fourier finite element (ffe) framework with hp an electromagnetics … · 2011. 2. 10. · vi eiec,...
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VI EIEC, Chiclana, Spain
An hp Fourier Finite Element (FFE) Framework withElectromagnetics and Multiphysics Applications
D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo,M. Paszynski, M.J. Nam
October 23, 2008
Basque Center for Applied Mathematics (BCAM)Promoting Technological Advances Through Mathematics
OVERVIEW
1. Motivation: Waveguide Design and Oil-IndustryApplications.
2. Method:
• Fourier finite element (FFE) method.
• Parallel implementation.
• Multi-physics framework.
3. Numerical Simulations:
• 2D resistivity logging measurements.
• Marine controlled source electromagnetic (CSEM) measurements.
• 3D resistivity logging measurements.
4. Conclusions and Future Work.
Basque Center for Applied Mathematics (BCAM)
D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
MOTIVATION (WAVEGUIDES)
Waveguide Design
Goal: Determine electric field intensity at the ports.
Basque Center for Applied Mathematics (BCAM)2
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
MOTIVATION (OIL-INDUSTRY)
Seismic Measurements
Figure from the USGS Science Center for Coastal and Marine Geol ogy
Basque Center for Applied Mathematics (BCAM)3
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
MOTIVATION (OIL-INDUSTRY)
Marine Controlled-Source Electromagnetics (CSEM)
Figure from the UCSD Institute of Oceanography
Basque Center for Applied Mathematics (BCAM)4
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
MOTIVATION (OIL-INDUSTRY)
Multiphysics Logging Measurements
OBJECTIVES: To determine payzones ( porosity ), amount ofoil/gas ( saturation ), and ability to extract oil/gas
(permeability ).
Basque Center for Applied Mathematics (BCAM)5
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
MOTIVATION (OIL-INDUSTRY)
Main Objective: To Solve a Multiphysics Inverse Problem
Given multi-frequency electromagnetic, acoustic, and nuclearmeasurements, the objective is to determine porosity, saturation, and
permeability distributions in the reservoir.
Basque Center for Applied Mathematics (BCAM)6
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
FOURIER FINITE ELEMENT METHOD
Deviated Wells (Forward Problem)Dip Angle
Invasion
Anisotropy
Triaxial Induction
Eccentricity
Laterolog
Through-Casing
Induction-LWD
Induction-Wireline
Inverse Problems
Multi-Physics
Objective: Find solution at the receiver antennas.
Basque Center for Applied Mathematics (BCAM)7
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
FOURIER FINITE ELEMENT METHOD
Non-Orthogonal System of Coordinates
ZZ~
ζ1
6
ζ3
ζ2
Material coefficients are constantwith respect to the quasi-azimuthaldirection ζ2
Fourier Series Expansion in ζ2
DC Problems: −∇σ∇u = f
u(ζ1, ζ2, ζ3) =l=∞∑
l=−∞
ul(ζ1, ζ3)ejlζ2
σ(ζ1, ζ2, ζ3) =m=∞∑
m=−∞
σm(ζ1, ζ3)ejmζ2
f(ζ1, ζ2, ζ3) =n=∞∑
n=−∞
fn(ζ1, ζ3)ejnζ2
Fourier modes ejlζ2 are orthogonalhigh-order basis functions that are(almost) invariant with respect tothe gradient operator.
Basque Center for Applied Mathematics (BCAM)8
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
FOURIER FINITE ELEMENT METHOD
Cartesian system of coordinates: x = (x1, x2, x3).New non-orthogonal system of coordinates: ζ = (ζ1, ζ2, ζ3).
