Advertisement

One Phase Darcy’s Flow In Double Porosity Media

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

Abstract

The basic object of examination of this book is associated with highly heterogeneous media where the permeability is contrasting in various elements of the domain. Construction of the averaged models in this case requires to solve several specific basic problems, as determination of the averaging method and a priori classification of media allowing to obtain constructive results of homogenization.

Keywords

Porous Medium Effective Permeability Cell Problem Heterogeneity Scale Exchange Kernel 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adler, P.M. (1992) Porous media: Geometry and transports. Butterworth-Heine-mann, N.Y., USA.Google Scholar
  2. 2.
    Ahaxony, A., Hinrichsen, E.L. and Hansen, A. (1991) Effective renormalization group algorithm for transport in oil reservoirs, Physica A., 177, no. 1–3, pp. 260–266.CrossRefGoogle Scholar
  3. 3.
    Ahiezer, A.I. and Peletinski, S.V. (1977) Methods of Statistical Physics. Nauka, Moscow.Google Scholar
  4. 4.
    Ahmadi, G. (1982) A continuum theory for two phase media, Acta mech., 44, no. 3–4, pp. 299–317.Google Scholar
  5. 5.
    Allaire, G. (1992) Homogenization and two-scale convergence, SIAM J. Math. Anal, 23, no.6, pp. 1482–1518.CrossRefGoogle Scholar
  6. 6.
    Allaire, G. (1997) Mathematical approaches and methods, In: Homogenization and Porous Media, Ed. U. Hornung, Springer, pp. 225–246.Google Scholar
  7. 7.
    An, L., Glimm, J., Sharp, D.H., and Zhang, Q. (1995) Scale-Up Flow in Porous Media, In: Mathematical Modeling of Flow Through Porous Media, Eds. A. Bourgeat et al., World Scientific Publishing, Singapore, 1995, pp. 26–43.Google Scholar
  8. 8.
    Arbogast, T. (1989) Analysis of the simulation of single phase flow through a naturally fractured reservoir, SIAM J. Math. Anal, 26, pp. 12–29.Google Scholar
  9. 9.
    Arbogast, T. and Douglas, J. (1988) Two-phase immiscible flow in naturally fractured reservoirs, Numer. simul. oil recov.: Proc. Symp., Munneapolis, Minn., Dec. 1–12, 1986. N. Y. etc., 1988. pp. 47–66.Google Scholar
  10. 10.
    Arbogast, T., Douglas, J. and Hornung, U. (1990) Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21, pp. 823–836.CrossRefGoogle Scholar
  11. 11.
    Axell, J. (1991) Conduction in two and three dimensions: effective medium theory, Phys. Lett, 153, no. 1, pp. 43–48.CrossRefGoogle Scholar
  12. 12.
    Bachmat, Y. and Bear, J. (1986) Macroscopic modelling of transport phenomena in porous media. 1. The continuum approach, Transport in Porous Media, no. 1, pp. 213–240.CrossRefGoogle Scholar
  13. 13.
    Bakhvalov, N.S. (1977) Averaging PDE With Fast Oscillating Coefficients. Nauka, Moscow.Google Scholar
  14. 14.
    Bakhvalov, N.S. (1975) Averaging of partial derivation equations with quiclky oscillating coefficiennts, Doklady Academii Nauk SSSR, 221, no. 3, 1975, pp. 516–519.Google Scholar
  15. 15.
    Bakhvalov, N.S. (1974) Averaged properties of bodies wiht periodic structure, Doklady Academii Nauk SSSR, 218, no. 5, pp. 1046–1048.Google Scholar
  16. 16.
    Bakhvalov, N.S. and Panasenko G.P. (1984) Homogenization of Processes In Periodic Media. Nauka, Moscow.Google Scholar
  17. 17.
    Bakhvalov, N.S. and Panasenko, G.P. (1989) Homogenization : Averaging Processes in Periodic Media. Kluwer Academic Publishers, Dordrecht.CrossRefGoogle Scholar
  18. 18.
    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
  19. 19.
    Barenblatt, G.I., Zheltov, Yu.P. and Kochina, I.N. (1960) On main representations in the theory of flow of homogeneous liquids in fractured rocks, Prikladnaya Mathematika i Mechanika, 24, no. 5, pp. 852–864.Google Scholar
  20. 20.
