Geological carbon dioxide (CO₂) storage in deep geological formations can only lead to significant reductions in anthropogenic CO₂ emissions if large amounts of CO₂ can be stored safely. Determining the storage capacity, which is the volume of CO₂ stored safely, is essential to determine the feasibility of geological CO₂ storage. One of the main constraints for the storage capacity is the physical mechanisms of fluid flow in heterogeneous formations, which has not been studied sufficiently. Therefore, I consider two related problems: a) the evolution of injection-induced overpressure that determines the area affected by CO₂ storage and b) the rate of buoyant fluid flow along faults that determines the leakage of CO₂. I use a layered model of a sandstone reservoir embedded in mudrocks to quantify the increase in storage capacity due to dissipation of overpressure into the mudrocks. I use a model of a fault surface with flow barriers to constrain the reduction in the buoyancy-driven leakage flux across the fault. Using the layered model with injection at constant rate, I show that the pressure evolution in the reservoir is controlled by the amount of overpressure dissipated into ambient mudrocks. A main result of this study is that the pressure dissipation in a layered reservoir is controlled by a single dissipation parameter, M, that is identified here for the first time. I also show that lateral pressure propagation in the storage formation follows a power-law governed by M. The quick evaluation of the power-law allows a determination of the uncertainty in the estimate of the storage capacity. To reduce this uncertainty it is important to characterize the petrophysical properties of the mudrocks surrounding the storage reservoir. The uncertainty in mudrock properties due to its extreme heterogeneity or limited data available can cause large variability in these estimates, which emphasizes that careful characterization of mudrock is required for a reliable estimate of the storage capacity. The cessation of the injection operation will reduce overpressure near the injector, but regional scale pressure will continue to diffuse throughout the whole formation. I have been able to show that the maximum radius of the pressure plume in the post-injection period is approximately 3.5 times the radius of the pressure plume at the cessation of injection. Two aquifers can be hydraulically connected by a fault cutting across the intermediate aquitard. If the upper aquifer contains denser fluid, an exchange flow across the fault will develop. The unstable density stratification leads to a fingering pattern with localized zones of upwelling and downwelling along the fault. Due to the small volume of the fault relative to the aquifers, the exchange-flow will quickly approach a quasi steady state. If the permeability of the fault plane is homogeneous, the average number of the quasi-steady plume fingers, (nu), scales with the square root of the Rayleigh number Ra and the exchange flux measured by dimensionless convective flux, the Sherwood number, Sh, is a linear function of Ra. The dispersive flux perpendicular to the flow direction induces the formation of wider fingers and subsequently the less convective flux parallel to the flow direction. In the flow system with larger Ra, even the same increase in transverse dispersivity [alpha]T causes stronger impact of the mechanical dispersion on the vertical exchange flow so that (nu) and Sh reduce more with larger [alpha]T . Both measured characteristics, however, follow the same scaling for the non-dispersive homogeneous case by using a modified Rayleigh number, Ra*, considering the mechanical dispersion. The presence of flow barriers along the fault triggers unsteady exchange flow and subsequently controls the growth of the plume fingers. If the barriers are sufficiently wide to dominate the flow system, they create preferential pathways for exchange flow that determines the distribution of the quasi-steady fingers, and (nu) converges to a constant value. In addition, wider barriers induce substantial lateral spreading and enhance the efficiency of structural trapping, and reduce the exchange rate but still follows a linear relationship function of the effective Rayleigh number, Raeff , defined by the vertical effective permeability. This study is motivated by geological CO₂ storage in brine-saturated aquifer, but the effect of geological heterogeneity is also important in many other geological and engineering applications, in particular the risk assessment of the injection operations or the migration of hydrocarbons in tectonic-driven or hydraulically developed faults in reservoirs. Better understanding of fluid flow in geologically heterogeneous formations will allow more precise estimate of the reservoir capacity as well as more efficient operation of injection or production wells. / text