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1 Velocity-porosity models: critical porosity and Nur's modified Voigt average

143

3.12 Anisotropy, dispersion, and attenuation

"

SV

#

1

1

1

ẳ o a BSV 0ị

ỵ BBB 0ị

ỵ BDD 0ị

2

2

1 ỵ 4a2 lb

1 ỵ a2 l

1 ỵ a2 l2ỵ

2

SH ẳ o2 aBSH 0ị

1

1 ỵ 4a2 l2b

The shear-wave splitting for exponentially correlated randomly layered media is

(

2lb

low freq

2

ỵ oa 0 Y

ẵBSV 0ị BSH 0ị

So; pị % S

1 ỵ 4a2 l2b

)

lỵ

l

BDD 0ị ỵ

BBB 0ị

1 ỵ a2 l2ỵ

1 ỵ a2 l2À

These equations reveal the general feature that the anisotropy (change in velocity

with angle) depends on the frequency, and the dispersion (change in velocity with

frequency) depends on the angle. Stratigraphic filtering causes the transmitted amplitudes to decay as expðÀ

LÞ, where L is the path length and g is the attenuation

coefficient for the different wave modes as described in the equations above.

One-dimensional layered poroelastic medium

The small-perturbation statistical theory has been extended to one-dimensional

layered poroelastic media (Gurevich and Lopatnikov, 1995; Gelinsky and Shapiro,

1997b; Gelinsky et al., 1998). In addition to the attenuation due to multiple scattering

in random elastic media, waves in a random porous saturated media cause inter-layer

flow of pore fluids, leading to additional attenuation and velocity dispersion. The

constituent poroelastic layers are governed by the Biot equations (see Section 6.1),

and can support two P-waves, the fast and the slow P-wave. The poroelastic parameters of the random one-dimensional medium consist of a homogeneous background

(denoted by subscript 0) upon which is superposed a zero-mean fluctuation. The

fluctuations are characterized by their variance and a normalized spatial correlation

function B(r/a), such that B(0) ¼ 1, where a is the correlation length. All parameters

of the medium are assumed to have the same normalized correlation function and the

same correlation length, but can have different variances. The poroelastic material

parameters include:

f, porosity; r, saturated bulk rock density; , permeability; f ; fluid density; ,

fluid viscosity, and Kf, fluid bulk modulus. Pd ¼ Kd ỵ 43 d is the dry (drained) Pwave modulus, with Kd and md being the dry bulk and shear moduli, respectively.

¼ 1 À Kd =K0 is the Biot coefficient (note that in this section, a denotes the

Biot coefficient, not the P-wave velocity). K0 is the mineral bulk modulus;

M ẳ ẵ=Kf ỵ ị=K0 1 ; H ẳ Pd ỵ 2 M is the saturated P-wave modulus

144

Seismic wave propagation

(equivalent to Gassmann’s equation). N ¼ MPd =H; o0 ¼ N=a2 is the characteristic frequency separating inter-layer-flow and no-flow regimes. oc ¼ =f is the

Biot critical frequency.

Plane P-waves are assumed to be propagating vertically (along the z-direction)

normal to the stack of horizontal layers. The fast P-wavenumber and attenuation

coefficient g are given by (Gelinsky et al., 1998):

ẳ k1R ỵ A

Z 1

h p

I

R

Bz=aị 2D expzk

ị coszk

=4ị

0

i

p

I

R

ỵ 2D expzkỵ

ị coszkỵ

=4ị ỵ C exp2zk1I ị sin2zk1R ị dz

ẳ k1I

Z

1

ỵ

0

h p

I

R

Bz=aị 2D expzk

ị coszk

ỵ =4ị

i

p

I

R

ị coszkỵ

ỵ =4ị ỵ C exp2zk1I ị cos2zk1R ị dz

2D expzkỵ

where superscripts R and I denote real and imaginary parts; k~1 ẳ k1R ỵ ik1I

and k~2 ẳ k2R ỵ ik2I are the complex wavenumbers for the Biot fast and slow PR

I

waves in the homogeneous background medium; k~ỵ ẳ kỵ

ỵ ikỵ

ẳ k~2 ỵ k~1 ;

R

I

~

~

~

k ẳ k ỵ ikÀ ¼ k2 À k1 . The quantities A, C, and D involve complicated functions

of frequency and linear combinations of the variances and covariances of the medium

fluctuations. Approximations for these quantities are discussed below. The above

expressions assume small fluctuations in the poroelastic parameters but are not limited

by any restriction on the relation between wavelength and the correlation length of the

medium fluctuations. The phase velocity for the fast P-wave is given as VP ¼ o= .

