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.