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Transport in Porous Media

, Volume 6, Issue 1, pp 35–69 | Cite as

Simulation of coning in bottom water-driven reservoirs

  • J. Bruining
  • C. J. Van Duijn
  • R. J. Schotting
Article

Abstract

Waterconing, as a result of oil recovery through a single well, is considered. It is assumed that a sharp transition, an interface, exists between the oil and the water and that the oil region between the cap rock and the bottom water in the lower half space is of infinite radial extent.

In this paper, we study the dynamical behavior of the oil-water interface starting from a horizontal position. We give a description in terms of the stream function and extend the vortex theory developed for fresh-salt groundwater flow problems. As a result, we find two singular integral equations: one for the shear flow along the interface and one for the flow component normal to the interface. We also give the algorithm to solve these equations and to determine the time evolution of the cone.

The problem is determined by two independent dimensionless quantities: the gravity number (G) [gravity forces/viscous forces] and the mobility ratio (M) [viscous forces in oil/viscous forces in water]. As is to be expected, these computations show that, for the schematization considered here, a stationary cone, and hence, a critical production rate (below which waterfree production is possible) does not exist. Breakthrough times are numerically determined for various values ofG andM.

Key words

Water-oil coning sharp interface vortex theory 

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References

  1. Addington, D. V., 1981, An approach to gas-coning correlations for a large grid reservoir simulator, JPT, Nov. 2267–2274Google Scholar
  2. Alt, H. W. and Duijn, C. J. van., 1990, A free boundary problem involving a cusp, publication in preparation.Google Scholar
  3. Amirat, Y., 1986, Approximation numérique d'un probléme á frontiére libre intervenant en milieu poreaux, Thésè de doctorat, Université Paris Dauphine.Google Scholar
  4. Bear, J., 1972,Dynamics of Fluids in Porous Media, American Elsevier, New York Chapter 9.Google Scholar
  5. Bear, J. and Verruijt, A., 1987,Modeling Groundwater Flow and Pollution, D. Reidel, Dordrecht, Chapter 13.Google Scholar
  6. Bruining, J., Duijn, C. J. van, Schotting, R. J., and Villeneuve, A. H., 1990, Interface model for coning in bottom water driven reservoirs. Elaborate version of this article to be published in Delft Progress Report (1990).Google Scholar
  7. Chavent, G. and Jaffré, J., 1986,Mathematical Models and Finite Elements for Reservoir Simulation. Studies in Mathematics and its Application 17, North Holland, Amsterdam.Google Scholar
  8. chierici, G. L. and Ciucci, G., 1964, A systematic study of gas and water coning by potentiometric models,SPEJ, Aug., 923–929.Google Scholar
  9. Dake, L. P., 1978,Fundamentals of Reservoir Engineering, Elsevier, Amsterdam Chapter 10.Google Scholar
  10. Dietz, D. N., 1983, The computation of the development of water or gas cones, private communication.Google Scholar
  11. Feynman, R., Leighton, R. B., and Sands, M., 1969, The Feynman Lectures on Physics, Addison-Wesley, Reading, Mass. Volume II, Chapter 14.Google Scholar
  12. Haitjema, H. M., 1977, Numerical application of vortices to multiple fluid flow in porous media, Delft Progress Report 2 pp. 139–146.Google Scholar
  13. Høyland, L. A., Papatzacos, P., and Skjaeveland, S. M., 1989, Critical rate for water coning: Correlation and analytical solution,SPE Reservoir Engng., Nov., 495–502.Google Scholar
  14. Huan-Zhang, C., 1983, Numerical simulation of coning behavior of a single well in a naturally fractured reservoir,SPEJ, Dec., 879–884.Google Scholar
  15. Josselin de Jong, G. de, 1960, Singularity distributions for the analysis of multiple fluid flow through porous media.J. Geothermal Res. 65, 3739–3758.Google Scholar
