The Stampriet Transboundary Aquifer System (STAS) is one of the well-investigated shared watercourses on the continent of Africa, located in the most arid region of southern Africa. It is the only permanent water source in the Stampriet Basin shared between Namibia, Botswana, and South Africa. However, in recent years, mineral resources exploration in the Stampriet basin has intensified, resulting in the discovery of an economic uranium ore body located in the second aquifer of the STAS called the Auob Aquifer. Uranium extractions will be conducted using the in situ leach mining technology, which is amenable to low-grade uranium deposits in saturated aquifers. In situ leaching for uranium will be the first of its kind in Namibia and has raised major concerns about possible groundwater contamination. This study will use mathematical models to simulate the spread of the contamination plume resulting from the kinetic reaction of the sulfuric acid with the uranium ore. For this, a mathematical model replicating the kinetic reaction between the UO₃ and the sulfuric acid was suggested and solved numerically using the Runge–Kutta method. The numerical solutions were depicted for 8 h. The condition of the existence of a unique 3-solution was presented. Partial differential equations for the transport of the uranyl sulfate, with linear and nonlinear iron exchange and multiple reactions, taking into account a fault, were proposed. These equations were solved numerically using the upwind numerical scheme. Suitable initial and boundary conditions were selected for the fault’s position to perform simulations. The numerical simulations suggest a possible rapid spread of uranyl under the condition that the solution escapes via the extraction boreholes. To our readers, we stress the fact that this study presents a scenario-based modeling framework, using hypothetical yet realistic assumptions, to evaluate potential risks of ISL operations under a range of plausible but unverified conditions.