Chemical or Heat Convection-Diffusion Transport Through Highly Heterogeneous Porous Media

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


The next step of the developed theory is related with the flow of miscible mixtures. The linear convection-diffusion equation written relatively to the concentration of a chemical component is a simplest model of mixture flow. In this chapter a more general model is studied when the field of convection transport velocity is not given and should be found as the solution of a parabolic equation for the velocity potential (the pressure).


Heat Transport Peclet Number Injection Well Diffusion Boundary Layer Block Boundary 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Allaire, G. (1992) Homogenization and two-scale convergence, SIAM J. Math. Anal, 23, no.6, pp. 1482–1518.CrossRefGoogle Scholar
  2. 2.
    Amaziane, B., Bourgeat A., Koebbe J.V. (1990) Proc. 2nd European Conf. Mathematics of Oil Recovery, Aries, France, Sept 11–14 , 1990, Ed.Technip, Paris, pp. 75–81.Google Scholar
  3. 3.
    Amirat, Y., Hamdache, K. and Ziani, A. Homogénéisation d’ équations hyperboliques du premier ordre: Application aux milieux poreux, Rapport I.N.R.I.A., no. 803.Google Scholar
  4. 4.
    Amirat, Y., Hamdache, K. and Ziani, A. (1989) Homogénéisation d’ équations hyperboliques du premier ordre et application aux écoulements miscibles en milieu poreux, Ann. Inst. H. Poincaré. Analyse non linéaire., 6, no. 5, pp. 397–417.Google Scholar
  5. 5.
    Amirat, Y., Hamdache, K. and Ziani, A. (1990) Comportemet limite de modèles d’ équations de convection-diffusion dégénérées, C. R. Acad. sei. Paris, Ser. A., 310, pp. 76–78.Google Scholar
  6. 6.
    Amirat, Y., Hamdache, K. and Ziani, A. (1990) Etude d’ une équation de transport à mémoire, C. R. Acad. sci. Paris, Ser. A., 311.Google Scholar
  7. 7.
    Amirat, Y., Hamdache, K. and Ziani, A. (1991) Kinetic formulation for a transport equation with memory, Commun. Partial Differ. Equations, 16, no. 8–9, pp. 1287–1311.CrossRefGoogle Scholar
  8. 8.
    Arbogast T., Douglas J., Hornung U. Derivation of The Double Porosity Model of Single Phase Flow via Homogenization Theory -SIAM J. Math. Anal., 1990, 21, pp. 823–836.CrossRefGoogle Scholar
  9. 9.
    Arbogast T., Douglas J., Hornung U. Modelling of Naturally Fractured Reservoirs by Formal Homogenization Thechniques- In: “Frontiers in Pure and Applied Mathematics”, Ed. R. Dautray, Elsevier, Amsterdam, 1991, pp. 1–19.Google Scholar
  10. 10.
    Bakhvalov N., Panasenko G. Homogénéisation:Averaging Processes in Periodic Media - Kluwer, Dordrecht, 1989.CrossRefGoogle Scholar
  11. 11.
    Barenblatt G.I., Entov V.M., Ryjik V.M. Theory of Transient Liquide and Gas Flow Through Porous Media - Moscow, Nedra, 1972, 288 p. (translated in Kluwer, Dordrecht).Google Scholar
  12. 12.
    Barenblatt G.I., Zheltov Iu.P., Kochina I.N. Basic Concepts in The Theory of Seepage of Homogeneous Liquids in Fissured Rocks -PMM, 24, 1960, pp. 852–864.Google Scholar
  13. 13.
    Barker J.A. Block-Geometry Functions Characterizing Transport in Densely Fissured Media- J. Hydrol, 1985, 77, pp. 263–279.CrossRefGoogle Scholar
  14. 14.
    Barker J.A. Modelling the Effects of Matrix Diffusion on Transport in Densely Fissured Media- Mem. of 18th Congress of the Inter. Assoc, of Hydrogeol., Cambridge, 1985, pp. 250–269.Google Scholar
  15. 15.
