Wavelet leader-based multifractal analysis for characterizing subsurface heterogeneities using velocity logs

In petroleum engineering, well logs are the key to characterize a hydrocarbon reservoir, and to extract meaningful features related to the lithology and the type of fluids present in the reservoir. Several approaches have been suggested to exploit well log data. In this view, this paper presents a wavelet leader-based multifractal analysis (WL), applied to velocity log data measured at two scientific deep boreholes: the pilot well (KTB-VB) and the ultra-deep main well (KTB-HB), to study the local behavior and the invariant scale properties of the investigated data. The suggested approach allows distinguishing different lithology types based on the wavelet leaders. First, local regularity profiles have been computed using the Peltier and Lévy-Véhél (PLV) algorithm from the different velocity logs, and a lithological segmentation has been carried out. Then, a multifractal analysis has been carried out on velocity logs using the WL algorithm. A clear correlation is shown between lithology and multifractal properties extracted from the investigated logs, specifically significant values singularity spectrum width $( \Delta h)$ correspond to the local filled fractures. To conclude, the singularity spectrum may then serve as a tool for characterizing subsurface heterogeneity and identifying a zone of macro- and micro-fractures.