A mathematical model is presented for describing the groundwater flow in a radial two-layer confined aquifer system with a constant-flux pumping well that has a wellbore skin and finite well radius. The Laplace-domain solution for the model is first derived by the Laplace transforms; and the time-domain solution in terms of the aquifer drawdown is then obtained form the Laplace inversion using the Bromwich integral method. When neglecting the well radius, our Laplace-domain solution is shown to reduce to a Laplace-domain solution given by Butler [J. Hydrol. 101 (1988) 15]. A unified numerical approach including a root search approach, the Gaussian quadrature, and the Shanks method is employed for evaluating this time-domain solution. The evaluated results of the solution agree well with those of the Laplace-domain solution estimated by the modified Crump algorithm. This new solution can be used either to predict the spatial and temporal drawdown distributions in both the skin and formation zones or to investigate the effects of the skin type, skin thickness and well radius on the drawdown distribution.