Three-Phase Displacement Theory: An Improved Description of Relative Permeabilities
- Ruben Juanes (Stanford U.) | Tad W. Patzek (U. of California)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 2004
- Document Type
- Journal Paper
- 302 - 313
- 2004. Society of Petroleum Engineers
- 4.1.2 Separation and Treating, 5.3.1 Flow in Porous Media, 5.4.6 Thermal Methods, 4.1.5 Processing Equipment, 5.5.7 Streamline Simulation, 5.5 Reservoir Simulation, 5.2.1 Phase Behavior and PVT Measurements, 5.8.8 Gas-condensate reservoirs, 5.3.2 Multiphase Flow, 5.4.1 Waterflooding, 5.4 Enhanced Recovery, 4.3.4 Scale, 5.4.2 Gas Injection Methods, 4.5.7 Controls and Umbilicals, 1.2.3 Rock properties
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In this paper, we revisit the displacement theory of three-phase flow and provide conditions for a relative permeability model to be physical anywhere in the saturation triangle. When capillarity is ignored, most relative permeability functions used today yield regions in the saturation space where the system of equations is locally elliptic, instead of hyperbolic. We are of the opinion that this behavior is not physical, and we identify necessary conditions that relative permeabilities must obey to preserve strict hyperbolicity. These conditions are in agreement with experimental observations and pore-scale physics.
We also present a general analytical solution to the Riemann problem (constant initial and injected states) for three-phase flow, when the system satisfies certain physical conditions that are natural extensions of the two-phase flow case. We describe the characteristic waves that may arise, concluding that only nine combi- nations of rarefactions, shocks, and rarefaction-shocks are possible. Some of these wave combinations may have been overlooked but can potentially be important in certain recovery processes.
The analytical developments presented here will be useful in the planning and interpretation of three-phase displacement experiments, in the formulation of consistent relative permeability models, and in the implementation of streamline simulators.
Three immiscible fluids - water, oil, and gas - may flow in many processes of great practical importance: in primary production below bubblepoint and with movable water; in waterfloods, manmade and natural; in immiscible CO2 floods; in steamfloods; in some gas condensate reservoirs; in gravity drainage of gas caps with oil and water; in WAG processes; and in contaminant intrusions into the shallow subsurface, just to name a few.
Relative permeabilities to water, oil, and gas are perhaps the most important rock/fluid descriptors in reservoir engineering. Currently, these permeabilities are routinely backed out from the theories of transient, high-rate displacements of inert and incompressible fluids that flow in short cores subjected to very high pressure gradients. More recently, the time evolution of area-averaged fluid saturations was measured with a CT scanner and, with several assumptions,1 used to estimate the respective relative permeabilities in gravity drainage. Superior precision of the latter approach allowed the determination of relative permeabilities as low as 10-6.
When the fractional-flow approach is used, flow of three immiscible incompressible fluids is described by a pressure equation and a 2×2 system of saturation equations.2 It was long believed (at least in the Western literature) that, in the absence of capillarity, the system of equations would be hyperbolic for any relative permeability functions. This is far from being the case and, in fact, loss of hyperbolicity occurs for most relative permeability models used today. In this paper, we argue that such a behavior is not physically based, and we show how to overcome this deficiency. To do so, we adopt an opposite viewpoint to that of the existing literature: strict hyperbolicity of the system is assumed, and the implications on the functional form of the relative permeabilities are analyzed.
There is a theory behind each quantitative experiment. Not only does any theory reduce and abstract experience, but it also overreaches it by extra assumptions made for definiteness. Theory, in its turn, predicts the results of some specific experiments. The body of theory furnishes the concepts and formul by which experiment can be interpreted as being in accord or discord with it. Experiment, indeed, is a necessary adjunct to a physical theory, but it is an adjunct, not the master.3 In other words, the relative permeability models are only as good as theories behind the displacement experiments from which these models have been obtained. If the theory is flawed, so are the relative permeabilities.
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