Random fractal models of heterogeneity: The Lévy-stable approach

Borehole measurements of petrophysical properties of sedimentary rock from contrasting geological settings are shown to be consistent with fractional Lévy motion. Specifically, successive increments in the measurements sequences are modeled accurately as having symmetric Lévy-stable distributions. The measurement sequences are statistically selfsimilar for a wide range of spatial scales, limited by the finite length of the measurement sequences at large scales and by the resolution limits of the logging devices at small scales. The measurements sequences display clear evidence of antipersistence (negative dependence in the increments). These results suggest that the fractional-Gaussian-noise model used in reservoir description is an inappropriate model for sedimentary rock, because the rock properties are not stationary and do not have a Gaussian distribution as required for fractional Gaussian noise. In contrast to those of Gaussian-based fractals, the sample paths of the new model increase and decrease in jumps of all magnitudes and mimic geological stratification. These results have fundamental implications for petroleum geostatistics, as many of the statistical techniques developed for Gaussian variables are not valid for Lévy-distributed variables.