Geophys. I. Znt. (1996) 127, 31-39
Surface-wave propagation in a cracked poroelastic half-space lying
under a uniform layer of fluid
M. D. Sharma
Department of Atmospheric and Oceanic Sciences, Kurukshetra University, Kurukshetra, 152119, India
Accepted 1996 April 9. Received 1995 December 29; in original form 1995 June 13
SUMMARY
The effect of cracks on the elastic properties of an isotropic elastic solid is studied
when the cracks are saturated with a soft fluid. A polynomial equation in effective
Poisson’s ratio is obtained, whose coefficients are functions of Poisson’s ratio of the
uncracked solid, crack density and saturating fluid parameter. Elastic and dynamical
constants used in Biot’s theory of wave propagation in poroelastic solids are modified
for the introduction of cracks. The effects of cracks on the velocities of three types of
waves are observed numerically. The frequency equation is derived for the propagation
of Rayleigh-type surface waves in a saturated poroelastic half-space lying under a
uniform layer of liquid. Dispersion curves for a particular model of oceanic crust
containing cracks are plotted. The effects of variations in crack density and saturation
on the phase and group velocity are also analysed.
Key words: cracks, porosity.
INTRODUCTION
The presence of internal cracks and pores in crustal rocks has
been recognized for some time, and the elastic properties of
such rocks and the wave speeds in them have been a subject
of continued interest. Biot (1956a, b) established a systematic
theory for the propagation of elastic waves in fluid-saturated
porous solids. He demonstrated the existence of three (two
dilatational and one shear) kinds of waves in such solids. In
subsequent years, using Biot’s theory, a large number of problems
of acoustics and seismic-wave propagation have been studied.
In the last few years, Philippacopoulos (1988) and Sharma
(1992) have studied Lamb’s problem for liquid-saturated
porous media. Wave propagation in anisotropic poroelastic
media has been discussed by Schmitt (1989) and Sharma &
Gogna (1991).
Nersesov, Semenov & Simibireva ( 1969) found that the
traveltime ratio of shear and compressional waves varied prior
to the occurrence of an earthquake in the Garm region of the
then USSR. Nur (1972) and Aggarwal et al. (1973) explained
these traveltime variations in terms of the changes in dilatancy
around the focal zone and the flow of pore fluid into
the dilatancy-formed cracks. The wave velocities for elastic
solids containing a dilute concentration of small cracks have
been approximated by Garvin & Knopoff (1973, 1975a, b).
O’Connell & Budiansky (1974) and Budiansky and O’Connell
(1976) calculated the effects of the introduction of cracks on
the elastic properties of an isotropic elastic solid using a selfconsistent procedure. Complete.and partial saturation of cracks
was also considered. O’Connell & Budiansky (1977) studied
01996 RAS
the viscoelastic properties of cracked viscoelastic solids. The
attenuation of elastic waves in cracked solids has been studied
by Chatterjee, Knopoff & Hudson (1980), Hudson (1981) and
Xu & King (1990).
For about the last 15 years, Stuart Crampin has been studying
the effects of dilatancy-formed cracks on the polarization of
shear waves. His large number of papers includes Crampin
(1978, 1984, 1985, 1987) and Crampin, McGonigle & Bamford
( 1980). Crampin (1985) suggested that aligned water-filled
cracks are widespread in the top 10-20 km of the crust.
Keeping in mind the co-existence of pores and cracks in the
Earth’s crust, I propose to study the effects of the introduction
of cracks on the elastic properties and wave velocities of
poroelastic solids. Oceanic crust is assumed to be a cracked
water-saturated porous solid, and the frequency equation for
Rayleigh-type surface waves is obtained. Dispersion is studied
numerically for a particular model of oceanic crust.
