Dispersion and attenuation of acoustic waves in randomly heterogeneous media

Abstract

We derive the effective displacement relation for acoustic waves in a spatially random heterogeneous one-dimensional medium. This relationship is expressed in terms of parameters σ R and σ A which represent the standard deviations of the randomly varying density ρ( x) and the randomly varying Young's modulus α( x), of the medium. In this way, we build the contributions into the total displacement relationship for the spatially random heterogeneous medium and apply this result to determine the dispersion and attenuation of acoustic waves propagating in the random heterogeneous medium. Attenuation and dispersion of waves propagating in media with randomly varying properties has been the subject of much study. Most of this work has neglected the effects of intrinsic dispersion and attenuation in order to concentrate on the effects of the medium inhomogeneities. We demonstrate how intrinsic attenuation may be easily included in the theoretical development, and explore the combined effects of scattering-based and intrinsic attenuation and dispersion on wave propagation. We apply the solution to model interwell acoustic waves propagating in the Kankakee formation at the Buckhorn Test Site, IL. The modeling results show that the strong dispersion in the frequency range of 500–2000 Hz is due to the reservoir heterogeneity. Alternatively, the velocity dispersion for frequencies greater than 2000 Hz corresponds to the intrinsic properties of the reservoir.