Geophys. J . Int. (1997) 128, Fl-F4
FAST-TRACK PAPER
Dynamic-equivalent medium approach for thinly layered saturated
sediments
S. Gelinsky and S. A. Shapiro
Wave Inversion Technology Group, Geophysical Institute, Karlsruhe Unioersity, Hertzstrasse 16, D-76187 Karlsruhe, Germany
Accepted 1996 October 29. Received 1996 October 17; in original form 1996 July 31
SUMMARY
The phase velocity and the attenuation coefficient of compressional seismic waves,
propagating in poroelastic, fluid-saturated, laminated sediments, are computed analytically from first principles. The wavefield is found to be strongly affected by the medium
heterogeneity. Impedance fluctuations lead to poroelastic scattering; variations of the
layer compressibilities cause inter-layer flow (a 1-D macroscopic local flow). These
effects result in significant attenuation and dispersion of the seismic wavefield, even in
the surface seismic frequency range, 10-100 Hz. The various attenuation mechanisms
are found to be approximately additive, dominated by inter-layer flow at very low
frequencies. Elastic scattering is important over a broad frequency range from seismic
to sonic frequencies. Biot's global flow (the relative displacement of solid frame and
fluid) contributes mainly in the range of ultrasonic frequencies. From the seismic
frequency range up to ultrasonic frequencies, attenuation due to heterogeneity is
strongly enhanced compared to homogeneous Biot models. Simple analytical
expressions for the P-wave phase velocity and attenuation coefficient are presented as
functions of frequency and of statistical medium parameters (correlation lengths,
variances). These results automatically include different asymptotic approximations,
such as poroelastic Backus averaging in the quasi-static and the no-flow limits,
geometrical optics, and intermediate frequency ranges.
Key words: attenuation, layered media, permeability, porosity, sediments, seismicwave propagation.
INTRODUCTION
A seismic wavefield that propagates through randomly heterogeneous elastic media faces multiple scattering and is exponentially attenuated due to coherent backscattering even without
the presence of any dissipative mechanism. Additionally, in
porous saturated rocks a propagating wave causes fluid flow
that leads to an extra attenuation. In homogeneous media this
dissipation is caused by the relative movement between the
solid matrix and the fluid, the Biot global flow (Biot 1962).
Biot theory predicts the existence of two compressional waves
in porous saturated rock: a normal PI mode and the highly
attenuated P, mode, the so-called slow wave. In heterogeneous
media, seismic waves that propagate through a stack of layers
with variable compliances may additionally cause inter-layer
flow of pore fluid across interfaces from more compliant into
stiffer layers (White 1983; Norris 1993;Gurevich & Lopatnikov
1995). Below a critical frequency wo = kN/rpZ (parameters
explained below) in the quasi-static limit the fluid pressure is
equilibrated between adjacent layers due to viscous fluid
0 1997 RAS
motion across the layer boundaries [excitation of diffusive
Biot slow waves at the interfaces, Chandler & Johnson (1981)l.
Above w,, in the no-flow limit, the layers behave as if they are
isolated, since the (propagating) Biot slow wave is highly
attenuated for these frequencies and thus the fluid pressure is
no longer equilibrated.
The method outlined here is a poroelastic extension of the
generalized O'Doherty-Anstey formalism for elastic waves in
multilayered media, which was introduced by Shapiro &
Hubral (1996). It is limited to the low-frequency range of Biot
theory, with frequencies well below w, = qi/kef (parameters
explained below). It is also a small-perturbation approach,
which describes the heterogeneous medium statistically. The
conmedium parameters [represented as X = X,( 1 + E,&)),
sisting of background X , and fluctuations E ~ ( z ) ]are the
poroelastic constants, rock density e, porosity p, permeability
k, and the fluid properties: viscosity q, density ef, and bulk
modulus K,. It is assumed that the first moment of the
fluctuations ( E ) = 0, and that moments ( 8 " ) higher than the
second are small and thus can be neglected. The theory,
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S. Gelinsky and S. A. Shapiro
however, can work remarkably well even for rather strong
fluctuations of more than 20 per cent [the limiting values are
the same as in the elastic case and are discussed in Shapiro &
Hubral (1996)l. Fluctuations of permeability can be much
larger than 20 per cent. These fluctuations, howeve;, affect
only the global-flow attenuation and are important only for
frequencies of the order of w, or higher [permeability is
included in the theory via the Darcy coefficient 4 = y/k, which
appears with an extra frequency factor in eq. (2)].
