We develop a high-order hybridizable discontinuous Galerkin (HDG) formulation to solve the immiscible and incompressible two-phase flow problem in a heterogeneous porous media. The HDG method is locally conservative, has fewer degrees of freedom than other discontinuous Galerkin methods due to the hybridization procedure, provides built-in stabilization for arbitrary polynomial degrees and, if the error of the temporal discretization is low enough, the pressure, the saturation and their fluxes converge with order P + 1 in L 2 -norm, being P the polynomial degree. In addition, an element-wise post-process can be applied to obtain a convergence rate of P + 2 in L 2 -norm for the scalar variables. All of these advantages make the HDG method suitable for solving multiphase flow trough porous media. We show numerical evidence of the convergences rates. Finally, to assess the capabilities of the proposed formulation, we apply it to several cases of water-flooding technique for oil recovery. KEYWORDSTwo-phase flow; immiscible; incompressible; hybridizable discontinuous Galerkin method; diagonally implicit Runge-Kutta method; differential algebraic equations.

We present a stable high-order hybridizable discontinuous Galerkin (HDG) formulation coupled with high-order diagonal implicit Runge-Kuta (DIRK) schemes to simulate slightly compressible one-phase flow through porous media. The HDG stability depends on the selection of a single parameter and its definition is crucial to ensure the stability and to achieve the high-order properties of the method. Thus, we extend the work of Nguyen et al. in J. Comput. Phys. 228:8841-8855, 2009 to deduce an analytical expression for the stabilization parameter using the material parameters of the problem and the Engquist-Osher monotone flux scheme. The formulation is high-order accurate for the pressure, the flux and the velocity with the same convergence rate of P+1, being P the polynomial degree of the approximation. This is important because high-order methods have the potential to reduce the computational cost while obtaining more accurate solutions with less dissipation and dispersion errors than low order methods. The formulation can use unstructured meshes to capture the heterogeneous properties of the reservoir. In addition, it is conservative at the element level, which is important when solving PDE's in conservative form. Moreover, a hybridization procedure can

We present a memory-efficient high-order hybridizable discontinuous Galerkin (HDG) formulation coupled with high-order fully implicit Runge-Kutta schemes for immiscible and incompressible two-phase flow through porous media. To obtain the same high-order accuracy in space and time, we propose using high-order temporal schemes that allow using large time steps. Therefore, we require unconditionally stable temporal schemes for any combination of element size, polynomial degree, and time step. Specifically, we use the Radau IIA and Gauss-Legendre schemes, which are unconditionally stable, achieve high-order accuracy with few stages, and do not suffer order reduction in this problem. To reduce the memory footprint of coupling these spatial and temporal high-order schemes, we rewrite the nonlinear system. In this way, we achieve a better sparsity pattern of the Jacobian matrix and less coupling between stages. Furthermore, we propose a fix-point iterative method to further reduce the memory consumption. The saturation solution may present sharp fronts. Thus, the high-order approximation may contain spurious oscillations.To reduce them, we introduce artificial viscosity. We detect the elements with high-oscillations using a computationally efficient shock sensor obtained from the saturation solution and the post-processed saturation of HDG. Finally, we present several examples to assess the capabilities of our formulation.

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