A new steady-state analytical solution to the two-dimensional radial-flow equation was developed for drawdown (head) conditions in an aquifer with constant transmissivity, no-flow conditions at the top and bottom, constant head conditions at a known radial distance, and a partially completed pumping well. The solution was evaluated for accuracy by comparison to numerical simulations using MODFLOW. The solution was then used to estimate the rise of the salt water-fresh water interface (upconing) that occurs under a pumping well, and to calculate the critical pumping rate at which the interface becomes unstable, allowing salt water to enter the pumping well. The analysis of salt water-fresh water interface rise assumed no significant effect on upconing by recharge; this assumption was tested and supported using results from a new steady-state analytical solution developed for recharge under two-dimensional radial-flow conditions. The upconing analysis results were evaluated for accuracy by comparison to those from numerical simulations using SEAWAT for salt water-fresh water interface positions under mild pumping conditions. The results from the equation were also compared with those of a published numerical sharp-interface model applied to a case on Cape Cod, Massachusetts. This comparison indicates that estimating the interface rise and maximum allowable pumping rate using the analytical method will likely be less conservative than the maximum allowable pumping rate and maximum stable interface rise from a numerical sharp-interface model.