A finite element method (FEM) based tool is used in this work to numerically modeling mineral-slurry like flows in a 3D lid-driven cavity. Accordingly, the context in which the referred FEM based tool is being developed is firstly emphasized. Both mathematical and numerical models utilized here are described next. A special emphasis is put on the flow governing equations and the particular FEM weighted residuals approach (Galerkin method) used to solve these equations. Since mineral-slurry flows both featuring relatively low flow velocities and containing large amounts of solid particles can be accounted for as laminar non-Newtonian flows, only laminar flows are discussed here. Indeed both Newtonian and non-Newtonian laminar flows are numerically studied using a 3D lid-driven cavity at two different Reynolds numbers. Two rheological models, power-law and Carreau-Yasuda, are utilized in the nonNewtonian flow simulations. When possible, the numerical results obtained here are compared with other numerical and experimental ones available in open literature. The associated averaged discrepancies from such comparisons are about 1%. The results obtained from the numerical simulations carried out here highlight the usefulness of the FEM based tool used in this work for realistically predicting the behavior of 3D Newtonian and non-Newtonian laminar flows. Multiphase turbulent flows including fluid-particle interaction models will be considered in future developments of this tool such to allow it properly representing the entire mineral-slurry transport phenomenon.