Skip to main content

Part of the book series: Theory and Applications of Transport in Porous Media ((TATP,volume 16))

  • 291 Accesses

Abstract

Problem of upscaling two-phase flow through highly heterogeneous media deals with a more complex case of transport equations, characterized by nonlinearity and degeneration of relative permeabilities, as well as by specific capillary phenomena which determine discontinuous behavior of the saturation, that has not anything similar in linear case.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ahmadi, A. (1992) Utilisation de Propriétés Equivalentes Dans les Modèles de Réservoir: Cas des Ecoulements Diphasiques Incompressibles, Thesis, Univ. Bordeaux-I.

    Google Scholar 

  2. Alhanai, W., Bertin, H. and Quintard M. (1992) Two-phase flow in nodular systems: laboratory experiments, Révue de l’Institut Français du Pétrole, 47, no. 1, pp. 29–44.

    Google Scholar 

  3. Amaziane, B. and Bourgeat, A. (1988) Effective behavior of two-phase flow in heterogeneous reservoir, numerical simulation in oil recovery, In: IMA, Mathematics and its Applications, M.F. Wheeler, Springer Verlag, 11, pp. 1–22.

    Google Scholar 

  4. Amaziane, B., Bourgeat A. and Koebbe, J. (1991) Numerical simulation and homo-genization of two-phase flow in heterogeneous porous media, Transport in Porous Media, no. 6, pp. 519–547.

    Google Scholar 

  5. Amirat, Y., Hamdache, K. and Ziani, A. (1988) Homogénéisation d’équations hyperboliques du premier ordre: Application aux milieux poreux, Rapport Scientifique I.N.R.I.A., no. 803.

    Google Scholar 

  6. Amirat, Y., Hamdache, K. and Ziani, A. (1989) Homogénéisation d’équations hyperboliques du premier ordre et application aux écoulements miscibles en milieu poreux, Annales Inst. H. Poincaré. Analyse non linéaire, 6, no. 5, pp. 397–417.

    Google Scholar 

  7. Amirat, Y., Hamdache, K. and Ziani, A. (1991) Kinetic formulation for a transport equation with memory, Commun. Partial Differential Equations, 16, no. 8–9, pp. 1287–1311.

    Article  Google Scholar 

  8. Anguy, Y. (1993) Application de la prise de moyenne volumique a l’étude de la relation entre le tenseur de perméabilité et la microgémetrie des milieux poreux naturels, Thesis, Univ. Bordeaux-I.

    Google Scholar 

  9. Arbogast, T., Douglas, J. and Hornung, U. (1990) Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Appl. Math., 21, pp. 823–836.

    Article  Google Scholar 

  10. Aronofsky, J.S., Masse, L. and Natanson, S.G. (1958) A model for the mechanism of oil recovery from the porous matrix due to water invasion in fractured reservoirs, Trans. AIME, 213, pp. 17–19.

    Google Scholar 

  11. Auriault, J.-L. (1983) Effective macroscopic description for heat conduction in periodic composites. Int. J. Heat and Mass Transfer, 26, no. 6, pp. 861–869.

    Article  Google Scholar 

  12. Badea, A. and Bourgeat, A. (1995) Homogenization of two phase flow through randomly heterogeneous porous media, In.: Mathematical Modelling of Flow through Porous media, Proc. Int. Conf., World Scientific Pub., Singapore, pp. 15–31.

    Google Scholar 

  13. Bakhvalov, N.S. and Panasenko, G.P. (1989) Homogenization of processes in periodic media. Ed. Nauka, Moscow. (Version in English: Bakhvalov N. and Panasenko, G. Homogenization: Averaging Processes in Periodic Media. Kluwer Academic Publishers, Dordrecht)

    Book  Google Scholar 

  14. Bertin, A., Alhanai, W. and Quintard, M. (1992) Ecoulement polyphasique dans un milieu poreux hétérogène de type nodulaire: drainage initial, C. R. Acad. Sci. Paris, Ser. II, 314, pp. 431–437.

