The effect of stresses on the sound velocity in rocks:
Theory of Acoustoelasticity and Experimental
Measurements
Xiaojun Huang, Daniel R. Burns, and M. Nafi Toksöz, ERL, MIT
Abstract
The theory of acoustoelasticity provides direct link between the change of elastic
wave velocities and residual stresses in solids. The general theory of acoustoelasticity is
reviewed. A number of experimental measurements of the effect of stresses on the sound
velocities in various types of rocks are compiled and compared to the acoustoelastic
theory. The theory of acoustoelasticity agrees within 1% of error with experiments
for stress levels that are representative for in-situ reservior conditions. With the measurements of Nur and Simmons, acoustoelastic theory is found to agree with Sayers’s
microcrack model within 2% of error, much smaller than experimental error which was
10%. We may safely conclude that the theory of acoustoelasticity is a macroscopic
version of the microcrack model and applicable to in-situ rocks.
1
Introduction
Knowledge of in-situ stresses is of great interest to the petroleum industry for many purposes.
In-situ stress is an important factor in engineering applications such as enhanced recovery of
hydrocarbons, prevention of sand production and borehole instability. In exploration, in-situ
stresses affect seismic anisotropy, caprock integrity, and whether or not fluids can flow along
or across faults.
The most accurate way by far for determining in-situ stresses is by conducting lowvolume hydraulic fracturing experiments (Zoback and Haimson, 1982). Apart from the
fact that such measurements are expensive and can be done at only a few locations along
the wellbore, hydraulic fracturing experiments are based on a number of assumptions that
barely hold in many real situations (Haimson, 1988). In recent years, some efforts have
focused on initiating an alternative mechanism to assess in-situ stresses using sonic well
logs. Following the establishment of a theoretical framework on wave propagating along a
borehole surrounded with prestressed formation (Sinha and Kostek, 1996), a multi-frequency
inversion scheme was developed and successfully applied to assess in-situ stress profiles using
a set of field sonic logs (Huang et al., 1999).
Both Sinha and Huang’s work are in the context of the theory of acoustoelasticity, the
basic theory that relates the change of sound wave velocities to initial stresses. This theory
invokes third-order elastic (TOE) constants to account for the nonlinear strain response to
1
stresses of finite magnitude. Developments in acoustoelasticity have been stimulated by
interest in the measurement of TOE constants of crystals and applied or residual stresses
in polycrystalline materials. Due to the presence of compliant mechanical defects (cracks,
microfractures, grain joints, etc), rocks in general display stronger nonlinear elastic behavior
than crystals or polycrystalline materials [Meegan et al. (1993), Johnson et al. (1993), and
Johnson and McCall (1994)]. In this paper, we will compile a number of experiments on
stress-induced changes on sound velocities in different types of rocks, and investigate how
well the theory of acoustoelasticity applies to rocks.
We shall first review the general theory of acoustoelasticity, and construct the theoretical
framework on the sound velocity change as a function of applied stresses. It is followed by a
discussion on many experiments on metal materials that confirm experimentally the theory
of acoustoelasticity. We then apply the theory of acoustoelasticity to a variety of published
experimental data on rocks. Finally, we shall conduct a comparison between the theory of
acoustoelasticity and Sayers’s microcrack model.
2
Theory of Acoustoelasticity
In the continuum theory, to establish a theoretical basis for the change of wave velocities by
initial stresses in a body, a nonlinear theory is required. The nonlinearity is caused by a large
deformation of the body and, in addition, a nonlinear stress-strain relation for the material.
The theory of acoustoelasticity was developed in the 1960s by introducing the second-order
effect of strain (third-order elastic stiffness) and separating the small motion superimposed
on a largely deformed body.
In this section, we review the derivation of basic equations of acoustoelasticity (Pao et al.,
1984). We first introduce three material configurations: the natural, the initial and the final
states. The natural and initial configurations refer to states when the material is free of
stresses and statically deformed, respectively. The final configuration denotes the material
state with wave-induced dynamic deformation superimposed on the static load (Fig 1). A
physical variable in the natural, initial, or final state is designated by a superscript label 0, i,
or f, respectively. The positions of a particle in the body at natural, initial, and final states
are measured by position vectors ξ, X, and x, respectively, all directed from the origin of a
common Cartesian coordinate system. The components of ξ and other physical quantities
which refer to the natural configuration are denoted by Greek subscripts; those of X and
others refer to the initial configuration by upper case Roman subscripts; and those of x and
others refer to the final configuration by lower case Roman subscripts. Thus ξα , XJ , and xj
(α, J, j = 1, 2, 3) are the components of position vectors in three respective configurations
(Fig 1).
The predeformation driven by T i , the initial Kirchhoff (the second Piola-Kirchhoff) stress
tensor that refers to the natural configuration, is static and T i must satisfy the equations of
equilibrium
∂
∂ui
i
[Tβγ
(δαγ + α )] = 0.
