In this paper we present a stabilized finite element method for the numerical solution of three-phase flow in porous media. The key idea of the proposed methodology is a multiscale decomposition into resolved (or grid) scales and unresolved (or subgrid) scales. In the context of subsurface flow and transport, the term multiscale
usually refers to subgrid heterogeneity and upscaling. In contrast, this paper deals with unresolved physics: multiple scales are present in the solution even if the medium is homogeneous.
We use the fractional flow approach for the mathematical description of the three-phase flow equations, which leads to a global pressure equation of elliptic type, and a system of conservation laws (the saturation equations). Numerical di culties in solving these equations include: high nonlinearity, advection-dominated flow, degenerate diffusion, sharpening near-shock solutions, boundary lay- ers, and convergence to nonphysical solutions.
The multiscale formalism allows one to split the original mathematical problem into a grid-scale problem and a subscale problem. The effect of the subgrid scales is then incorporated - in integral form - into the grid-scale equations. Accounting for the subgrid effects results in a finite element method that has enhanced stability properties and is not overly diffusive. Specific original contributions of the methodology proposed herein are: (1) the formulation is applied for the first time to the three-phase flow equations; (2) the fully nonlinear equations in conservation form are used, which is essential for correctly predicting the location of shocks; and (3) a novel expression of a discontinuity-capturing technique is proposed and compared with existing formulations.
The methodology is applied to the simulation of two problems of great practical interest: oil filtration in the vadose zone, and water-gas injection in a hydrocarbon reservoir. These numerical simulations clearly show the potential and applicability of the formulation for solving the highly nonlinear, nearly hyperbolic system of three-phase porous media flow on very coarse grids.