First Passage Time in a Two-Layer System

TL;DR

This paper analyzes the first passage time distribution in a two-layer stratified medium with different velocities, revealing exponential asymptotics and velocity-dependent crossover behaviors.

Contribution

It introduces a lattice generating function approach and a recursive method to study first passage times in stratified systems with analytical and numerical insights.

Findings

Asymptotic distribution is exponential in time for any velocities.

Decay constant relates to the largest eigenvalue of an operator.

Crossover from $L^{-2}$ to $L^{-1}$ behavior in anti-symmetric velocity case.

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 length $L$ crosses over from $L^{-2}$ behavior in diffusive limit to $L^{-1}$ behavior in the convective regime, where the crossover length $L^*$ 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.