Results

Images and Autocorrelation Functions

Figure 1 shows the first 50 (out of 100) slices for the original 40% porosity microstructure along with a magnified view of one of the slices used as input into the reconstruction process. At this porosity, the pore structure appears relatively open even in two dimensions. Figure 2 shows the first 50 slices for the reconstructed and modified three-dimensional microstructures generated based on the two-dimensional slice in Fig. 1b. The reconstructed system has some similarity to the original system but appears noisier with many isolated small solid areas breaking up larger porous regions. To the naked eye, the modified system appears much more similar to the original system than the reconstructed system does. This qualitative assessment is quantitatively verified in Figure 3 which shows the autocorrelation functions for all three systems. The original autocorrelation function was computed for the two-dimensional slice used to generate the reconstructed microstructure while the reconstructed and modified curves were computed for the entire new three-dimensional microstructures. The original and modified autocorrelation functions are seen to nearly overlap for values of distance up to 12 pixels. At longer lags, the functions no longer overlap as the reconstructed and modified systems are not able to match the long range order of the original microstructure. This could be due to the extent of the filtering operation used during the generation algorithm (30 pixels) or the random nature of the starting Gaussian noise image.

Figure 1: Original 40% porosity microstructure showing a) top 50 slices of 100³ system (slices proceed from left to right; top to bottom) and b) one of the slices selected for the reconstruction process.

 

Figure 2: Top 50 slices for a) reconstructed and b) modified three-dimensional microstructures for 40% porosity system.

Figure 3: Autocorrelation functions for original, reconstructed, and modified 40% porosity microstructures.

The autocorrelation function for the penetrable sphere model can be analytically determined as presented by Torquato and Stell [11]. Figure 4 shows a comparison of this analytical solution to the autocorrelation functions determined for a single two-dimensional slice and the overall three-dimensional microstructure for the 40% porosity system. For the three-dimensional image, minor variations between the analytical and computed autocorrelation function are observed. These are most likely due to the fact that the model microstructure is a digitized image and the analytical solution is for a continuum microstructure. Indeed, similar effects of digitization on the autocorrelation functions of the penetrable-sphere model have been observed by Berryman [12]. For the two-dimensional image, the variation is somewhat larger since a single two-dimensional slice will have a different porosity and surface area than the overall three-dimensional system, due to statistical variation.

Figure 4: Autocorrelation functions for two and three-dimensional images compared to analytical solution for penetrable spheres for 40% porosity microstructure.

Results for a lower porosity (19%) system are presented in Figures 5-7. For this lower porosity system, the pore space is discontinuous in two dimensions although it remains percolated in three dimensions. Once again, the visual similarity between the modified and original microstructures is striking and the autocorrelation functions perfectly overlap one another for distances up to 10 pixels. In fact, the modified reconstruction algorithm appears to visually reproduce porous microstructures for the entire range of porosities (15-40%) investigated in this study.

Figure 5: Original 19% porosity microstructure showing a) top 50 slices of 100³ system (slices proceed from left to right; top to bottom) and b) one of the slices selected for the reconstruction process.

Figure 6: Top 50 slices for a) reconstructed and b) modified three-dimensional microstructures for 19% porosity system.

Figure 7: Autocorrelation functions for original, reconstructed, and modified 19% porosity microstructures.

Characteristic Length Scales

Table I summarizes the results for the hydraulic radii and critical diameters measured for the original, reconstructed, and modified porous media. In general, the hydraulic radii of the reconstructed porous media were about half those of the original microstructures. In figure 2b, it appears that the curvature modification algorithm has opened the overall pore structure of the system. This is verified by the increase observed in the critical diameter between the reconstructed and modified porous media for porosities greater than 25%. For lower porosities, the critical diameter was not significantly affected by the curvature modification despite the increase in hydraulic radius. This may be due to the finite resolution of the 100³ lattice used in this study, as at these very low values of D_c, the effects of the underlying pixel lattice structure are more pronounced.

Table 1: Characteristic Length Scales in Pixel Units for
Reconstructed Porous Media

Porosity	Dc	Dc1	Dca	Rh	Rha	Rha
	orig	recon	mod	orig	recon	mod
0.400	8	4.0(0.0)2	8.8(0.5)	2.85	1.47(0.07)	2.71(0.17)
0.330	8	4.0(0.0)	7.2(0.8)	2.41	1.30(0.05)	2.45(0.12)
0.269	6	2.8(0.5)	6.0(0.6)	2.23	1.13(0.04)	2.34(0.11)
0.188	4	2.0(0.0)	2.4(0.4)	1.63	0.94(0.05)	1.60(0.03)
0.146	4	2.0(0.0)	1.6(0.4)	1.69	0.89(0.05)	1.70(0.08)

¹ Average of five reconstructed/modified systems

² Numbers in parentheses indicate standard error for five reconstructed systems

Transport Properties

Figures 8 and 9 provide plots of permeability and relative conductivity, respectively, for the original, reconstructed, and modified porous media. The agreement between transport properties of the original and reconstructed microstructures is summarized in Table II. The curvature modification is seen to significantly improve the agreement between the permeabilities of the original and reconstructed microstructures for porosities greater than 25%. In fact, for porosities greater than 30%, the average values for the modified systems are within 25% of the actual values for the original microstructures. These are the same systems that exhibited a significant change in the critical diameter after curvature modification, suggesting that D_c is a more relevant characteristic length for permeability than R_h, at least for the systems investigated in this study.

