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Three-dimensional Time-Domain Finite-Element
Modeling of Seismoelectric Waves
Jun Li, Changchun Yin, Yang Su, Bo Zhang, Xiuyan Ren, Yunhe Liu
Abstract—As the structure of the underground space becomes
increasingly complex, traditional two-dimensional seismoelectric
methods are no longer adequate for the comprehensive
exploration. To achieve precise imaging of the underground
space, it is in urgent need to develop three-dimensional full-
waveform modeling techniques. In this paper, we propose a
three-dimensional time-domain finite-element method to solve
the seismoelectric wave field in saturated porous media. Since the
electroosmotic feedback is very small, we can ignore the
mechanical disturbance caused by the electromagnetic fields
induced by seismic waves, and thereby can decouple the
electrokinetic coupling equations and separately solve the seismic
and electromagnetic waves. For the simulation of seismic
wavefield, we employ the explicit finite-element method, and
utilize a lumped mass matrix instead of a consistent mass matrix
to facilitate explicit recursion. Additionally, we apply the complex
frequency-shifted unsplit perfectly matched layer technique to
effectively handle seismic boundary conditions. Then, the velocity
fields obtained by solving the poroelastic equations serve as the
source term of the electromagnetic equations, and the finite-
element method is used to solve the electromagnetic wavefield.
Considering that the huge velocity difference exists between the
electromagnetic and seismic waves, we adopt an unconditionally
stable implicit method for the solution of the electromagnetic
wavefield. By combining explicit and implicit recursion, the
computational efficiency can be improved significantly. The
accuracy of our time-domain finite-element algorithm is
validated by checking our results against the analytical solutions
for a half-space model. Furthermore, we conduct numerical
simulations and analyses on a typical block model and a modified
SEG/EAEG salt dome model.
Index Terms—Finite-element method, numerical modeling,
electrokinetic effect, seisomoelectric waves, wave propagation.
I. INTRODUCTION
HE electrokinetic effect based on the double electric
layer in a fluid-saturated porous medium can cause
coupling between seismic and electromagnetic (EM)
waves [1], [2], [3], [4]. This coupling phenomenon has been
observed in both field experiments and laboratories [5], [6],
[7], [8], [9]. Based on this, the seismic EM exploration method
This paper is financially supported by the Major Research Project on
Scientific Instrument Development of National Natural Science Foundation of
China (42327901) and National Natural Science Foundation of China
(42030806, 42074120, 42174167, 42274093, 42304149). (Corresponding
author: Changchun Yin.)
Jun Li , Changchun Yin*, Yang Su, Bo Zhang, Xiuyan Ren, Yunhe Liu are
with the College of Geo-exploration Science and Technology, Jilin
University, Changchun, Jilin, 130021, China (e-mail: lij22@mails.jlu.edu.cn;
yinchangchun@jlu.edu.cn; suyangjlu@163.com; em_zhangbo@163.com;
jdrxy@hotmail.com; liuyunhe@jlu.edu.cn).
combines the advantages of seismic and EM methods, namely
seismic resolution and electromagnetic sensitivity to fluids.
This method has great application prospects in the exploration
of groundwater and oil & gas, the monitoring and advance
detection in coal mines [10], [11], [12], [13], [14].
Both theoretical and experimental studies have shown that
three types of EM waves are generated during the
seismoelectric conversion. The first is EM waves generated
directly by the earthquake source [15], [16]; the second is the
interface EM waves (also known as interface responses)
generated by the seismic wave at the interface between two
different media [10], [17]; the third is coseismic EM waves
generated along with seismic waves [18], [19]. These
seismoelectric signals are sensitive to the characteristics of
underground media such as porosity, permeability or salinity
[9], [20], and can be used to detect and characterize
underground structures. Therefore, as a theoretical basis and
technical means of seismic EM exploration, the quantitative
simulation is very important. Since Pride [21] derived a set of
governing equations for the propagation of seismic waves and
EM waves based on electrokinetic coupling coefficients in
1994, the quantitative numerical modeling of seismoelectric
waves has developed rapidly over the following three decades.
Initially, researchers proposed two analytical methods, with
one using a Green’s function [3], [15] and the other using a
reflectivity method [1], [20], [22], [23]. Since an analytical
method can only handle a half-space or layered earth model,
one needs to develop numerical methods like the finite
difference (FD) or finite element (FE) to handle complex
structures like oil and gas reservoirs.
Haines and Pride [10] proposed a time-domain FD (FDTD)
method in 2006 to simulate seismoelectric phenomena in an
arbitrary heterogeneity medium. Gao et al. [24] used the
frequency-domain FD method (FDFD) to simulate
seismoelectric waves in the two-dimensional SHTE mode.
They solved the Biot’s equations [25] and completed
Maxwell’s equations in the frequency domain to avoid
stability and quasi-static approximation problems. Tohti et al.
[26] proposed an FDTD algorithm to calculate the
seismoelectric response of a two-dimensional anisotropic
poroelastic medium. They analyzed the influence of medium
parameters on the propagation of slow longitudinal waves. Ji
et al. [27] proposed a high-order FDTD method to simulate the
seismoelectric response based on the time-varying full
Maxwell’s equations. Han et al. [28] proposed an FDTD
algorithm to simulate the seismoelectric response in porous
viscoelastic TTI media.
Compared with the FD method, the FE method has more
T
This article has been accepted for publication in IEEE Transactions on Geoscience and Remote Sensing. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TGRS.2024.3520236
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
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advantages in dealing with irregular underground anomalies
and free surfaces. Therefore, scholars also proposed FE
modeling of seismoelectric waves. In 2010, Zyserman et al.
[29] developed a FE method to solve 2D electroseismic
equations in two modes, and then in 2015, Zyserman et al.
[30] used FE method to solve the seismoelectric equations.
Wang et al. [31] proposed a frequency-domain FE (FEFD)
method to solve the seismoelectric and electroseismic
equations in 2D SHTE mode. They demonstrated the
feasibility of applying seismoelectric conversion signals for
the exploration and monitoring of pollutants. Li et al. [32]
proposed a time-domain FE (FETD) method based on
arbitrary quadrilateral meshes to simulate 2D SHTE
seismoelectric and electroseismic waves.
However, all these numerical algorithms are developed for
2D cases. When the medium distribution does not change in
one direction, the 2D forward modeling can accurately and
quickly calculate the seismoelectric responses. However,
when the model is very complex or the structural distribution
changes significantly in different directions, the 2D forward
algorithm is unable to effectively simulate the seismoelectric
response of a 3D model. At present, there are only a few
studies on 3D modeling algorithms. In 2013, Wang et al. [33]
proposed a 3D FDTD algorithm for modeling the strike-slip
faults in the underground, but they only studied the
seismoelectric response in half space and its accuracy was not
high enough due to approximations. Tohti et al. [34] further
extended their 2D FDTD algorithm to 3D orthotropic media.
