Public supply wells are commonly considered one of the most significant sources of freshwater on Earth. Therefore, potential well water contamination can conceivably be regarded as a crucial issue that is closely correlated with both environmental protection and water demand. In the present study, a three-dimensional numerical model is developed to simulate unsteady and spatially varying groundwater flow, along with contaminant migration. Besides, the proposed model is capable of investigating well water quality by the change of the wells’ pumping rates. The developed model uses a finite-volume time splitting numerical technique to solve governing groundwater flow and soluble contaminant transport equations. Comparison of the numerical simulation results with analytical solutions, as well as experimental and field data, clearly demonstrates the satisfactory performance of the present model. The fundamental aim of the study is to evaluate the effect of pumping rate and its variations on pollution migration through saturated porous media. To meet this purpose, contaminant concentrations and contaminants’ travel time were studied under different pump flow rate conditions. The modeling results revealed that choosing an optimum range for the pumping rate increases contaminant travel time and reduces aquifer vulnerability.