A generalization of the fractal/facies model |
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| Authors: | Fred J Molz Tomasz J Kozubowski Krzysztof Podgórski James W Castle |
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| Institution: | (1) Environmental Engineering and Science, Clemson University, Clemson, SC 29632, USA;(2) Mathematics and Statistics, University of Nevada, Reno, NV 89557, USA;(3) Mathematical Sciences, Indiana University-Purdue University, Indianapolis, IN 46202, USA;(4) Geological Sciences, Clemson University, Clemson, SC 29634, USA |
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| Abstract: | In order to generalize the fractal/facies concept, a new stochastic fractal model for ln(K) increment probability density functions (PDFs) is presented that produces non-Gaussian behavior at smaller lags and converges to Gaussian at larger lags. The model is based on the classical Laplace PDF. The new stochastic fractal family is called fractional Laplace motion (fLam) having stationary increments called fractional Laplace noise (fLan). This fractal is different from other fractals because the character of the underlying increment PDFs changes dramatically with lag size, which leads to lack of self-similarity. Data also appear to display this characteristic. In the larger lag size ranges, approximate self-affinity does hold. The basic field procedure for further testing of the fractional Laplace theory is to measure ln(K) increment distributions along transects, calculate frequency distributions from the data, and compare results to appropriate fLan family members. The variances of the frequency distributions should also change with lag size (scale) in a prescribed manner. There are mathematical reasons such as the geometric central limit theorem, for surmising that fLam/fLan may be more fundamental than other approaches that have been proposed for modeling ln(K) frequency distributions. |
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| Keywords: | Facies Fractal model Heterogeneity Groundwater hydraulics Geostatistics |
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