Detrended Fluctuation Analysis of Geophysical Well-Log Data

Abstract

Geophysical well-log data provide a unique description of the subsurface lithology, as they represent the depositional history of the subsurface formations, vis-à-vis the variation of their physical properties as a function of depth. However, a correct identification of depths to different lithostratigraphic units is possible only by using effective data analysis tools. In the present study, detrended fluctuation analysis (DFA) technique has been applied to gamma-ray log, sonic log and neutron porosity log of three different wells, A, B and C, located off the west-coast of India (i) to discuss the statistical characterization of different subsurface formation properties based on their fractal behavior and (ii) to identify the depths to the tops of formations by comparing the results of DFA with those of wavelet analysis. The DFA technique primarily facilitates to understand the intrinsic self-similarities in non-stationary signals like well-logs by determining the scaling exponent in a modified least-squares sense. In the present study, DFA was carried out in two ways using (i) non-overlapping window method to determine the global scaling exponent and (ii) overlapping window method to determine the local scaling exponent. In the non-overlapping window method, data segments of different windows, each having equal length, were first used to estimate the average fluctuations. The linear least-squares regression between the logarithm of average fluctuations and the logarithm of window lengths then defines the global scaling exponent. For gamma-ray logs of all the three wells, the non-overlapping window method shows two distinct ranges of global scaling exponents, in the ranges 0.5–1.0 and 1.0–1.6. While the former signifies the presence of persistent long-range power-law correlations, indicating the stochastic nature of the sedimentation pattern in the data, the latter indicates the existence of short-range correlations of non-stochastic nature but cease to be of power-law form. On the other hand, the sonic and neutron porosity logs of wells A and C show a single global scaling exponent value of greater than 1.0, signifying the non-stochastic nature of the interval transit time (primary porosity) and neutron porosity, respectively, in the entire data sequence as a function of depth. However, in case of well B, the sonic and neutron porosity logs show two distinct ranges of global scaling exponents, one in the range 0.5–1.0 and the other between 1.0 and 1.5, probably suggesting the effect of different diagenetic conditions in well B, compared to those in wells A and C. Choosing a particular window length and sliding it with unit shifts over the entire length of data for estimating the continuous variation of local scaling exponents as a function of depth defines the overlapping window method. This has been applied on all the log data sets of all the wells to generate the plots of variation of local scaling exponents as a function of depth. Comparison of such plots of variation of local scaling exponents of all the logs with the wavelet scalograms of respective logs revealed that the obtained depth estimates agree well with the known lithostratigraphy of the study region.