Solutions to the three-phase Buckley-Leverett problem

SPE Journal

Volumn 2, Issue 3, 1997, Pages 301-311

  Guzmán, Rafael E ^a,b,c   Fayers, F John ^d,e  

^a BP AMERICA INC   (United States)

^b Universidad de América   (Colombia)

^c STANFORD UNIVERSITY   (United States)

^d 8 The Dell ^*  (United Kingdom)

^e UNIVERSITY OF READING   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL METHODS; EQUATIONS OF MOTION; GRAVITATIONAL EFFECTS; PROBLEM SOLVING;

METHOD OF CHARACTERISTICS (MOC); THREE PHASE BUCKLEY LEVERETT PROBLEM;

MULTIPHASE FLOW;

EID: 0031237832     PISSN: 1086055X     EISSN: None     Source Type: Journal    
DOI: 10.2118/35156-PA     Document Type: Article

References (32)

1. Aanonsen, S.I.: "Application of Fractional-Flow Theory to 3-Phase, 1-Dimensional Surfactant Flooding," Proceedings of the First European Conference on the Mathematics of Oil Recovery, Cambridge, July, 1989, P. King (ed.), Cambridge University Press, Cambridge (1992).
2. Azevedo A.V. and Marchesin, D.: "Multiple Viscous Profile Riemann Solutions in Mixed Elliptic-Hyperbolic Models for Flow in Porous Media," Nonlinear Evolution Equations That Change Type, B. Keyfitz and M. Shearer (eds.), Springer-Verlag, New York (1990).
3. Bell, J.B., Trangenstein, J.A., and Shubin, G.R.: "Conservation Laws of Mixed Type Describing Three-Phase Flow in Porous Media," SIAM J. Applied Math. (December 1986) 46, No. 6, 1000-1017.
4. Buckley, S.E. and Leverett, M.C.: "Mechanism of Fluid Displacement in Sands," Trans., AIME (1941) 249, 107-116.
5. Falls, A.H., and Schulte, W.M.: "Theory of Three-Component, Three-Phase Displacement in Porous Media," Soc. Pet. Eng. Res. Eng. (August 1992a) 377-384.
6. Falls, A.M., and Schulte, W.M.: "Features of Three-Component, Three-Phase Displacement in Porous Media," Soc. Pet. Eng. Res. Eng. (November 1992b) 426-432.
7. Fayers, F.J.: "Extension of Stone's Method 1 and Conditions for Real Characteristics in Three-Phase Flow," Soc. Pet. Eng. Res. Eng. (November 1989) 437-445.
8. Glimm, J.: "The Intereaction of Nonlinear Hyperbolic Waves," Comm. Pure and Applied Math. (1988) 41, 569-590.
9. Gorell, S.B.: "Modeling the Effects of Trapping and Water Alternating Gas (WAG) Injection on Tertiary Miscible Displacements," paper SPE 17340 presented at the 1988 Sixth Symposium on Enhanced Oil Recovery, Tulsa, OK, April 17-20.
10. Grader, A.S., and O'Meara Jr., D.J.: "Dynamic Displacement Measurements of Three-Phase Relative Permeabilities Using Three Immiscible Liquids," paper SPE 18293 presented at the 1988 Annual Technical Conference and Exhibition, Houston, TX, October 2-5.
11. Guzmán, R.E.: Mathematics of Three-Phase Flow, PhD dissertation, Stanford University, Stanford, CA (July 1995).
12. Guzmán, R.E. and Fayers, F.J.: "Mathematical Properties of Three-Phase Flow Equations," (1997) 2, SPE 35154, Soc. Pet. Eng. J. (this issue).
13. Guzmán, R.E., Domenico, G., Fayers, F.J., Aziz, K., and Godi, A.: "Three-Phase Flow in Field Scale Simulations of Gas and WAG Injections," paper SPE 28897 presented at the 1994 European Petroleum Conference, London, U.K., October 25-27.
