# On a universal structure for immiscible three-phase flow in virgin reservoirs

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## Abstract

We discuss the solution for commonly used models of the flow resulting from the injection of any proportion of three immiscible fluids such as water, oil, and gas in a reservoir initially containing oil and residual water. The solutions supported in the universal structure generically belong to two classes, characterized by the location of the injection state in the saturation triangle. Each class of solutions occurs for injection states in one of the two regions, separated by a curve of states for most of which the interstitial speeds of water and gas are equal. This is a separatrix curve because on one side water appears at breakthrough, while gas appears for injection states on the other side. In other words, the behavior near breakthrough is flow of oil and of the dominant phase, either water or gas; the non-dominant phase is left behind. Our arguments are rigorous for the class of Corey models with convex relative permeability functions. They also hold for Stone’s interpolation I model [5]. This description of the universal structure of solutions for the injection problems is valid for any values of phase viscosities. The inevitable presence of an umbilic point (or of an elliptic region for the Stone model) seems to be the cause of this universal solution structure. This universal structure was perceived recently in the particular case of quadratic Corey relative permeability models and with the injected state consisting of a mixture of water and gas but no oil [5]. However, the results of the present paper are more general in two ways. First, they are valid for a set of permeability functions that is stable under perturbations, the set of convex permeabilities. Second, they are valid for the injection of any proportion of three rather than only two phases that were the scope of [5].

### Keywords

Wave curve method Riemann problem Umbilic point WAG injection Corey, Stone, and Brooks-Corey relative permeability models Immiscible flow## Preview

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