The compaction dynamics of a granular media subject to a sequence of vertical taps made of fluid pulses is investigated via molecular dynamics simulations. Our study focuses on three different levels: macroscopic (volume fraction), mesoscopic (Voronoi volumes, force distributions), and microscopic (grain displacements). We show that the compaction process has many characteristics which are reminiscent of the slow dynamics of glass forming systems, as previously suggested. For instance, the mean volume fraction slowly increases in time and approaches a stationary value following a stretched exponential law, and the associated compaction time diverges as the tapping intensity decreases. The study of microscopic quantities also put in evidence the existence of analogies with the dynamics of glass formers, as the existence of dynamical heterogeneities and spatially correlated motion of grains; however, it also shows that there are important qualitative differences, as, for instance, in the role of the cage effect. Correlations between geometry and dynamics of the system at the grain level are put in evidence by comparing a particle Voronoi volume and its displacement in a single tap.