Dependency of percolation critical exponents on the exponent of power law size distribution

The standard percolation theory uses objects of the same size. Moreover, it has long been observed that the percolation properties of the systems with a finite distribution of sizes are controlled by an effective size and consequently, the universality of the percolation theory is still valid. In this study, the effect of power law size distribution on the critical exponents of the percolation theory of the two dimensional models is investigated. Two different object shapes i.e., stick-shaped and square are considered. These two shapes are the representative of the fractures in fracture reservoirs and the sandbodies in clastic reservoirs. The finite size scaling arguments are used for the connectivity to determine the dependency of the critical exponents on the power law exponent. In particular, the deviations of percolation exponents from their universal values as well as the connectivity behavior of such systems are investigated numerically. As a result, this extends the applicability of the conventional percolation approach to study the connectivity of systems with a very broad size distribution.