The brackets denote averaging over the statistical ensemble,
i.e. over many realizations of the random medium under
consideration. In practice, there is only one set of logs providing
the parameters for the one medium that is studied, and spatial
averaging can be used to find the auto- and crosscorrelation
functions of the fluctuations. They can be written as the
product of a correlation function @ ( - [ / a ) and the variances
= 0) = a i Y ,and are defined by
&l)
+ 0)=
=<E~(z)E~(z
and by
Here, a is the correlation length of the fluctuations. A typical
stack of random layers is supposed to be thick enough to
cause self-averaging of the P-wave's vertical phase increment
and of its attenuation coefficient-the averaging is performed
in the process of wave propagation (references in Shapiro &
Hubral 1996).
THEORY
The starting point to describe layered poroelastic media is
the system of Biot's equations (Biot 1962). For vertically
propagating plane waves, these equations are transformed into
first-order differential matrix equations:
The wavefield parameters are contained in the vector
pf)T, and the matrix P, containing all
information about the medium, is given by
6 = (uz,w,,,z,
\ efw~
o
iwq
O
i
where P d = K , + 4 p d / 3 is the dry P-wave modulus; Kd and
pd are the bulk and shear moduli (throughout this paper,
the subscripts d,g,f denote properties of the dry frame, of
the grain material, and of the fluid, respectively); and
H = P d + a z M is the saturated P-wave modulus, with c( and
M defined by
The matrix P = Po+ P, is split into a homogeneous background
part Po, and a part P, that describes fluctuations. The
2, are the complex
eigenvalues of Po are ilz, and ilz, (t1,
wavenumbers of Biot's fast and slow compressional wave,
respectively). A Taylor expansion in the small fluctuations E ~
keeping only terms up to second order in E, yields the fluctuation matrix P,. Diagonalization of the matrix Po is achieved
by a linear transformation Pb = E;'PoEo. Here, Eo is the
eigenvector matrix of Po, and E;' its inverse. Next, P, is
replaced by Ph = E;' PEEo.The elements of Pi can be expressed
by eight known combinations of the medium fluctuations.
Finally, the wavefield vector is transformed, = EO'C. The
elements of = (d,, d,, u,, ti,)=, which is continuous across
interfaces, are for the homogeneous background medium the
displacements caused by down- and up-going waves of type 1
and 2, respectively. Thus eq. (1)is transformed to
<
<
dr
dz
-
+ (Pb + P:)<=
0.
(3)
In the following, it is assumed that the fluctuations and,
therefore, the elements of P: are small and that system ( 3 ) can
be solved by a small-perturbation expansion. This yields the
time-harmonic transmissivity T for a stack of layers with thickness
L as a function of the eigenvalues of Po and of the elements of
P:. For vertical incidence, T = exp[i(YL - w t ) - y L ] describes
the transmission of a compressional plane wave, propagating
through a stack of thin layers with a total thickness L and
which is inserted between two homogeneous half-spaces. The
P-wave phase velocity, V, = w / Y , and the attenuation
coefficient y can be calculated from T.
The next steps to derive the transmissivity are performed by
analogy with the purely elastic case (Shapiro & Hubral 1996).
However, now there is an interaction between the fast P, wave
and the strongly attenuated P, wave (due to the heterogeneity
of the layered medium) instead of the similar interaction
between P and SV waves for purely elastic heterogeneous
media. The following boundary conditions are considered:
d,(z = 0) = 1, d,(z = 0) = 0, U,(Z = L) = 0, U ~ ( Z= L)= 0. At
z=O, only a downgoing fast P, wave is given. In the depth
interval 0 < z < L, however, d,, u,, and u, are not equal to
zero because of the internal scattering processes caused by the
heterogeneity. Integrating eq. ( 3 ) , using the boundary conditions, and keeping only fluctuations of first order, the first
approximations for the down- and upgoing waves are obtained.
