Third-Order Elastic Moduli of the Dry and Water Saturated Rocks
Yanli Qu, Jinxia Liu
*
, Zhiwen Cui and Weiguo Lv
Department of Acoustics and Microwave Physics, College of Physics, Jilin Univercity, Changchun, China.
Keywords: Third-Order Elastic Moduli, water saturated rocks, acoustoelastic
Abstract: On the basis of acoustoelastic theory of elastic solid, on two different rocks acoustic velocities were
measured in dry state and water saturated state under uniaxial stress. Using the least-squares method and
error analysis, we obtained the third-order elastic moduli of dry rocks and saturated rocks. Then, by analogy
with Biot theory, we introduced the seepage strain to modify the acoustoelastic theory in the porous media,
and established an equivalent model of acoustoelastic theory in porous media. Using the above method, we
acquired the high precision third-order elastic moduli of fluid-solid coupling. By comparing the experiment
values with theoretical values of the acoustic velocities in rocks, the experiment values and theoretical
values show good agreement, and the feasibility of the equivalent model has been proved.
1 INTRODUCTION
The acoustoelasticity is the acoustic velocity of
elastic wave that changes with the stress (Pao et al.,
1984). Before the 1980s, the acoustoelastic theory is
mainly applied to metal media. Many scientists have
studied the acoustoelastic theory. For example,
Johnson and Shankland (1989), Meegan et al (1993),
Winkler and Liu (1996) studied acoustoelastic
effect. They found that the nonlinear effect of rock is
much more obvious than other media, and proved
the existence of acoustoelastic effect in rock through
a large number of experiments. The acoustoelastic
theory was extended to the porous rock with
compatibility conditions of acceleration waves by
Grinfeld and Norris (1996). Ba et al concluded the
method of Grinfeld and Norrisby by including solid
and fluid finite strains (Ba et al., 2013). Tian et al
studied the acoustoelastic theory of fluid-saturated
porous media in natural and initial coordinates (Tian,
2014).
Winkler and Liu measured third-order elastic
moduli in a variety of dry rocks (Winkler and Liu,
1996), and obtained the theoretical results on the
basis of the acoustoelastic theory of Thurston and
Brugger (1964), the equations are shown in Table
1.They found that this theory describes the relation
between acoustic velocities and stress. The porous
medium has seven independent third-order elastic
moduli, so there are certain limitations in laboratory
measurement (Grinfeld and Norris, 1996).
Currently, there are few experiments about
measuring the values of third-order moduli in porous
medium. Winkler and McGowan extended the work
of Winkler and Liu to water-saturated rocks
(Winkler and McGowan, 2004), and also obtained
the theoretical results on the basis of the
acoustoelasticity theory of Thurston and Brugger.
Winkler and McGowan found obvious difference
between theory and experiment. In this paper, we
also carried out an experiment to measure the third-
order elastic moduli of dry and saturated rocks under
uniaxial stress. On the basis of the acoustoelastic
theory of Winkler and Prioul, by analogy with Biot
theory, we established the equivalent acoustoelastic
model of water saturated rock. In addition, we
compared the experimental and theoretical values of
acoustic velocities.
2 PROCEDURE
The acoustoelastic formulas of dry rock under
uniaxial stress in Table 1 (Thurston and Brugger,
1964),where is the acoustic velocity in the
unstressed condition; is density; P is longitudinal
wave; S
//
is shear wave with polarization direction
parallel to stress; S
⊥
is shear wave with polarization
direction vertical to stress; is the acoustic velocity
of changing with stress. , and are strain
0
V
ρ
V
11
E
22
E
33
E
428
Qu, Y., Liu, J., Cui, Z. and Lv, W.
Third-Order Elastic Moduli of the Dry and Water Saturated Rocks.
In Proceedings of the International Workshop on Environment and Geoscience (IWEG 2018), pages 428-433
ISBN: 978-989-758-342-1
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

