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

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Abstract

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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Abbreviations

C i :

Overall volume fraction of component i

F i :

Overall fractional flow of component i

f j :

Fractional flow of phase j

f ij :

\(\partial{f_i}\slash\partial{S_j}\)

\(K_i^1\) :

K-value for partitioning of component i between the gas and oleic phase

\(K_i^2\) :

K-value for partitioning of component i between the aqueous and oleic phase

\(K_i^3\) :

K-value for partitioning of component i between the aqueous and gas phase

S j :

Saturation of phase j

x i :

Volume fraction of component i in the oleic phase

y i :

Volume fraction of component i in the gas phase

w i :

Volume fraction of component i in the aqueous phase

μ j :

Viscosity of phase j

λ :

Eigenvalue (characteristic velocity)

χ :

Nontie-triangle eigenvalue

τ :

Dimensionless time

ξ :

Dimensionless distance

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Authors and Affiliations

  1. Department of Chemical and Petroleum Engineering, University of Wyoming, Laramie, WY, 82070, USA

    T. LaForce

  2. Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, CA, 90089, USA

    K. Jessen

  3. Department of Energy Resources Engineering, Stanford University, Stanford, CA, 94305, USA

    F. M. Orr Jr.

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  1. T. LaForce
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  2. K. Jessen
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  3. F. M. Orr Jr.
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Correspondence to T. LaForce.

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LaForce, T., Jessen, K. & Orr, F.M. Four-component gas/water/oil displacements in one dimension: Part I. structure of the conservation law. Transp Porous Med 71, 199–216 (2008). https://doi.org/10.1007/s11242-007-9120-9

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