To calculate the transmissivity of the P, wave only the secondorder approximation of d, is needed. To obtain the phase
velocity and the attenuation coefficient of the dynamicequivalent medium, the logarithm of the transmissivity must
be analysed. Substitution of the first-order approximation
into the first equation of the transformed system eq. (3),
neglecting terms of order higher than O(E'), gives a solution
for the transmissivity. Using the assumption that the
thickness L of the stack of layers is much larger than the
wavelength and all correlation distances involved, the limits
Y = limL+m$&,[ln d,(L)]/L and y = -limL+m BB[ln d,(L)]/L
are evaluated for stationary random media. Because of the
self-averaging property, these limits are equal to their statistically averaged values. Thus it is possible to use properties of
ensemble averaging to express the attenuation coefficient and
the phase increment in terms of a restricted number of statistical
parameters of the medium fluctuations. Owing to the averaging,
random functions are replaced by their expectation values.
In the same way, the products of any two of these functions
0 1997 RAS, G J I 128, Fl-F4
,
Poroelastic multilayering: dynamic eflects
F3
are substituted by the respective auto- and crosscorrelation
functions. Since the medium fluctuations are stationary
functions of depth, their expectation values are independent of
depth and their correlation functions depend only on the depth
increment [ = z - z'. Performing the indi'cated calculations
gives both Y and y as functions of the auto- and crosscorrelations of the medium fluctuations:
The corresponding attenuation coefficient (integration of
eq. 5) reads
3
Our results are valid under the assumption of small
) higher, but they are
fluctuations, neglecting terms of O ( E ~and
valid for any relationship between the wavelength and the
correlation distance of the medium fluctuations and thus define
'dynamic-equivalent medium parameters'. Eqs (4) to (7) are
independent of any kind of low-frequency assumption, even
with respect to Riot's critical frequency [if a suitable Darcy
coefficient 9 is chosen, see Riot (1962)l.
Whereas the wavenumbers of Biot's fast and slow wave are
well known for all frequencies (see below), the new constants
A, B, and C are rather complicated functions of frequency and
of the numerous auto- and crosscorrelation functions. For a
better understanding of the implications of these results, we
introduce here an approximate solution, valid in the frequency
range below Biot's critical frequency w, = qqJ/kef (usually,
o,>> wo). Assuming that v = o / w , << 1, the small parameter v
can be used for a series expansions of the constants A, B, and
C. Since attenuation and dispersion due to inter-layer flow
and scattering are observable in most media for frequencies
w << wc, it is useful to neglect terms of O(v3)and higher (such
as permeability fluctuations) in the various series expansions.
Since for v << 1 it always holds that
Y = K:
+A -
d[@(-[/u)[\iiB exp(-jlc'-) cos
+ C exp(-2[~:)
([ K R
1
COS(~[K~)
.
--
(5)
In results (4) and (5) we introduced three new quantities, A,
B, and C (see eq. 8). They are combinations of the variances
of the medium fluctuations. The superscripts R and I define
the real and imaginary parts of the wavenumbers, for example
R, = K: k;. Also, R + = R2 El and R- = R, - R,. We assume
that there is only one correlation function @ ( - [ / a ) with
correlation length a for all fluctuations.
We choose an exponential correlation function, @([/a)=
exp(-[/a), and calculate the integrals (4) and ( 5 ) . Exponential
correlation functions of fluctuations of velocity and density,
and thus of the poroelastic constants, are often observed in
seismic practice (White, Sheng & Nair 1990). The integrations
(4) and (5) also can be performed for other correlation
functions (Gaussian, von Karman). As long as the correlation
function is rapidly decreasing for an increasing argument, any
kind of function yields basically the same low-frequency
behaviour for the inter-layer flow attenuation (yflowoc dS)
and
the scattering attenuation (y,,,,, x w 2, I-D Rayleigh scattering).
