Fractal Porous Media

Fractals are self-similar objects, e.g., Koch curve, Menger sponge, or Devil s staircase. The self-similarity of fractal objects is exact at every spatial scale of their construction (e.g., Avnir, 1989). Mathematically constructed fractal porous media, e.g., the Devil s staircase, can approximate the structures of metallic catalysts, which are considered to be disordered compact aggregates composed of imperfect crystallites with broken faces, steps, and kinks (Mougin et al., 1996). 

Transport of Water and Solutes in Soils as in Fractal Porous Media... 

Adler (1996) summarized results of theoretical studies of the Taylor dispersion in fractal porous media and numerical experiments with Taylor dispersion of particles in computer-generated porous space. He showed that the anomalous dispersion occurs when the material is fractal over all considered scales. Fickean dispersion was found when the material was homogeneous at the large scale, but contained a fractal microstructure. 

Adler, P.M. 1996. Transports in fractal porous media. J. Hydrol. 187 195-213. 

Lemaitre, R., and P.M. Adler. 1990. Fractal porous media. IV Three-dimensional Stokes flow through random media and regular fractals. Transp. Porous Media 5 325 340. 

Cushman, J.H. 1991. On diffusion in fractal porous media. Water Resour. Res. 27 643-644. 

Pachepsky, Y., D.A. Benson, and DJ. Timlin. 2001. Transport of water and solutes in soils as in fractal porous media, p. 51-75. In H.M. Selim and D.L. Sparks (ed.) Physical and chemical processes of water and solute transport /retention in soil. SSSA Special Publ. 56, Madison, WI. 

Zeng, Y., C.J. Gantzer, R.L. Payton, and S.H. Anderson. 1996. Fractal dimension and lacunarity of bulk density determined with x-ray computed tomography. Soil Sci. Soc. Am. J. 60 1718-1724. Zhan, H., and S.W. Wheatcraft. 1996. Macrodispersivity tensor for nonreactive solute transport in isotropic and anisotropic fractal porous media Analytical solutions. Water Resour. Res. 32 3461-3474. 

Adler P.M., Thovert J-F. 1993. Fractal porous media. Transport in Porous Media, 13 (1) 41—78. 

Zhan, H. and S.W. Wheatcraft. 1996. Macrodispersivity tensor for nonreactive solute transport in isotropic and anisotropic fractal porous media Analytical solutions. Water Resources Research 32(12) 3461-3474. 

A simple derivation shows that for the class of fractal porous media shown in Figure 5, the capillary pressure, Pc, and the saturation of the wetting phase, Swc, are linked by the following relation (cf. Winter, in preparation)... 

The shape of the distribution function of pore sizes is, of course, a crucial parauneter controlling wetting regimes in fractal porous media depending on its properties, the fraction of pores in one of the two wetting regimes varies affecting many important properties of the medium, such as, e.g. its ability of fluid transport. 

In the fractal porous medium, the diffusion is anomalous because the molecules are considerably hindered in their movements, cf. e.g., Andrade et al., 1997. For example, Knudsen diffusion depends on the size of the molecule and on the adsorption fractal dimension of the catalyst surface. One way to study the anomalous diffusion is the random walk approach (Coppens and Malek, 2003). The mean square displacement of the random walker (R2) is not proportional to the diffusion time t, but rather scales in an anomalous way ... 

The anomalous diffusivity described by Eq. [13] is due entirely to the fractal nature of the diffusing particle s trajectory in free space. In fractal and multifractal porous media, the diffusing particle s trajectory is further constrained by the geometry of the pore space (Cushman, 1991 Giona et al., 1996 Lovejoy et al., 1998). As a result, when fractional Brownian motion occurs in a two-dimensional fractal porous medium, De becomes scale-dependent, as described by the following equation (Orbach, 1986 Crawford et al., 1993),... 

In order to determine the distribution function of pore sizes consider a fractal porous medium consisting of polygonal grains and square-sectional pores in a certain scale range, 1- 1 < (see Figure 4). 

