Choice of representative image volume

The question of how the results depend on the total volume imaged has to be addressed. In previous work [Spanne 1994, Schwartz 1994, Auzerais 1996], calculation of transport properties on microtomographic images have been performed on either the full image or on a few subsets of the imaged data. Auzerais et al. [Auzerais 1996] found that cubes of size of much greater than 1 mm³ were required to estimate fluid permeability with acceptable accuracy. Unfortunately, the analysis of only a small number of subsets provides only a few datapoints to compare to experiment. The subsequent requirement to experimentally image many samples to obtain data across the full range of $\phi$ is both expensive and time consuming. In recent work [Lindquist 2000], the distributions of the flow relevant geometrical properties (e.g., pore size distribution, throat size, etc.) were measured on the same set of Fontainebleau sandstone cores considered here. They extracted, from the center of each core, a 256³ voxel image (3.09 mm³) and compared geometrical properties to the full core of seven times the volume. Even at this smaller scale, roughly ten grains on a side, the comparison showed good agreement for most blocks. The prospect of using smaller block sizes is an encouraging one. Rocks, even as homogeneous as Fontainebleau sandstone, exhibit local variability in the porosity [Thovert 2001, Arns 2001]. We illustrate this for the four Fontainebleau sandstone samples in Figure 1.

Figure 1: Variations in porosity of the four Fontainebleau sandstone samples measured along the core (480³ subvolumes).

By choosing independent subsamples ((L / N)³) of the original (L³) image one can obtain a larger ensemble (N ³) of samples which exhibit a wide range of porosity. The combination of an appropriately small window size on the imaged core and the natural heterogeneity of the rock allows properties of the rock to be derived over a wide range of porosities from a small number of core samples. In this section we use morphological measures to help define an "appropriate" window size.

A first test of the dependence of digitized data on image volume is the requirement to ensure that the geometrical and the topological descriptors of the image volume are consistent. A family of measures based on the Euler-Poincaré characteristic [Hadwiger 1957, Santaló 1953] has been shown to be very sensitive to the morphology of random materials [Arns 2001a]. In three dimensions, there are 4 measures related to the familiar measures of volume fraction, surface area, integral mean curvature and Euler characteristic (connectivity). Figure 2 illustrates the three latter measures [surface-to-volume (S/V), integral mean curvature (H), and Euler characteristic ($\chi$)] as functions of the porosity for the original image at 480³ and for cubic subsets of the image at scales of 240³ and 120³. Variability of the measures increases with decreasing window size but the values are consistent with the data for the larger volumes, suggesting that the averages obtained for the smaller blocks are meaningful. At low porosity, near the percolation threshold of the pore space, the scatter is greatest, but still acceptable at the scale of 120³.

One can also define the representative cell size by considering the two point correlation function -- the probability of finding two end points of a segment of length within the same phase. Defining a correlation length as the first zero of this two point function [Joshi 1974], we find that 130 µm. This is consistent with values reported in the literature [Oren and Bakke 2000]. For our system size of 120³ at 5.7 µm per voxel, this implies the cell spans more than five times the correlation length and good averaging behavior may be expected. Previous microtomographic work [Auzerais et al. 1996] found that for a system of similar size (112³ at 7.5 µm resolution), the averaging of the porosity was acceptable.

It is important to carefully select the number of samples over which the results are averaged in order to produce acceptable uncertainties. Use of samples of 120³ results in samples per core, provides a wide spread of porosities from each core, and is computationally realizable on common workstations. Example snapshots of 120³ subvolumes from the four cores are shown in Figure 3. Further assessment of the errors associated with the choice of the window size on the numerical computation of the elastic properties of the images is addressed in the next section.

Figure 2: Measures of morphology based on the Euler Poincaré characteristic for the Fontainebleau sandstone at the scale of the full 480³ image and for smaller cubic subsets [surface-to-volume (S/V), integral mean curvature (H), and Euler characteristic ($\chi$)].

Figure 3: Pore space images of a central 120³ subset of each of the four 480³ samples. The images are from the Fontainebleau cores at (a) 7.5% (7.04%), (b) 13% (12.6%), (c) 15% (19.6%), and (d) 22% (19.0%). The values in brackets note the porosities of the individual 120³ subsamples compared to the macroscopic porosity of the original core plugs. The differences in the bulk and subsample porosities again illustrate how one may obtain a large ensemble of samples with variable porosity from a single core plug.