In the case of exponential media, the attenuation above wo is
found to be ynow cc w0.' and yScatt= const. On the other hand,
periodic media behave completely differently. Their correlation
function is not a rapidly vanishing function of its argument,
and inter-layer flow attenuation for low frequencies is found
to be yflowx w2.There is no scattering attenuation or dispersion
in periodic media, but pass and stop bands can be observed.
+
+
1 +2a4)
+-I + 4 aCa(
4 + 4U2(KF2 + K : Z )
(7)
we assume that R + = 2, and R - 2,. In addition, R, z (1 + i ) ~ ,
and R, z K::= K , . The wavenumbers K , , K~ are defined by
x1 = W ( Q / H ) " ~and K~ = ( w w , ~ ~ / ~ ~ J , NFor
) " ~ convenience,
.
A,
B, and C keep their names after being expanded in a truncated
series. These approximations were proposed first by Gurevich,
Zyrianov & Lopatnikov (1995). As a specific example, we
study a medium that exhibits fluctuations of Pd, of a, and of
its fluid saturation contained in M . For these fluctuations the
constants A, B, and C are calculated as outlined above and
read
PHASE VELOCITY A N D ATTENUATION
The result for the phase velocity for an exponential correlation
function (integration of eq. 4) reads
w
-=lCf+AVP
Ba[ 1 + ~ ( K +
R .'-)I
1 + 2UK' + U"KR' + K12)
+
+
-
Ba[ 1
1 2aK:
-
2Ca2K f
1 4aK: + 4a2(i$
+
U(K:
K'+
)]
+ U'(K": + KI:)
+
0 1997 RAS, G J I 128, Fl-F4
+Kf)
With these constants, keeping in mind the frequency dependence of K ~ C C W ,~ , x , , h it, is easy to see that the phase
velocity has three limiting values which may be called a quasistatic velocity V,, = W [ K , + A ] - ' for w --* 0, an intermediate
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S. Gelinsky and S. A. Shapiro
no-flow velocity V,, = w [ q + A - 2 8 / ~ , ] for w >> wo but wavelengths smaller than the correlation length a, and finally a
ray-theoretical velocity Kay = W [ K ’ A - 2B/q - C / ( ~ K ~ ) ] - ’
above that limit (but necessarily below Biot’s wJ.
For illustration, we calculated these limiting velocities
and the frequency-dependent phase velocity and attenuation
Q-’= y / ( 2 ~ , in
) a model consisting of a heterogeneous stack
of layers with average properties of water-saturated Berea
sandstone (Norris 1993). P-velocity fluctuations of almost 15
per cent and the existence of zones with partial gas saturation
(2 per cent gas) yield the following variances (constant density):
o : ~= 0.080,
uGM= 0.065, 02, = 0.004, o ; ~= -0.0018,
ohm= 0.0002, u:, = -0.018. The corresponding average values
are Po = 2.95 x 10” Pa, M , = 7.33 x lo9 Pa, and ct0 = 0.79. The
resulting phase velocity (eq. 6) is plotted in Fig. 1, and the
attenuation Q-’ (eq. 7) in Fig. 2, both as functions of frequency,
normalized by Biot’s critical frequency w, = 2x x 1.6 x lo5 s-’.
The correlation lengths (exponential correlation function) considered in Figs 1 and 2 range from 3 m down to 3 cm (curves
from left to right: a = 3 m, 1 m, 0.3 m, 0.1 m, and 0.03 m). For
the thickest layers (a = 3.0 m, 1.0 m) the inter-layer flow cannot
equilibrate the pressure at seismic frequencies-the maximum
of inter-layer flow attenuation is below 1 Hz. For the thinnest
+
Vp [ d s ] ,heterogeneous Berea sandstone
,
3720
layers (a = 0.1 m, 0.03 m) there is a continuous change from
the quasi-static to the ray-theoretical velocity limit. This highfrequency limit is slightly affected by global-flow dispersion.
For the larger correlation lengths, the peaks of scattering and
inter-layer flow can be distinguished. Attenuation due to Biot
global flow becomes relevant above w/w, z lo-’.