They mainly studied the propagation characteristics of
seismoelectric signals in an orthogonal anisotropic medium
under the full-space model. The algorithms of the above
introduced are limited to half-space or full-space models,
without tackling 3D complex structures. To better understand
the propagation mechanism of the seismoelectric wavefield
and the distribution characteristics of seismoelectric waves in
complex underground media, we propose a FETD method to
solve seismoelectric waves in 3D earth.
In the sequence, we will first derive the seismoelectric
governing equations for 3D problems in time domain and then
introduce our FETD algorithm. After that, we compare our
modeling results with the analytical solution for a half-space
model to validate the accuracy of our algorithm. Finally, we
demonstrate the applicability of our algorithm to analyze the
seismoelectric response of 3D complex models by simulating
a typical underground block model and a modified
SEG/EAEG salt dome model.
II. METHODOLOGY
A. Governing Equations
In the following, we start with the complete seismoelectric
coupling equations [21] and convert them into the time
domain. According to Haines and Pride (2006) [10], due to the
weak electroosmotic feedback, the seismoelectric coupling
equations can be decoupled into Biot’s equations and
Maxwell’s equations that contain the seepage displacement
source term, i.e.
22
22
f
ρ ρ
tt
∂∂
∇⋅ + = +
∂∂
uw
τf
, (1)
22
22
ff
f
Pkt
tt
ρα η
ρϕ
∞
∂ ∂∂
−∇ + = + + ∂
∂∂
u ww
f
, (2)
[( 2 ) ] ( )
T
HG M G
α
= − ∇⋅ + ∇⋅ + ∇ +∇τu wI u u
, (3)
PM M
α
− = ∇⋅ + ∇⋅
uw
, (4)
f
L
t kt
η
σε
∂∂
∇× = + +
∂∂
Ew
HE
, (5)
t
µ
∂
∇× =− ∂
H
E
, (6)
where the source term f represents the average force density,
which is assumed to be evenly distributed in the porous
medium. Table I gives the terminology of parameters used in
the above equations. Note that we use the static bulk
conductivity, the permeability, and the electrokinetic coupling
coefficient in the modeling. Their specific expressions can be
found in Haartsen and Pride (1997) [1]. The poroelastic
constant α and bulk modulus H, M can be expressed by four
bulk moduli, namely
1b
s
K
αK
= −
, (7)
()
sf
sf
KK
MφKαφK
=+−
, (8)
2
4
3
b
HK GαM
=++
. (9)
TABLE I
PARAMETERS USED IN SEISMOELECTR IC EQUATIONS
Symbol
Meaning
Units
u
Solid displacement
m
w
Relative fluid-solid displacement
m
E
Electric field
V m-1
H
Magnetic field
A m-1
,,
xx yy zz
ττ τ
Bulk stress tensor Pa
,,
xy xz yz
τ τ τ
Shear stress tensor Pa
P
Pore fluid pressure
Pa
K
s
Solid bulk modulus
Pa
K
f
Fluid bulk modulus
Pa
K
b
Frame bulk modulus
Pa
G
Shear modulus
Pa
ε
Bulk electric permittivity
F m-1
ρ
Bulk density
(1 )
sf
ρ φ ρ φρ
=−+
kg m-3
ρ
s
Solid grain density
kg m-3
ρ
f
Pore fluid density
kg m-3
φ
Porosity
—
α
∞
Tortuosity
—
μ
Bulk magnetic permeability
H m-1
This article has been accepted for publication in IEEE Transactions on Geoscience and Remote Sensing. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TGRS.2024.3520236
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
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η
f
Pore fluid viscosity
Pa s
σ
Bulk conductivity
S m-1
k
Permeability
m2
L
Electrokinetic coupling coefficient
sC kg-1
C
f
Pore fluid salinity
mol L-1
B. Computational domain and boundary conditions
In seismoelectric modeling, we must impose boundary
conditions in bounded regions to absorb seismic and EM
waves. However, due to huge velocity differences between
seismic and EM waves, we take two different strategies to
absorb seismic and EM waves respectively. Refer to Fig. 1,
the brown area in the center is our computational area, and the
outside gray, pink, and red represent different areas of the
perfectly matching layer (PML) for absorbing seismic waves.
We apply ten expansion layers outside the PML (increasing
outward in multiples) to absorb EM waves, which are not
shown in the figure for simplicity. The light blue area at the
top is the air layers. Note that the seismic waves will be totally
reflected at the free surface, while EM waves can pass through
the free surface and propagate in the air layers.
Considering that the seismic waves cannot propagate in the
air layers, the boundary conditions need to be added at the
interface between the porous medium and the air layers. At the
interface, the stress satisfies
0
xz yz zz
ττ τ
= = =
. At the outer
boundary after the expansion, we impose the Dirichlet
boundary condition, that is
= = =uwE0
.
Fig. 1. Schematic diagram of the computational area for
seismoelectric modeling. The brown area is the inner
computational domain, the outer gray, pink, and red areas are
the faces, edges, and corners of PML, respectively. The top
light blue denotes the air layers.
C. FETD scheme
The assumption that the Biot’s and Maxwell’s equations are
decoupled allows us to solve the poroelastic equations (1)-(4)
first, and then use the solved seepage displacement fields as the
source in equation (5) to further solve the equations (5)-(6) for
EM fields. Fig. 2 shows the flow of our entire FETD algorithm.
In the following we will introduce our FETD algorithm for
seismoelectric modeling in details.
Fig. 2. Flowchart of FETD method for seismoelectric modeling.
