We study the enrichment of the interstellar medium with rapid neutron capture (r-process) elements produced in binary neutron star (BNS) mergers. We use a semi-analytic model to describe galactic evolution, with merger rates and time delay distributions of BNS mergers consistent with the latest population synthesis models. In order to study the dispersion of the relative abundances of r-process elements and iron, we applied a turbulent mixing scheme, where the freshly synthesized elements are gradually dispersed in the interstellar medium. We show that within our model the abundances observed in Milky-Way stars, in particular the scatter at low metallicities, can be entirely explained by BNS mergers. Our results suggest that binary neutron star mergers could be the dominant source of r-process elements in the Galaxy.