    Barker, J. A. (1985) Block-geometry functions characterizing transport in densely fissured media, J. Hydrol, 77, pp. 263–279.CrossRefGoogle Scholar
  21. 21.
    Barker, J.A. (1985) Modelling the effects of matrix diffusion on transport in densely fissured media, Mem. of XVIII Congr. of the Intern. Assoc, of Hydro-geol., Cambridge, pp. 250–269.Google Scholar
  22. 22.
    Basniev, K.S., Bedrikovetsky, P.G. and Dedinets, E.N. (1988) Detrmination of effective permeability of the porous-fractured medium, Inzhenerno-Phyzicheskii Journal, 55, no. 6, pp. 940–948.Google Scholar
  23. 23.
    Bazarov, LP., Gevorkian, E.V. and Nikolaev, P.N. (1989) Nonequilibrium Thermodynamics and Physical Kinetics. Ed. Moscow University, Moscow.Google Scholar
  24. 24.
    Berdichevski, V.L. (1983) Variational Principles of Continuum Mechanics. Nauka, Moscow.Google Scholar
  25. 25.
    Berdichevski, V.L. (1977) On averaging of periodic structures, Prikladnaya Math-ematika i Mechanika, 41, no. 6, pp. 993–1006.Google Scholar
  26. 26.
    Bensoussan, A., Lions, J.-L. and Papanicolaou, G. (1978) Asymptotic analysis for periodic structures, Amsterdam: North-Holland.Google Scholar
  27. 27.
    Bernasconi, J. (1978) Real space renormalization of bond-disordered conductance lattices, Phys. Rev. B., 18, pp. 2185–2191.CrossRefGoogle Scholar
  28. 28.
    Bleher, P.M. and Surgailis, D. (1983) Self-similar random fields, in : Issues of Science and Thechnology. Probability theory. Mathematical Statistics. Theoretical cibenetics, Ed. VINITI, Moscow, 20, pp. 3–51.Google Scholar
  29. 29.
    Böhm, M. and Showalter, R.E. (1985) Diffusion in fissured media, SIAM J. Math. Anal, 16, no. 3, pp. 500–509.CrossRefGoogle Scholar
  30. 30.
    Bolotin, V.V. and Moscalenko, V.N. (1969) To the calculation of strongly isotrope composite materials, Izvestiya Academii Nauk SSSR Ser. Mechanika Tverdogo Tela, no. 3.Google Scholar
  31. 31.
    Bourgat, J.F. (1978) Numerical experiments of the homogenization method for operators with periodic coefficients, Rapport de Recherche IRI, Rocquencourt, no. 277, pp. 212–245.Google Scholar
  32. 32.
    Bourgeat, A., Mikelic, A., and Piatnitski, A. (1998) Modeèle de double porosité aléatoire, C.R. Acad. Sci. Paris, Série 1, 327, pp. 99–104.CrossRefGoogle Scholar
  33. 33.
    Bruggeman, D.A.G. (1935) Berechnung verschiedenier physikalisher Konstanten von heterogenen Substanzen, I Dielektrizitäts Konstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen, Ann. Phys, 5, pp. 636–664.Google Scholar
  34. 34.
    Buevich, Y.A. and Markov, V.G. (1973) Continuum mechanics of monodisperse Suspension: reological equations of state for suspensions of moderate concentration, Prikladnaya Mathematika i Mechanika, 37, no. 6, pp. 1059–1077.Google Scholar
  35. 35.
    Buzinov, S.N. and Umrikhin, I.D. (1973) Gydrodynamic methods of well and field investigation, Nedra, Moscow.Google Scholar
  36. 36.
    Charlaix, E., Hulin, J.P. and Plona, T.J. (1987) Experimental study of tracer dispersion in sintered glass porous materials of variable compaction, Phys. Fluids, 30, pp. 1690–1698.CrossRefGoogle Scholar
  37. 37.
    Chen, Z.-X. (1990) Analytical solutions for double-porosity, double-permeability and layered systems, J. Petrol. Sci. and Eng., 5, no. 1, pp. 1–24.CrossRefGoogle Scholar
  38. 38.
    Coats, K.H. and Smith, B.D. (1964) Dead-end pore volume and dispersion in porous media, Trans. Soc. Petol. Eng., 231, no. 3, pp. 73–84.Google Scholar
  39. 39.