components. The , , and are
third-order elastic modulus, which are related to
third-order elastic modulus ,and:
,
,and
.
Table 1: The acoustoelastic formulas of dry rock under
uniaxial stress.
Mode
P(longitudinal)
S
//
(shear, parallel to stress)
S
⊥
(shear, vertical to stress)
The experiment whose results are presented on
this paper mainly measures the velocities of three
types of waves under uniaxial stress by ultrasonic
pulse transmission method. The three types of waves
are longitudinal wave and two transverse waves, one
with polarization direction vertical to stress and the
second one, with polarization direction parallel to
stress. The samples were selected from a quarry, and
the rocks are defined as rock A and rock B. Rock A
is yellow rust granite and rock B is granite 654.
The reference states (zero stress) of rocks are
shown in Table 2. Normal state denotes the natural
rocks under the indoor temperature. Dry state
denotes the rocks dried in the constant temperature
drying oven. Saturated state denotes the water
saturated rocks.
The schematic diagram of sample under pressure
is shown in Figure 1, which determines the direction
of coordinate axis.
Figure 1: Schematic diagram of sample under pressure.
Rock A is fragile in water saturated state, so, it
was loaded at 0-10MPa under dry state, and loaded
at 0-8MPa under saturated state. Rock B was loaded
at 0-10MPa under dry state and saturated state. The
waveforms about rock A and rock B have similar
patterns of change. Therefore, this paper only shows
rock A’s. The waveforms in dry state are shown in
Figure 2. The waveforms in saturated state are
shown in Figure 3. We can clearly see that the
waveforms move forward with the increase of the
stress.
The experimental measurements of acoustic
velocities of rock A and rock B are shown in Figure
4 and Figure 5, respectively. It can be seen that the
acoustic velocities in saturated rocks are faster than
that of dry rocks. In the reference state (i.e., zero
stress), the percentages of the increase of acoustic
velocities are for rock A, P: 25.91%, S
//
: 9.55%, S
⊥
:
17.77%; for rock B, P: 22.61%, S
//
: 7.93%, S
⊥
:
9.86%.
Table 2. The reference state properties of rock A and rock B.
RockA RockB
Normal Dr
y
Saturate
d
Normal Dr
y
Saturate
d
Densit
y(
k
g
/m
3
)
2575.12 2574.58 2584.98 2760.11 2755.56 2759.72
P-wave
(
m/s
)
3623.1 3441.9 4333.7 5036.2 4912.7 6023.4
S
//
-wave(m/s) 2473.2 2323.9 2545.8 3227.0 3133.1 3381.5
S
⊥
-wave(m/s)
2322.4 2052.6 2417.3 3142.1 3051.5 3352.3
Porosity 0.0104 0.00416
111
C
112
C
144
C
155
C
1
ν
2
ν
3
ν
321111
86 ννν ++= C
21112
2 νν += C
2144
ν = C
32155
2 νν += C
2
0
2
VV ρρ −
221123311211111
ECECEC ++
331551115522144
ECECEC ++
111552215533144
ECECEC ++
Third-Order Elastic Moduli of the Dry and Water Saturated Rocks
429

Figure 2a: P-waveform of dry rock A.
Figure 2b: S
//
-waveform of dry rock A.
Figure 2c: S
⊥
-waveform of dry rock A.
Figure 3a: P-waveform of saturated rock A.
Figure 3b: S
//
-waveform of saturated rock A.
Figure 3c: S
⊥
-waveform of saturated rock A.
Figure 4: Experimental values of acoustic velocities in dry
sample and saturated sample of rock A.
Figure 5: Experimental values of acoustic velocities in dry
sample and saturated sample of rock B.
IWEG 2018 - International Workshop on Environment and Geoscience
430

