Mathematical model and its solution for water-altering-gas (WAG) injection process incorporating the effect of miscibility, gravity, viscous fingering and permeability heterogeneity

Abstract

The oil recovery from the water-alternating-gas (WAG) injection process is significantly impacted by gravity, viscous fingering, and the permeability heterogeneity of the reservoir. Therefore, the combined effect of these parameters cannot be neglected in the WAG injection process. This article presents the development of a mathematical model of oil recovery and its solution for the WAG injection process that takes into account the combined effects of miscibility change, viscous fingering, gravity, and permeability heterogeneity in an inclined stratified reservoir. First, the governing equations and fractional flow functions were explained in relation to the effects of gravity, permeability heterogeneity, viscous fingering, and miscibility change in an inclined stratified porous medium. Then, a mathematical model was developed using fractional flow functions and conservation equations for both injected water and solvent. The model was generated in the form of a quasi-linear first-order partial differential equation, which was solved analytically in two dimensions (2-D) utilizing vector calculus. Next, this model was solved analytically by applying wave theory to practical constant pressure boundary conditions, which generate distinct waves at different times to provide pressure and saturation at the displacement front location. The total volumetric flux and breakthrough time are calculated from the analytical solution at various times. The presented analytical solutions can be used to predict different parameters for stratified porous media in a fast and efficient way. Finally, the results of analytical solution validated with high-resolution numerical simulation for a wide range of permeability heterogeneity, which shows excellent agreement for breakthrough time, saturation, and pressure versus displacement location of different waves at different times. This analytical solution will save time and money by offering guidance to engineers for analyzing the saturation and pressure distribution at different times and predicting oil recovery. It will also improve the understanding of the physics underlying the multiphase flow WAG injection process in heterogeneous reservoirs.