## 2 Formula Derivation  
Basic assumptions. 1. The rock formation is a homogeneous porous medium;
2. The permeability coefficient of the formation is the same in all directions;
3. The flow is steady flow;
4. The radius of influence is considered to be twice the distance from the stationary water surface to the bottom of the well according to Japanese data [34];
5. The primary derivative of the full head slope drop curve within the precipitation influence area approximates the slope of the line connecting the top plate of the water barrier to the center of the sphere. The conservation of mass and the continuity equation gives,  
\[\frac{\partial v_{x}}{\partial x}+\frac{\partial v_{y}}{\partial y}+\frac{ \partial v_{z}}{\partial z}=0, \tag{1}\]  
while the schematic diagram is shown in Figure 1. Introducing [PERSON]'s law, i.e.,  
\[\frac{\partial^{2}H}{\partial x^{2}}+\frac{\partial^{2}H}{\partial y^{2}}+ \frac{\partial^{2}H}{\partial z^{2}}=0. \tag{2}\]  
A spherical coordinate representation is used to Equation (2),  
\[\frac{1}{r^{2}}\cdot\frac{\partial}{\partial r}(r^{2}\cdot\frac{\partial H}{ \partial r})+\frac{1}{r^{2}\sin\theta}\cdot\frac{\partial}{\partial\theta}( \sin\theta\frac{\partial H}{\partial\theta})+\frac{1}{r^{2}\sin\theta}\cdot \frac{\partial^{2}H}{\partial\varphi^{2}}=0. \tag{3}\]  
The seepage field of the shaft excavation is symmetric about the \(Z\)-axis, i.e.,  
\[\frac{1}{r^{2}}\cdot\frac{\partial}{\partial r}(r^{2}\cdot\frac{\partial H}{ \partial r})+\frac{1}{r^{2}\sin\theta}\cdot\frac{\partial}{\partial\theta}( \sin\theta\frac{\partial H}{\partial\theta})=0. \tag{4}\]  
Taking \(H(r,\theta)=N(r)\zeta(\theta)=N(r)\psi(\theta)\), separate the variables twice to obtain,  
\[ \begin{split}(r^{2})\frac{d^{2}N}{dr^{2}}+(2r)\frac{dN}{dr}-(l+1) lN=0\\ (\frac{1}{\sin\theta})\frac{d}{d\theta}(\sin\theta\frac{d\psi}{d \theta})+(l+1)l\psi=0.\end{split}  \tag{5}\]  
Taking \(n=\cos\theta\), Equation (5) can be transformed into [PERSON]'s equation as follows,  
\[(1-n^{2})\frac{d^{2}P(n)}{dn^{2}}-2n\frac{dP(n)}{dn}+l(l+1)P(n)=0. \tag{6}\]  
Then the general solution of \(H\) is,  
\[H=\sum_{l=0}^{+\infty}(w_{l}r^{l}+\frac{z_{l}}{r^{l+1}})P_{l}(n)=\sum_{l=0}^{+ \infty}(w_{l}r^{l}+\frac{z_{l}}{r^{l+1}})\cdot\frac{1}{2^{l}l!}\cdot\frac{d^{ 4}(n^{2}-1)^{l}}{dn^{2}}(l=0,1,2,3,\cdots,n). \tag{7}\]  
Figure 1: Mathematical model schematic. The following two equations hold in the influence range and outside the influence range,  
\[ \begin{array}{l}H_{1}=\sum\limits_{l=0}^{+\infty}{(g_{l}r^{l}+\frac{h_{l}}{r^{l }+1})\cdot P_{l}(n)(l=0,1,2,3,\cdot\cdot\cdot,n)}\quad\text{Outside the ball}\\ H_{2}=\sum\limits_{l=0}^{+\infty}{(o_{l}r^{l}+\frac{q_{l}}{r^{l }+1})\cdot P_{l}(n)(l=0,1,2,3,\cdot\cdot\cdot,n)}\quad\quad\text{In the ball.}\end{array}  \tag{8}\]  
The boundary condition is,  
In this paper, we assume that the radius of influence of precipitation is \(2s_{w}\), while there are \(H_{1}=H_{2}\) and \(v_{1}=v_{2}\) for sphere. We can obtain,  
\[\left\{ \begin{array}{l}g_{0}P_{0}(b)+\big{[}\frac{\frac{Q}{2\pi K}\ln(r\sin \theta)+a-g_{0}}{\cos\theta}\big{]}P_{1}(n)+\sum\limits_{l=0}^{+\infty}\big{(} \frac{h_{l}}{r^{l}+1}\big{)}\cdot P_{l}(n)=\sum\limits_{l=0}^{+\infty}\big{(}o _{l}r^{l}\big{)}\cdot P_{l}(n)\\ \\ \frac{Q}{2\pi K\cos\theta}\frac{1}{r}\cdot P_{1}(n)-\sum\limits_{l=0}^{+\infty }\big{[}\frac{(r+1)h_{l}}{r^{l+2}}\big{]}\cdot P_{l}(n)=\sum\limits_{l=0}^{+ \infty}\big{(}lo_{l}r^{l-1}\big{)}\cdot P_{l}(n).\end{array} \right. \tag{15}\]  
i.e.,  
\[\left\{ \begin{array}{l}\frac{\frac{Q}{2\pi K}\ln(r\sin\theta)+a-g_{0}}{\cos \theta}+\frac{h_{1}}{r^{2}}=o_{1}r\\ \\ \frac{Q}{2\pi K\cos\theta}\frac{1}{r}-\frac{2h_{1}}{r^{3}}=o_{1}\\ \\ H_{0}=\frac{Q}{2\pi K}\ln(r\sin\theta)+a\\ \\ g_{n}=o_{n}=0,n\
eq 1\\ \\ g_{0}+\frac{h_{0}}{r}=o_{0}\\ \\ -\frac{h_{0}}{r^{2}}=0.\end{array} \right. \tag{16}\]  
We can derive,  
\[\left\{ \begin{array}{l}g_{0}=o_{0}\\ h_{1}=\frac{Q}{6\pi K\cos\theta}r^{2}-\frac{H_{0}-a_{0}}{3\cos\theta}r^{2}\\ \\ o_{1}=\frac{2(H_{0}-a_{0})}{3r\cos\theta}+\frac{Q}{6\pi Kr\cos\theta}.\end{array}