Barodiffusive Extension of Three-phase Flow Model for NegSat Method and Regularization of Three-phase Instability

Two fundamental problems of three-phase compositional flow in underground reservoirs have been solved by introducing the similar technique of barodiffusive extension of the classical three-phase model. It consists of introducing of the pseudo barodiffusion terms that are proportional to the weighted sum of the gradients of phase pressures, due to which one can change the direction of the fluxes of individual chemical components.

First of all, this technique enabled us to complete the method of negative saturations for three-phase flow, which was developed to describe the situations when various zones of reservoir contain different number of phases. The method consists of replacing the true fluid by a fictitious three-phase fluid having specific properties that satisfy the equivalence principle. Two fundamental problems, non resolved in preceding publications, concern (a) the replacement of a two-phase fluid by three phases, and (b) the extension to the case when overcritical zones appear. We have shown that the main difficulty in establishing the equivalence between two-phase and three-phase fluids consists of the non-colinear fluxes of chemical components in a two-phase flow. To reach the vectorial equivalence between fluxes, we have introduced the pseudo barodiffusion in the fictitious three-phase fluid. The barodiffusion coefficients and the directions of the fluxes result from the equivalence conditions in a unique way. The same technique provides the solution for the case when the flow contains the zones occupied by overcritical fluid.

In the case of ideal mixing within the phases without capillarity, the flow equations can be converted to the system of conservation laws with respect to the saturations or total concentrations. However the uniform flow equations are non-classical due to the terms of pseudo barodiffusion. The analysis has revealed that the barodiffusion terms ensure the hyperbolic character of the system. Consequently, the well known physical instability that arises in three-phase flow due the loss of hyperbolicity, does not appear in our extended barodiffusive model. Thus, the introduction of the small barodiffusion is the way to suppress the appearance of three-phase instability.

To ensure the numerical stability, we applied the monotone upwind high-order scheme for conservation laws with predictor-corrector. We have calculated several cases of miscible gas injection into the reservoir containing initially oil and water, and proved the good convergence of the result obtained compared to the simulations performed by Eclipse compositional and other techniques.