The geological storage of carbon dioxide (CO2), also referred to as CO2 geosequestration, represents one of the most promising options for reducing greenhouse gases in the atmosphere. However, most of the time, CO2 is captured with small amounts of other industrial gases such as sulphur dioxide (SO2) and hydrogen sulphide (H2S), which might be compressed together and stored in depleted petroleum reservoirs or aquifers. Moreover, during CO2 geosequestration in reservoirs, pressure variations during injection could force some amount of CO2 (with or without other acid gas impurities) into the caprock; thereby, altering the petrophysical, geochemical, and geomechanical properties of the caprock. Thus, the brittleness index of the reservoir and caprock might be impacted during CO2 geosequestration due to the chemical reactions between the rock minerals and the formation fluid. Furthermore, to meet the world net-zero carbon target, the promotion of CO2 utilization is paramount. This could be possible by developing an effective technology for cyclic CO2 geosequestration (with or without gas impurities). Therefore, studies on the co-injection of CO2 with other acid gases from industrial emissions, their withdrawal from the porous medium, and their impact on reservoir and caprock integrity are paramount. In this study, a dual-tubing string well completion technology was designed for cyclic injection and withdrawal of CO2 (with or without another acid gas), and numerical simulations were performed using TOUGHREACT codes, to model the cyclic process and investigate the co-injection of SO2 and H2S (separately) with CO2 in sandstone formations overlain by shale caprock. A novel technique of converting the volume fraction of minerals to their weight fraction was developed in this study, to evaluate the brittleness index of the sandstone reservoir and shale caprock during CO2 geosequestration. The findings of the study indicate that the porosity and permeability increase for the CO2 only and CO2–H2S injection cases, in the shale caprock; while for the CO2–SO2 injection case, porosity and permeability only decreased in the layers of the shale caprock contacted by SO2 and due to anhydrite precipitation. In all the injection cases, the porosity and permeability of the sandstone reservoir decreased in a few layers directly below the perforation interval of the production zone. However, in other regions in the sandstone reservoir, the porosity and permeability increased for the CO2 only and CO2–H2S injection cases. In contrast, for the CO2–SO2 co-injection case, porosity and permeability decreased in the layers of the sandstone rock contacted by SO2. In all the CO2 geosequestration cases, the brittleness of the shale and sandstone rocks investigated decreased slightly, except in the CO2–SO2 co-injection case where the brittleness of the sandstone rock decreased significantly. Based on the mineralogical composition of the formations in this study, co-injection of SO2 gas with CO2 gas, only decreased the brittleness index of the shale caprock slightly, but significantly decreased the brittleness of the sandstone reservoir.