∂ξβ
∂ξγ
2
(1)
Figure 1: Coordinates for a material point at the natural (ξ), initial (X) and final (x)
configuration of a predeformed body.
3
The equation of motion for the body at the final state is
f
∂ f
∂ 2 uf
f ∂uα
Tβα + Tβγ
= ρ0 2α ,
∂ξβ
∂ξγ
∂t
(2)
where T f is the final Kirchhoff stress tensor that refers to the natural configuration and ρ0
is the mass density which refers to the natural state.
Subtracting Eq (1) from Eq (2), we obtain the equation of motion for the incremental
displacement in natural coordinates u(ξ, t),
∂
∂ui
∂ 2 uα
i ∂uα
[Tαβ + Tβγ
+ Tβγ α ] = ρ0 2 .
∂ξβ
∂ξγ
∂ξγ
∂t
(3)
So far the only assumptions made in the derivations are that the predeformation is static
and that the dynamic disturbance is small. We have not asked how the particles are carried
from the position ξ to X, nor have we imposed restrictions on the constitutive property of
the material. Thus the equations of motion Eq (2) are applicable to a body in general form
of predeformation, finite or infinitesimal, elastic or inelastic.
Now we introduce the constitutive relationship in the context of acoustoelasticity. One
basic assumption in the theory is that the material is hyperelastic, i.e. the material remains
elastic throughout the deformation. A body in deformation is accompanied by a change of
internal energy W (per unit mass) or free energy F (per unit mass). The law of balance of
energy states the following:
dW = θdS + Tαβ dEαβ /ρ0 , dF = −Sdθ + Tαβ dEαβ /ρ0 ,
(4)
where θ is the temperature, S the entropy, and F = W − θS. For a hyperelastic body, W is
a function of strain E and S, and F is a function of E and θ. Therefore, we have
Tαβ = ρ0 (
∂W
∂F
)S = ρ0 (
)θ .
∂Eαβ
∂Eαβ
(5)
The subscript S indicates an adiabatic thermodynamic process, and θ an isothermal process.
The function W (E) may be expanded about the state of zero strain,
1
1
ρ0 W (E) = cαβγδ Eαβ Eγδ + cαβγδη Eαβ Eγδ Eη + · · · .
2
6
(6)
f
i
A constitutive equation for Tαβ
or Tαβ
is thus obtained by neglecting the higher order terms:
1
1
f
f
f
i
i
i
i
f
Eη
, Tαβ
= cαβγδ Eγδ
+ cαβγδη Eγδ
Eη
.
Tαβ
= cαβγδ Eγδ
+ cαβγδη Eγδ
2
2
(7)
From the difference of these two equations, a constitutive equation for the incremental
stress tensor Tαβ is derived,
Tαβ = cαβγδ Eγδ + cαβγδη eiγδ eη ,
4
(8)
where the infinitesimal strain tensor ei and e are used where 2eαβ =
∂uα
∂ξβ
∂u
+ ∂ξαβ . The difference
f
i
of the Lagrangian strain tensor in the final and initial states Eαβ
and Eαβ
respectively, is
given approximately by
1 ∂uα ∂uβ ∂uiλ ∂uλ ∂uiλ ∂uλ
f
i
+
+
+
).
Eαβ = Eαβ
− Eαβ
= (
2 ∂ξβ
∂ξα
∂ξα ∂ξβ
∂ξβ ∂ξα
(9)
In terms of displacement gradients, the constitutive equation is
Tαβ = cαβγδ (δργ +
∂uiρ ∂uρ
∂uiγ ∂u
)
+ cαβγδη
.
∂ξγ ∂ξδ
∂ξδ ∂ξη
(10)
i
In Eq (10), only terms linear in ∂u
or ∂u
are retained.
∂ξ
∂ξ
Substituting the constitutive equations for Tαβ into Eq (3), we obtain the equation of
motion for u(ξ, t),
2
∂
∂uγ
i ∂uα
0 ∂ uα
[T
+ Γαβγδ
]=ρ
.
∂ξβ γβ ∂ξγ
∂ξδ
∂t2
(11)
This is the equation for the dynamic disturbance with reference to the natural coordinates.
The initial stress can be arbitrarily distributed and the material can have intrinsic anisotropy.
The coefficient Γαβγδ = Γγδαβ is of lower order symmetry than cαβγδ . It is the effective elastic
moduli of the material after predeformation.
Γαβγδ
∂uiγ
∂ui
= cαβγδ + cαβρδ
+ cρβγδ α + cαβγδη eiη
∂ξρ
∂ξρ
(12)
Within the framework of the aforementioned theory, the assumptions made that lead to Eq
(3) are as follows:
• The predeformation is static and the body is at equilibrium in the initial state.
• The superposed dynamic motion is small.
