Underground carbon dioxide (CO2) injection may cause earthquakes by changing pore pressure and underground stress. Understanding the fundamental mechanisms of fault reactivation caused by fluid injection is of great significance for evaluating the risks associated with earthquakes and fluid leakage related to underground flow processes. This study presents the governing equations of two- phase flow (CO2 and water) in underground formations and faults. A numerical model simulates the entire process of fault activation and leakage induced by CO2 injection, investigating the effect of different fault properties and CO2 injection scenarios on fault activation and fluid leakage. Our numerical results show that fault permeability increases significantly after activation, with a maximum increase of up to 106 times. This transformation converts the fault slip zone from an impermeable barrier to a dominant seepage channel. The initial tangential permeability of the fault has a significant impact on the mechanical characteristics of fault activation. Higher initial tangential permeability leads to greater fault slip. Faults with higher shear stiffness show stronger resistance to injection- induced activation. In addition, the CO2 injection rate affects both earthquake occurrence and CO2 leakage caused by fault activation. At the same injection rate, low injection rates result in smaller fault slip but increased CO2 leakage. Conversely, higher injection rates lead to greater fault slip but reduced CO2 leakage.