BIOT’S THEORY FOR POROELASTIC
SOLIDS
The differential equations governing the displacement u of the
solid particles and U of the interstitial fluid in a homogeneous
poroelastic solid, in the absence of dissipation, are
p V z u + ( / Z + p + a 2 M ) V e + a M V e ‘ = i32(pu+p,w)/i3tz,
V(aMe + Me’)= d2(pfu+ rnw)/i3t2
(1)
Biot (1962), where w =p(U -u) represents the flow of fluid
relative to the solid measured in terms of volume per unit area
31
M . D.Sharma
32
of the bulk medium. These equations are also valid in the
presence of dissipation, but only when the viscosity of the
interstitial fluid is very small or when the frequency is very
high. When the interstitial fluid is non-viscous, the poroelastic
solid will be non-dissipative and eqs (1) are valid for all
frequencies.
A, p are LamC‘s constants for the solid skeleton; p is the
mass density of the bulk medium; in is Biot’s parameter, which
depends on the porosity p and fluid density pF ( m = Pf/p for
tube-like pores); e = V * u and e’ = V w represent dilatations;
s( and M are the elastic coefficients, which are related to the
coefficient of fluid content 7 , the bulk modulus K, of solid
grains and the bulk modulus K(=A + 2p/3) of the porous solid
skeleton (drained bulk modulus) by
x
= 1- K / K , ,
M
= K,/(yK,
+ a).
(2)
-
The coefficient of fluid content can be expressed as
Y = 8(1/Kf - l / K J >
K/K
+ a2M)e+ ~ M e ’ 1 +6 ~p(uiTj
~ + uj,J,
pf= -M(ae+e‘).
(4)
The displacements u and w are expressed in terms of velocity
potentials as
(5)
The mass densities pj(j = 1,2, 3) are given by
where
+
B =pM
+ mH - 2pfaM,
C = pm - p ; ,
and
H =I
+ 2p + a 2 M .
The potentials
16 (1 - v’)DE
1-9 1-22v ’
32
ii/p=l--(l-V)
45
where K(=/2+2p/3) is the bulk modulus, with 1 and p
Lam& constants. Poisson’s ratio, v, of the uncracked solid is
0.51/(A + p). The barred quantities represent the corresponding
elastic parameters for a saturated cracked solid. The crack
density parameter E and saturation parameter D are expressed
in terms of the elastic constants as
2-F
45 v--v
16 1 - V 2 D ( 1 + 3 v ) ( 2 - ~ ) - 2 ( 1 - 2 v ) ’
E=--
D=
A = (1 2p)M,
==
(3)
where K, denotes the bulk modulus of the interstitial fluid.
The stress components, z i j ,in the solid, and the fluid pressure
pFare expressed as
zij = [ ( A
fore the effects of cracks of any convex shape should be similar
to those for circular cracks, provided that the crack density E
is expressed in terms of area and perimeter of the cracks. Also,
the results for circular cracks may be used for more general
cases with negligible error. Following Budiansky & OConnell
(1976), the basic assumption of an isolated fluid in each crack
is valid for elastic waves of sufficiently high frequency. In such
cases, the moduli found are appropriate for stress changes that
occur sufficiently rapidly to prevent communication of fluid
pressure between the cracks.
Consider an elastic solid containing circular cracks
(a = b >> c) with a very small aspect ratio cia and saturated by
a fluid of bulk modulus R. Following Budiansky & OConnell
(1976), assuming that the fluid in each crack is isolated, the
elastic constants of the cracked solids are given by
l#J,, 4’ and Y
(8)
1-’
4 0 K (1 - S 2 )
I+-=3n K(1-2V)
[
’
where, for circular cracks R = (a/c)(R/K).The corresponding
relations for the cracked elastic solids with empty cracks are
obtained from (11)-( 14) by substituting D = 1 or S2 = 0.
The dependence of D and K / K on each other complicates
the evaluation of relations (11)-( 14). O’Connell & Budiansky
(1976) simplified these relations by restricting R to the extreme
values of 0 and co only. When the cracks are saturated with
some soft fluid having
in the range of intermediate to
moderately high values, and the aspect ratio c/a is small but
not negligible, then these extreme values of 0 are not appropriate. To solve the system of eqs (11)-( 14), we combine (11)
and (14) to obtain
satisfy the wave equations, i.e.
