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![Page 1: Computational electrodynamics in geophysical applications Epov M. I., Shurina E.P., Arhipov D.A., Mikhaylova E.I., Kutisheva A. Yu., Shtabel N.V.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e965503460f94b99e3e/html5/thumbnails/1.jpg)
Computational electrodynamics in geophysical applications
Epov M. I., Shurina E.P., Arhipov D.A., Mikhaylova E.I., Kutisheva A. Yu., Shtabel N.V.
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The main features of the geological media
• Heterogeneous media, fluid-saturated rocks.
• The complex geometry of objects.
• The complex configuration of interface boundary.
• The electrophysical properties: the contrast between separate fragments of media, anisotropy, polarization, the dispersion of the conductivity, permittivity and permeability.
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The Maxwell’s equations
0,
,
, 0.
t
t
B E
H J D
D B
0
, ,
.
B H D E
J J E
The Faraday's law
The Maxwell – Ampere law
The Gauss’s laws for electric and magnetic flux densities
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The Second order equations
• Hyperbolic equation
• Parabolic equation
21 0
2rot( rot )
t t t
JE E
E
1 0rot( rot )t t
JE
E
n 0
E0 gtE Ε
0t
gt
E
n 0
E0 gtE Ε
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Frequency domain. Helmholtz equation
1 2 2 20( ) ( )
( ) 0
k i k i
i
E E J
E
The boundary conditions
The charge conservation law
0
0
, PEC
e
n E 0
n E E
1 rot 0 PMCm
E n
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i j
The interface conditions
0,
0,
,
0,
0.
ij
ij
ij
ij
ij
E n
H n
D n
B n
E ni j i j
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12 2
2 2
2 2
{ : grad }
(rot, ) { ( ) : rot ( )}
(div, ) { ( ) : div ( )}
H u L u
H x x
H w w L
L
L L
L
2 2 2( ; )
2 2 2( ; )
2 2 2( ; )
,
,
.
H grad
H
H div
u u u
rotu u u
u u u
The functional spaces
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The functional subspaces and de Rham’s complex
0
0
0
(grad; ) (grad; ) | | 0 ,
(rot; ) (rot; ) | | 0 ,
(div; ) (div; )| | 0 .
H u H u
H H
H H
u u n
u u n
2( ; ) (rot; ) (div; ) ( )H grad H H L
20 0 0( ; ) (rot; ) (div; ) ( )H grad H H L
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Variational Formulations
For 30 2( ( )) J L find 0 (rot; ) E H such
that 0 (rot; ) W H the following is held2
1 02
( rot , rot ) ( , ) ( , ) ( , )t t t
JΕ E
Ε W W W W
1 0( rot , rot ) ( , ) ( , )t t
JE
E W W W
For 30 2( ( )) J L find 0 (rot; ) E H such
that 0 (rot; ) W H the following is held
Parabolic equation
Hyperbolic equation
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Time ApproximationWe introduce the following partition of the time ),0( T
Tttt M ...0 10
and function on -th time step
where ),,( zyxEu hjj is a solution on j-th time step
j step on j-th step of time scheme.
