Fractal Representation of Heterogeneous Properties: Characteristics of Heterogeneity

and Fractals

By

Fred J. Molz

School of the Environment

Clemson University(fredi@clemson.edu)

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

    

-1

0

1

2

3

0 20 40 60 80 100 120 140 160

measurement position (arbitrary unit)

prop

erty

val

ue (

arbi

trar

y un

it)

How a property distribution might look if it varies smoothly with position

measurement points

-1

0

1

2

3

0 20 40 60 80 100 120 140 160

measurement position (arbitrary unit)

prop

erty

val

ue (

arbi

trar

y un

it)

How an irregular (heterogeneous) property distribution might actually look

possible averaging volume of measuring device

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.

A more detailed example: exponentiated Brownian motion constructed from correlated Gaussian noise.

-4-3-2

-1012

34

0 200 400 600 800 1000

distance (arbitrary unit)

log

(K)

incre

me

nts fGn with H = 0.25

-6

-4

-2

0

2

4

0 200 400 600 800 1000

distance (arbitrary unit)

log

(K)

fBm with H=0.25

0

5

10

15

20

25

30

0 200 400 600 800 1000

distance (arbitrary unit)

K

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

0.1

0.2

0.3

0.4

-10 -5 0 5 10

Arbitrary Unit

Pro

babi

lity

=2.0

=0.8

=1.2

Properties of the Levy family of PDFs.

For a mean of zero, the Levy family is a 2-parameter family that may be represented in formula form as:

2

2

0

2exp

2

1

)()(2

;cosexp1

)(

C

x

C

xGPDFxLPDF

dkkxCkxLPDF

Properties of the Levy Family of PDFs(continued).

is the order of the highest statistical moment that exists for the Levy family (0 2).

= 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:

tsconsHr

rhCrhC H

tan,,

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/ negatively correlated increments. H = 1/ independent increments. 1/ < H 1 positively correlated increments.

(For the Gaussian case, = 2.)

For the Gaussian and Levy cases respectively, the correlated sets of increments are called:

– Fractional Levy Noise (fLn), and– Fractional Gaussian Noise (fGn).

The corresponding sums of the noises respectively are called:

– 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.

18

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:

– 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.

Stationary, auto-correlated Gaussian processes with finite correlation lengths:

– 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.)

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 self- affine 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.