    Google Scholar 

  15. Barenblatt, G.I. (1971) Flow of two immiscible liquids through the homogeneous porous medium, Izvestiya Academii Nauk SSSR, Mekhanika Zhidkosti i Gaza, no 5, pp. 144–151.

    Google Scholar 

  16. Barenblatt, G.I. and Vinichenko, A.P. (1980) Non-equilibrium flow of immiscible fluids in porous media, Uspekhi Mathematicheskikh Nauk, no 3, pp. 35–50.

    Google Scholar 

  17. Barenblatt, G.I., Entov, V.M. and Ryzhik, V.M. (1972) Theory of Non-Stationary Flow of Liquids and Gases Through Porous Media. Nedra, Moscow.

    Google Scholar 

  18. Barker, J.W. and Dupouy, P. (1996) An analysis of dynamic pseudo relative permeability methods, Proc. of 5th European Conf. of Mathematics of Oil Recovery, 3–6 Sept., Leoben, Austria.

    Google Scholar 

  19. Bougemaa, A. (1998) Private communication.

    Google Scholar 

  20. Bourgeat, A. (1996) Two-phase flow, in: Homogenization and Porous Media, Ed. Hornung, U., Springer, pp. 95–128.

    Google Scholar 

  21. Bourgeat, A., Hidani, A. (1995) Effective model of two-phase flow in a porous medium made of different rock types, Applicable Analysis, 56, pp. 381–399.

    Article  Google Scholar 

  22. Bourgeat, A., Luchkaus, S. and Mikelić, A. (1997) Convergence of the homogenization process for a double porosity model of immiscible two-phase flow, SIAM J. Appl. Math., to appear.

    Google Scholar 

  23. Bourgeat, A. and Mikelić, A. (1994) Homogenization of the two-phase immiscible flow in one dimensional porous medium, Asymptotic Analysis, 9, pp. 359–380.

    Google Scholar 

  24. Bourgeat, A. and Panfilov, M. (1998) Effective two-phase flow through highly heterogeneous porous media: Capillary nonequilibrium effects, Computational Geosciences, 2, no. 3, pp. 191–215.

    Article  Google Scholar 

  25. Bramble, J. H. and Shatz, A.H. (1970) Rayleight-Ritz-Galerkin method for Dirichlet’s problem using subspaces without boundary conditions, Comm. Pure Appl. Mathem, 23, pp. 653–675.

    Article  Google Scholar 

  26. Charpentier, I., and Maday, Y. (1995) Deux méthodes de décomposition de domaine pour la résolution d’équations aux dérivée partielles avec conditions de périodicité: application à la théorie de l’homogénéisation, C. R. Acad. Sci. Paris, Ser.I, 321, pp. 359–366.

    Google Scholar 

  27. Civan, F. (1993) Waterflooding of naturally fractured reservoirs: an efficient simulation approach, Paper SPE 25449, Proc. of the SPE Production Operation Sympos., 21–23 March, Oklahoma City, USA, pp. 395–407.

    Google Scholar 

  28. Civan, F. (1994) A theoretically derived transfer function for oil recovery from fractured reservoirs by waterflooding, Paper SPE/DOE 27745, Proc. of SPE/DOE 9th Sympos. on Improved Oil Recovery, 17–20 April, Tulsa, Oklahoma, USA, pp. 87–98.

    Google Scholar 

  29. Dale, M., Ekrann, S., Mykkeltveit, J. and Virnovsky, G. (1994) Effective relative permeability and capillary pressure for 1D heterogeneous media, Proc. of 4th European Conf. of Mathematics of Oil Recovery, 7–10 June, Røros, Norway.

    Google Scholar 

  30. deSwaan, A. (1978) Theory of waterflooding in fractured reservoirs, SPE J., pp. 87–98.

