Sidorov and Tarantin (2010) proposed that elastic wave propagation in fluid-saturated porous rocks obeying the Wyllie time-average equation can be described as a phonon-momentum-conservation process. This picture yields a shear-wave analogue of the time-average equation and a damping law α=Bk(Vliq−1+Vsol−1)f\alpha = Bk(V_{\mathrm{liq}}^{-1}+V_{\mathrm{sol}}^{-1})f with attenuation exactly proportional to frequency, hence a frequency-independent damping decrement and no velocity dispersion. The constant B≈0.12B\approx 0.12 was calibrated on a single laboratory study of water-saturated organic glass with cylindrical holes, and the mechanism was derived only for fully saturated media that satisfy the time-average equation. This problem takes these unresolved questions as its single objective: establish the empirical and theoretical domain of validity of the mechanism — whether BB is universal across lithologies and pore structures, whether the mechanism extends consistently to porous media outside the fully saturated Wyllie-valid class (partial saturation, complex pore geometry, heterogeneity, anisotropy, fractures), and whether the damping coefficient can be recovered in situ from borehole acoustic logging. The validation question is therefore open: the source paper has not been independently validated, and mainstream rock physics treats attenuation exclusively by causal relaxation mechanisms.

Background

Fluid-saturated porous rocks are two-component media: a solid matrix whose connected pore space is occupied by a liquid (typically brine). In borehole acoustic logging — the measurement of wave velocities and attenuation along a borehole with a sonic tool — and in seismic exploration, compressional (PP) and transverse/shear (SS) wave data are used to infer porosity and fluid content. The classic empirical velocity-porosity relation is the Wyllie time-average equation (Wyllie, Gregory and Gardner, 1956): the measured PP-wave slowness (inverse velocity) is the porosity-weighted average of the slownesses of the solid matrix and the pore liquid,

(1−k)/Vpsol+k/Vliq=1/Vp,(1-k)/V_{p_{\mathrm{sol}}} + k/V_{\mathrm{liq}} = 1/V_p,

where kk is the porosity coefficient (pore volume fraction). It holds approximately for well-cemented, fully liquid-saturated rocks but is known empirically not to be universal.

The standard physical theory of wave propagation in fluid-saturated porous solids is Biot's theory (Biot, 1956), in which coupled motion of the frame and pore fluid with viscous (global) flow produces intrinsic attenuation; squirt-flow and mesoscopic-flow refinements (e.g., Dvorkin and Nur's 1993 unified Biot-squirt model) attribute attenuation to local fluid flow at pore-scale inhomogeneities. All of these are relaxation mechanisms: attenuation is accompanied by velocity dispersion, and causality — the Kramers-Kronig relations for linear dissipative media (Futterman, 1962) — links the two. The empirical fact that rock attenuation is often nearly frequency-independent over the seismic and logging bands (constant-QQ behavior, treated by Kjartansson's 1979 constant-QQ viscoelastic model, which carries weak logarithmic dispersion) has long been discussed (Mavko, Kjartansson and Winkler, 1979).

Against this background, Sidorov and Tarantin (2010) proposed a different kinematic picture. They rewrote the time-average equation as a conservation of phonon momentum,

hν/Vsol−hνk/Vsol+hνk/Vliq=hν/V,h\nu/V_{\mathrm{sol}} - h\nu k/V_{\mathrm{sol}} + h\nu k/V_{\mathrm{liq}} = h\nu/V,

where a phonon — normally the quantized vibrational excitation of a crystal lattice — is used here as a quasi-particle label for the energy quantum hνh\nu of the elastic wave, and the phonon velocities are taken equal to the wave velocities in the corresponding components. Because this relation holds for every frequency, the wave velocity is frequency-independent (no dispersion), while the two porosity-proportional terms are identified with dissipation, giving a damping coefficient

α=Bk(Vliq−1+Vsol−1)f\alpha = Bk(V_{\mathrm{liq}}^{-1}+V_{\mathrm{sol}}^{-1})f

exactly proportional to frequency ff; the damping parameter K=α/fK=\alpha/f and the logarithmic damping decrement (the natural-logarithmic amplitude loss per cycle, i.e., attenuation per wavelength) are then frequency-independent, which the authors offer as an intrinsic explanation of the observed constant-decrement behavior that relaxation models cannot produce exactly. Replacing kk by 2k2k (two phonons for a transverse wave) yields a shear-wave analogue (1−2k)/Vssol+2k/Vliq=1/Vs(1-2k)/V_{s_{\mathrm{sol}}}+2k/V_{\mathrm{liq}}=1/V_s. From laboratory measurements by Garanin on water-saturated organic glass with cylindrical holes, the constant was calibrated as B≈0.12B\approx 0.12 (from the slope G=ΔK/Δk=125 μs/mG=\Delta K/\Delta k=125\,\mu\mathrm{s/m} of KK versus kk), and the shear-wave equation was checked qualitatively through the agreement of porosity coefficients computed from PP- and SS-wave logging velocities in one carbonate borehole interval.

