High order hybridizable discontinuous Galerkin method for three-phase flow in porous media

TL;DR

This work develops a high-order HDG method for incompressible, immiscible three-phase flow in porous media, using a semi-implicit operator-splitting to relax nonlinearities and allow large time steps. By introducing trace variables on the mesh skeleton and applying static condensation, the method achieves a substantial reduction in degrees of freedom while maintaining local conservation and compatibility with flow-transport coupling. The authors demonstrate optimal $k+1$ convergence for all variables, and a local postprocessing step yields potential $k+2$ superconvergence for smooth solutions, even in highly heterogeneous media. Numerical experiments across manufactured solutions and multiple heterogeneous scenarios confirm robustness, high-order accuracy, and practical efficiency, highlighting HDG as a competitive approach for complex multi-phase flow in porous media.

Abstract

We present a high-order hybridizable discontinuous Galerkin method for the numerical solution of time-dependent three-phase flow in heterogeneous porous media. The underlying algorithm is a semi-implicit operator splitting approach that relaxes the nonlinearity present in the governing equations. By treating the subsequent equations implicitly, we obtain solution that remain stable for large time steps. The hybridizable discontinuous Galerkin method allows for static condensation, which significantly reduces the total number of degrees of freedom, especially when compared to classical discontinuous Galerkin methods. Several numerical tests are given, for example, we verify analytic convergence rates for the method, as well as examine its robustness in both homogeneous and heterogeneous porous media.

Figures (18)

• Figure 1: Flow chart of semi-implicit method for three-phase flow. Subscript of $n$ refers to the $n$th time step.
• Figure 2: HDG nodes (of the Fekete type) for a mesh with four elements and polynomial degree $k=6$.
• Figure 3: Wetting phase saturation profile along the line $y=500$ for various polynomial orders. Simulation is terminated at $t=100$ days.
• Figure 4: Light oil phase saturation profile along the line $y=500$ for various polynomial orders. Simulation is terminated at $t=100$ days.
• Figure 5: Wetting and light oil phase saturation profile along $y=375$ for various polynomial orders. Simulation is terminated at $t=100$ days. Under $p$-refinement spurious oscillations are reduced, and the approximation converges.