For a homogeneously predeformed medium, i.e. T i and ∂ui /∂ξ are constant throughout
the body, the equation of motion [Eq (11)] is reduced to
∂ 2 uγ
∂ 2 uα
= ρ0 2 ,
∂ξβ ∂ξδ
∂t
(13)
i
Aαβγδ = Tβδ
δαγ + Γαβγδ
(14)
Aαβγδ
where
are constant coefficients.
A plane sinusoidal wave is represented by
uα = Uα exp[i(κνβ ξβ − ωt)],
5
(15)
where U is a constant complex vector, ω the angular frequency, κ(= 2π/wavelength) the
wave number, and ν (a unit vector) the wave normal. The wave speed is given by v = ω/κ.
On substituting Eq (15) into Eq. (13), we obtain a system of equations for the amplitude
vector U:
[Aαβγδ νβ νδ − ρ0 v 2 δαγ ]Uγ = 0.
(16)
The associated characteristic equation is
|Aαβγδ νβ νδ − ρ0 v 2 δαγ | = 0.
(17)
Once the initial stress and initial displacement gradients are specified, and the values of ρ0 ,
cαβγδ , and cαβγδη in a medium are given, the eigenvalues (wave velocities) and eigenvectors
(polarization directions) of Eq (16) can be solved for each direction ν of propagation.
3
Acoustoelastic Experiments on Metal Materials
An early review of the acoustoelastic experimental work and analytical foundations can be
found in an article by Smith (Smith, 1963). Another review of the subject by Pao (Pao
et al., 1984) summarized the theory in a more general form and various techniques which
had been used up to that time for making acoustoelastic measurements.
In a series of detailed experiments, Hsu (Hsu, 1974) utilized the ultrasonic pulse-echooverlap technique to make absolute wave-speed measurements of the two principal shear
waves propagating normal to the loading direction in aluminum and steel. Hsu denotes the
uniaxial stress direction as ξ3 . According to the theory of acoustoelasticity in Section 2, the
velocity of the shear wave polarized in ξ1 direction v13 is
2
i
v13
= (v13 )20 + ΠT33
,
(18)
where Π is a constant depending on the elastic moduli and the TOE constants of the material.
i
(v13 )20 is the shear velocity in the stress-free state and T33
is the applied uniaxial stress. Hsu
observed that for aluminum and steel, the relative change in wave speed with applied stresses
2
is very small; therefore, v13
− (v13 )20 = (v13 + (v13 )0 )(v13 − (v13 )0 ) ' 2(v13 )0 (v13 − (v13 )0 ), thus
Eq (18) becomes
v13 = (v13 )0 +
Π
Ti .
2(v13 )0 33
(19)
i
Similarly, v23 is also a linear function of the applied uniaxial stress T33
. For three different
types of steel and four different types of aluminum, Hsu found that the velocity changes were
linear functions of the applied stresses (Figure 2).
Additional wave-speed measurements using various wave modes were made by Egle and
Bray (Egle and Bray, 1976) on rail-steel specimens parallel as well as perpendicular to the
loading axis (Figure 3). Hirao et al conducted a surface wave experiment with a mild steel
specimen (Hirao et al., 1981). Both theoretical and experimental results for the change of
Rayleigh wave with respect to the uniaxial strain are plotted in Figure 4.
6
Figure 2: Shear velocity changes as function of applied compressive stress in three steel
specimens. Vij denotes the shear wave polarized in the j th direction propagating in the ith
direction (Hsu, 1974).
Figure 3: Relative change in wave speed ( ∆V
) as a function of strain for a rail specimen
V0
(Egle and Bray, 1976).
7
Figure 4: Relative variations of Rayleigh wave transit time and velocity versus the uniaxial
strain (Hirao et al., 1981).
Figure 5: Specimen geometry and scan lines for the aluminum panel with central hole measured by Kino et al (Kino et al., 1978).
Figure 6: Specimen geometry and scan line for the double edge-notched aluminum panel
measured by Kino et al (Kino et al., 1978).
8
Figure 7: Stress contour plots of results for the aluminum panel with central hole. Solid
curve, theory; dotted curve, acoustoelastic experiment (Kino et al., 1978).
Figure 8: Stress contour of the double edge-notched aluminum panel. Solid curve, photoelastic measurements; dotted curve, acoustoelastic measurements (Kino et al., 1978).
9
Kino et al (Kino et al., 1978) made use of the phase comparison technique to make precise
time-of-flight measurements with longitudinal waves while the transducer is scanned on a
two-dimensional grid across specimens of aluminum. Geometries of specimens and scan lines
are shown in Figure 5 and 6. Stress fields of both specimens are inverted from measured
longitudinal wave velocities using the aforementioned theory of acoustoelasticity, and compared with theoretical results and results by another measurement scheme (photoelasticity).
Stress-field contour plots are displayed in Figures 7 and 8.
These and many other experiments confirmed experimentally the theory of acoustoelasticity in the context of metal materials.
