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Journal of Statistical Physics

, Volume 79, Issue 5–6, pp 895–922 | Cite as

First passage time in a two-layer system

  • Jysoo Lee
  • Joel Koplik
Articles

Abstract

As a first step in the first passage problem for passive tracer in stratified porous media, we consider the case of a two-dimensional system consisting of two layers with different convection velocities. Using a lattice generating function formalism and a variety of analytic and numerical techniques, we calculate the asymptotic behavior of the first passage time probability distribution. We show analytically that the asymptotic distribution is a simple exponential in time for any choice of the velocities. The decay constant is given in terms of the largest eigenvalue of an operator related to a half-space Green's function. For the anti-symmetric case of opposite velocities in the layers, we show that the decay constant for system lengthL crosses over fromL−2 behavior in the diffusive limit toL−1 behavior in the convective regime, where the crossover lengthL* is given in terms of the velocities. We also have formulated a general self-consistency relation, from which we have developed a recursive approach which is useful for studying the short-time behavior.

Key Words

First passage problem convection-diffusion equation layered system asymptotic behavior 

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References

  1. 1.
    E. Guyon, J.-P. Nadal, and Y. Pomeau, eds.,Disorder and Mixing (Kluwer, Dordrecht, 1988).Google Scholar
  2. 2.
    G. Dagan,Annu. Rev. Fluid Mech. 19:183 (1987).Google Scholar
  3. 3.
    P. R. King,J. Phys. A 20:3935 (1987).Google Scholar
  4. 4.
    I. Webman,Phys. Rev. Lett. 47:1496 (1981).Google Scholar
  5. 5.
    M. Sahimi,Rev. Mod. Phys. 65:1393 (1993).Google Scholar
  6. 6.
    J. Bear,Dynamics of Fluids in Porous Media (Elsevier, Amsterdam, 1971).Google Scholar
  7. 7.
    F. A. L. Dullien,Porous Media: Structure and Fluid Transport, 2nd ed. (Academic Press, London, 1991).Google Scholar
  8. 8.
    J. R. L. Allen,Sedimentary Structures (Elsevier, Amsterdam, 1974).Google Scholar
  9. 9.
    G. Matheron and G. de Marsily,Water Resources Res. 16:901 (1980).Google Scholar
  10. 10.
    S. Redner,Physica D 38:287 (1989).Google Scholar
  11. 11.
    J.-P. Bouchaud, A. Georges, J. Koplik, A. Provata, and S. Redner,Phys. Rev. Lett. 64:2503 (1990).Google Scholar
  12. 12.
    G. Zumofen, J. Klafter, and A. Blumen,Phys. Rev. A 42:4601 (1990).Google Scholar
  13. 13.
    J. F. Willemsen and R. Burridge,Phys. Rev. A 43:735 (1991).Google Scholar
  14. 14.
    J. Ziman,Models of Disorder (Cambridge University Press, Cambridge, 1982).Google Scholar
  15. 15.
    S. Havlin and D. ben-Avraham,Adv. Phys. 36:695 (1987).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Jysoo Lee
    • 1
  • Joel Koplik
    • 1
  1. 1.Benjamin Levich Institute and Department of PhysicsCity College of the City University of New YorkNew York

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