## 3 Calculation of Groundwater Drawdown Difference
### Modeling a Partially Penetrating Well  
According to the above assumptions, the unsteady flow governing equation for a large-diameter partially penetrating well can be obtained as  
\[T\bigg{(}\frac{\partial^{2}S}{\partial r^{2}}+\frac{1}{r}\frac{\partial S}{ \partial r}\bigg{)}=S^{\prime}\frac{\partial S}{\partial t} \tag{1}\]  
where \(a=T/S^{\prime}\), \(T\) is the transmissivity; \(S^{\prime}\) is the coefficient of storage.  
The initial and boundary conditions are as follows  
\[S(r,0)=0,\ (0<r<\infty) \tag{2}\]  
\[S(\infty,t)=0,\ S(R_{w},t)=S_{w},\ (t>o) \tag{3}\]  
\[\lim_{r\to R_{w}}\bigg{(}lr\frac{\partial S}{\partial r}\bigg{)}=\frac{Q_{1}} {2\pi K},\ (l>z>0) \tag{4}\]  
where \(R_{w}\) is foundation pit radius; \(r\) is the distance from the waterproof curtain; \(S_{w}\) is groundwater drawdown outside the waterproof curtain; \(S_{h}\) is groundwater drawdown inside the foundation pit; \(S\) is groundwater drawdown at any location outside the pit; \(t\) is pumping time; \(Q_{1}\) is water inflow; \(K\) is the permeability coefficient. \(l\) is the distance between the bottom of the waterproof curtain and the impermeable layer.  
The governing equation of groundwater to the unsteady flow can be transformed into time and space problems by using Boltzmann transformation [47; 48]. The Boltzmann transformation reduces the equation for a contracting or expanding flow to the equation in spatially homogeneous system. This Boltzmann transformation method has also been applied in some studies on fully unconfined flow, and the partial differential equation can be transformed into ordinary differential equations, which simplify the groundwater flowing equation.  
\[u=\frac{r^{2}}{4 at} \tag{5}\]  
Equation (6) can be obtained by substituting Equation (5) into Equation (1).  
\[(1+u)\frac{dS}{du}+u\frac{d^{2}S}{du^{2}}=0 \tag{6}\]  
In order to calculate Equation (6), the boundary conditions of the ordinary differential equation can be obtained for a given time.  
\[u_{0}=\frac{{r_{w}}^{2}}{4 at} \tag{7}\]  
\[S\mid_{u\rightarrow\infty}=0 \tag{8}\]\[S\mid_{u\to u_{0}}=S_{w} \tag{9}\]  
\[\frac{dS}{du}=e^{-C_{1}}\frac{e^{-u}}{u}=A\frac{e^{-u}}{u} \tag{10}\]  
Equation (10) is calculated by integrals.  
\[\int_{u_{0}}^{\infty}\!\!\frac{dS}{du}du=0-S_{w}=A\!\int_{u_{0}}^{\infty}\!\! \frac{e^{-u}}{u}du=AW(u_{0}) \tag{11}\]  
\[A=-\frac{S_{w}}{W(u_{0})} \tag{12}\]  
\[\int_{u_{0}}^{u}\!\!\frac{dS}{du}du=S-S_{w}=A\!\int_{u_{0}}^{u}\frac{e^{-u}}{u }du=-\frac{S_{w}}{W(u_{0})}\!\int_{u_{0}}^{u}\!\!\frac{e^{-u}}{u}du \tag{13}\]  
\[W(u)=\int_{u}^{\infty}\!\!\frac{e^{-u}}{u}du,\;W(u_{0})=\int_{u_{0}}^{\infty} \!\!\frac{e^{-u}}{u}du \tag{14}\]  
where \(W(u)\) is the well function.  
At any time, the groundwater drawdown at different locations is shown in Equation (15).  
\[S=S_{w}-\frac{S_{w}}{W(u_{0})}\!\int_{u_{0}}^{u}\!\!\frac{e^{-u}}{u}du=S_{w} \frac{W(u)}{W(u_{0})} \tag{15}\]  
Based on [PERSON]'s law and [PERSON]'s assumption, the slope of the groundwater drawdown for a well is regarded as the hydraulic gradient. The multiplication of the slope and the soil permeability coefficient is considered to be the flow rate. The multiplication of the flow rate and the area of water passing a section of a partially penetrating well is the water inflow. Therefore, the groundwater inflow \(Q_{1}\) of a partially penetrating well with a large diameter can be obtained from Equation (16).  
\[Q_{1}=2K\pi rl\frac{\partial S}{\partial r}\mid_{r\to R_{w}}=4K\pi l \frac{S_{w}e^{-u_{0}}}{W(u_{0})} \tag{16}\]