# An Analytical Solution to Two-Region Flow Induced by Constant-Head Pumping in an Unconfined Aquifer
## 6 Conclusions  
This paper proposes an analytical solution for two-region flow induced by constant-head pumping tests. Based on the Dupuit assumption, the proposed model is available for an unconfined aquifer when the constant drawdown inside the pumping well is small. The nonlinear relationship between the specific discharge and hydraulic gradient in the non-Darcian region is expressed using [PERSON]'s equation, and the analytical solution is derived using the Boltzmann transform. The reliability of the analytical solution is verified through a numerical model via COMSOL Multiphysics and a set of hydraulic head\(-\)time data in a pumping test in Wisconsin. The non-Darcian effect on both the hydraulic head decline and dynamic development of the pumping rate is investigated during constant-head tests. The main conclusions are as follows:  
1. The proposed analytical solution can be used for drawdown simulations in the cases of two-region, pure Darcian, and pure non-Darcian flows induced by constant-head pumping tests. In the case of the two-regions flow, the proposed solution can also be used for the assessment of the dynamic development of the pumping flow rate and non-Darcian region.
2. The neglect of the finite expansion of the non-Darcian region may introduce errors to the hydraulic head simulation of a two-region flow. The turbulence flow in the finite non-Darcian region can lead to positive and negative deviations of the hydraulic head\(-\)time curves from pure non-Darcian and pure Darcian flows, respectively. The range of non-Darcian regions can decrease with time.  
Figure 7: Comparison of the hydraulic head increment\(-\)time curves of proposed and [PERSON] et al.’s [17] solutions.  
3. Using [PERSON]'s equation, the transient pumping rate\(-\)time behavior can be significantly controlled by the hydraulic head inside the pumping well \(h_{w}\) and non-Darcian coefficient \(n_{1}\). The decrease in \(h_{w}\) value or increase of \(n_{1}\) value can improve the pumping rate at a given time point. Such effect is gradually weakened and be disappeared towards the end of pumping time.  
Consequently, the proposed model provides a concise method for quantitatively investigating the good hydraulics around the pumping well under the combined effect of both non-Darcian and Darcian flows and offers a theoretical basis for designing the pumping rate during the constant-head test.  
Conceptualization, Y.Z. and L.X.; methodology, L.X.; software, Y.Z.; validation, Y.Z.; investigation, L.X.; resources, L.X.; writing\(-\)original draft preparation, Y.Z.; funding acquisition, L.X.; writing\(-\)review and editing, L.X. and Y.Z.; supervision, L.X. and Y.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by [National Natural Science Foundation of China] grant number [2017 YFC0405900], [Natural Science Foundation of Guangxi] grant number [2018 GXNSFAA138042] and [Hebei High-level Talent Funding Project] grant number [B2018003016].  
Not applicable.  
The data that support the findings of this study are available from the corresponding author upon reasonable request.  
The authors declare no conflict of interest.