Transport in Porous Media

, Volume 28, Issue 1, pp 109–124 | Cite as

Analytic Solution to a Problem of Seepage in a Chequer-Board Porous Massif

  • A. R. Kacimov
  • Yu. V. Obnosov


A study is made of steady two-dimensional seepage in a porous massif composed by a double-periodic system of ‘white’ and ‘black’ chequers of arbitrary conductivity. Rigorous matching of Darcy's flows in zones of different conductivity is accomplished. Using the methods of complex analysis, explicit formulae for specific discharge are derived. Stream lines, travel times, and effective conductivity are evaluated. Deflection of marked particles from the ‘natural’ direction of imposed gradient and stretching of prescribed composition of these particles enables the elucidation of the phenomena of transversal and longitudinal dispersion. A model of pure advection is related with the classical one-dimensional vective dispersion equation by selection of dispersivity which minimizes the difference between the breakthrough curves calculated from the two models.

seepage conductivity double-periodic structure advection dispersion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abramovitz, M. and Stegun, I. A.: 1965, Handbook of Mathematical Functions, Dover, New York.Google Scholar
  2. Bear, J.: 1972, Dynamics of Fluids in Porous Media, Elsevier, New York.Google Scholar
  3. Berdichevski, V.L.: 1985, The thermal conductivity of chess structures, Vestn. Mosk. Univ. Mat. Mech. 40(4), 56–63 (in Russian).Google Scholar
  4. Birkhölzer, J., Rubin, H., Daniels, H. and Rouve, J.: 1993, Contaminant advection and spreading in a fractured permeable formation: Part 1. Parametric evaluation and analytical solution, J. Hydrology 144, 1–33.Google Scholar
  5. Carera, J.: 1993, An overview of uncertainties in modeling groundwater solute transport, J. Contam. Hydrol., 13, 23–48.Google Scholar
  6. Davis, A. D.: 1986, Deterministic modeling of dispersion in heterogeneous permeable media, Ground Water 24(5), 609–615.Google Scholar
  7. Dykhne, A.M.: 1971, Conductivity of a two- dimensional two- phase system, Soviet. Phys. JETP 32, 63–65.Google Scholar
  8. Emets, Yu. P. and Obnosov, Yu. V.: 1990, Compact analog of a heterogeneous system with a checkerboard field structure, Soviet. Phys. Tech. Phys. 35, 907–913.Google Scholar
  9. Gachov, F. D.: 1966, Boundary Value Problems, Oxford and Addison.Google Scholar
  10. Goldshtein, R. V. and Entov, V. M.: 1994, Qualitative Methods in Continuum Mechanics, Wiley, New York.Google Scholar
  11. Golubeva, O. V. and Spilevoy, A. Y.: 1967, On plane seepage in media with continuously changing permeability along the curves of second order, Izv. Akad. Nauk SSSR Mekh. Zhid. Gaza 2, 174–179 (in Russian).Google Scholar
  12. Grinberg, G. A.: 1948, Selected Problems in Mathematical Theory of Electric and Magnetic Phenomena, Acad. of Sci., Moscow (in Russian).Google Scholar
  13. Kacimov, A. R.: 1997, Dynamics of ground water mounds: analytical solutions and integral characteristics, Hydro. Sci. J. (in press).Google Scholar
  14. Kacimov A. R. and Obnosov, Yu. V.: 1994, Minimization of ground water contamination by lining of a porous waste repository, Proc. Indian Natn. Sci. Acad. A 60, 783–792.Google Scholar
  15. Kasimov, A. R. and Obnosov, Yu. V.: 1995, Groundwater flow in a medium with periodic inclusions, Fluid Dynam. 30(5), 758–766.Google Scholar
  16. Kasimov, A. R. and Obnosov, Yu.V.: 1996, Rigorous values of efficiency of the Muskat well networks, DAN (Physi.-Dokl.), 353(2).Google Scholar
  17. Kacimov, A. R. and Obnosov, Yu. V.: 1996, Analytical solutions to a problem of sink-source flow in a porous medium, Arab Gulf J. Sci. Res. (in press).Google Scholar
