On the permeability of fractal tube bundles

The permeability of a porous medium is strongly affected by its local geometry and connectivity, the size distribution of the solid inclusions and the pores available for flow. Since direct measurements of the permeability are time consuming and require experiments that are not always possible, the reliable theoretical assessment of the permeability based on the medium structural characteristics alone is of importance. When the porosity approaches unity, the permeability-porosity relationships represented by the Kozeny-Carman equations and Archie's law predict that permeability tends to infinity and thus they yield unrealistic results if specific area of the porous media does not tend to zero. The goal of this paper is an evaluation of the relationships between porosity and permeability for a set of fractal models with porosity approaching unity and a finite permeability. It is shown that the tube bundles generated by finite iterations of the corresponding geometric fractals can be used to model porous media where the permeability-porosity relationships are derived analytically. Several examples of the tube bundles are constructed and relevance of the derived permeability-porosity relationships is discussed in connection with the permeability measurements of highly porous metal foams reported in the literature.