Geophys. J. Int. (1998) 133, 91–103
Scattering of a compressional wave in a poroelastic medium by an
ellipsoidal inclusion
B. Gurevich,1,* A. P. Sadovnichaja,1 S. L. Lopatnikov1,† and S. A. Shapiro2,3
1 Institute of Geosystems, Varshavskoe Shosse 8, Moscow 113105, Russia
2 Wave Inversion T echnology Group, Geophysical Institute, Karlsruhe University, Hertzstrasse 16, D-76187 Karlsruhe, Germany
3 Nancy School of Geology (ENSG/INPL) & CRPG, Batiment G, Computer Science Department, Rue du Doyen-Marcel-Roubault BP 40,
54501 Vandoeuvre L es Nancy, France
Accepted 1997 November 3. Received 1997 October 31; in original form 1997 July 4
SU MM A RY
We study the interaction of a plane elastic wave in a poroelastic medium with an
elliptical heterogeneity of another porous material. The behaviour of both the inclusion
and the host medium is described by Biot’s equations of poroelasticity with the
standard interface conditions of Deresiewicz and Skalak at the inclusion’s surface. The
scattering problem is studied in the Born approximation, which is valid for low contrast
of the inclusion’s properties with respect to the host medium. The resulting scattered
wavefield consists of the scattered normal compressional and shear waves and a Biot
slow wave, which attenuates rapidly with distance from the inclusion. The Born
approximation also allows us to derive explicit analytical formulae for the amplitudes
of these scattered waves and to compute the amount of energy scattered by the
inclusion into these waves. The amplitude and scattering cross-section for the Biot
slow wave depend on the relationship between the dimensions of the inclusion and the
wavelength of the Biot slow wave.
The analytical results for a single inclusion are used to estimate the effective
attenuation of a normal compressional wave in a poroelastic medium with randomly
distributed ellipsoidal inclusions. The effective attenuation due to the elastic scattering
of energy by the inclusions is compounded by an additional attenuation caused by
poroelasticity, i.e. by the scattering of the incident normal compressional wave into the
Biot slow wave. The frequency dependence of this so-called mode conversion attenuation
has the form of a relaxation peak, with the maximum of the dimensionless attenuation
(inverse quality factor) at a frequency at which the wavelength of the Biot slow wave is
approximately equal to the characteristic size of the inclusion. The width and the precise
shape of this relaxation peak depend on the aspect ratio of the ellipsoidal inclusion.
Physically, the mode conversion attenuation is associated with the wave-induced
flow of the pore fluid across the interfaces between the host medium and the inclusions.
The results of our study demonstrate how the local flow (or squirt) attenuation can be
effectively modelled within the context of the Biot theory of poroelasticity.
Key words: attenuation, cracked media, permeability, porosity, scattering, wave
propagation.
I NT R O DU C TI O N
The problems of elastic wave propagation in fluid-saturated
porous media have attracted increasing interest in recent
* Now at: Geophysical Institute of Israel, PO Box 2286, Holon 58122,
Israel. E-mail: [email protected].
† Now at: Chemical Faculty, Moscow State University, Vorobuovy
Gory, Moscow 119899, Russia.
© 1998 RAS
years. This interest is related to the demands of various applications as well as to certain specific features of the mechanical
properties of such media. As Biot’s theory of poroelasticity
shows, in macroscopically homogeneous poroelastic media, in
addition to normal compressional and shear waves, there exists
a so-called Biot slow wave (type II compressional wave), which
has a filtration nature and is characterized by a diffusion-type
dispersion equation. At low (seismic) frequencies this wave has
very high attenuation and, because of this, exists only near
91
92
B. Gurevich et al.
sources or in the vicinity of interfaces at which mode conversion
may occur. This phenomenon may result in specific features
of wave propagation in piecewise homogeneous saturated
porous media. In recent years these phenomena have been the
subject of intense studies (White 1983; Bourbie, Coussy &
Zinszner 1987; Coussy 1995).
One of the problems in which specific interface effects may
prove essential to include is the problem of scattering of a
normal compressional (or shear) elastic wave by an inclusion
of another poroelastic material. Such phenomena are accompanied by a loss of energy of the incident wave owing to the
mode conversion from the incident wave into the slow wave.
If there are many chaotically distributed heterogeneities in
the medium, this mode conversion may result in significant
attenuation of the incident (passing) normal compressional
wave. The idea of this slow-mode conversion mechanism of
attenuation, which is also known as the ‘transformational
mechanism’, was proposed by S. L. Lopatnikov (Lopatnikov
& Gurevich 1986, 1988; Gorbachev, Gurevich & Lopatnikov
1990; Gurevich & Lopatnikov 1995). Results of similar studies
for fluid-filled cavities (Yumatov & Markov 1984) as well as
analysis of the analogous 1-D problems (White, Mikhailova
& Lyakhovitsky 1975; Norris 1993; Gurevich, Zyrianov &
Lopatnikov 1997; Gelinsky & Shapiro 1997) show that
such phenomena may significantly affect the magnitude and
frequency dependence of passing and scattered waves.
Here we perform a theoretical study of the interaction of a
plane compressional elastic wave in a fluid-saturated porous
medium with an isolated ellipsoidal inclusion of another porous
material. We study both the amplitudes of scattered waves and
their energy characteristics. We put particular emphasis on the
difference of the scattering phenomenon in poroelasticity from
the classical elastic case (Gubernatis, Domany & Krumhansl
1977a; Gubernatis et al. 1977b; Truell, Elbaum & Chick 1969;
Yamakawa 1962; Korneev & Johnson 1993a,b). From this
point of view, we are primarily interested in the behaviour of
the scattering cross-section from the incident wave into the
slow wave, which is absent in elastic media. By studying the
effect of the medium porosity and permeability on the scattered
field we shall be able to concentrate on the specific features of
the scattering phenomenon in poroelastic media.
In the course of our study, we shall confine ourselves to low
frequencies (in terms of Biot theory), where the Biot slow wave
is characterized by diffusion-type behaviour. At higher frequencies the Biot slow wave behaves similarly to any other
propagating wave, and thus scattering at such frequencies does
not differ so drastically from the classical elastic case (Biot
1956b; Bourbie et al. 1987).
In solving the scattering problem, we follow the technique
used by Gubernatis et al. (1977b) in their treatment of the
elastic scattering problem. The key point of the method of
Gubernatis et al. is the use of an integral representation, which
relates the displacement field outside the inclusion to the
incident field and the field inside the inclusion. This integral
representation is constructed from the dynamic equations
for homogeneous elastic media and boundary conditions at
interfaces between elastic materials. For porous media such
integral representations have been considered by a number
of authors (Sadovnichaja et al. 1990; Manolis & Beskos
1989; Norris 1985; Boutin, Bonnet & Bard 1987). Such
representations are derived using the Biot equations of poroelasticity as dynamic equations for a homogeneous material
and the boundary conditions of Deresiewicz & Skalak (1963)
as interface conditions.
