Effect of water-saturated elastic porous layers on intensity of seismic waves


Mardonov, B.

Soil Mechanics and Foundation Engineering 13(3): 221-224

1976


7.
K.
E.
Egorov,
"Analysis
of
bases
under
circular
foundation
of
finite
thickness,"
Pro-
ceedings
of
the
Seventh
International
Conference
on
Soil
Mechanics
[in
Russian],
Stroiizdat,
Moscow
(1969).
8.
A.
M.
Gorlov
and
R.
V.
Serebryanyi,
Automated
Analysis
of
Rectangular
Plates
on
Elastic
Bases
[in
Russian],
Stroiizdat,
Moscow
(1968).
9.
E.
A.
Neustroev
and
A.
I.
Tseitlin,
"Analysis
of
elastic
plates
on
elastic
bases,"
Osnovaniya,
Fundamenty
i
Mekhan.
Gruntov,
No.
5
(1971).
10.
T.
A.
Malikova,
"Computer
analysis
of
large
rectangular
plates,"
Osnovaniya,
Fundamenty
i
Mekhan.
Gruntov,
No.
4
(1968).
11.
S.
N.
Yagolkovskii,
"Effect
of
consideration
of
cohesion
between
elastic
layer
and
underlying
base
on
results
of
beam
analysis,"
Osnovaniya,
Fundamenty
i
Mekhan.
Gruntov,
No.
4
(1969).
EFFECT
OF
WATER-SATURATED
ELASTIC
POROUS
LAYERS
ON
INTENSITY
OF
SEISMIC
WAVES
B.
Mardonov
UDC
624.042.7:624.131.3
Different
models
of
porous
water-saturated
media
and
the
problem
of
the
propagation
of
seismic
waves
in
them
have
been
discussed
in
[1-7].
In
this
article,
the
writer
describes
an
investigation
of
wave
propagation
in
water-saturated
porous
media
containing
a
small
quantity
of
gas
(air)
pores
fixed
with
respect
to
the
solid
skeleton,
and
examines
the
prob-
lem
of
the
passage
of
seismic
waves
through
a
soil
layer
located
between
an
elastic
layer
and
a
semiinfinite
mass.
Such
a
problem
arises,
for
example,
in
the
investigation
of
the
effect
of
the
water
table
on
the
intensity
of
seismic
waves.
The
process
of
wave
propagation
in
a
water-saturated
soil
is
studied
here
on
the
basis
of
M.
A.
Biot's
model
medium
[2],
the
equations
for
whose
motion
in
Cartesian
coordinates
x
and
y
are
written
in
the
form:
(
Pit
at2
+fin
p.
029)
+b
at
at
0-1-
2
(DPI
0c2)=2N)
Atpv-I-Q
Aqh;
ao
82(pi. ,
82
92
b
(--51
8
a
=Q
R
q):
;
P22
T
a
1
2
P22
at
dot
,
a=
1
,
2
,
.
(
a
ih
a
ih
pii
-
r-
P12
T
0
---)
=
N
A
Ilh
;
1
2
01
2
at
at
a2%
,
82%
_
b
(
a
14
_
a,,,
\
=
0;
PISS
-
r
P22
0
i
2
a
is
at
at
I
as
as
A
=
a
x:
+
ay
,
,
in
which
VI
and
4
0
2
are
the
longitudinal
potentials;
tp
l
and
4,
2
are
the
transverse
potentials;
Pit
=
M
Ps
P12
;
17
22
(
1
t7)
p
i -
p
L2
;
Pis
is
the
effective
mass;
m
is
the
porosity;
and
p
s
and
pf
are
the
densities
of
the
solid
skeleton
material
and
of
the
fluid,
respectively.
The
coefficient
b
characterizes
the
dis-
sipative
properties
of
the
medium.
The
elastic
constants
A,
N,
Q,
and
R
are
expressed
in
terms
of
the
mechanical
parameters
of
the
phases
by
the
equations
presented
in
[8]:
A
+
2
EL
±
N
Q
a
Ro
RRo
13; 13
-
Roe
m
K
o
m
m
Ko
+
a
R
o
,
a
=
1
-
m
- 7
0
-
Moscow
State
University.
Translated
from
Osnovaniya,
Fundamenty
i
Mekhanika
Gruntov,
No.
3,
pp.
42-43,
May-June,
1976.
This
material
is
protected
by
copyright
registered
in
the
name
of
Plenum
Publishing
Corporation,
227
West
17th
Street,
New
York,
N.Y.
10011.
No
part
of
this
publication
may
be
reproduced,
stored
in
a
retrieval
system,
or
transmitted,
in
any
form
or
by
any
means,
electronic,
mechanical,
photocopying,
microfilming,
recording
or
otherwise,
without
written
permission
of
the
publisher.
A
copy
of
this
article
is
available
from
the
publisher
for
$7.50.
