The paper addresses the problem of geomechanical effects in the vicinity of production/injection wells and their impacts on the processes of enhanced oil recovery by CO2 injection and CO2 sequestration in a partially depleted oil reservoir. In particular, it focuses on natural fracture systems and their dynamics caused by variations in the rock geomechanical state due to reservoir pressure changes during production/injection processes. The comprehensive approach to the problem requires the combined modeling of both geomechanical and flow phenomena associated with effective coupling simulations of their evolution. The paper applies such an approach to a real, partially depleted oil reservoir in Poland. An effective method of coupled geomechanical and dynamic simulations was used together with the natural boundary and initial conditions for both simulation types. In addition, typical operating conditions were applied in analyzing the processes of enhanced oil recovery by CO2 injection and CO2 sequestration. The detailed results of relevant modeling and simulations are presented and discussed focusing on various scale consequences, including the reservoir, well, and completion ones. Both general conclusions as well as the ones specific to the analyzed geological structure are drawn; they confirm the significant dependence of well performance on geomechanical effects and point out several key factors for this dependence. The conclusions specific to the analyzed structure concern fracture reactivation in tensile/hybrid failure mode caused by pressure build-up during CO2 injection and the importance of the fracture-induced aperture changes resulting from the normal stress, while the shear stress is found to be negligible.