( v z + o ” u ; ) ~ j = o ; ( j = 1,2),
(V‘
+ 0”u:)Y = 0 ,
(9)
where w is the angular frequency. The velocities of propagation
u l , u2, and u3 of the fast compressional (or Pf) wave, slow
compressional (or P , ) wave and shear wave, respectively, are
given by
9
+ -16
E(
1 - 2V)( 1 - V2)
= 0,
and from (13), we have
45 v - v
2E( 1 - 2v)( 1 - V2)
ED(^ -Vz) = - __
16 1 + 3 ~ ( 1 + 3 ~ ) ( 2 - S )
+
C R A C K E D ELASTIC SOLID
‘
Substituting (16) in (15) yields a fifth-order equation in 5,
i.e.
O’Connell & Budiansky (1974) suggested that the effects for
circular and elliptical cracks are nearly coincident, and there0 1996 RAS, GJI 127, 31-39
Surface-wave propagation in a cracked poroelastic halj-space
The coefficients aj ( j = 1,2, . .. , 6) are functions of v,
and are given by
+
a, =(1- ~ v ) E [ ~1E ’ (3v)]
+ (1+ 3v)
(i
-E
::
--E
E
and Q,
,
I )
i
45
1 - 2V)E 4E( 1 - 2V) - 4
a,=(
405
81
2025
[
- :~‘(4
2025
64
-~ v(v
1215
256
- 3v) - -(4
I assume that the increase in stress in an elastic solid causes
the introduction of cracks (Nur 1972), and the pores at the
boundary of such a crack may squeeze shut. This may prevent
the communication of fluid pressure between the cracks in a
poroelastic solid. Furthermore, the fluid in cracks can be
considered to be isolated for elastic waves of high frequency.
Biot’s theory formulated earlier is applicable to both these
situations. Therefore, to represent the co-existence of pores
and cracks, the elastic constants and dynamical parameters
used in Biot’s theory are modified in accordance with
Budiansky & O’Connell(l976).
The modifications are as follows.
(1) The term porosity in Biot’s theory refers to the effective
porosity that encompasses only intercommunicating void
space. The sealed pores were considered to be part of the solid.
Hence the introduction of isolated cracks does not change the
effective porosity. However, to calculate the density of the
cracked poroelastic aggregate, the porosity is assumed
to be
2025
+ -(v2
+ 8~ + 4) + (1 + 3 ~ )
256
x
33
+ 3v) +
where Pc(= Nu,) represents the crack porosity. For circular
(a = b >> c) cracks, the volume of a sample crack is equal to
4xa2c/3. Hence
P,=N
+ 2) + (1 + 3v)
(; )
-na2c
=;n;E.
(2) If ps and pf denote the densities of the solid grain and
fluid, respectively, then the density ( p ) of the bulk poroelastic
material containing cracks is
+ 4E( 1- 2V) + (1+ 3V)( 9
405
45
+ ( 1 + 3 ~ ) --V--VE’+-E(1+3V)
[ 6 4
4
4
-4E‘)l
+
FV’
1
,
where
E’
=E
+ 0.75Qln.
Eq.(17) is solved for if, lying between 0 and 0.5, with the
given values of v, E and Q. The value of 2a thus found will be
used to calculate K / K and p/p. Note that the values of ii, K/K
and ji/p can be obtained for all values of R, and there is no
need to apply any restriction on Q or any other variable. The
variations of ii with v, E and R are shown in Fig. 1. Fig. 2
shows the variations of KIK and p/p with E and R, but v is
fixed at 113.
CRACKED POROELASTIC SOLID
The widespread distribution of aligned water-filled cracks,
suggested by Crampin (1985), and the role of pore fluid
diffusion in explaining the velocity anomalies prior to some
earthquakes (Nur 1972; Aggarwal et al. 1973) indicate the
co-existence of water-saturated pores and cracks in the Earth’s
crust. To define this co-existence, I consider the introduction
of cracks in a fluid-saturated porous solid.
