The pressure variation induced by CO2 injection into deep geological formations and the resulting stress perturbation may activate pre-existing faults or cause the generation of new ones. These fractures may represent a preferential path for the escape of the injected gas, damage or even break the injection/extraction wells, give rise to microseismic phenomena and have an impact on the ground deformations. Numerical simulations by traditional finite elements (FE) cannot address this problem due to the compatibility condition which prevents relative movements between adjacent elements. In this PhD thesis, an appopriate class of FE, called interface elements (IE), is developed in order to simulate discontinuous displacement fields in the porous medium. Several numerical problems are related to this formulation because the IE add strong non-linearities to the equilibrium equations and the arising algebraic system typically has a large condition number. These issues have been deeply investigated developing special time-stepping techniques to deal with the non-linearity and studying ad hoc preconditioners for the iterative solution to the linearized systems. The numerical tools have been finally used in a realistic field example of CO2 sequestration in a depleted gas reservoir. The efficiency of the proposed numerical schemes has been tested with several aspects concerning the possible enviromental impact analysed.