10
4
Stress-induced Velocity Changes in Rocks
The stress-induced velocity change in rocks is a well-established observation. It is classically
modeled by a change in the alignments of cracks or any other alignment of micro-structural
flaws or defects, induced by a changing stress (e.g. (Sayers et al., 1990)). Only a few studies
take the approach of acoustoelasticity (Johnson and Rasolofosaon, 1996). It it advantageous
to employ the theory of acoustoelastic theory in estimating in-situ stresses, for it provides
direct links between stresses and the change of elastic wave velocities. In this section, we
shall first test the theory of acoustoelasticity against experimental measurements of different
types of rocks. We then make a connection between Sayers’s microcrack model and the
theory of acoustoelasticity.
Our goal here is not to present an exhaustive analysis of the experimental data available
from the literature, but to illustrate the relevance of the above analysis. We analyze high
precision data by Dillen (2000). We also take experimental measurements by Johnson (1993)
and Lo et al (1986) as an example of velocity changes with confining pressures, and Nur and
Simons (1969) with uniaxial stresses.
4.1
Colton Sandstone
In Dillen’s thesis, the ultrasonic experiment is carried out on a cubic block of Colton sandstone. The sample consists of lithic quartz and feldspar. It is fairly homogeneous and
has a porosity of about 3%. At zero stress state, the measured P-wave velocities in the
X− and Z−directions are approximately equal and differed by 5% from the velocity in
the Y −direction. We assume that the Colton sandstone is transversely isotropic with the
symmetric axis in the Y −direction. Figure 9 shows the load cycle ABCD as a function of
experiment time. The entire ABCD stress path has equal normal stresses in the X− and
Z−directions.
According to the characteristic equation (Eq (17)), when a plane wave is propagating in
the X−direction, ν1 = 1, ν2 = 0 and ν3 = 0; therefore, Eq (17) becomes
|Aα1γ1 − ρ0 v 2 δαγ | = 0.
(20)
In matrix form, it is
where
A11 − ρ0 v 2
A16
A15
0 2
A61
A66 − ρ v
A65
A51
A56
A55 − ρ0 v 2
= 0,
i
A11 = c11 + T11
+ (2c11 + c111 )ei11 + c112 ei22 + c113 ei33 ,
i
A55 = c55 + T33
+ (2c55 + c155 )ei11 + c144 ei22 + c344 ei33 ,
i
A66 = c66 + T22
+ (2c66 + c166 )ei11 + c266 ei22 + c366 ei33 .
(21)
(22)
(23)
(24)
In Dillen’s experiment, one of the principal initial strains is in the X−direction; therefore,
ei31 = ei21 = 0, thus A51 = A15 = A16 = A61 = 0. From Eq (21) we may obtain the
11
12
A
10
B
C
D
stress X = stress Z
stress (MPa)
8
6
4
2
0
0
stress Y
200
400
600
800
1000
1200
1400
experiment time (s)
Figure
5.1: cycle
Loading
cycle of
ABCD
of the tri-axial
pressure
machine.
To preserve
Figure 9:
Loading
ABCD
the tri-axial
pressure
machine.
To preserve
the intrinsic
the isotropy
intrinsic of
transverse
isotropy
of the insample,
the stress inis the
X-direction
is in the
transverse
the sample,
the stress
the X−direction
equal
to the stress
equal
to
the
stress
in
the
Z-direction.
Z−direction (Dillen, 2000).
the X- velocity
and Z-directions.
During
parts A, B,as
C,follows
and D, the X-force and the
compressional
propagating
in X−direction
Z-force increase. Part A shows a decrease of the stress in the Y -direction,
A11 the Y -direction was kept conwhereas during parts B and C the
.
(25)
vp2x stress
= 0 in
stant at 2 MPa and 4 MPa, respectively.ρFinally part D simulates increasing
hydrostatic stress conditions up to 10 MPa.
A11 is determined in Eq (22). According to the acoustoelasticity theory, initial strains ei11 ,
i
i
i
ei22 and ei33 are linearly related with applied stresses T11
, T22
and T33
through Hook’s Law.
i T
i results
i
i T
−1
i
i
Let E =5.2.3
[ei11 ei22 eExperimental
]
and
T
=
[T
T
T
]
,
we
have
E
=
C
T.
Noting
that T11
= T22
33
11 11 33
in the experiment.
C is the
of elasticsignals
moduli,
Fig. (5.2) shows
thematrix
full waveform
of compressional-waves as func
tions of experiment time (top horizontal
axis)
c11 c12 c13 and transmission travel time
(vertical axis). The bottom horizontal
axis indicates
the segments A, B, C,
C = c12 c22 c12 (26)
and D similar to those in Fig. (5.1).
source and receiver are both P-wave
c13 The
c12 c11 transducers aligned in the X-direction. The transducer-sample coupling, described
in the
previous
section, produces
traces
easily
Substituting
initial
strains
as a function
of appliedclean
stresses
intowith
Eq (22)
anddiscernible
then substituting
first
arrivals
and
amplitudes.