An exact expression for the distribution function can be found by applying an analog of the transfer matrix method (cf. Mandelbrot et al., 1985, Mosolov et al., 1987). At each step of the iterative procedure the porosity, 0, of the fractal porous medium changes as follows ... 

Figure 4. Fractal porous medium with square-sectional pores and polygonal grains.

Fig. 6. Cross-section through a fractal porous medium showing the configuration of the thin film phase in the pores. Only pores with sizes below the critical threshold level are entirely wetted. In the remaining pores the presence of the film phase is controlled by the magnitude of the adverse pressure (a) the porewails are entirely wetted (b) the film phase appears in corners only.

Obviously, the diffusion coefficient of molecules in a porous medium depends on the density of obstacles that restrict the molecular motion. For self-similar structures, the fractal dimension df is a measure for the fraction of sites that belong... 

Daccord G, Lenormand R (1987) Fractal patterns from chemical dissolution. Nature 325 41 3 Daccord G, Lietard O, Lenormand R (1993) Chemical dissolution of a porous medium by a reactive fluid, 2, Convection vs. reaction behavior diagram. Chem Eng Sci 48 179-186 Darmody RG, Thorn CE, Harder RL, Schlyter JPL, Dixon JC (2000) Weathering implications of water chemistry in an arctic-alpine environment, north Sweden. Geomorphology 34 89-100 Dijk P, Berkowitz B (1998) Precipitation and dissolution of reactive solutes in fractures. Water Resour Res 34 457-470... 

The third relaxation process is located in the low-frequency region and the temperature interval 50°C to 100°C. The amplitude of this process essentially decreases when the frequency increases, and the maximum of the dielectric permittivity versus temperature has almost no temperature dependence (Fig 15). Finally, the low-frequency ac-conductivity ct demonstrates an S-shape dependency with increasing temperature (Fig. 16), which is typical of percolation [2,143,154]. Note in this regard that at the lowest-frequency limit of the covered frequency band the ac-conductivity can be associated with dc-conductivity cio usually measured at a fixed frequency by traditional conductometry. The dielectric relaxation process here is due to percolation of the apparent dipole moment excitation within the developed fractal structure of the connected pores [153,154,156]. This excitation is associated with the selfdiffusion of the charge carriers in the porous net. Note that as distinct from dynamic percolation in ionic microemulsions, the percolation in porous glasses appears via the transport of the excitation through the geometrical static fractal structure of the porous medium. 

In general, in order to include more types of porous media the random fractal model can be considered [2,154,216]. Randomness can be introduced in the fractal model of a porous medium by the assumption that the ratio of the scaling parameters c X/A is random in the interval [c0,1 ], but the fractal dimension I) in this interval is a determined constant. Hence, after statistical averaging, (66) reads as follows ... 

As pointed out in the introductory section evaluation of wettability at the level of a network presuposes knowledge of the distribution function of pore sizes. This function can be derived theoretically provided that some assumptions can be made about the structure of the porous medium. One possibility is to assume that the medium is a (deterministic) fractal. A number of recent papers support the hypothesis, that at least some reservoir rocks are indeed fractal (cf. Sen et al., 1981 Katz et al., 1985 Wong, 1987 Hansen et al., 1988). 

Fig. 5. Three-dimensional view of the fractal construct used as a model porous medium.

Note that in our approximation, due to the randomized character of the fractal medium the average porosity of the disordered porous glasses determined by (70) depends only on the fractal dimension Dr and does not exhibit any scaling behavior. In general, the magnitude of the fractal dimension may also depend on the length scale of a measurement extending from X to over... 

In brief, we have observed the scale-dependent diffusion of a tracer particle in a porous silica gel, although the explored medium is probably not fractal. At small scales, the diffusion is free at large scales it is slowed due to the tortuous paths of the porous structure. The crossover region corresponds to sizes comparable with the mesh size of the silica gels. The data support a structural description in terms of double porosity. 

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