-finite element hp self-adaptive goal-oriented simulations of … · 2011. 2. 10. · c....

Joint Industry Research Consortium on Formation Evaluation

Self-Adaptive Goal-Oriented hp-Finite ElementSimulations of Induction and Laterolog Measurements

in the Presence of Steel Casing”

David Pardo ([email protected]),C. Torres-Verdin, L. Demkowicz, L. Tabarovsky

Collaborators: Science Department of Baker-Hughes,

J. Kurtz, M. Paszynski, D. Xue (Cynthia)

August 17, 2005

Department of Petroleum and Geosystems Engineering, andInstitute for Computational Engineering and Sciences (ICES)

The University of Texas at Austin

OVERVIEW

1. Motivation

2. Numerical Methodology• hp-Finite Elements (Exponential convergence)

• Automatic Goal-Oriented Refinements (in the quantity of interest)

3. Current Stage of the 2D High Performance FE Software• Flexibility

• Reliability

• Accuracy

• Performance

4. Simulations in Presence of Metal Casing of:• Induction Instruments

• Laterolog Measurements

5. Conclusions and Future Work

The University of Texas at Austin

D. Pardo, C. Torres-Verdin, L. Demkowicz 17 Aug 2005

MOTIVATION

10

0 c

m5

0 c

m

100 Ohm m

10000 Ohm m

1 Ohm m

100 Ohm m

15

cm

10

0 c

m

0.0

00

00

1 O

hm

m0

.1 O

hm

mDescription of Antennas

10

cm

10000 Ohm m

Borehole0.1 Ohm m

10000 Rel. Perme.

0.0

00

00

1 O

hm

mM

an

dre

l

Radius=10.8 cm

Radius 7.6 cm

Magnetic Buffer

Goal: To Study the Effectof Invasion, Anistotropy, andMagnetic Permeability.

The University of Texas at Austin2

High Performance Finite Element Software


MOTIVATION

First Vert. Diff. Hφ for different antennas

10−5

100

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Amplitude First Vert. Diff. Magnetic Field (A/m2)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)No Invasion

Vert. Dipoles (Ring)Toroid

−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

Vert. Dipoles (Ring)Toroid

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

In LWD instruments, we obtain similar results using toroids or a ring of vert. dipoles




MOTIVATION

First Vert. Diff. Ez for a toroid antenna

10−4

10−2

100

102

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Amplitude First Vert. Diff. Ez (V/m2)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

No Invasion

Toroid

−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

Toroid

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

Toroids are adequate for identifying highly resistive layers




MOTIVATION

First Vert. Diff. Eφ for a solenoid antenna

0.04 0.08 0.12 0.16 0.2−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

First Vert. Diff. Amplitude of Electric Field (V/m2)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)No Invasion

Solenoid

−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

Solenoid

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

Solenoids are adequate for identifying low resistive layers




MOTIVATION

Use of Magnetic Buffers ( Eφ for a solenoid)

10−4

10−2

100

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

First Vert. Diff. Amplitude of Electric Field (V/m2)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)No Invasion

With Magnetic BuffersWithout Magnetic Buffers

−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

Solenoid


100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

Use of magnetic buffers strengthen the signal in combination with solenoids




MOTIVATION

Use of Magnetic Buffers ( Hφ for a toroid)

10−5

100

105

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

First Vert. Diff. Amplitude of Magnetic Field (A/m2)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)No Invasion


−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

Toroid


100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

However, magnetic buffers weaken the signal in combination with toroids




MOTIVATION

Invasion study ( Eφ for a solenoid)

10−1

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Amplitude First Vert. Diff. Electric Field (V/m2)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)Invasion Study

No Invasion0.15m Invasion0.4m Invasion0.9m Invasion

−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

Solenoid


100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm − m 100 Ohm − m

Large invasion effects can be sensed using solenoids




MOTIVATION

Invasion study ( Hφ for a toroid)

10−5

100

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3


Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)Invasion Study


−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

Toroid


100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm − m 100 Ohm − m

Small invasion effects can be sensed using toroids




MOTIVATION

Invasion study ( Eφ for a solenoid)

0.04 0.08 0.12 0.16 0.2−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3


Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)Invasion Study


−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

Solenoid


100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm − m 100 Ohm − m

Invasion in resistive layers cannot be sensed using solenoids




MOTIVATION

Invasion study ( Hφ for a toroid)

