Geologic carbon storage is considered to be one of the main solutions to significantly reduce CO2 emissions to the atmosphere to mitigate climate change. CO2 injection in deep geological formations entails a two-phase flow, being CO2 the non-wetting phase. One of the main concerns of geologic carbon storage is whether the overpressure induced by CO2 injection may compromise the caprock integrity and faults stability. We numerically investigate the two-phase flow effects that govern the overpressure evolution generated by CO2 injection and how this overpressure affects the caprock geomechanical stability. We find that fluid pressure increases sharply at the beginning of injection because CO2 has to displace the brine that fills the pores around the injection well, which reduces the relative permeability. However, overpressure decreases subsequently because once CO2 fills the pores around the injection well, CO2 can flow easily due to its low viscosity and because the relative permeability to CO2 increases. Furthermore, the pressure drop that occurs in the capillary fringe due to two-phase flow interference decreases as the CO2 plume becomes larger. This overpressure evolution induced by CO2 injection, which remains practically constant with time after the initial peak, is very beneficial for maintaining caprock stability. Thus, the sealing capacity of the caprock will be maintained, preventing CO2 leakage to occur across the caprock.