عنوان مقاله [English]
Mass conservation equations of three-phase incompressible, immiscible flow in porous media, called Buckley-Leverett (BL) equations, are of great importance in the study of flow in hydrocarbon reservoirs. For two-phase flow, these equations reduce to a scalar BL equation, in which the flux function is non-convex, as its convexity changes with respect to the conserved variable. Similarly, in three-phase flow, the non-convexity of flux functions leads to a linearly degenerate system of equations. In both non-convex and degenerate problems, it is necessary that the so-called Oleinik entropy condition is satisfied, in order to have physically admissible answers.
Moreover, when gravity is included in the BL equations, the non-convexity of the flux function increases and some of the wave speeds of the system of equations become zero. Such a situation refers to a sonic point in gas
dynamics. Many numerical methods have difficulty dealing with these points, especially when sonic points occur on an expansion wave. In such cases, some numerical methods produce non-physical expansion shocks instead of expansion fans. The latter case occurs when there is a counter current multi-phase flow in porous media.
In order to avoid this kind of problem, entropy correction procedures should be used. These procedures usually generate additional numerical diffusion at sonic points to disperse expansion shocks.
In this work, a relatively new numerical method, called the dominant wave method, is used to simulate immiscible flow in porous media. The dominant wave method is represented in the form of central methods. This method is locally conservative and uses a finite volume numerical flux approximation. The dominant wave method eliminates the necessity of characteristic decomposition by detecting the dominant wave speed of the system. On the other hand, this method generates less numerical diffusion in comparison with conventional central methods. Here, slope limiters are used to obtain a higher order of accuracy. Fromms and minmod limiters are used in this paper, and a comparison between these two limiters is provided. To avoid expansion shocks at sonic points, Hartens entropy correction is used.
In this work, gravitational effects are studied in a homogenous, one-dimensional reservoir using the Buckley-Leverett model. Three different models for relative permeability functions are considered in this study, including Corey, Extended Corey and Stone type models.