The 21st International Congress on Sound and Vibration
13-17 July, 2014, Beijing/China
MODELLING ELASTIC WAVE PROPAGATION IN 3D
STRESSED FORMATION USING 3D FINITE DIFFERENCE ALGORITHM
Jinxia Liu, Zhiwen Cui, Ruolong Song, Weiguo Lv and Kexie Wang
Department of Acoustics and Microwave Physics, College of Physics, Jilin University,
Changchun, China, 130012
e-mail:[email protected]
The investigation of subsurface stress is the most important for geophysical exploration. 3D
subsurface stress situation is able to provide some significant information for exploiting oil
and gas and reducing the risk of various hazards. Acoustoelastic theory provides an opportunity to compute the exact seismic signatures of interest. We simulate the wave field with 3D
subsurface stress information by 3D finite difference algorithm and study elastic wave propagation in anisotropic medium induced by stress within nonlinear acoustic theory. We obtain
the Piola-Kirchhoff equation for anisotropic medium induced by 3D subsurface stress and
simulate elastic wave field to know the change of wave field in 3D stressed medium and 3D
stressed two-layered medium. The results show that the morphology of wavefront varies obviously, and is elliptic in 3D subsurface stress information. The reflection and transmission of
acoustic waves at the interface are more complicated than that in unstressed medium. The
shear wave splitting occurs when the recorded plane deviates from the principal stress plane.
The investigation will help us to have insight into the influences of 3D subsurface stress on
the properties of elastic waves and is of significance in practical application.
1.
Introduction
Study on elastic wave propagation in 3D stressed media is of particular interest in seismology
and exploration geophysics.3-D simulations seem necessary for comprehensive understanding of
the seismic wave propagation across the 3D stressed media. In recent years, some experiments1,2
have proved that the nonlinear response of sedimentary rocks subject to applied stresses is more
obvious than that of other media. This phenomenon spurs many researchers in geophysics and
seismology to study on elastic waves in rocks subject to statically elastic deformations based on
acoustoelastic theory3-5.The acoustoelastic effect, i.e., the fact that the velocity of sound in a material varies with the externally applied stress, has been investigated for more than 60 years6. Acoustoelastic theory supports the intuitive expectation that an isotropic solid in the presence of uniaxial
stress exhibits symmetry close to hexagonal, while an isotropic solid in non-equal biaxial stresses
exhibits anisotropy close to orthorhombic. In other words, stress coverts an isotropic medium into
an anisotropic medium. Based on acoustoelastic theory, Liu and Sinha 7,8developed a 2.5D finitedifference time-domain (FDTD) method and a perfectly matched layer (PML) absorbing boundary
condition (ABC) in 3D cylindrical coordinates to simulate elastic wave propagation in fluid-filled
boreholes in biaxially7 and triaxially8 stressed formations, respectively. Elastic wave propagation at
ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
a plane boundary between two media is one of the most fundamental problems, and has been addressed abroad in the intrinsic anisotropic case. 9 However, very few quantitative results have been
published on the modelling propagation of elastic waves from an interface between 3D stressed
media. In this work, we apply a non-perfectly matched layer10 to simulate elastic wave propagation
in stressed medium and stressed two-layered medium in 3D Cartesian coordinates.
2.
Theoretical background
In this paper, we will follow the acoustoelastic theory described by Pao et al.11. The main results
of the theory are recorded for the isothermal (purely mechanical) case. Three configurations of the
body play crucial roles in the theory: reference, intermediate (deformation), and current configuration. We choose the intermediate (deformation) configuration to describe wave propagation in rock
of elastic deformation. Let the vectors X,x and xc with components XA,xi and xic represent the reference, intermediate and current states, respectively. A subscript comma indicates partial differentiation with respect to X or x depending on the following subscript, and summation convention applies
to repeated subscripts, also when these subscripts are capitals. The constitutive equation in the intermediate coordinates can be obtained from the references11
σ ijc = Tij + C ijkl ekl + u i ,k Tkj
(1)
where
Cijkl = cijkl (1 − Enn ) + cijklmn Emn
+ cmjkl
∂u i
∂uii
∂u i
∂u i
+ cimkl j + cijml k + cijkm l
∂xm
∂xm
∂xm
∂xm
ρ~ ≅ ρ 0 (1 − Enn )
(2)
(3)
where σ ijc is the first Piola-Kirchhoff stress tensor. The Cauchy stress tensor Til in the intermediate
state and the stiffness tensor Cijkl are functions of the static strain Emn and static displacement gradi∂uii
ent
. The Superscripts i denotes the intermediate state. ekl = (u k ,l + ul ,k ) / 2 is the elastic strain
∂xm
due to wave small–amplitude propagation, ui , k is the displacement gradient, cijkl and cijklmn are sec-
ond-order and third-order elastic constants respectively, the isotropic material has two independent
second-order elastic constants and three independent third-order elastic constants. ρ~ and ρ 0 are the
mass density in intermediate and reference states, respectively. For an anisotropic medium induced
by stresses characterized by the stress Til and the effective stiffness tensor Cijkl , the first –order partial differential equations for the particle velocity vector V and first Piola-Kirchhoff stress tensor
σ c are given by
ρ~V& = σ c
i,j=1,2,3
(4)
i
ij , j
σ& ijc = Cijkl e&kl + Vi , kTkji
i,j=1,2,3
(5)
The partial differential Eqs in (4)-(5) are approximated using a central differencing scheme with
staggered grids.12 For numerical computation, the infinite physical domain has to be truncated to a
finite computational domain. To minimize the reflection of waves from such an artificial outer
boundary, a nonsplitting perfectly matched layer(NPML) method10 is used.
ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
3.
Numerical examples and discussions
The source time function is
f (t ) = ω0 t e
2 2
−
ω0
3
t
sin (ω0t )
(6)
ω0 = 2πf 0 , f 0 is center frequency. The medium parameters13 including the density ρ, compressional
velocity VP ,shear velocity VS and third-order elastic constants(c111, c112, c123) are listed in Table 1.
The media A and B are isotropic when the stresses are inexistence. The model consists of stressed
medium and stressed two-layered media, respectively. The stressed medium is an infinite medium B.
The upper layer of two-layered media is the medium A half-space; the lower half-space is the medium B. The acoustic sources used in this study operates at a center frequency of 2KHz. The FD
cell size is Δx = Δy = Δz = 0.1 m, and the time step is Δt = 5 × 10−6 s. The computation domain consists of N x × N y × N z = 140 × 140 × 140 . A point body force in x direction is located at
( x, y, z )=(70,70,50).We first compare FD results with elastic wave field in anisotropic medium using the reference’s parameters14.The agreement of our snapshots with the reference13is excellent.
Then, we model the elastic wave fields in infinite medium and two anisotropic layers, respectively.
Table 1. Formation properties.
Formation
A
B
VP (m/s)
2000
2127
VS (m/s)
1275
1418
ρ (kg/m3)
c111(GPa)
c112(GPa)
c123(GPa)
2000
2062
-3797.3
-9550
-420.9
-1370
-95.9
1062
Figure 1. Snapshots of velocity in Z direction(t=3ms) for stressed medium.
(a) and (b) in principal stress plane under biaxial stresses;(c)and (d)under 3D stresses.
Fig. 1. shows the snapshots of velocity in z direction for infinite stressed medium B. The snapshot times are 3 milliseconds. Fig.1 (a) and (b) are medium B applied by biaxial stresses (T11=10MPa，T22=-20MPa). Fig.1 (c) and (d). are medium B applied by triaxial stresses (T11=-10MPa，
T22=-20MPa, T33=-5MPa). Fig.1 (a) and (c) are recorded in principal stress plane. Fig.1 (b) and (d)
are recorded in non-principal stress plane. The results show that the morphology of wavefront varies obviously in 3D subsurface stress information by comparing snapshots of biaxial stresses with
those of triaxial stresses. The morphology of wavefront (can be found in Fig.1 (d)) become elliptic
when stress T33 appear. Shear wave splitting are found when the snapshots are recorded in nonprincipal stress plane. But shear wave splitting (Fig.1(d)) is not more obvious than that (Fig.1(b)) of
the biaxial stresses because the existence of the stress T33 decreases the anisotropy induced by biaxial stresses.
ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Figure 2. Snapshots of velocity in Z direction(t=4ms) for stressed two-layered medium.
(a)and(b)under biaxial stresses;(c)and (d)under triaxial stresses.
Figure 3. Snapshots of velocity in Z direction(t=5ms) for stressed two-layered medium.
(a) and (b)under biaxial stresses;(c) and (d)under 3D stresses.
Fig. 2. and Fig.3. show the snapshots of velocity in z direction. The snapshot times are 4 and 5
milliseconds, respectively. In Fig.2 and Fig.3, (a) and (b) are both upper and lower layers applied by
biaxial stresses (T11=-20MPa，T22=-5MPa), (c) and (d) are both upper and lower layers applied by
triaxial stresses (T11=-20MPa，T22=-5MPa, T33=-5MPa). Fig.(a) and Fig.(c) are recorded in principal stress plane. Fig.(b) and Fig.(d) are recorded in non-principal stress plane. We note the wave
types and wavefront shapes become complex when the two layers are applied by stresses. We can
find reflection and transmission of compressional and shear waves, and shear wave splitting in reflection waves when the snapshots are recorded in non- principal stress plane. Moreover, we can
also find the wave field is more complex under biaxial stresses than that under triaxial stresses. The
results also show the anisotropy induced by biaxial stressed decreases because the vertical stressed
T33 appears.
4.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grants (nos.
11134011,41004044 and 40974067) and State Key Laboratory of Acoustics (CAS) (Grant
no.SKLOA201108)
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