Characterization of long-range correlations in complex distributions and profiles

Abstract

Characterizing long-range correlations in complex distributions, such as the porosity logs of field-scale porous media, and profiles, such as the fracture surfaces of rock and materials, is an important problem. We carry out an extensive analysis of such distributions represented by synthetic and real data to determine which method provides the most efficient and accurate tool for characterizing them. The synthetic data and profiles are generated by a fractional Brownian motion (FBM) and the real data analyzed are a porosity log of an oil reservoir and time variations of the pressure fluctuations in three-phase flow in a fluidized bed. The FBM is generated by three different numerical methods and the data are analyzed by seven different techniques. Our analysis indicates that the size of the data array greatly influences the accuracy of characterization of its long-range correlations. We also find that if the size of the data array is large enough, the commonly used rescaled-range $(R/S)$ method of analyzing FBM series fails to provide accurate estimates of the Hurst exponent, although it can provide a reasonably accurate analysis of a data array that is generated by a fractional Gaussian noise. In contrast, the maximum entropy and wavelet decomposition methods offer highly accurate and efficient tools of characterizing long-range correlations in complex distributions and profiles. New methods that are somewhat similar to the $R/S$ method are also suggested.