Evidence for Non‐Gaussian Scaling Behavior in Heterogeneous Sedimentary Formations

Vertical and horizontal fluctuations in permeability and porosity in sedimentary formations are analyzed and are found to be consistent with scaling models based on Levy-stable probability distributions. The approach avoids the assumption of Gaussian behavior and is supported by evidence from horizontal and vertical well logs and from permeability measurements on a sandstone outcrop and segments of core from a heterogeneous formation. The incremental values in these measurement sequences are accurately modeled as having Levy-stable distributions. The width of the distribution of increments depends on the spatial scale in a manner consistent with scaling behavior. The width of the distribution is smaller for horizontal increments than for vertical increments, reflecting the reduced variability in the horizontal direction. The scaling parameters are in the range associated with antipersistence and are roughly the same magnitude in the vertical and horizontal directions. The relationships between different physical properties are briefly studied, and it is suggested that they be quantified through off-diagonal terms in a multivariate Levy width matrix. Simulations designed to reproduce the observed statistical features are also described. These results have some fundamental implications, as Levy-stable distributions require a different set of statistical tools and theoretical methods compared to finite-variance distributions.

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