This dissertation presents analytical solutions to address several unresolved issues on the modeling of CO₂ flow in permeable media. Analytical solutions are important as numerical simulations do not yield explicit expressions in terms of the model parameters. In addition, simulations that provide the most comprehensive solutions to multiphase flow problems are computationally intensive. Accordingly, we address the following topics in this dissertation. The method of characteristics (MOC) solution of the overall mass conservation equation of CO₂ in two-phase flow through permeable media is derived in the presence of compressibility. The formally developed MOC solutions rely on the incompressible fluid and rock assumptions that are rarely met in practice; hence, the incompressible assumption is relaxed and the first semi-analytic MOC solution for compressible flow is derived. The analytical solution is verified by simulation results. Fractional flow theory is applied to evaluate the CO2 storage capacity of one-dimensional (1D) saline aquifers. Lack of an accurate estimation of the CO₂ storage capacity stands in the way of the fully implementation of CO₂ storage in aquifers. The notion of optimal solvent-water-slug size is incorporated into the graphical solution of combined geochemical front propagation and fractional flow theory to determine the CO₂ storage capacity of aquifers. The analytical solution is verified by simulation results. The limits of the Walsh and Lake (WL) method to predict the performance of CO₂ injection is examined when miscibility is not achieved. The idea of an analogous first-contact miscible flood is implemented into the WL method to study miscibly-degraded simultaneous water and gas (SWAG) displacements. The simulation verifies the WL solutions. For the two-dimensional (2D) displacements, the predicted optimal SWAG ratio is accurate when the permeable medium is fairly homogeneous with a small cross-flow or heterogeneous with a large lateral correlation length (the same size or greater than the interwell spacing). We conclude that the WL solution is accurate when the mixing zone grows linearly with time. We examine decoupling of large and small-scale heterogeneity in multilayered reservoirs. In addition, using an analytical solution derived in this research, the fraction of layers in which the channeling occurs is determined as a function of the Koval factor and input dispersivity. We successfully present a simulation configuration to verify the off-diagonal elements of the numerical dispersion tensor. Numerical dispersion is inevitably introduced into the finite difference approximations of the 2D convection-dispersion equation. We show that the off-diagonal elements of the numerical dispersion tensor double when the flow velocity changes with distance. In addition, the simulation results reveal that the flow becomes more dispersive with distance travelled if there is convective cross-flow. In addition, local mixing increases with the convective cross-flow between layers. A numerical indicator is presented to describe the nature of CO₂ miscible displacements in heterogeneous permeable media. Hence, the quantitative distinction between flow patterns becomes possible despite the traditionally qualitative approach. The correlation coefficient function is adopted to assign numerical values to flow patterns. The simulation results confirm the accuracy of the descriptive flow pattern values. The order-of-one scaling analysis procedure is implemented to provide a unique set of dimensionless scaling groups of 2D SWAG displacements. The order-of-one scaling analysis is a strong mathematical approach to determine approximations that are allowed for a particular transport phenomenon. For the first time, we implement the scaling analysis of miscible displacements while considering effects of water salinity, dissolution of CO₂ in the aqueous phase, and complex configurations of injection and production wells. / text