# An Analytical Solution to Two-Region Flow Induced by Constant-Head Pumping in an Unconfined Aquifer
## 4 Verification  
In order to investigate the acceptability of the proposed analytical solution, Case A in Table 1 is conducted to compare the simulation results of the hydraulic head and development of region N by the analytical solution with that of the numerical solution. In addition, a series of drawdown data from the constant-head pumping test in Wisconsin, USA, has been presented to demonstrate the reliability of the proposed solution in practice. The numerical solution is performed by using the finite element method via COMSOL Multiphysics, which has proven to be one of the most useful software for multiphysics simulations [38]. The finite element solution is derived based on Equations (3), (4), and (9a) for the region N and Equations (3), (6), and (9b) for the region D. It should be noted that the distance of infinity boundary in Equation (6) is depicted by \(r=2000\) m in the COMSOL solution. A MATLAB program is developed to automatically solve the proposed analytical solutions of Equations (12a,b) and (15a,b). A hypothetical unconfined aquifer is assumed to be a sandy medium. Generally, the range of storativity in an unconfined aquifer is 0.01-0.3, and the hydraulic conductivity of fine sand unconfined aquifer ranges from 1 to 5 m\({}^{3}\)/ day [23]. Hence, a set of parameters is reasonably selected for Case A, as listed in Table 1. Considering the requirement of the Dupuit assumption, the drawdown inside the pumping well needs to take a relatively smaller value so that the parameters of \(b=12\) m and \(h_{w}=9\) m are chosen. The comparison of the hydraulic head\(-\)time curves of the two-region flow (\(R\) is a variable), pure Darcian flow (\(R=0.2\) m), and pure non-Darcian flow (\(R=\infty\)) are demonstrated in Figure 2. The pure non-Darcian COMSOL solution is approximated by \(R=2000\) m. It is noticed that \(K_{1}\) and \(S_{1}\) are meaningless for the simulation of pure Darcian flow, when \(K_{2}\) and \(S_{2}\) are meaningless for the simulation of pure non-Darcian flow. Such comparison clearly indicates that the deviations between the simulation results by both proposed and finite element solutions are small under all three flow conditions. Certain deviations in the first hour after pumping starts are assumed to be induced by the neglect of the vertical flow of the proposed analytical solution, which can gradually disappeared with time. Such errors should be acceptable in case the drawdown inside the pumping well is relatively small. It verifies the validity of the use of the proposed analytical solution for the simulation of the hydraulic head for the two-region flow. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameters** & **Case A** & **Case B** & **Case C** \\ \hline \(h_{w}\) (m) & 9 & \(7,8,9\) & 9 \\ \(K_{1}\)\(\left[\left(\text{m}/\text{day}\right)^{\text{m}_{1}}\right]\) & 1.1 & 1.1 & 1.1 \\ \(K_{2}\) (m/day) & 1.1 & 1.1 & 1.1 \\ \(n_{1}\) & 1.1 & 1.2 & 1,1.2, 1.4 \\ \(S_{1}\) & 0.01 & 0.01 & 0.01 \\ \(S_{2}\) & 0.01 & 0.01 & 0.01 \\ \(b\) (m) & 12 & 10 & 12 \\ \(r_{w}\) (m) & 0.2 & 0.2 & 0.2 \\ \(r\) (m) & 3 & 5 & 2 \\ \(d\) (mm) & 0.3 & 0.3 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values for hypothetical cases. Figure 2: Comparison of the hydraulic head\(-\)time curves of analytical and numerical solutions. Moreover, the hydraulic head of the two-region flow is found to be substantially greater than that of pure Darcian one and to be smaller than that of pure non-Darcian one (Figure 2), suggesting a negative non-Darcian effect on the decline of the hydraulic head. It is because the greater \(n_{1}\) and \(R\) values lead to a greater turbulence degree in the pumped unconfined aquifer. A greater degree of turbulence means a relatively larger specific discharge so that the recharge from the far-away region can be enhanced and the release of gravity storage can be improved [15].