The displacement of a fluid caused by another one, inside annular spaces, is commonly found in the oil industry and most of these rearrangements involve non-Newtonian materials. The annular space often shows irregularities caused by erosion, in which considerable amounts of drilling fluid can be left behind during the displacement process, compromising the cementing operation efficiency. Motivated by that industrial process, fluid-fluid displacement tests at constant flow rate were performed in annular spaces in which their exterior walls displayed - in a determined axial position - an abrupt expansion followed by an abrupt contraction. The purpose of the tests were to determine the displacement efficiency as a function of flow rate, rheological properties and geometric cavity. The results revealed a strong influence of these parameters on the displacement efficiency. At the same time, a numerical research was developed. Numerical simulations of the Navier-Stokes equations in axisymmetric geometry for incompressible fluids were coupled to the Level-Set method to capture the interface. Fluids with constant viscosity and the generalized Newtonian model with viscosity function of Carreau-Yasuda were used. That allowed to simulate displacements between two Newtonian fluids and a Newtonian and a non-Newtonian fluid. This was used both as a displacer and as a displaced fluid. Simulations were performed for several diameters and viscosities ratios, relaxatation time, capilar and Reynolds numbers. We identified when the approximation of the annular space by two parallel plates can be applied and calculated how the shape of the interface depends on the investigated parameters.