1})0}+\sum_{m=1}^{\infty }\left(a_{(\mathrm{p,1})m}\cos m\varphi_{\mathrm{p}}+b_{(\mathrm{p,1})m}\sin m \varphi_{\mathrm{p}}\right) \tag{15}\]  
where \(a_{(\mathrm{p,1})0}=\frac{1}{2\pi}\int_{0}^{2\pi}f_{\mathrm{p,1}}(\varphi_{ \mathrm{p}})\mathrm{d}\varphi_{\mathrm{p}}\), \(a_{(\mathrm{p,1})m}=\frac{1}{\pi}\int_{0}^{2\pi}f_{\mathrm{p,1}}(\varphi_{ \mathrm{p}})\cos m\varphi_{\mathrm{p}}\mathrm{d}\varphi_{\mathrm{p}}\), \(b_{(\mathrm{p,1})m}=\frac{1}{\pi}\int_{0}^{2\pi}f_{\mathrm{p,1}}(\varphi_{ \mathrm{p}})\sin m\varphi_{\mathrm{p}}\mathrm{d}\varphi_{\mathrm{p}}\).  
After substituting Equation (7) into Equations (13) and (14), the coefficients can be obtained:  
\[E_{(\mathrm{p,1})} =0,\ F_{(\mathrm{p,1})}=a_{(\mathrm{p,1})0}a_{\mathrm{p}},\ A_{( \mathrm{p,1})m}=-C_{(\mathrm{p,1})m}=\frac{a_{(\mathrm{p,1})m}}{ma_{\mathrm{p }}^{m-1}+ma_{\mathrm{p}}^{m-1}},\] \[B_{(\mathrm{p,1})m}=-D_{(\mathrm{p,1})m}=\frac{b_{(\mathrm{p,1} )m}}{ma_{\mathrm{p}}^{m-1}+ma_{\mathrm{p}}^{m-1}}\]  
Thus, \(h_{\mathrm{p,1}}(x,y)\) can be obtained by substituting Equations (8) and (9) into \(h_{\mathrm{p,1}}(\rho_{j}\varphi_{j})\). The additional radial hydraulic gradient \(i_{\mathrm{t,1}}\) at the tunnel boundary can then be determined using \(h_{\mathrm{p,1}}(x,y)\).  
Boundary conditions for step 2 of the first iteration (in the \(\zeta_{\mathrm{t}}\) plane):  
At the water table:  
\[h_{\mathrm{t,1}}\left|{}_{\mathrm{p}=1}\right.=0 \tag{16}\]  
At the tunnel circumference:  
\[\frac{\partial h_{\mathrm{t,1}}}{\partial\rho_{\mathrm{t}}}\left|{}_{\mathrm{ p}=a_{\mathrm{t}}}=(i_{\mathrm{t}}-i_{\mathrm{t,1}})/\lambda_{\mathrm{t}}=f_{ \mathrm{t,1}}(\varphi_{\mathrm{t}})\right. \tag{17}\]  
The boundary condition (17) can be expanded by the Fourier series:  
\[f_{\mathrm{t,1}}(\varphi_{\mathrm{t}})=a_{(\mathrm{t,1})0}+\sum_{m=1}^{\infty }\left(a_{(\mathrm{t,1})m}\cos m\varphi_{\mathrm{t}}+b_{(\mathrm{t,1})m}\sin m \varphi_{\mathrm{t}}\right) \tag{18}\]  
where \(a_{(\mathrm{t,1})0}=\frac{1}{2\pi}\int_{0}^{2\pi}f_{\mathrm{t,1}}(\varphi_{ \mathrm{t}})d\varphi_{\mathrm{t}}\), \(aBoundary conditions for step 1 of the \(k\)th (\(k\) = 2,3,4... ) iteration (in the \(\zeta_{\rm p}\) plane):  
At the water table:  
\[h_{\rm p,\dot{k}}\left|{}_{\rho_{\rm p}=1}\right.=0 \tag{19}\]  
At the parallel adit circumference:  
\[\left.\frac{\partial h_{\rm p,\dot{k}}}{\partial\rho_{\rm p}}\right|_{\rho_{\rm p }=\alpha_{\rm p}}\,=\,-i_{\rm p,\dot{k}-1}/\lambda_{\rm p} \tag{20}\]  
Boundary conditions for step 2 of the \(k\)th (\(k\) = 2,3,4... ) iteration (in the \(\zeta_{\rm t}\) plane):  
At the water table:  
\[h_{\rm t,\dot{k}}\left|{}_{\rho=1}\right.=0 \tag{21}\]  
At the tunnel circumference:  
\[\left.\frac{\partial h_{\rm t,\dot{k}}}{\partial\rho_{\rm t}}\right|_{\rho_{ \rm t}=\alpha_{\rm t}}\,=\,-i_{\rm t,\dot{k}}/\lambda_{\rm t} \tag{22}\]  
The coefficients \(E_{(j,k)}\), \(\Gamma_{(j,k)}\), \(A_{(j,k)m}\), \(C_{(j,k)m}\), \(B_{(j,k)m}\), and \(D_{(j,k)m}\) can be obtained using the boundary condition Equations (19)-(22). After iterations are completed, all the water head \(h_{\rm j,\dot{k}}\) in the iteration process are superposed to obtain the final total water head \(h(x,y)\). Finally, the pressure head around the tunnel is:  
\[H_{\rm t}=\left.h(x,y)\right|_{(x-\dot{l})^{2}+(y+c_{\rm t})^{2}=r_{\rm t}^{2} }-y \tag{23}\]