Transcript Molz PTTC 1 - Clemson University

Fractal Representation of Heterogeneous
Properties: Characteristics of Heterogeneity
and Fractals
By
Fred J. Molz
School of the Environment
Clemson University
([email protected])
Sections of Presentation.

Characteristics of natural heterogeneity.

Theory of non-stationary processes with stationary
increments (stochastic fractals).

Relation to the historical development of stochastic
subsurface hydrology.

Conclusions.
Based on detailed measurements, it appears
that property distributions in natural porous
media are defined by:
Irregular Functions
3
property value (arbitrary unit)
How a property distribution might look
if it varies smoothly with position
2
1
0
measurement points
-1
0
20
40
60
80
100
120
140
160
measurement position (arbitrary unit)
property value (arbitrary unit)
3
How an irregular (heterogeneous) property
distribution might actually look
2
1
0
possible averaging
volume of measuring device
-1
0
20
40
60
80
100
120
measurement position (arbitrary unit)
140
160
In addition to irregularity, natural
heterogeneity, similar to coastlines, displays:
Structure Across a Variety of Scales.
As shown by borehole flowmeter
measurements in wells, hydraulic
conductivity (K) also displays structure across
a variety of scales.
An approach for defining and studying the
fundamental properties of irregular functions
derives from the following observation:
Below is an example of an increment
distribution obtained from an irregular
function using a lag (measurement separation)
of 250.
Illustration of the process whereby stationary
random numbers are summed to form an
approximation of the non-stationary random
function Brownian motion.
log(K) increments
A more detailed example: exponentiated
Brownian motion constructed from correlated
Gaussian noise.
4
3
2
1
0
-1
-2
-3
-4
fGn with H = 0.25
0
200
400
600
800
1000
distance (arbitrary unit)
4
fBm with H=0.25
log(K)
2
0
-2
-4
-6
0
200
400
600
800
1000
800
1000
distance (arbitrary unit)
30
25
K
20
15
10
5
0
0
200
400
600
distance (arbitrary unit)
Sections of Talk
(continued)



Characteristics of natural heterogeneity.
Theory of non-stationary processes with
stationary increments (stochastic fractals).
Relation to the historical development of stochastic
subsurface hydrology.

Conclusions.
The presence of spatially non-stationary
property distributions, and variability across
many scales, leads one to study:
Non-Stationary Stochastic Processes
With Stationary Increments






A mathematical theory was developed during the
early-to-mid Twentieth Century [Feller, 1968].
One deals with the increments of a property
distribution (e.g. Log permeability) rather that the
property distribution itself.
A set of increments are collected for a constant
measurement separation, often called “lag”.
Different, but constant, lags result in different
increment sets.
The theory is developed by studying the statistics of
the increment sets (means, variances, etc.), and how
the parameters of the increment probability
distributions vary with lag.
Properties of certain distributions, such as Central
Limit Theorems, play an important role.
Numerous data sets have now shown that the
probability density functions (PDFs) of Log
permeability increments seem to fall within
the Levy family of PDFs or CDFs.
0.4
a=0.8
Probability
0.3
a=1.2
0.2
a=2.0
0.1
0
-10
-5
0
Arbitrary Unit
5
10
Properties of the Levy family of PDFs.

For a mean of zero, the Levy family is a 2parameter family that may be represented in
formula form as:
LPDF ( x) 
exp  Ck


1

a
coskxdk ;
0
a  2  LPDF ( x)  GPDF ( x)
 x2 

exp 
2 
C 2
 2C 
1
Properties of the Levy Family of PDFs
(continued).






a is the order of the highest statistical moment that
exists for the Levy family (0  a  2).
a = 2 results in a Gaussian distribution.
C is a width parameter that is analogous to the
standard deviation of the Gaussian case.
Except for the Gaussian special case, the variance
of a Levy PDF is infinite.
All members of the Levy family obey a generalized
Central Limit Theorem, that make them suitable
candidates for developing a stochastic fractal theory
of heterogeneity.
The general scaling rule, as a function of lag, h, that
results is:
C rh   C h r
r ,a , H  cons tan ts
a
a
aH
Another interesting property that may be
derived for increment sets governed by a
Levy PDF is that:
In General the Increments are Correlated.




0 < H < 1/a  negatively correlated increments.
H = 1/a  independent increments.
1/a < H  1  positively correlated increments.
 (For the Gaussian case, a = 2.)
For the Gaussian and Levy cases respectively, the
correlated sets of increments are called:
–
–

The corresponding sums of the noises respectively
are called:
–
–

Fractional Levy Noise (fLn), and
Fractional Gaussian Noise (fGn).
Fractional Levy Motion (fLm), and
Fractional Brownian Motion (fBm).
These constitute the Levy/Gaussian class of
stochastic fractals.
What do the data show?
Data Show That in Most Cases Increment
PDF’s Have a Non-Gaussian Appearance
(After Painter, WRR, 2001)
Sections of Talk
(continued)


Characteristics of natural heterogeneity.
Theory of non-stationary processes with stationary
increments (stochastic fractals).

Relation to the historical development of
stochastic subsurface hydrology.

Conclusions.
A Short Review of History:
In subsurface hydrology, use of
stochastic processes to represent
property distributions was introduced
by Freeze [1975, WRR].

Stationary, uncorrelated Gaussian
processes:
–
–

Stationary, auto-correlated Gaussian
processes with finite correlation lengths:
–

Property values follow a normal or log-normal
PDF with no auto-correlation.
Parameters of the distribution (e.g. mean and
variance) are independent of position.
Same as above, but property values are
correlated over a finite distance [Gelhar and
Axness, 1983, WRR].
( The above may now be viewed as the
“classical” stochastic processes.)
18
A Short Review of History
(Continued)

Stationary, auto-correlated, Gaussian or Levy
processes with infinite correlation lengths [Molz
and Boman, 1993, WRR; Painter and Patterson,
1994, GRL]:
–
–

These are the so-called fractional noises.
They are a type of stationary stochastic fractal.
Non-stationary Gaussian or Levy processes with
stationary, auto-correlated increments [Neuman,
1990, WRR; Molz and Boman, 1993, WRR; Painter
and Paterson, 1994, JRL]:
–
–
–
–
Only the statistical parameters of the property
increment distributions have meaning.
Increments are the fractional noises described above.
These models of heterogeneity have the strongest
data-based support.
Actual property distributions show multi-fractal
character [Liu and Molz, 1997, WRR; Painter and
Mathinthakumar, 1999; AWR].
Examples of various stochastic processes.
Stationary, Uncorrelated, Gaussian:
Examples of various stochastic processes.
Stationary, Correlated Gaussian
Examples of various stochastic processes.
Unstationary, with Stationary, Correlated,
Gaussian Increments:
Examples of various stochastic processes.
Unstationary, with Stationary, Correlated,
Levy Increments:
Small-scale gas permeability measurements
made on vertical cores of sandstone.
Hydraulic conductivity measurements made
at the Savannah River Site using the
electromagnetic, borehole flowmeter.
Conclusions.

Data show that many natural property distributions
are irregular.

Logic leads one to study irregular functions through
their increment distributions.

Increment distributions described by the
Levy/Gaussian family of PDFs have innate scaling
properties characteristic of what are called selfaffine stochastic fractals.

Data show that natural systems display at least
some of the statistics and scaling of this PDF family

The theory of non-stationary stochastic processes
with stationary increments is a natural
extension of traditional stochastic hydrology.

As stochastic hydrology has been generalized by
necessity, the theory has become more realistic but
less predictive in a traditional sense.