First, we use FE method to solve the poroelastic equations
[35], [36]. Substituting (3) and (4) into (1) and (2), we obtain
22
22
[( 2 ) ] ( )
T
f
HG αMG ρ ρ
tt
∂∂
∇⋅ − ∇⋅ + ∇⋅ + ∇⋅ ∇ +∇ + = +
∂∂
uw
u w uu f
,(10)
22
22
()
ff
f
Mkt
tt
ρα η
αρ
ϕ
∞
∂ ∂∂
∇ ∇⋅ +∇⋅ + = + + ∂
∂∂
u ww
u wf
. (11)
Multiplying both sides of the above two equations with a test
vector N (same as the node basis function), we obtain the
following weak form:
ΩΩ
22
22
Ω Ω Ω
[( 2 ) ] Ω()Ω
Ω Ω Ω
T
f
HG αMd G d
ρdρdd
tt
∇ − ∇⋅ + ∇⋅ + ∇ ∇ +∇ +
∂∂
⋅+⋅ =⋅
∂∂
∫∫
∫∫ ∫
u w uu
uw
f
NN
N NN
,(12)
2
2
2
2
()
f
ff
M dd
t
d dd
kt
t
αρ
ρα η
ϕ
ΩΩ
∞
Ω ΩΩ
∂
∇ ∇⋅ +∇⋅ Ω+ ⋅ Ω+
∂
∂∂
⋅ Ω+ ⋅ Ω= ⋅ Ω
∂
∂
∫∫
∫ ∫∫
u
uw
ww
f
NN
N NN
, (13)
where Ω represents the model volume. After expanding the
displacement fields in (12) and (13) along the x, y, and z
directions and integrating the equations, we obtain
1 2 11 14 15 21 24 25
1 2 17 12 16 27 22 26
1 2 18 19 13 28 29 23
+
+
+
x x y z x y zx
y x y z x y zy
z x y z x y zz
x
y
z
u u u u w w wF
uu
w
w
wu u w w wF
u uu Fwuw w
+++ + =
+ ++ + =
+
+
+
+
++
++++ =
MM K K K K K K
MM K K K K K K
MM K K K K K K
, (14)
3 4 21 24 25 31 34 35
3 4 27 22 26 37 32 36
3 4 28 29 23 38 39 33
+
+
+
xx
yy
zz
x x y z x y zx
y x y z x y zy
z x y z x y zz
u u u u w w wF
u u u u w w wF
u uuuwwww
ww
ww
Fw
++ +++ + =
+
+
+++ + ++ + =
+ + + ++ + =
M M DK K K K K K
M M DK K K K K K
M M DK K K K K K
, (15)
where
, = ,, ,
ii
w i xyzu
represent the solid acceleration and
relative fluid-solid acceleration in three directions, while
, = ,
, ,
ii
w i xyzu
represent the solid velocity and relative fluid-
solid velocity in the three directions, respectively.
=,,,
ii xyzF
This article has been accepted for publication in IEEE Transactions on Geoscience and Remote Sensing. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TGRS.2024.3520236
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
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represent the source in the three directions. M, D, and K are
the mass matrix, damping matrix, and stiffness matrix,
respectively. The specific forms of the matrices and source
terms are given in the Appendix.
Note that the mass matrix obtained by integration here is
non-diagonal, which is called a consistent mass matrix.
Considering that the amount of computation in 3D forward
modeling is very large, the computational efficiency will be
very low if the equation is solved directly. In order to improve
the computational efficiency of our FE method, we use the
mass concentration technology to obtain the diagonal mass
matrix [37], [38], and to simplify the solution so that the
equation can be explicitly recursive.
After obtaining the diagonalized lumped mass matrix, we
can rewrite (14) and (15) in matrix form, i.e.
1 2 12
+++ =u u wFM Mw K K
, (16)
3 4 23
+++ =
+M Mw D K Ku w u wF
. (17)
Using the basic matrix algorithm, (16) and (17) can further
be written as
00
00
ss
ss s
sf
f f ff f
sf
uu u
ww w
++ =
KK
MD F
M D KK F
, (18)
where
11
1 2 4 3 4 31 2
1
24
11
1 24 2 2 24 3
11
2 31 1 3 31 2
11
2 4 31
, ,
, ,
, ,
, ,
, .
sf
sf
ss
sf
ff
sf
sf
−−
−
−−
−−
−−
=−=−
=−=
=−=−
=−=−
=−=−
M M MM M M M MM M
D MM DD D
K K MM K K K MM K
K K MM K K K MM K
F FMM FF FMMF
(19)
It is obvious that the global mass matrix of (18) is diagonal,
so we can use explicit time recursion to solve it. In this paper,
we adopt the long-term stability and energy-preserving
Newmark integration format [39]. Taking the solid
displacement u as an example, the implementation process of
the specific scheme is given by
12 1
11
11
[(0 5 ) ]
[(1 ) ]
()
.
nn n nn
n n nn
n nn
tt
t
ββ
γγ
++
++
+−
= +∆ +∆ − +
= +∆ − +
= −−
uu u uu
u u uu
u M F Du Ku
, (20)
where β = 0, γ = 0.5, the superscript n represents the nth time
step. Using this scheme, we can calculate the displacement
and velocity fields at all times by explicit recursion.
Furthermore, we use the calculated relative fluid-solid
velocity field as the source in (5)-(6) to further solve EM
fields. We only study the electrostatic near field of EM
disturbance, with the influence of induction being ignored.
Therefore, the displacement current in (5) can be neglected,
and (6) can be written as
0∇× =E
. Furthermore, we
introduce the electric potential Φ that satisfies
= −∇ΦE
.
Taking the divergence of (5), we get the Poisson equation for
the electric potential as
( )= ( )
f
Lkt
η
σ
∂
∇⋅ ∇Φ ∇⋅ ∂
w
. (21)
We use the nodal finite-element method to solve (21).
Similarly, multiplying both sides of the equation by the test
vector N, we obtain its weak form as
Ω Ω
Ω = Ω(Φ)NN
f
η
σLdd
k
∇⋅∇ ∇⋅
∫∫
w
, (22)
where Ω represents the entire model domain. We further use
the nodal basis function to approximate displacement and
potential, integrate both sides of the above equation, and apply
boundary conditions to obtain a global equation system, i.e.
=ΦK b
, (23)
where
.
,()
jjj
iii
fi ii
jx jy jz
NNN
NNN
xx yy zz
NNN
L Nw Ndw Nw
kx
d
d
yz
d
σ
η
Ω
Ω ΩΩ
∂∂∂
∂∂∂
++
∂∂ ∂∂ ∂∂
∂∂∂
=
∂∂∂
= Ω
Ω+ Ω+ Ω
∫
∫∫∫
b
K
(24)
The electric potential Φ can be solved by the MUMPS
direct solver [40], which allows us to decompose the
coefficient matrix K only once, and then continue to back-
replace the source term b in time recursively. Finally, we use
the element interpolation and calculate the electric fields from
potential Φ.
D. Stability and absorbing boundary condition
Since the explicit finite-element method is used to solve for
the seismic waves, to ensure the stability of the numerical
solution, the size of spatial step must be much smaller than the
shortest wavelength. According to our tests, when the ratio of
the mesh size to the shortest wavelength is less than 0.05, we
can achieve good modeling accuracy. Meanwhile, to ensure
the stability of the solution, the time step needs to meet the
Courant-Friedrichs-Lewy (CFL) stability condition, which is
determined based on the minimum grid size and the maximum
velocity.
