The aim of this work was to develop the exact groundwater flow model within a confined aquifer. We argued that, the Theis groundwater flow model is an approximation of the real formulation of the model as Theis removed some components of the equation to have a simple model. Initially, we derived an exact groundwater flow equation for a confined aquifer so as to include all high order terms that were removed by Theis and also to take into account the assumptions that were used during the derivation of the groundwater flow by Theis. Thereafter, we proved that the new groundwater flow equation has a unique solution. We then derived a new numerical scheme for a singular partial differential equation that combines the Mellin transform and the Lagrange approximation of a continuous function. The Mellin transform was used to remove the singularity in the newly developed exact groundwater flow equation for a confined aquifer. The equation became ordinary, wherein we used the Adam Bashforth method to the ordinary differential equation in the Mellin space. The inverse of Mellin was then used to get the exact numerical scheme in real space. We present the stability analysis of the new numerical scheme using the von Neumann method. Lastly, numerical simulations using experimental field data are presented. Our solution is compared to that of Theis. Our simulations show the importance of the scaling factor which was removed from the Theis groundwater flow equation. The simulations also show that the change in drawdown dependdepends on the scaling factor.