    Crapiste, G.H., Rotstein, E. and Whitaker, S. (1986) A general closure scheme for the method of volume averaging, Chem. Eng. Sci., 41, pp. 227–235.CrossRefGoogle Scholar
  40. 40.
    Danian, F. (1986) Interpretation des essais de puits: Les méthodes nouvelles. Technip, Paris.Google Scholar
  41. 41.
    De Genne, P. Ideas of Scaling in Polymer Physics. Mir, Moscow.Google Scholar
  42. 42.
    De Swaan, A.O. (1976) Analytic solutions for determining naturally fractured reservoir properties by well testing, SPE J., no. 6, pp. 117–122.Google Scholar
  43. 43.
    Dean, R.H. and Lo, L.L. (1988) Simulations of naturally fractured reservoirs, SPE Reservoir Eng., 3, no. 2, pp. 638–648.Google Scholar
  44. 44.
    Deans, H.A. (1963) A mathematical model for dispersion in the direction of flow in porous media, Trans. Soc. Petrol. Eng., 288, pp. 49–52.Google Scholar
  45. 45.
    Deich, M.E. and Philippov, G.A. (1968) Gas Dynamics of Two-Phase Media. Energyia, Moscow.Google Scholar
  46. 46.
    Desbarats, A.J. (1978) Numerical estimation of effective permeability in sand-shale formations, Water Resour. Res., 23, no. 2, pp. 273–286.CrossRefGoogle Scholar
  47. 47.
    Douglas, J., Paes Lerne, P.J., Arbogast, T. and Schmitt T. (1987) Simulation of flow in naturally fractured reserviors, Proc. IX SPE symp. on reservoir simulation, Soc. of Petrol Eng. Dallas (Tex.), Paper SPE 16019, pp. 271–279.Google Scholar
  48. 48.
    Duguid, I.O. and Lee, P.C. (1977) Flow in fractured porous media, Water Resour. Res., 13, no. 3, pp. 558–566.CrossRefGoogle Scholar
  49. 49.
    Dykhne, A.M. (1970) Conductivity of a two-dimentional two-phase system, Soviet Physics JETP 59, no. 1(7), pp. 111–115.Google Scholar
  50. 50.
    Emets, Y.P. (1986) Electric Properties of Composite Materials With Regular Structure. Naukova Dumka, Kiev.Google Scholar
  51. 51.
    Evans, R.D. (1983) A proposed model for multiphase flow through naturally fractured reservoir, Sos. Petr. Eng. J., 22, no. 5, pp. 669–680.Google Scholar
  52. 52.
    Fllgelman, H., Cinco-Ley, H., Ramey H.J., et al. (1989) Pressure drawdown test analysis of a gas well-application of new correlations, SPE Formation Evaluation, Sept. pp. 406–453.Google Scholar
  53. 53.
    Geld, P.V. and Mitushov, E.A. (1990) Generalized method of self-similar field to determine elastic properties of heterogeneous materials, J. Prikladnoi Mathem-atiki i Technicheskoi Physiki, no. 1, pp. 96–100.Google Scholar
  54. 54.
    Geld, P.V., Sachkov, I.N., Gofman, A.G. and Sidorenko, F.A. (1990) Conductivity of heterogeneous systems: finit element method, Doklady Academii Nauk SSSR, 315, no. 3, pp. 604–607.Google Scholar
  55. 55.
    Golf-Rakht, T.D. (1986) Basis of oil geology and development of fractured reservoirs, Nedra, Moscow.Google Scholar
  56. 56.
    Happel, G. and Brenner, G. (1976) Hydrodynamics at low Reynolds Numbers. Mir, Moscow.Google Scholar
  57. 57.
    Hashin, Z. and Shtikmen, S. (1962) On some variational priciples inanisotropic and homogeneous elasticity, J. Mech. Phys. Solids., 10, no. 4, pp. 343.CrossRefGoogle Scholar
  58. 58.
    Hill, R. (1963) Elastic properties of reinforced solids: some theoretical principles, J. Mech. Phys. Solids., 11, no. 5.Google Scholar
  59. 59.
    Hinch, E.J. (1977) An averaged-equation approach to particle interactions in a fluid suspension, J. Fluid Mech., 83, pt. 4, pp. 695–720.CrossRefGoogle Scholar
  60. 60.
    Hornung, U. and Jäger, W. (1987) A model for chemical reactions in porous media, In: Complex chemical reactions, modelling and simulation, B., Springer, New-York.Google Scholar
  61. 61.
