This thesis proposal explores e cient computational methods for the approximation of solutions to partial di erential equations that model ow and transport phenomena in porous media. These problems can be challenging to solve as the governing equations are coupled, nonlinear, and material properties are often highly varying and discontinuous. The high-order implicit hybridizable discontinuous method (HDG) is utilized for the discretization, which signi cantly reduces the computational cost. To our knowledge, HDG methods have not been previously applied to this class of complex problems in porous media. The HDG method is high-order accurate, locally mass-conservative, allows us to use unstructured complicated meshes, and enables the use of static condensation. We demonstrate that the HDG method is able to e ciently generate high- delity simulations of ow and transport phenomena in porous media. Several challenging benchmarks are used to verify and validate the method in heterogeneous porous media. High-order methods give rise to less sparse discretization matrices, which is problematic for linear solvers. To address the issue of less sparse discretization matrices (compared to low-order methods), we develop and deployed a novel nested multigrid method. It is based on a combination of p-multgrid, h-multigrid and algebraic multigrid. The method is demonstrated to be algorithmically e cient, achieving convergences rates of at most 0:2. We also show how to implement the multigrid technique in many-core parallel architectures. Parallel computing is a critical step in the simulation process, as it allows us to consider larger problems, and potentially generate simulations faster. Traditional performance measures like FLOPs or run-time are not entirely appropriate for nite element problems, as they ignore solution accuracy. A new accuracy-inclusive performance measure has been investigated as a part of my research. This performance measure, called the Time-Accuracy-Size spectrum (TAS), allows us to have a more complete assessment of how e cient our algorithms are. Utilizing TAS also enables a systematic way of determining which discretization is best suited for a given application.