# An Analytical Solution to Two-Region Flow Induced by Constant-Head Pumping in an Unconfined Aquifer
###### Abstract  
This paper proposes an analytical solution to the problem of two-region flow induced by constant-head pumping tests in an unconfined aquifer based on the Dupuit assumption. The two-region flow includes a finite non-Darcian pattern near the pumping well and a semi-finite Darcian pattern in the rest region. By [PERSON]'s equation, the solution to the two-region flow is derived using the Boltzmann method. The reliability of the proposed solution is investigated through comparisons with the numerical solution by COMSOL Multiphysics and the hydraulic head\(-\)time data from the pumping test in Wisconsin. The influences of the finite non-Darcian flow pattern on the hydraulic head and pumping rate are also discussed. The results demonstrate that a greater degree of non-Darcian turbulence can effectively increase the pumping flow rate, although such influence can be reduced over time. A method for determining the range of the non-Darcian region has been developed and validated by the specific discharge and Reynolds number. constant-head test; unconfined aquifer; two-region flow; non-Darcian effect; analytical solution +
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conducted a constant-head test in a saturated clay aquifer using an improved permeability test device and proposed a numerical solution using the finite difference method to simulate the flow induced by constant-head pumping. The results indicated an obvious non-Darcian characteristic of the pumping-induced flow. These studies have also noticed that neglecting the non-Darcian effect can introduce severe errors in pumping-flow simulation and aquifer parameter determination. Thus far, few studies have been conducted to analytically investigate such non-Darcian flows in response to constant-head pumping. To our knowledge, [PERSON] et al. [17] proposed the first analytical solution to the non-Darcian flow induced by the constant-head test using an aquifer injection well. Their model with time-dependent conductivity was developed using the Laplace transform based on Izbash's equation with a constant non-Darcian coefficient in the entire confined aquifer. Based on the careful literature review, the authors noted that the existing solutions for the constant-head test are generally proposed based on the assumption that the pumped aquifer is confined. However, many pumping wells are located in unconfined aquifers worldwide [18; 19; 20], and there has been a lack of studies on flow mechanics related to constant-head pumping in unconfined aquifers. So far, existing pumping flow models in the unconfined aquifer are mainly proposed by considering (1) with and (2) without vertical flow components. With the vertical flow component, the release of gravity storage takes a long time to complete, which further leads to the formation of an unsaturated region above the hydraulic head and to the vertical replenishment of the aquifer [21]. Without the vertical flow component, the flow is horizontal and uniform everywhere in a vertical section. Gravity storage is considered to be released instantaneously. The solution considering the vertical recharge flow is more accurate than that without vertical flow. However, such solutions are extremely complex and have empirical parameters that cannot be tested, which inhibits their applicability. In a constant-head test with a small drawdown inside the pumping well or long enough pumping time, the influence of the vertical flow is limited. In this case, the use of the Dupuit assumption assuming that the gravity storage is an instantaneous release, is reasonable [22; 23]. The Dupuit assumption insists that the effect of vertical flow can be ignored when the drawdown of the unconfined aquifer changes slowly and considers that all recharge flow comes from the horizontal direction. Hence, it can be used to simplify the unconfined flow problem and makes it analytically tractable [23].