Abstract
Abstract. Previous studies on the non-Darcian flow into a pumping well assumed that critical radius (RCD) was a constant or infinity, where RCD represents the location of the interface between the non-Darcian flow region and Darcian flow region. In this study, a two-region model considering time-dependent RCD was established, where the non-Darcian flow was described by the Forchheimer equation. A new iteration method was proposed to estimate RCD based on the finite-difference method. The results showed that RCD increased with time until reaching the quasi steady-state flow, and the asymptotic value of RCD only depended on the critical specific discharge beyond which flow became non-Darcian. A larger inertial force would reduce the change rate of RCD with time, and resulted in a smaller RCD at a specific time during the transient flow. The difference between the new solution and previous solutions were obvious in the early pumping stage. The new solution agreed very well with the solution of the previous two-region model with a constant RCD under quasi steady flow. It agreed with the solution of the fully Darcian flow model in the Darcian flow region.
Highlights
Darcy’s law indicates a linear relationship between the fluid velocity and the hydraulic gradient (Bear, 1972), which is a basic assumption used to handle a great deal of problems related to flow in porous and fractured media
We develop a MATLAB program named as two-region model with moving critical radius (MTRM) to facilitate the computation
A new two-region flow model considering the time-dependent critical radius (RCD) is established to investigate the groundwater flow into a pumping well, and a new iteration method is proposed to estimate RCD, based on the finite-difference method. Results show that this iteration method is convergence it has not been analytical verified using rigorous mathematic model
Summary
Darcy’s law indicates a linear relationship between the fluid velocity and the hydraulic gradient (Bear, 1972), which is a basic assumption used to handle a great deal of problems related to flow in porous and fractured media. Giorgi (1997) and Chen et al (2001) analytically derived the Forchheimer law from the Navier–Stokes equation Another widely used model describing the non-Darcian flow was the Izbash equation (Izbash, 1931). To test the accuracy of the semi-analytical solutions (Wen et al, 2008a; Sen, 2000), Mathias et al (2008) and Wen et al (2009) employed the finite-difference method to study the non-Darcian flow problems, and their results showed that the semi-analytical solution only agreed very well with the numerical solution at late pumping stage. We will investigate non-Darcian flow into a fully penetrating pumping well considering a time-dependent critical radius using the finite-difference method. This new model reduces to the F-ND model when the critical radius is infinite and it becomes the fully Darcian flow model when the critical radius is 0
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