The mechanism was thus derived and tested only for fully liquid-saturated media satisfying the time-average equation, and its damping constant was calibrated on a single artificial porous medium. What therefore remains unresolved is: whether BB is universal across lithologies and pore structures; whether the phonon mechanism applies to porous media for which the time-average equation fails; the extension to heterogeneous media with complex pore geometry and partial saturation, and to anisotropic or fractured media; and the practical difficulty of measuring the damping coefficient in situ from borehole logging data, which is needed for direct verification.

Problem Statement

Determine the domain of validity — empirical and theoretical — of the phonon-momentum-conservation mechanism for elastic wave propagation in fluid-saturated porous rocks proposed by Sidorov and Tarantin (2010). The single overarching objective is to establish whether the mechanism validly describes wave propagation and damping outside the narrow setting in which it was derived and calibrated, and, if it does not, to identify where it fails. Concretely, an answer must settle the following jointly linked questions.

(1) The damping law and the constant BB. Establish whether, on natural lithologies and pore structures — not only the single water-saturated organic-glass model — the damping coefficient obeys α=Bk(Vliq−1+Vsol−1)f\alpha = Bk(V_{\mathrm{liq}}^{-1}+V_{\mathrm{sol}}^{-1})f with a single constant B≈0.12B\approx 0.12, or characterize how BB varies with lithology, pore geometry, and saturation state. The linear dependence of the damping parameter K=α/fK=\alpha/f on the porosity coefficient kk, and the proportionality of α\alpha to ff over the measured frequency band, must be tested, not assumed.

(2) The class of media. Establish whether the mechanism, which was derived only for fully liquid-saturated media satisfying the Wyllie time-average equation, extends consistently to porous media outside this class — heterogeneous media with complex pore geometries, partially saturated media, and anisotropic or fractured media — or fails there. All three listed media categories are in scope: the mechanism must be assessed for each, and a result for one category does not by itself determine the others.

(3) In-situ verification. Establish whether the damping coefficient can be recovered from borehole acoustic logging data with sufficient fidelity — given the need to separate intrinsic attenuation from scattering, geometric spreading, and tool effects — to verify or falsify the damping relations directly in rock.

References

  1. V. K. Sidorov and M. V. Tarantin, 'A possible mechanism for elastic wave propagation in porous rocks', Doklady Earth Sciences 434(1), 1253-1256 (2010), doi:10.1134/S1028334X10090242, https://doi.org/10.1134/S1028334X10090242
  2. V. K. Sidorov and M. V. Tarantin, 'Mechanism of Elastic Waves Reflection in Geological Media', International Journal of Geosciences 3(1), 175-178 (2012), doi:10.4236/IJG.2012.31019, https://doi.org/10.4236/IJG.2012.31019
  3. V. K. Sidorov and M. V. Tarantin, 'The origin of low-frequency maxima in the spectra of seismic noise', Izvestiya, Physics of the Solid Earth 49(1), 59-62 (2013), doi:10.1134/S106935131301014X, https://doi.org/10.1134/S106935131301014X
  4. V. K. Sidorov, M. V. Tarantin and N. L. Beltyukov, 'Relationship between the dependences of dielectric permittivity and conductivity of moistened rocks', Doklady Earth Sciences 465(1), 1172-1176 (2015), doi:10.1134/S1028334X15110148, https://doi.org/10.1134/S1028334X15110148
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  7. M. A. Biot, 'Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low-Frequency Range', The Journal of the Acoustical Society of America 28(2), 168-178 (1956), doi:10.1121/1.1908239, https://doi.org/10.1121/1.1908239
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  9. W. I. Futterman, 'Dispersive body waves', Journal of Geophysical Research 67(13), 5279-5291 (1962), doi:10.1029/JZ067i013p05279, https://doi.org/10.1029/JZ067i013p05279
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  11. G. Mavko, E. Kjartansson and K. Winkler, 'Seismic wave attenuation in rocks', Reviews of Geophysics 17(6), 1155-1164 (1979), doi:10.1029/RG017i006p01155, https://doi.org/10.1029/RG017i006p01155
  12. B. Gurevich and J. M. Carcione (eds.), 'Attenuation and Dispersion of Elastic Waves in Porous Rocks: Mechanisms and models', Society of Exploration Geophysicists (2022), ISBN 9781560803904, doi:10.1190/1.9781560803911, https://doi.org/10.1190/1.9781560803911
  13. V. A. Garanin, in: Prikladnaya geofizika (Applied Geophysics), No. 60, pp. 44-52 (Nedra, Moscow, 1970) [in Russian], no DOI