    Bedrikovetski P.G., Istomin G.V., Kniazeva M.B. Miscible Displacement From The Fractured Porous Media - Izvstiya Academii Nauk SSSR, Ser. MJG, 1986, N6, pp. 100–110 (translated in: Fluid Mechanics, 1987).Google Scholar
  16. 16.
    Böhm M., Showalter R.E. Diffusion in Fissured Media- SIAM J. Math. Anal., 1985, 16, N 3, pp. 500–509.CrossRefGoogle Scholar
  17. 17.
    Bourgeat A. Homogenization Method Applied to the Behaviour of a Naturally Fissured Reservoir- Mathematical Method in Energy-Research, Ed. SIAM (K.J. Kross), 1984, pp. 181–193.Google Scholar
  18. 18.
    Bourgeat A. Homogenized Behaviour of Diphasic Flow in a Naturally Fissured Reservoir with Uniform Fractures - Comput. Methods in Applied Mechanics and Engineering, 1984, N 47, pp. 205–217.Google Scholar
  19. 19.
    Businov S.N., Umrihin I.D. Gydrodynamics Methods of Well and Field Investigations - Moscow, Nedra, 1973, 243 p. (in Russian).Google Scholar
  20. 20.
    Charlaix E., Hulin J.R, Plona T.J. Experimental Study of Tracer Dispersion in Sintered Glass Porous Materials of Variable Compaction - Phys. Fluids, 1987, 30, pp. 1690–1698.CrossRefGoogle Scholar
  21. 21.
    Coats K.H., Smith B.D. Dead-End Pore Volume and Dispersion in Porous Media - Trans. Soc. Pet. Eng., 1964, 231, N 3, pp. 73–84.Google Scholar
  22. 22.
    Deans H.A. A Mathematical Model For Dispersion in the Direction of Flow in Porous Media - Trans. Soc. Pet. Eng., 1963, v.288, pp. 49–52.Google Scholar
  23. 23.
    De Swaan A.O. Analytic Solutions for Determining Naturally Fractured Reservoir Properties by Well Testing - SPE J., 1976, June, pp. 117–122.Google Scholar
  24. 24.
    Douglas J., Arbogast T. Dual Porosity Models for Flow in Naturally Fractured Reservoirs - In: “Dynamics of Fluids in Hierarchical Porous Media”, Ed. J.H. Cushman, Academic Press, 1990, pp. 177–220.Google Scholar
  25. 25.
    Duguid I.O., Lee P.C., Water Resour. Res., 13, N3, (1977), pp. 558–566.CrossRefGoogle Scholar
  26. 26.
    Egorov A.G., Pudovkin M.A., in: .Applied Problems of Math. Phys., Riga, Ed. Latvien State University, (1983), pp. 98–107 (in Russian).Google Scholar
  27. 27.
    Egorov A.G., Salamatin A.N., Teplophysika Vysokikh Temperatur, 22, N5, (1984), pp. 919–923 (in Russian).Google Scholar
  28. 28.
    Evans R.D., SPE J., 22, N5, (1982), pp. 669–680.Google Scholar
  29. 29.
    Hornung U., in -.Nonlinear Partial Differential Equations(P. File, R. Bates, Eds.), Springer-Verlag, Berlin/New York, (1988).Google Scholar
  30. 30.
    Hornung U., Jager W., in :Complex Chemical Reactions, Modelling and Simulation(P. Denflhard, W. Jager, Eds.), Springer - Verlag, Berlin/New York, (1987).Google Scholar
  31. 31.
    Hornung U., Showalter R.E., J. Math. Analysis and Applications, 147, Nl, (1990), pp. 69–80.CrossRefGoogle Scholar
  32. 32.
    Hornung U., Showalter R.E., Walkington N.J., J. Math. Analysis and Applications, 155, Nl, (1991), pp. 1–20.Google Scholar
  33. 33.
    Kazemi H., Seth Thomas G.W., SPE J., (1969), Dec, pp. 463–472.Google Scholar
  34. 34.
    Khruslov E.Ya., Averaged Model of Non Stationnary Flow through Fractured Porous Media, USSR, Ed. Acad. Sei. Ukraine, Preprint N50, (1988), 34p. (in Russian).Google Scholar
  35. 35.
    Khruslov E.Ya., Uspekhi Math. Nauk, 45, Nl(277), (1990), pp. 197–198 (in Russian).Google Scholar
  36. 36.
    Kutliarov V.S., J. PMTF, (1967), Nl, pp. 84–88 (in Russian).Google Scholar
  37. 37.
    Levy T., European J. of Mechanics. Ser. B : “Fluids”, 9, N4, (1990), pp. 309–327.Google Scholar