    Hornung, U. and Showalter, R.E. (1990) Diffusion models for fractured media, J. Math. Anal, and Appl., 147, no. 1, pp. 69–80.CrossRefGoogle Scholar
  62. 62.
    Hornung, U., Showalter, R.E. and Walkington, N.J. (1991) Microstructure models of diffuion in fissured media, J. Math. Anal, and Appl., 155, no. 1, pp. 1–20.CrossRefGoogle Scholar
  63. 63.
    Ibraguimov, A.I. (1984) Some problems of qualitative theory of elliptic and parabolic equations. Dr. Thesis, Steklov Mathematical Institute, oscow.Google Scholar
  64. 64.
    Jäger, W. and Mikelic, A. (1994) On the flow conditions at the boundary between a porous medium and an imprevious solid, In: Progress in partial Differential Equations, vol. 314 of Pitman Research Notes in Mathematics, Eds. M. Chipot, J. Saint Jean Paulin and I. Shafrir, Longman Scientific and Technical, London.Google Scholar
  65. 65.
    Jäger, W. and Mikelic, A. (1995) On the boundary conditions at the contact interface between a porous medium and a free fluid, Annali della Scuola Normale Supriore di Piza, Classe di Scienze. Google Scholar
  66. 66.
    Kalinin, V.A. and Baiuk, N.O. (1990) Energy limitations for the effective modulus of anisotropic microheterogeneous media, Doklady Academii Nauk SSSR, 313, no. 5, pp. 1090–1094.Google Scholar
  67. 67.
    Kartashov, E.M. (1979) Analytical methods in thermo-conductivity of the solid bodies. Ed. Superior School, Moscow.Google Scholar
  68. 68.
    Kazemi, H. Seth and Thomas, G. W. (1969) The interpretation of interference tests in naturally fractured reservoirs with uniform fracture distribution, SPE J.no. 12, pp. 463–472.Google Scholar
  69. 69.
    Kent, T.L., Dixon, T.N. and Pierson, R.G. (1983) Fractured reservoirs simulation, Ibid. 23, no. 1, pp. 42–54.Google Scholar
  70. 70.
    Khrouslov, E.Ya. (1988) Hommohenized model of non-stationary flow in fractured porous media, Preprint Ukrainian Academy of Sciences, Kiev, no. 50.Google Scholar
  71. 71.
    Khrouslov, E.Ya. (1990) Hommohenized model of highly heterogeneous medium with memory, Uspekhi Mathematicheskikh Nauk, 45, no. 1(297), pp. 197–198.Google Scholar
  72. 72.
    King P.R. (1988) The use of renormalization for calculating effective permeability, Transport in Porous Media, p. 643Google Scholar
  73. 73.
    King P.R., Muggeridge, A.H. and Price, W.G. (1993) Renormalization calculations of immiscible flow, Transport in Porous Media, bf 12, pp. 237–260.CrossRefGoogle Scholar
  74. 74.
    Kirkpatrick, S. (1973) Percolation and conduction, Rev. Mod. phys., 45, no. 4, pp. 574–588.CrossRefGoogle Scholar
  75. 75.
    Kochin, N.E., Kibel, I.A. and Rose, N.V. (1963) Theoretical Hydromechanics. Part 2. Fizmatgiz, Moscow.Google Scholar
  76. 76.
    Kohring, G.A. (1991) Calculation of the permeability of porous media using hy-drodynamic cellular automata, J. Stat.Phys., 63, no. 1/2, pp. 411–418.CrossRefGoogle Scholar
  77. 77.
    Kolesnichenko, V.V. (1978) Main equations of hydro-thermodynamics of the multiphase multicomponent continuum with chemical reactions: (phenomenological theory), Institut Prikladnoi Mathematiki Academii Nauk SSSR, Preprint, no. 125.Google Scholar
  78. 78.
    Kozlov, S.M. (1978) Homogenization of differential operators with closely periodic quickly oscillating coefficients, Mathematicheskii Sbornik. Novaya Seriia, 107(149), no. 2(10), pp. 199–217.Google Scholar
  79. 79.
    Kozlov, S.M. (1978) Homogenization of random structures, Doklady Academii Nauk SSSR, 241, no. 5, pp. 1016–1019.Google Scholar
  80. 80.
