# An Analytical Solution to Two-Region Flow Induced by Constant-Head Pumping in an Unconfined Aquifer
## 2 Problem Statement  
Figure 1 shows a schematic of the constant-head pumping test in an unconfined aquifer considering two-region, non-Darcian and Darcian, flow. Since the hydraulic gradient near the pumping well is usually large, a faster specific discharge can make the Reynolds number greater than the critical Reynolds number, which further converts this region of the Darcian pattern into a non-Darcian pattern. Such non-Darcian turbulence effect is diminished with distance from the pumping well. Therefore, the assumption of two-region flow is preferred to simulate the pumping flow under the influence of such non-Darcian turbulence. For convenience, the regions of the non-Darcian pattern and Darcian pattern are hereinafter referred to as the region N and region D, respectively. The assumptions of the mathematical model are mainly mentioned as follows: (1) the aquifer is unconfined, homogeneous, and isotropic; the initial hydraulic head is identical in the entire aquifer; (2) the pumping well fully penetrates the unconfined aquifer and the hydraulic head inside the pumping well is invariable from the start of pumping; (3) the finite non-Darcian region, such as the region N, is mainly developed around the pumping well, and the semi-infinite Darcian region, such as the region D, is sequentially located near the region N; and (4) the radius of the wellbore is the same as the effective radius with a fixed positive number [31]. Based on the law of mass conservation and Dupuit assumption, the Boussinesq equation is used to describe the pumping flow in the regions N and D as Equations (1a) and (1b), respectively. \[\frac{\partial q_{1}}{\partial r}+\frac{q_{1}}{r}=\frac{S_{1}}{h_{1}}\frac{ \partial h_{1}}{\partial t}, \tag{1a}\] \[\frac{\partial q_{2}}{\partial r}+\frac{q_{2}}{r}=\frac{S_{2}}{h_{2}}\frac{ \partial h_{2}}{\partial t}, \tag{1b}\]  
where \(q_{1}\) and \(q_{2}\) are the specific discharges in the regions N and D [L/T], respectively, \(h_{1}(r,t)\) and \(h_{2}(r,t)\) are the hydraulic heads in the regions N and D [L], respectively, and the above parameters are functions of time \(t\) [T] and distance \(r\) [L] from the pumping well. \(S_{1}\) and \(S_{2}\) are the storativity in the regions N and D, respectively. Considering that the values of storativity are determined by the aquifer medium, in general, it should be considered that \(S_{1}=S_{2}\). Figure 1: Schematic of the constant-head test in an unconfined aquifer considering the two-region transform. Assuming that the groundwater in the regions N and D are under the non-Darcian and Darcian conditions, respectively [28], the specific discharges in the regions N and D are expressed using the [PERSON]'s and Darcian equations, respectively. \[q_{1}=\left(-K_{1}\frac{\partial h_{1}}{\partial r}\right)^{\frac{1}{n_{1}}}, \tag{2a}\]  
\[q_{2}=-K_{2}\frac{\partial h_{2}}{\partial r}, \tag{2b}\]  
where \(K_{1}\) is the quasi-hydraulic conductivity in the region N [\(\text{L}^{\text{n}_{1}}/\text{T}^{\text{n}_{1}}\)], \(K_{2}\) is the hydraulic conductivity in the region D [L/T], and \(n_{1}\) is the non-Darcian coefficient representing the turbulent flow in the region N. Based on [PERSON]'s equation, the flow is non-Darcian when \(1<n_{1}\leq 2\), which means that the Reynolds number of the flow is greater than the critical Reynolds number [24]. When \(n_{1}=1\), Equation (2a) becomes an expression of [PERSON]'s law, and \(K_{1}\) is the hydraulic conductivity [12]. The values of \(K_{1}\) and \(n_{1}\) are both constant empirical parameters, which can be evaluated by approximating aquifer drawdown-time measured data [32; 33; 34; 35; 36]. For example, [PERSON] [34] derived the value of the non-Darcian coefficient, \(n_{1}\), by an automatic program according to [PERSON]'s rule through a set of constant rate pumping data in a leakage aquifer located in the Kingdom of Saudi Arabia, and the quasi-hydraulic conductivity, \(K_{1}\), was obtained by further substituting other parameters into the proposed analytical solution.