Multiscale Statistical Distribution of Porous Media Transport Behaviour: A Fractal Geometry Approach

The transport properties of porous media exhibit complex multiscale behaviours, which are governed by nonlinear interaction of structural heterogeneity, and which present significant challenges for theoretical understanding and practical modelling. To address this complexity, we propose a fractal-based framework to quantitatively link structural parameters with transport behaviours, focusing specifically on electrical current flow in porous media. Our approach develops a tortuosity model based on self-similarity principles in order to describe the geometric structure, and to assess the transport properties, such as permeability and electrical conductivity.

At the single-capillary level, key microstructural properties, such as pore geometry and connectivity, and transport properties, including permeability and electrical conductivity, can be quantified using metrics such as fractal dimension, tube number, and characteristic length. These parameters capture both structural complexity and scaling behaviour. Taking electrical conductivity as an example, a two-dimensional porous medium with a grid resolution of 16,384 × 16,384 is generated using the Quartet Structure Generation Set (QSGS) method and partitioned into smaller scales (e.g., 1024, 512, 256, and 128) to explore multiscale behaviour and scaling effects. Finite difference methods are employed to calculate the electrical field distributions and derive the effective electrical conductivity. These results are then mapped to the parameters of the self-similar tortuosity model, providing insights into its ability to capture the complex relationships between structure and transport properties.

Statistical analysis reveals that the measured fractal dimensions follow a Weibull distribution across scales, characterised by its distinctive shape and scale parameters. By contrast, characteristic length and tube number values exhibit scale-dependent variations that influence their respective distribution patterns. Tube number conforms to a lognormal distribution, reflecting its intrinsic variability. These findings enable the development of more accurate and computationally efficient multiscale models, with potential applications in areas such as fluid flow, heat transfer, and the design of advanced porous materials.