A fractal modeling approach for effective thermal conductivity in unsaturated granular porous media with rough surfaces

Heat transport phenomena in unsaturated porous media have constituted a significant research focus across scientific and engineering disciplines over past decades. The inherent complexity of pore-solid microstructures has hindered the development of precise theoretical models. While numerous models have been proposed to predict effective thermal conductivity, many conventional approaches rely on empirical parameters lacking physical justification. To address this limitation, this study establishes a novel theoretical framework derived from Laplace's equation with incorporation of rough surface effects. The conceptual framework represents the porous system as fractal-distributed solid particles, where interstitial spaces are occupied by water-air phases. The model incorporates key structural parameters including porosity, saturation, fractal dimension for particle radius size, representative length, maximum particle radius, relative roughness in rough porous media, and second Archie exponent. A systematic parametric analysis is conducted to quantify the functional dependence of effective thermal conductivity on key geometric parameters. The model demonstrates good agreement with multiple experimental datasets, confirming its validity and accuracy across varied granular porous media configurations and conditions. This fractal-based model advances fundamental understanding of heat transfer mechanisms in unsaturated granular porous media, offering enhanced physical insights for geoscience and thermophysics applications. • A fractal model of effective thermal conductivity in unsaturated granular porous is derived originated from the Laplace’s equation with rough surfaces. • A systematic parametric analysis is conducted to elucidate the functional relationships between proposed model and key geometrical parameters. • The predictions of proposed model demonstrate strong agreement with multiple experimental datasets, confirming its validity and accuracy across varied porous media configurations and conditions. • The proposed fractal model provides the fundamental understanding of heat transfer mechanisms in unsaturated granular porous media.