# An Analytical Solution to Two-Region Flow Induced by Constant-Head Pumping in an Unconfined Aquifer
## 3 Solution Derivation  
Substituting Equations (2a) and (2b) into Equations (1a) and (1b) yields  
\[\frac{\partial^{2}h_{1}}{\partial r^{2}}+\frac{n_{1}}{r}\frac{\partial h_{1}}{ \partial r}=\frac{S_{1}}{h_{1}}\frac{n_{1}}{K_{1}\frac{1}{n_{1}}}\left(-\frac{ \partial h_{1}}{\partial r}\right)^{\frac{n_{1}-1}{n_{1}}}\frac{\partial h_{1} }{\partial t}, \tag{9a}\] \[\frac{\partial^{2}h_{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial h_{2}}{ \partial r}=\frac{S_{2}}{h_{2}K_{2}}\frac{\partial h_{2}}{\partial t}. \tag{9b}\]  
The flow rate in region N is assumed to be equal to the specific discharge multiplied by the discharge section area as described in the following equation. \[Q\approx-Aq_{1}=2\pi rh_{1}\left(-K_{1}\frac{\partial h_{1}}{\partial r} \right)^{\frac{1}{n_{1}}}, \tag{9c}\]  
where \(q_{1}=\left(K_{1}\frac{\partial h_{1}}{\partial r}\right)^{\frac{1}{n_{1}}}\) is the specific discharge [L/T], and \(A=2\pi rh_{1}\) is the discharge section area [L\({}^{2}\)]. To simplify the derivation of the solution, the items of \(\frac{S_{1}}{h_{1}}\frac{n_{1}}{K_{1}\frac{1}{n_{1}}}\approx\frac{S_{1}}{b} \frac{n_{1}}{K_{1}\frac{1}{n_{1}}}\), \(\frac{S_{2}}{h_{2}K_{2}}\approx\frac{S_{2}}{hK_{2}}\) and \(2\pi rh_{1}\approx 2\pi rb\) are introduced to Equations (9a), (9b), and (9c), respectively, which can be used to linearize the Dupuit equation. The errors caused by such approximations have been found to be relatively small. Hence, they can be neglected for modeling purposes [36]. Thus, Equations (9a), (9b), and (9c) can be approximately written as  
\[\frac{\partial^{2}h_{1}}{\partial r^{2}}+\frac{n_{1}}{r}\frac{\partial h_{1} }{\partial r}=\frac{S_{1}}{b}\frac{n_{1}}{K_{1}\frac{1}{n_{1}}}\left(-\frac{ \partial h_{1}}{\partial r}\right)^{\frac{n_{1}-1  
where  
\[F_{1}=\frac{Q^{n_{1}}\exp\left(\frac{\varepsilon_{1}\tau_{w}{}^{1-n_{1}}}{4}\eta _{w}{}^{2}\right)r^{1-n_{1}}}{\left(2\pi\right)^{n_{1}}h_{w}{}^{n_{1}-1}bK_{1}}, \tag{13a}\] \[F_{2}=\frac{Q\exp\left(\frac{\varepsilon_{1}\tau_{w}{}^{1-n_{1}}}{4a_{1}}\eta _{w}{}^{2}\right)}{4\pi h_{w}{}^{\frac{n_{1}-1}{n_{1}}}b^{\frac{1}{n_{1}}}K_{2} \exp\left(-\frac{\varepsilon_{2}}{4}\eta^{2}\right)}\exp\left(-\frac{ \varepsilon_{1}R^{1-n_{1}}}{4n_{1}}\eta_{R}{}^{2}\right),\] (13b) \[v=\frac{\varepsilon_{2}}{4}\eta^{2},\] (13c) \[v_{R}=\frac{\varepsilon_{2}}{4}\eta_{R}{}^{2},\] (13d) \[\eta=rt^{-\frac{1}{2}},\] (13e) \[\eta_{R}=Rt^{-\frac{1}{2}},\] (13f) \[\eta_{w}=r_{w}t^{-\frac{1}{2}},\] (13g) \[\frac{\partial^{2}h_{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial h_{2}}{ \partial r}=\frac{S_{2}}{h_{2}K_{2}}\frac{\partial h_{2}}{\partial t}. \tag{9b}\]0  
The constant-head pumping test can make the hydraulic head near the pumping well declines rapidly at the early pumping time and approaches to \(h_{w}\) in a short time. After that, the magnitude of the hydraulic head change can slow down considerably. Therefore, the proposed analytical solution based on the Dupuit assumption is acceptable except for a short period of time when the pumping starts. In a constant-head test, since the turbulence of the flow can be weakened with increasing radial distance from the pumping well, the region N can be defined as the area with Reynolds number, \(Re\), greater than the critical Reynolds number, \(Re_{c}\), and the \(R\) value can be determined by investigating the position of \(Re=Re_{c}|_{r\to R}\) at any interesting time. In addition, it should be noted that the proposed solution can also be used to assess the drawdown behavior of pure Darcian flow with \(R=r_{w}\) and pure non-Darcian flow with \(R=\infty\).