Using a high-resolution primitive equation model of the western Mediterranean Sea, we analyzed the dispersion properties of a set of homogeneously distributed, passive particle pairs. These particles were initially separated by different distances D0 (D0 = 5.55, 11.1 and 16.65 km), and were seeded in the model at initial depths of 44 and 500 m. This realistic ocean model, which reproduces the main features of the regional circulation, puts into evidence the three well-known regimes of relative dispersion. The first regime due to the chaotic advection at small scales lasts only a few days (3 days at 44 m depth, a duration comparable with the integral timescale), and the relative dispersion is then exponential. In the second regime, extending from 3 to 20 days, the relative dispersion has a power law tα where α tends to 3 as D0 becomes small. In the third regime, a linear growth of the relative dispersion is observed starting from the twentieth day. For the relative diffusivity, the D2 growth is followed by the Richardson regime D4/3. At large scales, where particle velocities are decorrelated, the relative diffusivity is constant. At 500 m depth, the integral timescale increases (> 4 days) and the intermediate regime becomes narrower than that at 44 m depth due to the weaker effect of vortices (this effect decreases with depth). The turbulent properties become less intermittent and more homogeneous and the Richardson law takes place.