Page 211 - gas transport in porous media

Tidwell
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the permeability decreased with increasing sample volume. This is a common finding
reflecting the fact that larger sample volumes integrate over more heterogeneity. The
investigators also found the sample mean of the permeability decreased slightly with
increasing sample volume. This is consistent with theory that suggests the effective
permeability of a medium with short-range spatial correlation, as was the case here,
approaches that of the geometric mean of the smaller scale measurements (e.g., Gelhar
and Axness, 1983).
An alternative approach to scaling investigations is to collect samples from the field
that are subsequently analyzed in the laboratory. For example, Parker and Albrecht
3
(1987) acquired soil cores of three different volumes (92, 471, and 1770 cm ) taken
from two different soil layers along closely spaced transects. Saturated hydraulic con-
ductivitiesandsolutedispersivitieswerethenmeasuredinthelaboratory.Asexpected,
the variance of the natural-log conductivity and dispersivity decreased with increasing
core volume. However, in this case the mean permeability was found to increase with
increasing sample volume, which was inconsistent with the short-range spatial cor-
relation and linear flow imposed in the test. Upon closer inspection the investigators
concluded that the integrity of the smaller core samples had been compromised during
collection. This highlights but one of the difficulties with conducting scaling experi-
ments. That is, care must be taken to avoid introducing bias into the experiment due
to changes in sample integrity, sample density, or measurement precision/accuracy
for different scales of measurement.
In an effort to avoid such biasing, Tidwell and Wilson (1997, 1999a, 1999b, 2000)
employed a consistent measurement device and sampling strategy to acquire per-
meability data over a range of different scales. Specifically, they used a computer
automated minipermeameter test system with six different size tip seals, each provid-
ing approximately an order of magnitude larger sample volume than the next smaller.
Over 150,000 permeability values were collected from three, meter-scale blocks of
rock; including, two cross-bedded sandstones and one volcanic tuff. Characterization
of each block face involved high-resolution mapping of the heterogeneous permeabil-
ity field with each of five different size tip seals, plus the collection of a single large tip
sealmeasurementdesignedtointerrogatemostofthesamplingdomain. Theseexhaus-
tive data sets, measured under consistent experimental conditions yielded empirical
evidence of permeability scaling (e.g., Figure 11.1). Specifically, as the sample sup-
port increased the sample variance decreased, the semivariogram range increased
linearly, while the small-scale (i.e., smaller than the tip seal) spatial structure was
preferentially filtered from the permeability maps and semivariograms. Although all
three-rock samples exhibit similar qualitative scaling trends, distinct differences were
also noted. These differences were most evident in the quantitative characteristics of
the aforementioned trends and in the scaling of the mean permeability. These differ-
ences in part can be explained on the basis of the spatial characteristics of the three
rock samples and the divergent flow geometry imposed by the minipermeameter
tip seal.
Beyond the work of Tidwell and Wilson, there is a growing body of evidence
demonstrating the role of flow geometry on scaling processes. One example is the