SUB
DO
MA
IN I
I
SUB
DO
MA
IN I
ISU
BD
OM
AIN
I
0
SUBDOMAIN III
SUBDOMAIN III
PARAMETERIZATIONZ
ZZ~ ζ1
6
ζ3
ζ2
ZZZ~ ζ1
6
ζ3
Subdomain I ; Subdomain II ; Subdomain III
x1 = ζ1 cos ζ2
x2 = ζ1 sin ζ2
x3 = ζ3
;
x1 = ζ1 cos ζ2
x2 = ζ1 sin ζ2
x3 = ζ3 + tan θ0
ζ1 − ρ1
ρ2 − ρ1
ρ2
;
x1 = ζ1 cos ζ2
x2 = ζ1 sin ζ2
x3 = ζ3 + tan θ0ζ1
Basque Center for Applied Mathematics (BCAM)9
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
FOURIER FINITE ELEMENT METHOD
Final Variational Formulation
We define the Jacobian matrix J =∂(x1, x2, x3)
∂(ζ1, ζ2, ζ3)and its
determinant |J | = det(J ).
Variational formulation in the new system of coordinates:
Find u ∈ uD + H1D(Ω) such that:
⟨∂v
∂ζ, σ
∂u
∂ζ
⟩
L2(Ω)
=⟨
v , f⟩
L2(Ω)∀v ∈ H1
D(Ω) ,
where:σ := J −1σJ −1T
|J | ; f := f |J | .
Same variational formulation with new materials and load data
Basque Center for Applied Mathematics (BCAM)10
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
FOURIER FINITE ELEMENT METHOD
Five Fourier modes are enough to represent EXACTLY the newmaterial coefficients.
Direct Current:
Find u ∈ uD + H1D(Ω) such that:
n=k+2∑
n=k−2
⟨(∂v
∂ζ
)
k
, σk−n
(∂u
∂ζ
)
n
⟩
L2(Ω2D)
=⟨
vk , fk
⟩
L2(Ω2D)∀vk
Alternate Current:
Find (E)s ∈ HΓE(curl; Ω) such that:
n=s+2∑
n=s−2
⟨(∇ζ×F)s , (µ−1)s−n (∇ζ×E)l
⟩
L2(Ω2D)
−⟨
Fs , (k2)s−nEl
⟩
L2(Ω2D)= −jω
⟨
Fs , (Jimp)s
⟩
L2(Ω2D)∀ Fs
Basque Center for Applied Mathematics (BCAM)11
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
FOURIER FINITE ELEMENT METHOD
Example (7 Fourier Modes)
n=k+2∑
n=k−2
⟨(∂v
∂ζ
)
k
, σk−n
(∂u
∂ζ
)
n
⟩
L2(Ω2D)︸ ︷︷ ︸
(k, k − n, n)
=⟨
vk , fk
⟩
L2(Ω2D)
Stiffness Matrix:
(−3,0,−3) (−3,−1,−2) (−3,−2,−1) 0 0 0 0
(−2,1,−3) (−2,0,−2) (−2,−1,−1) (−2,−2,0) 0 0 0
(−1,2,−3) (−1,1,−2) (−1,0,−1) (−1,−1,0) (−1,−2,1) 0 0
0 (0,2,−2) (0,1,−1) (0,0,0) (0,−1,1) (0,−2,2) 0
0 0 (1,2,−1) (1,1,0) (1,0,1) (1,−1,2) (1,−2,3)
0 0 0 (2,2,0) (2,1,1) (2,0,2) (2,−1,3)
0 0 0 0 (3,2,1) (3,1,2) (3,0,3)
Basque Center for Applied Mathematics (BCAM)12
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
FOURIER FINITE ELEMENT METHOD
A Self-Adaptive Goal-Oriented hp-FEMOptimal 2D Grid
(Through Casing Resistivity Problem)
p=1
p=2
p=3
p=4
p=5
p=6
p=7
p=8
ddd
We vary locally the element sizeh and the polynomial order ofapproximation p throughout thegrid.
Optimal grids are automaticallygenerated by the computer.
The self-adaptive goal-orientedhp-FEM provides exponentialconvergence rates in terms ofthe CPU time vs. the error ina user prescribed quantity ofinterest.