To
visualize
shear-wave
splitting,
shear-wave
Eq (22) into Eq (25), we find that the square of compressional velocity
propagating in
transducers
were
aligned
in
the
X-direction
and
polarized
in
the
Y
Z-plane
X−direction is a linear function of applied stresses T11 and T22 :
in a direction of 45 degrees to the symmetry axis. Fig. (5.3) displays the
i
i
vp2x = (vpx )20 + AT11
+ BT22
,
(27)
81
where (vpx )0 denotes the compressional wave propagating in the X−direction in the natural
state. A and B are constants that are completely determined by elastic moduli and TOE
constants of the rock.
i
i
T22
is constant in loading period B and C, thus vp2x is a linear function of T11
. In period
i
A, normal stresses in X and Z−directions T11 raise from 5 MPa to 7 Mpa, and for the same
12
i
period of time, T22
decreases from 5 MPa to 1 MPa. We may work out the relation between
i
i
T11 and T22 in period A as
i
i
T22
= 5 − 2T11
.
(28)
i
Substituting Eq (28) into Eq (27), we may also find that vp2x is a linear function of T11
in loading period A, only with a different slope from that in periods B and C. Similar
i
dependence of vp2x on T11
is found in load period D with yet another slope.
Similar analysis holds for dependence of P wave propagating in the Y −direction and for
shear waves.
Figures 10, 11, 12 and 13 show both experimental and theoretical results of compressional
and shear velocities versus the normal stress in the X−direction during cycle ABCD. The
fact that very small root mean square errors between theory and experiments suggests that
the theory of acoustoelasticity holds for rocks in the stress range of the experiment. We may
infer that in the regime of small stresses which is below 10 MPa in the above experiment, there
is no permenant deformation in the rock, i.e., cracks will reopen when stresses are removed.
This satisfied the assumption associated with the theory of acoustoelasticity which requires
the rock to be elastic.
4.2
Chelmsford Granite, Chicopee Shale and Berea Sandstone
A second experimental data set is analyzed for the following reason. The stress range up
to 10 MPa to which the Colton sandstone was subjected, as described above, is too low to
encompass in-situ stresses that occur in a hydrocarbon reservoir. An exception is overpressured reservoirs, showing anomalously high pore fluid pressures, resulting in correspondingly
low effective stresses. Therefore, Johnson and Christensen (1993) and Lo’s (1986) experimental data with confining pressures up to 200 MPa on Millboro and Braillier shales and
up to 100 MPa on Berea sandstone, Chicopee shale and Chelmsford granite are analyzed in
the following.
In both Johnson and Lo’s experiments, three compressional and six shear velocities are
measured for each of the rock samples under vacuum dry condition (Johnson and Christensen,
1993) and Lo et al. (1986). Figure 14 illustrates velocities measured and symmetry planes
of rock samples each of which is measured transversely isotropic in the natural state.
Figures 15, 16 and 17 show that root mean square errors between experiments and theoretical predictions are below 1% for every shale sample at all confining pressures. The
excellent agreement between theory and experiment is not surprising, for all shale samples
have very low porosities thus crack growth and coalescence, primary factors attributing to the
highly nonlinear and inelastic behavior in rocks, are very inactive. For the Berea sandstone
sample, experiments and theory show sound agreement at confining pressure levels that are
higher than 30 MPa (Figure 19). A significant portion of cracks in this rock are closed at the
confining pressure of 30 MPa therefore the rock starts to have a similiar behavior to shales.
The corresponding error between experiments and theory is also winthin 1%. For the granite
sample, the required confining pressure to close the majority of cracks is 40 MPa (Figure
18). When stresses are applied to a rock, elastic and inelastic deformations are competing
each other. The inelastic deformation includes permenant closure of cracks, development of
13
Period A
Period B
PX − wave velocity (m/s)
3150
PX − wave velocity (m/s)
3090
3080
3100
SRME=1.0709e−04 m/s
3070
3060
3050
3050
Experiment
Theory
3040
3030
SRME=0.0054 m/s
5
5.5
6
6.5
stress T11 (MPa)
Experiment
Theory
3000
2950
7
2
4
Period C
10
Period D
PX − wave velocity (m/s)
3200
PX − wave velocity (m/s)
3160
3140
3150
3120
SRME=0.0309 m/s
3100
SRME=2.7841e−04 m/s
3100
3050
3080
3000
Experiment
Theory
3060
3040
6
8
stress T11 (MPa)
4
6
8
stress T11 (MPa)
Experiment
Theory
2950
2900
10
0
2
4
6
stress T11 (MPa)
8
10
Figure 10: Theoretical and experimental results of velocity of the compressional wave propagating in the X−direction versus the normal stress in the X−direction during the cycle
ABCD (Dillen, 2000).
