Formation factors for a class of deterministic models of pre-fractal pore-fracture networks

The main aim of this paper is to reveal effects of fractal features on the formation factors associated with different transport processes in scale invariant pore-fracture networks. Accordingly, we explore a class of deterministic infinitely ramified networks associated with pre-fractal standard Sierpinski carpets (including Sierpinski cubes and inverse Menger sponges). The focus is placed on the effects of network ramification, connectivity, and loopiness on the transport streamline constriction and tortuosity of transmission paths. The differences between the formation factors associated with the diffusibility, electrical conductivity, and hydraulic permeability are elucidated. Explicit expressions for the constrictivity and formation factors of deterministic pre-fractal networks are derived. In this regard, we stress that the electrical formation factor obeys the Archie law only if the random walk in the pre-fractal pore-fracture network is recurrent. We also note that the Archie's exponent can be either equal to, or less than the power-law exponent characterizing the scaling behavior of diffusibility. The notion of the structural formation factor is introduced. The values of scaling exponents characterizing the formation factors associated with different transport properties of the model networks are found to be within the ranges of field observations.