finite-volume modelling of geophysical electromagnetic

Finite-volume modelling of geophysicalelectromagnetic data using potentials on

unstructured staggered grids

Hormoz Jahandari and Colin G. Farquharson

Department of Earth SciencesSt. John’s, Newfoundland, Canada

SEG 85th Annual Meeting, New Orleans

October 20, 2015

Hormoz Jahandari and Colin G. Farquharson 1 / 35

Contents

1 Unstructured grids

2 Finite-volume discretization of Maxwell’s equations (directEM-field and potential formulation)

3 Example for a grounded wire source

4 Example for a helicopter EM survey

5 Conclusions


Unstructured grids

Model irregular structures


Unstructured grids

Topographical features

Geological interfaces

Local refinement (at observation points, sources, interfaces)


Dual tetrahedral-Voronoı grids

Grid generator: TetGen (Si, 2004)

tetrahedral grid Voronoı grid


Staggered finite-volume schemes

Magnetic field divergence free

Easy for implementingboundary conditions

Satisfies the continuity oftangential E

Physically meaningful



Direct EM-field method

Unknowns are E and/or H

Simpler

Smaller system of equations

Ill-conditioned

EM Potential (A� φ) method

Unknowns are A and φ

Larger system of equations

Well-conditioned

Allows studying the galvanic and inductive parts


Direct EM-field formulation of Maxwell’s equations

Maxwell’s equations:

∇� E � �iωµ0H� iωµ0Mp

∇�H � σ E� Jp

Helmholtz equation for electric field

∇�∇� E� iωµ0σE � �iωµ0Jp � iωµ0p∇�Mpq

Homogeneous Dirichlet boundary condition:

E � 0 at8

orE � τ � 0 on Γ


EM-field formulation: Discretization

Integral form of Maxwell’s equations:

¾

BSD

E � d lD � �iµ0ω

¼

SD

H � dSD � iµ0ω

¼

SD

Mp � dSD

¾

BSV

H � d lV � σ

¼

SV

E � dSV �

¼

SV

Jp � dSV



Discretized form of Maxwell’s equations:

WDj

q�1

Eipj,qq lDipj,qq � �iµ0ωHj S

Dj � iµ0ωMpj S

Dj

WVi

k�1

Hjpi,kqlVjpi,kq � σEi S

Vi � Jp i S

Vi .



Discretized form of Helmholtz equation:

WVi

k�1

��

WDj

q�1

Eipj,qq lDipj,qq

� lVjpi,kq

SDjpi,kq

� � iωµ0σEi S

Vi

� �iωµ0

WVi

k�1

Mpjpi,kq

lVjpi,kq

SDjpi,kq

� iωµ0Jp i


EM potential (A-φ) formulation of Maxwell’s equations

Magnetic vector and electric scalar potentials:

E � �iωA�∇φµ0H � ∇� A

Helmholtz equation in terms of potentials with Coulomb gauge

∇�∇� A�∇ p∇ � Aq � iωµ0σA� σµ0∇φ � µ0Jp � µ0∇�Mp

Conservation of charge

�∇ � J � ∇ � Jp

iω∇ � σA�∇ � σ∇φ � ∇ � Jp

Homogeneous Dirichlet boundary condition:

pA, φq � 0 at8

orpA � τ, φq � 0 on Γ


EM potential (A-φ) formulation of Maxwell’s equations

Relations used for the finite-volume discretization

∇�H� µ�10 ∇ψ � iωσA� σ∇φ � Jp �∇�Mp (1)

µ0H � ∇� A (2)

ψ � ∇ � A (3)

iω∇ � σA�∇ � σ∇φ � ∇ � Jp (4)


The system of equations for the A-φ method

Decompose A and φ into real and imaginary parts:

A � Are � iAim ; φ � φre � iφim

Resulting block matrix equation:��

A �ωB C 0ωB A 0 �C0 �ωD E 0ωD 0 0 E

��

��

Are

Aim

Φre

Φim

��

��

S1

0S2

0

��

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

Sparse coefficient matrix, A−φ method

(a) refinement at

observation points


Solution of the finite-volume schemes

Direct solution: MUMPS sparse direct solver (Amestoy et. al, 2006)

Iterative solution: BiCGSTAB and GMRES solvers from SPARSKIT(Saad, 1990)


Example 1: grounded wire

100 m wire along the x axis operating at 3 Hz

Dimensions of the prism: 120 � 200 � 400 m

σground � 0.02 S{m ; σprism � 0.2 S{m

Observation points along the x axis

−400

0

z (m)

0 500 1000

x (m)

100 m



Dimensions of the domain: 40 � 40 � 40 km

Number of tetrahedra: 162,689



(on Apple Mac Pro; 2.26 GHz Quad-Core Intel Xeon processor)



Scattered electric field

10−11

10−10

10−9

10−8

10−7

10−6

Sec

ondar

y E

x (

V/m

)

800 1200

x (m)

In phase

IE

FE ; potential method (Α−φ)

FV ; direct E−field method

FV ; potential method (Α−φ)10−11

10−10

10−9

10−8

10−7

10−6

800 1200

x (m)

Quadrature

IE

FE ; potential method (Α−φ)

FV ; direct E−field method

FV ; potential method (Α−φ)



Galvanic part (�∇φ)

Inductive part (�iωA)

Total electric field:E � �∇φ� iωA

−100

0

100

y (m)

900 1000 1100

In phase

−∇φ

−100

0

100

900 1000 1100

Quadrature

−∇φ

−100

0

100

y (m)

900 1000 1100

−iωA

−100

0

100

900 1000 1100

−iωA

−100

0

100

y (m)

900 1000 1100

x (m)

E

−100

0

100

900 1000 1100

x (m)

E10−9

10−8

10−7

10−6

E (V/m)



