3D multi-fractal analysis of porous media using 3D digital images: considerations for heterogeneity evaluation

Pore structure heterogeneity is a critical parameter controlling mechanical, electrical and flow transport behaviour of rock. Multi-fractal analysis method was used for a heterogeneity comparison of three-dimensional rock samples with different lithology. Six real digital samples, containing three sandstones and three carbonates, were used. Based on the mercury injection capillary pressure test on these samples, we found that the carbonate samples are more heterogeneous than sandstones, but primary results of multi-fractal behaviours for all samples were similar. We show that if multi-fractal is used to evaluate and compare heterogeneity of different samples, one needs to follow some considerations such as (1) all samples must have the same size in pixel, (2) samples volume must be bigger than representative volume element, (3) multi-fractal dimensions should be firstly normalized to a determined porosity value and (4) multi-fractal results should be interpreted based on resolution of the imaging tool (effects of fine scale sub-resolution pores are missed). Results revealed that using normalized fractal dimensions, the real samples were divided to less and high heterogeneous groups. Moreover, the study of scale effect also showed that porous structures of these samples are scale invariant in a wide range of scales (from one to eight times bigger).