    Google Scholar 

  31. Douglas, J. and Arbogast, T. (1990) Dual porosity models for flow in naturally fractured reservoirs, in: Dynamics of Fluids in Hierarchical Porous Media, Ed. Cushman, J., Academic Press, pp. 177–206.

    Google Scholar 

  32. Douglas, J., Peacemen, D.W. and Rachford, Jr. (1959) A method for calculating multidimensional immiscible displacement, Trans. of AIME, 216, pp. 297–308.

    Google Scholar 

  33. Durlofsky, L.J. (1998) Coarse scale models of two phase flow in heterogeneous reservoirs: volume averaged equations and their relationship to existing upscaling techniques, Computational Geosciences, 2, no. 2, pp. 73–92.

    Article  Google Scholar 

  34. Hansen, A., Roux, S., Aharony, A., Feder, J., Jøssang, T. and Hardy, H.H. (1997) Real-space renormalization estimates for two-phase flow in porous media, Transport in Porous Media, 29, pp. 247–279.

    Article  Google Scholar 

  35. Hassanizadeh, S.M., and Gray, W.G. (1993) Thermodynamic basis of capillary pressure in porous media, Water Resour. Res., 29, pp. 3389–3405.

    Article  Google Scholar 

  36. Hassanizadeh, S.M., and Gray, W.G. (1993) Toward an improved description of the physics of two-phase flow, Adv. Water Resour., 16, pp. 53–67.

    Article  Google Scholar 

  37. Hassanizadeh, S.M., and Gray, W.G. (1990) Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries, Adv. Water Resources, 13, pp. 169–186.

    Article  Google Scholar 

  38. Hassanizadeh S.M. (1997) Dynamic effects in the capillary pressure-saturation relationship, Proc. 4th Int. Conf. Civ. Eng., Sharif Univ. Technol., Tehran, Iran, 4–6 May, 1997, V.4: Water Resources and Environmental Engineering, pp. 141–149.

    Google Scholar 

  39. Kalaydjian, F. (1992) Dynamic capillary pressure curve for water/oil displacement in porous media, theory vs. experiment, SPE Conf., Washington, DC, October 4–7, Paper 24813, pp. 491–506.

    Google Scholar 

  40. Kazemi, H., Gilman, J.R. and El-Shharkaway, A.M. (1992) Analytical and numerical solution of oil recovery from fractured reservoirs with empirical transfer functions, SPE Reservoir Engineering, no. 7(2), pp. 219–227.

    Google Scholar 

  41. King, P., Muggeridge, A.H. and Price, W.G. (1993) Renormalization calculation of immiscible flow, Transport in Porous Media, 12, pp. 237–260.

    Article  Google Scholar 

  42. Konovalov, A.N. (1988) Problems of Multiphase Incompressible Flow in Porous Media, Novosibirsk, Nauka, Siberian Department (in Russian).

    Google Scholar 

  43. Kumar, A., Farmer, L., Jerauld, G.R. and Li, D. (1997) Efficient upscaling from cores to simulation models, Paper SPE 38744, Proc. 1997 SPE Annual Technical Conf. and Exhibition, 5–8 Oct., San Antonio, Texas, USA, pp. 257–271.

    Google Scholar 

  44. Kutliarov, V.S. (1967) Convective Diffusion in Fractured Porous Reservoirs, J. Prikladnoi Mathematiki and Tekhnicheskoi Phyziki, no. 1, pp. 84–88, (in Russian).

    Google Scholar 

  45. Kyte, J.R. and Berry, D.W. (1975) New pseudo-functions to control numerical dispersion, SPE J., August, p. 269–275.

    Google Scholar 

  46. Lemouzy, P., Kruel Romeu, R. and Morelon, I. (1993) A new scaling-up method to compute relative permeability and capillary pressure for simulation of heterogeneous reservoirs, Paper SPE 26660, Proc. 68th SPE Annual Technical Conf. and Exhibition, 3–6 October, Houston, Texas, USA.