In Fig. 9, the curvature modification is seen to have very little effect on the relative conductivity of the reconstructed porous media. This suggests that the modification is changing the pore sizes much more than the overall tortuosity of the pore system. As shown in Table II, for porosities greater than 25%, however, the reconstructed and modified porous media exhibit relative conductivities within a factor of 2.5 of those of the original microstructures.

Table 2: Average Transport Properties of Reconstructed Porous Media

Original	k	k/k*	k/k*	$(\sigma / \sigma_0)$	$\frac{(\sigma / \sigma_0)}{(\sigma / \sigma_0)^*}$	$\frac{(\sigma / \sigma_0)}{(\sigma / \sigma_0)^*}$
Porosity	(orig)
	pixels2	k*=Recon1	k*=Moda		*=Recona	*=Moda
0.400	0.41	5.7(0.27)2	1.24(0.29)	0.2	1.64(0.05)	1.50(0.09)
0.330	0.21	6.65(0.13)	1.33(0.38)	0.14	2.1(0.13)	1.97(0.21)
0.269	0.11	9.38(0.12)	2.71(0.25)	0.093	2.5(0.04)	2.51(0.11)
0.188	0.02	8.73(0.33)	4.31(0.53)	0.035	3.41(0.22)	3.5(0.37)
0.146	0.0125	18.94(0.43)	19.84(0.75)	0.026	7.22(0.22)	10.6(0.58)

¹ Average of five reconstructed/modified systems

² Numbers in parentheses indicate coefficient of variation in k^* or (σ / σ₀)* for the 5 reconstructed systems

Figure 8: Calculated permeabilities for original, reconstructed, and modified microstructures for five different porosities.

Figure 9: Calculated relative conductivities for original, reconstructed, and modified microstructures for five different porosities.

To adequately reconstruct a three-dimensional porous medium, it is vital to capture the three-dimensional percolation properties of the pore space as well as the two-dimensional characteristics of the pores. Since the starting point in this study is a two-dimensional image of the pore system, capturing the three-dimensional connectivity is indeed the research challenge. For the penetrable sphere model, it is well known that the pore (matrix) phase has a percolation threshold of about 3% [27]. From our studies, the reconstructed microstructures appear to exhibit a percolation threshold closer to 10% porosity. This difference in connectivity between original and reconstructed microstructures was alluded to by Adler et al. [9] as one of the main reasons for differences between the transport properties of the two types of media. In this study, while good agreement has been obtained between transport properties for porosities far away from the apparent percolation thresholds of the model and reconstructed media (i.e.>25%), large differences have been observed in values for porosities nearer to these percolation thresholds (i.e. <20%).

The percolation threshold of about 10% observed for the reconstructed systems may be an inherent limitation of the reconstruction technique employed in this study, as a similar threshold has been obtained for systems based on thresholding three-dimensional images of Gaussian-filtered white noise [5]. Furthermore, Renault [28] has observed a reduction in the percolation threshold for site percolation on a three-dimensional network from a value of 0.31 to a value between 0.1 and 0.2 when a variety of different spatial correlations were introduced. To obtain a significantly lower percolation threshold, it may be necessary to utilize higher order information such as a three-point correlation function or start with a different initial three-dimensional image structure than the image of Gaussian noise employed in this study.

Martys et al. [29] have developed a universal scaling relationship for the permeability of a porous medium as a function of its porosity and percolation threshold. The basic equation is of the form

$$\frac{ks^{2}}{2 \phi _s} \: \: \alpha \: \: (\phi-\phi_c)^{n}$$ (8)

where s is the specific surface of the porous medium, φ_s is the solids fraction, φ is the porosity, φ_c is the percolation threshold for porosity, and n is a critical exponent. Figure 10 provides a plot of this universal scaling for values of φ_c of 0.03 and 0.08 for the original and modified reconstructed porous media respectively. A value of 0.08 was selected as the percolation threshold for porosity in the reconstructed media on the basis that the porosity was connected for reconstructed systems with 10.5% porosity but disconnected for systems with 7.2% porosity. In figure 10, all of the data points are seen to lie on a single line, with some scatter, further confirming that the differences in permeability between the original and modified systems are largely due to their different percolation thresholds.

Figure 10: Scaled permeabilities for original and modified reconstructed microstructures for five different porosities. Vertical bars for modified systems indicate standard error in computed scaled permeabilities.