0 1996 RAS, GJI 127,31-39
P = (1 - PIP* + PPf = P - P J P S - Pf)
’
(22)
(3) On the introduction of saturated cracks, the elastic
constants for the poroelastic solid are modified as follows:
-
K/K=l---
16 1 - V2
DE
9 1-22a
9
32
,&’p=l--(l-ij)
45
The bulk moduli of solid and fluid parts, i.e., K , and K ,
remain unchanged. The coefficient of fluid content, y, characterizing the volume of fluid entering the pores depends on
the effective porosity and hence remains unchanged. Biot’s
parameter also remains unchanged. The constants a and M
are modified using the modifications given by (23).
We now consider a more general case - partial saturation
of cracks in a liquid-saturated porous solid. For such a
situation, we define the saturation parameter 5 as the fraction
of cracks that are fully saturated in a unit volume of the bulk
material. The elastic constants for a partially saturated poroelastic cracked solid are obtained from (23) by replacing D
with 1- 5 + D5.The porosity fl changes to
34
M . D.Sharma
0.5 1
0.5 1
I
0.4
qv,,
-- --_________-----
0.3
I>
0.2
0.1
0.0
0.0
0.0
0.4
0.2
0.6
1.0
0.8
0.0
CRACK DENSITY
0.5
-
0.4
-
0.3
-
,
,/,/’
,.’/,,.’
,.*’
-
,:7*..,,
0.5
-
0.4
-
0.3
-
0.2
0.4
0.6
0.8
CRACK DENSITY
1.0
I>
4.Y
0.2
,y ,.*-
-
0.0
,C’
v=o
I>
0.1
v = 0.25
,(/*..,
,,v
y-..
-
1
.-*
..--1
1
1
,
1
,
1
1
1
Figure 1. Variation of effective Poisson’s ratio with crack density.
0.2
I
0.0
0.0
0.2
0.6
CRACK DENSITY
0.4
0.8
1.0
0.0
I
0.2
I
j
I
(
I
0.4
0.6
CRACK DENSITY
I
0.8
I
I
1.0
Figure 2. Variation of modulus ratio with crack density.
Similar to in Biot’s theory, in the cracked poroelastic solids
two compressional (Pc and P,) and one shear wave are
propagated. To observe the effect of the presence and
saturation of cracks on the velocities of these waves I restrict
the numerical study to a particular model. Following the
experimental results of Yew & Jogi (1976), assuming that the
laboratory samples of water-saturated limestone contain no
cracks, we choose the following values for the relevant
Q 1996 RAS, GJI 127, 31-39
Surface-wave propagation in a cracked poroelastic half-space
A=
1.444 x 10" dyne ern-',
35
p s = 2.5 g ~ m - ~ ,
LIQUID LAYER
p = 1.20 x 10" dyne cm-',
pf= 1.0 g ~ m - ~ ,
K , = 3.0 x 10" dyne cm-',
p = 0.144.
(25)
H
The bulk modulus of water, K f = K = 0.214 x 10" dyne ern-',
is used to calculate y and Q. The aspect ratio (cia) for the
cracks is assumedO to be equal to 0.01.
=
=
Using the above parameters, the variations of velocities of
Pf, P, and S waves with crack density are shown in Fig. 3.
The saturation parameter 5 is assigned the values 0, 0.25, 0.5,
0.75 and 1.
(MEDIUM I)
-x
.
CRACKED POROELASTIC
HALF SPACE
(MEDIUM II)
tz
DISPERSION OF SURFACE WAVES
Figure 4. Geometry of the model considered.
Geometry of the medium
The medium considered consists of a uniform layer of liquid
of thickness H resting on a cracked poroelastic solid halfspace. In the rectangular coordinate system (x, y, z) with the
z-axis in the direction of increasing depth in the solid, the
plane z = O is taken as the interface between the liquid
layer and the poroelastic solid (Fig. 4). Hence, the liquid layer
(medium I) occupies the region - H < z < 0 and the region
z > 0 is occupied by the cracked poroelastic solid (medium 11).
