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

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 nondominant 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].

Uniqueness of the Riemann Solution for Three-Phase Flow in a Porous Medium

“…The separatrix B -O is also invariant for system (2.1)-(2.2). Indeed, introducing the net water/gas saturation s wg := s w + s g = 1 − s o and net viscosity ratio r wg := r w +r g , this separatrix is conveniently parametrized as s w = (r w /r wg ) s wg and s g = (r g /r wg ) s wg for 0 ≤ s wg ≤ 1; substituting these expressions for s w and s g in system (2.1)-(2.2) both equations reduce to the single BL equation In the case when more general permeabilities are used instead of quadratic ones the role of the separatrix changes and is played by a different curve reaching O [7].…”

Computing numerical solutions of the pseudo-parabolic Buckley–Leverett equation with dynamic capillary pressure

“…The multidimensional case is the subject of an ongoing work. We expect this approach might be used to other applications as discussed in the introduction, along with the pertinent high order approximation methods (see [2,5,14,27]). In the work [7], the authors use the global pressure and a complementary pressure by using the Kirchhoff transformation as primary unknown.…”

“…Of course, is it possible to overcome this difficulty, but for very special cases as discussed in the works [22,23]; see also [1], where there is a comprehensive discussion on these issues based on the so-called phase-formulation for a three-phase flow problem. Furthermore, for three-phase flow problems in homogeneous medium the arguments in [14] are rigorous for the class of Corey models with convex relative permeability functions (see also [9]). …”

“…Clearly, it depends upon the underlying problem at hand. For instance, see [1-3, 25,27,34] and the references cited therein, for a comprehensive list of works where distinct and successful procedures were developed on separating the underlying physical processes and treating each such process appropriately, which includes multidimensional problems [39,3,32,2] and the situation with the occurrence of nonclassical structure (or resonance effects [5] see also [6]) linked to the three-phase model [1]; see also [14,22,23].…”

“…In [2] the authors solve a convection-diffusion problem linked to the delicate case of three-phase problems (see also [1,14]) by splitting diffusive and convective effects. For each time step, the hyperbolic problem is solved first and next it is solved the parabolic system; the nonlinear diffusive term makes use of solution for convection equation.…”