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Fractal heterogeneities and cycles in sediments

O'Doherty and Anstey (1971) and Anstey and O'Doherty (2002a) described variations in well logs by the superposition of different types of deposition, leading to ``layers inside layers''. Their classification includes

A large number of small thickness layers ($\leq 1$ m) for weakly transitional depositions with small reflection coefficients;
Cyclic layers of thicknesses from 1 to 10 m with sharp interfaces, corresponding to fine layering depositions inside a facies for short-period sea cycles; and
Horizons imaged by seismic reflection, i.e. different facies for a small number of thicker blocky layers associated with low-order cycles.
They suggested that transmission losses could be compensated by multiple reflections, depending on seismic wavelength. This classification is in agreement with the fact that high exponents, $H$, appear for shorter scales, $b$, in Table 4. The estimation performed on the sonic logs indicates fractal properties for distances shorter than $b\simeq 5$m. Acccording to Anstey and O'Doherty (2002a), well log signals are the superposition of several processes with different scales. The von Kármán model captures part of it. Parameters extracted by our analysis describe heterogeneities corresponding to type 2 of the O'Doherty-Anstey classification, which is a fractal behavior inside major geological units, at least from 10 down to 1 m, with a correlation length of 5 m. Previous estimations of the correlation length on well logs were produced by direct calculation of the spatial autocorrelation (Shiomi et al., 1997; White et al., 1990). White et al. (1990) suggested the possibility of superposition of two correlation lengths at 5 and 20 m. The wavelet detection analysis of gamma-ray and resistivity well logs for a sandstone confirmed the strong evidence of local cyclicity in the stratigraphic sequences (Rivera et al., 2004). We think that direct estimation of correlation distance $b$ using the autocorrelation function, or our estimation method, captures the shortest dominant cycle in the sedimentary layers. This would explain why the fractal behavior seems to hold for larger scales in Figures 3(e) and 3(f).

Parameter $H$, estimated from well logs, is $0< H\leq0.5$ and consistent with an antipersistent fractional Brownian motion characteristic of cyclicity (see Table 1). The Hurst exponent commonly exhibits some antipersistence in sediments with values from 0.2 to 0.5 for sandstones (Lu et al., 2002; Dolan et al., 1998). High values of 0.5 and 0.6 could be interpreted, in a clastic context, to be caused by a transitional deposition involving persistency, as in natural floods. Natural flood records exhibit a Hurst exponent, $0.5\leq Hu \leq 1.0$, associated with so-called black noise (Mandelbrot and Wallis, 1969; Hurst, 1951).

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