As in the classical elastic case, an exact solution of the
scattering problem for porous media is not feasible. Following
the approach of Gubernatis et al. (1977b), we tackle the
problem for porous media with the so-called Born approximation, which allows us to obtain explicit analytical results
for moderate frequencies and for inclusions with small relative
contrast of properties (with respect to the host medium)
(Gurevich et al. 1992). The results are obtained for a wide
range of frequencies and sizes of inclusions. For the spherical
case at very small frequencies, the results of our study show
good agreement with the exact solution (Berryman 1985).
FO R M U LAT IO N OF PR O B LEM
Consider a homogeneous fluid-saturated isotropic porous
medium described by a Biot system of linear differential
equations and containing an inclusion of another porous
material described by the same equations but with different
coefficients. The host medium consists of solid grains characterized by the Lamé constants l and m , bulk modulus
s
s
K =l +2 m and density r , and a pore fluid with bulk
s
s
s 3 s
modulus K , dynamic viscosity g and density r . The elastic
f
f
grains form an elastic matrix, which is characterized by porosity
w, permeability k, Lamé constants l and m and bulk modulus
K=l+2 m. For time-harmonic compressional waves of fre3
quency v=2pf in the host medium, the Biot equations of
poroelasticity can be written in the following form (Biot 1962):
VΩt+v2(ru+r w)=0 ,
f
giv
w=0 .
−Vp+v2(r u+qw)+
f
k
(1a)
(1b)
Here u is the average solid displacement; w is the average
fluid displacement relative to the solid, which is related to the
absolute fluid displacement U by
w=w( U−u) ;
(2)
t is the stress tensor in the porous continuum; p is the pore
fluid pressure; the symbol V is the Hamilton operator; r is the
average density of the saturated porous medium,
r=wr +(1−w)r ;
(3)
f
s
and q=mr /w, where m is the tortuosity coefficient, which
f
depends on the geometry of the pore space. The fluid pressure
and the components t of the stress tensor are related to the
ij
components u and w of the solid and fluid displacement
i
j
vectors by
t =m(u +u )+d (l u +Cw ) ,
(4a)
ij
i,j
j,i
ij c k,k
k,k
−p=Cu +Mw .
(4b)
k,k
k,k
The coefficients in eqs (4) are related to the material properties
by
l =l+bC ,
c
C=bM ,
NA
(5a)
(5b)
B
b−w
w
,
(5c)
+
K
K
s
f
b=1−K/K .
(5d)
s
Here and below the time-harmonic factor e−ivt is implicit.
M=1
© 1998 RAS, GJI 133, 91–103
Scattering in a poroelastic medium
Seeking the solution of eqs (1) in the form of a plane wave
propagating along the x-axis, (u, w)=(u , w ) exp(ikx), we can
0 0
obtain a standard Biot dispersion equation for plane compressional waves in a poroelastic medium (Biot 1956a,b;
Bourbie et al. 1987). This equation has two different roots,
corresponding to the normal (fast) and type II (slow) compressional waves. The existence of the compressional waves of
the second kind is a unique feature of Biot’s theory. At high
(ultrasonic) frequencies this wave behaves as a wave in the
pore fluid and propagates with a velocity independent of the
elastic properties of the rock matrix. On the other hand, at
low (seismic) frequencies the Biot slow wave is actually a
filtration wave, or a kind of a diffusion process. At these
frequencies it propagates with a very slow velocity and
an attenuation so large, that it cannot be observed in any
imaginable seismic experiment. Despite this fact, the slow wave
contributes to the energy balance in a poroelastic medium,
and thus has to be taken into account for correct modelling
of wave propagation in such media.
At low frequencies that obey the condition
v%v ,
(6)
c
where v is a so-called Biot characteristic frequency,
c
v =gw/kr ,
(7)
c
f
the wavenumbers k and k of the fast and slow compressional
1
2
waves may be expressed in the form
AB
AB A B
r 1/2
v
+ia = (1+iQ−1 /2) ,
k =v/c +ia =v
B c
B
1
1
B
H
1
q 1/2
ivg 1/2
=
,
k =v
2
N
kN
(8a)
(8b)
where
c =(H/r)1/2
(9a)
1
is the velocity of the normal (fast) compressional wave, and
a =vQ−1 /2c
B
B
1
is its amplitude attenuation coefficient. Here
H=l +2m
c
and
N=(MH−C2)/H=
(9b)
93
to be negligible and its wavenumber k to be a real number,
1
r 1/2
k #v/c =v
.
(12)
1
1
H
AB
The equations given above describe the wave dynamics of
the host medium. We will further assume that the porous
inclusion is described by the same equations but with different
coefficients, which we will denote by primed letters, w∞, k∞, l∞ ,
c
etc.
At the interface between the host medium and the inclusion
we assume the fulfilment of the interface conditions of
Deresiewicz & Skalak (1963), i.e. the continuity of the following
field quantities:
(1)
(2)
(3)
(4)
total normal and tangential stresses;
pore fluid pressure;
solid displacement vector;
normal component of the relative fluid displacement.
Let the origin of a Cartesian coordinate system Ox x x be
1 2 3
situated inside the inclusion (Fig. 1). Consider a plane, timeharmonic, fast compressional wave of frequency v propagating
parallel to the Ox axis with the displacement given by
1
u0=exp(ik x ) .
(13)
1 1
The factor exp(−ivt) is implicit. We seek to determine the
displacements fields us, ws created as a result of the interaction
of the incident wave u0 with the inclusion, at a large distance
r from the inclusion (as compared with the characteristic size
d of the inclusion and the wavelength l =2p/k ). The total
1
1
wavefield u in the far-field region will be equal to u0+us.
GE NE R A L EQ U ATI O NS FO R TH E
S CAT TER E D FI EL D
Integral representation
The formulation of the problem given above is similar
to corresponding problems in electrodynamics, optics and
acoustics (Truell et al. 1969; Ishimaru 1978). One of the
(10a)
A
B
4
M
K+ m .
H
3
(10b)
It turns out from the Biot theory that at low frequencies the
compressional type II wave is indeed slow compared with the
fast compressional wave, which, in fact, is a normal P wave
with very small attenuation, so that
|k |%|k |
1
2
and
(11a)
A
B
r2M
r v
rC
r2
# f
−2
w%1 .
(11b)
Q−1 = f 1+
B
r|q|
r2 H
rH
r v
f
f
c
In this paper we confine ourselves to low frequencies, and
assume that the condition (6) and, hence, conditions (11) are
fulfilled. In particular, we will assume the attenuation of the
fast compressional wave in a homogeneous poroelastic medium
© 1998 RAS, GJI 133, 91–103
Figure 1. Geometry of the scattering problem. P is the normal fast
1
compressional wave, P is the Biot slow wave and S is the shear wave.
2
94
B. Gurevich et al.
established ways of tackling these problems is based on integral
representations of the wavefield. For waves in elastic media
this approach has been developed by Gubernatis et al.
(1977a,b). For poroelastic media it is natural to use an
analogous integral representation based on the Biot equations
of poroelasticity (Sadovnichaja et al. 1990; Manolis & Beskos
1989; Norris 1985; Boutin et al. 1987). Norris (1985) has shown
that the scattered displacement field of the solid us may be
written in the form
us(r)=A
eik1r
eik2r
eik3r
+A
+A
,
I r
II r
III r
ws(r)=−
(14a)
eik2r
H
A
.
