Method of negative saturations for flow with variable number of phases in porous media: extension to three-phase multi-component case
Various EOR methods lead to the appearance of specific macroscopic surfaces called interfaces of phase transition (IPT) such that the number of phases on two sides of an IPT is different, and fluids separated by an IPT are in non-equilibrium. Therefore, the flow equations are also different on two sides of an IPT and cannot be deduced from each other by a continuous degeneration, which imposes difficulties in numerical modelling. To describe such systems, we developed a new conceptual mathematical method based on the replacement of the real single-phase fluid by an imaginary multi-phase multi-component continuum having fictitious properties. As the result, the fluid over all zones becomes multi-phase and can be described by uniform multi-phase hydro- and thermodynamic equations, which allows applying the direct numerical simulation. The equivalence principle determines the physical properties of the fictitious multi-phase fluid, as well as the structure of the uniform multi-phase equations. It also proves that the saturation of each phase becomes an extended function negative or higher than unity in non-equilibrium zones, which becomes the efficient method of tracking the interfaces, the number of phases at any point, and their degree of disequilibrium. The method was developed in [1, 2] for the two-phase case. In the present paper, the new version of the method is developed for the three-phase case with gravity, diffusion, and capillarity. We have obtained the new equivalent uniform multi-phase equations which contain additional non-classical terms responsible for the diffusion and gravity across an IPT. The comparison with classical method is presented. The presentation is illustrated by several examples of simulation by means of the code developed by the research group; their concern: EOR by miscible methods and CO 2 bubble raising in aquifer.