    Kozlov, S.M. (1979) Homogenization of random operators, Mathematicheskii Sbornik Novaya Seriia, 109(151), no. 2(6), pp. 188–202.Google Scholar
  81. 81.
    Kozlov, S.M. (1979) Conductivity of two-dimentional random media, Uspekhi Mathematicheskikh Nauk, 34, no. 4(208), pp. 193–194.Google Scholar
  82. 82.
    Kozlov, S.M. (1985) Homogenization method and random walk in heterogeneous media, Uspekhi Mathematicheskikh Nauk, 40, no. 2(242), pp. 61–120.Google Scholar
  83. 83.
    Kozlov, S.M. (1989) Geometric aspects of homogenization, Uspekhi Mathematicheskikh Nauk, 44, no. 2, pp. 79–120.Google Scholar
  84. 84.
    Landau, L.D. and Lifhitz, E.M. (1960), Electodynamics of Continuum Media. Pergamon Press, Oxford.Google Scholar
  85. 85.
    Landau, L.D. and Lifshits, E.M. Continuum Mechanics. Fizmatgiz, Moscow.Google Scholar
  86. 86.
    Landauer, R. (1952) The electrical resistance of binary metallic mixtures, J. Appl. Phys., 23, pp. 779–784.CrossRefGoogle Scholar
  87. 87.
    Levy, T. (1990) Filtration in a porous fissured rock: influence of the fissures con-nexity, Europ. J. Mech. B., 9, no. 4, pp. 309–327.Google Scholar
  88. 88.
    Livshits, I.M. and Rozentsveig, L.N. (1946) To the theory of elastic properties of liquid cristals, Russian J. Experimental and Theoretical Physics, 16, no. 11, p. 967.Google Scholar
  89. 89.
    Lifshits, E.M. and Pitaevsky, L.P. (1979) Physical Kinetics. Nauka, Moscow.Google Scholar
  90. 90.
    Lin, S.H. (1992) Transient conduction in heterogeneous media, Intern. Commun. Heat and Mass Transfer., 19, no. 2, pp. 165–174.CrossRefGoogle Scholar
  91. 91.
    Lindstrom, F.T. and Narasimham, N.L. (1973) Mathematical theory of a kinetic model for dispersion of previously distributed chemicals in a sorbing porous medium, SI AM J. Appl. Math., 24, pp. 496–510.CrossRefGoogle Scholar
  92. 92.
    Lions, J.-L. (1978) Notes on somes computational aspects of homogenization in composite materials, Numerical methods in mathematical physics and optimal control, Ed. Marchuk G.N. and J.-L. Lions, Nauka, Novosibirsk, pp. 5–19.Google Scholar
  93. 93.
    Marinbakh, M.A. and Lyusternik, V.E. (1985) Effect of macroheterogeneity on effective permeability of porous media, Dynamika mnogofaznykh sred, Nauka, Novosibirsk, pp. 123–127.Google Scholar
  94. 94.
    Maries, C.M. (1982) On macroscopic equations governing multiphase flow with difuusion and chemical reactions in porous media, Int. J. Eng. Sei., 20, pp. 643–662.CrossRefGoogle Scholar
  95. 95.
    Mikelic, A. (1997) On the transmission conditions at the contact interface between a porous medium and a free fluid, In: Homogenization and Porous Media, Ed. U. Hornung, Springer, pp. 69–76.Google Scholar
  96. 96.
    Mohanty, S. and Sharma, M.M. (1991) A Monte Carlo RSRG method for the percolation/conduction properties of correlated lattices, Phys. Lett, 154, no. 9, pp. 475–481.CrossRefGoogle Scholar
  97. 97.
    Monin, A.Kh. and Yaglom, F.M. (1975) Statistical Theoty of Turbulence. Nauka, Moscow.Google Scholar
  98. 98.
    Neale, G. and Jacob, H.N. (1975) Flow perpendicular to mats of randomly arranged cilindrical fibers: (Importance of cell models), AIChE J., 21, no. 4, pp. 805–807.CrossRefGoogle Scholar
  99. 99.
    Nigmatulin, R.I. (1978) Basis for Mechanics of Heterogeneous Media. Nauka, Moscow.Google Scholar
  100. 100.
    Nigmatulin, R.I. (1987) Dynamics of Multiphase Media. Nauka, Moscow.Google Scholar
  101. 101.