(1)
221
in
which
A,
p,
and
K
are
Lame's
coefficients
and
the
modulus
of
dilatation
of
the
porous
skeleton;
R
o
is
the
modulus
of
compressibility
of
the
fluid;
and
K
o
is
the
true
modulus
of
compressibility
of
the
solid
skeleton.
Equations
(1)
describe
the
process
of
propagation
of
damped
waves,
two
of
them
being
longitudinal
and
one
transverse.
As
shown
in
[8],
in
the
case
of
propagation
of
monochro-
matic
waves
of
very
low
frequency,
the
longitudinal
wave
of
the
first
type
experiences
a
negligible
damping,
whereas
the
wave
of
the
second
type
vanishes
for
all
practical
purposes
because
of
considerable
damping.
If
the
propagating
wave
has
a
very
high
frequency,
the
effect
of
the
viscosity
of
the
fluid
can
be
neglected.
In
the
last-mentioned
case,
undamped
waves
are
propagated
in
the
medium,
two
of
them
being
longitudinal
and
one
transverse,
whose
velocities,
according
to
[2],
are
equal
to:
C
1.
2
D
(P
R
—Q
2
)
M
,
22
2
M
in
which
D.--
-P9
22
+Rp
i1
-2Qp
12
M
p
n
p
n
--
(4
2
;
P
A
-1-
2
N.
It
should
be
noted
that
the
system
of
equations
(1)
is
valid
for
media
consisting
of
two
components
and
rules
out
the
presence
of
a
third.
If
in
addition
to
the
liquid
component
(water),
the
medium
contains
a
certain
amount
of
air,
then
for
the
investigation
of
the
wave
processes
it
is
necessary
to
replace
in
Eqs.
(1)
the
value
of
R
o
by
the
equivalent
modulus
of
compressibility
R
eq
,
according
to
the
equation
Ro
Rw
Req
(
—a)
R
w
+
8
Ro
in
which
R
w
is
the
modulus
of
compressibility
of
the
air;
and
c
is
its
content
by
volume.
K
o
3
()
--
vo
Figure
1
shows
the
relations
of
the
values
c
i
=
/c
o
(in
which
c
o
=
v
,
Ps
I
I-
vo
i
=
1,
2;
and
v
o
is
Poisson's
ratio
of
the
solid
skeleton
material)
to
the
porosity
m
for
different
values
of
c.
For
the
analyses,
it
was
assumed
that
R
°
/K
°
=
0.2,
of/o
s
=
0.4,
and
v
o
=
0.2,
and
use
was
made
of
the
experimental
curve
for
the
variation
of
the
ratio
K/K
o
as
a
function
of
the
porosity
presented
in
[9].
Poisson's
ratio
for
the
water-saturated
soil
was
assumed
to
be
equal
to
0.3.
The
curves
obtained
show
that
in
the
absence
of
air
the
velocity
of
a
longitudinal
wave
of
the
first
type
is
always
greater
than
the
propagation
velocity
of
sound
in
the
fluid
(dashed
line).
Even
a
small
air
content
leads
to
a
substantial
decrease
in
this
velocity
(curves
2
and
3).
The
curves
for
c
2
show
that
the
velocity
of
a
longitudinal
wave
of
the
second
type
de-
pends
little
on
the
air
content.
Let
a
plane
nonstationary
wave
impinge
at
an
angle
0
0
on
a
water-saturated
layer
of
an
elastic
semiinfinite
mass.
We
will
denote
by
(ux
(0)
,
uy
(o)
)
.
and
(u
x
(
2
),
uy(
2
))
the
compo-
nents
of
the
travel
vector
of
displacement
in
the
lower
semiinfinite
mass
and
in
the
elastic
layer,
respectively,
which
satisfy
Lame's
equations,
and
by
(u
x
(
1
),
lly(
1)
)
and
(v
x
(
1
),
vy(
1
))
the
components
of
the
vector
of
shift
of
the
solid
and
liquid
phases
in
the
water-saturated
layer.
Let
us
direct
the
x
axis
along
the
boundary
between
the
water-saturated
layer
and
the
lower
elastic
semiinfinite
mass,
and
the
y
axis
vertically
upward.
At
the
boundary
points,
the following
conditions
should
be
satisfied:
0)
0)
0)__
u
0).
0)__
v
0).
ux
==
ux
;
uy
,
Y
a(0
YU
)
=
a(I)
+
a
;
crT
y
)
=
(24
1
y
)
for
y
=
0
;
YY
Ux
(I)
llx
(2)
;
tty
(1)
U
-(
y
2
)
.
a
yd
i
(r(2)
a
=
0;
e
xy
l)
=
a
(
x
2
y
)
for
H;
a
(2
ify
)
0;
13
(2
xy
)
=0
for
y=.
