## Introduction

The permeabilities for porous media, both saturated and unsaturated, have received much attention (De Wiest, 1969; Bear, 1972; Bowles, 1984; Jumikis, 1984; Kaviany, 1995; Panfilov, 2000) due to practical applications, including chemical engineering, soil science and engineering, oil production, polymer composite molding, and heat pipes. Since the microstructures of porous media are usually disordered and extremely complicated, this makes it very difficult to analytically find the permeability of the media, especially for unsaturated (or multiphase) porous media.

Conventionally, the permeabilities of porous media were found by experiments (Levec et al., 1986; Sasaki et al., 1987; Wang et al., 1994; Wu et al., 1994; Shih and Lee, 1998; Chen et al., 2000). Besides, much effort was also devoted to numerical simulations of permeabilities for porous media. Simacek and Advani (1996) performed the numerical solution by reducing a two-dimensional problem to a one-dimensional equation. Adler and Thovert (1998) applied a fourth-order finite difference scheme for permeabilities of real Fontainebleau sandstone. Although no adjustable parameter was involved, a large discrepancy between the average numerical permeability (plotted against porosity) and the experimental data was observed. Ngo and Tamma (2001) applied the finite-element method to calculate the permeability for the porous fiber mat by assuming the Stokes flow in the intertow region and the Brinkman's flow inside the tow region. Compared with single-phase (or saturated) flow in porous media, the multiphase (or unsaturated) immiscible flows in porous media are not well understood. The multiphase immiscible flows in porous media are very important in practical applications such as the petroleum industry, chemical engineering, and soil engineering. The lattice Boltzmann method (LBM) (Benzi et al., 1992; Sahimi and Mukhopadhyay, 1996; Martys and Chen, 1996; Chen and Doolen, 1998), based on the Navier-Stokes equation coupled with Darcy's law, has been extensively used to simulate multiphase flows through porous media in order to understand the fundamental physics associated with enhanced oil recovery, including relative permeabilities. The LBM is particularly useful for complex geometrical boundary conditions and varying physical parameters. However, the results either from numerical simulations or from experiments are usually expressed as correlations with one or more empirical constants, or as curves, and the mechanisms behind the phenomena are thus often ignored. In order to get a better understanding of the mechanisms for permeability, the analytical solution for permeability of porous media becomes a challenging task.

Recently, Yu and Lee (2000) developed a simplified analytical model for evaluating the permeabilities of porous fabrics used in liquid composite molding. This permeability model, which is related to porosity and architectural structures of porous fabrics, is based on the one-dimensional (1-D) Stokes flow in macropores between fiber tows and on the 1-D Brinkman flow in micropores inside fiber tows. Good agreement between theoretical predictions and experimental results was found. However, this model may only apply to those media whose macropores can be simplified as one-dimensional channels. So this and several other models may not be applicable to random/disordered porous media. In addition, this model is only suitable for saturated porous media.

Ransohoff and Radke (1988) applied the circular and triangle capillary models to numerically simulate the flow resistance for laminar flow through unsaturated porous media and studied the dependence of flow resistance on corner geometry, surface shear viscosity, and contact angle.

Katz and Thompson (1985) presented experimental evidence indicating that the pore spaces of a set of porous sandstone samples (in nature) are fractals and are self-similar over 3 to 4 orders of magnitude in length extending from 10 Å to 100 μm. They argued that the pore volume is a fractal with the same fractal dimension as the pore–rock interface. This conclusion was supported by correctly predicting the porosity from the fractal dimension, which was measured by a log-log plot of a number of pores vs. the pore size, and a fractal correlation, ϕ = *C*(*l*_{1}/*l*_{2}), is presented to correlate the measurements on a variety of porous sandstone samples (pores). In the correlation, ϕ is the porosity of porous sandstone, *D*_{f} (= 2 ∼ 3 in three dimensions) is the fractal dimension of pores, *C* is a constant of order one, and *l*_{1} and *l*_{2} are the lower and upper limits, respectively, of the self-similar regions. Krohn and Thompson (1986) also carried out the experiments on sandstone, and the fractal dimensions of the five sandstone pores were found to be in the range of 2.55–2.85 in three dimensions. Figure 1 (Krohn and Thompson, 1986) displays one of the fractal scaling laws of five sandstone pores. In the figure, the negative slope of the solid line gives the fractal dimension *D* = 2.75 obtained by fitting the measurements from the automatic technique. This figure shows that the sandstone pores are fractal objects in nature. Readers may also consult the paper by Krohn and Thompson (1986) for more evidence that the porous media are fractals in nature.

Smidt and Monro (1998) performed experimental investigations on the images of laboratory-made synthetic sandstone and on modeled sandstone. Their results showed that the pore space of both the synthetic and the modeled sandstone was found to be fractals and the fractal scaling laws were obtained by the box-counting method (see Figure 2). Figure 2 presents the fractal scaling law for the laboratory synthetic stone pores, and the slope of the log(*N*_{d}) ∼ log(1/*d*) plot shows fractal dimension 1.89 (in two dimensions) obtained by the box-counting method (count the number, *N*_{d}, of boxes of side length *d* for covering the pore space). This suggests that the laboratory synthetic stone pores are also fractal objects.

According to the fractal character of real porous media, Yu and Cheng (2002) developed a fractal permeability model for bidispersed saturated porous media (see Figure 3), and this fractal model is also applicable to porous fabrics (Yu et al., 2002) (see Figure 4; Yu et al., 2001). It is seen that the random porous fabric is also a fractal medium. For more fractal microstructures of fabrics, readers may consult the paper by Yu et al. (2001). Although this model does not contain any empirical constant, and good agreement is found between the model predictions and experimental data, it does not apply to unsaturated porous media. The saturated porous medium is, in fact, only the special case of the unsaturated porous medium. It is therefore more meaningful for practical applications to develop an analytical solution for the permeability of unsaturated (or multiphase) porous media. Once the credible permeability is obtained, it also can be used to analyze the heat and mass transfer in unsaturated porous media such as soil (Liu et al., 1995, 1998).

In this article, we focus our attention on the derivation of an analytical fractal model for both the phase and relative permeability of two-phase porous media based on the available evidence that porous media in nature are fractal objects (Katz and Thompson, 1985; Krohn and Thompson, 1986; Young and Crawford, 1991; Perfect and Kay, 1991; Smidt and Monro, 1998; Yu and Li, 2001; Yu et al., 2001, 2002; Yu and Cheng, 2002). This article is organized as follows: the following section describes the fractal characteristics of microstructures of porous media, which are the theoretical bases for the present fractal analysis of permeability for porous media. The complete fractal permeability models for both saturated and unsaturated porous media are given in the third section. The results and discussions are arranged in the fourth section, and then come the concluding remarks.