C II r
(14b)
Eqs (14) show that in the far field the scattered wavefield is a
superposition of the scattered fast compressional, slow compressional and shear waves, with A , A , A denoting their
I II III
respective vector scattering amplitudes. These vectors depend
on the direction of scattering, and not on the distance r. Note
that, unlike the two other similar terms, the function exp(ik r)/r
2
in eq. (14b) for the scattering amplitude into the Biot slow
wave is exponentially decaying with r. Furthermore, the vectors
A , A are parallel to the direction of propagation of the
I II
scattered wave, whereas A is orthogonal to it, i.e.
III
A =A r̂ ;
(15a)
I
I
A =A r̂ ;
(15b)
II
II
A =A r̂ ,
(15c)
III
III 3
where the scalars A , A , A are so-called scalar scattering
I II III
amplitudes, and r̂ is a unit vector orthogonal to r̂, the unit
3
vector in the direction from the origin to the observation
point. In our notation the Cartesian components Ak , Ak , Ak
I II III
of the scattering amplitudes are given by
A BPC
D
k 2
1
1
1
r̂ ts +B(1) (r̂)us +B(2) (r̂)ws n
Ak =− r̂
k
ijk
j
ijk
j i
I
4p
k
m j ij
3
S
×exp(−ik Ωr∞) dS(x∞) ,
1
v wr C2 k2
1
H
i
2
r̂ ts + ps
Ak =− r̂
II
4p k v r H2 k2
m j ij C
c f
3 S
PC A
D
B
+B(3) (r̂)us +B(4) (r̂)ws n exp(−ik Ωr∞) dS(x∞) ,
ijk
j
ijk
j i
2
1
Ak =−
III
4p
PC
(16a)
D
1
(d −r̂ r̂ )ts +B(5) (r̂)us n
j k ij
ijk
j i
m ij
(16b)
S
×exp(−ik Ωr∞) dS(x∞) .
(16c)
3
In eqs (16) the integration is performed over the surface S of
the inclusion using the values of the field variables on the
outer side of S, i.e. in the host medium; n are the Cartesian
i
components of the outer normal vector to S; r̂ , i=1, 2, 3, are
i
the Cartesian components of a unit vector r̂ in the direction
of scattering; k =k r̂ are the wavenumber vectors in the
i
i
direction of scattering; and B(m) are functions of position and
ijk
material properties, which are defined in Norris (1985).
In the original paper of Norris (1985) the equations for the
scattering amplitudes are derived using the formulation of Biot
(1956a,b), whereas we are using the Biot (1962) formulation.
The advantage of the latter notation is that all the field
variables, u , w , t , p, are continuous across interfaces. Thus
i i ij
the integrals in eqs (16) may be thought of as performed over
the inner surface of the inclusion, i.e. over the values of field
variables in the inclusion material. By applying the Green
theorem we can transform the surface integrals in eqs (16) to
volume integrals,
A B P GC
H
k 2
1
1
Ak =− r̂
k
I
4p
k
3
V
D
1
r̂ ts +B(1) (r̂)us +B(2) (r̂)ws
ijk
j
ijk
j
m j ij
×exp(−ik Ωr∞)
1
dV (x∞) ,
,i
i
v wr C2 k2
1
H
2
Ak =− r̂
r̂ ts + ps
II
4p k v r H2 k2
m j ij C
c f
3 V
(17a)
+B(3) (r̂)us +B(4) (r̂)ws exp(−ik Ωr∞)
ijk
j
ijk
j
2
(17b)
1
Ak =−
III
4p
P GC
V
P GC A
B
D
H
D
dV (x∞) ,
,i
1
(d −r̂ r̂ )ts +B(5) (r̂)us
j k ij
ijk
j
m ij
H
×exp(−ik Ωr∞)
3
dV (x∞) .
(17c)
,i
Eqs (14) and (17) give an integral formulation of the
scattering problem. These equations are to be solved for the
components us of the scattered displacement field of the solid
m
us. Note that eqs (17), along with the displacements, involve
the stress components ts =t −t0 , the pressure ps=p−p0
ij
ij
ij
and their derivatives with respect to x . We can express the
i
stress components and fluid pressure through the displacements
by using the constitutive equations (4) with the host medium
constants for the incident field and with the primed (inclusion)
constants for the total field:
ts =m∞(u +u )+d (l∞ u +C∞w )
ij
i,j
j,i
ij c k,k
k,k
−[m(u0 +u0 )+d (l u0 +Cw0 )] ,
(18a)
i,j
j,i
ij c k,k
k,k
−ps=C∞u +M∞w −(Cu0 +Mw0 ) .
(18b)
k,k
k,k
k,k
k,k
Similarly, the spatial derivatives of the stress components and
pressure may be expressed through the displacements using
the equations of motion (1):
ts =−v2(r∞u +r∞ w −ru0 −r w0 ) ,
(19a)
ij,j
i
f i
i
f i
ps =v2(r∞ u +q̃∞w −r u0 −q̃w0 )=0 .
(19b)
i
f i
i
f i
i
Eqs (14) and (17)–(19) provide a complete integral formulation
of the scattering problem. These equations may be solved by
iteration. The first iteration is usually called the Born
approximation.
Born approximation
Eqs (14) show the spatial structure of the scattered field far
from the inclusion, but do not give its amplitude, since the
latter is expressed (through eqs 17) in terms of the integral
of the yet unknown wavefield inside the inclusion. It is
conventional to solve such equations by iteration. The first
iteration consists of replacing the total wavefield inside the
inclusion with the incident wavefield u0, w0. It is clear from
the physical point of view that this replacement, the so-called
Born approximation (Gubernatis et al. 1977b; Ishimaru 1978),
is accurate if the properties of the material of the inclusion are
not too different from those of the host medium. Estimates for
waves of different nature show that the validity condition for
© 1998 RAS, GJI 133, 91–103
Scattering in a poroelastic medium
the Born approximation has the form k dD%1, where k
0
0
is the wavenumber of the incident wave, d is the characteristic
size of the inclusion and D is a dimensionless measure of the
contrast of the inclusion properties with respect to the host
medium.
Here and below we assume that this condition is satisfied
in our case. Thus, we can replace the displacement components
inside the inclusion u in eqs (17), (18) and (19) by the incident
i
wavefield, which in our case is a plane wave exp(ik x∞ ),
1 1
propagating along the Ox axis. Then eqs (17) simplify to
1
k2 dl +2dm cos h dr
c
− cos h S(k ) ,
(20a)
A= 1
1
I 4p
H
r
A
B
k k3 Ck
(Cdl −HdC−2Cdm cos h)S(k ) ,
A = 1 2
c
2
II
4p ivgH2
A
(20b)
B
dr
k2 k dm
(20c)
A = 3 1 sin 2h− sin h S(k ) ,
3
III 4p k m
r
3
where h is the scattering angle, i.e. the angle between the
propagation directions of the incident and scattered waves, so
that cos h=r̂ , and the factor S is given by
1
P
exp[−i(k r∞−k x∞ )] dV (r∞) ,
(21)
j
1 1
V
for j=1, 2 and 3.
Eqs (14) and (20) provide explicit expressions, in the Born
approximation, for the scattered wavefield in the far field of
the inclusion, caused by the incidence of a plane normal (fast)
compressional wave propagating along the Ox axis.
