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.