    Nikolaevski, V.N. (1996) Geomechanics and Fluidodynamics With Applications to Reservoir Engineering. Kluwer Academic Publishers, Dordrecht.Google Scholar
  102. 102.
    Nikolaevskii, V.N., Bondarev, E.A., Mirkin M.I. et al, (1968) Motion of Hydrocarbon Mixtures in Porous Media. Nedra, Moscow.Google Scholar
  103. 103.
    Nguetseng, G. (1989) A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal, 20, pp. 608–629.CrossRefGoogle Scholar
  104. 104.
    Odeh, A.S. (1965) Unsteady-state behaviour of fractured reservoirs, SPE J., no. 1, pp. 60–66.Google Scholar
  105. 105.
    Olarewaju, J.S. and Lee, W.J. (1989) New pressure-transient analysis model for dual-porosuty reservoir, SPE Form. Evaluation, no. 9, pp. 384–390.Google Scholar
  106. 106.
    Ootani, I., Ohashi, Y.H., Ohashi, K. and Fukuchi M. (1992) Breakdown properties of random systems with distributed conductances, J. Phys. Soc. Jap., 61, no. 4, pp. 1399–1407.CrossRefGoogle Scholar
  107. 107.
    Palatnik, B.. (1991) Numerical simulation of multiphase flow through highly heterogeneous periodic media, In: Proc. 7-th All Russian Congress on Theoretics and Applied Mechanics, Moscow, Aug. 15–21.Google Scholar
  108. 108.
    Panasenko, G.P. (1983) Avaeraging the fields in composite materials with highly modulus structure, Vestnik MGU, Ser.15, Vychislitelnaya matematika i kibernetika, no. 2, pp. 20–27.Google Scholar
  109. 109.
    Panasenko, G.P. (1988) Homogenization of processes in strongly heterogeneous media, Doklady AN SSSR, 298, no. 1, pp. 76–79 (translated in: Soviet Phys. Dokl., 1988.)Google Scholar
  110. 110.
    Panasenko, G.P. (1988) Averaging the carcas constructions with random structures, Doklady AN SSSR, 244, no. 2. pp. 335–337 (translated in: Soviet Phys. Dokl, 1988.)Google Scholar
  111. 111.
    Panasenko, G.P. (1990) Numerical-Analytical Method of Multicomponent Homogenization for Equations with Contrast Parameters, J. Vychislitelnoi Mathematiki i Mathematicheskoi Physiki, 30, no. 2, pp. 243–253 (translated on English).Google Scholar
  112. 112.
    Panasenko, G.P. (1992) Principle of decomposition for the homogenized operator for nonlinear system of equations in periodic and carcas constructions, Doklady AN SSSR, 263, no. 1, pp. 35–40.Google Scholar
  113. 113.
    Panasenko, G.P. and Sysuev, V.V. (1986) Averaging the parameters of mathema-tic models of heat, water and mass transport in an aggregated random-periodic porous medium, Mathematical methods of rock description and calculation of their effective propeties, Moscow, MOIP, pp. 33–42.Google Scholar
  114. 114.
    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., World Scientific Pub., Singapore, pp. 276–300.Google Scholar
  115. 115.
    Panfilov, M.B. (1995) Homogenised Models of Convection-Diffusion Transfer in Periodic Network Media, Preprint Oil & Gas Research Institute, Russian Acad. Sci., no. 26.Google Scholar
  116. 116.
    Panfilov, M. (1994) Averaged Model-Type Transition In Flows Through MultipleGoogle Scholar
  117. 118.
    Heterogeneous Porous Media, CR. Acad. Sei. Paris, Ser.II, 318, pp. 1437–1444.Google Scholar
  118. 117.
    Panfilov, M. (1993) Structural Averaging of Porous Flow Processes in Heterogeneous Media, Fluid Dynamics, no. 3, pp. 112–120.Google Scholar
  119. 118.
    Panfilov, M.B. (1991) Averaging transport through heterogeneous porous media, In: Proc. 7-th All Russian Congress on Theoretics and Applied Mechanics, Moscow, Aug. 15–21.Google Scholar
  120. 119.
    Panfilov, M. (1990) Mean mode of porous flow in highly inhomogeneous media, Soviet Physics Doklady, 35, pp. 225–227.Google Scholar
  121. 120.
