A combined mixed finite element method and discontinuous Galerkin method for hybrid-dimensional fracture models of two-phase flow

Abstract

In this paper, a combined numerical method consisting of the mixed finite element method (MFE) for the pressure equation and the discontinuous Galerkin (DG) method for the saturation equation is proposed to solve hybrid-dimensional fracture models of incompressible two-phase flow in porous media. The hybrid-dimensional fracture models treat fractures as $(d-1)$-dimensional interfaces immersed in $d$-dimensional matrix domains and take fluid exchange between fractures and surrounding matrix into account. Fully implicit approximation schemes combining the MFE-DG method with the backward Euler time discretization for the models with both a single fracture and an intersecting fractures network are all formulated successfully. The stability of the discrete solution is analyzed, and optimal error estimates in $H\left(div\right)$-norm for the velocity and in $L^2$-norm for the pressure are derived, as well as in the discrete $H^1$-norm for the saturation. Numerical experiments with a single fracture and a T-junction intersecting fractures network are conducted to verify the accuracy of our theoretical analysis.