#### Fractal permeability for saturated porous media

Consider a unit cell consisting of a bundle of tortuous capillary tubes with variable cross-sectional area. The total volumetric flow rate *Q* through the unit cell is a sum of the flow rates through all the individual capillaries. The flow rate through a single tortuous capillary is given by modifying the well-known Hagen-Poiseulle equation (Denn, 1980) to give

- (11)

where *G* = π/128 is the geometry factor for flow through a circular capillary, λ is the hydraulic diameter of a single capillary tube, μ is the viscosity of the fluid, Δ*P* is the pressure gradient, and *L*_{t} is the length of the tortuous capillary tube. The total flow rate *Q* can be obtained by integrating the individual flow rate, *q*(λ), over the entire range of pore sizes from the minimum pore λ_{min} to the maximum pore λ_{max} in a unit cell. According to Eqs. 3, 10, and 11, we have

- (12)

where *D*_{f} is the pore-area fractal dimension, and 1 < *D*_{f} < 2 in two dimensions. Since 1 < *D*_{T} < 2 and 1 < *D*_{f} < 2, the exponent 3 + *D*_{T} − 2*D*_{f} > 0 and 0 < (λ_{min}/λ_{max}) < 1. Also, according to the Yu and Li's criterion (Yu and Li, 2001), (λ_{min}/λ_{max}) ≅ 0 (because (λ_{min}/λ_{max}) ∼ 10^{−2}). It follows that Eq. 12 can be reduced to

- (13)

Using Darcy's law, we obtain the expression for the permeability of a porous medium as follows

- (14)

which indicates that the permeability is a function of the pore-area fractal dimension *D*_{f}, tortuosity fractal dimension *D*_{T} and structural parameters, *A*, *L*_{0} and λ_{max}. Equations 13 and 14 indicate that the total flow rate and total permeability are very sensitive to the macropore λ_{max}, and the total flow rate and total permeability are mainly determined by the maximum/macro pore λ_{max}. This is consistent with the practical situation.

For straight capillaries, *D*_{T} = 1, Eqs. 13 and 14 can be reduced to

- (15)

- (16)

respectively. Equations 13–16 present the single-phase flow rates and single-phase permeabilities (also called absolute permeabilities). Equations 13–16 indicate that the flow rate and permeability are very sensitive to the maximum pore size λ_{max}. It is also shown that the higher the fractal dimension *D*_{f}, the larger the flow rate and the permeability value. From Eqs. 13–16, it can be seen that the flow rate and the permeability will reach the maximum possible values as the pore-area fractal dimension approaches its maximum possible value of 2. This is consistent with fractal theory. Equations 13–16 are valid not only for isotropic porous media but also for anisotropic porous media. For anisotropic porous media, we only need to calculate the principal permeabilities (Yu et al., 2002) in the principal directions by separately using Eqs. 14 or 16, then finding the other permeability components.

#### Fractal permeability for unsaturated porous media

We now extend the preceding analysis to the permeability for unsaturated porous media. The unique difference between the saturated and unsaturated porous media is that for saturated porous media there is only a single fluid such as water filled with pores or capillary pathways. In an unsaturated porous media there are at least two different fluids, such as water and gas. This means that pores are partially filled with water and gas. Figures 5a and 5b display those typical pores, and Figure 5c shows a simplified model for the cross section of a capillary tube partially filled with water and gas. From Figure 5c, we can obtain the pore volume, *V*_{p}, and the volume, *V*_{w}, occupied by water or wetting phase as

- (17)

and

- (18)

respectively, where λ and λ_{g} are the diameter of a capillary pathway and the diameter of the nonwetting (such as gas) phase pathway, and *V*_{g} is the volume occupied by a nonwetting fluid (such as gas). According to the definition for saturation, *S*_{w}, we have

- (19)

and

- (20)

Obviously, Eqs. 19 and 20 satisfy *S*_{w} + *S*_{g} = 1, which is expected. From Eqs. 19 and 20 the diameter for nonwetting fluid (such as gas) can be expressed as

- (21)

Equation 21 denotes that λ_{g} = 0 as *S*_{w} = 1 and λ_{g} = λ as *S*_{w} = 0, and vice versa. This is expected and is consistent with the physical situation.

