^{1}and Darrell R. Jackson

^{1}

### Abstract

While Biot theory can successfully account for the dispersion observed in sand sediments, the attenuation at high frequencies has been observed to increase more rapidly than Biot theory would predict. In an effort to account for this additional loss, perturbation theory is applied to Biot’s poroelastic equations to model the loss due to the scattering of energy from heterogeneities in the sediment. A general theory for propagation loss is developed and applied to a medium with a randomly varying frame bulk modulus. The theory predicts that these heterogeneities produce an overall softening of the medium as well as scattering of energy from the mean fast compressional wave into incoherent fast and slow compressional waves. This theory is applied to two poroelastic media: a weakly consolidated sand sediment and a consolidated sintered glass bead pack. The random variations in the frame modulus do not have significant effects on the propagation through the sand sediment but do play an important role in the propagation through the consolidated medium.

This work was supported by the Office of Naval Research.

I. INTRODUCTION

II. HETEROGENOUS BIOT’S POROELASTIC EQUATIONS

III. PERTURBATION THEORY

IV. GENERAL PROPERTIES OF COMPRESSIONAL SOUND SPEED AND ATTENUATION

A. Softening of the poroelastic medium

B. Limits of applicability

C. Causality

V. SOUND SPEED AND ATTENUATION FOR SPECIFIC CORRELATION FUNCTIONS

VI. DISCUSSION

VII. CONCLUSIONS

### Key Topics

- Wave attenuation
- 43.0
- Elastic moduli
- 31.0
- Speed of sound
- 24.0
- Sintering
- 13.0
- Correlation functions
- 11.0

## Figures

(a) Fast compressional sound speed and (b) attenuation determined from Eq. (41) with given by Eq. (54) as a function of the variance of the moduli variations normalized by . Both the sound speed and the attenuation have been normalized by their respective values when . These results were calculated for using the SAX99 sediment parameters in Table I, while the value of the mean bulk modulus was increased from for the solid line, to for the short-dashed line, and to for the long-dashed line.

(a) Fast compressional sound speed and (b) attenuation determined from Eq. (41) with given by Eq. (54) as a function of the variance of the moduli variations normalized by . Both the sound speed and the attenuation have been normalized by their respective values when . These results were calculated for using the SAX99 sediment parameters in Table I, while the value of the mean bulk modulus was increased from for the solid line, to for the short-dashed line, and to for the long-dashed line.

Values of for which remains equal to for the SAX99 sediment parameters (solid line) and the sintered glass bead parameters (dash-dot line) as is increased by three orders of magnitude.

Values of for which remains equal to for the SAX99 sediment parameters (solid line) and the sintered glass bead parameters (dash-dot line) as is increased by three orders of magnitude.

Examples of (a) sound speed ratio and (b) attenuation for the SAX99 sediment, assuming an exponential correlation for the frame bulk modulus variations. The correlation length is held fixed at , while the variance and the mean bulk modulus are increased. The variance is shown in the figures and the mean bulk modulus is (long-dashed line), (short-dashed line), and (long-short dashed line). In (a) the sound speed ratio is defined as the sound speed of the sediment normalized by the sound speed of the pore fluid. In (b) the lower set of curves is the contribution of the scattering from the fast compressional waves into the slow compressional waves to the attenuation of the mean fast compressional wave. The solid line in each figure is the prediction of Biot theory with no variation in the bulk modulus and .

Examples of (a) sound speed ratio and (b) attenuation for the SAX99 sediment, assuming an exponential correlation for the frame bulk modulus variations. The correlation length is held fixed at , while the variance and the mean bulk modulus are increased. The variance is shown in the figures and the mean bulk modulus is (long-dashed line), (short-dashed line), and (long-short dashed line). In (a) the sound speed ratio is defined as the sound speed of the sediment normalized by the sound speed of the pore fluid. In (b) the lower set of curves is the contribution of the scattering from the fast compressional waves into the slow compressional waves to the attenuation of the mean fast compressional wave. The solid line in each figure is the prediction of Biot theory with no variation in the bulk modulus and .

Examples of sound speed ratio and attenuation for the sintered glass beads, assuming an exponential correlation for the frame bulk modulus variations. In (a) and (b), the correlation length is held fixed at , while the variance and the mean bulk modulus are increased. The variance is shown in the figures and the mean bulk modulus is (long-dashed line), (short-dashed line), and (long-short dashed line). In (c) and (d), the variance and mean bulk modulus are held fixed at and . The solid line in each figure is the prediction of Biot theory for the sintered glass beads with no variation in the bulk modulus and . In (a) and (c) the sound speed ratio is defined as the sound speed of the sediment normalized by the sound speed of the pore fluid.

Examples of sound speed ratio and attenuation for the sintered glass beads, assuming an exponential correlation for the frame bulk modulus variations. In (a) and (b), the correlation length is held fixed at , while the variance and the mean bulk modulus are increased. The variance is shown in the figures and the mean bulk modulus is (long-dashed line), (short-dashed line), and (long-short dashed line). In (c) and (d), the variance and mean bulk modulus are held fixed at and . The solid line in each figure is the prediction of Biot theory for the sintered glass beads with no variation in the bulk modulus and . In (a) and (c) the sound speed ratio is defined as the sound speed of the sediment normalized by the sound speed of the pore fluid.

Examples of (a) sound speed ratio and (b) attenuation for the sintered glass bead pack, assuming a von Kármán function for the frame bulk modulus variations. The correlation length for each curve is held fixed at , the variance is , and the mean bulk modulus is . The Hurst coefficient is (long-dashed line), (short-dashed line), and (long-short dashed line). In (a) the sound speed ratio is defined as the sound speed of the sediment normalized by the sound speed of the pore fluid.

Examples of (a) sound speed ratio and (b) attenuation for the sintered glass bead pack, assuming a von Kármán function for the frame bulk modulus variations. The correlation length for each curve is held fixed at , the variance is , and the mean bulk modulus is . The Hurst coefficient is (long-dashed line), (short-dashed line), and (long-short dashed line). In (a) the sound speed ratio is defined as the sound speed of the sediment normalized by the sound speed of the pore fluid.

## Tables

Parameters used in calculating Biot results unless otherwise stated. The values in the column of parameters labeled “SAX99 Sand” correspond to the values in Table I of Ref. 6 for the best fit to the sound speed and attenuation measured during SAX99. Note that for the frame moduli, the imaginary parts have been removed and the moduli are assumed to be purely real. The values in the column of parameters labeled “Sintered Glass Beads” were determined from the parameters listed for the sintered glass bead sample B in Table III of Ref. 30.

Parameters used in calculating Biot results unless otherwise stated. The values in the column of parameters labeled “SAX99 Sand” correspond to the values in Table I of Ref. 6 for the best fit to the sound speed and attenuation measured during SAX99. Note that for the frame moduli, the imaginary parts have been removed and the moduli are assumed to be purely real. The values in the column of parameters labeled “Sintered Glass Beads” were determined from the parameters listed for the sintered glass bead sample B in Table III of Ref. 30.

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