  16. Josselin de Jong, G de, 1977, Review of vortex theory for multiple fluid flow, Delft Progress Report 2, pp. 225–236.Google Scholar
  17. Josselin de Jong, G de, 1979, Vortex theory for multiple fluid flow in three dimensions, Delft Progress Report 4, pp. 87–102.Google Scholar
  18. Josselin de Jong, G. de, 1981, The simultaneous flow of fresh and salt groundwater in aquifers of large horizontal extension determined by shear flow and vortex theory, in A. Verruijt and F. B. J. Barends (eds.)Proc. Euromech. 143, 75–82, Balkema, Rotterdam.Google Scholar
  19. Josselin de Jong, G de, 1988, The Hodograph method in two-dimensional groundwater flow problems, private communication.Google Scholar
  20. Kuo, M. C. T. and DesBrisay, C. L., 1983, Simplified method for water coning predictions, SPE paper 12067.Google Scholar
  21. Landau, L. D. and Lifshitz, E. M., 1960,Electrodynamics of Continuous Media, Pergamon, Oxford, Chapter IV, pp. 124–125.Google Scholar
  22. Lehner, F. K., 1986, Critical water free production rates and gravity drainage rates for multiwell systems,SPE Reservoir Engng., July, 363–370.Google Scholar
  23. Lucas, S. K., Blake, J. R., and Kucera, A., 1990, Boundary integral method applied to water coning in oil reservoirs, submitted toJ. Austral. Math. Soc. Ser. B.Google Scholar
  24. Meng, J. C. S. and Thomson, J. A. L., 1978, Numerical studies of some nonlinear hydrodynamic problems by discrete vortex element methods,J. Fluid Mech. 84, 433–453.Google Scholar
  25. Muskat, M. and Wyckoff, R. D., 1935, An approximate theory of water coning in oil production.Trans. AIME 114, 144–161.Google Scholar
  26. Peters J. H., 1983, The movement of fresh water injected in salaquifers, Rapport SWE 83-007 Keuringsinstituut voor Waterleiding artikelen KIWA N.V., Postbox 1072, 3430 BB Nieuwegein, The Netherlands Waterworks Testing and Research Institute (ibid.) Proc. Salt Water Intrusion Meeting (Bari), Estratto da Geologia Applicata e Idrologeologia, Bari, 1983 - Volume XVIII, parte II.Google Scholar
  27. Richardson, J. G., Sangree, J. B., and Sneider, R. M., 1987, Coning,J. Petrol. Technol. Aug., 883–884.Google Scholar
  28. Saffman, P. G. and Taylor, G. F. R. S., 1958, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liduid,Proc. R. Soc. London Ser. A 246, 312–331.Google Scholar
  29. Schols, R. S., 1972, Water coning - an empirical formula for the critical oil production rate,Erdol-Erdgas-Zeitschrift 88, 6–11.Google Scholar
  30. Sobocinski, D. P. and Cornelius, A. J., 1965, A correlation for predicting water coning time,J. Petrol. Techno. May, 594–600.Google Scholar
  31. Stegemeier, G. L., 1977, Oil entrapment and mobilization in porous media, in (eds), D. O. Shah and R. S. Schechter,Improved Oil Recovery by Surfactant and Polymer Flooding, Academic Press, New York.Google Scholar
  32. Van Domselaar, H. R., 1972, Critical gas-flow ratio for a cluster of wells over bottom water, SPE paper 4036 prepared for the SOE 47th Annual Fall Meeting San Antonio, TX.Google Scholar
  33. Weinstein, H. G., Chappelear, J. E., and Nolen, J. S., 1986, Second comparative solution project: A three-phase coning study,J. Petrol. Technol., Mar., 345–353.Google Scholar
  34. Wheatley, M. J., 1985, An approximate theory of oil/water coning, SPE paper 14210 presented at the SPE 60th Annual Technical Conference and Exhibition, Las Vegas.Google Scholar
  35. Yih, C. S., 1964, A transformation for free surface flow in porous media,Phys. Fluids 7, 20–24.Google Scholar

Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • J. Bruining
    • 1
  • C. J. Van Duijn
    • 2
  • R. J. Schotting
    • 2
  1. 1.Dietz Laboratory, Faculty of Mining and Petroleum Engineering, Delft University of TechnologyGA DelftThe Netherlands
  2. 2.Faculty of Technical Mathematics and InformaticsDelft University of TechnologyAJ DelftThe Netherlands

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