Fluid flow through porous media is a subject of common interest in many branches of engineering as well as applied natural science. In this work, we investigate the behavior and numerical treatment of multiphase flow in porous media. To be more specific, we take the sequestration of CO₂ in geological media as an example. Mathematical modeling and numerical study of carbon sequestration helps to predict both short and long-term behavior of CO₂ storage in geological media, which can be a benefit in many ways. This work aims at developing accurate and efficient numerical treatment for problems in porous media on non-rectangular geometries. Numerical treatment of Darcy flow and transport have been developed for many years on rectangular and simplical meshes. However, extra effort is required to extend them to general non-rectangular meshes. In this dissertation work, for flow simulation, we develop new H(div)- conforming mixed finite elements (AT and AT [superscript red] ) which are accurate on cuboidal hexahedra. We also develop the new direct serendipity finite element (DS [subscript r] ), which is H¹ -conforming and accurate on quadrilaterals and a special family of hexahedra called truncated cubes. The use of the direct serendipity finite element reduces the number of degrees of freedom significantly and therefore accelerates numerical simulations. For transport, we use the newly developed direct serendipity elements in an enriched Galerkin method (EG), which is locally conservative. The entropy viscosity stabilization is applied to eliminate spurious oscillations. We test the EG-DS [subscript r] method on problems with diffusion, transport, and coupled flow and transport. Finally, we study two-phase flow in heterogeneous porous media with capillary pressure. We work on a new formulation of the problem and force the continuity of the capillary flux with a modification to conquer the degeneracy. The numerical simulation of two-phase flow is conducted on non-rectangular grids and uses the new elements. Advisors/Committee Members: Arbogast, Todd James, 1957- (advisor), Wheeler, Mary F (committee member), Ghattas, Omar (committee member), Demkowicz, Leszek F (committee member), Hesse, Marc A (committee member).