14
Period A
Period B
2910
2870
PY − wave velocity (m/s)
PY − wave velocity (m/s)
SRME=0.0029 m/s
2900
2860
2890
2850
2880
2870
2840
2860
Experiment
Theory
2850
2840
SRME=3.1357e−04 m/s
5
5.5
Experiment
Theory
2830
6
6.5
stress T11 (MPa)
2820
7
2
4
Period C
10
Period D
PY − wave velocity (m/s)
3050
PY − wave velocity (m/s)
2915
2910
3000
SRME=3.1716e−04 m/s
2905
SRME=0.0376 m/s
2950
2900
2900
2895
2850
Experiment
Theory
2890
2885
6
8
stress T11 (MPa)
4
6
8
stress T11 (MPa)
Experiment
Theory
2800
2750
10
0
2
4
6
stress T11 (MPa)
8
10
Figure 11: Theoretical and experimental results of velocity of the compressional wave propagating in the Y −direction versus the normal stress in the X−direction during the cycle
ABCD.
15
Period A
Period B
SXZ − wave velocity (m/s)
2080
SXZ − wave velocity (m/s)
2040
2035
2060
SRME=3.5727e−05 m/s
2030
2040
2025
2020
2020
2000
Experiment
Theory
2015
2010
SRME=0.0027 m/s
5
5.5
6
6.5
stress T11 (MPa)
Experiment
Theory
1980
1960
7
2
4
Period C
10
Period D
SXZ − wave velocity (m/s)
2100
SXZ − wave velocity (m/s)
2100
2080
2050
SRME=2.8773e−04 m/s
2060
SRME=0.0078 m/s
2000
2040
Experiment
Theory
2020
2000
6
8
stress T11 (MPa)
4
6
8
stress T11 (MPa)
Experiment
Theory
1950
1900
10
0
2
4
6
stress T11 (MPa)
8
10
Figure 12: Theoretical and experimental results of velocity of the shear wave propagating in
the X−direction and polarizing in the Z−direction versus the normal in the X−direction
during the cycle ABCD (experiment by Dillen).
16
Period A
Period B
SXY − wave velocity (m/s)
2020
SXY − wave velocity (m/s)
2030
2020
2010
SRME=2.7087e−04 m/s
2010
2000
1990
1990
1980
1980
Experiment
Theory
1970
Experiment
Theory
1970
1960
SRME=6.7788e−04 m/s
2000
1960
5
5.5
6
6.5
stress T11 (MPa)
1950
7
2
4
Period C
10
Period D
2030
SXY − wave velocity (m/s)
2100
SXY − wave velocity (m/s)
2040
SRME=7.6680e−05 m/s
SRME=0.0208 m/s
2050
2020
2000
2010
Experiment
Theory
2000
1990
6
8
stress T11 (MPa)
4
6
8
stress T11 (MPa)
Experiment
Theory
1950
1900
10
0
2
4
6
stress T11 (MPa)
8
10
Figure 13: Theoretical and experimental results of velocity of the shear wave propagating in
the X−direction and polarizing in the Y −direction versus the normal in the X−direction
during the cycle ABCD (experiment by Dillen).
17
VP33 VS3a VS3b
VP11 VSV1 VSH1
VP45 VSV45 VSH45
Figure 14: Velocities measured and symmetry planes of rock samples each of which is measured transversely isotropic in the natural state (both Johnson and Lo’s experiments).
18
new cracks that results from local failures and permenant relative movement between rock
grains. In order to account for inelastic deformation, the theory of acoustoelasticity has to
be modified. From the above analysis, we find that the theory of acoustoelasticity applies to
all shale samples of low porosity. For the sandstone and granite samples, in the intermediate
stress regime which is about 10 MPa to 30 or 40 PMa, they undergo primarily inelastic
deformations. When the stress level is 30 or 40 MPa higher, the theory of acoustoelasticity
works well with all types of rocks that are studied in this paper, because most cracks are
closed at that confining pressure level and thus elastic deformation becomes primary. Rocks
are under high confining pressures when they are kilometers beneath the surface of the earth
where we are interested to measure in-situ stresses. So for the purpose of estimating in-situ
stresses, the theory of acoustoelasticity is applicable to all types of rocks.
Braillier Shale
6
6
Vpaa
Vpbb
Vpcc
5.8
4
Vpd
VSac
VSbc
5.5
VSca
VSdh
VSdv
3.5
5.6
5
RMSE (km/s):
4.5
5
4.8
RMSE (km/s):
0.0144
0.0131
3
0.0064
4
3.5
0.0045
3
velocity (km/s)
5.2
velocity (km/s)
velocity (km/s)
5.4
0.0043
2.5
2
4.6
RMSE (km/s):
4.4
0.0456
0.0203
0.0120
4.2
4
2.5
0.0114
2
1.5
1.5
0
100
200
confining pressure (MPa)
1
0
100
200
confining pressure (MPa)
1
0
100
200
confining pressure (MPa)
Figure 15: Theoretical and experimental results for Braillier shale (experiment by Johnson
and Christensen (1993). ∆: experiment; solid line: theory. RMSE denotes root mean square
error between theory and experiment.
