Wavelet based multiscale analysis of geophysical downhole measurements: Application to a clayey siliclastic sequence

In order to reveal the structural properties at small scale within a geological unit apparently homogeneous at large scale, we explore the stochastic components of petrophysical measurements recorded in a borehole drilled through the Cretaceous siliclastic units of the South-eastern Basin of France. First, we analyse the instantaneous fluctuations of gamma ray log measurements searching for scaling laws in its power spectrum. Power spectra indicate a scaling behaviour with average exponent βa values mostly falling in the range 1 < βa < 3, a typical fingerprint of fractional Brownian motion. We show that the changes in scaling exponents extracted from the natural gamma ray measurement by a Continuous Wavelet Transform present a close correlation with the stochastic behaviour of radioactive decay processes described by the Poissonʹs law. Secondly, we investigate the stochastic component of three different electrical resistivity logs. In the space/frequency domain explored by a Continuous Wavelet Transform, their local spectral densities show self-affine proprieties. An integrated statistical analysis of the local variations of the three pseudo-logs extracted from these measurements allows to point out new petrophysical heterogeneity within the silty unit.