Modeling of Acoustic Attenuation in Porous Media

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Modeling of Acoustic Attenuation in Porous Media

Steven Pride, Yder J. Masson

To cite this version:

Steven Pride, Yder J. Masson. Modeling of Acoustic Attenuation in Porous Media . IV European Conference on Computational Mechanics, May 2010, Paris, France. �hal-01654150�

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ECCM 2010

IV European Conference on Computational Mechanics Palais des Congrès, Paris, France, May 16-21, 2010

Modeling of Acoustic Attenuation in Porous Media

S.R. Pride1, Y. J. Masson2

1 _{Lawrence Berkeley National Laboratory, Berkeley, CA, USA, srpride@lbl.gov}

2 _{Dept. of Earth & Planetary Science, University of California at Berkeley, Berkeley, CA, yder_masson@berkeley.edu}

In recent years [1-3], the authors have been trying to develop a quantitative understanding of what seismic attenuation in rocks and sediments is primarily due to. Although many attenuation mechanisms have been proposed in the literature, only a few more recent models have both begun to predict attenuation levels that are consistent with field measurements in the seismic band (10 Hz to a few kHz) and been based on a realistic physical description of rocks. These models assume that rock contains “mesoscopic-scale” heterogeneity, which is defined as heterogeneity over scales larger than the grains, but smaller than the seismic wavelengths. When rocks containing mesoscopic structure are stressed, the more compliant parts of the rock respond with a larger fluid pressure change than do the stiffer parts, which results in fluid flow and wave attenuation.

One approach we have taken to better understand the mechanism of seismic attenuation in the presence of mesoscopic heterogeneity is to perform numerical simulations [2-3]. Computer-generated synthetic rock samples are created that have spatially variable poroelastic moduli and porous-continuum properties (e.g., porosity and permeability) distributed over the pixels. The numerical experiments consist of applying a time-varying stress to the sample surface, and measuring the resultant sample strain (defined as the average local strain throughout a sample). The Fourier transform of the stress and strain determine the sample’s complex frequency-dependent moduli, while the ratios of the imaginary and real parts of these moduli define the inverse quality factor 1/Q for the compressional and shear modes. The local response within such synthetic samples is obtained using finite-difference approximations of Biot’s poroelasticity equations. The finite-difference algorithm we employ [4-5] is an explicit time-stepping scheme on a staggered grid that can allow for the full-range of dynamics contained in Biot’s equations including the onset of viscous-boundary layers in the pores at high-enough frequency.

Using both numerical and analytical modeling, the level of attenuation in samples containing mesoscopic-scale heterogeneity is shown to be proportional to the square of the contrast in elastic moduli present. Stated differently, in materials possessing random fluctuations, the attenuation is proportional to the variance of the probability distribution used to randomly distribute the local elastic moduli. Further, when the heterogeneity is smoothly varying (i.e., not placed in the form of “patches” that are characterized by step function changes), the high frequency limit of seismic Q -1 (a measure of attenuation) is not the often assumed (and modeled) f -1/2 dependence, but is instead f -1 where f is wave frequency. Further, when the poroelastic moduli are distributed as self-affine fractals having a Hurst exponent H, both numerical results and analytical arguments demonstrate the scaling law Q(f) = f H. Last, in pure shear, it is demonstrated that contrasts in the bulk modulus can be responsible for local fluid pressure changes (some positive and some negative) that average to zero in pure shear, but that are responsible for a considerable amount of mesoscopic-scale flow and associated shear attenuation. In the laboratory, we have been measuring the mesoscale fluctuation in the elastic properties of actual rocks using a novel servo-controlled indentor. The device measures the fluctuations of the “surface” or “indention” modulus on the surface of a planar rock slab. We will present several maps of the

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mesoscale fluctuations in elastic moduli actually measured on real rock samples.

Last, in the special case of wave propagation through unconsolidated grain packs, there is experimentally observed to be more attenuation than can be explained using existing models such as those due to Biot (viscous flow between the peaks and troughs of a compressional wave) or those due to mesoscopic-scale fluctuations in the frame properties as discussed above. We have focused on rattler grains to explain the missing amount of attenuation. Rattlers are grains that have not been jammed into immobile contact with the surrounding grains. They typically occupy 10 to 15% of the grain pack at low confining stress, and become jammed as the stress levels on the pack are increased. When the grain pack is shook by a passing seismic wave, a rattler will experience relative motion between itself and the surrounding jammed grains. This creates an enhanced amount of shearing in the viscous fluid of the pores that will attenuate more energy than either flow at the macroscopic (wavelength) or mesoscopic patch scale. This mechanism has been analytically modeled over the past year and can explain the attenuation data on unconsolidated sediments.

References

[1] Pride, S.R., Berryman, J.G. and J.M. Harris, Seismic attenuation due to wave-induced flow. Journal of Geophysical Research, 109, B01201, 2004.

[2] Pride, S.R. and Y.J. Masson, Acoustic attenuation in self-affine porous structures. Physical

Review Letters, 97, 184301, 2006

[3] Masson, Y.J. and S.R. Pride, Poroelastic finite-difference modeling of seismic attenuation and dispersion due to mesoscopic-scale heterogeneity. Journal of Geophysical Research,

112, B03204, 2007

[4] Masson, Y.J., Pride, S.R., and K.T. Nihei, Finite difference modeling of Biot’s poroelastic equations at seismic frequencies. Journal of Geophysical Research, 111, B10305, 2006

[5] Masson, Y.J. and S.R. Pride, Finite-difference modeling of Biot’s poroelastic equations across all frequencies. Geophysics (in press), 2009.