The Solution by the Wave Curve Method of Three-Phase Flow in Virgin Reservoirs

Abstract

There are two goals of this study. The first is to provide an introduction to the wave curve method for finding the analytic solution of a porous medium injection problem. Similar to fractional and chromatographic flow theory, the wave curve method is based on the method of characteristics, but it is applicable to an expanded range of physical processes in porous medium flow. The second goal is to solve injection problems for immiscible threephase flow, as described by Corey's model, in which a mixture of gas and water is injected into a porous medium containing oil and irreducible water. In particular we determine, for any 123 100 A. V. Azevedo et al.choice of the phase viscosities, the proportion of the injected fluids that maximizes recovery around breakthrough time. Numerical simulations are performed to compare our solutions for Corey's model with those of other models. For the injection problems we consider, solutions for Corey's model are very similar to those for Stone's model, despite the presence of an elliptic region in the latter; and they are very different from those for the Juanes-Patzek model, which preserves strict hyperbolicity. A nice feature of our analytical method is that it facilitates explaining both differences and similarities among the solutions for the three models considered.

Bifurcation under parameter change of Riemann solutions for nonstrictly hyperbolic systems

“…This remarkable fact was observed for left and right data on the boundary of the saturation triangle [4,5].…”

“…Following [45], we classify discontinuities in terms of the nature of the equilibrium points U − and U + of the system of ODE's (5). In [45], the following types of equilibria were studied: repeller, repeller saddle, saddle, saddle attractor and attractor.…”

“…The current work shows how the bifurcation between types I and II for the local solution in three-phase flow is governed by a delicate balance between quadratic and cubic terms in the Taylor expansion of the flux at the umbilic point. On the other hand, away from the umbilic point, the solution seems insensitive to parameter change [5].…”

Negative Saturation Approach for Non-Isothermal Compositional Two-Phase Flow Simulations

“…For other cases discussed later, we have used the input data that are shown in Table 1. For details of the wave-curve method, we refer to Wahanik et al (2010) and Azevedo et al (2010). Figure 1 compares the numerical NegSat solution to the analytical solution obtained with the wave-curve method.…”

Recovery of Light Oil by Air Injection

“…This wave represents the slowest saturation wave. Recall that several saturation waves typically appear for three-phase flow [4,46] and the faster saturation waves were only observed at earlier times and moved away to the right.…”