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.
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Lee, J., Koplik, J. First passage time in a two-layer system. J Stat Phys 79, 895–922 (1995). https://doi.org/10.1007/BF02181208
- First passage problem
- convection-diffusion equation
- layered system
- asymptotic behavior