,))((
)(,)(
))(()(
1
21
1
120
jj
jjj
jjj
jjj ttttt
ttttt
j
11
12 )(
))(()(
jjj
jjj ttttt
Then the function of interest is
)()()(),,,( 22110 tutututzyxE jj
jj
jj
h
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Newmark-beta Scheme
where jJ0 the value of right hand side on j-th
time step, parameter of the scheme. 4/1
)W,)21(
()W,)(
)(
2()W,
)(
22
)(
2()W),)21(((
h2
01
0
0h2
111
1
1
1
1h2
111
1
1
h21
1
t
J
t
J
t
Juu
uuu
uuuu
jj
j
jjjj
jj
jj
jj
jjjj
jjj
jjjj
jj
jjjj
jjj
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The variational formulation
The following propertyallows to fulfill the variational analog of the charge conservation law
0 0( ; ) , ( ; )H grad H rot
1 20( , ) ( , ) ( , )k i
E W E W J W
20(( ) , ) 0 (grad, )i H E
For 30 2( ( )) J L to find 0 (rot; ) E H such
that 0 (rot; ) W H the following is held
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Geometric domain decomposition
Difficulties:• Local source of the field (the source should be in one
subdomain and can’t touch its boundaries)• Balancing the dimensions of subdomains matrices
(CPU time should be comparable in different subdomains)
• The geometry of the computational domain should be taken into account
Decomposition approaches:• Custom decomposition (effective, but time-consuming)• Automatic decomposition
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Automatic Decomposition
• Decomposition by enclosed “spheres”
• Decomposition by layers
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EM Logging
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Borehole - Inclined bed
1 – 3-coil probe, 2 – borehole with mud, 3 – host formation, 4 – low-conductive bed, Г – generator coil, И1, И2 – receiver coils
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ElectroPhysical Properties
Operating frequency 14 МHz,
amperage J=1 А.
Domain
1 1 1 0
2 1 1 5
3 1 1 0.1
4 1 1 0.01
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Re Ex (X0Y)
00
Zenith angle
450 750
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Re Ez (X0Y)
00
Zenith angle
450 750
Ez=0
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Surface Soundings
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Transmitter loop 40 x 40 m²Receiver loop 20 x 20 m²Impulse length 5 µsSimulation time 10 msMesh: 335666 edges, 49244 nodes, 281342 tetrahedrons Computation one time step 30 sec, after current is turn offSolver: Multilevel iterative solver with V-cycle
Anisotropic layer
Isotropic layer
Zenith Angle 0, 30, 60, 902 2
11 z
12 21 23 32
22 y
13 31
2 233
' = cos θ sin θ
' = ' =0 ' = ' =0
' =
' = ' =( - )sinθcosθ
' = sin θ+ cos θ
z x
x z
x
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x
y
-500 0 500-600
-400
-200
0
200
400
600
x
y
-500 0 500-600
-400
-200
0
200
400
600
Transversal isotropic medium θ=0°
x
y
-500 0 500-600
-400
-200
0
200
400
600
x
y
-500 0 500-600
-400
-200
0
200
400
600
x
y
-500 0 500-600
-400
-200
0
200
400
600
Ex, z=-50
Ey, z=0
Ez, z=-50Ey, z=-50
Ex, z=0
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x
y
-500 0 500-600
-400
-200
0
200
400
600
x
y
-500 0 500-600
-400
-200
0
200
400
600Ex z=0 Ey z=0
x
y
-500 0 500-600
-400
-200
0
200
400
600
x
y
-500 0 500-600
-400
-200
0
200
400
600
x
y
-500 0 500-600
-400
-200
0
200
400
600Ex z=-50 Ey z=-50 Ez z=-50
Transversal isotropic medium rotated for zenith angle θ=60°
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The anisotropic object in the isotropic halfspace
5.12 -0.75 0.27
= -0.75 1.03 -0.05
0.27 -0.05 1.02
10 0 0
0 1 0
0 0 1
1 0 0
0 1 0
0 0 10
24
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Re Ex, Ez for vertical object
The cross-section x=3.4 m
The isotropic object The anisotropic object
The conductivity of the medium is=0.01 Sm
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Re Ex, Ez for horizontal object
The cross-section z= -1 m
The isotropic object The anisotropic object
The conductivity of the medium is=0.01 Sm
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The multiscale modeling in media with microinclusions
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10div grad u
0 1 1 2
1| gu u
2
10
u
n
1 , 0u
1 , 1u
21
0
1 , 0u
2
10
1 , 1u The problem is stated in the domain
and governed by
the following equation:
1
electric potential
electrical conductivity
electrical resistivity
u
Problem definition
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1 2( ) , ( ) : ( ( ), ( )) ( ) ( ) ,H u v L u x v x u x v x d x
