Nowadays, numerical simulation of oil reservoirs plays a major role in petroleum engineering. During the initial stages of hydrocarbon production (primary recovery), the pressure difference between the reservoir and the separator at the surface is high enough to move the hydrocarbons. Afterwards, as the pressure of the reservoir decreases, a fluid (usually water) is injected to maintain the flow rate. This is known as secondary recovery.
If a single hydrocarbon is considered and the pressure is above its bubble point pressure, the two-phase immiscible flow model is widely used in industry.
We assume that the two phases, water and oil, and the rocks are incompressible, the phases are immiscible and completely fill the voids of the porous media. In this scenario, the governing equations are obtained from combining, for each phase, mass conservation with Darcy’s law. This leads to a coupled nonlinear system of transient PDE’s . Improving the efficiency and the accuracy of numerical methods to solve these equations
is of the major importance during the planning, management and environmental analysis of oilfields [2, 3].
We present a high-order hybridizable discontinuous Galerkin formulation (HDG)  to solve the two-phase flow problem in a heterogeneous porous media. In particular, we apply the HDG formulation in two different schemes, the fully implicit, and the implicit pressure, explicit saturation (IMPES). According to , when a time integration algorithm of order p + 1 is used in conjunction with element-wise polynomials of degree p = 0, the scalar unknowns and their fluxes converge with order p + 1 in the L 2 -norm. In addition, we apply a local
post-processing technique to obtain a new approximation of the scalar unknowns that converge with order p + 2. Finally, we will present several examples in order to illustrate and discuss the main features of the proposed formulation.