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