10−5

100

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3


Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)Invasion Study


−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

Toroid


100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm − m 100 Ohm − m

Invasion in resistive layers should be studied using toroids




MOTIVATION

Invasion and mandrel magnetic permeab. ( Eφ)

0.04 0.08 0.12 0.16 0.2−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3


Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)Invasion Study

No Invasion0.4/0.9 m Invasion0.4/0.9m Invasion, Perm. Mandrel=100

−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

Solenoid

No Invasion0.4/0.9 m Invasion0.4/0.9m Invasion, Perm. Mandrel=100

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

2 Ohm−m (0.4m) 2 Ohm−m (0.4m)

5 Ohm−m (0.9m) 5 Ohm−m (0.9m)

The effect of magnetic permeability on the mandrel is similar to the effect of magnetic buffers




MOTIVATION

Anisotropy ( Hφ)

10−5

100

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3


Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)Anisotropy Study

No Anisotropy1x5 Ω ⋅ m1x5 Ω ⋅ m ; 1000x10000 Ω ⋅ m

−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

Toroid

No Anisotropy1x5 Ω ⋅ m1x5 Ω ⋅ m ; 1000x10000 Ω ⋅ m

100 Ohm−m

1000x10000 Ohm−m

1x5 Ohm−m

100 Ohm−m

100 Ohm−m

1000x10000 Ohm−m

1x5 Ohm−m

100 Ohm−m

Anisotropy effects may be important when studying resistive layers




MOTIVATION

0.8 Ohm−m0.8 Ohm−m0.8 Ohm−m0.8 Ohm−m0.8 Ohm−m0.8 Ohm−m0.8 Ohm−m0.8 Ohm−m0.8 Ohm−m0.8 Ohm−m

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5.5"6"

8"

30

00

fe

et

30

00

fe

et

1500 feet

Ca

sin

g:

0.0

00

01

Oh

m−

m4

fe

et

1 f

oo

t

Ce

me

nt:

2 O

hm

−m

Bo

reh

ole

:1

Oh

m−

m

5 Ohm−m

5 Ohm−m

10 Ohm−m10 Ohm−m10 Ohm−m10 Ohm−m10 Ohm−m10 Ohm−m10 Ohm−m10 Ohm−m10 Ohm−m10 Ohm−m

1 Ohm−m

20

fe

et

1.8 feet

0.2 feet

Axisymmetric 3D problem.

Seven different materials.

Through casing resistivityinstrument.

Large variations on resistivity.

Objective: Study the effectof invasion THROUGH CASINGon laminated sands.




MOTIVATION

10−7

10−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

Amplitude of first difference of Ez (V/m2)

Po

sitio

n o

f th

e r

ece

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

NO WATER INVASION

1 Hz100 Hz10000 Hz

Variations due to frequency are small (below 5%)




MOTIVATION

10−7

10−6

−5

−4

−3

−2

−1

0

1

2

3

4

5


Po

sitio

n o

f th

e r

ece

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

1 Hz

NO WATER INVASION6" WATER INVASION12" WATER INVASION24" WATER INVASION36" WATER INVASION

Variations due to water ivasion are large




MOTIVATION

10−7

10−6

−5

−4

−3

−2

−1

0

1

2

3

4

5


Po

sitio

n o

f th

e r

ece

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

100 Hz






MOTIVATION

10−7

10−6

−5

−4

−3

−2

−1

0

1

2

3

4

5


Po

sitio

n o

f th

e r

ece

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

10000 Hz






MOTIVATION

10−7.25

10−7.24

10−7.23

10−7.22

10−7.21

10−7.2

10−7.19

−5

−4

−3

−2

−1

0

1

2

3

4

5


Po

sitio

n o

f th

e r

ece

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

NO WATER INVASION

10−6 Ohm−m, 1 Hz

10−6 Ohm−m, 10000 Hz

Casing resistivity can be analyzed from different frequency measureme nts




RESISTIVITY LOGGING PROBLEMS

Type of Problems We Can Solve with 2Dhp90 (v.7.2)

Physical Magnetic Insulators DisplacementDevices Buffers Currents

Casing Casing CombinationImperfections of all

Materials Isotropic Anisotropic*Sources Toroidal Solenoidal Dipoles in

Antennas Antennas Any DirectionElectrodes Finite Combination

Size Antennas of AllLogging LWD/MWD Laterolog NormalInstruments

Induction Dielectric Cross-wellInstruments

Frequency 0-10 Ghz.Invasion Water Oil etc.