Galvanic part (�σ∇φ)

Inductive part (�iωσA)

Total current density:J � �σ∇φ� iωσA

−100

0

100

y (m)

900 1000 1100

In phase

−σ∇φ

−100

0

100

900 1000 1100

Quadrature

−σ∇φ

−100

0

100

y (m)

900 1000 1100

−iωσA

−100

0

100

900 1000 1100

−iωσA

−100

0

100

y (m)

900 1000 1100

x (m)

J

−100

0

100

900 1000 1100

x (m)

J

10−10

10−9

10−8

10−7

J (A/m2)



Cumulative error versus the changing cell size at the observation points

1e−08

1e−07

1e−06

1e−05

Cum

ula

tive

erro

r (V

/m)

10

Cell size (m)

in phase, no interpolation (E−field method)

quadrature, no interpolation (E−field method)

in phase, no interpolation (A− φ method)

quadrature, no interpolation (A− φ method)

full line: 1st order

dashed line: 0.5th order



10−16

10−12

10−8

10−4

100

Resid

ual

no

rm

0 500 1000 1500 2000

BCGSTAB

a without Coulomb gauge

1

3

10−16

10−12

10−8

10−4

100

Resid

ual

no

rm

0 500 1000 1500 2000

Iteration

c with Coulomb gauge

1

3

5

10−16

10−12

10−8

10−4

100

0 500 1000 1500 2000

GMRES

b without Coulomb gauge

0x50

0x200

0x500

3x50

3x200

3x500

10−16

10−12

10−8

10−4

100

0 500 1000 1500 2000

Iteration

d with Coulomb gauge

0x50

0x500

3x200

3x500

5x500


Example 2: Ovoid, HEM survey

Ovoid: massive sulfide ore body, Voisey’s Bay, Labrador, Canada

HEM survey of the region has been simulated

Transmitter and receiver towed below the helicopter 30 m above ground

Transmitter-receiver separation was 8 m and the frequencies were 900and 7200 Hz

σground � 0.001 and σovoid � 100 S/m were chosen by try-and-error

Number of tetrahedra: 240, 692



Topography of the region



White dots show the observation points



The plan view



Grid refined at the sources and observation points



FV results (circles) vs real HEM data (lines)

0

200

400

Hz(p

pm

)

6242800 6243000 6243200 6243400

Northing (m)

blue: 900 Hz

red: 7200 Hz

dashed line: HEM ; in phase

full line: HEM ; quadrature

FV ; in phase

FV ; quadrature



Amplitude of the horizontal component of total current density at avertical section along the observation profile

−100

−50

0

50

100

z (m)

In phase

900 Hz

Quadrature

900 Hz

−100

−50

0

50

100

z (m)

6243000 6243200

y (m)

7200 Hz

6243000 6243200

y (m)

7200 Hz

10−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

J (A/m2)



Galvanic part (�σ∇φ)

Inductive part (�iωσA)

Total current density:J � �σ∇φ� iωσA

6243000

6243200

6243400

y (m)

In phase

−σ∇φ

Quadrature

−σ∇φ

6243000

6243200

6243400

y (m)

−iωσA −iωσA

6243000

6243200

6243400

y (m)

555600 555800 556000

x (m)

J

555600 555800 556000

x (m)

J

10−16

10−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

J (A/m2)


Conclusions

A finite-volume approach was used for modelling total field EM data. Itused staggered tetrahedral-Voronoı grids.

The potential formulation of Maxwell’s equation was discretized andsolved and compared to the solution of the EM-field formulation.

Accuracy and versatility were tested using two examples: one with agrounded wire source and a small conductivity contrast; another one witha realistic body, magnetic sources and a large conductivity contrast.

Unlike the EM-field scheme, the A� φ scheme could be solved usinggeneric iterative solvers.

The gauged problems were harder to solve than the ungauged problems.

Solutions were decomposed into galvanic and inductive parts. The resultswere in good agreement with the type of sources that were used.

While both EM-field and potential schemes possessed the same trends ofaccuracy, the potential approach showed lower cumulative errors.


Acknowledgements

ACOA(Atlantic Canada Opportunities Agency)

NSERC(Natural Sciences and Engineering Research Council ofCanada)

Vale


References

Amestoy, P. R., Guermouche, A., L’Excellent, J. -Y. and S. Pralet, 2006. Hybrid scheduling for the parallelsolution of linear systems, Parallel Computing, 32, 136156.

Ansari, S. M., and C. G. Farquharson, 2013, Three-dimensional modeling of controlled-sourceelectromagnetic response for inductive and galvanic components: Presented at the 5th InternationalSymposium on Three-Dimensional Electromagnetics.

Farquharson, C. G., and D. W. Oldenburg, 2002, An integral-equation solution to the geophysicalelectromagnetic forward-modelling problem, in Three-Dimensional Electromagnetics: Proceedings of theSecond International Symposium: Elsevier, 319.

Farquharson, C. G., Duckworth, K. and D. W. Oldenburg, 2006. Comparison of integral equation andphysical scale modeling of the electromagnetic responses of models with large conductivity contrasts,Geophysics, 71, G169-G177.

Madsen, N. K., and R. W. Ziolkowski, 1990, A three-dimensional modified finite volume technique forMaxwell’s equations: Electromagnetics, 10, 147-161.

Si, H., 2004. TetGen, a quality tetrahedral mesh generator and three-dimensional delaunay triangulator,v1.3 (Technical Report No. 9). Weierstrass Institute for Applied Analysis and Stochastics.

Saad, Y., 1990. SPARSKIT: a basic tool kit for sparse matrix computations. Report RIACS 90-20,Research Institute for Advanced Computer Science. NASA Ames Research Center.


finite-volume modelling of geophysical electromagnetic

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