HDG Methods for incompressible and immiscible two-phase flow in porous media: the existence and convergence of discrete solutions

In this paper, we investigate the numerical approximation of incompressible, immiscible two-phase flow in porous media using backward Euler methods for time discretization and hybridized discontinuous Galerkin (HDG) methods for saturation and pressure discretizations. Using the Brouwer fixed-point theorem, we establish the existence of a discrete solution for the proposed scheme. Furthermore, the sequence of discrete solutions solved by the discrete scheme converges, potentially after passing to a subsequence, to a weak solution of the continuous problem. Finally, two numerical examples are presented to validate both the accuracy and performance of the discrete scheme.