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

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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.

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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].

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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.

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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.

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Empirical fits to irregular property data that are Levy-like around the mean, but non-Levy in the tails. (After Painter, WRR, 2001)

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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.

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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.

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The alluvial fan studied was composed of four facies: flood plain, channel, levee, and debris flow deposits.

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A realization of facies structure only using the transition probability / Markov chain indicator approach of Carle and Fogg [1996,1997] is shown below:

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

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

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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.

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

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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.

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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.

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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

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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.

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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. (www.elsevier.com/locate/cageo)www.elsevier.com/locate/cageo I. Two programs are available: A. One for generating fractal structure based on SRA. B. One for detecting fractal structure based on dispersional analysis.

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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

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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.

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