Basque Center for Applied Mathematics (BCAM)13
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
FOURIER FINITE ELEMENT METHOD
Axisymmetric Logging-While-Drilling (LWD) SimulationGOAL-ORIENTED HP-ADAPTIVITY (Quadrilateral Elements)
p=1
p=2
p=3
p=4
p=5
p=6
p=7
p=8
-0.7222222 2.870370 -1.137037
1.085185 2Dhp90: A Fully automatic hp-adaptive Finite Element code
Basque Center for Applied Mathematics (BCAM)14
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
FOURIER FINITE ELEMENT METHOD
Axisymmetric Logging-While-Drilling (LWD) SimulationGOAL-ORIENTED HP-ADAPTIVITY (ZOOM TOWARDS FIRST RECEIVER
ANTENNA)
p=1
p=2
p=3
p=4
p=5
p=6
p=7
p=8
1.7160494E-02 0.1369136 0.3603704
0.4344445 2Dhp90: A Fully automatic hp-adaptive Finite Element code
Basque Center for Applied Mathematics (BCAM)15
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
PARALLEL IMPLEMENTATION
We Use Shared Domain Decomposition
Basque Center for Applied Mathematics (BCAM)16
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
MULTIPHYSICS FRAMEWORK
We are generating new data structures based on:
The addition of one new module/library for solving inverseproblems.
The use of different number of equations for each element/node.
• Enables to consider different physics for each element/node.
• Enables the use of different number of Fourier modes for eachelement/node.
The combination of different types of elements used for eachphysics:
• Continous H1-elements (DC problems, elasticity, etc.)
• Nedelec (edge) H(curl)-elements (Electromagnetics).
• Raviart-Thomas (face) H(div)-elements (Fluid-dynamics)
• Discontinuous L2-elements.
Basque Center for Applied Mathematics (BCAM)17
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
MULTIPHYSICS FRAMEWORK
Final hp-grid and solution
Basque Center for Applied Mathematics (BCAM)18
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Model Problem I: UNIFORM FORMATION
1 Ohm−m
z=0 mz=50 mTX(HED)
Basque Center for Applied Mathematics (BCAM)19
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Model Problem I: UNIFORM FORMATION — 0.25 Hz —
0 2000 4000 6000 8000 1000010
−17
10−16
10−15
10−14
10−13
10−12
10−11
10−10
10−9
Am
plitu
de o
f Ele
ctric
Fie
ld (
V/(
A m
2 )
Horizontal Distance between TX and RX (m)
Uniform Material, 0.25 Hz
1 MODE5 MODES9 MODESEXACT
0 2000 4000 6000 8000 10000−600
−500
−400
−300
−200
−100
0
Pha
se o
f Ele
ctric
Fie
ld (
degr
ees)
Horizontal Distance between TX and RX (m)
Uniform Material, 0.25 Hz
1 MODE5 MODES9 MODESEXACT
Basque Center for Applied Mathematics (BCAM)20
For more info, visit: www.ices.utexas.edu/ ∼pardo
D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Model Problem I: UNIFORM FORMATION — 0.75 Hz —
0 2000 4000 6000 8000 1000010
−20
10−18
10−16
10−14
10−12
10−10
Am
plitu
de o
f Ele
ctric
Fie
ld (
V/(
A m
2 )
Horizontal Distance between TX and RX (m)
Uniform Material, 0.75 Hz
1 MODE5 MODES9 MODESEXACT
0 2000 4000 6000 8000 10000−1000
−900
−800
−700
−600
−500
−400
−300
−200
−100
0
Pha
se o
f Ele
ctric
Fie
ld (
degr
ees)
Horizontal Distance between TX and RX (m)
Uniform Material, 0.75 Hz
1 MODE5 MODES9 MODESEXACT
