Scaling theory of hydrodynamic dispersion in percolation networks

Real-space renormalization-group arguments are used to derive scaling relations for the mean displacement 〈R〉 and the variance 〈(R-〈R〉${)}^2$〉 of a tracer particle in a fluid flowing through a heterogeneous material which is near a percolation threshold $p_c$; both small- and large-Peclet-number regions are studied. The existence of a noninteger-power-law dependence of 〈R〉 and 〈(R-〈R〉${)}^2$〉 on time and the strength of flow, which cannot be described by a convection-diffusion equation, is revealed. Particularly at large Peclet numbers, the variance exhibits anomalously fast time dependence and an associated divergence near $p_c$. As p→$p_c$, the region dominated by convection extends prominently, while the region controlled by diffusion shrinks.