We present a numerical investigation of laminar miscible displacement flows in narrow, vertical, eccentric annuli. This study is motivated by the primary cementing stage of oil and gas well production, where successful displacement of drilling mud is crucial for the well integrity and zonal isolation. The large number of characterizing parameters makes a complete description of such flows challenging. In turn, this means that the design of effective strategies for primary cementing is a difficult task. As a result the existing literature is mostly based on non-inertial Hele-Shaw models and experiments in narrow annuli, where the dimensionality of the problem is reduced. In this preliminary study, we run a series of three-dimensional numerical simulations, using a Volume of Fluid (VOF) method to capture the interface between the fluids. Both Newtonian and non-Newtonian fluids are considered, and a variety of different phenomena are observed, e.g. dispersive spikes, static layers, instabilities and secondary flows. The range of flow parameters used in the simulations are similar to existing experimental data to allow for a preliminary comparison. The results show qualitative agreement with the experiments and gap-averaged models.