Combining stochastic facies and fractal models for representing natural heterogeneity

Sedimentary deposits are often characterized by various distinct facies, with facies structure relating to the depositional and post-depositional environments. Permeability (k) varies within each facies, and mean values in one facies may be several orders of magnitude larger or smaller than those in another facies. Empirical probability density functions (PDFs) of log(k) increments from multi-facies structures often exhibit properties well modeled by the Levy PDF, which appears unrealistic physically. It is probable that the statistical properties of log(k) variations within a facies are very different from those between facies. Thus, it may not make sense to perform a single statistical analysis on permeability values taken from a mix of distinct facies. As an alternative, we employed an indicator simulation approach to generate large-scale facies distributions, and a monofractal model, fractional Brownian motion (fBm), to generate the log(k) increments within facies. Analyses show that the simulated log(k) distributions for the entire multi-facies domain produce apparent non-Gaussian log(k) increment distributions similar to those observed in field measurements. An important implication is that Levy-like behavior is not real in a statistical sense and that rigorous statistical measures of the log(k) increments will have to be extracted from within each individual facies.