Transport and dispersion in random networks with percolation disorder

The transport of dynamically neutral tracer in flow through a random network of tubes with percolation disorder is investigated. For each bond of the system, the tracer motion is governed by a convection-diffusion equation, with the drift term accounting for the local bond velocity. This represents an extension of random walks in disordered media to the important case of finite macroscopic flow rates. A calculational technique is developed which provides, for a given configuration of the disorder, the exact moments of the distribution of transit times for the tracer to traverse the system. This approach is implemented numerically for L×L random networks at the bond percolation threshold. When the total fluid flux Q vanishes, the kth moment of the transit time is found to scale with sample size as $L^{k(2+\theta )}$, where θ=t-β/ν is the exponent describing the scale dependence of the diffusion coefficient of the ‘‘ant in the labyrinth,’’ $D_{\mathrm{ant}}$. By contrast, when Q becomes large, the kth moment is found to scale as $Q^{\mathrm{-}1}$ $L^{2\mathrm{-}\beta /\nu +\left(k\mathrm{-}1\right)(2+\theta )}$. This behavior of the moments at large Q is explained on the basis of a simple heuristic argument and from a more detailed analytical calculation. Furthermore, a scaling ansatz for the transit-time moments is postulated which describes our data for all flow rates. The large-Q behavior of the moments leads to a longitudinal dispersion coefficient which scales as $U^2$ $L^2$/$D_{\mathrm{ant}}$, where U is the average flow velocity, in agreement with a prediction of de Gennes.