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# Test of length scales for simple geometries

Certain porous media with simple geometries allow analytical calculations of the quantities studied in this paper. Before results are presented for flow in the more complex random geometries, it is instructive to first examine these simpler cases. These resulys are summarized in Table I, and discussed below.

For a simple 3-D tube with radius R [22], k ~ R2, h, Λ, and dc scale as R, and Γ = 1, so that the relation k = c Γl 2 , with c a constant, holds exactly for all three length scales.

Now consider a periodic square array of insulating solid circular inclusions in two dimensions where L is the spacing between nearest-neighbor circles, d is the circle diameter, w = Ld is the width of the narrowest part of the pore space between the disks, p = πd 2/(4L2 ) is the inclusion area fraction, and the porosity φ = 1 − p. We note that φ and w are related by φ = 1 − π(1 − w/L)2 /4 .

We have used exact solutions for this system in various limits as a check of the fluid-flow algorithm, using an L = 301 pixel square unit cell. In the limit w << L, a result from lubrication theory [5,25] is that the permeability k is given by

and in the limit d << L, the permeability is given by [25]

Figure 4 shows the computed quantity log10(L2/k) plotted vs. p = (1 − φ). The solid line is Eq. (8), and the dashed line is Eq. (9). For large values of 1 − φ (small values of φ), w << L and the data points agree well with Eq. (8). The agreement is also still surprisingly good for intermediate values of 1 − φ, and thus fairly large values of w. The data points at small values of 1 - (d << L) agree well with Eq. (9).

Figure 4: Permeability k vs. porosity φ for the periodic square array of circles system. The solid line is Eq. (8) and the dashed line is Eq. (9).

Similar "lubrication-theory" calculations can be done for the electrical conductivity [5], and thus Λ, in the limit w → 0. The technique for calculating Λ using the conductivity is taken from Ref. [3]. The results are, in the limit w − 0,

Figure 5 shows Λ/(2w) and Γ = σ/σo vs. w. The numerical results for conductivity at small w/L do not agree as well with Eq. (10) as the permeability data did with Eq. (8). This is at least partly becasue the flow of electrical current is not as dominated by the neck as is fluid flow, since the tangential electrical current is not forced to be zero at the edges of the neck like the fluid velocity. This is seen clearly in the differing power laws in w for k and Γ, 5/2 and 1/2. Therefore the range of w over which the lubrication theory result holds for the electrical conductivity is much smaller than for the fluid permeability. The numerical results for Λ/(2w) are seen to be heading toward the exact asymptotic limit of 1, as w approaches zero. In the limit w/L → 1, both σ/σo and Λ agree well with the exact asymptotic expressions given in Sec. 3, and shown in Fig. 5.

Figure 5: Quantities Λ/(2d) and Γ = σ/σo vs. the scaled neck width w/L for the peridic square array of circles model. The lines are the exact asymptotic formulas from Eq. (10) (small dashes), L 2 / (πd) (solid), and (2 φ − 1) (large dashes).

Using the above results, we examine in what regimes the relation k = cΓl 2 holds (see Table 1). When w << L, k ~ w5/2, Γ ~ w1/2 , and Λ and dc ~ w, so that the relation k = cl2Γ again holds exactly. However, h goes to a constant for w << L, and hence this relation fails for l = h. This failure is indicative of the paramter h not accurately representing the dynamically connected pore space, as suggested by Johnson, Koplik, and Schwartz [3]. We note that for the case of Λ, c = 1/18, which is different from the value of 1/12 that has been proposed for general pore structures [5]. The value of 1/12 comes from the case of circular cylindrical tube [22]. In the case where the circle size is small compared with the spacing between circles (d << L), k ~ L2 ln(L/d), dc ~ L, h ~ L2 /d, and Λ ~ L2 / d, and all three length scales fail to satisfy the relation k = c Γl 2.

Table 1: The relationships between kh, Λ, and dc for a single tube, and for a periodic square array of circles. The tube radius is R, the circle diameter is d, L is the spacing between circle centers, w = Ld is the width of the neck between adjacent circles, and c stands for an arbitrary constant.
Geometry k Γ k/(Γh2) k/(Γ Λ2) k/(Γ dc2)
Tube
Periodic
Circles
R2 1 R R R

w << L w5/2 w1/2 w2 c c
d << L L2 ln (L/d) 1 ln (L/d)/(L/d)2 ln (L/d)/(L/d)2 ln (L/d)

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