Basque Center for Applied Mathematics (BCAM)21
For more info, visit: www.ices.utexas.edu/ ∼pardo
D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Model Problem I: UNIFORM FORMATION — 1.25 Hz —
0 2000 4000 6000 8000 1000010
−22
10−20
10−18
10−16
10−14
10−12
10−10
Am
plitu
de o
f Ele
ctric
Fie
ld (
V/(
A m
2 )
Horizontal Distance between TX and RX (m)
Uniform Material, 1.25 Hz
1 MODE5 MODES9 MODESEXACT
0 2000 4000 6000 8000 10000−1400
−1200
−1000
−800
−600
−400
−200
0
Pha
se o
f Ele
ctric
Fie
ld (
degr
ees)
Horizontal Distance between TX and RX (m)
Uniform Material, 1.25 Hz
1 MODE5 MODES9 MODESEXACT
Basque Center for Applied Mathematics (BCAM)22
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Model Problem II: CSEM SCENARIO WITHOUT OIL
1 Ohm−m
0.3 Ohm−m
z=0 m
z=1000 m
z=50 m
10 Ohm−m
TX(HED)
6
Basque Center for Applied Mathematics (BCAM)23
For more info, visit: www.ices.utexas.edu/ ∼pardo
D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Model Problem II: WITHOUT OIL — 0.25 Hz —
0 2000 4000 6000 8000 1000010
−17
10−16
10−15
10−14
10−13
10−12
10−11
10−10
Am
plitu
de o
f Ele
ctric
Fie
ld (
V/(
A m
2 )
Horizontal Distance between TX and RX (m)
Without Oil, 0.25 Hz
1 MODE5 MODES9 MODESEXACT
0 2000 4000 6000 8000 10000−600
−500
−400
−300
−200
−100
0
Pha
se o
f Ele
ctric
Fie
ld (
degr
ees)
Horizontal Distance between TX and RX (m)
Without Oil, 0.25 Hz
1 MODE5 MODES9 MODESEXACT
Basque Center for Applied Mathematics (BCAM)24
For more info, visit: www.ices.utexas.edu/ ∼pardo
D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Model Problem II: WITHOUT OIL — 0.75 Hz —
0 2000 4000 6000 8000 1000010
−20
10−18
10−16
10−14
10−12
10−10
Am
plitu
de o
f Ele
ctric
Fie
ld (
V/(
A m
2 )
Horizontal Distance between TX and RX (m)
Without Oil, 0.75 Hz
1 MODE5 MODES9 MODESEXACT
0 2000 4000 6000 8000 10000−800
−700
−600
−500
−400
−300
−200
−100
0
Pha
se o
f Ele
ctric
Fie
ld (
degr
ees)
Horizontal Distance between TX and RX (m)
Without Oil, 0.75 Hz
1 MODE5 MODES9 MODESEXACT
Basque Center for Applied Mathematics (BCAM)25
For more info, visit: www.ices.utexas.edu/ ∼pardo
D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Model Problem II: WITHOUT OIL — 1.25 Hz —
0 2000 4000 6000 8000 1000010
−20
10−18
10−16
10−14
10−12
10−10
Am
plitu
de o
f Ele
ctric
Fie
ld (
V/(
A m
2 )
Horizontal Distance between TX and RX (m)
Without Oil, 1.25 Hz
1 MODE5 MODES9 MODESEXACT
0 2000 4000 6000 8000 10000−900
−800
−700
−600
−500
−400
−300
−200
−100
0
Pha
se o
f Ele
ctric
Fie
ld (
degr
ees)
Horizontal Distance between TX and RX (m)
Without Oil, 1.25 Hz
1 MODE5 MODES9 MODESEXACT
Basque Center for Applied Mathematics (BCAM)26
For more info, visit: www.ices.utexas.edu/ ∼pardo
D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Model Problem III: CSEM SCENARIO WITH OIL
1 Ohm−m
1 Ohm−m
0.3 Ohm−m
100 Ohm−m
z=0 m
z=1000 m
z=−1000 m
z=−1100 m
z=50 m
10 Ohm−m6
TX(HED)
Basque Center for Applied Mathematics (BCAM)27
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Model Problem III: WITH OIL — 0.25 Hz —
0 2000 4000 6000 8000 1000010
−15
10−14
10−13
10−12
10−11
10−10
Am
plitu
de o