The volume, *V*_{w}, occupied by the wetting fluid (such as water) can be written as

- (22)

where λ_{w} is the effective diameter of wetting fluid occupying the cross section of a capillary pathway. Again, according to the definition of saturation

- (23)

This results in

- (24)

Equation 24 indicates that λ_{w} = 0, as *S*_{w} = 0, and λ_{w} = λ, as *S*_{w} = 1, and vice versa. This is again expected and is consistent with the physical situation. Usually, 0 < *S*_{w} < 1, such as saturation in soil, in which pores are partially filled with fluid such as water, that is, the two phases (such as water and gas) coexist in soil. From Eqs. 21 and 24, we can directly write the effective maximum and the smallest diameters for wetting and nonwetting fluids in the largest and smallest pores (or capillary tubes) as

- (25)

- (26)

- (27)

- (28)

The volume fractions, ϕ_{w} and ϕ_{g}, for the wetting phase and the nonwetting phase fluids in a unit cell are given by (Bear, 1972)

- (29)

- (30)

respectively, and clearly ϕ_{w} + ϕ_{g} = ϕ.

The permeations of both wetting and nonwetting fluids play important roles in unsaturated (or multiphase) porous media. Muskat and Meres (1936) recommended that the phase permeabilities *K*_{w} and *K*_{g} be treated as isotropic and given by

- (31)

- (32)

or

- (33)

- (34)

where *K* is the absolute permeability (given by Eq. 14) used in the single-phase flows, and *k*_{rw} and *k*_{rg} are the relative permeabilities of the *w* and *g* phases, respectively.

In this work, the single-phase fractal permeability Eq. 14 is extended to the phase fractal permeabilities, *K*_{w} and *K*_{g} (*k*_{rw} and *k*_{rg}), under the following assumptions similar to those by Kaviany (1995):

- 1
The Darcy (Stokes) flow is applicable with a negligible interfacial drag in two-phase porous media.

- 2
The body force is neglected.

- 3
The liquid flow is not coupled with gas flow.

- 4
The viscosities of the liquid and gas phases are independent of each other.

The tortuosity fractal dimension, *D*_{T}, is usually determined by the box-counting method or Monte Carlo method, and an analytical expression for *D*_{T} has not yet been developed. In this work, we also assume that the wetting and the nonwetting phases flow through tortuous paths have approximately the same tortuosity as the single-phase flow, that is, *D*_{T} = *D*_{T,w} = *D*_{T,g} = 1.10 (Yu and Cheng, 2002) measured using the box-counting method (see Figure 6). Figure 6a depicts one of the possible tortuous streamlines passing through the bidispersed porous medium, and Figure 6b is the fractal scaling law measured by the box-counting method applied to the tortuous streamline. Using the Monte Carlo simulation, Wheatcraft and Tyler (1988) obtained the averaged tortuous streamline fractal dimension *D*_{T} = 1.081 for flow through heterogeneous media (see Figure 7). Figure 7a demonstrates the simulation results for tortuous streamlines by a fractal random-walk model, and Figure 7b is a plot of fractal travel distance, *L*_{F}, vs. the scale of observation, *L*_{s}, for the fractal random-walk model. Their Monte Carlo simulation result, *D*_{T} = 1.081, is very close to the averaged result, *D*_{T} = 1.10, obtained by the box-counting method (Yu and Cheng, 2002). This work uses *D*_{T} = 1.10, in calculating permeability.

Applying the analogy between the Darcean single-phase and two-phase flows, and modifying the single-phase fractal permeability model, Eq. 14, by replacing λ_{max} in Eq. 14 with Eqs. 25–28, and fractal dimension *D*_{f} (Eq. 9) with *D*_{f,w} and *D*_{f,g}, we can obtain the phase fractal permeabilities for the wetting and nonwetting fluids

- (35)

- (36)

respectively. Equations 35 and 36 reveal that the phase permeability for wetting and nonwetting fluids in porous media is a function of saturation (*S*_{w}), fractal dimensions (*D*_{T}, *D*_{f,w}, or *D*_{f,g}), and microstructure parameters (λ_{max}, *A*, and *L*_{0}). It can be seen that the present fractal phase permeabilities do not contain any empirical constant.

In Eqs. 35 and 36, the fractal dimensions *D*_{f,w} and *D*_{f,g} can be obtained by extending Eq. 9 and inserting Eqs. 25–30 as

- (37)

- (38)

Compared with Eq. 37, it is seen that Eq. 9 is only a specific case of *S*_{w} = 1, and Eq. 37 is a more general form for area fractal dimension in porous media, including the saturated and unsaturated porous media.

We can now turn our attention again on the relative permeabilities for the wetting and nonwetting phases. Noting Eqs. 14, 25–28, 33–36, we arrive at

- (39)

- (40)

It is evident that the relative permeability is a function of saturation *S*_{w} and fractal dimensions *D*_{f}, *D*_{T}, and *D*_{f,w} (or *D*_{f,g}), and there is no empirical constant in this fractal relative permeability model.