Safe sequestration of CO2 in subsurface formations requires a caprock that inhibits the unwanted flow of the stored fluids. However, the preexisting fractures and the induced fractures initiated by overpressure and thermal stresses may act as potential leakage pathways. In this study, we describe a physical problem where the caprock with tiny embedded microfractures is subjected to overpressurization at the storage layer. The average effect of the existing and induced fractures is accounted for by employing a spatially variant permeability field. Also, the alteration of effective local stress is considered by applying Pedrosa’s stress-sensitive model. The coupled model for the pressure diffusion in the caprock and monitoring aquifer is solved using a general Bessel function solution and Laplace transform. The nonlinearity due to stress sensitivity was attenuated by Pedrosa’s transform, and a perturbation solution was obtained. The obtained solution was verified analytically and compared against classical solutions. The spatial variability of the caprock permeability field is effectively represented by the caprock’s two endpoint permeability values, storage/caprock interface permeability, and caprock/monitoring aquifer interface permeability. The ratio between the caprock’s two endpoint permeability values, intactness ratio, is observed to decrease with the increase in the permeability modulus, indicating permeability enhancement due to pressure buildup in the caprock. We identified that the temporal change in the spatially variant permeability field due to stress sensitivity is negligible. The results revealed that all averaging methods underestimate the pressure inside the caprock compared to a spatially variant case. This underestimation is minimum at the interface of caprock and monitoring formation for the harmonic average. The pressure evolution in the monitoring aquifer shows an overestimation of pressure when arithmetic and geometric average permeabilities are considered, while the results obtained using a harmonic average are similar to those of the spatially variant case. The reported work is unique as it accounts for pressure diffusion through preexisting and induced fractures and provides a coupled solution for pressure evolution in monitoring aquifers. This analytical model can be extended for double porosity formations.