DISCUSSION A N D CONCLUSIONS
We determined analytically the attenuation and dispersion for
seismic waves in saturated layered sediments, applying a
generalization of the O’Doherty-Anstey formalism for poroelasticity. They depend on frequency and on fluctuations of
the poroelastic parameters, permeability, and of fluid properties
such as the bulk modulus, density and viscosity. The resulting
dynamic-equivalent medium model gives the P-wave phase
velocity and the attenuation coefficient in the full frequency
range from the quasi-static to the ray-theoretical limit, including
the no-flow regime for intermediate frequencies. For the first
time, the combined effects of scattering and fluid flow are
treated together from first principles. The attenuation in comparison to homogeneous systems is strongly enhanced. In
addition to Biot’s global flow, P waves are attenuated in
heterogeneous media due to scattering and inter-layer flow.
This study suggests that a greater part of the attenuation that
can be observed in porous saturated sedimentary rocks at
seismic frequencies is caused by scattering and local flow on
macroscopic length-scales defined by the layering. The interlayer flow effect in the seismic frequency range is dominant for
thin layers (correlation lengths of centimetres) with high permeability. Scattering contributes more significantly to the total
attenuation at seismic frequencies for heterogeneities on larger
scales (correlation lengths of several metres).
ACKNOWLEDGMENTS
-5
-4
-3
-2
-1
log(frequency / Biot-critical frequency)
-6
Figure 1. Phase velocity as a function of normalized frequency
(f,= 1.6 x lo5 Hz) for various correlation lengths. Vp was calculated
for a heterogeneous stack of layers with average properties of Berea
sandstone. The dashed lines show the limiting velocities: from top to
bottom, V,,,, Vnf, and V&. The solid lines were calculated for correlation
lengths a = 3.0 m, 1.0 m, 0.3 m, 0.1 m, 0.03 m (from upper left to lower
right). The arrows mark the seismic frequency range (15-150 Hz).
UQ, heterogeneous Berea sandstone
O’OI2
0.01
We acknowledge many helpful discussions with Boris Gurevich,
proof-reading by Ulrich Werner, and the financial support of
STATOIL (Norway) and the German BMBF in the framework
of the German-Norwegian Project, Phase 11.
REFERENCES
Biot, M., 1962. Mechanics of deformation and acoustic propagation
in porous media, J. appl. Phys., 3 3 , 1482-1498.
Chandler, R. & Johnson, D.L., 1981. The equivalence of quasi-static
flow in fluid saturated porous media and Biot’s slow wave in the
limit of zero frequency, J . appl. Phys., 52, 3391-3395.
Gurevich, B. & Lopatnikov, S.L., 1995. Velocity and attenuation of
elastic waves in finely layered porous rocks, Geophys. J. lnt.,
121, 933-947.
Gurevich, B., Zyrianov, V.B. & Lopatnikov, S.L., 1995. Seismic
attenuation in finely-layered porous rock: Cumulative effect of fluid
flow and scattering, in 65th Ann. lnt. Mtg. Soc. Expl. Geophys.,
Extended Abstracts, pp. 874-877.
Norris, A.N., 1993. Low-frequency dispersion and attenuation in
partially saturated rocks, J. acoust. Soc. Am., 94, 359-370.
Shapiro, S.A. & Hubral, P., 1996. Elastic waves in thinly layered
sediments: The equivalent medium and generalized ODohertyAnstey formulas, Geophysics, 61, 1282-1300.
White, J., 1983. Underground Sound, Application of Seismic Waves,
Elsevier, Amsterdam.
White, B., Sheng, P. & Nair, B., 1990. Localization and backscattering
spectrum of seismic waves in stratified lithology, Geophysics, 55,
1158-1 165.
1
0.008
0.006
0.004
0.002
0
-6
-5
-4
-3
-2
-1
log(frequency/ Biot-critical frequency)
Figure2. Reciprocal quality factor Q-’ for the same model as in
Fig. 1. From the left to the right peak the correlation lengths are
a = 3 . 0 m , 1.0m,0.3m,0.1m,0.03m.
0 1997 RAS, GJI 128, Fl-F4