Mine tailings are commonly stored in off-stream reservoirs and are usually composed of water with high concentrations of fine particles (microns). The rupture of a mine-tailings pond promotes, depending on the characteristics of the stored material, the fluidization and release of hyper-concentrated flows that typically behave as non–Newtonian fluids. The simulation of non–Newtonian fluid dynamics using numerical modelling tools is based on the solution of mass and momentum conservation equations, particularizing the shear stress terms by means of a rheological model that accounts for the properties of the fluid. This document presents the extension of Iber, a two-dimensional hydrodynamic numerical tool, for the simulation of non–Newtonian shallow flows, especially those related to mine tailings. The performance of the numerical tool was tested throughout benchmarks and real study cases. The results agreed with the analytical and theoretical solutions in the benchmark tests; additionally, the numerical tool also revealed itself to be adequate for simulating the dynamic and static phases under real conditions. The outputs of this numerical tool provide valuable information, allowing researchers to assess flood hazard and risk in mine-tailings spill propagation scenarios.