1
S(k )=
j
E NE R GY C HA R A C T ER I ST ICS O F TH E
S CATTE R ED WAVEF IE LD
To estimate the energy loss by the incident wave due to the
scattering into waves of different kinds, we have to compute
energy characteristics of the scattered field, the so-called
scattering cross-sections.
For a given frequency v, corresponding to time period T ,
the total scattering cross-section s(v) is defined as the ratio
of the average power flux scattered in all directions from
the inclusions to the average intensity of the incident wave,
s(v)=ps/J0, where
f (t)=
1
T
P
T
f (t) dt .
0
Thus, the power scattered into a differential element of
solid angle dV is ds(v)=r̂ Js dS/ J0, where dS=r2dV
i i
is a differential surface element normal to the propagation
direction, Js are the Cartesian components of the intensity
i
vector of the scattered wave and J0 is the intensity of the
incident field.
The differential cross-section s denotes the amount of
d
energy scattered into a unit solid angle,
ds(v) r2r̂ Js i i .
s =
=
d
dV
J0
(22)
The differential cross-section can be related to the scattering
amplitudes by calculating the stress field from the displacement
field given by eqs (14) and (20), and then computing the
intensities of the incident and scattered fields from the displacements and stresses. If the incident wave is a unit-amplitude,
© 1998 RAS, GJI 133, 91–103
95
plane, fast compressional wave exp(ik x ), this procedure leads
1 1
to the relationship (Sadovnichaja et al. 1990)
k m
Im k (l+2m)
2
A2 exp(−2 Im k r)+ 3 mA2 .
s =A2 +
II
2
d
I
k
bC
k H III
1
1
(23)
Note that only the term associated with the type II wave
depends on the radial distance r. The other two terms are
independent of r because the normal (fast) compressional
wave and the shear wave have zero attenuation, and thus
their energy for a given solid angle is preserved during the
propagation along the radial distance. This is not the case for
the type II wave, since it has large attenuation, and hence its
energy decays with r. This results in the presence of the
exponential term on the right-hand side of eq. (23).
Eq. (23) shows the energy (per unit solid angle) present
in the scattered wavefield at a large (compared with the
characteristic size d) distance r from the inclusion. However,
if we want to estimate the amount of energy carried away
from the incident wave, we have to drop the exponential term,
since we want to estimate the energy initially scattered into a
certain angle, not just a portion that remains there at a distance
r after all the propagation and attenuation of the type II wave.
For zero porosity w, eq. (23) leads to the classical relationship
for elastic media (Gubernatis et al. 1977a),
k (l+2m)A2 +k mA2
I
3 III .
s = 1
d
k (l+2m)
1
Integration of the differential cross-section, as defined by eqs
(22) and (23), over the full solid angle yields the total scattering
cross-section,
P
s dV .
(24)
d
4p
In many physical situations the characteristics of scattering
by a single inclusion (amplitudes and cross-sections) are used
to determine effective characteristics (elastic velocities and
attenuation constants) of materials containing an ensemble
of randomly distributed inclusions (Fig. 2). In a general
formulation, the problem of the determination of effective
properties leads to a many-body problem, whose exact solution
is not possible. Thus the problem is usually handled by various
approximation methods, such as the single-scattering approximation, the Foldy–Twersky approximation, etc. (Ishimaru
1978). In the limit of low volume concentration of inclusions,
all these approximations lead to the same simple result for the
effective attenuation coefficient aeff of a propagating wave
due to scattering, aeff=(1/2)nsS. This equation says that the
effective attenuation coefficient aeff is proportional to the total
scattering cross-section for a single inclusion times the number
n of inclusions per unit volume. For a poroelastic medium aeff
may be expressed in the form
s=
1
1
aeff= ns= n(s +s +s ) ,
1
2
3
2
2
(25)
where the subscript indicates the type of the scattered wave.
For the specific dimensionless attenuation (inverse quality
factor) Q−1=2aeff/Re k , we have
1
s
(26)
Q−1#n =nc s/v ,
1
k
1
96
B. Gurevich et al.
geophysics and rock mechanics to model the effect of natural
heterogeneities such as microcracks, caverns, etc. The general
spheroidal case includes the particular case of a sphere, the
only situation for which an exact solution of the poroelastic
problem is available (Berryman 1985).
To obtain closed-form expressions for the scattering amplitudes in the case of an ellipsoidal inclusion we need to compute
the integral in eq. (21). To simplify the calculations, we will
assume that the size d of the inclusion is small compared with
the wavelength of the normal (fast) compressional wave l , i.e.
1
d
(27)
2p =k d%1 .
1
l
1
In the elastic case such an assumption leads to the following
simple formulae for the scattering amplitudes of the
compressional and shear elastic waves A and A respectively:
P
S
dl+2dm cos2 h
ab2k2 dr
1
cos h−
,
(28a)
A =
P
3
r
l+2m
A
A
B
B
dr
ab2k2 k dm
3 1 sin 2h− sin h ,
(28b)
3
k m
r
3
and for the differential scattering cross-section (compressional
and shear combined),
A =
S
CA
s = k4
1
d
+k k3
1 3
Figure 2. A porous medium with randomly distributed inclusions.
(a) Anisotropic (aligned) orientation, (b) isotropic (random) orientation.
where the velocity c of the fast compressional wave in the
1
host medium is given by eq. (9a). Thus, from scattering crosssections for a single inclusion we can estimate the effective
attenuation in a medium with chaotically distributed identical
inclusions as a function of frequency and of parameters of the
host medium and the inclusion.
E LLI P S O ID A L IN CLU S IO N
Conditions for long-wave asymptotic
In the previous sections we have derived general relationships
for amplitude and energy characteristics of the scattering of
an elastic wave in a poroelastic medium by an inclusion. In
particular, eqs (20) and (23), giving the scattering amplitudes
and cross-sections in the Born approximation, are explicit and
may be easily programmed to compute the amplitudes and
cross-sections for any given shape of the inclusion.
Besides this, it is also interesting to obtain analytical results
for some typical simple shapes. These results may then be
compared with known results, e.g. with the corresponding
results for the elastic case (Gubernatis et al. 1977b). Here and
below we will consider the particular case of a spheroidal
inclusion (ellipsoid of revolution). Spheroids are often used in
B
BD
dr cos h dl+2dm cos2 h 2
−
r
l+2m
A
k dm sin 2h dr sin h 2
1
S2 ,
−
m
r
(29)
where l and m are the Lamé constants of the host elastic
medium, l+dl and m+dm are the Lamé constants of the
inclusion material and S is the volume of the inclusion.
After obtaining the corresponding formulae for the scattering
amplitudes, we will be able to compare them with the elastic
expressions (28) and (29) and, thus, investigate the specific
properties of the scattering phenomenon in porous media.
