With the recent GIEC report about global warming urging humanity to limit the global temperature increase to 2°C maximum, research on the geological storage of carbon dioxide appears more important than ever. However, the injection in geological formations (such as deep saline aquifers and depleted gas/oil fields) of supercritical CO2, stores it in the porosity of the host rock raising legitimate concern about the safety and long-term behavior of such dynamic multiphase hydrosystems. Additionally, the economic and energetic weight of such storage complicates its development at the world scale without strong political incentives. The storage of CO2 in ultramafic formations in some specific contexts appears, on the contrary, as a very appealing technology since it involves the safe mineralization of the carbon by precipitation of carbonates with the major alkaline earth metals (i.e. Mg, Ca…) leached from the formation itself. Moreover, as these rocks contain high amounts of ferrous iron, its oxidation by the water co-injected with CO2 produces dihydrogen, which can be economically valuable rendering the whole process more viable. Large ophiolite formations (Oman, Papua New Guinea, east coast of Adriatic Sea…) are expected to have a storage capacity of several billion tons of CO2 and could produce similar amounts of clean dihydrogen. We present experimental results on the mineral carbonation of natural cores of serpentinites by the continuous percolation of carbon-saturated water. We show that the dimensionless Péclet (relative importance of diffusion and convection processes), and Damköhler (relative importance of convection and chemical processes) numbers as well as the initial geometry of the porosity and permeability control the localization of the silicate dissolution and the carbonate precipitation in the porous medium. We also show that the chemical behavior is principally controlled by the reactivity of calcium-bearing silicates (wollastonite, diopside) and the precipitation of calcite as well as the initial iron content of the different phases. Such results are particularly interesting for the design and the optimization of pilot sites and the development of this technology at industrial scale.