For a medium with an exponential correlation function with correlation length a, the

expressions after carrying out the integrations are (Gelinsky et al., 1998):

R

I

Da 1 ỵ ak

ỵ k

ị

o

R

ẳ k1 ỵ A

I ỵ a2 ẵkR ị2 ỵ kI ị2

VP

1 ỵ 2ak

R

I

Da 1 ỵ akỵ

ỵ kỵ

ị

2

2Ca2 k1R

2

I þ a2 ðk

I

1 þ 2akþ

1 þ 4ak1I þ 4a2 ðk1R þ k1I Þ

þ þ kþ Þ

Â

Ã

R

I

Da 1 À aðkÀ

À kÀ

Þ

I

ẳ k1 ỵ

I ỵ a2 kR2 ỵ kI2 ị

1 ỵ 2ak

R

I

Da 1 akỵ

kỵ

ị

Ca1 ỵ 2ak1I ị

ỵ

ỵ

2

2

2

I ỵ a2 k R2 ỵ k I ị

1 ỵ 4ak1I ỵ 4a2 k1R ỵ k1I ị

1 ỵ 2akỵ

ỵ

ỵ

R2

2

145

3.12 Anisotropy, dispersion, and attenuation

Gelinsky et al. (1998) introduce approximate expressions for A, C, and D, valid

in the frequency range below Biots critical frequency. Other

approximations

p

~ỵ % k~ % k~2 ; k~2 % 1 ỵ iịk2 ; k~1 % kR ¼ k1 ; k1 ¼ o =H ; and k2 ẳ

used

are:

k

1

p

ooc f =2Nị:

o

Aẳ

2

!

r

40 M02 2

Pd0 2

20 M0 2

2

ỵ

MM ỵ 2P ỵ 2

Pd0

H0 H0 PP

Pd0

2D o

¼

k2

2

"

rﬃﬃﬃﬃﬃﬃ

Á

Pd0 20 M0 2

Pd0 À 20 M0 2

2

2

2

ỵ

2

P 2M

PP

PM

MM

2

Pd0

H0 H0

#

2

Pd0 20 M0

2

ỵ

P2d0

C

o

ẳ

2k1 8

!

r 2

Á 40 M02 À 2

Á

Pd0 2

20 M0 À 2

2

2

2

þ2

PM þ 2P þ 2 MM þ 4 þ 4M

Pd0

H0 H02 PP

Pd0

The phase velocity has three limiting values: a quasi-static value for o ( o0 and

o ! 0, Vqs ¼ oẵk1 ỵ A1 ; an intermediate no-flow velocity for o ) o0 , Vnf ẳ

oẵk1 ỵ A 2D=k2 1 , and a ray-theoretical limit Vray ẳ oẵk1 ỵ A À 2D=k2 À C=

ð2k1 ÞÀ1 . The quasi-static limit is equivalent to the poroelastic Backus average (see

Section 4.15). The inter-layer flow effect contributes significantly to the total attenuation in the seismic frequency range for highly permeable thin layers with correlation

lengths of a few centimeters.

Uses

The equations described in this section can be used to estimate velocity dispersion

and frequency-dependent anisotropy for plane-wave propagation at any angle in

randomly layered, one-dimensional media. They can also be used to apply angledependent amplitude corrections to correct for the effect of stratigraphic filtering in

amplitude-versus-offset (AVO) modeling of a target horizon below a thinly layered

overburden. The corrected amplitudes are obtained by multiplying the transmissivity

by expð

LÞ for the down-going and up-going ray paths (Widmaier et al., 1996). The

equations for poroelastic media can be used to compute velocity dispersion and

attenuation in heterogeneous, fluid-saturated one-dimensional porous media.

Assumptions and limitations

The results described in this section are based on the following assumptions:

layers are isotropic, linear elastic or poroelastic with no lateral variation;

the layered medium is statistically stationary with small fluctuations (<30%) in the

material properties;

## 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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1 Velocity-porosity models: critical porosity and Nur's modified Voigt average