Due to the limitation of computing resources, we often need
to introduce absorbing boundary conditions at the artificial
truncation boundaries to reduce reflections. Since EM waves
can be directly absorbed by expanding the edge, we only
consider the absorbing boundary conditions of seismic waves
here. Currently, the perfectly matched layer (PML) proposed
by Berenger [41] is recognized as the boundary with the most
ideal absorptions. In this paper, we use an unsplit complex-
frequency-shifted perfectly matched layer (CFS-UPML) based
on the second-order displacement equations [42], [43], [44].
For the CFS-UPML boundary condition, we need to
transform the governing equations (10) and (11) into the
frequency domain first, and replace the differential
n∂∂
by
1
n
sn⋅∂ ∂
(n=x,y,z), and finally we can transform them into
the time domain for solutions. Note that sn is the complex
stretch function that can be expressed as
n
nn
n
d
sk i
αω
= + +
, (25)
where kn and dn are the scaling and attenuation factors in three
directions, respectively. αn is the frequency-shifted factor, i is
the imaginary unit, while
ω
is the angular frequency. The
This article has been accepted for publication in IEEE Transactions on Geoscience and Remote Sensing. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TGRS.2024.3520236
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
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values of sn in three areas divided by CFS-UPML in Fig. 1 are
(sx, 1, 1), (1, sy, 1), and (1, 1, sz) in the surface area; (sx, sy, 1),
(sx, 1, sz), and (1, sy, sz) in the edge area; and (sx, sy, sz) in the
corner area.
Based on previous researches [45], [46], to obtain good
absorption, we take the x-direction as example and select the
following parameters for our CFS-UPML layer, i.e.
2
1 ( 1)( )
x max
x
kk L
=+−
, (26)
2
max
3( ) log
2
x
vx
dR
LL
= −
, (27)
0
(1 )
x
x
fL
απ
= −
, (28)
where kmax is the maximum scaling factor, we take kmax = 7
here; x is the distance from PML to the internal interface, L is
the total thickness of the PML, vmax is the maximum velocity
in the model that is assumed to be the fast longitudinal wave
velocity, R is the theoretical reflection coefficient; f0 is the
dominant frequency of the seismic wavelet.
We consider a homogeneous full-space of the dimension
320 m×320 m×320 m with a PML zone of 60 m in the cube.
Table II gives the physical parameters for the porous medium.
We take an explosion source located at the center of the model,
with the source time function being a 30 Hz Ricker wavelet.
Fig. 3 shows the wavefield snapshots of three solid
displacement components at two different times. It is seen that
there is no visible diffraction and boundary reflection at the
PML boundary. This implies that our PML absorbing
boundary condition works effectively.
Fig. 3. Snapshots of three components of the solid
displacement at two moments (t = 40, 60 ms). The outer black
lines distinguish the computational domain with the PML
region.
III. ACCURACY VERIFICATION
To verify the accuracy of the algorithm presented above, we
compare our FETD results with the analytical solutions from
Gao and Hu [3] for a half-space model.
Fig. 4. A half-space model. The top blue area is the air layer,
the yellow star represents the source, while the red triangle
represents the receiving point.
Fig. 4 shows a schematic diagram of the half-space model.
The entire model is divided into 220×220×170 elements. The
top part contains 10 air layers, while the other five surfaces
have 10 layers of EM expansion grids and 30 layers of seismic
absorption grids. The grid size of the internal calculation
domain is 2 m. Since the explicit FE method is used to solve
the seismic waves, it is necessary to ensure the numerical
stability of time discretization. We choose a time step of 0.02
ms, which meets the CFL criterion. Meanwhile, for EM field
simulation, since an implicit method is employed, we can use
a larger time step for efficiency. Here, we take a time step that
is 10 times the seismic time step, i.e., 0.2 ms.
To simulate the wavefield generated by an explosion source,
we use the moment tensor source M to represent the force
source in (10) and (11), namely
() ( )
s
st
δ
=− ∇−f M rr
, (29)
where s(t) is the source time function, rs represents the
location of the source and
()
s
δ
−rr
is the Dirac delta function.
We take an explosion source with M0 = 1000 N·m for our
simulation and assume that the source time function is a 30 Hz
Ricker wavelet. The source is located at (1 m, 1 m, 81 m) and
the receiving point is located at (31 m, 31 m, 141 m). The
parameters for the underground porous medium are given in
Table II, the conductivity of the air layer is taken as 10-10 S m-1.
TABLE II
PARAMETERS OF POROUS MEDIA FOR SEISMOELECTRIC
MODELING
Symbol
Porous medium
Sandstone
Reservoir
ρ
s
(kg m-3)
2650
2500
2400
ρ
f
(kg m-3)
1000
1000
980
φ
0.2
0.2
0.4
α
∞
3
3
3
K
s
(GPa)
12.2
7.31
4.69
K
f
(GPa)
1.985
2.0
2.25
K
b
(GPa)
9.6
5.48
1.06
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content may change prior to final publication. Citation information: DOI 10.1109/TGRS.2024.3520236
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G (GPa)
5.1
4.95
5.94
v
p
(m s-1)
2694.7
2400
2484
v
s
(m s-1)
1482.7
1500
1800
η
f
(Pa s)
0.001
0.001
0.001
k (m2)
10-10
10-10
10-10
C
f
(mol L-1)
0.01
0.082
0.00082
σ (S m-1)
0.0062
0.05
0.001
L (sC kg-1)
2.07
×
10-9
9.59
×
10-10
6.77
×
10-9
In Fig. 5, we show the comparison of the solid displacement
and electric field calculated using our FETD method (black
dashed lines) and the analytical solutions (red solid lines). Fig.
5(a), 5(b), and 5(c) show respectively three components of the
solid displacement, in which the symbols ‘P’, ‘P-PF’, and ‘P-
SF’ represent the direct P wave, the P wave and S wave
reflected by the free surface. Fig. 5(d), 5(e), and 5(f) show
three components of the electric field, where the symbols ‘P’
and ‘P-PF’ represent the coseismic electric fields generated by
the direct P wave and the reflected P wave generated by the
free surface. It is seen that our FETD results match well the
analytical solutions either for the solid displacements or the
electric fields. This verifies the effectiveness of our FETD
method in simulating the seismoelectric waves.
Fig. 5. Comparison of our FETD results (black dashed lines)
with analytical solutions (red solid lines) for a half-space
model. (a) Solid displacement ux; (b) solid displacement uy; (c)
solid displacement uz; (d) electric field Ex; (e) electric field Ey;
(f) electric field Ez.