19
Millboro Shale
6
6
Vpaa
Vpbb
Vpcc
5.8
4
Vpd
VSac
VSbc
5.5
VSca
VSdh
VSdv
3.5
5.6
5
RMSE (km/s):
0.0124
0.0188
5
4.8
3
0.0271
velocity (km/s)
velocity (km/s)
0.0375
5.2
RMSE (km/s):
RMSE (km/s):
4.5
4
3.5
0.0045
3
0.0159
0.0065
velocity (km/s)
5.4
0.0154
2.5
2
4.6
2.5
4.4
2
0.0140
1.5
4.2
4
1.5
0
100
200
confining pressure (MPa)
1
0
100
200
confining pressure (MPa)
1
0
100
200
confining pressure (MPa)
Figure 16: Theoretical and experimental results for Millboro shale (experiment by Johnson
and Christensen (1993). ∆: experiment; solid line: theory.
20
Chicopee Shale
7
4
4
RMSE (km/s):
RMSE (km/s):
RMSE (km/s):
3.8
3.8
0.0049
0.0294
6.5
0.0294
0.0182
0.0101
3.6
0.0324
0.0082
3.6
0.0124
0.0063
3.4
3.4
3.2
3.2
5.5
velocity (km/s)
velocity (km/s)
velocity (km/s)
6
3
2.8
3
2.8
5
2.6
Vp11
Vp45
V
4.5
p33
4
0
50
100
confining pressure (MPa)
2.6
2.4
VSH1
VSV1
VSH45
2.2
2
0
50
100
confining pressure (MPa)
2.4
VSV45
VS3a
VS3b
2.2
2
0
50
100
confining pressure (MPa)
Figure 17: Theoretical and experimental results for Chicopee Shale (experiment by Lo et al
(1986). ∆: experiment; solid line: theory.
21
Chelmsford Granite
7
4
RMSE (km/s):
3.9
0.0223
6.5
3.8
RMSE (km/s):
0.0190
RMSE (km/s):
3.7
0.0110
0.0137
0.0137
3.8
0.0232
0.0110
3.6
0.0110
3.7
3.5
3.6
3.4
0.0110
5.5
velocity (km/s)
velocity (km/s)
velocity (km/s)
6
3.5
3.4
3.3
3.2
5
3.3
4.5
4
40
Vp11
Vp45
Vp33
60
80
100
confining pressure (MPa)
3.1
3.2
VSH1
VSV1
VSH45
3.1
3
40
60
80
100
confining pressure (MPa)
3
2.9
2.8
40
VSV45
VS3a
VS3b
60
80
100
confining pressure (MPa)
Figure 18: Theoretical and experimental results and their relative errors for Chelmsford
granite (experiment by Lo et al (1986). ∆: experiment; solid line: theory.
22
Berea Sandstone
5
3
Vp11
Vp45
Vp33
2.9
2.8
4.4
2.7
2.7
4.2
2.6
2.6
3.8
2.5
2.4
RMSE (km/s):
3.6
0.0177
3
20
RMSE (km/s):
2.3
2.2
0.0145
0.0082
2.2
0.0117
2.1
40
60
80 100
confining pressure (MPa)
2.4
0.0145
0.0282
3.2
2.5
RMSE (km/s):
2.3
0.0193
3.4
velocity (km/s)
2.8
4
2
20
VSV45
VS3a
VS3b
2.9
4.6
velocity (km/s)
velocity (km/s)
4.8
3
VSH1
VSV1
VSH45
0.0164
0.0164
2.1
40
60
80 100
confining pressure (MPa)
2
20
40
60
80 100
confining pressure (MPa)
Figure 19: Theoretical and experimental results and their relative errors for Berea sandstone
(experiment by Lo et al (1986). ∆: experiment; solid line: theory.
23
4.3
Barre Granite: Acoustoelastic Approach and Sayers’s Microcrack Model
Nur and Simmons (Nur and Simmons, 1969) subject a cylinder of Barre granite to a uniaxial
compressive stress normal to the axis of the cylinder. Compressional and shear wave velocities
were measured for propagation normal to the cylinder axis at various angles β to the stress
axis.
It is more convenient to work with cylindrical coordinate given the geometry of the experimental setup. Note that in cylindrical coordinate, T = [Ti11 Ti22 Ti33 ]T = [Tirr Tiββ Tizz ]T ,
and E = [ei11 ei22 ei33 ]T = [eirr eiββ eizz ]T . The initial strain E is linearly related to the initial
stress T through elastic moduli as mentioned previously. Components of E and T in Eq
(22) can be those in cylindrical coordinates as defined above. Suppose the applied uniaxial
stress, σ, coincides with β = 0o ; therefore, at angle β, the three normal stresses components
of T in cylindrical coordinates are


σcos2 β
T =  σsin2 β  .
