  We present a new method for approximating solutions to the incompressible
miscible displacement problem in porous media. At the discrete level, the
coupled nonlinear system has been split into two linear systems that are solved
sequentially. The method is based on a hybridizable discontinuous Galerkin
method for the Darcy flow, which produces a mass--conservative flux
approximation, and a hybridizable discontinuous Galerkin method for the
transport equation. The resulting method is high order accurate. Due to the
implicit treatment of the system of partial differential equations, we observe
computationally that no slope limiters are needed. Numerical experiments are
provided that show that the method converges optimally and is robust for highly
heterogeneous porous media in 2D and 3D.