Moreover, we also analyze the calculation time and memory
consumption for this model. In our numerical experiment, we
run all calculations at a Dell workstation with one CPU of
Intel Xeon Gold 6246R @3.40 GHz and 512 GB memory. For
the model in Fig. 4, when solving the seismic waves, we
divide it into 220×220×170 elements and obtain 50110866
unknowns. For such a large number of unknowns, we cannot
use a direct solver to calculate the seismic wavefield in our
workstation. However, as we use a lumped mass matrix, the
amount of data and calculation are largely reduced, we can use
the explicit recursion to solve it efficiently. Additionally,
when solving EM waves, we need only to solve one electric
potential at each node. This will yield 8351811 unknowns that
can be solved directly. The calculation of the seismic and EM
wavefields for each time step takes approximately 7.8 s, while
the total memory consumption for this model is approximately
471.4 GB.
IV. NUMERICAL EXPERIMENTS
When the underground structure has a limited extension, we
have to take it as a 3D model and cannot use the traditional 2D
method to simulate its seismoelectric responses. To better
understand the propagation characteristic of seismoelectric
waves in the underground, we design in this section a typical
block model and one with an embedded complex anomalous
body and calculate their seismoelectric responses.
A. A typical block model
We first set up a model with a block reservoir embedded in
the sandstone background. The parameters of the sandstone
and reservoir are given in Table II. As shown in Fig. 6, the
model is divided into 240×240×150 elements, in which the top
air layer contains 10 expansion layers, while the remaining
surfaces contain 30 absorbing layers and 10 expansion layers.
The grid size of the internal calculation area and the absorbing
layer is set to 2 m, the expansion layer is expanded three times
of the internal grid size. The number of grids in the block
reservoir is 80 m×80 m×40 m, with the center located at (0 m,
0 m, 100 m). We take an explosion source with M0 = 2.54×107
N·m (approximately the energy generated by one kilogram of
explosives [20]) located at (1 m, 1 m, 3 m). The source time
function is assumed to be a 40 Hz Ricker wavelet. Considering
that the effective signal of the borehole measurement is
stronger and thus a better record can be produced [1], we lay
out a survey line in the well, located from (61 m, 1 m, 1 m) to
(61 m, 1 m, 199 m) with a total of 100 receivers. For this
model, the simulation time of each time step takes
approximately 8.1 s.
Fig. 6. A block underground model used for our FETD
simulation with an 80 m×80 m×40 m reservoir buried 80 m
below the free surface.
In order to analyze the propagation characteristics of 3D
seismoelectric waves, we show snapshots of the solid
displacement and electric field at three different moments in
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Figs. 7-8 and Figs. 9-10, respectively. Fig. 7 shows the sliced
snapshot of the solid displacement along three planes centered
at the source location. It can be seen that due to the symmetry
of the block model in the x and y directions, the slices along
the xz- and yz-planes are symmetrical. In order to observe the
propagation of seismoelectric waves in the underground
medium more clearly, we show the slices along the xz plane
separately in Fig. 8. Figs. 8 (a)-(c) show three components of
solid displacement at t = 36 ms, respectively. It is seen that the
P wave (a combination of the direct P wave and the P wave
generated by the free interface) has just arrived at the upper
interface of the block reservoir, while the reflected S wave
generated by the free interface is also visible. Due to the
explosion source used, the direct P wave has no component in
the y-direction, and the converted S wave reflected from the
free interface has weak energy in the y-direction and is almost
invisible from Fig. 8(b). At t = 56 ms, as shown in Fig. 8(d),
the P wave has traversed the upper interface of the block
reservoir, generating both reflected and transmitted waves,
along with diffracted waves at the reservoir’s corners.
Simultaneously, as can be seen from Fig. 8(d), the Rayleigh
waves propagating near the free surface are apparent, and a
weaker transmitted S wave is visible in the y component of the
solid displacement in Fig. 8(e). At t = 120 ms, the reflected P
and S waves generated by the free interface pass through the
block reservoir (Fig. 8g and 8i), once again producing
reflected, transmitted, and diffracted waves. The Rayleigh
waves near the free surface are also distinctly observable both
in the slices along the xy-plane in Figs. 7 (g)-(i) and in Fig. 8g
and 8i. Fig. 8(h) reveals that the reflected and transmitted S
waves generated by the reservoir’s upper interface further
generate reflected and transmitted waves at other reservoir
interfaces.
Fig. 7. Snapshots of three components of the solid
displacement at the moments t = 36, 56, 120 ms.
Fig. 8. Snapshots of three components of the solid
displacement at the moments t = 36, 56, 120 ms in the xz-
plane. The outer black lines distinguish the computational
domain and the PML region, while the inner black boxes
represent the xz-slice of the blocky reservoir across the source.
We further analyze the wavefield snapshots of the electric
field. Similar to the solid displacement snapshot, we show the
sliced snapshot of the electric field along three planes centered
at the source location in Fig. 9. For the convenience of
analysis, we show the slices along the xz plane in Fig. 10.
Among them, Figs. 10 (a)-(c) show the snapshots of three
components of the electric field at t = 36 ms. It is seen that the
accompanying electric fields are generated by the direct P
wave and the reflected P wave at the free interface. Due to the
mechanical and salinity contrasts between the reservoir and
the background sandstone, when the P wave arrives at the
upper interface of the reservoir, an interface EM wave is
generated. Since the velocity of the EM wave is very high, the
interface EM wave will immediately propagate everywhere, so
we can see the spotlight inside the reservoir. Notably, the
interface electric field is also present in the y-component (the
signal is relatively weak, so we use a different color bar to
show Ey), which is a phenomenon not observed in 2D
seismoelectric simulations. At t = 56 ms, the coseismic electric
field caused by the transmitted P wave, after it passes through
the reservoir, along with the interface EM wave generated as
the transmitted P wave reaches the reservoir’s lower interface
can be seen. Additionally, the darker coloration of the
accompanying electric field within the reservoir compared to
the background sandstone can be observed. This is due to the
larger electrokinetic coefficient of the reservoir that enhances
its ability to induce stronger electric fields. Figs. 10 (g)-(i)
show that at t = 120 ms, the coseismic electric fields are
generated by the reflected, transmitted, and diffracted P waves.
The coseismic electric fields accompanying the Rayleigh wave
propagating near the free surface in the x and z directions are
also clearly visible.
This article has been accepted for publication in IEEE Transactions on Geoscience and Remote Sensing. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TGRS.2024.3520236
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Fig. 9. Snapshots of three components of the electric field at
the moments t = 36, 56, 120 ms.
