Formulations and numerical methods of the black oil model in porous media

The black oil model for describing the hydrocarbon equilibrium in porous media is considered in this paper. Various formulations of the governing equations that describe this model, including phase, weighted fluid, global, and pseudoglobal pressure-saturation formulations with the total velocity and flux, are rst constructed. These formulations are more suitable for their mathematical and numerical analysis. Finite element approximate procedures are then analyzed. These procedures are based on the use of mixed finite element methods for the pressure equations and Galerkin finite element methods for the saturation equations. Error estimates are stated first for the case where capillary diffusion coefficients are assumed to be uniformly positive. Then an error analysis is carried out in detail for a degenerate case where these coefficients can be zero.