Scattering amplitudes in the long-wave approximation
If the condition (27) is satisfied and the frequency of the
incident wave satisfies condition (6) of the low-frequency limit
of the Biot theory, the scattering amplitudes, as given by
eqs (20), may be expressed in a closed form. For an ellipsoid
of revolution (spheroid) with semi-axes a =a, a =a =b,
1
2
3
a≤b, these expressions may be written in the form
A
B
dl +2dm cos2 h
ab2k2 dr
1
cos h− c
,
3
r
l +2m
c
ab2k k C2 dC dl +2dm cos2 h
1 2
− c
A =
II
C
l +2m
4
c
M K+ m
3
A=
I
A
B
A
(30a)
B
sin(k r̃)−(k r̃) cos(k r̃)
2
2
2 ,
(k r̃)3
2
dr
ab2k2 k dm
3 1 sin 2h− sin h .
A =
III
3
k m
r
3
Here
×
A
B
r̃= √a2 cos2 Q+b2 sin2 Q=b √c2 cos2 Q+sin2 Q ,
(30b)
(30c)
(31)
© 1998 RAS, GJI 133, 91–103
Scattering in a poroelastic medium
97
where Q is the angle between the smaller semi-axis a of the
ellipsoid and the direction of scattering, and c=a/b is the
spheroid’s aspect ratio.
We see that eqs (30a) and (30c) for scattering into the normal
compressional and shear waves are identical to those for the
elastic case (with the obvious substitution l  l, dl  dl). In
c
c
other words, the amplitudes of the normal compressional and
shear waves in a poroelastic medium scattered by an inclusion
of another porous material are the same as those in an elastic
medium with the elastic constants of the host and inclusion
calculated from the Gassmann formula (eq. 5a). We shall,
therefore, focus our attention on the expression for the
scattering amplitude of the slow wave, eq. (30b). We see that
the frequency dependence of this amplitude A depends
II
essentially on the parameter |k r̃|, i.e. on the relation between
2
the size of the inclusion d and the wavelength of the slow wave
l =2p/|k |. To analyse this dependence in an invariant form,
2
2
we introduce a dimensionless parameter
r̃
j= = √c cos2 Q+sin2 Q .
b
(32)
Then, eq. (30b) may be rewritten in the form
A =d̃F(x, Q) exp(−ik r̃) ,
II
2
where
d̃=
k C2
1
A
4
k2 M K+ m
2
3
B
A
dC dl +2dm cos2 h
− c
C
l +2m
c
(33)
B
(34)
is a frequency-independent proportionality constant, defined
by the properties of the host and inclusion, and F(x, Q) is a
dimensionless function which defines the dependence of the
scattering amplitude on frequency and on the size and shape
(aspect ratio) of the inclusion,
Figure 3. Normalized scattering amplitude of the Biot slow wave for
inclusions of different aspect ratio c, (a) versus normalized frequency,
( b) versus angle w (at frequency x=105).
sin( √ixj)−( √ixj) cos( √ixj)
exp(i √ixj) .
j3
In the opposite case, when the wavelength of the second
wave is small compared with the smaller semi-axis a,
F(x, Q)=c
(35)
In eq. (35), x=|k b|2=vgb2/kN denotes a dimensionless
2
measure of frequency (normalized frequency). The presence of
the exponential term in eq. (35) means that the scattered field
at a distance r from the centre of the inclusion is proportional
to the expression
exp[ik (r−r̃)]
2
.
r
c(ix)3/2 c(k b)3 k3 ab2
= 2 = 2
.
3
3
3
(37)
Therefore, in the low-frequency asymptotics,
A =
II
ab2k k C2
1 2
4
3M K+ m
3
A
c(ix)1/2
ck b
k a
k ab2
=− 2 =− 2 =− 2
,
2j2
2j2
2j2
2r̃2
B
A
B
dC dl +2dm cos2 h
.
− c
C
l +2m
c
© 1998 RAS, GJI 133, 91–103
(38)
(39)
and the scattering amplitude is given by
A =−
II
(36)
This corresponds to the fact that the scattered wave is generated
(and hence begins to attenuate) at the surface of the inclusion,
not in its centre.
Fig. 3 shows the absolute value of the normalized scattering
amplitude F as a function of the normalized frequency x
and angle Q. If the larger semi-axis of the ellipsoid is small
compared with l , then at any angle Q the parameter |k r̃| is
2
2
small and
F=
F=−
×
ab2k C2
1
4
2k M K+ m
2
3
A
exp(−ik r̃)
2 .
r̃2
B
A
dC dl +2dm cos2 h
− c
C
l +2m
c
B
(40)
We see that if the inclusion is ‘large’ (|k a|&1 or a&l )
2
2
and thin (c%1), then A strongly depends on the angle Q. If
II
Q=0 or Q=p, then j=c and
F=−k b/2c=−k b2/2a .
(41)
2
2
If, on the other hand, sin Q differs from zero or, more precisely,
if |sin Q|&c, then j=sin Q and
F=−k cb/2 sin2 Q=−k a/2 sin2 Q .
(42)
2
2
For intermediate frequencies satisfying the condition a<l <b,
2
the behaviour of F follows the low-frequency formula (37) at
Q=0 or Q=p, and the high-frequency formula (42) at other
angles. In other words, for thin ellipsoids (c%1) the frequency
at which the transition between the low- and high-frequency
98
B. Gurevich et al.
behaviour takes place depends on the angle Q. For Q=0 and
Q=p this transition occurs at l #a, and at l #b for other
2
2
angles. In the case of a sphere, a=b and c=j¬1, and, as
expected, the scattering amplitude does not depend on Q for
any frequency.
As we see in Fig. 3( b), for inclusions with small aspect
ratio the dependence of F on the angle Q at sufficiently large
normalized frequencies x has sharp peaks at Q=0 and Q=p.
This means that the wavefield scattered by a thin (oblate)
inclusion exhibits strong directionality parallel to the axis of
revolution. This fact can be explained as follows. The effect of
a thin ellipsoidal inclusion is similar to that of a thin porous
layer (Gurevich, Marschall & Shapiro 1994). As the velocity
of the scattered slow wave is much smaller than that of the
incident normal compressional wave, according to Snell’s law
the reflection and refraction angles must be close to a right
angle. That is exactly what we observe.
the following expression for the total scattering cross-section:
s=
P
4p
×
Scattering cross-sections in the long-wave approximation
A
s =s +s +s =k4
d
d1
d2
d3
1
+
k bC
1
A
4
3
B
√ 2|k |3 K+ m
2
B
dr cos h dl +2dm cos2 h 2
S2
− c
r
H
A
A
B
B
4
3
dC dl +2dm cos2 h 2
|F|2 dV .
− c
C
H
√ 2pk bC
1
4
|k |3 K+ m
2
3
A
PA
B
B
p dC dl +2dm cos2 Q 2
− c
|F|2 sin Q dQ ,
(45)
C
H
0
with F given by eq. (35). To simplify the integral in eq. (45)
we recall that F is independent of Q for small or spherical
inclusions, and strongly directional for thin or ‘large’ ones.
