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

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

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0 = 0 ị 2 0 0 p2 ;

C8 ẳ 1 À 8p2 02 Y 2 ;



C9 ¼ 4Y 2 Z 02 = 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

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

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