The plane z = - H represents the free surface of the liquid.
Particle motion is restricted to the 2-D x-z plane, and therefore
all the quantities are independent of the y-coordinate and
components in the y-direction vanish. The displacement vector
is u = (ux,0, uz), and the stress at the surface perpendicular to
the z-axis is (B,,, 0, oZJ.
where 0, = (Ao/po)'iz is the velocity of a dilatational wave in
liquid. 1, and pa are the bulk modulus and density of the
liquid, respectively.
For surface-wave propagation in the x-z plane, the solution
of (26) is
4,
= LAOexp(kz5,)
+ Bo exp(-kzt,)I
exPC%
-
ct)l,
(27)
where k denotes the horizontal wavenumber, c is the phase
The displacement and stress
velocity and 5, = (1 components are
Solution of basic equations
(1) For the fluid layer (medium I), the equation of motion in
terms of the potential 4, is
(2) For 2-D motion in the x-z plane, the displacements
I
\
I
\
i
0.61 I I
0.0 0.2
I
I I I I I I I
0.4 0.6 0.8 1.0
0.6!
0.0
CRACK DENSITY
Figure 3. Variation of velocity ratio with crack density.
0 1996 RAS, GJI 127, 31-39
I
1
I
i
;
0.6
I
1
0.8
CRACK DENSITY
0.2
0.4
I
I
I
1
1.0
0.0
0.4 0.6 0.8
CRACK DENSITY
0.2
1.0
36
M . D.Sharma
u = (ux,0, u,) and w = (wx,
0, w,) in the poroelastic solid are
Pl@l
w,=-+-
ax
P2@2
ax
+-P3@3
aZ ,
w,=-
P l a h +--P2W2
aZ
aZ
where
P3%3
ax
(35)
(29)
where & is the component of Y in the y-direction.
The potentials 4 j ( j = 1,2, 3) satisfy the wave equations (9).
Solutions for dl, 42 and 4, representing surface-wave propagation in the x-z plane and satisfying the decay condition at
z --* cc are
4j = A
exp [- kzCj
+ ik(x - ct)]
( j = 1,2, 3),
where t j = ( 1 - c2/v;)1/2 and the phase velocity
<min(v,, v2, v3).
Non-zero stress components at the x-y plane are
Zzx=p
(30)
c
is
Eq. (34) is the frequency equation for the propagation of
Rayleigh-type surface waves in a poroelastic solid half-space
lying under a uniform layer of liquid. This equation can be
written as F(c, kH) = 0, and hence indicates the dispersive
nature of propagating surface waves.
(2
-+- 2)
- 2 4 3
,
++ __
aax,
242
azax
Special cases
Boundary conditions
At the free surface of the liquid, i.e. at z = - H , we have
0.
this stress-free condition in (28) yields
( G ~=~ ) Applying
~
B,
=
(1) When H -+ co, we have So = 1 and Sb = to, and eq. (34)
reduces to the frequency equation for Stoneley waves at the
interface of a liquid and liquid-saturated porous solid (Hazra
1984).
(2) When H -+ 0, we have So= 0 and Sb = 2, and eq. (34)
reduces to
C2
- A o exp(- 2kHt0),
(a
and hence
C2
+ P ~ 02) ? R ~ - ( E+ p i ) v:- R
=O.
(37)
This is the frequency equation for Rayleigh waves in a
~ o = ~ o ~ ~ P ~ - ~ ~ 5 o -exPC-kto(z+H))
~ ~ ~ ~ P C ~ S o ~ poroelastic
~ + ~ ~ half-space
1
(Deresiewicz 1962).
x exp[ik(x
-ct)].
