The tube wave, or low-frequency manifestation of the Stoneley wave, has been modeled previously using the quasi-static approximation; I extend this method to include the effect of the formation matrix compressibility, which tends to marginally increase the tube-wave attenuation. Using the Biot theory of poroelasticity, I develop a fully dynamic description of the Stoneley wave. The dispersion relation derived from Biot's equations reduces in the low-frequency limit to the quasi-static dispersion relation. Comparisons of the quasi-static and dynamic theories for typical sandstones indicate the former to be a good approximation to at least 1 kHz for oil and water infiltration. At higher frequencies, usually between 5 and 20 kHz for the formations considered, a maximum in the Stoneley Q is predicted by the dynamic theory. This phenomenon cannot be explained by the quasi-static approximation, which predicts a constantly increasing Q with frequency. Instead, the peak in Q may be understood as a transition from dispersion dominated by bore curvature to a higher frequency regime in which the Stoneley wave behaves like a wave on a flat fluid-porous interface. This hypothesis is supported by analytical and numerical results.
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Andrew N. Norris (1989) studied this question.