Thus, we can write the following approximate formula:
p
(43)
The first and third terms on the right-hand side of eq. (43)
correspond to the scattering into the normal compressional
and shear waves. As expected, they are identical to the
analogous expressions for an elastic medium, eq. (29). The
second term, which we denote by s , corresponds to the
d2
scattering into the Biot slow wave. Recalling the preceding
analysis of the frequency dependence of F (eqs 37 and 39), we
note that s is proportional to k2 km , with the exponent m
d2
1 2
varying from 1 to 3, depending on the relation between the
wavelength of the slow wave and the size of the inclusion. In
either case, the frequency dependence of s is less steep than
d2
that of s and s . This fact is the result of our longd1
d3
wavelength assumption (eq. 27). The scattering into the normal
compressional and shear waves is most efficient when the
size of the inclusion is comparable with their respective wavelengths. By assuming the condition (27) we have restricted
ourselves to much smaller inclusions. Since the wavelength of
the Biot slow wave is much smaller than that of the normal
compressional wave, the scattering into the slow wave is much
more efficient for these small inclusions.
We will therefore concentrate on the analysis of the energy
characteristics of the scattered Biot slow wave. Eq. (24) gives
|F|2 cos2m Q sin Q dQ=q (x)
m
P
p
|F|2 sin Q dQ ,
0
0
where the function q (x) is given by
m
1−exp[−(2m+1)/√x(1−c2)]
1
q (x)=
m
2m+1
1−exp[−1/√x(1−c2)]
B
B
A
√ 2|k |3 K+ m
2
PA
×
P
dC dl +2dm cos2 h 2
− c
C
H
×|F|2 exp[−2 Im k (R−r̃)]
2
dm sin 2h dr sin h 2
−
S2 .
+k k3
1 3
m
r
k bC
1
(44)
4p
In eq. (44) the exponential term has been dropped in order
to account for all the energy scattered by the inclusion, not
just what remains at a distance R. Consider the scattering
cross-section for an incident wave propagating along the
spheroid’s axis of revolution. In this case the scattering angle
h equals the polar angle Q, and the integrand is rotationally
symmetric. Taking the solid angle element in a standard
form, dV=sin Q dQ dz (where z is the azimuthal angle), and
integrating with respect to z, we obtain
s =
2
Having obtained the scattering amplitudes, we can now derive
expressions for the energy characteristics of the scattered
wavefield, as defined by eqs (22)–(26). Substitution of eqs
(30a,c) and (33) for the scattering amplitudes gives the following
expression for the differential cross-section:
s dV=
d2
(46)
(47)
and m takes the value 0, 1 or 2. Note that q (x)#1/(2m+1)
m
for small frequencies or when c=1, and q (x)#1 for large
m
frequencies and oblate inclusions. Thus q (x) is constant for a
m
spherical inclusion, while for a non-spherical one it changes
over a frequency range of several decades by only one order
of magnitude. Using eq. (46) we can rewrite eq. (45) in the
form
s =d̂
2
√ 2pk bC
A
1
B
4
|k |3 K+ m
2
3
P
p
|F|2 sin Q dQ ,
(48)
0
where
d̂=
CA
B A
B
A B D
dC dl 2
dC dl dm
dm 2
q .
− c −4
− c
q +4
2
1
C
H
C
H H
H
(49)
Using eq. (48), we may express the dimensionless attenuation
Q−1, as defined by eq. (26), in the form
Q−1=nd̂
√ 2pbC
A
4
|k |3 K+ m
2
3
B
P
p
|F|2 sin Q dQ ,
(50)
0
where d̂ and F are given by eqs (49) and (35) respectively. The
integral in eq. (50) cannot be expressed in terms of elementary
© 1998 RAS, GJI 133, 91–103
Scattering in a poroelastic medium
functions (unless the inclusion is spherical, which means c=1,
see eqs 35 and 32), and has to be evaluated numerically. To
do so in a normalized (dimensionless) form, we first define a
normalized attenuation factor as
1
Y=
cx3/2
P
p/2
|F|2 sin Q dQ ,
0
and rewrite eq. (50) in the form
Q−1=2√ 2pcb3nd̂
bC
bC
3
Y=
Y,
n d̂
4
4
√2 v
K+ m
K+ m
3
3
decreases with frequency as Y#1/4x1/2, and
Q−1=
3n d̂bC
3n d̂x−1/2bC
v
v
=
.
4
4
4 √ 2 K+ m
4 √2|k |b K+ m
2
3
3
A
B
A
(56)
B
(51)
Recalling that in the case of a sphere q (x)¬1, q (x)¬1/3,
0
1
q (x)¬1/5, we note that in eqs (55) and (56) the parameter d̂
2
is given by
(52)
d̂=
CA
B A
B
A BD
dC dl 2 4 dC dl dm 4 dm 2
.
− c −
− c
+
C
H
3 C
H H 5 H
(57)
As can be seen in Fig. 4, slightly oblate inclusions with aspect
ratios of 1/2 and the like exhibit similar behaviour.
where
4
4
n = pnab2= pncb3
v 3
3
99
(53)
Strongly oblate (‘penny-shaped’) inclusions
is the volume density of inclusions.
Fig. 4 shows the normalized attenuation Y as a function of
frequency for different aspect ratios. The results are distinctly
different for spherical and near-spherical inclusions (c#1) on
the one hand, and for strongly oblate (thin) inclusions (c%1)
on the other.
For inclusions with small aspect ratio (c%1), the normalized
attenuation reveals more complex behaviour. Looking at Fig. 4,
we identify three distinct regions.
Spherical inclusions
n d̂cx3/2
Q−1= v
3
For spherical inclusions c=1, and the function F does not
depend on Q. Thus, eq. (51) gives an explicit formula
Y=
|[sin( √ ix)−( √ ix) cos( √ ix)] exp(i √ ix)|2
.
x3/2
(54)
The frequency dependence of Y has the form of a relaxation
peak at normalized frequency x#1 (Fig. 4). At lower
frequencies (x%1), when the inclusion is much smaller than
the wavelength of the Biot slow wave, we have Y#x3/2/9, so
that
n d̂x3/2
Q−1= v
3 √2
bC
.
4
K+ m
3
A
B
(55)
In the other limit ( x&1), when the inclusion is large compared
with the wavelength of the Biot slow wave, attenuation
(1) Low frequencies ( x%1). This means that the larger
semi-axis of the ellipsoid is small compared with l . In this
2
case F is given by the approximation (37) and for attenuation
we get Y#cx3/2/9, so that
n d̂|k3 |ab2x3/2
bC
bC
= v 2
,
3
4
4
K+ m
K+ m
3
3
A
B
A
B
(58)
an expression that for a fixed inclusion volume is identical to
that for the spherical case (eq. 55). This was to be expected,
since the inclusion is small compared with any of the
wavelengths, and thus its shape does not affect the result.
(2) Intermediate frequencies ( xc%1%x). In this case
a<l <b, and Y is proportional to x1/2. In other words, in
2
this region attenuation still increases with frequency, but the
increase is much less steep.
(3) Higher frequencies ( x&1). When l becomes smaller
2
than both dimensions of the inclusion, F is given by eq. (39),
and eq. (51) reads
P
Y=
c
4x1/2
Y=
1
,
8cx1/2
p/2
sin Q
dQ .
(c2 cos2 Q+sin2 Q)2
(59)
0
The integral in eq. (59) can be evaluated analytically. For c%1
the result is
(60)
and, recalling that for large frequencies q (x)¬1 for m=0, 1
m
and 2,
Q−1=
=
Figure 4. Effective attenuation due to scattering into the Biot slow
wave versus frequency for inclusions of different aspect ratio c.