(32)
Following Deresiewicz & Skalak (1963), the boundary
conditions appropriate for the interface between liquid and
a liquid-saturated porous solid are the continuity of stress
components, liquid pressure and normal component of
displacement, i.e. at z = 0
(~,z)ll= (GZJl >
(33)
Applying the boundary conditions (33) we obtain a system
of four homogeneous equations in A , , A,, A , and Ao. A nontrivial solution of this system of equations requires a frequency
equation to be satisfied. The equation is
Numerical results
To solve the frequency equation (34) and to study the dispersion of surface waves, numerical work is restricted to a
particular model. To model the oceanic crust, we consider
water-saturated limestone lying under a uniform layer of water.
The relevant constants for water-saturated limestone are the
same as in the previous section. The bulk modulus and density
of oceanic water are assumed to be
do = 0.214 x 10" dyne cm-2,
po = 1 g ~ m - ~ .
(38)
Using the numerical values of all the elastic and dynamical
constants, the frequency equation (34) is solved for c/v' with
given values of the non-dimensional number kH. v' is a fixed
value. When 5 = 1 it represents the velocity of a P, wave with
E = 0, and when E = 0.2 it is the velocity of P , wave with 5 = 0.
The group velocity ( U ) is obtained numerically using the
formula
Fundamental modes of phase velocity, for different values of
varying with kH are shown in Fig. 5. Phase-velocity
and group-velocity curves for the fully saturated case are
shown in Fig. 6.
5 and E ,
0 1996 RAS, GJI 127, 31-39
31
Surface-wave propagation in a cracked poroelastic half-space
Figure 5. Variation of phase velocity with dimensionless wavenumber. (a) E = 0.2; (b)
1.0
= 1.
- PHASE VELOCITY
- - - - GROUP VELOCITY
-
- - - - PHASE VELOCITY
GROUP VELOCITY
-
1.0
-
-
0.8 -
0.8
-
-
cG O.= 9
w
>
-
0.4
-
>
\
\
\
0.4
\
\
\
\
-
\
\
\
\
\
0.2 -
\
,,
'------
-
__-c
0.2
-
(4
0.0
I I I I I I I I I I I I I I I " I I I
0.0
0.5
1 .o
1.5
2.0
DISCUSSION O F N U M E R I C A L RESULTS
Fig. 1 shows that, if the Poisson's ratio of an uncracked solid is
zero, it will remain unchanged even after the introduction of any
number of dry cracks. For E = 9/16, the effective Poisson's ratio V
vanishes for any value of v provided that the cracks are completely
dry. V = v at E = 0 indicates that there is no change in the elastic
properties of the solid in the absence of cracks. For a fixed value
of v, V increases with increasing Q which implies that the effective
0 1996 RAS, G J I 127, 31-39
.\._____-----
(b)
0.0
1
0.0
1
1
1
0.5
~
1
1
1
1 .o
1
~
1
1
1.5
'
1
~
1
1
1
1
~
2.0
Poisson's ratio increases with increasing bulk modulus l? of the
saturating fluid and/or decreasing aspect ratio c/u. In a dry cracked
solid, V is independent of non-zero v when E 2 9/16. For any fixed
E the value of V does not change considerably when C2 assumes a
value greater than 100. Therefore the case R + 00 can be approximated by C2 = 100 (Fig. 1). For large values of C2 ( 210) the
variations of V with E are approximately linear, whatever the value
of v. So for values of iz 2 10 we can write V = v + constant x E.
Fig. 2 shows that the introduction of cracks decreases the
38
M . D.Sharma
bulk modulus and rigidity modulus of an elastic solid. The
increase in E results in the decrease of K and p. The effective
moduli K and p increase with increasing R. This implies that
an increase in the bulk modulus of the saturating fluid and/or
a decrease in the aspect ratio c/a increases the elastic moduli
of the cracked solid. These effective moduli vanish at E = 9/16,
and become negative for ~ > 9 / 1 6 ,only when cracks are dry.