    Panfilov, M. (1990) Irregular averaging of filtration transfer processes in heterogeneous media, Proc. 2nd Europ. Conf. Mathematics of Oil Recovery, Arie, pp. 347–350.Google Scholar
  122. 121.
    Panfilov, M.B. (1990) Averaged irreditary models of flow through fracturedporous media, Proc. Int. Symp. on Development Oil Fields with Fractured Rocks, Varna, Oct. 22–25, Varna, book 3, pp. 52–57.Google Scholar
  123. 122.
    Panfilov, M.B. (1990) Averaged irreditary models of flow with highly heterogeneous interior structure, Proc. Int. Conf. Development of G as-Condensate Reservoirs, Krasnodar, May 29 - June 2, Section 6, pp. 83–87.Google Scholar
  124. 123.
    Panfilov, M.B. (1989) Unlocal models and hydrodynamic irrevirsibility of flow through porous media in oil and gas reservoirs, In: Proc. 3 rd Republican Conf. Ukrainian Acad. Sei. “Integral Equations in Applied Modeling”, Odessa, Nov. 14–16, Ed. Naukova Dumka, Kiev, pp. 102–103.Google Scholar
  125. 124.
    Panfilov, M.B. (1987) Regularized asymptotics and averaged models of flow in periodic heterogeneous media, In: Thechnology and Technique of Enhanced Oil, Gas and Condensate Recovery, Ed. Moscow Oil & Gas Institute, Moscow, no. 199.Google Scholar
  126. 125.
    Panfilov, M.B. (1986) Flow through porous media with fast oscillating properties, Proc. VI All Russian, Congress on Theor. and Appl. Mechanics, Tashkent, Sept. 24–30, Ed. Russian Acad. Sci., Tashkent, pp. 503–504.Google Scholar
  127. 126.
    Panfilov, M.B. (1986) Quasi-equilibrium asymptotic of underground depletion in oil and gas reservoirs, Soviet Physics Doklady, 31(5), pp. 225–227.Google Scholar
  128. 127.
    Panfilov, M. and Panfilova, I. (1996) Averaged models of flows with heterogeneous internal structure. Nauka, Moscow.Google Scholar
  129. 128.
    Prues, K. and Narasimhan, T.N. (1985) A practical method for modelling fluid and heat flow in fractured porous media, SPE J., no. 2, pp. 14–26.Google Scholar
  130. 129.
    Raats, P. A.C. (1968) Transport in soils: The balance of momentum, SoilSei. Sor. Amer. Proc, 32, pp. 452–456.CrossRefGoogle Scholar
  131. 130.
    Rakhmatulin, Kh.A. (1956) Basis of gas dynamics for coexisting motions of compressible media, Prikladnaya Mathematika and Mekhanika, 20, no. 2, pp. 184–195.Google Scholar
  132. 131.
    Reynolds, P.J., Klein, W. and Stanley, H.E. (1977) A real space renormalization group for site and bond percolation, J. Phys. C, 10, pp. L167-L172.CrossRefGoogle Scholar
  133. 132.
    Rojdestvensky, B.L. and Yanenko, N.N. (1978) Systems of QuasilinearEquations. Nauka, Moscow.Google Scholar
  134. 133.
    Ross, B. (1986) Dispersion in fractal networks, Water Resour. Res., 22, no. 5, pp. 823–827.CrossRefGoogle Scholar
  135. 134.
    Sahimi, M., Hughes, B.D., Scriven, L.E. and Davis, H.T. (1983) Real-space renormalization and effective medium approximation to the percolation conduction problem, Chem. Eng. Sei., 28, pp. 307–311.Google Scholar
  136. 135.
    Salamatin, A.N. (1987) Mathematicak Models of Disperse Flows. Ed. Kazan University, Kazan.Google Scholar
  137. 136.
    Sanchez-Palensia, E. (1980) Non-homogeneous media and vibration theory. VoluGoogle Scholar
  138. of multiophase media, Nauka, Novosibirsk, pp. 199–203.Google Scholar
  139. 140.
    Shah, N. and Ottino, J.M. (1986) Effective transport properties of disordered multiphase composites: Application of real-space renormalization group theory, Chem. Eng. Sci., 41, pp. 283–296.CrossRefGoogle Scholar
  140. 141.
    Sheidegger, A. (1960) Physics of flow through porous media. Gostoptekhizdat, Moscow.Google Scholar
  141. 142.