© 1998 RAS, GJI 133, 91–103
3n x−1/2bC
v
4
√
8 2c K+ m
3
A
B
3n bC
v
A
A
B
dC dH 2
−
C
H
B
4
8 √ 2|k a| K+ m
2
3
A
B
dC dH 2
−
.
C
H
(61)
The attenuation factor as given by eq. (61) is larger than
that in the spherical case (eq. 56) by a factor of 1/2c. This
reflects the fact that at high frequencies ( x=|k |a&1), the
2
Biot slow wave exists within just a thin boundary layer around
100
B. Gurevich et al.
the surface of the inclusion. Thus it is the surface area of the
inclusion that matters at these frequencies, rather than its
volume. If a sphere and a penny-shaped body have equal
volumes, the thin inclusion’s surface area is obviously larger
than that of the sphere.
C O M PA R I S O N W IT H K NO W N R E SU LTS
The results for the scattering amplitudes and cross-sections in
poroelastic media can be compared with the exact solution
obtained by Berryman (1985, 1986) for spherical inclusions.
The solution of Berryman (1985) is given in the form of a
series expansion. An explicit analytical result is only available
for the case of small inclusions (k a  0, |k a|  0). For the
1
2
radial solid displacement u , in the direction of incidence
r
(h=0) at large distances from the inclusion, this result reads
u =B(+) exp(ik r)/k2 r+B(−) exp(ik r)/k2 r ,
r
0
1
1
0
2
2
with the coefficient B(−) given by
0
i(k a)3A C
2
0
B(−) =
0
4
3M∞H K∞+ m
3
A
CA
B
B A
B
(62)
A
C
C∞
4
4
−
× C K∞+ m −C∞ K+ m +CC∞
3
3
M∞ M
BD
DI S C U SS I O N
.
(63)
Here, as in our notation, primed quantities refer to the material
of the inclusion. Eqs (62) and (63) are written for an incident
wave with the displacement amplitude or the solid equal
to A /ik . By dividing the amplitude of the second wave,
0 1
B(−) /k2 , by A /ik , we can obtain the amplitude of the scattered
0
2
0 1
Biot slow wave in our notation (i.e. for an incident wave with
unit amplitude). Taking the limit of low contrast of the
inclusion properties with respect to the host medium, i.e.
M∞−M
%1,
M
K∞−K
%1 ,
K
C∞−C
%1 ,
C
(64)
we obtain
a3k k C2
B(−) ik
1 2
1=
AB = 0
II
A k2
4
0 2
3M K+ m
3
A
B
A
B
H∞−H C∞−C
−
.
H
C
(Urick 1948; Hay & Mercer 1985). In such a situation, a plane
acoustic wave is scattered into an acoustic wave as well as
into a viscous wave (shear wave in a viscous fluid), which,
similarly to the Biot slow wave, is described by the diffusion
equation. As shown by Urick (1948) and Hay & Mercer
(1989), in the limit |k a|&1 (where k is the wavenumber of
s
s
the viscous wave) the attenuation factor Q−1 is proportional
to v−1/2, which is equivalent to our result.
The frequency dependence of the attenuation for oblate
inclusions may be compared with the result for a layer in a
poroelastic medium (Gurevich et al. 1994), or for a randomly
layered porous medium (Gurevich & Lopatnikov 1995). In
such media the attenuation also has a relaxation peak when
the wavelength of the Biot slow wave is approximately equal
to the thickness of the layer. The attenuation factor Q−1 is
proportional to v1/2 in the low-frequency limit (|k a|%1) and
2
to v−1/2 at higher frequencies. This is equivalent to our results
for inclusions with c%1, when the wavelength of the Biot slow
wave becomes smaller than the length of the inclusion 2b. This
is an obvious requirement for the effect of the oblate spheroid
to be similar to that of a layer.
(65)
Eq. (65) is identical to our low-frequency result (eq. 38) for
forward scattering from a spherical inclusion (a=b, cos h=1).
This agreement of the results demonstrates the applicability of
the Born approximation to poroelastic media.
We note, however, that our result is, in certain aspects, more
general than that of Berryman (1985) since it covers a wide
range of frequencies, not restricted by the condition |k a|%1.
2
We also are able to model the scattering for a more general
inclusion shape, namely a spheroidal shape, which may be
used to simulate wave propagation in a poroelastic medium
with microcracks. This substantial generalization has been
achieved at the expense of accuracy. Our results have been
obtained in the Born approximation, which is only valid when
condition (64) is fulfilled. The higher the contrast of the
inclusion with respect to the host material, the less the accuracy
of our results.
We can also compare our results with those for the scattering
of an acoustic wave by an elastic inclusion in a viscous fluid
In the preceding sections we have studied the effect of an
isolated spheroidal inclusion on the wavefield of a normal
compressional wave in a poroelastic medium. Of particular
interest are the analytical and numerical results concerning the
attenuation of a plane elastic wave due to the presence of an
ensemble of such inclusions.
We have seen that at certain frequencies the elastic wave
attenuation may be dominated by the scattering of the incident
compressional wave into the Biot slow wave, which has very
high attenuation. We call this specific attenuation mechanism
‘slow-mode-conversion attenuation’. As shown in the preceding
analysis, such attenuation occurs in a poroelastic medium
with randomly distributed macroscopic (with respect to the
individual pore scale) inclusions of another poroelastic
material, in which the standard Biot attenuation Q−1 , as well
B
as attenuation caused by ordinary elastic scattering Q−1 , are
sc
also present. However, slow-mode-conversion attenuation
should not be confused with either of these phenomena. In
fact, it differs both from the standard Biot attenuation and
from the elastic scattering by the physical nature, dominant
frequency range and frequency dependence of the attenuation
coefficient. In a real situation the total attenuation Q−1 is the
T
sum of the attenuation constants for all three mechanisms.
Q−1 =Q−1 +Q−1 +Q−1 .
(66)
T
B
sc
In the limit of low frequency Q−1 varies linearly with frequency
B
(eq. 11), while Q−1 and Q−1 are proportional to v3 and v3/2
sc
respectively (see eqs 55 and 58). Thus, at sufficiently low
frequencies the standard Biot attenuation will dominate in the
total attenuation of the normal compressional wave. This
means that for waves of very low frequencies, at which the size
of the heterogeneities is much smaller than the wavelengths of
both the fast and the slow waves, the heterogeneous porous
medium is equivalent to an (effective) homogeneous porous
medium.
An important class of macroscopic inhomogeneities in
porous rocks is cracks and lenses, which are often modelled
as thin (strongly oblate) spheroidal inclusions. Our analysis
© 1998 RAS, GJI 133, 91–103
Scattering in a poroelastic medium
shows that for inclusions with the larger dimension (‘length’)
longer than the wavelength of the slow wave, the scattered
wavefield of the slow wave is qualitatively similar to that
caused by the presence of a thin, flat layer. The attenuation
in a thinly layered poroelastic medium has been studied
in detail (White et al. 1975; Norris 1993; Gurevich &
Lopatnikov 1995; Gurevich et al. 1994, 1997). It has been
shown that fine layering of the poroelastic medium results in
a specific attenuation of compressional and shear waves. This
attenuation is caused by the flow of the pore fluid between
different layers, the flow that results from the difference in
compliances between the adjacent layers. When a medium
composed of layers with different compliances is subjected
by the incident wave to compression (or extension), the pore
fluid tends to flow from the more compliant into the less
compliant layer (or vice versa). The attenuation caused by this
phenomenon is called ‘interlayer flow attenuation’.
