Approximations for transport parameters and self-averaging properties for point-like injections in heterogeneous media

We focus on transport parameters in heterogeneous media with a flow modelled by an ensemble of periodic and Gaussian random fields. The parameters are determined by ensemble averages. We study to what extent these averages represent the behaviour in a single realization. We calculate the centre-of-mass velocity and the dispersion coefficient using approximations based on a perturbative expansion for the transport equation, and on the iterative solution of the Langevin equation. Compared with simulations, the perturbation theory reproduces the numerical results only poorly, whereas the iterative solution yields good results. Using these approximations, we investigate the self-averaging properties. The ensemble average of the velocity characterizes the behaviour of a realization for large times in both ensembles. The dispersion coefficient is not self-averaging in the ensemble of periodic fields. For the Gaussian ensemble the asymptotic dispersion coefficient is self-averaging. For finite times, however, the fluctuations are so large that the average does not represent the behaviour in a single realization.