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Transport in Porous Media

, Volume 72, Issue 1, pp 83–96 | Cite as

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

  • Tara LaForceEmail author
  • Kristian Jessen
  • Franklin M. OrrJr.
Article

Abstract

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.

Keywords

Three-phase flow Compositional model Four-component system Conservation laws Hyperbolic system Constant K-value 

Nomenclature

\(\overline{\overline D}(S_1,S_1)\)

The dispersion tensor

q

“Saturation” of phase one on the tie-triangle extension

r

“Saturation” of phase two on the tie-triangle extension

Λ

Shock velocity

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References

  1. Azevedo A.V. and Marchesin D. (1995). Multiple viscous solutions for systems of conservation laws. Trans. Am. Math. Soc. 347: 3061–3077 CrossRefGoogle Scholar
  2. Azevedo A.V., Marchesin D., Plohr B. and Zumbrun K. (1996). Nonuniqueness of solutions of Riemann problems. Z. Angew. Math. Phys. 47: 977–998 CrossRefGoogle Scholar
  3. Falls A.H. and Schulte W.M. (1992). Features of three-component, three-phase displacement in porous media. SPE Reservoir Eng. 7: 426–432 Google Scholar
  4. Guzman R.E. and Fayers F.J. (1997). Solutions to the three-phase Buckley-Leverett problem. SPE J. 2: 301–311 Google Scholar
  5. Helfferich, F.G.: General theory of multicomponent, multiphase displacement. Presented at the Society of Petroleum Engineers Annual Technical Conference and Exhibition, Las Vegas, NV, 23–25 Sept 1979Google Scholar
  6. Isaacson E., Marchesin D. and Plohr B. (1990). Transitional waves for conservation laws. SIAM J. Math. Anal. 21: 837–866CrossRefGoogle Scholar
  7. Jessen K., Stenby E.H. and Orr F.M. (2004). Interplay of phase behavior and numerical dispersion in finite-difference compositional simulation. SPE J. 9: 193–201 Google Scholar
  8. Johansen T., Wang Y., Orr F.M. and Dindoruk B. (2005). Four-component gas/oil displacements in one dimension: part I: global triangular structure. Transport Porous Media 61: 59–76 CrossRefGoogle Scholar
  9. Johns R.T., Dindoruk B. and Orr F.M. (1993). Analytical theory of combined condensing/vaporizing gas drives. SPE Adv. Tech. Series 1: 7–16 Google Scholar
  10. LaForce T. and Johns R.T. (2005a). Composition routes for three-phase partially miscible flow in ternary systems. SPE J. 10: 161–174 Google Scholar
  11. LaForce, T., Johns, R.T.: Analytical solutions for surfactant enhanced remediation of non-aqueous phase liquids (NAPLs). Water Resour. Res. 41, W10420, doi:10.1029/2004WR003862 (2005b)Google Scholar
  12. LaForce, T., Jessen, K., Orr, F.M.: Four-component gas/water/oil displacements in one dimension: part I, structure of the conservation law. TIPM, doi: 10.1007/s11242-007-9120-9 (2007)Google Scholar
  13. LaForce, T., Cinar, Y., Johns, R.T., Orr, F.M.: Experimental confirmation for analytical composition routes in three-phase partially miscible flow. Presented at the 2006 SPE/DOE Symposium on Improved Oil Recovery, Tulsa, Oklahoma, 22–26 April 2006Google Scholar
  14. Lax P.D. (1957). Hyperbolic systems of conservation laws II. Comm. Pure App. Math. 10: 537–566 CrossRefGoogle Scholar
  15. Marchesin D., Plohr B. and Schecter S. (1997). An organizing center for wave bifurcation in multiphase flow models. SIAM J. Appl. Math. 57: 1189–1215 CrossRefGoogle Scholar
  16. Schecter S., Marchesin D. and Plohr B.J. (1996). Structurally stable Riemann solutions. J. Differ. Eq. 126: 303–354 CrossRefGoogle Scholar
  17. de Souza A.J. (1995). Wave structure for a nonstrictly hyperbolic system of three conservation laws. Math. Comput. Modelling 22: 1–29 CrossRefGoogle Scholar
  18. Wang, Y.: Analytical calculation of minimum miscibility pressure. Dissertation, Stanford University (1998)Google Scholar
  19. Wang Y., Dindoruk B., Johansen T. and Orr F.M. (2005). Four-component gas/oil displacements in one dimension: part II: analytical solutions for constant equilibrium ratios. Trans. Porous Media 61: 177–192 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  • Tara LaForce
    • 1
    Email author
  • Kristian Jessen
    • 2
  • Franklin M. OrrJr.
    • 3
  1. 1.Department of Chemical and Petroleum EngineeringUniversity of WyomingLaramieUSA
  2. 2.Department of Chemical Engineering and Materials ScienceUniversity of Southern CaliforniaLos AngelesUSA
  3. 3.Department of Energy Resources EngineeringStanford UniversityStanfordUSA

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