f Ele
ctric
Fie
ld (
V/(
A m
2 )
Horizontal Distance between TX and RX (m)
With Oil, 0.25 Hz
1 MODE5 MODES9 MODESEXACT
0 2000 4000 6000 8000 10000−180
−160
−140
−120
−100
−80
−60
−40
Pha
se o
f Ele
ctric
Fie
ld (
degr
ees)
Horizontal Distance between TX and RX (m)
With Oil, 0.25 Hz
1 MODE5 MODES9 MODESEXACT
Basque Center for Applied Mathematics (BCAM)28
For more info, visit: www.ices.utexas.edu/ ∼pardo
D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Model Problem III: WITH OIL — 0.75 Hz —
0 2000 4000 6000 8000 1000010
−16
10−15
10−14
10−13
10−12
10−11
10−10
Am
plitu
de o
f Ele
ctric
Fie
ld (
V/(
A m
2 )
Horizontal Distance between TX and RX (m)
With Oil, 0.75 Hz
1 MODE5 MODES9 MODESEXACT
0 2000 4000 6000 8000 10000−350
−300
−250
−200
−150
−100
−50
Pha
se o
f Ele
ctric
Fie
ld (
degr
ees)
Horizontal Distance between TX and RX (m)
With Oil, 0.75 Hz
1 MODE5 MODES9 MODESEXACT
Basque Center for Applied Mathematics (BCAM)29
For more info, visit: www.ices.utexas.edu/ ∼pardo
D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Model Problem III: WITH OIL — 1.25 Hz —
0 2000 4000 6000 8000 1000010
−17
10−16
10−15
10−14
10−13
10−12
10−11
10−10
Am
plitu
de o
f Ele
ctric
Fie
ld (
V/(
A m
2 )
Horizontal Distance between TX and RX (m)
With Oil, 1.25 Hz
1 MODE5 MODES9 MODESEXACT
0 2000 4000 6000 8000 10000−450
−400
−350
−300
−250
−200
−150
−100
−50
Pha
se o
f Ele
ctric
Fie
ld (
degr
ees)
Horizontal Distance between TX and RX (m)
With Oil, 1.25 Hz
1 MODE5 MODES9 MODESEXACT
Basque Center for Applied Mathematics (BCAM)30
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
NO OIL
1 Ohm−m
0.3 Ohm−m
z=0 m
z=1000 m
z=50 m
10 Ohm−m
TX(HED)
6
Basque Center for Applied Mathematics (BCAM)31
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
INFINITE LAYER OF OIL
1 Ohm−m
1 Ohm−m
0.3 Ohm−m
100 Ohm−m
z=0 m
z=1000 m
z=−1000 m
z=−1100 m
z=50 m
10 Ohm−m6
TX(HED)
Basque Center for Applied Mathematics (BCAM)32
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
FINITE LAYER OF OIL
1 Ohm−m
0.3 Ohm−m
z=0 m
z=1000 m
z=−1000 m
z=−1100 m
z=50 m
100 Ohm−m
x=2000 m x=5000 m
1 Ohm−m 1 Ohm−m
10 Ohm−m6
TX(HED)
1 Ohm−m
Basque Center for Applied Mathematics (BCAM)33
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
FINITE LAYER OF OIL + ANISOTROPY
0.3 Ohm−m
z=0 m
z=1000 m
z=−1000 m
z=−1100 m
z=50 m
100 Ohm−m
x=2000 m x=5000 m
10 Ohm−m6
TX(HED)
1 Ohm−m
3 Ohm−m
1 Ohm−m
3 Ohm−m
3 Ohm−m 3 Ohm−m
1 Ohm−m 1 Ohm−m
Basque Center for Applied Mathematics (BCAM)34
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Comparison — 0.25 Hz —
0 2000 4000 6000 8000 1000010
−15
10−14
10−13
10−12
10−11
10−10
Am
plitu
de o
f Ele
ctric
Fie
ld (
V/(
A m
2 )
Horizontal Distance between TX and RX (m)
0.25 Hz
NO OILINFINITE LAYER OILFINITE LAYER OILANISOTROPY
0 2000 4000 6000 8000 10000−250
−200
−150
−100
−50
0
Pha
se o
f Ele
ctric
Fie
ld (
degr
ees)
Horizontal Distance between TX and RX (m)
0.25 Hz
NO OILINFINITE LAYER OILFINITE LAYER OILANISOTROPY
The finite layer of oil is clearly identified, and it is different fr om thesolution for the infinite layer of oil. To consider anisotropy is e ssential.
