Conservation laws of mixed type describing three-phase flow in porous media
SIAM Journal on Applied Mathematics
Proceedings of the Symposium on Numerical Simulation in Oil Recovery on Numerical simulation in oil recovery
SIAM Journal on Applied Mathematics
Admissibility conditions for shocks in conservation laws that change type
SIAM Journal on Mathematical Analysis
Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type
Zeitschrift für Angewandte Mathematik und Physik (ZAMP)
Nonuniqueness of solutions of Riemann problems
Zeitschrift für Angewandte Mathematik und Physik (ZAMP)
On the oscillatory solutions in hyperbolic conservation laws
Nonlinear Analysis: Real World Applications
Fundamentals of Numerical Reservoir Simulation
Fundamentals of Numerical Reservoir Simulation
Nonexistence of Riemann solutions for a quadratic model deriving from petroleum engineering
Nonlinear Analysis: Real World Applications
Three-phase immiscible displacement in heterogeneous petroleum reservoirs
Mathematics and Computers in Simulation - Special issue: Applied and computational mathematics - selected papers of the fifth PanAmerican workshop - June 21-25, 2004, Tegucigalpa, Honduras
Numerical Modeling of Degenerate Equations in Porous Media Flow
Journal of Scientific Computing
Mathematics and Computers in Simulation
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Standard models for immiscible three-phase flow in porous media exhibit unusual behavior associated with loss of strict hyperbolicity. Anomalies were at one time thought to be confined to the region of nonhyperbolicity, where the purely convective form of the model is ill-posed. However, recent abstract results have revealed that diffusion terms, which are usually neglected, can have a significant effect. The delicate interplay between convection and diffusion determines a larger region of diffusive linear instability. For artificial and numerical diffusion, these two regions usually coincide, but in general they do not.Accordingly, in this paper, we investigate models of immiscible three-phase flow that account for the physical diffusive effects caused by capillary pressure differences among the phases. Our results indicate that, indeed, the locus of instability is enlarged by the effects of capillarity, which therefore entails complicated behavior even in the region of strict hyperbolicity. More precisely, we demonstrate the following results. (1) For general immiscible three-phase flow models, if there is stability near the boundary of the saturation triangle, then there exists a Dumortier-Roussarie-Sotomayor (DRS) bifurcation point within the region of strict hyperbolicity. Such a point lies on the boundary of the diffusive linear instability region. Moreover, as we have shown in previous works, existence of a DRS point (satisfying certain nondegeracy conditions) implies nonuniqueness of Riemann solutions, with corresponding nontrivial asymptotic dynamics at the diffusive level and ill-posedness for the purely convective form of the equations. (2) Models employing the interpolation formula of Stone (1970) to define the relative permeabilities can be linearly unstable near a corner of the saturation triangle. We illustrate this instability with an example in which the two-phase permeabilities are quadratic.Results (1) and (2) are obtained as consequences of more general theory concerning Majda-Pego stability and existence of DRS points, developed for any two-component system and applied to three-phase flow. These results establish the need for properly modelling capillary diffusion terms, for they have a significant influence on the well-posedness of the initial-value problem. They also suggest that generic immiscible three-phase flow models, such as those employing Stone permeabilities, are inadequate for describing three-phase flow.