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19 Velocity dispersion, attenuation, and dynamic permeability in heterogeneous poroelastic media

0 =0 ị 20 0 p2 ;

C8 ẳ 1 À 8p2 02 Y 2 ;

C9 ¼ 4Y 2 Z02 =20 ;

C7 ẳ Y 2 1 4p2 02 ị

C10 ¼ 1 À 2p2 02

141

3.12 Anisotropy, dispersion, and attenuation

For multimode propagation (P–SV), neglecting higher-order terms restricts the range

of applicable pathlengths L. This range is approximately given as

maxfl; ag < L < Y 2 maxfl; ag=2

where l is the wavelength, a is the correlation length of the medium, and s2 is the

variance of the fluctuations. The equations are valid for the whole frequency range,

and there is no restriction on the wavelength to correlation length ratio.

The angle- and frequency-dependent phase and group velocities are given by

o

1

cphase ẳ p ẳ p

2

2

2

kx ỵ kz

p þ kz2 =o2

c

group

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

s

2ﬃ

@o 2

@o

¼

þ

@kx

@kz

In the low- and high-frequency limits the phase and group velocities are the same.

The low-frequency limit for pressure waves in acoustic media is

h

i

freq

2

2

2

%

c

1

À

=2

cos

À

clow

0

fluid

and for elastic waves

freq

ẳ 0 1 AP =2ị

clow

P

freq

clow

ẳ 0 1 ASV =2ị

SV

freq

ẳ 0 1 ASH =2ị

clow

SH

These limits are the same as the result obtained from Backus averaging with

higher-order terms in the medium fluctuations neglected. The high-frequency limit

of phase and group velocities is

freq

chigh

ẳ c0 1 ỵ 2 =2 cos2

fluid

for fluids, and

chigh

P

freq

À

Á À

Â

ÁÃ

¼ 0 1 À 2 1 À 3p2 20 =2 = 1 À p2 20

À

Á À

Â

ÁÃ

freq

chigh

¼ 0 1 À 2

## The rock physics handbook, second edition

## 4 Tsvankin's extended Thomsen parameters for orthorhombic media

## 2 Phase, group, and energy velocities

## 4 Impedance, reflectivity, and transmissivity

## 5 Reflectivity and amplitude variations with offset (AVO) in isotropic media

## 11 Waves in layered media: stratigraphic filtering and velocity dispersion

## 12 Waves in layered media: frequency-dependent anisotropy, dispersion, and attenuation

## 7 Kuster and Toksöz formulation for effective moduli

## 10 Hudson's model for cracked media

## 4 Random spherical grain packings: contact models and effective moduli

## 3 Gassmann's relations: isotropic form

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19 Velocity dispersion, attenuation, and dynamic permeability in heterogeneous poroelastic media