LONG RANGE CORRELATED PERCOLATION
1. Correlated
2. Uncorrelated
We incorporate the spatial correlation properties of real rocks into
the framework of the percolation problem, to investigate
the effects that this has on the various quantities of interest, and to
consider the consequent implications.
As an example, imagine an oil reservoir
made from a river system. The old river
channels represent good sand with high permeability. The other rock
(shale) has poor permeability. Hence, for many purposes it can be
modeled by a conductor/insulator or percolation system. The sand bodies
may be thought of as some shapes distributed in space. They may tend to
avoid each other or stack next to each other. Fortunately for the
petroleum industry, they may also overlap, so it is possible for large
``clusters'' of sand bodies to exist.
In order to quantify these ideas, we consider the correlated
percolation model [H. A. Makse, S. Havlin, M. Schwartz, and H. E. Stanley,
Method
for Generating Long-Range Correlations for Large Systems,
Phys. Rev. E 53, 5445-5449 (1996);
H. A. Makse, S. Havlin, P.-Ch. Ivanov, P. R. King, S. Prakash, and
H. E. Stanley, "Permeability Fluctuations in Sedimentary Rocks:
Connectivity, Permeability, and Spatial Correlations", [Proc. Int'l
Conf. on Pattern Formation, Australia], Physica A 233, 587--605
(1996)]. In the limit
where correlations are so small as to be negligible a site in the square
lattice is occupied at random with a probability p.
However, the fact that we find spatial correlations in the rock
suggest that the process can be better modeled using the correlated
percolation model where each site is not independently occupied, but is
occupied with a probability that depends on the occupancy of the
neighborhood. For a method of generating long-range correlations.
We analyse the structural and dynamic properties of the
resulting connected structure. It is worth noting that the percolation
model applies not only to the scale of the pore structure but also to
larger scales such as the lamination scale. For both the discrete
(sand/shale) and continuous systems (permeability), it is important to
know how long-range correlations influence the macroscopic connectivity
and flow.
The impact of correlations is apparent from the Figures above.
Figure 1 is for percolation with long-range
scale-invariant correlations, and Figure 2 is for
conventional uncorrelated percolation. Both figures are plotted
at the critical
concentration p_c, above which fluid can flow since there exists an
``incipient infinite cluster'' that forms just when a connected path
breaks through. The occupancy probability p corresponds to the net
to gross or volume fraction of good sand in actual sand systems. It is
apparent by visual inspection that the clustering properties for the two
cases differ dramatically. For example, by comparing
the Figures, we see that the
clusters are much larger and more compact in the case of long-range
correlations. This implies that there are fewer dead-ends and hence
less unswept oil. Therefore, the recovery percentage increases for such
strongly correlated systems. Our preliminary results indicate an
increase of about 10% in the recovery percentage of correlated systems
in comparison with uncorrelated systems.
SPATIAL PATTERNS IN PERMEABLE ROCKS
Sedimentary rocks have complicated permeability patterns arising from
the geological processes that formed them. We concentrate on pattern
formation in one particular geological process, avalanches (grainflow)
in windblown or fluvial sands.
We analyze
data on two sandstone samples from different, but similar, geological
environments, and find that the permeability fluctuations display
long-range power-law correlations characterized by an exponent H. For
both samples, we find H ~ 0.82-0.90. These permeability
fluctuations significantly affect the flow of fluids through the rocks.
[ H. A. Makse, G. Davies, S. Havlin, P.-Ch. Ivanov,
P. R. King, and H. E. Stanley,
Long-range Correlations in
Permeability Fluctuations in Porous Rock, Phys. Rev. E 54,
3129-3134 (1996);
H. A. Makse, S. Havlin, P.-Ch. Ivanov, P. R. King, S. Prakash, and
H. E. Stanley, "Permeability Fluctuations in Sedimentary Rocks:
Connectivity, Permeability, and Spatial Correlations", [Proc. Int'l
Conf. on Pattern Formation, Australia], Physica A 233, 587--605
(1996)].
Typical Example of pattern in a sample of Triassic, fluvial trough
cross bedded sandstone from Hollington, near Stafford in the East
Midlands of England. (Courtesy of BP)
AN EFFICIENT WAY TO GENERATE RANDOM SEQUENCES WITH LONG-RANGE CORRELATIONS
One of the most used methods to generate a sequence of random numbers
with power-law correlations is the Fourier filtering method (Ffm).
It consists of filtering the Fourier
components of a uncorrelated sequence of random numbers with a suitable
power-law filter in order to introduce correlations among the variables.
This method has the disadvantage of presenting a finite cutoff in the
range over which the variables are actually correlated.
Other methods present similar problems (see for instance Chapter 9 in
Feder's book). As a consequence, one must generate a very
large sequence of numbers, and then use only the small fraction of them
that are actually correlated (this fraction can be as small as 0.1%
of the initial length of the sequence. This limitation
makes the Ffm not suitable for the study of scaling properties in the
limit of large systems.
We have modified the Ffm in order to remove the cutoff in the range of
correlations. We show that in the modified method the actual
correlations extend to the whole system
[H. A. Makse, S. Havlin, M. Schwartz, and H. E. Stanley,
Method
for Generating Long-Range Correlations for Large Systems,
Phys. Rev. E 53, 5445-5449 (1996)].
Collaborators
S. Havlin and M. Schwartz (Bar-Ilan), P. R. King and G. Davies
(BP),
S. Prakash (ESPCI, Paris), P.-Ch. Ivanov and
H. E. Stanley (BU).
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