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Intrinsic seismic attenuation is often quantified by the inverse quality factor Q-1 of sedimentary rock within the seismic band of frequencies, which we loosely define as 1 to 104 Hz. For transmission experiments (earthquake recordings, VSP, cross-well tomography, sonic logs), the total measured attenuation can be decomposed as $Q_{\rm total}^{-1} = Q_{\rm scat}^{-1} + Q^{-1}$where both contributions to the total attenuation are necessarily positive. The inverse quality factor Q-1 for the intrinsic attenuation represents the fraction of wave energy irreversibly lost to heat during a wave period as normalized by the strain energy. For refletion seismic prospecting, there are other wave energy losses due to reflection and transmission effects at interfaces. These effects are neglected here for reflection seismic by exclusion. For the types of transmission experiments that we do consider here, we justify neglecting them since we are basically studying the losses within any single block of material, and not treating effects at interfaces between blocks since these can be handled independently with well-known methods.

Crosswell experiments in horizontally-stratified sediments produce negligible amounts of scattering loss so that essentially all loss is attributable to intrinsic attenuation. Quan and Harris (1997) use tomography to invert for the amplitudes of crosswell P-wave first arrivals to obtain the Q-1 for the layers of a stratified sequence of shaly sandstones and limestones (depths ranging from 500-900 m). The center frequency of their measurements is roughly 1750 Hz and they find that 10-2 < Q-1 < 10-1 for all the layers in the sequence. Sams et al. (1997) also measure the intrinsic loss in a stratified sequence of water-saturated sandstones, siltstones and limestones (depths ranging from 50-250 m) using VSP (30-280 Hz), crosswell (200-2300 Hz), sonic logs (8-24 kHz), and ultrasonic laboratory (500-900 kHz) measurements. Sams et al. (1997) calculate (with some inevitable uncertainty) that in the VSP experiments, $Q^{-1}/Q_{\rm scat}^{-1} \approx 4$, while in the sonic experiments, $Q^{-1}/Q_{\rm scat}^{-1} \approx 19$; i.e., for this sequence of sediments, the intrinsic loss dominates the scattering loss at all frequencies. Sams et al. (1997) also find 10-2 < Q-1 < 10-1 across the seismic band.

We demonstrate here that wave-induced fluid flow generates enough heat to explain these measured levels of intrinsic attenuation. Other attenuation mechanisms need not be considered, although they may in fact be present but contribute much smaller fractions of the overall observed attenuation. The induced flow occurs at many different spatial scales that can broadly be categorized as ``macroscopic'', ``mesoscopic'', and ``microscopic.''

The macroscopic flow is the wavelength-scale equilibration occuring between the peaks and troughs of a P-wave. This mechanism was first treated by Biot (1956a) and is often simply called ``Biot loss.'' However, the flow at such macro scales drastically underestimates the measured loss in the seismic band (by as much as 5 orders of magnitude). Mavko and Nur (1979) therefore proposed a microscopic mechanism due to microcracks in the grains and/or broken grain contacts. When a seismic wave squeezes a rock having such grain-scale damage, the cracks respond with a greater fluid pressure than the main porespace resulting in a flow from crack to pore that Mavko and Nur (1979) named ``squirt flow.'' Dvorkin et al. (1995) have given a squirt-flow model applicable to liquid-saturated rocks. Although squirt flow seems entirely capable of explaining much of the measured attenuation in the laboratory at ultrasonic frequencies and may also turn out to be important for propagation in ocean sediments at ultrasonic frequencies (Williams et al., 2002) as well, we show here that this mechanism cannot explain the attenuation in the seismic band.

Thus, a third mechanism based on mesoscopic-scale heterogeneity seems required to explain seismic attenuation. Mesoscopic length scales are those larger than grain sizes but smaller than wavelengths. Heterogeneity across these scales may be due to lithological variations or to patches of different immiscible fluids. When a compressional wave squeezes a material containing mesoscopic heterogeneity, the effect is similar to squirt with the more compliant portions of the material responding with a greater fluid pressure than the stiffer portions. There is a subsequent flow of fluid capable of generating significant amounts of loss in the seismic band.

Prior models of such mesoscopic loss have focused on flow between the layers of a stratified material due to P-waves propagating normal to the layering [e.g., White et al. (1975), Norris (1993), Gurevich and Lopatnikov (1995), and Gelinksy and Shapiro (1997)]. The present study seeks to model the flow for arbitrary mesoscopic geometry, albeit under the restriction that only two porous phases are mixed together in each averaging volume.

In the next section, we review a recent model of Pride and Berryman (2003a,b) treating the mesoscopic loss created by lithological patches having, for example, different degrees of consolidation. This so-called ``double-porosity'' model provides the theoretical framework that will be used throughout. Then, we derive a new patchy-saturation variant of the model and, in following section, a new squirt-flow variant. The results are then discussed in the concluding section. The main point of the paper is to derive models for patchy-saturation and squirt using the same notation and approach as the double-porosity theory; in so doing, we aim to draw conclusions about the nature of attenuation in the seismic band of frequencies.

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