Basque Center for Applied Mathematics (BCAM)35
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Comparison — 0.75 Hz —
0 2000 4000 6000 8000 1000010
−16
10−15
10−14
10−13
10−12
10−11
10−10
Am
plitu
de o
f Ele
ctric
Fie
ld (
V/(
A m
2 )
Horizontal Distance between TX and RX (m)
0.75 Hz
NO OILINFINITE LAYER OILFINITE LAYER OILANISOTROPY
0 2000 4000 6000 8000 10000−450
−400
−350
−300
−250
−200
−150
−100
−50
0
Pha
se o
f Ele
ctric
Fie
ld (
degr
ees)
Horizontal Distance between TX and RX (m)
0.75 Hz
NO OILINFINITE LAYER OILFINITE LAYER OILANISOTROPY
As we increase the frequency, the effect of oil becomes morelocalized.
Basque Center for Applied Mathematics (BCAM)36
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM
Comparison — 1.25 Hz —
0 2000 4000 6000 8000 1000010
−18
10−17
10−16
10−15
10−14
10−13
10−12
10−11
10−10
Am
plitu
de o
f Ele
ctric
Fie
ld (
V/(
A m
2 )
Horizontal Distance between TX and RX (m)
1.25 Hz
NO OILINFINITE LAYER OILFINITE LAYER OILANISOTROPY
0 2000 4000 6000 8000 10000−600
−500
−400
−300
−200
−100
0
Pha
se o
f Ele
ctric
Fie
ld (
degr
ees)
Horizontal Distance between TX and RX (m)
1.25 Hz
NO OILINFINITE LAYER OILFINITE LAYER OILANISOTROPY
As we increase the frequency, the effect of oil becomes morelocalized.
Basque Center for Applied Mathematics (BCAM)37
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM0.75 Hz (FINITE LAYER OF OIL)
TX: x = 0 m ; RX: x = 2000 m.
Basque Center for Applied Mathematics (BCAM)38
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM0.75 Hz (FINITE LAYER OF OIL)
TX: x = 0 m ; RX: x = 5000 m.
Basque Center for Applied Mathematics (BCAM)39
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 2D MARINE CSEM0.75 Hz (FINITE LAYER OF OIL)
TX: x = 0 m ; RX: x = 8000 m.
Basque Center for Applied Mathematics (BCAM)40
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Model Problem
0.05 m
mΩ .
0.2
5 m
1.0
m
150 kHz (Wireline)
Fiber Glass Mandrel
Mandrel Radius: 0.04 m
105
Finite SizeLoop Antenna
mΩ .
mΩ .
mΩ .
mΩ .
mΩ .
mΩ .
mΩ .
mΩ .20
mΩ .30mΩ .50
Oil−Based M
ud (1000 m)
1.5 m
Ω
.
0.5 m0.65 m
3 m
SHALE
SAND
SANDSAND
SAND
OIL
SHALE
3
200.05
0.1
100
1
1
4 m
Anisotropy (10:1)
Anisotropy (10:1)
Anisotropy (10:1)
Anisotropy (8:1)
Basque Center for Applied Mathematics (BCAM)41
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Verification
Logging Instrument in a Homogeneous Formation
1 3 5 7 9
10−2
100
102
104
Number of Fourier Modes
Re
lativ
e E
rro
r (in
%)
20 KHz (30 degrees)20 KHz (60 degrees)150 KHz (30 degrees)150 KHz (60 degrees)
Basque Center for Applied Mathematics (BCAM)42
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Verification
Logging Instrument in a Homogeneous Formation
10−2
100
102
−6
−4
−2
0
2
4
6
Resistivity (Ω−m)−2 −1 0 1
x 10−3
Wireline, 150 Khz
Real Part (V/m)
−0.02 −0.01 0Imag Part (V/m)
1 1
Basque Center for Applied Mathematics (BCAM)43
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Verification
Logging Instrument in a Homogeneous Formation
10−2
100
102
−6
−4
−2
0
2
4
6
Resistivity (Ω−m)−2 −1 0 1
x 10−3
Wireline, 150 Khz
