Extension of Stone's Method 1 and Conditions for Real Characteristics in Three-Phase Flow

Fayers, F.J.

doi:10.2118/16965-pa

Cited by 63 publications

(66 citation statements)

References 12 publications

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“…In particular, we do not allow for mathematical singularities such as elliptic regions (regions of the saturation space where the system is elliptic in character) and umbilic points (isolated saturation states where the system is nonstrictly hyperbolic) inside the saturation triangle. We support the view that these singularities are artifacts of an incorrect mathematical model, rather than a necessary consequence dictated by physics [10,11,15,20,21,32,40].…”

Section: Character Of the System Of Equationssupporting

confidence: 76%

“…It suffices to say that most of them give rise to elliptic regions [8,15,19,40]. In [20] we show it is possible to formulate models which are strictly hyperbolic everywhere in the three-phase flow region, even when using the usual multiphase form of Darcy's equation, and relative permeabilities which are functions of the current fluid saturations alone.…”

Section: Relative Permeabilitiesmentioning

confidence: 94%

“…It was long believed that, for negligible capillary forces, the system of equations would be strictly hyperbolic for any relative permeability functions. This is far from being the case and, in fact, most relative permeability models used today give rise to systems which are not strictly hyperbolic for the entire range of admissible saturations [8,15,18,19,39,40]. Loss of strict hyperbolicity typically occurs at bounded regions of the saturation triangle (the so-called elliptic regions), where the system is elliptic in character.…”

Section: Introductionmentioning

confidence: 99%

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Analytical Solution to the Riemann Problem of Three-Phase Flow in Porous Media

Juanes

Patzek

2004

Transport in Porous Media

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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.

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Section: Character Of the System Of Equationssupporting

confidence: 76%

Section: Relative Permeabilitiesmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

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Analytical Solution to the Riemann Problem of Three-Phase Flow in Porous Media

Juanes

Patzek

2004

Transport in Porous Media

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show abstract

“…It was long believed that, when capillarity is ignored, this system of equations would be strictly hyperbolic for any relative permeability functions. This is far from being the case and, in fact, most relative permeability models used today give rise to systems which are not strictly hyperbolic for the entire range of admissible saturations [7,19,31,36,65,66]. Loss of strict hyperbolicity typically occurs at bounded regions of the saturation triangle (the so-called elliptic regions), where the system is elliptic in character.…”

Section: Introductionmentioning

confidence: 99%

Relative Permeabilities for Strictly Hyperbolic Models of Three-Phase Flow in Porous Media

Juanes¹,

Patzek²

2004

Transport in Porous Media

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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.

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“…The experimental two-phase data of Saraf and Fatt (1967) and tbe UKLB data set (Fayers, 1989), togetber witb Stone's model I were used to construct tbe following solutions. As such, tbree-pbase relative permeabilities calculated from two-phase Corey-k, and two-phase extended Corey-k, are botb treated.…”

Section: Klmentioning

confidence: 99%

Solutions to the Three-Phase Buckley-Leverett Problem

Guzman¹,

Fayers²

1996

ECMOR v - 5th European Conference on the Mathematics of Oil Recovery

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This paper presents method of characteristics (MOC) solutions to the three-phase Buckley-Leverett problem with and without gravity, for the three classes of equations described by Guzmán and Fayers (1996). Thus, solutions for equations with single and multiple umbilic points, and mixed elliptic and hyperbolic equations are constructed. A close relation between the mathematical properties of the equations (in terms of location and quantity of umbilic points) and the physics of flow is descrihed. This relationship can be observed by the effect of gravity on the efficiency of the displacement, the formation of oil and water banks, and the presence of a unique or infinite solution paths within the three-phase flow region for a cIass of solutions. Elliptic regions did not generate physically inadmissible effects and stabie shocks with saturations on opposite sides of elliptic regions were constructed.

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Extension of Stone's Method 1 and Conditions for Real Characteristics in Three-Phase Flow

Cited by 63 publications

References 12 publications

Analytical Solution to the Riemann Problem of Three-Phase Flow in Porous Media

Analytical Solution to the Riemann Problem of Three-Phase Flow in Porous Media

Relative Permeabilities for Strictly Hyperbolic Models of Three-Phase Flow in Porous Media

Solutions to the Three-Phase Buckley-Leverett Problem

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