Fig. 10. Snapshots of three components of the electric field at
the moments t = 36, 56, 120 ms in the xz-plane. The outer
black lines distinguish the computational domain and the PML
region, while the inner black boxes represent the xz-slice of
the blocky reservoir across the source.
Fig. 11 shows the solid displacement and electric field
excited by the explosion source at an array of 100 receivers
located from (61 m, 1 m, 1 m) to (61 m, 1 m, 199 m). The
symbols ‘P’, ‘S’, and ‘R’ in Figs. 11(a) and 11(c) denote the
superposition of the direct P waves and the reflected P waves
generated at the free interface, the reflected S waves generated
at the free interface, and the Rayleigh wave, respectively. The
symbol ‘P1’ in Figs. 11(a) and 11(c) represent the combined
reflected and transmitted P waves generated by P and S waves
passing through the interface of the block reservoir, whereas
‘P1’ in Fig. 11(b) refers specifically to the reflected and
transmitted P waves generated by S waves passing through the
reservoir interface. The symbol ‘P2’ in Figs. 11 (a)-(c)
indicates the transmitted P waves produced when the reflected
wave from the lower reservoir interface propagates to other
interfaces, while the symbol ‘S1’ denotes the transmitted S
wave generated by the reservoir interface. Since we take the
quasi-static approximation when calculating EM fields, the
coseismic electric field accompanying S wave will be missing
[47]. From Figs. 11 (d)-(f), we can see the coseismic electric
field accompanying displacement fields, excluding S wave.
The symbol ‘EMF’ represents the interface EM wave
generated by the direct P wave at the free interface. The
symbol ‘EM1’ represents the interface EM wave generated by
the direct P wave and P wave reflected at the free interface
upon reaching the reservoir’s top interface, while the symbol
‘EM2’ represents the interface EM wave generated by the
transmitted P wave after it passes through the upper interface
of the reservoir and reaches other interfaces. Notably, the
interface responses are observed in all three components of the
electric field, with the y-component clearly revealing the
complete interface EM wave generated by the underground
interfaces. Although the interface responses of the y-
component are an order of magnitude smaller than those in the
x- or z-component, they are observable signals [48] and can
provide valuable supplementary information for the
interpretation of the seismoelectric signals in contrast to 2D
seismoelectric modeling. Moreover, if the velocity of the
background sandstone is obtained in advance, the interface
depth can be inferred from the observed interface responses.
Fig. 11. Solid displacements and electric fields at an array of
100 receivers located from (61 m, 1 m, 1 m) to (61 m, 1 m,
199 m). (a)-(c) represent three components of the solid
displacement, while (d)-(f) represent three components of the
electric field.
B. Modified SEG/EAEG salt dome model
In this section, we consider a complex model that is modified
from the SEG/EAEG salt dome model proposed by Aminzadeh
et al. [49]. The grid number of the entire model is 240×240×150,
and the grid size for the internal computational domain is 2 m.
The salt dome is buried 60 m below the free interface. Fig. 12
shows its shape and location. The poroelastic parameters and
resistivity information are given in the ‘Reservoir’ of Table II.
The parameters of the background sandstone are the same as
those in the previous section. The settings of the external
expansion layers and the absorbing layers, the source and source
time function are also the same as in the last section. In addition,
we lay out two survey lines in the well, one is located from (61
m, 1 m,1 m) to (61 m, 1 m,199 m) with a total of 100 receivers,
and the other is located from (1 m, 61 m,1 m) to (1 m, 61 m,199
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content may change prior to final publication. Citation information: DOI 10.1109/TGRS.2024.3520236
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m) with a total of 100 receivers. For this model, the simulation
time of each time step takes approximately 8.0 s.
Fig. 12. The modified SEG/EAEG salt dome model.
To analyze the propagation characteristics of 3D
seismoelectric waves in complex media, we present the
wavefield snapshots of the solid displacement and electric
field at three times, as shown in Figs. 13-15 and Figs. 16-18,
respectively. Figs. 13 and 16 show the sliced snapshots of the
solid displacement and electric field on three planes centered
at the source location, respectively. It can be seen that the
response of the salt dome model is different from that of the
block model. It is asymmetric in the x- and y-directions, so the
slices along xz- and yz-planes centered at the source are
different. To further analyze the propagation of seismoelectric
waves more clearly, we separately show slices along xz- and
yz-planes with the source as the center. Figs. 14 and 15 show
the solid displacement snapshots for the xz- and yz-slices,
while Figs. 17 and 18 show the corresponding electric field
snapshots for these slices at t = 40, 56, 104 ms.
Refer to Figs. 14 (a)-(c), from the x- and z-components at t
= 40 ms, one can see that the P wave (the direct P wave and
the reflected P wave generated at the free interface are
superimposed together) has just arrived at the top interface of
the salt dome, while the reflected S wave generated at the free
interface is also visible. A similar case can be observed in the
y and z components in Fig. 15 (a)-(c). At t = 56 ms, one can
see from Figs. 14(d) and 14(f) that the P wave passes through
the upper interface of the salt dome, generating both reflected
and transmitted waves, one can also see that the Rayleigh
waves are generated at the free interface. In Fig. 14(e), the y
component shows a weak transmitted S wave, similar to the x
component in Fig. 15(d). At t = 104 ms, the reflected P wave
and S wave generated at the free interface pass through the salt
dome (Figs. 14 g-i), generating again the reflected and
transmitted waves. Additionally, the Rayleigh waves
propagating near the free surface are also clearly visible,
which is more obvious in the slice along the xy-plane in Figs.
13 (g)-(i). Fig. 14(h) shows that the transmitted S wave
generated at the upper interface of the salt dome further
generates reflected and transmitted waves at the boundary of
the salt dome. Figs. 15 (g)-(i) show consistent observations.
Fig. 13. Snapshots of three components of the solid
displacement at the moments t = 40, 56, 104 ms.
Fig. 14. Snapshots of three components of the solid
displacement at the moments t = 40, 56, 104 ms in the xz-slice.
The outer black lines distinguish the computational domain
and the PML region, while the inner black frames represent
the xz-slice of the salt dome model across the source.
Fig. 15. Snapshots of three components of the solid
displacement at the moments t = 40, 56, 104 ms in the yz-slice.
The outer black lines distinguish the computational domain
and the PML region, while the inner black frames represent
the yz-slice of the salt dome model across the source.
This article has been accepted for publication in IEEE Transactions on Geoscience and Remote Sensing. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TGRS.2024.3520236
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Furthermore, we analyze the electric field snapshots in the
xz- and yz-slices. Figs. 17 (a)-(c) show the snapshots of three
components of the electric field at t = 40 ms. The
accompanying electric field generated by the direct P wave
and the reflected P wave generated at the free interface can be
observed in the x- and z-components, the interface EM waves
are visible in all three components within the salt dome
reservoir. A similar pattern can be seen from Figs. 18 (a)-(c).