1
1 10( ) ( ) : | 0H v H v
1 10 0 0
1
( )
( ) ( ) , ( ) held
( ) ( ) 0x
Find u H u such that v H the following is
u x v x d in
Variational problem
We introduce the Hilbert space
Then the variational problem of the homogeneous elliptic problem states:
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Discrete variational problem
Let's consider a partition in the area Ω. Element is a tetrahedron.h hK
Let's introduce the spaces
Then the variational problem of the homogeneous elliptic problem states:
10 ( )( ) : 1,...,n ;K
i Kh hV Hspan i K
0, 10 ( )( ) : 1,...,n ;K
i Kh hW Hspan i K
,
,
0
1
( )
( ) ( ) , ( )
( ) ( ) 0
h h h
h h
x
Find u V u such that v V the following is held
u x v x d in
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Taking into account the partition we introduce the following statements:h
where and – quadrature points and weights respectively. lx l
Discrete variational problem
1
( )
1( ) ( ) 0
( )
( ) ( ) 0, ,
, ,
K Ki j
K
K Kl i l j l
K l l
x
K x xx
x x d x K
x K
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The basic principles
The local functions 0 , 1,...,3j j
The local multiscale “form functions” , 1,...,3i i
The global multiscale “form functions” , 1,...,i i N
FEM
Assemble according degrees offreedom associated with nodes of the coarse mesh
The integration points
Heterogeneous Finite Element Method
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015 mm
40 mm
X
ZY
15 mm
Scalability
InclusionsVolume of inclusions, %
Number of Cores
1 2 45х10х10 3.81 823 455 246
CPU time (sec)
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1.4e 7inclusion Оhm m 3.13matrix Оhm m
Method The error
Physical experiment
2.77 -
Maxwell's approach
2.84 2.35%
Bruggeman's approach
2.82 1.64%
Approach of coherent potential
2.80 1.24%
Numerical Modeling
2.79 0.72%
,eff Оhm m
2мм
Comparison with the physical experiment
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1.8e 8inclusion Оhm m
6.15matrix Оhm m
Method The error
Physical experiment
4.25 -
Numerical Modeling
4.00 5.80%
,eff Оhm m
Comparison with the physical experiment
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В)horizontal
The size of the inclusions: _*100%
_
V inclusionsC
V domain
a) vertical b) arbitrary directed
{1e 3, 1e 3}
1
inclusion
matrix
Оhm m
Оhm m
г) spheres
5.8мм
The cylinder with inclusions
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The influence of the geometry and orientation of the inclusions
Horizontal plates
Arbitrary oriented plates
Vertical plates
Spheres
Horizontal plates
Arbitrary oriented plates
Vertical plates
Spheres
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0
81
1.391
1.8 10
Ohm m the matrix
Ohm m theinslusions
The percolation
The size of the inclusions
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The calculation of the effective tensor coefficients
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E. Shurina, M. Epov, N. Shtabel and E. Mikhaylova. The Calculation of the Effective Tensor Coefficient of the Medium for the Objects with Microinclusions // Engineering, Vol. 6 No. 3, 2014, pp. 101-112.
The main steps of the algorithm
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Mathematical modelThe Helmholtz equation
in Ω
Boundary conditions
0
0
, PEC
n E 0
n E Ee
1 PMCE n gm
i
2 2k i is the wave number
1 2( 0E) + Ek
The direct problem
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Calculation of the effective coefficient
Z is a complex-valued second rank tensor, which can be interpreted as the analog of i
Scalar
Tensor
(i ) rot E H
rotZ E H
11 12 13
21 22 23
31 32 33
z z z
Z z z z
z z z
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The 1-st method of calculating tensor Z
2 2( ) ( )/ij i jZ RH E
L L
Re( ) Re( ) / Re( )ij i jZ RH E
Im( ) Im( ) / Im( )ij i jZ RH E
where , , ,i j x y z
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1
2
3
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
x i y i z i l l i
x j y j z j l l j
x k y k z k l l k
E x E x E x z RH x
E x E x E x z RH x
E x E x E x z RH x
where , , 1,i j k N , ,l x y z
The 2-nd method of calculating tensor Z
Fields E and rot H are calculated in N points of the domain (for example, in barycentres of tetrahedral finite elements). We obtain the set of tensors Z {Zm, m=1,..,N-2}, by running over the points xi, xj, xk. The effective tensor coefficient of the medium is calculated as an average of {Zm, m=1,..,N-2}.
