Fractal heterogeneities in sonic logs and low-frequency scattering attenuation |

The high-frequency quasi-cyclic variations of seismic velocities can be described as an antipersistent fractional Brownian motion as demonstrated by our sonic-log data from a clastic reservoir. The correlation length, estimated for von Kármán's model, is about 5m, but the sonic logs contain larger local cycles at 10 and 20 m : our method extracts one dominant cycle of deposition. Conventional seismic surveys contain frequencies as high as 100 Hz, with typical peak frequencies at 25 Hz. Our statistical description of geological heterogeneities below 10 m can thus be used consistently in our low-frequency scattering theory to estimate the scattering loss caused by small-scale heterogeneities.

Shear waves have shorter wavelengths than compressional waves and can be more attenuated because they are more sensitive to heterogeneities. We showed the existence of a high-frequency cutoff for the depth of penetration of waves, whose position in frequency depends on the maximum size of fractal heterogeneities. The dominant frequency of a wavelet decreases faster for higher fractal exponents and for larger characteristic sizes of heterogeneities. This loss of high-frequency content influences resolution in seismic imaging. Our study recommends using low-frequency P-waves for deep targets under a strongly heterogeneous overburden.

Agreement of our results with the Backus limit and the Rayleigh diffusion regime is due to the use of velocity fluctuations. Nevertheless, proper connection with the O'Doherty-Anstey formula requires use of the logarithm of impedance. Moreover, more complex, multiple scattering occurs when sizes of heterogeneities are similar to that of the seismic wavelength. Large-scale local cycles, present in the sonic-log data, call for incorporation of resonant scattering effects into high-frequency scattering theories.

Fractal heterogeneities in sonic logs and low-frequency scattering attenuation |

2013-03-02