Note that although the P wave has not yet arrived at the upper
interface of the salt dome in the yz-slice, it has propagated to
the top bulge of the salt dome. As a result, the generated
interface response spreads rapidly, allowing the interface wave
to become visible in the yz-slice. At t = 56 ms and 120 ms, the
accompanying coseismic electric field caused by the reflected
and transmitted P waves, as well as the interface EM waves
generated when the P wave arrives at the salt dome boundary,
can be observed in Figs. 17 and 18. Additionally, the
coseismic electric field accompanying the Rayleigh wave
propagating near the free surface can be observed in the xz-
and yz-slices.
Fig. 16. Snapshots of three components of the electric field at
the moments t = 40, 56, 104 ms.
Fig. 17. Snapshots of three components of the electric field at
the moments t = 40, 56, 104 ms in the xz-slice. The outer black
lines distinguish the computational domain and the PML
region, while the inner black frames represent the xz-slice of
the salt dome model across the source.
Fig. 18. Snapshots of three components of the electric field at
the moments t = 40, 56, 104 ms in the yz-slice. The outer black
lines distinguish the computational domain and the PML
region, while the inner black frames represent the yz-slice of
the salt dome model across the source.
It is worth noting that although the parameters used in the
salt dome model and the block model in the previous section
are consistent, their burial depths are almost the same, the y
component of the electric field in the xz-slice or the x
component in the yz-slice shows that the interface responses in
the salt dome model are almost two orders of magnitude larger
than those in the block model. This may be due to the salt
dome’s relatively thin structure in the horizontal direction at
the top and in the vertical direction at the bottom (both
dimensions are much smaller than the wavelength), causing
waves to reflect and interfere multiple times within the salt
dome. These repeated interactions at the boundaries of the salt
dome continuously amplify the interface responses.
Furthermore, the amplitude of overall interface responses in
this salt dome model is comparable to that of the coseismic
electric field, which is crucial for the identification of
underground reservoirs.
Fig. 19 shows the travel time diagrams of the solid
displacement along two survey lines, with (a)-(c) showing a
downhole survey on the right side of the source (referred to as
Survey line 1), and (d)-(f) showing another downhole survey
in front of the source (Survey line 2). Similar to the block
model discussed in the previous section, the symbols ‘P’, ‘S’,
and ‘R’ in Fig. 19 represent the superposition of direct P
waves and reflected P waves generated at the free interface,
the reflected S waves generated at the free interface, and the
Rayleigh waves, respectively. The symbol ‘P1’ in Fig. 19(c)
and 19(f) indicates the superposition of reflected and
transmitted P waves generated by P and S waves passing
through the salt dome reservoir interface. In contrast, the
symbol ‘P1’ in Fig. 19(b) and 19(d) represents the
superposition of reflected and transmitted P waves generated
by S waves passing through the reservoir interface. Due to the
asymmetric structure of the salt dome, the energy of the
reflected wave is primarily concentrated on the left side in the
xz-slice, while the energy on the right side is relatively weaker.
This article has been accepted for publication in IEEE Transactions on Geoscience and Remote Sensing. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TGRS.2024.3520236
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As a result, the color of ‘P1’ in Fig. 19(c) appears lighter. In
contrast, the model is relatively symmetrical in the yz-slice,
leading to stronger reflected and transmitted P waves in Fig.
19(f). The symbol ‘P2’ represents the transmission P wave
generated by the reflected wave (including reflections that
undergo multiple interactions within the salt dome) from the
bottom interface when it propagates to other interfaces. The
symbol ‘S1’ denotes the transmission S wave generated at the
reservoir interface.
Fig. 19. Three components of the solid displacement. (a)-(c)
Array of 100 receivers located from (61 m, 1 m, 1 m) to (61 m,
1 m, 199 m); (d)-(f) array of 100 receivers located from (1 m,
61 m, 1 m) to (1 m, 61 m, 199 m).
Fig. 20 shows the travel time diagram of the electric field
along the two survey lines, where (a)-(c) show the electric
field along Survey line 1, while (d)-(f) show the electric field
along Survey line 2. From Fig. 20, we can see the coseismic
electric fields accompanying the above displacement fields.
Due to the irregular shape of the salt dome, there are
noticeable differences in the arrival time and amplitude of the
coseismic electric field generated by the reflected P wave
between Survey lines 1 and 2. These differences are very
useful for identifying 3D underground anomalies. The symbol
‘EMF’ represents the interface EM wave generated by the
direct P wave at the free interface. The symbol ‘EM1’ denotes
the interface EM wave generated by the direct P wave and the
reflected P wave generated at the free interface arriving at the
upper interface of the salt dome. The symbol ‘EM2’ indicates
the interface EM wave generated by the transmitted P wave
inside the reservoir as it reaches the boundary of the salt dome.
Note that the interface EM wave can be observed in three
components of the electric field along both Survey line 1 and 2.
In the case of known wave velocity, we can roughly infer the
burial depth and thickness of the underground reservoir from
these interface responses. The amplitudes of these interface
responses are comparable to those of the coseismic electric
fields, and are significantly larger than those observed in the
block reservoir model in the previous section. This is
attributed to the presence of many thin layers in the salt dome
model, which amplify the interface response when the waves
pass through these thin layers. This highlights the advantage
of seismoelectric signals in distinguishing thin layers. From
Figs. 20(b) and 20(d), one can see that the interface response
is relatively strong in the y-component of Survey line 1 and
the x-component of Survey line 2. Compared to Fig. 20(a),
20(c), 20(e), and 20(f), their coseismic electric field signal
masks part of the interface electric field signal (they are
difficult to separate as their arrival times are consistent), while
the y-component in Fig. 20(b) and the x-component in Fig.
20(d) clearly show the arrival time of the interface EM waves.
This demonstrates that 3D seismoelectric modeling can
provide more useful information than 2D modeling. It is more
comprehensive in the identification and interpretation of
seismoelectric signals.
Fig. 20. Three components of the electric field. (a)-(c) Array
of 100 receivers located from (61 m, 1 m, 1 m) to (61 m, 1 m,
199 m); (d)-(f) array of 100 receivers located from (1 m, 61 m,
1 m) to (1 m, 61 m, 199 m).
