Highly gravity-driven flow of a NAPL in water-saturated porous media using the discontinuous Galerkin finite-element method with a generalised Godunov schemeHighly gravity-driven flow of a NAPL in water-saturated porous media

In this paper, we develop an implementation of gravity effects within a Global Pressure formulation in a numerical scheme based on the implicit pressure explicit saturation (IMPES) approach. We use the Discontinuous Galerkin Finite-Element Method (DGFEM) combined with a generalised Godunov scheme to model an immiscible two-phase flow with predominant gravity effects. The saturation profile of a displacing non-aqueous phase liquid (NAPL) in an initially water-saturated porous medium depends strongly on the ratio between the total specific discharge and the density difference between the NAPL and water. We discuss the solution of the nonlinear Buckley-Leverett equation for the general case in which the flux function is non-monotonic. Using a detailed functional analysis of the characteristics of the given hyperbolic equation, three limit cases are identified as significant for modelling the shock and rarefaction regions. The derived maximum (or entry) and front saturations of NAPL are functions of the viscosity ratio M and the gravity number G. We first test the developed numerical model in the case of a one-dimensional highly gravity-driven flow of NAPL within a homogeneous porous medium. The numerically calculated NAPL entry and front saturations of NAPL agree well with the theoretical values. Furthermore, the numerical diffusion of the shock front is lower than that of the calculated using a first-order Finite Volume method, which is generally used in reservoir engineering because of its robustness. Finally, we apply the developed DGFEM scheme to a 2D heterogeneous porous medium and analyse its capability of modelling the non-uniform saturation field using spatial moment analysis.