SPE 77539 Three-Phase Displacement Theory: An Improved Description of Relative Permeabilities

A quantitative understanding of three-phase flow in porous media is required to address many diverse processes in the subsurface, e.g., improved oil recovery, CO2 sequestration, and aquifer clean-up. In turn, all predictive models of three-phase flow originate from interpretations of onedimensional laboratory experiments; when these interpretations are flawed, so are the models. In this paper we revisit the foundations of displacement theory in three-phase flow and provide the most general conditions for any relative permeability model to be physical anywhere in the saturation triangle. In doing so, we put to rest a controversy that has persisted in petroleum literature for the better part of the last six decades. When capillarity is ignored, the system of conservation laws describing incompressible immiscible flow should be strictly hyperbolic. This natural property of the system fails for most relative permeability models used today. 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 the most general analytical solution to the Riemann problem (constant initial and injected states) for three-phase flow, and describe the characteristic waves that may arise, concluding that only 9 combinations of rarefactions, shocks and rarefaction-shocks are possible. Some of these wave combinations have been overlooked by many because of the associated conceptual and mathematical difficulties. 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 streamtube simulators. Introduction Three immiscible fluids, water, oil and gas, may flow in many processes of great practical importance: in primary production below bubble point and with movable water; in waterfloods, man-made and natural; immiscible CO2 floods; steam floods; in some gas condensate reservoirs; in gravity drainage of gas caps with oil and water; 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. Nowadays, 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, 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. 2 THREE-PHASE DISPLACEMENT THEORY: . . . SPE 77539 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. It was long believed that, in the absence of capillarity, the system of equations would be strictly hyperbolic for any relative permeability functions. This is far from being the case and, in fact, loss of strict hyperbolicity occurs for virtually all relative permeability models used today. In this paper we argue that such a behavior is not physicallybased, and 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 for 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 formulæ 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. 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. Displacement Theory in Three-Phase Flow Governing Equations. We outline the mathematical formulation of multiphase flow in porous media under the following assumptions: (1) one-dimensional flow; (2) immiscible flow; (3) incompressible fluids; (4) homogeneous rigid porous medium; (5) multiphase flow extension of Darcy’s law; (6) negligible gravitational effects; and (7) negligible capillary pressure effects. A detailed derivation of the governing equations can be found elsewhere. 5 Assumption 2 prevents mass transfer between phases and, therefore, one can identify components with phases. The one dimensional mass conservation equation for the α-phase is, in the absence of source terms: ∂t(mα) + ∂x(Fα) = 0, (1) where mα is the mass density, Fα is the mass flux of the α-phase, and ∂t(·), ∂x(·) denote partial derivatives with respect to time and space, respectively. For three-phase flow the system consists of aqueous, vapor and liquid phases, corresponding to water (w), gas (g) and oil (o) components, respectively. The mass density of each phase is the mass per unit bulk volume of porous medium: