Transcript Molz PTTC 2 - Clemson University

Fractal-Facies Concept: Motivation,
Application and Computer Codes
By
Fred J. Molz
School of the Environment
Clemson University
([email protected]
Sections of the Presentation

Problems with the Levy model and proposed
solutions.



The fractal / facies hypothesis.
Data supporting the fractal / facies model.
Software for generating fractal / facies structure.

Conclusions.
While Initial Analyses Suggested That the
Levy Fractal Model Fit Data Better Than the
Gaussian Fractal Model,
Problems Began to Surface:





Levy distributions are known as “Fat Tailed”
PDF’s. This means that tail decay is much slower
than the exponential Gaussian case.
Thus as one gets far from the mean, the probability
of rare events can be millions or billions of times
greater than the Gaussian case.
This leads to generated property distributions that
are too heterogeneous even for the real world.
This problem has been gotten around by rejecting
random numbers in the generation process that are
outside pre-set bounds (Truncating the PDF.).
Finally, as one would expect, careful examination
of the tail behavior of data-based increment PDF’s
shows that the tails of the data are not Levy [Lu and
Molz, 2001].
What was the response to the basic problems
with the Levy PDF?


Painter [2001, WRR], the original Levy proponent,
was motivated to propose his “flexible scaling
model” that allowed one to tune between Gaussian
and Levy behavior using continuous variance
subordination.
Field-oriented considerations led Lu, Molz, Fogg
and Castle [2002, HJ] to consider the neglected
implications of facies structure in many of the past
k data sets that were collected.
–
This motivated what we now call the fractal / facies
model of natural heterogeneity.
Illustration of the motivation for Painter’s
[WRR, 2001] flexible scaling model.
Empirical increment log[R] frequency distribution
(dots) and possible PDF fits. R = electrical resistivity.
Empirical fits to irregular property data that
are Levy-like around the mean, but non-Levy
in the tails.
(After Painter, WRR, 2001)
Sections of the Presentations.
(Continued)
Problems with the Levy model and proposed
solutions.




The fractal / facies hypothesis.
Data supporting the fractal / facies model.
Software for generating fractal / facies structure.

Conclusions.
What motivated the present version of the
fractal/facies concept?

It seems logical that the statistics of a property
distribution should be facies dependent:
–
–
–




Different depositional processes.
Different materials deposited.
Vastly different periods of time.
Thus, determining statistics across facies may be
like mixing apples and oranges.
It was realized that superimposing and renormalizing a set of Gaussian PDF’s with zero
means and different variances, produced a Levylike PDF with Gaussian tails.
This suggested that the apparent Levy behavior of
increment Log(k) PDF’s could be the result of
mixing several different Gaussian distributions.
The concept was first illustrated and tested using
data and a facies distribution from an alluvial fan
environment in Livermore, CA.
The alluvial fan studied was composed of
four facies:
flood plain, channel, levee, and
debris flow deposits.
A realization of facies structure only using the
transition probability / Markov chain
indicator approach of Carle and Fogg
[1996,1997] is shown below:
Facies
Channel
Levee
Floodplain
Debris Flow
0
80
60
20
40
Z
20
40
0
0
X
20
Y
60
40
60
80
80
Increment Log(K) variances for each facies
were selected so that the overall Log(K)
frequency distribution was reproduced
reasonably well.
80
60
40
20
log(K) (m/day)
1.46
0.49
-0.49
-1.46
-2.44
-3.41
-4.39
-5.36
-6.34
0
-7.32
Frequency
100
Then realizations were constructed with
different fractal structure within each facies
type using a developed computer code based
on successive random additions (SRA).
K (m/day)
Channel
Debris Flow
80
0
60
Levee
20
40
Z
20
Floodplain
40
0
0
X
20
Y
60
40
60
80
80
9.849910
5.184000
2.808000
0.432000
0.302500
0.173000
0.086520
0.000043
0.000022
0.000000
Synthetic horizontal Log(K) data were then
determined along vertical transects.
Best fitting Levy and Gaussian PDF’s were
then fitted to the increment Log(K) data.
0.08
sample
Probability
Gaussian
0.06
Levy
0.04
0.02
0
-8
-6
-4
-2
0
2
4
Increments in Log(K) (m/day)
6
8
Cumulative Distribution
However, careful examination of the tail
behavior showed once again that the behavior
was not Levy.
0.10
sample
0.05
Gaussian
Levy
0.00
-8
-6
-4
-2
0
2
4
Increments in log(K) (m/day)
6
8
The resulting Levy-like increment Log(K)
PDF was shown to derive from the
superposition of four Gaussian PDF’s, one
corresponding to each respective facies.
1

Probability
0.8

0.6
0.4


0.2
0









Arbitrary Unit
1
Probability
0.8
0.6
0.4
0.2
0
-8
-6
-4
-2
0
2
Arbitrary Unit
4
6
8
Sections of the Presentation.
(Continued)

Problems with the Levy model and proposed
solutions.



The fractal / facies hypothesis.
Data supporting the fractal / facies model.
Software for generating fractal / facies structure.

Conclusions.
Increment Log(k) values from the inter-dune
facies of Goggin’s [1988] Page sandstone
data appear much more Gaussian than the
entire data set (wind-ripple & grain-flow).
log(k) (md)
4
3
2
1
0
50
100
150
200
0.4
0.8
Cumulative distribution
No. of data points
1
0.8
0.6
0.4
Sample CDF
(interdune)
Gaussian
CDF
0.2
0
-0.8
-0.4
0
Increments in log(k) (md)
In order to determine the validity and
limitations of the fractal/facies concept, more
hard data are needed.
Cumulative Distribution
Increment Log(k) data from the present project
collected from a well-defined, bioturbated
sandstone facies yield Gaussian behavior.
1
Sample CDF
0.8
Gaussian
CDF
0.6
0.4
0.2
0
-1
-0.5
0
0.5
Increments in log(k) (m/day)
1
Software for Generating Fractal / Facies
Structure.

I.
FORTRAN computer programs associated
with a paper entitled “An efficient, threedimensional, anisotropic, fractional
Brownian motion and truncated fractional
Levy motion simulation algorithm based on
successive random additions” is available
from the Computers &Geosciences web site.
(www.elsevier.com/locate/cageo)
Two programs are available:
A. One for generating fractal structure based
on SRA.
B. One for detecting fractal structure based
on dispersional analysis.
An fBm realization with H= 0.41. Vertical
increment variance is 4 times horizontal
variance. The correct scaling is verified by
dispersional analysis with  = H-1 = -0.59.
A
B
log(DA)
0
-1
y = -0.5898x - 0.575
-2
-3
0
1
2
log(lag)
3
4
Conclusions

Increment log(property) PDFs usually appear nonGaussian

The Levy PDF, unless truncated, yields property
distributions that are too variable.

The fractal / facies hypothesis proposes that:
–
–
Data from different facies should not be mixed.
Levy-like increment PDFs result from the
superposition of several independent Gaussian PDFs,
each associated with a different facies.

This concept may be viewed as a discrete version of
Painter’s [2001] continuous subordination model.

Limited data support Gaussian increment PDFs
within individual facies, and more data are needed.