  18. Kacimov, A. R. and Obnosov, Yu. V.: 1997, Explicit, rigorous solutions to 2-D heat transfer: Two component media and optimization of cooling fins. Int. J. Heat Mass Trans. 40(5), 1191–1196.Google Scholar
  19. Kitanidis, P. K: 1994, The concept of dilution index, Water Resour. Res. 30(7), 2011–2026.Google Scholar
  20. Magnico, P., Leroy, C., Bouchaud, J. P., Gauthier, C. and Hulin, J. P.: 1993, Tracer dispersion in porous media with a double porosity. Phys. Fluids A, 5(1), 46–57.Google Scholar
  21. Milton, G. W., McPhedran, R. C. and McKenzie, D. R.: 1981, Transport properties of arrays of intersecting cylinders, Appl. Phys. 25, 23–30.Google Scholar
  22. Morel-Seytoux, H. J. and Nachabe, M.: 1992, An effective scale-dependent dispersivity deduced from a purely convective flow field, Hydrol. Sci. J. 37, 93–104.Google Scholar
  23. Muskat, M.: 1946, Physical Principles of Oil Production, McGraw-Hill, New York.Google Scholar
  24. Neuman, S. P.: 1990, Universal scaling of hydraulic conductivities and dispersivities in geologic media, Water Resour. Res. 26(8), 1749–1758.Google Scholar
  25. Obdam, A. N. M. and Veling, E. J. M.: 1987, Elliptical inhomogeneities in groundwater flow - an analytical description, J. Hydrology 95, 87–96.Google Scholar
  26. Obnosov, Yu. V.: 1996, Exact solution of a problem of R-linear conjugation for a rectangular checkerboard field. Proc. Royal Soc. London, A 452, N1954, 2423–2442.Google Scholar
  27. Oleinik, O. A. Kozlov, S.M. and Zhikov, V.V.: 1994, Trans. Homogenization of Differential Operators and Integral Functionals, Springer, Berlin.Google Scholar
  28. Ottino, J. M.: 1989, The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge University Press, New York.Google Scholar
  29. Pandey, R. N. and Gupta, S. K.: 1983, Evaluation of the hydrodynamic dispersion coefficient using leached fraction equation, J. Hydrology 62, 313–319.Google Scholar
  30. Pickup, G. E., Ringrose, P. S., Jensen, J. L. and Sorbie, K. S.: 1994, Permeability tensors for sedimentary structures, Math. Geol. 26(2), 227–250.Google Scholar
  31. Polubarinova-Kochina, P.Ya.: 1977, Theory of Ground Water Movement, Nauka, Moscow (in Russian).Google Scholar
  32. Rayleigh, Lord: 1892, On the influence of obstacles arranged in rectangular order upon the properties of medium, Phil. Mag. 34, 481–502.Google Scholar
  33. Romeu, R. K. and Noetinger, B.: Calculation of internodal transmissivities in finite difference models of flow in heterogeneous porous media, Water Resour. Res. 31, 943–959.Google Scholar
  34. Romm, E. S.: 1985, Structural Models of Porous Media of Rocks, Nedra, Leningrad (in Russian).Google Scholar
  35. Schwartz, F.W.: 1977, Macroscopic dispersion in porous media: the controlling factors, Water Resour. Res. 13, 743–752.Google Scholar
  36. Strack, O. D. L.: 1992, A mathematical model for dispersion with a moving front in groundwater, Water Resour. Res. 28, 2973–2980.Google Scholar
  37. Yarmitsky A. G.:1986, A seepage theorem on two inclusions, Izv. AN SSSR Mekh. Zhid. Gasa, 20(4), 76–82 (in Russian).Google Scholar
  38. Youngs E. G.:1986, The analysis of groundwater flows in unconfined aquifers with nonuniform hydraulic conductivity, Transport in Porous Media 1, 399–417.Google Scholar
  39. Zheng C., Bradbury K. R. and Anderson, M.:1992, A computer model for calculation of groundwater paths and travel times in transient three-dimensional flows, Wisconsin Geol. Nat. History Survey, Information Circular 70.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • A. R. Kacimov
    • 1
  • Yu. V. Obnosov
    • 1
  1. 1.Institute of Mathematics and Mechanics, Kazan UniversityKazanRussia

Personalised recommendations