Fractal-Facies Concept: Motivation, Application and Computer Codes By Fred J. Molz School of the Environment Clemson University

Sections of the Presentation n Problems with the Levy model and proposed solutions. n The fractal / facies hypothesis. n Data supporting the fractal / facies model. n Software for generating fractal / facies structure. n Conclusions.

While Initial Analyses Suggested That the Levy Fractal Model Fit Data Better Than the Gaussian Fractal Model, Problems Began to Surface: n Levy distributions are known as “Fat Tailed” PDF’s. This means that tail decay is much slower than the exponential Gaussian case. n 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. n This leads to generated property distributions that are too heterogeneous even for the real world. n This problem has been gotten around by rejecting random numbers in the generation process that are outside pre-set bounds (Truncating the PDF.). n 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? n 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. n 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) n Problems with the Levy model and proposed solutions. n The fractal / facies hypothesis. n Data supporting the fractal / facies model. n Software for generating fractal / facies structure. n Conclusions.

What motivated the present version of the fractal/facies concept? n 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. n Thus, determining statistics across facies may be like mixing apples and oranges. n It was realized that superimposing and re- normalizing a set of Gaussian PDF’s with zero means and different variances, produced a Levy- like PDF with Gaussian tails. n This suggested that the apparent Levy behavior of increment Log(k) PDF’s could be the result of mixing several different Gaussian distributions. n 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:

Increment Log(K) variances for each facies were selected so that the overall Log(K) frequency distribution was reproduced reasonably well.

Then realizations were constructed with different fractal structure within each facies type using a developed computer code based on successive random additions (SRA).

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.

However, careful examination of the tail behavior showed once again that the behavior was not Levy.

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.

Sections of the Presentation. (Continued) n Problems with the Levy model and proposed solutions. n The fractal / facies hypothesis. n Data supporting the fractal / facies model. n Software for generating fractal / facies structure. n 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) Increments in log(k) (md) Cumulative distribution Sample CDF (interdune) Gaussian CDF

In order to determine the validity and limitations of the fractal/facies concept, more hard data are needed. Increment Log(k) data from the present project collected from a well-defined, bioturbated sandstone facies yield Gaussian behavior.

Software for Generating Fractal / Facies Structure. n FORTRAN computer programs associated with a paper entitled “An efficient, three- dimensional, anisotropic, fractional Brownian motion and truncated fractional Levy motion simulation algorithm based on successive random additions” is available from the Computers &Geosciences web site. ( I. 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= Vertical increment variance is 4 times horizontal variance. The correct scaling is verified by dispersional analysis with  = H-1 = A B

Conclusions n Increment log(property) PDFs usually appear non- Gaussian n The Levy PDF, unless truncated, yields property distributions that are too variable. n 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. n This concept may be viewed as a discrete version of Painter’s [2001] continuous subordination model. n Limited data support Gaussian increment PDFs within individual facies, and more data are needed.