Reservoir characterization using multifractal detrended fluctuation analysis of geophysical well-log data

The spatio-temporal variations in geophysical well-log signals, which often reflect their scale invariant properties, can be well studied with multifractal analysis. In this study, we have carried out fractal and multifractal studies using detrended fluctuation analysis (DFA) and multifractal DFA (MFDFA) respectively. While the DFA primarily facilitates to understand the intrinsic self-similarities in non-stationary signals like well-logs by determining the fractal scaling exponents in a modified least-squares sense, the MFDFA, which in fact, is a generalization of DFA, provides a comprehensive understanding of the multifractal behaviour of the signals through multifractal singularity spectrum as well as the Hurst exponents. DFA and MFDFA have been applied to gamma-ray log and neutron porosity logs of two wells (well B and well C), located in the western offshore basin, India, to study the nature of the subsurface formation properties, vis-à-vis their multifractal behaviour. The estimated DFA fractal scaling exponents, represented in the form of contour plots enable easy identification of the depths to the tops of reservoir zones. On the other hand, the multifractal singularity spectra provide a unique platform for an improved interpretation of logs in terms of their sedimentation pattern and lithological differences. This has been tested with gamma-ray log data of wells B and C. We show that the multifractal behaviour of gamma-ray log is largely influenced by the presence of shale and variations in the subsurface sedimentation pattern. Similarly, the role of gas in a pay zone on the multifractal behaviour was established by comparing the multifractal singularity spectra of the original neutron porosity log and a synthetic neutron log (which we call gas-corrected log), generated using density log. The MFDFA of only that portion of the original neutron log representing the pay zone and its gas-corrected equivalent unequivocally suggest that the presence of gas in the reservoir zones weakens the multifractal behaviour of neutron porosity logs. This emphasizes the significance of multifractal studies of well-logs for effective reservoir characterization. The observed multifractal behaviour in all logs is found to be due to the presence of long-range correlations in the data.