Stratified disordered media: Exact solutions for transport parameters and their self-averaging properties

We investigate the transport of a passive tracer in a two-dimensional stratified random medium with flow parallel and perpendicular to the strata. Assuming a Gaussian random flow with a Gaussian correlation function, it is not only possible to derive exact expressions for the temporal behaviour of the dispersion coefficients characterising the tracer transport, but also to investigate the self-averaging properties of these quantities. We give explicit results for the dispersion coefficient and its mean square fluctuations as a function of time. As it turns out, the sample to sample fluctuations which are encoded in the latter quantity remain finite even in the asymptotic limit of infinite times, which implies that the given two-dimensional model is not self-averaging.