14. Helfferich, P.O.: "Theory of Multicomponent, Multiphase Displacement in Porous Media," Soc. Pet. Eng. J. (February 1981) 271, 51-62.
15. Holden, H.: "On the Riemann Problem for Prototype of a Mixed Type Conservation Law," Comm. on Pure and Applied Math. (1987) 40, 229-264.
16. Holden, H., Holden, L., and Risebro, N.H.: "Some Qualitative Properties of 2 × 2 Systems of Conservation Laws of Mixed Type," Nonlinear Evolution Equations That Change Type, B. Keyfitz and M. Shearer (eds.), Springer-Verlag, New York (1990).
17. Holden, L., and Hoegh-Krohn, R.: "A Class of Nonlinear Hyperbolic Conservation Laws," J. Diff. Eqn. (1990) 84, 73-99.
18. Isaacson, E.L., Marchesin, D., and Plohr, B.J.: "The Structure of the Riemann Solution for Non-Strictly Hyperbolic Conservation Laws," (March, 14-18 1988) Second International Conference on Nonlinear Hyperbolic Problems, Aachen, FGR.
19. Isaacson, E.L., Marchesin, D., and Plohr, B.J.: "Transitional Shock Waves," Contemporary Mathematics (1989) 100, 125-145.
20. Keyfitz, B.L.: "Change of Type in Three-Phase Flow: A Simple Analogue," J. Diff. Eqs. (1989) 80, 280-305.
21. Lax, P.D.: "Hyperbolic Conservation Laws II," Comm. Pure and Applied Math. (1957) 10, 537-566.
22. LeVeque, R.J.: Numerical Methods for Conservation Laws, second edition, Birkhäuser Verlag, Basel (1992).
23. Liu, T.P.: "The Riemann Problem for General General Systems of Conservation Laws," J. Diff. Eqn. (1975) 18, 218-234.
24. Marchesin, D., and Madeiros, H.B.,: "A Note on the Stability of Eigenvalue Degeneracy in Nonlinear Conservation Laws of Multiphase Flow," Contemporary Mathematics (1989) 100, 215-224.
25. Oleinik, O.A.: "Discontinuous Solutions of Non-Linear Differential Equations," Uspekhi Mat. Nauk (N.S) (1957) 12, No. 3 (75), 3-73, English translation in: Amer. Math. Soc. Transi., Ser. 2, v. 26, 95-172, (1963).
26. Oleinik, O.A.: "Discontinuous Solutions of Non-Linear Differential Equations," Uspekhi Mat. Nauk (N.S) (1957) 12, No. 3 (75), 3-73, English translation in: Amer. Math. Soc. Transi., Ser. 2, v. 26, 95-172, (1963).
27. Saraf, D.N., and Fatt, I.: "Three-Phase Relative Permeability Measurement Using a Nuclear Magnetic Resonance Technique for Estimating Fluid Saturation," J. Pet. Tech., Trans., AIME (September 1967) 240, 235-242.
28. Shalimov, B.V.: "Flow of a Three-Phase Fluid in a Porous Medium (Buckley-Leverett Model)" Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza (1972) 39, No. 1, 39-44.
29. Stone, H.L.: "Probability Model for Estimating Three-Phase Relative Permeability," J. Pet. Tech., Trans., AIME (February 1970) 249, 214-218.
30. Trangenstein, J.A.: "Three-Phase Flow with Gravity," Contemporary Mathematics (1989) 100, 147-159.
31. Welge, H.J., Johnson, E.F., Hicks, A.L., and Brinkman, F.H.: "An Analysis for Predicting the Performance of Cone-Shaped Reservoirs Receiving Gas or Water Injection," J. Pet. Tech. (August 1962) 894-898.
32. Wingard, J.S. and Orr, F.M., Jr.: "An Analytical Solution for Steam/Oil/Water Displacements," Soc. Pet. Eng. Advanced Technology Series (1994) 2, No. 2, 167-176.