    Shermegor, T.D. (1977) Elasticity Theory for Micro-heterogeneous Media. Nauka, Moscow.Google Scholar
  142. 143.
    Shvidler, M.I. (1985) Statistical Hydrodynamics of Porous Media. Nedra, Moscow.Google Scholar
  143. 144.
    Showalter, R.E. and Walkington, N.J. (1991) Micro-structure models of diffusion in fissured media, J. Math. Anal, and Appl, 155, no. 1, pp. 1–20.CrossRefGoogle Scholar
  144. 145.
    Stinchcombe, R.B. (1977) Conductivity and spin-move stiffness in disordered systems: An exactly soluble model, J. Phys. C, 7, pp. 179–203.CrossRefGoogle Scholar
  145. 146.
    Toffoli, T. and Marolus, N. (1991) Cellular Automata Machines. Mir, Moscow.Google Scholar
  146. 147.
    Tsybulsky, G.P. (1977) Equations of nonequilibrium two-phase flow through porous media, Numerical solution in problems of multiphase noncompressible liquid flow through porous media, Ed. Siberian Branch of Acad. Sei., Novosibirsk, pp. 203–213.Google Scholar
  147. 148.
    Tsybulsky, G.P. (1985) Equations of macro-nonequilibrium flow through porous media, Numerical methods in continuum mechanics, Ed. Siberian Branch of Acad. Sci., Novosibirsk, 16, no. 5, pp. 133–140.Google Scholar
  148. 149.
    Tuvaeva, I.V. and Panfilov, M.B. (1990) Convective dispersion of the front n heterogeneous medium and the problem of its description, Preprint, Oil & Gas Research Institute, Moscow, no. 6.Google Scholar
  149. 150.
    Van Dyke, M. (1964) Perturbation methods in fluid mechanics. Academic Press, New York - London.Google Scholar
  150. 151.
    Vasiliev, L.L. and Tanaeva, S.A. (1971) Thermophysic properties of Porous Materials. Nauka i Technika, Minsk.Google Scholar
  151. 152.
    Vladimirov, V.S. (1983) Equations of Mathematical Physics. Nauka, Moscow.Google Scholar
  152. 153.
    Volkov, S.D. and Stavrov, V.P. (1978) Statistical Mechanics of Composite Materials. Ed. Belorussian University, Minsk.Google Scholar
  153. 154.
    Warren, J.E. and Root, J. (1963) The behavior of naturally fractured reservoirs, SPE J, 3, no. 9, pp. 245–255.Google Scholar
  154. 155.
    Wilson, W.G. and Loidlaw, W.G. (1992) Microscopic-based flow invasion simulations, J. Stat. Phys., 66, no. 3/4, pp. 1165–1176.CrossRefGoogle Scholar
  155. 156.
    Wu, Y.-S. and Prues, K. (1988) A multiple-porosity method for simulation of naturally fractured petroleum reservoirs, SPE Res. Eng., 3, no. 1, pp. 327–336.Google Scholar
  156. 157.
    Yu Boming and Yao, K.L. (1991) Computation of heat conduction in self-similar porous structures, Phys. Rev. A., 44, no. 6, pp. 3664–3668.CrossRefGoogle Scholar
  157. 158.
    Yurinsky, V.V. (1980) On averading of an elliptic boundary-value problem with random coefficients, Syberian Math. J., 21, no. 3, pp. 209–223.Google Scholar
  158. 159.
    Yurinsky, V.V. (1982) On averaging og the non-divergent equations of the second order with random coefficients, Syberian Math. J., 23, no. 2, pp. 176–188.Google Scholar
  159. 160.
    Zhikov, V.V. (1991) On the estimations for the averaged matrix and averaged tensor, Uspekhi Mathematicheskikh Nauk, 46, no. 3(279), pp. 49–109.Google Scholar
  160. 161.
    Zhikov, V.V. and Oleinik, O.A. (1991) Homogenization of Differential Operators. Nauka, Moscow.Google Scholar
  161. 162.
    Zhikov, V.V., Kozlov, S.M., Oleinik, O.A. and T’ en Ngoan, K. (1979) Averaging and G-convergence of differential operators, Russian Math. Surveys, 34, no. 5, pp. 65–133.CrossRefGoogle Scholar
  162. 163.
    Zubarev, D.N. (1977) Nonequilibrium Statistical Thermodynamics. Nauka, Moscow.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

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

Personalised recommendations