Real Part (V/m)
−3 −2 −1 0
x 10−3Imag Part (V/m)
13
13
Basque Center for Applied Mathematics (BCAM)43
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Verification
Logging Instrument in a Homogeneous Formation
10−2
100
102
−6
−4
−2
0
2
4
6
Resistivity (Ω−m)−2 −1 0 1
x 10−3
Wireline, 150 Khz
Real Part (V/m)
−3 −2 −1 0
x 10−3Imag Part (V/m)
135
135
Basque Center for Applied Mathematics (BCAM)43
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Verification
Logging Instrument in a Homogeneous Formation
10−2
100
102
−6
−4
−2
0
2
4
6
Resistivity (Ω−m)−2 −1 0 1
x 10−3
Wireline, 150 Khz
Real Part (V/m)
−3 −2 −1 0
x 10−3Imag Part (V/m)
1359
1359
Basque Center for Applied Mathematics (BCAM)43
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Verification
Logging Instrument in a Homogeneous Formation
10−2
100
102
−6
−4
−2
0
2
4
6
Resistivity (Ω−m)−2 −1 0 1
x 10−3
Wireline, 150 Khz
Real Part (V/m)
−3 −2 −1 0
x 10−3Imag Part (V/m)
135913
135913
Basque Center for Applied Mathematics (BCAM)43
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Dip Angle
10−2
100
102
−6
−4
−2
0
2
4
6
8
10
Resistivity (Ω−m)−2 −1 0 1
x 10−3
Wireline, 150 Khz
Real Part (V/m)
−3 −2 −1 0
x 10−3Imag Part (V/m)
0 degrees30 degrees60 degrees
0 degrees30 degrees60 degrees
Basque Center for Applied Mathematics (BCAM)44
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Dip Angle + Invasion
10−2
100
102
−6
−4
−2
0
2
4
6
8
10
Resistivity (Ω−m)−2 −1 0 1
x 10−3
Wireline, 150 Khz
Real Part (V/m)
−3 −2 −1 0
x 10−3Imag Part (V/m)
0 degrees30 degrees60 degrees
0 degrees30 degrees60 degrees
INVASION
NO INVASION
Basque Center for Applied Mathematics (BCAM)45
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Dip Angle + Anisotropy
10−2
100
102
−6
−4
−2
0
2
4
6
8
10
Resistivity (Ω−m)−2 −1 0 1
x 10−3
Wireline, 150 Khz
Real Part (V/m)
−3 −2 −1 0
x 10−3Imag Part (V/m)
0 degrees30 degrees60 degrees
0 degrees30 degrees60 degrees
VERT. ρ
HORIZ. ρ
Basque Center for Applied Mathematics (BCAM)46
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Dip Angle + Invasion + Anisotropy
10−2
100
102
−6
−4
−2
0
2
4
6
8
10
Resistivity (Ω−m)−2 −1 0 1
x 10−3
Wireline, 150 Khz
Real Part (V/m)
−3 −2 −1 0
x 10−3Imag Part (V/m)
0 degrees30 degrees60 degrees
0 degrees30 degrees60 degrees
INVASION
NO INVASION
VERT. ρ
HORIZ. ρ
Basque Center for Applied Mathematics (BCAM)47
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
60-Degree Deviated Well
10−2
100
102
−6
−4
−2
0
2
4
6
8
10
Resistivity (Ω−m)−2 −1 0 1
x 10−3
Wireline, 150 Khz
Real Part (V/m)
−3 −2 −1 0
x 10−3Imag Part (V/m)
NO INV.NO INV + ANI.INVINV + ANI
NO INV.NO INV + ANI.INVINV + ANI
INVASION
NO INVASION
VERT. ρ
HORIZ. ρ
Basque Center for Applied Mathematics (BCAM)48
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Vertical Well with 0.03 m Eccentricity
10−2
100
102
−6
−4
−2
0
2
4
6
8
10
Resistivity (Ω−m)−2 −1 0 1
x 10−3
Wireline, 150 Khz
Real Part (V/m)
−3 −2 −1 0
x 10−3Imag Part (V/m)
NO INV.NO INV + ECC.INVINV + ECC
NO INV.NO INV + ECC.INVINV + ECC
INVASION
NO INVASION