C. Sensitive analysis
In the past, many researchers have conducted sensitivity
analysis of seismoelectric responses, including the effects of
properties such as salinity and permeability on coseismic EM
responses [50], [51], direct radiation EM responses [19], [52],
and interface responses [9], [20], [51]. Here, we mainly
consider the effects of the conductivity, permeability, and
porosity on seismoelectric conversion for complex models.
Taking again the SEG/EAEG salt dome model as example and
considering that 3D modeling can provide more information
on the interface responses than 2D modeling, we focus on the
y-component of the interface EM wave (which cannot be
observed in 2D modeling) on Survey line 1.
We analyze the effect on the interface EM wave by
changing conductivity, permeability or porosity of the salt
dome while keeping other parameters unchanged. Fig. 21
shows the travel time of Ey when the conductivity changes
from 0.01 S/m to 0.0001 S/m, the permeability changes from
10-10 m2 to 10-12 m2, and the porosity changes from 0.2 to 0.4.
It can be seen that when the conductivity decreases (a-c), the
permeability decreases (d-f) or the porosity increases (g-i), the
maximum amplitude of Ey will increase, which is consistent
with previous analyses regarding the effect of these
parameters on interface responses [20]. Furthermore, we
analyze the effect of medium parameters on the amplitude and
time shift of the electric field. Considering that the results
observed at different depths are similar, we choose the
receiving point at a depth of 121 m where the interface
response is clearly visible. Fig. 22 shows the y-component of
the electric field recorded at the receiving point (61 m, 1 m,
This article has been accepted for publication in IEEE Transactions on Geoscience and Remote Sensing. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TGRS.2024.3520236
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121 m) for different conductivities, permeabilities or
porosities. Again we can see that when the conductivity
decreases, the permeability decreases, or the porosity increases,
the amplitude of Ey increases. Note that in Fig. 22(c), as the
porosity increases, Ey not only changes its amplitude, but also
the time shift. This is because the change in porosity will
change the velocity of seismic wave in the salt dome, thereby
affecting the arrival time of P wave passing through the salt
dome reservoir interface.
Through sensitivity analysis, we find that the interface
responses are very sensitive to the conductivity, permeability
and porosity of the underground media. The sensitivity of the
interface responses to conductivity can help track or monitor
pollutants, while the sensitivity to permeability and porosity
can help detect or monitor aquifers or cracks filled with fluids.
Fig. 21. The y-component of electric field recorded for an
array of 100 receivers located from (61 m, 1 m, 1 m) to (61 m,
1 m, 199 m) for different conductivities, permeabilities and
porosities. The maximum electric fields are given in each plot.
Fig. 22. The y-component of electric field recorded at (61 m, 1
m, 121 m) for different conductivities, permeabilities and
porosities.
V. CONCLUSION
Based on the electrokinetic coupling equations in saturated
porous media, we have successfully developed a time-domain
finite-element (FETD) algorithm for the simulation of
seismoelectric waves for 3D heterogeneous anomalies. By
decoupling the seismoelectric coupling equations, we can
separately solve the seismic waves and EM waves. In this case,
we can use the explicit FE method to solve seismic waves first
and then use the obtained relative fluid-solid velocity fields as
source to do EM modeling. By combining explicit and implicit
recursion, we can largely improve the modeling efficiency.
The application of our algorithm to the block model and the
modified SEG/EAEG salt dome model showed that our
algorithm can offer accurate numerical simulations of
seismoelectric responses for complex 3D structures. By
analyzing the seismoelectric responses of two inhomogeneous
models, we found that the seismoelectric interface responses
are very helpful in the identification of underground structures,
especially thin layers. We believe that compared to 2D
modeling, the three components of coseismic signal and
interface response obtained from our 3D seismoelectric
modeling contain more comprehensive information on the
propagation of seismoelectric waves in the underground. The
seismoelectric signals obtained from 3D full-waveform
simulations have the potential for the characterization of
subsurface properties due to the sensitivity of seismoelectric
signals to the conductivity, permeability, and porosity. This
brings the potentials for application in detecting oil and gas,
underground water, and industrial pollutant. Furthermore, the
development of 3D full-waveform modeling of seismoelectric
waves also lays the foundation for future inversion and
interpretation of seismoelectric data.
APPENDIX
The mass matrix, damping matrix, stiffness matrix, and
source term matrix in (14) and (15) can be written as
1 02 03 0
40 0
, , ,
, ,
ff
ff
ρ ρ ρ
ρ α η
φk
∞
= = =
= =
M KM KM K
M KD K
(30)
11 1 2 3 12 1 2 3
13 1 2 3
TT
14 4 4 15 5 5
T
16 6 6
TT
17 4 4 18 5 5
T
19 6 6
, ,
,
, ,
,
,
2
,
,
( ) ( 2)
( 2)
( 2) ( 2)
( 2)
HGG GHG
GG H
HG G HG G
HG G
HG G HG G
HG G
+ ++
++
−+ −+=
−
−
=+=
=
=
=
=
=
+
+ −+
−
=
+
K KKKK KKK
K KK K
K K KK K K
K KK
K KKK KK
K KK
(31)
This article has been accepted for publication in IEEE Transactions on Geoscience and Remote Sensing. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TGRS.2024.3520236
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21 1 22 2 23 3
24 4 25 5 26 6
TTT
27 4 28 5 29 6
, , ,
, , ,
, , ,
αMαMαM
αMαMαM
αMαMαM
= = =
= = =
= = =
K KK K K K
KKKKKK
K KK KK K
(32)
31 1 32 2 33 3
34 4 35 5 36 6
TTT
37 4 38 5 39 6
, , ,
, , ,
, , ,
MMM
MMM
MMM
= = =
= = =
= = =
K KK K K K
KKKKKK
K KK KK K
(33)
=,,
i
i
zF fd i xy
Ω
= ⋅Ω
∫
N,
, (34)
where Ω denotes the entire model domain, the superscript ‘T’
represents the matrix transpose, and the matrices K0-K6 are
given by
0Ω
12
Ω Ω
34
Ω Ω
56
ΩΩ
Ω,
Ω, Ω,
Ω, Ω,
Ω, Ω
ij
jj
ii
jj
ii
jj
ii
NNd
NN
NN
dd
xx yy
NN
NN
dd
zz xy
NN
NN
dd
xz yz
=
∂∂
∂∂
= =
∂∂ ∂∂
∂∂
∂∂
= =
∂∂ ∂∂
∂∂
∂∂
= =
∂∂ ∂∂
∫
∫∫
∫∫
∫∫
K
KK
KK
KK.
(35)
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content may change prior to final publication. Citation information: DOI 10.1109/TGRS.2024.3520236
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This article has been accepted for publication in IEEE Transactions on Geoscience and Remote Sensing. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TGRS.2024.3520236
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