Traditional mathematical models of multiphase flow in porous media use a straightforward extension of Darcy's equation, and the key element of these models is the appropriate formulation of the relative permeability functions. It is well known that for one-dimensional flow of three immiscible incompressible fluids, when capillarity is neglected, most relative permeability models used today give rise to regions in the saturation space with elliptic behavior (the so-called elliptic regions). We believe that this behavior is not physical, but rather the result of an incorrect (or incomplete) mathematical model. In this paper we identify necessary conditions that must be satisfied by the relative permeability functions, so that the system of equations 1 R. Juanes and T. W. Patzek: Strictly hyperbolic models of three-phase flow 2 describing three-phase flow is strictly hyperbolic everywhere in the saturation triangle. These conditions seem to be in good agreement with pore-scale physics and experimental data.

In this paper we study one-dimensional three-phase flow of immiscible, incompressible fluids through porous media. The model uses the common multiphase flow extension of Darcy's equation, and does not include gravity and capillarity effects. Under these conditions, the mathematical problem reduces to a 2 × 2 system of conservation laws, whose essential features are: (1) the system is strictly hyperbolic; (2) both characteristic fields are nongenuinely nonlinear, with single, connected inflection loci. We argue that these are necessary properties for the solution to be physically sensible, and show they are natural extensions of the two-phase flow model. We present the complete analytical solution to the Riemann problem (constant initial and 1 R. Juanes and T. W. Patzek: Analytical solution to the Riemann problem . . . 2 injected states) in detail, and describe the characteristic waves that may arise, concluding that only 9 combinations of rarefactions, shocks and rarefaction-shocks are possible. We demonstrate that assuming the saturation paths of the solution are straight lines may result in very inaccurate predictions for some realistic systems. Efficient algorithms for computing the exact solution are also given, making the analytical developments presented here readily applicable to the interpretation of lab displacement experiments, and to the implementation in streamline simulators.

No abstract

The complex physics of multiphase flow in porous media are usually modeled at the field scale using Darcy-type formulations. The key descriptors of such models are the relative permeabilities to each of the flowing phases. It is well known that, whenever the fluid saturations undergo a cyclic process, relative permeabilities display hysteresis effects. In this paper we investigate hysteresis in the relative permeability of the hydrocarbon phase in a two-phase system. We propose a new model of trapping and waterflood relative permeability, which is applicable for the entire range of rock wettability conditions. The proposed formulation overcomes some of the limitations of existing trapping and relative permeability models. The new model is validated by means of pore-network simulation of primary drainage and waterflooding. We study the dependence of trapped (residual) hydrocarbon saturation and waterflood relative permeability on several fluid/rock properties, most notably the wettability and the initial water saturation. The relevance of relative permeability hysteresis is then evaluated for modeling geological CO 2 sequestration processes. Here we concentrate on CO 2 injection in saline aquifers. In this setting, the CO 2 is the nonwetting phase, and trapping of the CO 2 is an essential mechanism after the injection phase, during the lateral and upward migration of the CO 2 plume. We demonstrate the importance of accounting for CO 2 trapping in the relative permeability model for predicting the distribution and mobility of CO 2 in the formation. We conclude that a proper treatment of the nonwetting phase trapping leads to a higher estimate of the amount of CO 2 that it is safe to inject.

No abstract