  38. 38.
    Nguetseng G., SIAM J. Math. Anal., 20, N3, (1989), pp. 608–628.CrossRefGoogle Scholar
  39. 39.
    Odeh A.S., SPE J., (1965), Janv., pp. 60–66.Google Scholar
  40. 40.
    Palatnik B.., Proc. 1th All Russian Congress on Theoretics and Applied Mechanics, Moscow, Aug. 15–21, (1991), p. 275 (in Russian).Google Scholar
  41. 41.
    Panasenko G.P. Averaging Fields in Composite Materials with High-Modulus Reinforcement- Vestnik MGU, Ser. 15, N 2,1983, pp. 20–27 (in Russian).Google Scholar
  42. 42.
    Panasenko G.P. Averaging of Processes in Highly Heterogeneous Structures - Dokladi AN SSSR, 298, 1988, pp. 76–79 (translated in: Soviet Phys. Dokl., 1988.)Google Scholar
  43. 43.
    Panasenko G.P., J. Vychislitelnoi Mathematiki i Mathematicheskoi Physiki, 30, N2, (1990), pp. 243–253 (translated on English)Google Scholar
  44. 44.
    Panfilov M., Proc. 6th All Russian Congress on Theoretical and Applied Mechanics, Tashkent, Sept. 24–30, (1986), pp. 503–504 (in Russian).Google Scholar
  45. 45.
    Panfilov M., Proc. 3rd Ukraine Conf. of Ukraine Acad. Sei. “Integral Equations in Applied Modelling”, Odessa, Nov. 1–16, (1989), part.2, pp. 102–103 (in Russian).Google Scholar
  46. 46.
    Panfilov M., Soviet Physics Dokl, 35(3), (1990), pp. 225–227.Google Scholar
  47. 47.
    Panfilov M.B., Proc. 2nd European Conference on the Mathematics of Oil Recovery, Aries, France, Sept. 11–14, 1990, Ed. Technip, (1990), pp. 347–350.Google Scholar
  48. 48.
    Panfilov M., Izvestiya Academii Nauk (Russia), Mekhanika Zhid-kosti i Gasa, N6, (1992), pp. 103–116; (translated in Fluid Dynamics, N3, (1993), pp. 112–120).Google Scholar
  49. 49.
    Panfilov M., CR. Acad. Sei. Paris, 318, Serie II, (1994), pp. 1437–1443.Google Scholar
  50. 50.
    Panfilov M., Homogenised Models of Convection-Diffusion transport in Periodic Network Media, Moscow, Ed. Oil h Gas Research Insti- tute, Russian Acad.Sci., preprint N26, (1995), 72p.Google Scholar
  51. 51.
    Panfilov M., Averaged Models of Convection-Diffusion Transfer Through Highly Heterogeneous Porous Media, in : Math. Modelling of Flow Through Porous Media, Proc. Int. Conf., World Scientific Pub., Singapore, pp. 276–300.Google Scholar
  52. 52.
    Panfilov, M. and Panfilova, I. (1996) Averaged models of flows with heterogeneous internal structure. Ed. Nauka, Moscow (in Russian).Google Scholar
  53. 53.
    Prues K., Narasimhan T.N., SPE J., (1985), Feb., pp. 14–26.Google Scholar
  54. 54.
    Showalter R.E., Walkington N.J., J. of Mathematical Analysis and Applications, 155, Nl, (1991), pp. 1–20.CrossRefGoogle Scholar
  55. 55.
    Volkov I.A., in :.Problems of Appl. Math. Geometr. Modelling, Leningrad, Ed. Ingenerno-Stroitelnogo Instituta, (1967), pp. 33–35 (in Russian).Google Scholar
  56. 56.
    Volkov I.A., in :Theor. and Experim. Research of Hydrocarbon Migration Laws, Leningrad, Ed. VNIIGRI, (1980), pp. 104–117 (in Russian).Google Scholar
  57. 57.
    Warren J.E., Root J., SPE J., 3, Sept., (1963), pp. 245–255.Google Scholar
  58. 58.
    Wu Y.-S., Prues K., SPE Res. Eng., 3, N1, (1988), pp. 327–336.Google Scholar
  59. 59.
    Teodorovich E.V., Fluid Dynamics, no. 4, (1994).Google Scholar

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© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Mikhail Panfilov
    • 1
  1. 1.Russian Academy of SciencesOil & Gas Research InstituteMoscowRussia

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