Basque Center for Applied Mathematics (BCAM)49
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Model Problem and Verification
Ω . mµr
µr
Ω . m
0.25
m
0.15 m
0.05 m0.1 m
0.7
m
Magnetic Buffer:
=10000=10000ρ
Metallic Mandrel:ρ =10
−5
=50
Loop AntennaFinite Size
1 3 5 7 910
−2
10−1
100
101
102
LWD PROBLEM (30 degrees)
Number of Fourier Modes
Re
lative
Err
or
(in
%)
Basque Center for Applied Mathematics (BCAM)50
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Dip Angle
10−2
100
102
−6
−4
−2
0
2
4
6
8
10
Resistivity (Ω−m)−0.4 −0.2 0 0.2
LWD, 2 Mhz
Real Part (V/m)
−0.4 −0.2 0 0.2Imag Part (V/m)
0 degrees30 degrees60 degrees
0 degrees30 degrees60 degrees
Basque Center for Applied Mathematics (BCAM)51
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Dip Angle + Invasion
10−2
100
102
−6
−4
−2
0
2
4
6
8
10
Resistivity (Ω−m)−0.4 −0.2 0 0.2
LWD, 2 Mhz
Real Part (V/m)
−0.4 −0.2 0 0.2Imag Part (V/m)
0 degrees30 degrees60 degrees
0 degrees30 degrees60 degrees
INVASION
NO INVASION
Basque Center for Applied Mathematics (BCAM)52
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Dip Angle + Anisotropy
10−2
100
102
−6
−4
−2
0
2
4
6
8
10
Resistivity (Ω−m)−0.4 −0.2 0 0.2
LWD, 2 Mhz
Real Part (V/m)
−0.4 −0.2 0 0.2Imag Part (V/m)
0 degrees30 degrees60 degrees
0 degrees30 degrees60 degrees
VERT. ρ
HORIZ. ρ
Basque Center for Applied Mathematics (BCAM)53
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
Dip Angle + Invasion + Anisotropy
10−2
100
102
−6
−4
−2
0
2
4
6
8
10
Resistivity (Ω−m)−0.4 −0.2 0 0.2
LWD, 2 Mhz
Real Part (V/m)
−0.4 −0.2 0 0.2Imag Part (V/m)
0 degrees30 degrees60 degrees
0 degrees30 degrees60 degrees
INVASION
NO INVASION
VERT. ρ
HORIZ. ρ
Basque Center for Applied Mathematics (BCAM)54
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
RESULTS: 3D RESISTIVITY LOGGING
60-Degree Deviated Well
10−2
100
102
−6
−4
−2
0
2
4
6
8
10
Resistivity (Ω−m)−0.4 −0.2 0 0.2
LWD, 2 Mhz
Real Part (V/m)
−0.4 −0.2 0 0.2Imag Part (V/m)
NO INV.NO INV + ANI.INVINV + ANI
NO INV.NO INV + ANI.INVINV + ANI
INVASION
NO INVASION
VERT. ρ
HORIZ. ρ
Basque Center for Applied Mathematics (BCAM)55
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D. Pardo, C. Torres-Verdın, L.E. Garcıa-Castillo, M. Pasz ynski, M. J. Nam 23 Oct 2008
CONCLUSIONS AND FUTURE WORK
• A Fourier-Finite-Element method provides a suitable formulation forsimulation of resistivity geophysical applications.
• Goal-oriented refinements are essential in marine CSEM geophysi calapplications due to the dissipative nature of the earth.
• A parallel implementation based on a shared domain-decompositi on issimple and provides additional performance for a moderate num ber ofprocessors.
• We are developing a multiphysics framework for the joint-inversi on ofmultiphysics measurements.
• We are looking for Ph.D. students and postdoctoral fellows to furthe rdevelop this software and work on the joint-inversion of multiphysi csmeasurements.
Basque Center for Applied Mathematics (BCAM)
Basque Center for Applied Mathematics (BCAM)56
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