The slow-mode-conversion attenuation in a poroelastic
medium with inclusions is of the same physical nature. When
an ordinary compressional (or shear) wave propagates in a
porous rock containing small but macroscopic inclusions, and
the inclusion’s solid matrix has, say, lower elastic constants
than the host medium, the pore fluid flows from the inclusions
into the host medium during compression cycles, and in the
opposite direction during extension cycles. The pore fluid
flow, in turn, results in a viscous absorption, causing effective
attenuation of the propagating wave. This is very similar to
the local flow attenuation mechanism proposed by Mavko &
Nur (1975, 1979) and examined theoretically by many authors
(O’Connell & Budiansky 1977; O’Connell 1984; Murphy,
Winkler & Kleinberg 1986). In essence, our results demonstrate
that the local flow or squirt attenuation may be effectively
modelled fully within the context of Biot’s theory of poroelasticity, with the assumption of the presence of macroscopic
heterogeneities. All real rocks are heterogeneous by their
nature, so this assumption is quite realistic. It serves as a basic
concept for the so-called double-porosity description, which
provides an alternative approach to the investigation of highly
heterogeneous porous materials (Wilson & Aifantis 1982;
Beskos & Aifantis 1986; Auriault & Boutin 1994; Berryman &
Wang 1995). Although the double porosity description is more
general, our formulation has the advantage of enabling one to
specify explicitly the shape of the heterogeneities.
The model considered here has some similarity with the
BISQ model of Dvorkin & Nur (1993). The difference between
the two lies in the fact that the BISQ model assumes the
existence in the Biot medium of surfaces with zero fluid
pressure, whereas our model is free of such heuristic assumptions.
The local flow attenuation is so-called in contrast to the
global flow attenuation described by the standard Biot theory
for homogeneous poroelastic materials (we also call the latter
mechanism ‘standard Biot attenuation’). The global flow
attenuation is also associated with the wave-induced flow of the
pore fluid, but that flow takes place due to pressure gradients
between regions of compression and extension in the passing
wave. The lower the frequency, the larger the distance between
regions of compression and extension, and the weaker the
flow. In reality, the peak frequency for global flow attenuation,
the so-called Biot characteristic frequency f =v /2p, as given
c
c
by eq. (7), depends strongly on the permeability of the rock,
and for permeabilities of the sort found in real rocks it lies in
the megahertz or submegahertz range. In contrast, the local
© 1998 RAS, GJI 133, 91–103
101
flow attenuation, as modelled by our theory, happens between
regions of different compliance within the rock, and at a much
more local scale. Its peak frequency is determined by the
relationship between the wavelength of the Biot slow wave
(fluid diffusion length) and the thickness (smaller dimension)
of the spheroidal inclusion, |k (v )|a¬a/l (v )=1, which
2 0
2 0
gives f =v /2p=kN/ga2. For rocks with a stiff frame
0
0
(K &K ) this yields the following approximation for the peak
s
f
frequency: f =kK /gwa2. Thus the typical frequency range of
0
f
this mechanism depends on the scale of the heterogeneities
considered. But it is always much lower than both the Biot
characteristic frequency f and the peak frequency for ordinary
c
scattering, at which the size of the inclusion equals the wavelength of the normal compressional wave. This wavelength is
always much larger than that of the Biot slow wave.
By treating the complex poroelastic scattering phenomenon
using the Born approximation, we were able to obtain explicit
and relatively simple analytical formulae for a number of
common cases. However, this simplicity and clarity has been
achieved at the expense of accuracy, and was only possible
under a number of assumptions, which impose limitations on
the class of situations for which our results are valid. Among
these assumptions are
(1) the assumption of low frequencies in terms of Biot’s
theory (eq. 6);
(2) low-frequency (or long-wavelength) assumption with
respect to elastic scattering (eq. 27);
(3) assumption of low contrast of the inclusion material
with respect to the host medium (validity condition for the
Born approximation);
(4) the condition of low volume concentration of inclusions
[validity condition for the linear relationship (eq. 25) between
the effective attenuation and scattering cross-section for a
single inclusion].
The first two of these assumptions are mainly technical, and
may be overcome, leading to more complex formulae, which
would incorporate explicitly three mechanisms of attenuation
(standard Biot and ordinary scattering as well as slow-mode
conversion). However, the other two assumptions are more
fundamental and cannot be overcome so easily. Precise
quantitative results for large contrast (e.g. for fluid-filled cracks)
can be obtained using numerical algorithms for modelling
wave propagation in 3-D poroelastic media (Dai, Vafidis &
Kanasewich 1995; Carcione 1996). To waive the limitation
upon the volume concentration, techniques developed recently
in wave physics may be employed. These techniques, however,
require detailed information on the statistical distribution of
heterogeneities in a rock.
The above remarks provide only a very general, qualitative
understanding of the range of validity of our model. A more
comprehensive analysis of the range of validity requires a
comparison with a full 3-D numerical simulation of wave
propagation in a poroelastic medium, which will be the subject
of another paper. However, it is clear enough that the
assumptions discussed above are rather strict, and are not
fulfilled in many situations. In those cases the simple and
intuitive results obtained in this paper should be regarded as
estimates showing general trends of the attenuation, rather
than precise quantitative predictions. Further development of
the model presented would require application of more
elaborate mathematical methods, and should be based on a
102
B. Gurevich et al.
detailed analysis of the typical distributions of heterogeneities
in porous rocks (see e.g. Murphy et al. 1984).
CON CLU SION S
In this study we have developed a quantitative model of
scattering of elastic waves by ellipsoidal inclusions in a poroelastic medium. We have studied the problem using the Born
approximation, which is applicable for inclusions with a small
relative contrast of properties with respect to the host medium.
By applying the Born approximation to the Biot equations
of poroelasticity we have obtained explicit analytical formulae
for the amplitudes and energy characteristics of the compressional and shear waves and the Biot slow compressional
wave scattered by the inclusion. The results for the Biot
slow wave depend heavily on the relationship between the
dimensions of the inclusion and the wavelength of the Biot
slow wave.
The analytical results have been used to estimate the effective
attenuation of a normal compressional wave in a poroelastic
medium with randomly distributed ellipsoidal inclusions. The
frequency dependence of such attenuation coefficients is consistent with the results for horizontally layered porous rocks.
The additional attenuation in a poroelastic medium with
inclusions, in comparison with a similar elastic continuum, is
caused by the wave-induced flow of the pore fluid from the
host medium into and out of the inclusion material. In other
words, our results demonstrate how the local flow attenuation
can be effectively modelled within the context of the Biot
theory of poroelasticity.
A CK NO W L ED GM EN TS
The work reported in this paper was supported in part by
Grant RWG000 from the International Science Foundation.
The work of Boris Gurevich was also partly supported by
project 754/28/96 of the Earth Science Research Administration
of the Israel Ministry of National Infrastructures. The authors
thank V. V. Ivanov for valuable comments and help with
computations.
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