Heterogeneities characterization from velocity logs using multifractional Brownian motion

Well logs provide additional information from the borehole that cannot be derived from other subsurface investigations. Their analysis may bring supplementary features about the earth’s heterogeneities. In previous researches, the logs were modeled using fractional Brownian motions indexed by Hurst (or Hölder) exponents quantifying their global regularity degrees. Indeed, these monofractal models, characterized by the same Hurst exponent, do not allow to study the depth-evolution of the local Hurst parameter. In order to overcome this problem, we propose to model the logs using multifractional Brownian motions and suggest an algorithm, developed previously for financial time series, to estimate the local Hurst exponent function. First, the potential of this algorithm is assessed through its application on synthetic sonic log data simulated by the successive random additions method. We note that the estimated Hurst functions (or regularity profiles) are very close to the theoretical Hurst functions. Second, this analysis is extended to sonic logs data from the KTB pilot borehole (Germany). Using the resulted regularity profiles, we carry out a lithological segmentation and fault identification on the geological layers crossed by the well. We derived also a correlation between the Hurst value variation and the lithological changes.