Carbon sequestration involves capturing CO2 from an exhaust source, compressing it to a supercritical fluid state, and injecting it underground where it can be stored. The environmental engineering goal of carbon sequestration is to prevent further increases in atmospheric CO2 concentration levels. This dissertation examines three aspects of carbon sequestration pertaining to the fluid dynamics of injection and post-injection within the geologic subsurface. First, a time dependent injection strategy of brine alternating with CO2 is proposed as a method to reduce the CO2 mobility by increasing the rates of residual trapping and dissolution. After making some assumptions, the equations governing the dynamics of CO2 mass transport become a coupled set of 1D wave equations, whose wave speeds provide insight into the relative permeability conditions required for this injection strategy to be most effective. Numerical solutions using the method of characteristics are then compared against 3D TOUGH2 simulations and comparable favorably to one another. Both models predict that alternating brine injection can reduce the mass fraction of mobile CO2 to less than 10% using a volume ratio brine:CO2 of less than 2.75 and on time scales that are 100 – 10,000 times faster than would occur with a continuous injection of CO2. Second, the stability of residually trapped CO2 is analyzed with respect to its susceptibility to become remobilized. Here, a reservoir containing a region of homogeneously dispersed pockets of residually trapped CO2 is considered. Should the pore pressure decrease, the CO2 will expand, remobilize, rise within the domain, and potentially spread along the caprock of the formation. The dynamics of this process are predicted using two different relative permeability models: Brooks-Corey, and a modified Brooks-Corey that incorporates percolation theory. Experimental data justifying this latter model is presented. More importantly, the time scales of remobilized CO2 motion are shown to be vastly different for these two models and suggest the need for further experimental data. Finally, as dry CO2 is injected into brine occupied reservoirs, there exists the potential for mechanical tension forces to be generated within the brine which can cause cavitation to occur. These cavitation dynamics are modeled here using explicit and averaged equations and compared to experimental data from drying an idealized synthetic heterogeneous porous media.