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 mm^{3} 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 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 256^{3} voxel image
(3.09 mm^{3}) 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.

By choosing independent subsamples ((*L* / *N*)^{3}) of the
original (*L*^{3})
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 ()] as functions of the porosity for the original
image at 480^{3} and for cubic subsets of the image at scales of
240^{3}
and 120^{3}. 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 120^{3}.

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 120^{3} 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 (112^{3} 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 120^{3} results in samples per core, provides a wide
spread of
porosities from each core, and is computationally realizable on common
workstations. Example snapshots of 120^{3} 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.