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

Abstract. 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.