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# Acoustic velocity dispersion and attenuation

As seismic acoustic waves pass through rock, some of their energy will be lost to heat. For tight, hard rocks, this loss can be negligible. However, for most sedimentary rocks, this loss will be significant, particularly on seismic scales. In reality, all rocks are inelastic to some degree. This article discusses the calculations to account for this energy loss.

## Attenuation

To account for the attenuation that occurs, we must rewrite our wave equation to include this energy or amplitude loss with depth, z. ....................(1)

where:

• A(z,t) is the amplitude at some point of depth and time
• A0 is the initial amplitude
• k* is the complex wave number (k* = k + l).

Note that here αl is a loss parameter, and not an aspect ratio. Therefore, we can rewrite Eq. 1 as ....................(2)

Another common measure is the loss decrement δ: ....................(3)

where the wavelength λ depends on the velocity V and frequency f: λ = V/f. However, the most common measure of attenuation is 1/Q. ....................(4)

## Relationship to velocity

One of the most straightforward descriptions of the relation of velocity to attenuation was developed by Cole and Cole and applied to attenuation measurements by Spencer. The Cole-Cole relationships involve both:

• Peak frequency or characteristic relaxation time, τ, for the attenuation mechanism
• Spread factor, b, which determines the distribution of relaxation times.

The real and imaginary components, B′ and B", of a general modulus, B = B′ + iB", are ....................(5) ....................(6)

where y = ln(ωτ), B0 and B are the zero and infinite frequency moduli.

This would lead to a general attenuation of ....................(7)

These relations connecting velocity and attenuation are plotted in Fig. 1. This figure represents losses and velocity dispersion (frequency dependence) caused by a single relaxation mechanism. At high frequencies, the material is unrelaxed and stiffer, and it has a higher velocity. At low frequencies, the material has time to relax, and velocities are lower.

## Impact of fluid mobility

Fluid mobility also influences rock inelastic properties. Most of the observed losses are caused by relative motion of fluid in the pore space. For a constant pore fluid type, permeability will control the motion and dissipation, thus making attenuation a permeability indicator. For variations in viscosities, mobility also will be dependent on frequency, and attenuation and dispersion may indicate fluid type.

Many models have been proposed, such as those of:

• Biot,
• O’Connell and Budiansky,
• Walsh,
• Dvorcik and Nur.

Unfortunately, the different mechanisms proposed often give contradictory results.

Wave attenuation and dispersion in vacuum dry rock is relatively negligible. Porous rocks containing fluids show a strong frequency-dependent attenuation. Variations in fluid properties such as modulus, viscosity, and polarity have a strong influence on 1/Q

These results indicate that the dominant 1/Q mechanism is the interaction and motion of fluid in the rock frame rather than intrinsic losses either in the frame or the fluids themselves. Squirt flow is believed to be the primary loss mechanism in consolidated rocks, although the inertial Biot mechanism may be important in highly permeable rocks (Vo-Thant, Yamamato et al.).

Fluid motion and pressure control velocity changes and seismic sensitivity to pore fluid types. One obvious factor is viscosity. The two most commonly used theoretical concepts are the inertial coupling of Biot and the squirt-flow mechanism (see, for example, O’Connell and Budiansky, or Dvorcik and Nur). Biot gives a characteristic frequency, ωc (roughly, the boundary between high and low frequency range) with the viscosity dependence, η, in the numerator: ....................(8)

Here, Φ is porosity, k is permeability, and ρ is fluid density. However, squirt-flow mechanisms lead to viscosity dependence in the denominator: ....................(9)

Here, K is frame modulus, and α is crack aspect ratio. These contrasting dependencies indicate that viscosity can be an obvious test to ascertain which theory is applicable.

Compressional (Vp) and shear (Vs) velocities for a sample of the Upper Fox Hills Sandstone (Heather well) are shown in Fig. 2. Several features should be noted. For the dry sample (open symbols), Vp and Vs show little frequency or temperature influence. This confirms that the primary dispersive and temperature effects are dependent on pore fluids. When saturated with glycerine, strong temperature and frequency dependence is obvious. Shear velocity is not independent of the fluid, but increases with increasing fluid viscosity, indicating a viscous contribution to the shear modulus. Vp increases with viscosity also. More importantly, the dispersion curve shows a systematic shift to lower frequencies with increasing velocities, consistent with squirt flow.

Attenuation (1/Q) and velocity dispersion are strongly dependent on pore phase and compressibility, particularly as controlled by partial gas saturation. Attenuation could become a valuable direct hydrocarbon indicator (e.g., Tanner and Sheriff). More recently, Klimentos used the ratio of compressional to shear attenuations as a hydrocarbon indicator in well logs. Unfortunately, application of these properties is not frequent because of incomplete understanding of the phenomena and lack of appropriate tools to extract the information. Laboratory measurements at frequencies and amplitudes encompassing the seismic range have confirmed the strong dependence on partial gas saturation (Fig. 3a). However, attenuation is decreased by confining pressure, dropping rapidly as pressure increases (Fig. 3b). Attenuation peaks will also depend on specific rock characteristics. Absorption peaks seen in one frequency band may not be apparent in others.

With the improving quality of seismic data, maps of the estimated attenuation are becoming a common displayed attribute. The relative values of 1/Q measured through time-lapse reservoir monitoring are becoming robust. As indicated in Fig. 3a, 1/Q will be sensitive to many of the common recovery processes.

## Nomenclature

 A(z,t) = wave amplitude with distance and time Bo = rock modulus, zero frequency, GPa or MPa Boo = rock modulus, infinite frequency, GPa or MPa B′ = rock modulus, real component, GPa or MPa B" = rock modulus, imaginary component, GPa or MPa αl = logarithmic decrement (loss), nepers/m ω = frequency (radian), s–1 (radians/s)