Large-scale deep disposal of CO2 is viewed as a promising tool to curb the increasing streak of anthropogenic carbon emissions. Therefore, a good understanding of the scientific and engineering challenges related to CO2 storage is required to carry out the extensive deployment of CO2 storage. Analytical and semi-analytical approaches provide an easy-to-use tool to verify the success of long-term and large-scale deployment of subsurface CO2 storage. The classical solutions lack significant features that are essential in the interpretation of the spatiotemporal evolution of pressure plume. In this dissertation, obviating the limiting assumptions, I report a set of mathematical models and their solutions.The physical models considered in this thesis include storage-caprock-overburden system, storage sites with existing abandoned wells, and caprock-monitoring aquifer system where caprock has a permeability field with both spatial and temporal variability. The solution methods used in this thesis are semi-analytical in nature where the exact Laplace space solutions are inverted to time domain by using a numerical inversion scheme. The obtained solutions were verified analytically and compared with the classical solutions.The results indicate that the behavior of pressure plume evolution can be a useful tool to safeguard the caprock integrity. The nature of temporal behavior of leakage rates from the storage-caprock interface is investigated and several scaling relations are identified. I also presented benchmark pressure trends to detect the pressure anomaly signaling potential CO2 leakage through storage-caprock interface and through abandoned wells. The results also showed that the average effect of pre-existing natural fractures and/or induced fractures on caprock permeability field can be represented by an intactness ratio, and the usefulness of varying flow capacity in caprocks is found prominent when compared to the classical solutions with uniform permeability fields.The presented solutions and scaling relations will find their usefulness in numerous engineering problems such as CO2 and waste disposal in deep geological formations and geothermal energy extraction. Theories studied in this thesis will also be invoked as benchmark analytical studies in fields where diffusion is of immense interest.