Multifractal analysis of the pore- and solid-phases in binary two-dimensional images of natural porous structures

Abstract

We use multifractal analysis (MFA) to investigate how the Rényi dimensions of the solid mass and the pore space in porous structures are related to each other. To our knowledge, there is no investigation about the relationship of Rényi or generalized dimensions of two phases of the same structure. Images of three different natural porous structures covering three orders of magnitude were investigated: a microscopic soil structure, a soil void system visible without magnification and a mineral dendrite. Image size was always 1024 × 1024 pixels and box sizes were chosen as powers of 2. MFA was carried out according to the method of moments, i.e., the probability distribution was estimated for moments ranging from − 10 < q < 10 and the Rényi dimensions were calculated from the log/log slope of the probability distribution for the respective moments over box sizes. A meaningful interval of box sizes was determined by estimating the characteristic length of the pore space and taking the next higher power of 2 value as the smallest box size, whereas the greatest box size was determined by optimizing the coefficients of determination of the log/log fits for all q. The optimized box size range spans from 32 to 1024 pixels for all images. Good generalized dimension (Dq) spectra were obtained for this box size range, which are capable of characterizing heterogeneous spatial porous structure. They are alike for all images and phases which the exception of the solid mass of the soil void system, which shows a rather flat D q behavior. A closer examination reveals that similar patterns of structure gain similar spectra of generalized dimensions. The capacity dimension for q = 0 is close to the Euclidian dimension 2 for all investigated images and phases.