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Attenuation and velocity dispersion are leading edge subjects for research of seismic rock physics. In reality, there are various dissipative mechanisms contributing to seismic reflection reflection degration, including anelastic attenuation, geometrical spreading, spherical divergence, transmission loss, mode conversions, intrabed multiples, refractions, and scattering. To interest of geophysical exploration for petroleum and gas, much work focuses on inelastic attenuation and dispersion of body waves (P- and S-waves) resulting from the presence of fluids in the pore space of rocks[1]. Researches show that seismic attenuation is more sensitive to fluid related parameters like saturation, porosity, permeability and viscosity than velocity.

It is commonly accepted that the presence of fluids in the pore space of rocks causes attenuation and dispersion by the mechanism broadly known as wave-induced fluid flow (WIFF). Rock physics theories based on WIFF have developed into 3 categories according to research scale: macroscopic mechanism based on Biot theory, mesoscopic mechanism based on White model and microscopic mechanism based on squirt model. Three models apply to attenuation and dispersion in, respectively, homogeneous saturated rocks with high permeability and surrounding pressure within supersonic frequency band, patchy saturated rock within seismic frequency band and poorly permeable homogeneous saturated rock with low pressure. Mesoscopic flow is a significant mechanism of fluid-related attenuation in the seismic-exploration band[2].

In fact, all three dissipation mechanisms are all related to fluid viscosity. Despite this common feature, there is as yet no unified theoretical model of all these mechanisms[3]. This is because attenuation mechanisms at different scales are dependent on different factors. For example, attenuation model at microscopic scale cannot explain attenuation and velocity dispersion in seismic frequency band. Yet in geophysical exploration, geophysical data of different frequency bands (surface seismic and VSP are 101~102Hz, sonic log is 103~104Hz, and laboratory supersonic measurement is over105Hz) for joint utilization have to match each other. Therefore it is necessary to expand rock physics model of single frequency band to full frequency band, and integrate multi-scale geophysical data at one consistent scale. This is fundamental for horizon calibration, logging constrained seismic inversion, seismic interpretation and attribute analysis. So my initial inspiration is to construct a full-band attenuation model.

Literature Review

Dvorkin and Nur (1993) and Dvorkin et al. (1994) introduced a unified Biot–squirt (BISQ) model. The BISQ formulas can be used to calculate partially saturated rock velocities and attenuation (at high pressure) at any frequency[4]. It combines global flow and local flow, while still failed to account for attenuation in seismic band and not consistent with Gassmann predictions in the low-frequency limit. Dvorkin et al. (1995) extend Mavko-Jizba squirt relations for all frequencies to calculate these moduli, velocities, and attenuations at any intermediate frequency[5]. Chapman et al. (2002) presented a squirt or local flow model that considers frequency-dependent, wave-induced exchange of fluids between pores and cracks, as well as between cracks of different orientations[6]. The formulas can be used to calculate squirt-related velocity dispersion and attenuation at any frequency. Generally speaking, most theoretical models of squirt-flow attenuation focus on poro structure but cannot explain attenuation in seismic band.

Gurevich et al., (2010) proposed a new model of squirt-flow attenuation that uses a pressure relaxation approach[7]. Moduli are expressed in terms of frequency and pressure. The resulting model is consistent with the Gassmann and Mavko–Jizba equations at low and high frequencies, respectively. It can also be naturally incorporated into Biot’s theory of poroelasticity to obtain velocity and attenuation prediction in a broad frequency range. Here I find some frequency-dependent parameters may bridge the gap between models at different scales.

Tang et al.(2012) incorporated microscale squirt in cracks into the classic Biot poroelastic wave theory, and adopted crack density and aspect ratio parameters combine the two types of theories to form a unified theory for modeling wave propagation through a cracked porous medium[8]. The frequency-dependent dynamic permeability bridged the gap again. One advantage of the work is that, by using specific pore-crack geometry, the relaxation parameter is no longer a necessary parameter because it can be expressed by the crack geometry parameter or aspect ratio. So I think it is promising to consider pore and crack geometry to extend established attenuation model from certain frequency band to full frequency. In practice, Wang and Sun (2010) extended Kuster-Toksöz model from very high frequency to full frequency, realizing the full-frequency band velocity modeling, based on Chapman’s (2002) crack-pore microstructure model. Furthermore, they proposed a full-frequency velocity prediction model by coupling differential effective medium(DEM) model with a dispersion theoretical model (microstructure model)[9].

Proposed Research

As we can see above, quite a few full-frequency attenuation models are based on microscopic mechanism. Yet to better analyze attenuation in seismic band, mesoscopic mechanism is properly a better alternative. For instance, Vogelaar et al.(2007) expanded the low-frequency layered White model to the full frequency range[10], using dynamic permeability[11](the dynamic permeability of Muller and Gurevich[12], which is related to WIFF at frequencies much lower than the Biot characteristic frequency, is different from the dynamic permeability of Johnson et al. (1987), which is frequency-dependent permeability arising from inertial effects at frequencies of the order of Biot’s characteristic frequency). This work provides an idea to set up full frequency model based on White model or WIFF theory. For instance, it is likely to expand the spherical White model to full frequency range.

