Application of Percolation Theory to Microtomography of Structured Media: Percolation Threshold, Critical Exponents and Upscaling

摘要

Percolation theory provides a tool for linking between microstructure and macroscopic material properties. In this paper, percolation theory is applied to the analysis of microtomographic images for the purpose of deriving scaling laws for upscaling of properties. We have tested the acquisition of quantities such as percolation threshold, crossover length, fractal dimension, critical exponent of correlation length from microtomography. By inflating or deflating the target phase and percolation analysis, we can get a critical model and the estimation of percolation threshold. Crossover length is determined from the critical model by numerical simulation. Fractal dimension can be obtained either from the critical model or from the relative size distribution of clusters. Local probabilities of percolation are used to extract critical exponent of correlation length. For near isotropic samples such as sandstone and bread the approach works very well. For strongly anisotropic samples, such as highly deformed rock (mylonite) and a tree branch, the percolation threshold and fractal dimension can be assessed with accuracy. However, the uncertainty of correlation length makes it difficult to accurately extract the critical exponents of correlation length. This aspect of percolation theory can therefore not be reliably used for upscaling properties of strongly anisotropic media. Other methods of upscaling have to be used for such media.