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# Results for random systems

Figure 6 shows the permeability k (in units of pixels squared) and Fig. 7 shows the relative conductivity for 1000x1000 unit cell random systems, plotted vs. porosity. For porosities 50% or below, the number ratio of small to large circles was fixed at 1.5. Each data point in Figs. 6 and 7 has been averaged over the x and y directions and over four random structures. The permeability changes by more than two orders of magnitude from φ = 0.8 down to φ = 0.35, with a typical standard deviation computed over the different configurations of about 10% at the higher porosities, increasing to about 30% at the lower porosities. The standard deviation in the conductivity data was never more than a few percent for all porosities.

Figure 6: Permeability k, in units of pixels squared, vs. porosity for the 1000x1000 pixel random systems, averaged over the x and y directions and four configurations.

Figure 7: Showing the relative conductivity Γ vs. porosity for the 1000x1000 pixel random systems, averaged over the x and y directions and four configurations.

Figure 8 shows the three length scales, Λ, h, and dc plotted against porosity. Note that h/2 is really plotted in Fig. 8, not h itself. Figure 9 shows Λ vs. dc. The Λ vs. dc plot in Fig. 9 is roughly linear, in general agreement with the predictions of Banavar and Johnson [12].

Figure 8: Three length scales Λ, h/2, and dc plotted vs. porosity for the 1000x1000 pixel random systems. Each point represents an average over four configurations, and for Λ and dc, an additional average over the x and y directions. Note that h/2 is plotted in the figure, not the value of h itself.

Figure 9: Λ plotted vs. dc using the data of Fig. 8. The slope of the dashed line is fit to the data, and the line is constrained to go through the origin.

Figure 10 shows the scaled permeabilities for the 1000x1000 pixel unit cell random models. The permeabilities are scaled by the factor c' / (Γ l 2), where the choices of l are (a) l = Λ, (b) l = h, and (c) l = dc, and c' is a constant chosen to make the high porosity end of each curve be approximately equal to one. Figure 10a is quite flat over the whole range of porosity shown, implying that the parameter Λ is giving a good estimate of the pore length scale that is controlling permeability. The value of c' used was 6.5. References [3] and [5] used c' = 1/c = 12 in the relation k = c Γl 2.

Figure 10b, where the permeability is scaled by the hydraulic radius h, with c' = 32, shows a systematic dip toward lower values as porosity decreases. This is consistent with the results obtained for the periodic model, since the permeability depends sensitively on the pore necks, while h is determined by all the pore area and volume. Therefore as the critical necks controlling the flow become small while the porosity still remains fairly large, the value of h will increasingly overestimate the permeability length scale.

Figure 10c shows how the parameter dc serves to scale the permeability, with c' = 12. Reasonably flat behavior is shown across the whole range of porosity, with some rise at lower porosity. The lowest porosity, 0.35, has a significantly narrower pore-size distribution than does the 0.60 porosity system, for example, as can be seen in Fig. 2. Since a basic assumption behind the Katz-Thompson equation is a broad pore-size distribution [4,9], Fig. 10c may be showing a systematic trend in the constant c' as the pores size distribution becomes narrower. The constant c' is expected to get smaller for narrower pore-size distributions [26], which is consistent with our data.

Figure 10: Permeabilities of Fig. 6, scaled by the values of Γ from Fig. 7 and the length scales of Fig. 8, plotted against porosity. A different value of c' was used in each graph, in order to make the high porosity data points fall on the y = 1 line:
(a) c' = 6.5, (b) c' = 32, and (c) c' = 12.

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