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Variational formulation
1( 0i Z E) + E
Helmholtz equation in anisotropic media
Variational formulation:
1 0d i Z d
E W E W
The problem in anisotropic media
0 0H ( , ) E rot E
Find such that 0 (rot; ) W H
the following is held
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Boundary conditions
The domain with one side boundary
conditions
0
0
,
e
n E 0
n E E
1 0m
E n
0e
n E E
1 0m
E n
The domain with boundary conditions
given by the closed path
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• The size of the computational domain:
15 mm 40 mm 15 mm• The diameter of the
inclusions d = 2 mm• The number of the inclusions
is different
Domains
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The electrophysical properties of the computational domain
The matrix The inclusions
ε [F/m] 4.5 ε0 1 ε0
σ [Sm/m] 0.001 0.1
µ [H/m] 1 µ0 1 µ0ε0 = 8,85 ×10-12 F/m µ0 = 4π ×10-7 H/m
The mesh (40 inclusions)
The results of numerical experiment
Number of the inclusions
Volume of the
inclusions
The size of SLAE
40 regular 2% 171 872
40 chaotically 2% 169 412
176 regular 27% 296 070
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The homogeneous medium. The one size boundary conditions. The frequency 10 kHz
1
0.001 5.352 8 0.001
1.868 1 0.001 1.869 1
9.999 4 5.352 8 0.001
9.293 2 9.328 4 9.294 2
9.963 2 0.001 9.963 2 .
9.385 2 9.42 4 9.386 2
E
Z E E
E E
E E E
i E E
E E E
2
1 3 9.023 21 2.297 26
1.91 21 1 3 3.404 27
3.007 16 1.62 15 1 3
2.502 6 1.457 21 1.196 26
6.404 21 2.502 6 2.8 26 .
1.25 16 5.38 16 2.502 6
e e e
Z e e e
e e e
e e e
i e e e
e e e
EzR – Re Ez computed for homogeneous medium (=0.001Sm/m) with inclusionsEzR tensor – Re Ez computed for the medium with tensor coefficient Z2
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1
2.103 3 1.880 6 2.112 3
3.987 3.563 3 4.004
2.096 3 1.873 6 2.105 3
2.701 3 1.425 2.710 3
2.130 6 1.124 3 2.137 6
2.690 3 1.419 2. 3
.
699
E E E
Z E
E E E
E E E
i E E E
E E E
2
1 3 1.405 16 2.322 20
5.211 18 1 3 1.001 20
1.824 14 1.567 13 1 3
1.7516 1.7957 16 2.843 20
9.857 18 1.7516 2.353 21 .
4.418 14 4.998 13 1.7516
e e e
Z e e e
e e e
e e
i e e
e e
The homogeneous medium. The one size boundary conditions. The frequency 7 GHz
EzR – Re Ez computed in homogeneous medium (=0.001Sm/m) with inclusionsEzR tensor - Re Ez computed in the medium with tensor coefficient Z2
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176 inclusions
10 kHz 7 GHz
In the medium with inclusionsIn anisotropic mediumIn uniform medium, 0.001 Sm/m
In the medium with inclusionsIn anisotropic mediumIn uniform medium, 0.001 Sm/mIn uniform medium, 0.1 Sm/m