# An Analytical Solution to Two-Region Flow Induced by Constant-Head Pumping in an Unconfined Aquifer
## 5 Discussion  
Accurate evaluation of the dynamic development of the pumping rate \(Q\) is vital for designing a practical constant-head test. In this section, the effect of the hydraulic head inside the pumping well, \(h_{w}\), and the degree of turbulence in the non-Darcian flow, represented by \(n_{1}\), on the dynamic development of the pumping rate is investigated using hypothetical case studies: Case B and Case C. The parameters for the two case studies are listed in Table 1.  
Figure 5 shows the simulation results for \(Q\)-\(t\) curves ith different \(h_{w}\). The results indicate that the development of \(Q\) is negatively correlated with the \(h_{w}\) value and it confirms the fact that a small decrease in \(h_{w}\) value can greatly increase the flow rate. It is due to that the flow rate is essentially caused by the hydraulic gradient resulting from the hydraulic head decline inside the pumping well. Therefore, a greater \(h_{w}\) means a slower specific discharge, which can lead to a lower flow rate. Figure 4: Comparison of the hydraulic head\(-\)time curves of the analytical solution and measured data. Figure 5: Dynamic development of the flow rates with different hydraulic heads inside the pumping well. Figure 6 represents the effect of non-Darcian turbulence on the pumping flow according to Case C. The \(Q\)-\(t\) curves at a fixed radial distance \(r=2\) m are shown in Figure 6a), which indicates that the flow rate at a given time point is found to be increased by \(n_{1}\) at the early pumping time. As the pumping continues, the deviations of the flow rates with different \(n_{1}\) values can be decreased. Theoretically, it is understandable. The flow rate is dominated by the recharge from gravity storage, which is assumed to be equal to the specific discharge multiplied by the discharge section area as described in Equation (9c). The higher the specific discharge is, the higher the pumping rate is. Thus, at the early pumping time, the release of gravity storage of the aquifer gradually replenishes the water level nearby the pumping well, causing a greater turbulence flow with a greater \(n_{1}\) value can result in a faster specific discharge and a larger flow rate in the unconfined aquifer. As the pumping continues, the range of the region N is degenerated, which further reduces the influence of \(n_{1}\) and the gradual agreement of the \(Q\)-\(t\) curves. Figure 6b presents the \(h\)-\(r\) curves at the same pumping time \(t=1\) day to investigate the development characteristics of non-Darcian turbulence with radial distance. It shows that there is a positive correlation between the non-Darcian coefficient and the hydraulic head, of which effect is suppressed at a larger radial distance. The larger \(n_{1}\) value means the larger the hydraulic gradient, leading to a larger flow rate. Nearby the pumping well, the hydraulic head fluctuated greatly, but there is no unreasonable mutation, which can prove the continuity of the proposed solution. Subsequently, a comparison of the proposed model with [PERSON] et al. 's [17] solution has been demonstrated in Figure 7. It shows the results for the proposed models in an unconfined aquifer based on Equation (12a,b), labeled as the Proposed solution in U, and in a confined aquifer based on Equation (17a,b), entitled as the Proposed solution in C. By [PERSON] et al. 's [17] hypothetical case, a set of parameters is given as \(h_{w}=40\) m, \(r=0.2\) m, \(b=B=20\) m and \(S_{1}=S_{2}=0.004\). [PERSON] et al. [17] propose a semi-analytical model for a constant-head pumping test in the confined aquifer by considering the hydraulic conductivity as a variable related exponentially with time and assuming that the pumping flow is obeyed by non-Darcian law. Therefore, to allow the proposed solution in the confined aquifer to be fully consistent with the actual situation of [PERSON] et al. 's [17] research, it can be assumed that \(R=\infty\) and \(K_{1}=0.006+0.054\mathrm{e}^{-0.05t}\) m/hr.