Variational bounds for first-passage-time problems in stratified porous media

We examine the first-passage-time problem for passive tracer transport in flow through porous media. The simplified model used [G. Matheron and G. de Marsily, Water Resources Res. 16, 901 (1980)] pertains especially to groundwater flow, and assumes that the medium is fully stratified. Transport normal to the layering is governed by diffusion alone; transport parallel to the layering is governed by both diffusion and convection. The fluid velocity varies randomly from layer to layer. The region of interest is vertically infinite but horizontally finite (of length 2L), with a source inside and sinks on the boundaries. We average a path-integral expression for the Green function over velocity fluctuations and approximate the result in the limits of long distance and long time via Feynman’s variational method. We calculate the exit time distribution and the mean first passage time. The latter is proportional to L^4/3, consistent with previous work.