Multiple Viscous Solutions for Systems of Conservation Laws

Azevedo, A. V.; Marchesin, D.

doi:10.2307/2154771

Cited by 9 publications

(19 citation statements)

References 9 publications

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“…Other authors have made the point that loss of hyperbolicity causes severe difficulties; see Azevedo and Marchesin (1995); Azevedo et al (1996Azevedo et al ( , 1999Azevedo et al ( , 2002; Bell et al (1986);Čanić (2003); Frid and Liu (1998). The difficulties entailed by loss of hyperbolicity have been mentioned as supporting Juanes and Patzek's model.…”

Section: Discussionmentioning

confidence: 96%

“…Our conclusions do not contradict Bell et al (1986), as these authors deliberately chose parameters in Stone's model that maximize the size of the elliptic region. All the strange phenomena observed by many authors (see Holden (1987); Azevedo and Marchesin (1995); Azevedo et al (1996Azevedo et al ( , 1999Azevedo et al ( , 2002; Bell et al (1986);Čanić (2003); Frid and Liu (1998)) occur in an area about 10 −3 of the saturation triangle. In Fig.…”

Section: Comparison Between the Models Of Corey And Of Stonementioning

confidence: 92%

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The Solution by the Wave Curve Method of Three-Phase Flow in Virgin Reservoirs

et al. 2009

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There are two goals of this study. The first is to provide an introduction to the wave curve method for finding the analytic solution of a porous medium injection problem. Similar to fractional and chromatographic flow theory, the wave curve method is based on the method of characteristics, but it is applicable to an expanded range of physical processes in porous medium flow. The second goal is to solve injection problems for immiscible threephase flow, as described by Corey's model, in which a mixture of gas and water is injected into a porous medium containing oil and irreducible water. In particular we determine, for any 123 100 A. V. Azevedo et al.choice of the phase viscosities, the proportion of the injected fluids that maximizes recovery around breakthrough time. Numerical simulations are performed to compare our solutions for Corey's model with those of other models. For the injection problems we consider, solutions for Corey's model are very similar to those for Stone's model, despite the presence of an elliptic region in the latter; and they are very different from those for the Juanes-Patzek model, which preserves strict hyperbolicity. A nice feature of our analytical method is that it facilitates explaining both differences and similarities among the solutions for the three models considered.

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Section: Discussionmentioning

confidence: 96%

Section: Comparison Between the Models Of Corey And Of Stonementioning

confidence: 92%

The Solution by the Wave Curve Method of Three-Phase Flow in Virgin Reservoirs

et al. 2009

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“…2. In three-phase, immiscible flow the loss of hyperbolicity may cause nonunique solutions (Azevedo and Marchesin 1995;Azevedo et al 1996). When umbilic points are present, it is necessary to introduce transitional waves that do not meet Lax and Liu entropy conditions in order to construct solutions (Lax 1957;Isaacson et al 1990;Schecter et al 1996).…”

Section: Loss Of Hyperbolicitymentioning

confidence: 99%

“…As a consequence, the eigenvalues of Eq. 11 may become complex conjugates (Azevedo and Marchesin 1995;Azevedo et al 1996;Falls and Schulte 1992a, b;Guzman and Fayers 1997a, b). This would result in two complex and one real characteristic.…”

Section: Loss Of Hyperbolicitymentioning

confidence: 99%

Four-component gas/water/oil displacements in one dimension: Part I. structure of the conservation law

2007

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This paper presents the first analysis of the mathematical structure of a system of conservation laws modeling compositional flow of four components in three phases. The phase behavior that results from assuming the equilibrium volume ratios of the components in the phases are fixed (constant K-values) when up to three phases may form, is studied. We parameterize the equations in the three-phase region and show that within the three-phase region two of the characteristic curves can be found using three-phase immiscible flow theory. The third eigenvalue can also be found analytically when the K-values are constant. We show that the eigenvalue problem given by the conservation law has a discontinuity at the boundary of the two-and three-phase regions. Finally, the loss of strict hyperbolicity is discussed.

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“…In threephase immiscible flow it has been shown that it is not always possible to construct solutions to Riemann problems using Lax entropy constraints (Azevedo and Marchesin 1995;Azevedo et al 1996;Isaacson et al 1990;Marchesin et al 1997;Schecter et al 1996). A new entropy condition is necessary.…”

Section: Fast Slow and Transitional Shocksmentioning

confidence: 99%

Four-component gas/water/oil displacements in one dimension: part II, example solutions

2007

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In this article analytical solutions are constructed for a system of conservation laws modeling compositional flow of four components in three phases using the method of characteristics (MOC). Every component partitions between all phases present, and the equilibrium volume ratios of the components in each phase are fixed. Riemann problems modeling the injection of carbon dioxide and water in a depleted oil reservoir are studied, and the sensitivity of the solutions to changes in boundary conditions is analyzed. Finally the MOC solutions are compared to simulated displacements.

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Multiple Viscous Solutions for Systems of Conservation Laws

Cited by 9 publications

References 9 publications

The Solution by the Wave Curve Method of Three-Phase Flow in Virgin Reservoirs

The Solution by the Wave Curve Method of Three-Phase Flow in Virgin Reservoirs

Four-component gas/water/oil displacements in one dimension: Part I. structure of the conservation law

Four-component gas/water/oil displacements in one dimension: part II, example solutions

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