H
h,
(
2
)
and
ayy
(2)
are
the
components
of
the
stress
tensor
in
the
are
the
same
components
for
the
solid
skeleton
in
the
water-
in
which
a
yy
(
0
),
a
x
(
0
),
a
yy
(
1
),
elastic
media;
a
yy
(')
and
a
(
1
)
222
2
3
A,
2
A
o
3
_ _
_
_
3
a18
0.32
0.
.
5
4/6
432
0,48
1354
4
8
m
Bc
Bo
2
3
432
448
454
48
m
0
,
5
Fig.
1
Fig.
2
Fig.
1.
Graphs
for
relations
of
wave
propagation
velocities
e
l
and
c
2
to
porosity
m
for
different
values
of
e.
1)
e
=
0;
2)
0.006%;
3)
0.02%.
Fig.
2.
Graphs
for
relations
of
wave
refraction
coefficients
to
poros-
ity
m
for
different
values
of
y.
1)
0.4;
2)
0.8;
3)
1.2.
saturated
layer;
c
=
-mpf;
pf
is
the
pressure
in
the
fluid;
and
h
and
H
are
the
thicknesses
of
the
elastic
and
water-saturated
layers,
respectively.
In
Eqs.
(2),
a
=
0
means
that
the
fluid
cannot
flow
freely
through
the
boundary
y
into
the
elastic
layer.
Figure
2a
shows
the
curves
for
the
relation
of
the
refraction
coefficients
of
longitudi-
nal
waves
of
the
first,
A
l
/A
0
,
and
second,
A2/A0,
types
in
the
elastic
layer
to
the
porosity
in
the
case
of
normal
incidence
of
a
longitudinal
wave
with
an
amplitude
A
0
,
and
Fig.
2b
shows
similar
curves
for
the
refraction
coefficient
of
a
transverse
wave
B
i
/B
o
in
the
case
of
normal
incidence
of
a
transverse
wave.
The
curves
were
constructed
for
instants
of
time
when
account
is
not
taken
of
the
effect
of
the
reflected
waves
from
the
boundaries
y
=
H
and
y
=
H
h.
Here,
the
thickness
of
the
elastic
layer
(or
the
water
table)
does
not
affect
the
amplitude
of
the
refracted
waves.
For
construction
of
the
graphs,
the
notation
y
=
/a
2
/a
0
was
introduced,
in
which
a
o
and
a
2
are
the
propagation
velocities
of
the
longitudinal
waves
in
the
lower
semiinfinite
mass
and in
the
elastic
layer,
respectively.
For
the
elas-
tic
layer,
v
=
0.3.
It
follows
from
Fig.
2
that
the
presence
of
a
water-saturated
layer
be-
tween
the
elastic
media
may
lead
to
a
decrease
in
the
amplitude
of
the
refracted
waves.
LITERATURE
CITED
1.
Ya.
I.
Frankel',
"Toward
a
theory
of
seismic
and
seismoelectric
phenomena
in
wet
soil,"
Izvestiya
Akad.
Nauk
SSSR,
Ser.
Georgraf.
i
Geofiz.,
8,
No.
4
(1944).
2.
M.
A.
Blot,
"Mechanics
of
deformation
and
propagation
of
acoustic
waves
in
a
porous
medium,"
in:
Mechanics
[Russian
translation],
Collection
of
Translations
No.
3
(1962).
3.
Kh.
A.
Rakhmatulin,
"Principles
of
the
gasdynamics
of
interpenetrating
media,"
Prikl.
Matem.
i
Mekhan.,
20,
No.
2
(1956).
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V.
N.
Nikolaevskii,
"Linear
approximation
in
the
mechanics
of
compacted
media,"
Izves-
tiya
Akad.
Nauk
SSSR,
Otd.
Mekhanika
i
Mashinostroenie,
No.
5
(1962).
5.
V.
N.
Nikolaevskii,
K.
S.
Baskiev,
A.
T.
Gorbunov,
and
T.
A.
Zotov,
Mechanics
of
Saturated
Porous
Media
[in
Russian],
Nedra,
Moscow
(1970).
6.
V.
P.
Stepanov,
"Reflection
of
sound
waves
from
surfaces
dividing
different
two-compo-
nent
media,"
Trudy
VNllneftegaza,
No.
42,
Nedra,
Moscow
(1965).
223
7.
P.
P.
Zolotarev,
"Investigation
of
processes
of
elastic
deformation
and
heat
transfer
in
porous
media,"
Author's
Abstract
of
Candidate's
Dissertation
[in
Russian],
VNIInef-
tegaza
(1965).
8.
L.
Ya.
Kosachevskii,
"Propagation
of
elastic
waves
in
two-component
media,"
Prikl.
Matem.
i
Mekhan.,
23,
No.
3
(1959).
9.
Physicomechanical
Properties
of
Rocks,
Soils,
and
Minerals
under
High
Pressures
and
Temperatures
[in
Russian],
Akad.
Nauk
SSSR,
Moscow
(1974).
224