Advection of an impurity in percolation media with a finite correlation length
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TICAL, NONLINEAR, AND SOFT MATTER PHYSICS
Advection of an Impurity in Percolation Media with a Finite Correlation Length L. V. Matveev Nuclear Safety Institute, Russian Academy of Sciences, Moscow, 115191 Russia Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudny, Moscow oblast, 141700 Russia email: [email protected] Received September 10, 2013
Abstract—The impurity transport regimes in percolation media with a finite correlation length, which are caused by advection and diffusion mechanisms, have been analyzed. It has been shown that the change in the transport characteristics of a medium from the selfsimilar type to the statistically homogeneous type occurs through two stages because of the structural features of percolation clusters (presence of a backbone and dead ends). As a result, new anomalous transport regimes appear in the system. The quasiisotropic and moder ately and strongly anisotropic media have been considered. DOI: 10.1134/S1063776114030054
1. INTRODUCTION The transport problem in percolation media has been studied for more than five decades [1]. There are various approaches such as continuoustime random walks [2], equations with fractional derivatives [3], scalingbased descriptions [4], comb models [5], and renormalization models [6], as well as numerical mod els, for description of nonclassical transport in indi cated media. The nonclassical transport character is manifested in the exponent γ ≠ 1 of the time depen dence R2 ∝ tγ of the variance R2 of impurity. When impurity transport in a percolation medium is due to the infiltration flux, a fruitful approach is a random advection model [7, 8], where the features of the medium were taken into account in a slow (power law) decrease in the correlation function of the veloc ity with an increase in the distance. Such a behavior of the velocity correlation function is due to the fact that percolation media in a sufficiently large spatial region are selfsimilar because of the fractal properties of per colation clusters forming infiltration paths. The character of transport in percolation media is determined by two factors: first, the structure of the network of channels (clusters) through which particles migrate and, second, the transport mechanism through these channels (advection or diffusion). The structure of the percolation cluster has the following features as compared to usual statistically homoge neous media. The correlation length ξ is an important characteristic of the cluster. At scales smaller than ξ, the cluster has a fractal structure, which promotes the appearance of the anomalous transport regime. At scales larger than ξ, the migration medium becomes statistically homogeneous (the dimension of the net work of conducting channels is equal to the dimension
of the containing space) and it can be expected that transport will be described by classical laws in which the average displacement of particles (at a nonzero average velocity) and variance increase linearly with the time (i
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