Since there are few unified theories and models to connect attenuation mechanisms of microscale, mesoscale and macroscale, my proposed project is mesocopic flow based multi-scale attenuation model incorporating effective medium theory. The hypothetical model can obtain module, attenuation, dispersion and other information at any intermediate frequency from fluid effects, and by taking poro and crack texture (stiff porosity, crack porosity, crack aspect ratio and etc.) into consideration the information is more accurate. Work has to done to overcome high frequency limitation of effective medium theory. Actually Guverich and Makarynska (2012) extend Hashin-Shtrikman(HS) bounds to quasi-static moduli of composite viscoelastic media like porous medium, a mixture of an elastic solid and a linear Newtonian fluid[3]. In my opinion, the work took a trial to combination of effective medium theory and poroelastic theory. But the bounds are quasi-static, the frequency must be much lower than Biot’s characteristic frequency, and these bounds account for viscous shear relaxation and squirt-flow dispersion instead of Biot’s global flow dispersion.

My hypothetical model can be used for numerical simulation of practical seismic wave field, velocity dispersion correction formula, attenuation compensation seismic inversion and wavelet extraction.

Methodology and Procedures

Inspired by [7, 10], the first step is to modify existent attenuation models or set up new full-frequency models, starting from mesoscopic mechanism. Frequency-dependent dynamic permeability, moduli and fluid pressure are all feasible intermediaries between different frequency bands. A practicable way is to express these parameters in each scale at first, and find a unified formula to connect these bands.

Secondly, inspired by the full-frequency velocity prediction model[9], I can combine certain model of effective medium theory with mesoscopic mechanism at first, since mesoscopic mechanism is more capable to explain seismic attenuation and dispersion. For example, Endres et al.(1997) has showed a direction in a way, although the inclusion-based model are irrelevant to attenuation[13]. Modification can be to rewrite the fluid pressure pf as frequency-dependent form, and thus integrate pore geometry into attenuation mechanism due to WIFF.

The next step is to extend the poro structure incorporated mesocopic mechanism to full band, with the experience of full-frequency attenuation model in the first step. We have to deal with several problems: How does fluid in given the poro and crack structure flow at each scale? How can we apply effective medium theory to high frequency?

Proper rock physics experiments can provide intuitive guidance. It is beneficial to measure compressional velocity, shear velocity, elastic moduli and other parameters in different environments, so as to gradually theoretical models and enhance fitting precision.


  1. [1] Müller, Tobias M., Boris Gurevich, and Maxim Lebedev. "Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks—A review." Geophysics 75, no. 5 (2010): 75A147-75A164.
  2. [2] Pride, Steven R., James G. Berryman, and Jerry M. Harris. "Seismic attenuation due to wave‐induced flow." Journal of Geophysical Research: Solid Earth 109, no. B1 (2004).
  3. [3] Gurevich, Boris, and Dina Makarynska. "Rigorous bounds for seismic dispersion and attenuation due to wave-induced fluid flow in porous rocks." Geophysics 77, no. 6 (2012): L45-L51.
  4. [4] Mavko, Gary, Tapan Mukerji, and Jack Dvorkin. The rock physics handbook: Tools for seismic analysis of porous media. Cambridge university press, 2009.
  5. [5] Dvorkin, Jack, Gary Mavko, and Amos Nur. "Squirt flow in fully saturated rocks." Geophysics 60, no. 1 (1995): 97-107.
  6. [6] Chapman, Mark, Sergei V. Zatsepin, and Stuart Crampin. "Derivation of a microstructural poroelastic model." Geophysical Journal International 151, no. 2 (2002): 427-451.
  7. [7] Gurevich, Boris, Dina Makarynska, Osni Bastos de Paula, and Marina Pervukhina. "A simple model for squirt-flow dispersion and attenuation in fluid-saturated granular rocks." Geophysics 75, no. 6 (2010): N109-N120.
  8. [8] Tang, Xiao-Ming, Xue-lian Chen, and Xiao-kai Xu. "A cracked porous medium elastic wave theory and its application to interpreting acoustic data from tight formations." Geophysics 77, no. 6 (2012): D245-D252.
  9. [9] Wang, H., Z. Sun, and Y. Xiao. "Full-frequency Band Velocity Prediction Model and Its Application on Seismic Reservoir Prediction." In 73rd EAGE Conference and Exhibition incorporating SPE EUROPEC 2011. 2011.
  10. [10] Vogelaar, Bouko, and David Smeulders. "Extension of White's layered model to the full frequency range." Geophysical Prospecting 55, no. 5 (2007): 685-695.
  11. [11] Johnson, David Linton, Joel Koplik, and Roger Dashen. "Theory of dynamic permeability and tortuosity in fluid-saturated porous media." Journal of fluid mechanics 176 (1987): 379-402.
  12. [12] Müller, Tobias M., and Boris Gurevich. "Effective hydraulic conductivity and diffusivity of randomly heterogeneous porous solids with compressible constituents." Applied physics letters 88, no. 12 (2006): 121924.
  13. [13] Endres, Anthony L., and Rosemary J. Knight. "Incorporating pore geometry and fluid pressure communication into modeling the elastic behavior of porous rocks." Geophysics 62, no. 1 (1997): 106-117.