(29)
0
Substituting Eq (29) into Eq (25), we obtain the compressional wave velocity as a function
of applied stress σ and the angle β as
vp2 = (vp )20 + Aσ + Bσcos2 β,
(30)
where A and B are constants determined by elastic moduli and TOE constants of the rock.
(vp )0 is the compressional velocity in the natural state. We choose the average stress-free
velocity (vp )0 at all angles to be 3.79 km/s, and then invert Nur’s data (10 MPa, 20 MPa
and 30 MPa) for constants A and B. The results are A = 0.008 and B = 0.0225.
An alternative way of analyzing stress-induced velocity changes in rocks is to think microscopically which has long been well received in geophysical community[e.g.Sayers (1988a),
Sayers et al. (1990)]. Since the theory of acoustoelasticity, which is a macroscopic approach,
also works reasonably well with rocks, as shown previously, we shall make comparison between the two approaches.
Sayers applied the microcrack theory to the measurements of Nur and Simmons (Sayers,
1988b). His formula for the compressional velocity at angle β is
vp = A + Bcos2β,
(31)
where A and B are unknown variables that depend not only on the elastic properties of the
rock, but also the applied stress. He evaluated A and B by fitting the measurements of Nur
and Simmons with the following results:
Stress (MPa)
A
B
10
4.052 0.199
20
4.271 0.301
30
4.414 0.322
24
Figure 20 exhibits compressional wave velocity measurements of Nur and Simmons for Barre
granite compared with the acoustoelastic theory and microcrack model prediction. An error
analysis shows errors between experiment measurements and both models are mostly below
2% (Figure 21). Relative errors between the two models also suggest the two models agree
with each other quite well (Figure 22). Note that the uniaxial assumption has 10% error in
the experiment (Nur and Simmons, 1969).
The microcrack model implicitly deals with the velocity dependence on the applied stress.
A and B are inverted for each applied stress level and thus are dependents of applied stress.
On the other hand, in the acoustoelastic approach, A and B are constants that depend only
on the elastic properties of the rock. So it is not surprising that microcrack model fits the
experiments slightly better than acoustoelastic model. However, for the ultimate purpose
of inverting measured velocity changes for formation stresses, it is a disadvantage of the
microcrack model not to work explicitly with the velocity dependence on the applied stress.
5
compressional wave velocity (km/s)
30 MPa
20 MPa
4.5
10 MPa
4
3.5
0
10
20
30
40
50
β (degrees)
60
70
80
90
Figure 20: Compressional wave velocity measurements of Nur and Simmons (Nur and Simmons, 1969) for Barre granite compared with the acoustoelastic theory and microcrack model
prediction (Sayers, 1988b). Solid line: acoustoelasticity, Dash line: microcrack model.
25
Relative errors between theory and experiments
4
10 MPa
20 MPa
30 MPa
3.5
3
% error
2.5
2
1.5
1
0.5
0
0
10
20
30
40
50
β (degrees)
60
70
80
90
Figure 21: Relative error between experiment measurements and theory for Barre sandstone.
Solid line: acoustoelasticity, Dash line: microcrack model.
26
Relative errors between acoustoelastic theory and microcrack model
3.5
30 MPa
3
2.5
% error
2
10 MPa
1.5
1
20 MPa
0.5
0
0
10
20
30
40
50
β (degrees)
60
70
80
90
Figure 22: Relative error between acoustoelatic theory and microcrack model for Barre
sandstone.
27
5
Conclusion
Rocks exhibit strong nolinear stress-strain behavior. As a result, applied or residual stresses
in rocks affect sound velocity considerably. In the context of the widely accepted microcrack
approach, velocity changes are considered to result from crack closure driven by stresses.
Because the direct dependence of velocities changes on stresses, the relationship necessary
to invert sound velocity change for formation stresses, is not available. On the other hand,
in the framework of acoustoelasticity, velocity changes are explicitly related with stresses. In
the past 40 years, the theory of acoustoelasticity has been confirmed and widely employed to
evaluate initial stresses for metal materials. Very few works have been conducted for rocks.
We compiled various rock measurements and have found that the theory of acoustoelasticity
agrees within 1% of error with experiments using confining pressures higher than certain
levels for most rocks. The necessary confining pressure level is easily achieved under insitu conditions; therefore, the theory of acoustoelasticity is applicable to rocks in estimating
formation stresses through in-situ sound velocity measurements. Using the measurements of
Nur and Simmons, we compared acoustoelastic theory with the microcrak model and found
they agree with each other within 2% of error, much smaller than experimental error (10%).
We may safely conclude that the theory of acoustoelasticity is a macroscopic version of the
microcrack model.
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29