    Google Scholar 

  47. Lenormand, R. (1995) Transport equations for fluid displacements in heterogeneous porous media: the MHD model, in: Math. Modelling of Flow Through Porous Media, Proc. Int. Conf., Eds. Bourgeat, A., Carasso, C, Luckhaus, S. and Mikelic, A., World Scientific Publishing, Singapore, pp. 248–265.

    Google Scholar 

  48. Nikolaevskii, V.N., Bondarev, E.A., Mirkin M.I. et al, (1968) Motion of Hydrocarbon Mixtures in Porous Media. Nedra, Moscow (in Russian).

    Google Scholar 

  49. Panfilov, M. (1990) Mean mode of porous flow in highly inhomogeneous media, Soviet Physics Doklady, 35, pp. 225–227.

    Google Scholar 

  50. Panfilov, M. (1995) Averaged Models of Convection-Diffusion Transfer Through Highly Heterogeneous Porous Media, in: Math. Modelling of Flow Through Porous Media, Proc. Int. Conf., Eds. Bourgeat, A., Carasso, C., Luckhaus, S. and Mikelic, A., World Scientific Publishing, Singapore, pp. 276–300.

    Google Scholar 

  51. Panfilov, M. (1996) Homogenized model with capillary nonequilibrium for two-phase flow through highly heterogeneous porous media, C. R. Acad. Sci. Paris, Ser.II, no. 3.

    Google Scholar 

  52. Panfilov, M., and Bourgeat, A. (1996) Capillary Relaxation Model for Two-Phase Flow Through Dual-Porosity Media Proc, 4th Int. Sympos. On Evaluation of Reservoir Wettability and its Effects on Oil Recovery, Sept. 11–13, 1996, Montpellier, France.

    Google Scholar 

  53. Panfilov, M. (1998) Upscaling Two-Phase Flow In Double Porosity Media: Nonuniform Homogenization, in: “Recent Advances in Problems of Flow and Transport in Porous Media” (Proc. Int. Conf.), Eds. J.M. Crolet and M.E. El-Hatri, Kluwer Acad. Publ., Dordrecht, pp. 195–215.

    Google Scholar 

  54. Panfilov, M. (1998) Upscaling two-phase flow through double porosity media, in: Proc. of 6-th European Conf. on the Mathematics of Oil Recovery, Peebles, Scotland, 8–11 Sept., paper C-28.

    Google Scholar 

  55. Panfilov, M. and Panfilova, I. (1996) Averaged models of flows with heterogeneous internal structure. Ed. Nauka, Moscow (in Russian).

    Google Scholar 

  56. Quintard, M. and Whitaker, S. (1988) Two Phase Flow in Heterogeneous Porous Media: The Method of Large-Scale Averaging, Transport in Porous Media, no. 3, pp. 357–413.

    Google Scholar 

  57. Smith, E.H. (1991) The influence of small-scale heterogeneity on average relative permeability, in: Reservoir Characterization II, Eds. Lake L.W., Caroll H.B. and Wesson T.C., Academic Press.

    Google Scholar 

  58. Saez, A. E., Otero, C. J. and Rusinek, I. (1989) The Effective Homogeneous Behaviour of Heterogeneous Porous Media, Transport in Porous Media, no. 4, pp. 212–238.

    Google Scholar 

  59. Suquet, P. (1990) Une métode simplifiée pour des propriétés élastiques de matériaux hétérogènes à structure périodique, C. R. Acad. Sei. Paris, Ser.II, 311, pp. 769–774.

    Google Scholar 

  60. Stone, H.L. (1991) Paper SPE 21207.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Panfilov, M. (2000). Two-Phase Flow in Double Porosity Media. In: Macroscale Models of Flow Through Highly Heterogeneous Porous Media. Theory and Applications of Transport in Porous Media, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9582-7_4

Download citation

  • DOI: https://doi.org/10.1007/978-94-015-9582-7_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-5401-2

  • Online ISBN: 978-94-015-9582-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics