Progress Report (Baker-Atlas)
A Fully Automatic Goal-Oriented hp-AdaptiveFinite Element Strategy for
Simulations of Resistivity Logging Instruments.
David Pardo ([email protected]),C. Torres-Verdin, L. Demkowicz, L. Tabarovsky, A. Bespalov
Collaborators: Science Department of Baker-Atlas, J. Kurtz, M. Paszynski
February 22, 2005
Institute for Computational Engineering and Sciences (ICES)The University of Texas at Austin
OVERVIEW
1. Overview2. Mathematical Formulation of Electrodynamic Problems
• Maxwell’s Equations and Boundary Conditions
• Variational Formulation
• Axial Symmetry
3. Error Estimation: Towards Certified Solutions• Control of Modeling and Discretization Errors
• Automatic Built-in Error Estimation
• Benchmarking Problems
4. Numerical Results
5. Conclusions and Future Work
The University of Texas at Austin
David Pardo 22 Feb 2005
MAXWELL’S EQUATIONS (FREQUENCY DOMAIN)
Time Harmonic Maxwell’s Equations:
∇× E = −jωµH
∇× H = (σ + jω²)E + Jimp
Reduced Wave Equation:E-Formulation H-Formulation
∇×
(
1
µ∇ × E
)
−(ω2²−jωσ)E = −jωJ imp ; ∇×
(
1
σ + jω²∇ × H
)
+jωµH = ∇×1
σ + jω²J imp
Boundary Conditions (BC):
• Perfect Electric Conductor Surface:
n × E = 0 ; n · H = 0
• Idealized Antennas (Impressed Surface Electric Current):
n ×1
µ∇× E = −jωJimpS ; n × H = J
impS
The University of Texas at Austin2
High Performance Finite Element Software
David Pardo 22 Feb 2005
MAXWELL’S EQUATIONS (FREQUENCY DOMAIN)
Variational formulation
The reduced wave equation in Ω,
E-Formulation: ∇ ×(
1
µ∇ × E
)
− (ω2² − jωσ)E = −jωJ imp
H-Formulation: ∇ ×(
1
σ + jω²∇ × H
)
+ jωµH = ∇ ×1
σ + jω²J imp
Variational formulation:
E-Formulation:
Find E ∈ HD(curl; Ω) such that:∫
Ω
1
µ(∇ × E)(∇ × F̄) dV −
∫
Ω
(ω2² − jωσ)E · F̄ dV =
−jω
∫
Ω
Jimp · F̄ dV + jω
∫
ΓN
JimpS · F̄ dS ∀ F ∈ HD(curl; Ω)
H-Formulation:
Find H ∈ H̃S + HD(curl; Ω) with J̃impS = n × H|S and such that:
∫
Ω
1
σ + jω²(∇ × H)(∇ × F̄) dV + jω
∫
Ω
µH · F̄ dV =∫
Ω
∇ × (1
σ + jω²Jimp) · F̄ dV ∀ F ∈ HD(curl; Ω)
The University of Texas at Austin3
High Performance Finite Element Software
David Pardo 22 Feb 2005
MAXWELL’S EQUATIONS (FREQUENCY DOMAIN)
Variational formulation in cylindrical coordinatesUsing cylindrical coordinates (ρ, φ, z):
Eφ-Formulation:
Find Eφ ∈ H̃1D(Ω) such that:∫
Ω
1
µ(∂Eφ
∂z
∂F̄φ
∂z+
1
ρ2∂(ρEφ)
∂ρ
∂(ρF̄φ)
∂ρ) dV −
∫
Ω
k2Eφ · F̄φ dV =
−jω
∫
Ω
J impφ · F̄φ dV + jω
∫
ΓN
J impφ,S · F̄φ dS ∀ Fφ ∈ H̃1
D(Ω) .
Eρ,z-Formulation:
Find E = (Eρ, 0, Ez) ∈ H̃D(curl; Ω) such that:∫
Ω
1
µ(∂Eρ
∂z
∂F̄ρ
∂z+
∂Ez
∂ρ
∂F̄z
∂ρ) − k2
∫
Ω
EρF̄ρ + EzF̄z dV = −jω
∫
Ω
J impρ
F̄ρ + Jimpz
F̄z dV +
jω
∫
ΓN
J impρ,S F̄ρ + Jimpz,S F̄z dS ∀ F = (Fρ, 0, Fz) ∈ H̃D(curl; Ω) .
Hφ-Formulation:
Find Hφ ∈ f1(Jimpρ,S , J
impz,S ) + H̃
1
D(Ω) such that:
∫
Ω
1
σ + jω²(∂Hφ
∂z
∂F̄φ
∂z+
1
ρ2∂(ρHφ)
∂ρ
∂(ρF̄φ)
∂ρ) dV − jω
∫
Ω
µHφ · F̄φ dV =
∫
Ω
1
σ + jω²(∂J imp
ρ
∂z−
∂J impz
∂ρ)F̄φdV ∀ Fφ ∈ H̃
1
D(Ω) .
Hρ,z-Formulation:
Find H = Hρ, 0, Hz) ∈ f2(Jimpφ,S ) + H̃D(curl; Ω) such that:
∫
Ω
1
σ + jω²(∂Hρ
∂z
∂F̄ρ
∂z+
∂Hz
∂ρ
∂F̄z
∂ρ) − jω
∫
Ω
µ(HρF̄ρ + HzF̄z) dV =
∫
Ω
1
σ + jω²(∂Jφ
∂zF̄ρ +
1
ρ
∂(ρJ impφ )
∂ρF̄z) dV ∀ F = (Fρ, 0, Fz) ∈ H̃D(curl; Ω) .
The University of Texas at Austin4
High Performance Finite Element Software
David Pardo 22 Feb 2005
TOWARDS CERTIFIED SOLUTIONS
Sources of Error
Physical (Real) ProblemExact GeometryExact PhysicsExact InstrumentsNo Boundaries
No Errors in the Solution
−→
Mathematical ProblemApprox. GeometryMaxwell’s EquationsPerfect InstrumentsSome Boundaries
Errors Associated to:GeometryApproximationsCoupling PhysicsMaterial Coefficients
−→
Computational ProblemApprox. GeometryMaxwell’s EquationsPerfect InstrumentsBoundary Conditions
Errors Associated to:Mathematical ModelingGeometry and BC’sThe Code (Bugs)Discretization
The University of Texas at Austin5
High Performance Finite Element Software
David Pardo 22 Feb 2005
TOWARDS CERTIFIED SOLUTIONS
Sources of Error
Computational ProblemApprox. GeometryMaxwell’s EquationsPerfect InstrumentsBoundary Conditions
Solution is Not Exact Due toErrors in:
• Mathematical Modeling
• Geometry and BC’s
• The Code (Bugs)
• Discretization
The University of Texas at Austin5
High Performance Finite Element Software
David Pardo 22 Feb 2005
TOWARDS CERTIFIED SOLUTIONS
Discretization Error
The self-adaptive goal-oriented hp-adaptive strategy provides a veryaccurate built-in discretization error estimator.
Coarse grid Fine grid(hp) (h/2, p + 1)
-
global hp-refinement
Discretization Error Estimate for Coarse Grid =Fine Grid Solution - Coarse Grid Solution
The University of Texas at Austin6
High Performance Finite Element Software
David Pardo 22 Feb 2005
TOWARDS CERTIFIED SOLUTIONS
Sources of Error
Computational ProblemApprox. GeometryMaxwell’s EquationsPerfect InstrumentsBoundary Conditions
Solution is Not Exact Due toErrors in:
• Mathematical Modeling
• Geometry and BC’s
• The Code (Bugs)
• Discretization
The University of Texas at Austin6
High Performance Finite Element Software
David Pardo 22 Feb 2005
TOWARDS CERTIFIED SOLUTIONS
Avoiding Errors in the Code Using Benchmarking Examples
Solutions in a Homogeneous Lossy (1 Ω m) Media (2 Mhz)
Solenoid Antenna Toroid Antenna
10−80
10−60
10−40
10−20
100
0
5
10
15
20
25
30
35
40
45
50
55
Amplitude (V/m)
Dis
tanc
e in
z−a
xis
from
tran
smitt
er to
rece
iver
(in
m)
−180 −90 0 90 1800
5
10
15
20
25
30
35
40
45
50
55
Phase (degrees)
Dis
tanc
e in
z−a
xis
from
tran
smitt
er to
rece
iver
(in
m)
Analytical solutionNumerical solution
Analytical solutionNumerical solution
10−80
10−60
10−40
10−20
100
0
5
10
15
20
25
30
35
40
45
50
55
Amplitude (A/m)
Dis
tanc
e in
z−a
xis
from
tran
smitt
er to
rece
iver
(in
m)
−180 −90 0 90 1800
5
10
15
20
25
30
35
40
45
50
55
Phase (degrees)
Dis
tanc
e in
z−a
xis
from
tran
smitt
er to
rece
iver
(in
m)
Analytical solutionNumerical solution
Analytical solutionNumerical solution
Electric Field Magnetic Field
The University of Texas at Austin7
High Performance Finite Element Software
David Pardo 22 Feb 2005
TOWARDS CERTIFIED SOLUTIONS
Sources of Error
Computational ProblemApprox. GeometryMaxwell’s EquationsPerfect InstrumentsBoundary Conditions
Solution is Not Exact Due toErrors in:
• Mathematical Modeling
• Geometry and BC’s
• The Code (Bugs)
• Discretization
The University of Texas at Austin7
High Performance Finite Element Software
David Pardo 22 Feb 2005
TOWARDS CERTIFIED SOLUTIONS
Avoiding Errors in the Code Using Benchmarking Examples
Solutions in a Homogeneous Lossy (1 Ω m) Media (2 Mhz) in Presence ofa Conductive Mandrel
Solenoid Antenna Toroid Antenna
10−10
10−5
100
0.5
1
1.5
2
2.5
3
3.5
Amplitude (V/m)
Dis
tan
ce
in
z−
axis
fro
m tra
nsm
itte
r to
re
ce
ive
r (in
m)
Analytical solution
Mandrel Resisitivity: 10−7
Mandrel Resisitivity: 10−5
Mandrel Resisitivity: 10−3
Mandrel Resisitivity: 1
−180 −90 0 90 1800.5
1
1.5
2
2.5
3
3.5
Phase (degrees)
Dis
tan
ce
in
z−
axis
fro
m tra
nsm
itte
r to
re
ce
ive
r (in
m)
Analytical solution
Mandrel Resisitivity: 10−7
Mandrel Resisitivity: 10−5
Mandrel Resisitivity: 10−3
Mandrel Resisitivity: 1
10−5
100
105
0.5
1
1.5
2
2.5
3
3.5
Amplitude (A/m)
Dis
tan
ce
in
z−
axis
fro
m tra
nsm
itte
r to
re
ce
ive
r (in
m)
−180 −90 0 90 1800.5
1
1.5
2
2.5
3
3.5
Phase (degrees)
Dis
tan
ce
in
z−
axis
fro
m tra
nsm
itte
r to
re
ce
ive
r (in
m)
Analytical solution
Mandrel Resisitivity: 10−7
Mandrel Resisitivity: 10−5
Mandrel Resisitivity: 10−3
Mandrel Resisitivity: 1
Analytical solution
Mandrel Resisitivity: 10−7
Mandrel Resisitivity: 10−5
Mandrel Resisitivity: 10−3
Mandrel Resisitivity: 1
Electric Field Magnetic Field
The University of Texas at Austin8
High Performance Finite Element Software
David Pardo 22 Feb 2005
TOWARDS CERTIFIED SOLUTIONS
Summary
Computational ProblemApprox. GeometryMaxwell’s EquationsPerfect InstrumentsBoundary Conditions
Solution is Not Exact Due toErrors in:
• Mathematical Modeling
• Geometry and BC’s
• The Code (Bugs)
• Discretization
The University of Texas at Austin8
High Performance Finite Element Software
David Pardo 22 Feb 2005
THROUGH CASING RESISTIVITY INSTRUMENTS
10 cm.
50 c
m.
100
cm.
10000 Ohm m
1 Ohm m
100 Ohm m
100 Ohm m0.1
Ohm
m
z
25 c
m.
25 c
m.
150
cm.
CASI
NG 0
.000
001
Ohm
m
Axisymmetric 3D problem.
Five different materials.
Size of computational domain:SEVERAL MILES.
Material properties varying byup to TEN orders of magnitude(10000000000!!!).
Objective: DetermineSecond Difference of PotentialReceiving Electrodes.
The University of Texas at Austin9
High Performance Finite Element Software
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THROUGH CASING RESISTIVITY INSTRUMENTS
Logging Through Casing (Benchmark Problem)Rock Formation: Homogeneous Media
10000100010010110
−9
10−8
10−7
10−6
Resistivity (Ohm m)
Sec
ond
Diff
eren
ce o
f Pot
entia
l (no
rmal
ized
) in
Vol
ts
Kaufmann Approx. FormulaExact 2nd DerivativeExact 2nd Diff. of Pot.Numerical 2nd Diff. of Pot.
The second vertical difference of the Electric Potential is proportional to the formationconductivity.
The University of Texas at Austin10
High Performance Finite Element Software
David Pardo 22 Feb 2005
THROUGH CASING RESISTIVITY INSTRUMENTS
Final Log Obtained by Our Finite Element Software
0.001 0.01 0.1 1 10−1
−0.5
0
0.5
1
1.5
2
Second Difference of Potential (normalized) in micro Volts
Po
sitio
n o
f R
ece
ivin
g E
lectr
od
es (
z−
axis
) in
m
Electrodes at the borehole centerElectrodes at the borehole wallExact solution for one layer formation
0.0001 0.001 0.01 0.1 1−1
−0.5
0
0.5
1
1.5
2
Second Difference of Potential (normalized) in micro Volts
Po
sitio
n o
f R
ece
ivin
g E
lectr
od
es (
z−
axis
) in
m
Electrodes at the borehole centerElectrodes at the borehole wallExact solution for one layer formation
Resistivity of casing = 10−6 Resistivity of casing = 10−7
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THROUGH CASING RESISTIVITY INSTRUMENTS
Approximation Error
0 0.05 0.1 0.15 0.2 0.25−1.5
−1
−0.5
0
0.5
1
1.5
2
Relative Error of Second Difference of Potential (in %)
Po
sitio
n o
f R
ece
ivin
g E
lectr
od
es (
z−
axis
)
5500 6000 6500 7000 7500 8000 8500−1.5
−1
−0.5
0
0.5
1
1.5
2
Number of Unknowns
Po
sitio
n o
f R
ece
ivin
g E
lectr
od
es (
z−
axis
)
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THROUGH CASING RESISTIVITY INSTRUMENTS
Damaged Casing
HALF THICKNESSCASING AREA
0.00001 Ohm m
0.000001 Ohm mCASING RESISTIVITY
LESS RESISTIVE AREA
0.1
m0.1
m
Imperfect Casing
0.2m
0.2m
1.3m
1.3 m
FO
RM
AT
ION
FO
RM
AT
ION
BOREHOLE
CASING
ELECTRODES
The University of Texas at Austin13
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THROUGH CASING RESISTIVITY INSTRUMENTS
Damaged Casing
0.0001 0.01 1 100 10000−1
−0.5
0
0.5
1
1.5
2
Second Difference of Potential (normalized) in micro Volts
Pos
ition
of R
ecei
ving
Ele
ctro
des
(z−a
xis)
in m
Perfect casingDamaged casingDamaged casing using a calibrated toolExact solution for one layer formation
In the presenceof damagedcasing, the useof calibratedinstruments isessential.
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THROUGH CASING RESISTIVITY INSTRUMENTS
10 cm.
50 c
m.
100
cm.
10000 Ohm m
1 Ohm m
100 Ohm m
100 Ohm m0.1
Ohm
m
z
CASI
NG 0
.000
01 O
hm m
165
cm.
20 c
m.
Axisymmetric 3D problem.
Toroid Antennas.
Size of computational domain:SEVERAL MILES.
Different frequencies.
Material properties varying byup to NINE orders ofmagnitude (1000000000!!!).
Objective: DetermineFirst Difference of Electric andMagnetic Fields.
The University of Texas at Austin15
High Performance Finite Element Software
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THROUGH CASING RESISTIVITY INSTRUMENTS
First Difference of Electric Field at Different Frequencies
10−15
10−12
10−9
10−6
−1
−0.5
0
0.5
1
1.5
2
Amplitude of Electric Field (V/m)
First Difference of E Field Vertical Derivative
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
−180 −90 0 90 180−1
−0.5
0
0.5
1
1.5
2
Phase (degrees)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
Antennas located at the Borehole CENTER
0.1 Hz10 Hz1 Khz10 Khz100 Khz1 Mhz
0.1 Hz10 Hz1 Khz10 Khz100 Khz1 Mhz
10−15
10−12
10−9
10−6
−1
−0.5
0
0.5
1
1.5
2
Amplitude of Electric Field (V/m)
First Difference of E Field Vertical Derivative
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
−180 −90 0 90 180−1
−0.5
0
0.5
1
1.5
2
Phase (degrees)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
Antennas located at the Borehole WALL
0.1 Hz10 Hz1 Khz10 Khz100 Khz1 Mhz
0.1 Hz10 Hz1 Khz10 Khz100 Khz1 Mhz
Toroid antennas are more sensitive to the rock formation resistivity whenlocated on the borehole’s wall
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High Performance Finite Element Software
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THROUGH CASING RESISTIVITY INSTRUMENTS
First Difference of Magnetic Field at Different Frequencies
10−15
10−12
10−9
10−6
−1
−0.5
0
0.5
1
1.5
2
Amplitude of Magnetic Field (A/m)
First Difference of Magnetic Field (Normalized)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
−180 −90 0 90 180−1
−0.5
0
0.5
1
1.5
2
Phase (degrees)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
Antennas located at the Borehole CENTER
0.1 Hz10 Hz1 Khz10 Khz100 Khz1 Mhz
0.1 Hz10 Hz1 Khz10 Khz100 Khz1 Mhz
10−15
10−12
10−9
10−6
−1
−0.5
0
0.5
1
1.5
2
Amplitude of Magnetic Field (A/m)
First Difference of Magnetic Field (Normalized)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
−180 −90 0 90 180−1
−0.5
0
0.5
1
1.5
2
Phase (degrees)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
Antennas located at the Borehole WALL
0.1 Hz10 Hz1 Khz10 Khz100 Khz1 Mhz
0.1 Hz10 Hz1 Khz10 Khz100 Khz1 Mhz
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THROUGH CASING RESISTIVITY INSTRUMENTS
Electromagnetic Fields at Different Frequencies
0.001 0.01 0.1 1 10 100 100010
−12
10−11
10−10
10−9
10−8
10−7
10−6
Am
plitu
de
of E
lectr
om
ag
ne
tic F
ield
s (
A/m
an
d V
/m) First Difference of EM Fields
Frequency (in Khz)0.001 0.01 0.1 1 10 100 1000
−150
−100
−50
0
50
100
150
Ph
ase
(d
eg
ree
s)
Frequency (in Khz)
Magnetic FieldElectric Field (z−component)
Magnetic FieldElectric Field (z−component)
ElectromagneticFields are al-most constantfor frequenciesbelow 1 kHz. Asudden drop inthe amplitudeoccurs at fre-quencies above20 kHz.
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THROUGH CASING RESISTIVITY INSTRUMENTS
� � �� � �
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� �� �
� �� �
� �� �
� � �� � �
� � �� � �
� � �� � �
� � �� � �
� � �� � �
� � �� � �
�
� ����� � � �
� � �� � �
5.5" Borehole radio ; 0.5" Casing ; 2" Cement
175m
=58
0’50
m=
165’
475m
=16
00’
0.00
001
Oh
m m
1 O
hm
m
2 O
hm
m
20 Ohm m
5 Ohm m
5 Ohm m
Axisymmetric 3D problem.
Five different materials.
Different positions of receivingantenna.
Toroidal Transmitterand receiver separated by upto half a mile.
Objective: DetermineFirst Difference of PotentialReceiving Electrodes.
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High Performance Finite Element Software
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THROUGH CASING RESISTIVITY INSTRUMENTS
Final Log Obtained by Our Finite Element Software
Magnetic Field Electric Field
10−12
10−11
10−10
10−9
−100
−80
−60
−40
−20
0
20
40
60
80
100Distance from Casing: 0 m
Amplitude of Magnetic Field (A/m)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
−180 −90 0 90 180−100
−80
−60
−40
−20
0
20
40
60
80
100
Phase (degrees)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
0.1 Hz NO INV0.1 Hz 60m INV100 Hz NO INV100 Hz 60m INV
0.1 Hz NO INV0.1 Hz 60m INV100 Hz NO INV100 Hz 60m INV
10−11
10−10
10−9
10−8
−100
−80
−60
−40
−20
0
20
40
60
80
100Distance from Casing: 0 m
Amplitude of Electric Field (V/m)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
−180 −90 0 90 180−100
−80
−60
−40
−20
0
20
40
60
80
100
Phase (degrees)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
0.1 Hz NO INV0.1 Hz 60m INV100 Hz NO INV100 Hz 60m INV
0.1 Hz NO INV0.1 Hz 60m INV100 Hz NO INV100 Hz 60m INV
If the transmitter antenna is located on the surface of a cased well, thereceived EM signal within the borehole is too weak.
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THROUGH CASING RESISTIVITY INSTRUMENTS
Final Log Obtained by Our Finite Element Software
Magnetic Field Electric Field
10−9
10−8
10−7
−100
−80
−60
−40
−20
0
20
40
60
80
100Frequency: 1 Hz
Amplitude of Magnetic Field (A/m)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
−180 −90 0 90 180−100
−80
−60
−40
−20
0
20
40
60
80
100
Phase (degrees)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
50 m250 m500 m
50 m250 m500 m
10−12
10−10
10−8
−100
−80
−60
−40
−20
0
20
40
60
80
100Frequency: 1 Hz
Amplitude of Electric Field (V/m)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
−180 −90 0 90 180−100
−80
−60
−40
−20
0
20
40
60
80
100
Phase (degrees)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
50 m250 m500 m
50 m250 m500 m
If the transmitter antenna is located on the surface of a cased well, thereceived cross-well EM signal is too weak.
The University of Texas at Austin21
High Performance Finite Element Software
David Pardo 22 Feb 2005
THROUGH CASING RESISTIVITY INSTRUMENTS
Final Log Obtained by Our Finite Element Software
Magnetic Field Electric Field
10−11
10−10
10−9
10−8
−100
−80
−60
−40
−20
0
20
40
60
80
100Distance from Casing: 500 m
Amplitude of Magnetic Field (A/m)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
−180 −90 0 90 180−100
−80
−60
−40
−20
0
20
40
60
80
100
Phase (degrees)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
0.1 Hz NO INV0.1 Hz 60m INV100 Hz NO INV100 Hz 60m INV
0.1 Hz NO INV0.1 Hz 60m INV100 Hz NO INV100 Hz 60m INV
10−12
10−11
10−10
−100
−80
−60
−40
−20
0
20
40
60
80
100Distance from Casing: 500 m
Amplitude of Electric Field (V/m)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
−180 −90 0 90 180−100
−80
−60
−40
−20
0
20
40
60
80
100
Phase (degrees)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
0.1 Hz NO INV0.1 Hz 60m INV100 Hz NO INV100 Hz 60m INV
0.1 Hz NO INV0.1 Hz 60m INV100 Hz NO INV100 Hz 60m INV
If the transmitter antenna is located on the surface of a cased well, a 60meters layer of water cannot be detected by using cross-well EM signals.
The University of Texas at Austin22
High Performance Finite Element Software
David Pardo 22 Feb 2005
THROUGH CASING RESISTIVITY INSTRUMENTS
Final Log Obtained by Our Finite Element Software
Magnetic Field Electric Field
10−8
10−7
10−6
−100
−80
−60
−40
−20
0
20
40
60
80
100Distance from Casing: 500 m
Amplitude of Magnetic Field (A/m)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
−180 −90 0 90 180−100
−80
−60
−40
−20
0
20
40
60
80
100
Phase (degrees)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
0.1 Hz NO INV0.1 Hz 60m INV100 Hz NO INV100 Hz 60m INV
0.1 Hz NO INV0.1 Hz 60m INV100 Hz NO INV100 Hz 60m INV
10−9
10−8
10−7
−100
−80
−60
−40
−20
0
20
40
60
80
100Distance from Casing: 500 m
Amplitude of Electric Field (V/m)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
−180 −90 0 90 180−100
−80
−60
−40
−20
0
20
40
60
80
100
Phase (degrees)
Po
sitio
n o
f re
ce
ive
r a
nte
nn
a in
th
e z
−a
xis
(in
m)
0.1 Hz NO INV0.1 Hz 60m INV100 Hz NO INV100 Hz 60m INV
0.1 Hz NO INV0.1 Hz 60m INV100 Hz NO INV100 Hz 60m INV
If the transmitter antenna is located downhole a cased well, it is feasibleto perform meaningful cross-well EM measurements.
