Ž . Journal of Applied Geophysics 42 1999 99–115 www.elsevier.nlrlocaterjappgeo Dispersion and attenuation of acoustic waves in randomly heterogeneous media J.O. Parra ) , C.L. Hackert, R. Ababou, M.J. Sablik Instrumentation and Space Research DiÕision, Southwest Research Institute, P.O. Drawer 28510, San Antonio, TX, 78228-0510, USA Received 25 August 1998; received in revised form 24 November 1998; accepted 21 June 1999 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 s and s which represent the standard deviations of the R A Ž . Ž . randomly varying density r x and the randomly varying Young’s modulus a 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. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Dispersion; Attenuation; Acoustic waves; Heterogeneous media 1. Introduction As geophysicists obtain and require more accurate information about acoustic velocities, the dispersion, or frequency dependence of these velocities becomes more important. Dispersion is known to fall into two basic classes: intrinsic, which is based on anelasticity; and scattering, ) Corresponding author. Tel.: q1-210-522-3284; fax: q1-210-647-4325; E-mail: jparra@swri.edu which is based on local wavelength-scale varia- tions in the rock formation. Intrinsic dispersion is a local property of the rock. Scattering disper- sion is a property of a neighborhood of rocks, and includes the effects of reflections, refrac- tions, and the law requiring continuity of dis- placement. This paper builds on previous one-dimen- sional and plane wave stochastic random media solutions to provide a more complete theory. Ž . Backus 1962 was the first to examine how waves propagate through small scale variations, 0926-9851r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. Ž . PII: S0926-9851 99 00027-0