Carbon dioxide (CO2) storage and sequestration is regarded as an effective approach to mitigate greenhouse gas emissions. While injecting an enormous amount of CO2 into carbonate-rich aquifers, CO2 dissolves in the formation brine under the large pressure and the subsequently formed CO2-enriched brine reacts with the calcite. Reaction-induced changes in pore structure and fracture geometry alter the porosity and permeability, giving rise to the concerns of CO2 storage capacity and security. Especially in the reservoir or aquifer with natural fractures, the fractures provide highly permeable pathways for fluid flow. This study aims to analyze the acid-rock interaction and subsequent permeability evolution in the systems with complex fracture configurations during CO2 injection by implementing a pore-scale Darcy-Brinkman-Stokes (DBS) reactive transport model. The model has been developed by expanding the functionality of OpenFOAM, which is an open-source code for computational fluid dynamics. A series of partial differential equations are discretized by applying the finite volume method and then solved sequentially. Different fracture configurations, such as fracture length, density, connection, and mineral components, have been considered to investigate their impacts on the dynamic porosity-permeability relationship, dissolution rate, and reactant transport characteristics during CO2 storage. Various mineralogical compositions were investigated to compare their effects on the reactive transport and subsequent system behavior induced by water-rock interactions. The investigation revealed several interesting findings. We found that calcium (Ca) ions concentration was low in the poorly connected area at the initial time. Because CO2-enriched brine saturated the system and reacted with calcite, Ca ions started accumulating in the system. However, Ca ions barely flowed out of the poorly connected area, and the concentration became high. Lengths of branches mainly influenced the dissolution rates, while they had slight impacts on the porosity-permeability relationship. While fracture connectivity had an apparent influence on the porosity-permeability relationship, it showed a weak relevance on the dissolution rate. The calcite-clay fracture-matrix system showed a slow increase of permeability because CO2-enriched brine was injected. These microscopic insights can help enhance the CO2 sealing capacity to guarantee environmental safety.