Forchheimer flow to a well-considering time-dependent critical radius

Previous studies on the non-Darcian flow into a pumping well assumed that critical radius (R-CD) was a constant or infinity, where R-CD 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 R-CD was established, where the non-Darcian flow was described by the Forchheimer equation. A new iteration method was proposed to estimate R-CD based on the finite-difference method. The results showed that R-CD increased with time until reaching the quasi steady-state flow, and the asymptotic value of R-CD only depended on the critical specific discharge beyond which flow became non-Darcian. A larger inertial force would reduce the change rate of R-CD with time, and resulted in a smaller R-CD 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 R-CD under quasi steady flow. It agreed with the solution of the fully Darcian flow model in the Darcian flow region.