The University of Texas at Austin23
High Performance Finite Element Software
David Pardo 22 Feb 2005
LOGGING INSTRUMENTS WITH A MANDREL
100 c
m50 c
m
100 Ohm m
10000 Ohm m
1 Ohm m
100 Ohm m
15 c
m100 c
m
0.0
00001 O
hm
m0.1
Oh
m m
Description of Antennas
10 c
m
10000 Ohm m
Borehole0.1 Ohm m
10000 Rel. Perme.
0.0
00001 O
hm
mM
an
dre
l
Radius=10.8 cm
Radius 7.6 cm
Magnetic Buffer
Goal: To Compute FirstDifference of Potentialon Receiving Antennas
The University of Texas at Austin24
High Performance Finite Element Software
David Pardo 22 Feb 2005
LOGGING INSTRUMENTS WITH A MANDREL
Eφ (normalized) for a solenoid antenna
0.1 0.18 0.3−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Amplitude of Electric Field (V/m)
Pos
ition
of r
ecei
ver a
nten
na in
the
z−ax
is (i
n m
)
−180 −90 0 90 180−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Phase (degrees)
Pos
ition
of r
ecei
ver a
nten
na in
the
z−ax
is (i
n m
)
100 Ohm m
1 Ohm m
10000 Ohm m
100 Ohm m
Frequency: 2 Mhz
The University of Texas at Austin25
High Performance Finite Element Software
David Pardo 22 Feb 2005
LOGGING INSTRUMENTS WITH A MANDREL
First Difference of Eφ (normalized) for a solenoid antenna
0.05 0.075 0.1 0.125−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Amplitude of Electric Field (V/m)
Pos
ition
of r
ecei
ver a
nten
na in
the
z−ax
is (i
n m
)
−180 −90 0 90 180−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Phase (degrees)
Pos
ition
of r
ecei
ver a
nten
na in
the
z−ax
is (i
n m
)
100 Ohm m
1 Ohm m
10000 Ohm m
100 Ohm m
Frequency: 2 Mhz
The University of Texas at Austin26
High Performance Finite Element Software
David Pardo 22 Feb 2005
LOGGING INSTRUMENTS WITH A MANDREL
First Difference of Eφ (normalized) for a solenoid antenna
0.0003 0.0008−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Amplitude of Electric Field (V/m)
Pos
ition
of r
ecei
ver
ante
nna
in th
e z−
axis
(in
m)
−180 −90 0 90 180−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Phase (degrees)
Pos
ition
of r
ecei
ver
ante
nna
in th
e z−
axis
(in
m)
100 Ohm m
1 Ohm m
10000 Ohm m
100 Ohm m
Frequency: 2 Mhz, NO MAGNETIC BUFFER
The University of Texas at Austin27
High Performance Finite Element Software
David Pardo 22 Feb 2005
LOGGING INSTRUMENTS WITH A MANDREL
First Difference of Hφ (normalized) for a toroid antenna
10−6
10−4
10−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Amplitude of Magnetic Field (A/m)
Pos
ition
of r
ecei
ver
ante
nna
in th
e z−
axis
(in
m)
−180 −90 0 90 180−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Phase (degrees)
Pos
ition
of r
ecei
ver
ante
nna
in th
e z−
axis
(in
m)
100 Ohm m
1 Ohm m
10000 Ohm m
100 Ohm m
Frequency: 2 Mhz
The University of Texas at Austin28
High Performance Finite Element Software
David Pardo 22 Feb 2005
LOGGING INSTRUMENTS WITH A MANDREL
First Difference of Ez (normalized) for a toroid antenna
10−4
10−2
100
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Amplitude of Electric Field (V/m)
Pos
ition
of r
ecei
ver a
nten
na in
the
z−ax
is (i
n m
)
−180 −90 0 90 180−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Phase (degrees)
Pos
ition
of r
ecei
ver a
nten
na in
the
z−ax
is (i
n m
)
100 Ohm m
1 Ohm m
10000 Ohm m
100 Ohm m
Frequency: 2 Mhz
The University of Texas at Austin29
High Performance Finite Element Software
David Pardo 22 Feb 2005
CONCLUSIONS
• It is possible to simulate a variety of EM logging instrumentsby using the self-adaptive goal-oriented hp-FEM.
• For TCRT, preliminary simulations suggest to
1. Place antennas on the borehole’s wall,2. use calibrated instruments,3. use low frequencies (typically below 5 kHz),4. use downhole antennas for cross-well EM measurements, and5. avoid the use of cross-well EM measurements for assesment of
water injection.
• For LWD instruments, preliminary simulations suggest to
1. Use both solenoid and toroid antennas,2. use magnetic buffers to amplify EM signals, and3. compute first differences of EM fields.
Institute for Computational Engineering and Sciences
The University of Texas at Austin30
High Performance Finite Element Software
FUTURE WORK
Within the next 3 months
• Implementation of the goal-oriented self-adaptive algorithmfor 2D edge elements. It would allow us to solve 2.5 Dproblems.
• Parallel implementation of the 2D code (Maciek Paszynski).
Long term goals
• Solve 3D problems with casing at DC.
• Solve 3D problems with mandrel at AC.
• Invert coupled sonic and EM measurements in 2D.
The University of Texas at Austin
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