Variations in the values of Q > 1 show more effect on l? than
on p . The variation of p / g with E is approximately linear for
R 2 10 and l?/K zz 1 for Q 2 100. The change in the values of
K/K and p/g is not very significant for large values of Q
( 2 LOO), and therefore R -+ 03 can be approximated by 52 = 100.
Fig. 3 shows that the velocities of all three waves in watersaturated cracked limestone decrease with increasing E. An
increase in the value of the saturation parameter increases
the velocities of all three waves. The velocity of the shear
waves is least affected by a change in the value of the saturation
parameter, whereas the velocity of the P, waves is most affected.
A very small change from complete saturation to partial
saturation of cracks affects the velocities of both the dilatational
waves to a large extent. The velocities of the Pf and S waves
vary approximately linearly with E for all values o f t ; however,
the variation in the velocity of the P, wave with E is linear
only when 5 is very near to 1. In the case of partial saturation,
i.e. 5 < 1, the velocity curves are truncated at different values
of E. For given 5, the truncation occurs at the value of E at
which the effective bulk modulus of water-saturated limestone
becomes negative. In the present problem, these values of E
are 0.24, 0.30, 0.40 and 0.61 for t = O , 0.25, 0.5 and 0.75,
respectively. For 5 = 1, the velocity of shear waves decreases
more rapidly with E than it does for dilatational waves.
Fig. 5 shows that, for a fixed value of E = 0.2, the phase
velocity decreases with increasing saturation parameter t. For
higher values of E and for 5 < 1, the surface wave does not
exist because, for these combinations of E and 5, the effective
bulk modulus of water-saturated limestone becomes negative.
The phase velocity decreases with increasing k H , and the
gradient of c with kH is so large that the group velocity
becomes negative. For a fixed value of g, surface waves exist
only in a finite part of the kH domain, which varies with the
choice of elastic parameters and the value of E. For a fixed
value of 5 = 1, the phase velocity decreases with increasing E
when the value of kH is small (i.e. < l ) . For higher values of
k H , the phase velocity increases with increasing crack density E.
Fig. 6 shows that the group velocity is positive only when 5 =
1 and the value of E is greater than about 0.7. Up to kH z 1.3,
the group velocity decreases with increasing k H , and after that
it increases to touch the phase velocity at very large values
of kH.
It is concluded that the parameter R representing the bulk
modulus of the saturating fluid and the aspect ratio of cracks
have a definite effect on the effective Poisson’s ratio and elastic
moduli of an elastic solid. The extreme values of R as 0 and
co are not sufficient to characterize the effect of cracks. For a
cracked solid where the cracks have a very small aspect ratio
and/or are saturated with a hard fluid, we can write
<
8= v
+ CIE,
K/Kzl,
p / p = 1+ C Z E ,
where c1 and c2 are constants.
An increase of saturation increases the velocity of all three
waves propagating in a poroelastic solid. The effect of the
degree of saturation of cracks is much greater on the velocities
of both the dilatational waves than it is on that of the shear
wave. The velocity of the fast P wave and S wave varies
linearly with crack density. The existence of surface waves in
a cracked water-saturated limestone half-space under a uniform
layer of water depends upon the degree of saturation and the
crack density. The phase velocity decreases with increasing
saturation. The group velocity can assume positive values only
when the limestone contains fully saturated cracks and the
crack density assumes high values.
In this study, a more realistic model of the focal region of
an earthquake has been given, and a general relation to find
the effective Poisson’s ratio and effective elastic constants of
any cracked material has been derived. The knowledge of the
effect of crack density and saturation on the velocities of
propagation of waves in a poroelastic solid will lead to a
clearer understanding of the velocity anomalies observed prior
to some earthquakes. I hope this piece of work will be useful
in the study of the prediction of earthquakes by dilatancydiffusion processes.
ACKNOWLEDGMENTS
This work is a contribution towards a project entitled ‘Wave
Propagation in Cracked Solids in Relation to Earthquake
Prediction’, funded by the Department of Science &
Technology, Government of India.
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