Analytical tortuosity–porosity correlations for Sierpinski carpet fractal geometries

•A new analytical tortuosity–porosity correlation was found in the Sierpinski carpet.•A new analytical tortuosity–porosity correlation was found in a circular-based Sierpinski carpet.•In both cases, the correlations were found to be linear with distinct slopes.•The correlation is expected to be linear for any other shaped repeating element.•These tortuosity–porosity correlations are applicable for the infinite generation.

Naturally-occurring porous media, such as sedimentary rock, rarely consist of mono-sized particles, but rather tend to consist of distributions of particle sizes (poorly-sorted porous media). In this study, deterministic fractal geometries including a Sierpinski carpet and a slightly altered version of the Sierpinski carpet with a generator that has a circular inclusion were used to provide insight into the poorly-sorted porous media found in sedimentary rock. The relationships between tortuosity and porosity within these fractal geometries were investigated by presenting and applying a novel mathematical approach. We found a new correlation between the tortuosity, τ, and porosity, ϕ  , within the Sierpinski carpet (τ=32−ϕ2), which agrees well with previous empirical observations reported in the literature. We also found an analytical tortuosity–porosity correlation within the circular-based Sierpinski carpet (τ=(1−4π)ϕ+4π), which is to the best of the authors’ knowledge, the first tortuosity–porosity relationship proposed for such fractal geometry.