According to the formulas in Table 1, we can
obtain multiple sets of third-order elastic moduli (
, and ) under different stress. Then, using
to determine the
minimum deviation of third-order elastic moduli
values, where is experimental value; is the
predicted value based on acoustoelastic theory
( )(Prioul et al., 2004).The values of third-
order elastic moduli( , and )were determined
and are presented on as Table 3.
Referring to the paper of Prioul et al (Prioul et al.,
2004), we analyzed the errors of the third-order
elastic moduli. We assume that the third-order
elastic moduli with are ,
and the disturbance away from is . The
increment of is . We
obtained . The results are shown in
table 3. Comparing the values of third-order elastic
moduli( , and ), we found that the values of
in dry rocks and saturated rocks are quite
different, and the changes of and are not
obvious. We know that and are related to shear
waves. In fact, the shear moduli of dry rocks and
saturated rocks should be equivalent under ideal
condition.
Table 3: The third-order elastic moduli of rocks.
State Rock A Rock B
, GPa
, GPa
Dry
-28109 45 -7923.3 59.5
Saturated
-81899 170 -72507 500
, GPa
Dry
-11849 3 -7702.2 41.5
, GPa
Saturated
-18907 7.5 -10043 27.5
, GPa
Dry
-2515.7 7.8 -2657.0 6.6
, GPa
Saturated
-2144.6 13.9 -1086.8 30.4
3 THE EQUIVALENT MODEL OF
ACOUSTOELASTIC THEORY
OF WATER SATURATED
MEDIA
Similar to Winkler and McGowan’s work, we dealt
with the experimental data based on the
acoustoelastic theory of elastic solid. In fact, from
the Biot theory (1972), the deformation of seepage
strain is inevitable when the rock is immersed in
water. Therefore, by analogy with the Biot theory,
we proposed the seepage strain and established the
formulas of acoustic velocities in saturated rocks.
The equations are shown in table 4. is the density
of water saturated rocks; , and are the third-
order acoustoelastic moduli of fluid-solid coupling;
is the relative seepage strain, ( ;
, is the proportion of fluid content
in body strain; , is total fluid strain; , is
skeleton compression modulus; and , is the
particle compression modulus (Dupuy and Stovas,
2014)).
Table 4: The equivalent model of acoustoelastic
theory of water saturated rocks.
Mode
P(longitud
inal
)
S
//
(shear,
parallel to
stress
)
S
⊥
(shear,
vertical to
stress
)
In a limited experimental environment, in order
to verify the equivalent model of the acoustoelastic
theory of porous media, here we assumed that the
rocks are rigid. So the values of third-order elastic
moduli( , and )in Table 4 are approximately
equal to the values of dry rocks. Then, we used the
values of third-order elastic moduli( , and )in
Table 3 and formulas in Table 4, acquired the values
of , and as Table 5.
1
ν
2
ν
3
ν
[]
∑
−=
ij
pred
ij
mes
ij
CC
2
2
χ
mes
ij
C
pred
ij
C
2
VC
ij
ρ =
1
ν
2
ν
3
ν
2
min
χ
[]
1231121110
,, CCCa =
0
a
a
2
χ
2
min
22
χχχ −=Δ
63.6
2
≤Δ χ
1
ν
2
ν
3
ν
1
ν
2
ν
3
ν
2
ν
3
ν
1
ν
1
ν
±±
±±
2
ν
±±
2
ν
±±
3
ν
±±
3
ν
±±
s
ρ
H
55
h
66
h
s
ξ
ss
eαζ =
sb
kk −= 1 α
s
e
b
k
s
k
2
V
s
ρ
s
s
HECECECV ξρ ⋅++++
221123311211111
2
0
s
s
hECECECV ξρ ⋅++++
55331551115522144
2
0
s
s
hECECECV ξρ ⋅++++
66111552215533144
2
0
1
ν
2
ν
3
ν
1
ν
2
ν
3
ν
H
55
h
66
h
Third-Order Elastic Moduli of the Dry and Water Saturated Rocks
431

Table 5: The third-order acoustoelastic moduli of
fluid-solid coupling.
Rock A,
GPa
1176.7
8.6
-8282.4
46.5
-3401.7
40
Rock B,
GPa
42970
450
-12450
140
-1682.8
135
4 DISCUSSION
We measured the acoustic velocities in dry rocks
and saturated rocks under uniaxial stress and
obtained third-order moduli of dry rocks by the least
squares method. Second, on the theory of acoustic
elasticity of dry rock, we established the equivalent
model of acoustoelastic theory of porous media, and
acquired the values of the third-order elastic moduli
of fluid-solid coupling. Third, we compared the
experimental values and the theoretical values of
acoustic velocities in dry rocks and saturated rocks.
The comparisons between theoretical and
experimental values of dry rocks are shown in
Figure 6 and Figure 7, and the comparisons between
the theoretical and experimental values of water
saturated rocks are shown in Figure 8 and Figure 9.
Through the above results, one can find that the
values of (third-order elastic modulus) in dry
rocks and saturated rocks are quite different,
however, the changes of and (third-order
elastic modulus) are not obvious. It indicates that
contains the contribution of fluid-solid coupling.
Figure 6: Comparison of experimental values and
theoretical values of acoustic velocities in dry rock A.
Figure 7: Comparison of experimental values and
theoretical values of acoustic velocities in dry rock B.
Figure 8: Comparison of experimental values and
theoretical values of acoustic velocities in saturated rock
A.
Figure 9: Comparison of experimental values and
theoretical values of acoustic velocities in saturated rock
B.
H
55
h
66
h
±± ±
±± ±
1
ν
2
ν
3
ν
1
ν
IWEG 2018 - International Workshop on Environment and Geoscience
432

5 CONCLUSIONS
In this study, there are several conclusions: (1)The
values of third-order moduli have differences in dry
rocks and saturated rocks.(2)The acoustic velocities
are increasing with the increase of stress in dry rocks
and saturated rocks.(3)Through the comparisons
between theories and experiments, the equivalent
model of acoustoelastic theory of porous media can
describe the relationship between acoustic
velocitiesand stress. The feasibility of the equivalent
model of acoustoelastic theory has been proved. It
has a certain significance for the study of the
acoustoelastic effectin porous media.
ACKNOWLEDGMENTS
This work is supported by the National Natural
Science Foundation of China (Grant No. 41474098),
Natural Science Foundation of Jilin Province of
China (Grant No.20180101282JC) and the State Key
Laboratory of Acoustics, Institute of Acoustics,
Chinese Academy of Sciences (Grant
No.SKLA201608).
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