Yu. 2009. On the Physical Properties of Apparent Two Phase Fractal Porous Media

On the Physical ProperƟes of Apparent Two-Phase Fractal Porous Media
Boming Yu,* Jianchao Cai, and Mingqing Zou
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In this study, we summarized some basic characters of fractal porous media, including the fractal pore or par cle size distribu on, pore or par cle density func on, the fractal dimensions for the pore and solid phases, and their rela ons. The geometric porosi es vs. the fractal dimensions and microstructures of porous media were reviewed and discussed in two and three dimensions. The specific surface areas of fractal porous media in two and three dimensions were derived and were expressed as a func on of the fractal dimensions and microstructural parameters. The fluid velocies in fractal porous media were also derived and found to be a func on of the fractal dimensions and microstructural parameters of the medium. The parameters presented are the fundamental ones and may have poten al in analysis of transport proper es in fractal porous media.
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· Vol. 8, No. 1, February 2009
review of the fractal models of fragmented and aggregated soils based on the Sierpinski carpet geometry and applied two different models to represent a fractal porous medium, as shown in Fig. 1, which shows third-iteration Sierpinski carpets. In Fig. 1a, the black represents the solid phase, which is fractal, while the white gaps represent pores, which are not fractal. In Fig. 1b, the white represents the pore phase, which is fractal, while the black spots represent solids, which are not fractal. Perrier et al. (1999) presented an independent and similar study. It is worth pointing out that the Sierpinski carpet, as shown in Fig. 1, is a single-phase fractal medium. Figure 1 will be discussed further below. Compared with single-phase (or saturated) transport phenomena in porous media, multiphase (or two-phase) immiscible flows (or transport) in porous media are not well understood yet. Multiphase immiscible flows in porous media are very important in practical applications such as the petroleum industry, chemical engineering, and soil engineering. The single (saturated) porous medium is, in fact, only a special case of a two-phase (unsaturated) porous medium. The fractal dimensions for two-phase (unsaturated) porous media might be more fundamental than some macroscopic parameters such as permeability and porosity. The reviews by Rieu and Perrier (1998) and Perrier et al. (1999), however, did not involve a two-phase fractal medium. This work summarizes some fundamentals about singlephase and two-phase fractal porous media in a more general way. The relationships among the fractal dimensions, porosities, specific surface areas, and fractal velocity with the microstructure of porous media are derived and discussed below.
particles, and geometric structures. Some researchers (Bear, 1972; Kaviany, 1995) have modified the Kozeny–Carman equation as a function of porosity φ, specific surface area S, and tortuosity τ (defined as the ratio of the real length to the straight-line length). The Kozeny–Carman equation has been hotly debated since its inception due to its dependence on the microstructures of porous media. Another example is the thermal conductivity of porous media; available results (Hsu et al., 1995; Feng et al., 2004) show that the thermal conductivity of a porous medium is related not only to the porosity but also to the microstructures of the porous medium. The dielectric constant of porous ultra-low-k (dielectric constant) dielectrics has also been found to be a function of porosity and pore size distributions (Nakano et al., 1993; Beckers and de Leeuw, 2001). Since its establishment by Mandelbrot (1982), fractal geometry has attracted considerable interest in many disciplines for its ability to characterize irregular and disordered objects such as, for example, porous media (Katz and Thompson, 1985; Gimenez et al., 1997; Yu and Li, 2001; Yu and Cheng, 2002; Perfect, 2005) and the roughness of surfaces (Majumdar, 1992), rocks, and fractures (Sahimi, 1993, 1995; Adler and Thovert, 1998; Berkowitz, 2002). Katz and Thompson (1985) might have been the first to present experimental evidence indicating that the pore spaces of a set of sandstone samples are fractals and are self-similar across three to four orders of magnitude in length extending from 1 nm to 100 μm. They argued that the pore volume is a fractal with the same fractal dimension as the pore–rock interface. They also obtained an empirical correlation for the fractal dimension of pore spaces. Rieu and Perrier (1998) made a comprehensive