Finite element modeling of the electric field for geophysical application

Trofimuk Institute of Petroleum Geology and Geophysics SB RAS

Shtabel Nadezhda, Antonov Eugeniy

Outline

1. The types of the geophysical problems:

• mathematical models

2. Finite element method:

• spaces,

• variational formulations,

• discretization

3. The impulse sounding problem in time domain

4. The sounding problem in frequency domain

The types of the geophysical problems

Problems in time domain Problems in frequency domain

• The surface sounding• The marine geoelectric• The impulse sounding

•The borehole logging•The frequency sounding

Time dependent second order equations for electric field

Helmholtz equation for electric field

Second order equations

• Hyperbolic equation

• Parabolic equation

21 0

2

JErot( rot E)

E

t t t

1 0Jrot( rot E)

E

t t

E n 0

0

E gt

0

E

t

gt

E n 0

0

E gt

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

or

n E 0

n E E

1 rot 0 PMCm

E n

Advantages of applying the vector finite element method to the electromagnetic problems

• Solves equations in nature quantities as 3D vector field

• Keeps the tangential component of the electric field continuous on the interface boundary

• The normal component of the electric field has jump on the interface boundary

• The FEM solution fulfils to the charge conservation law

• Any type and geometry of the field source are taking into account

2 2

2 2

2 2

{grad, : 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

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

Variational Formulations

For 320 ))(( LJ find 0E (rot; )H such

that 0W (rot; )H the following is held2

1 02

( rot , rot W) ( , W) ( , W) ( , W)JE E

Et t t

1 0( rot , rot W) ( , W) ( , W)JE

Et t

For 320 ))(( LJ find 0E (rot; )H such

that 0W (rot; )H the following is held

Parabolic equation

Hyperbolic equation

The variational formulation

The following propertyallows to fulfill the variational analog of the charge conservation law

0 0( ; ) , ( ; )H H grad rot

1 20( , W) ( ,W) ( ,W)k i

E E J

20(( ) , ) 0 (grad, )i H E

For 320 ))(( LJ to find 0E (rot; )H such

that 0W (rot; )H the following is held

Discretization by time and space

1

,eN

k k ik

E e W W

Basic function of the space H (rot,Ω)

2 1

2 2

2n n n

t

E E E E 1n n

t

E E E

2 1 1 1 1

1

2

( )

n n n n n

n n n

Be Ce Se Ce Ce

Se F F

1

0

( rot , rot ) ( , )

( , ) ( , )

i j i jij ij

i j iij i

B C

S F J

W W W W

W W W

Discretization of Helmholtz equation

1

,eN

k k ik

E e W W

Basic function of the space H (rot,Ω)

Be Ce iwF

1

0

( rot , rot ) ( , )

( , )

i j i jij ij

ii

B C k

F J

W W W W

W

Im0( , ), ( ,0)R Re e e J J

O.V. Nechaev, E.P. Shurina, M.A. Botchev. Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation // Computers and Mathematics with Applications. - 2008 - Vol. 55 - Pp. 2346-2362

Mesh generation

Complex structure of the investigated media requires to use meshes that have good approximation of the curvilinear boundaries, have local thickening.

C. Geuzaine and J.-F. Remacle. Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering 79(11), pp. 1309-1331, 2009.

Subdomain Ω1: airσ = 1е-6 1/(Оm·m)

Subdomain Ω2:Conductive ground

Field source – the current loop with impulse signal

Impulse sounding

EMF, V

Time, s

=10

=1

=0,1

The features of the EMF graphs for the impulse sounding problem

1. Domain 5000 m х 5000 m – 50 sizes of source2. Domain 10000 м х 10000 м – 100 sizes of source3. Domain 25000 м х 25000 м – 250 sizes of source

The size of the domain should be not less then 25 km for the source loop with the size 500 m

The limitation of the computational domain

EMF, V

t

The limitation for computational domain

Domain: 25 km х 25 km х 10 km

Source loop: 500 m х 500 m

Layers – 5 km, 300 m, 3.5 km

The size of the mesh: 332`760 nodes2`288`590 edges

Domain: 25 km х 25 km х 16 km

Source loop: 500 м х 500 м

Layers – 5 km, 1.5 km, 2.5 km,1 km, 2 km, 4 km

The size of the mesh: 84`355 nodes574`035 edges

=10=5

=1000

Hz

w

The features of the EMF graphs for the frequency impulse sounding problem

Hz

w

The features of the EMF graphs for the frequency impulse sounding problem

Hz

w

The features of the EMF graphs for the frequency impulse sounding problem