ALL AXISYMMETRIC RESISTIVITY LOGGING PROBLEMS




THE hp-FINITE ELEMENT METHOD

The h-Finite Element Method1. Convergence limited by the polynomial degree, and large material

contrasts.

2. Optimal h-grids do NOT converge exponentially in real applications.

3. They may “lock” (100 % error).

The p-Finite Element Method1. Exponential convergence feasible for analytical (“nice”) solutions.

2. Optimal p-grids do NOT converge exponentially in real applications.

3. If initial h-grid is not adequate, the p-method will fail miserably.

The hp-Finite Element Method1. Exponential convergence feasible for ALL solutions.

2. Optimal hp-grids DO converge exponentially in real applications.

3. If initial hp-grid is not adequate, results will still be great.




GOAL-ORIENTED ADAPTIVITY

Mathematical Formulation (Goal-Oriented Adaptivity)

DIRECT PROBLEM DUAL PROBLEM

L(Ψ)=b(Ψ,G)




GOAL-ORIENTED ADAPTIVITY

Movie Presentation (Sensitivity Functions)

We want to study resolution and depth of investigation of a logginginstrument.

We have: |L(Ψ)| = |∫

S dV | ≤

∫

|S| dV .

In the next movies, we display: log10 |S|.

Scales:

• Red → |S| = |L(Ψ)| ∗ 104.

• Blue → |S| = |L(Ψ)| ∗ 10−2.

Direct Current




SELF-ADAPTIVE GOAL-ORIENTED hp-FEM

Algorithm for Goal-Oriented Adaptivity

Solve DIRECT andDUAL problems onGrid hp.

−→Solve DIRECT andDUAL problems onGrid h/2, p + 1.

Compute e = Ψh/2,p+1 − Ψhp, and ẽ = Ψh/2,p+1 − ΠhpΨh/2,p+1.Compute ǫ = Gh/2,p+1 − Ghp, and ǫ̃ = Gh/2,p+1 − ΠhpGh/2,p+1.

|L(e)| = |b(e, ǫ)| ∼ |b(ẽ, ǫ̃)| ≤∑

K |bK(ẽ, ǫ̃)| ≤∑

K ‖ ẽ ‖E,K‖ ǫ̃ ‖E,K.

Apply the fully automatic hp-adaptive algorithm.

Solve DIRECT andDUAL problems onGrid hp.

−→Solve DIRECT andDUAL problems onGrid h/2, p + 1.





First. Vert. Diff. Eφ (solenoid). Position: 0.475m

0 1000 8000 27000 6400010

−5

10−4

10−3

10−2

10−1

100

101

102

103

Number of Unknowns N (scale N1/3)

Re

lativ

e E

rro

r in

%

Goal−Oriented hp−Adaptivity

Upper bound for |L(e)|/|L(u)||L(e)|/|L(u)|

0 1000 8000 27000 6400010

−5

10−4

10−3

10−2

10−1

100

101

102

103

Number of Unknowns N (scale N1/3)

Re

lativ

e E

rro

r in

%

Energy−norm hp−Adaptivity

Energy−norm error|L(e)|/|L(u)|





Goal-Oriented vs. Energy-norm hp-Adaptivity

Problem with Mandrel at 2 Mhz.

Continuous Elements (Goal-Oriented Adaptivity)

Quantity of Interest Real Part Imag PartCOARSE GRID -0.1629862203E-01 -0.4016944732E-02

FINE GRID -0.1629862347E-01 -0.4016944223E-02

Continuous Elements (Energy-norm Adaptivity)

Quantity of Interest Real Part Imag Part0.01% ENERGY ERROR -0.1382759158E-01 -0.2989492851E-02

It is critical to use GOAL-ORIENTED adaptivity.





First. Vert. Diff. Eφ (solenoid). Position: 0.475mGOAL-ORIENTED HP-ADAPTIVITY

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





First. Vert. Diff. Eφ (solenoid). Position: 0.475mGOAL-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




CURRENT STAGE OF THE 2D hp-FE SOFTWARE

Model Problem with Steel Casing

100 Ohm m

100 Ohm m

0.1

Ohm

m

10000 Ohm m

0.0

00

00

1 O

hm

m

1 Ohm m

z

0.15 m0.20 m

0.65 m

0.5 m

0.5 m

0.1m

Frequency: 10 Hz - 10 kHz.

Casing resistivity: 10−6 Ohm · m.

Casing width: 0.01127 m

Discretization error < 0.1 %

Toroidal antennas

Size (domain): 500m x 4000m





Flexibility (What Problems Can We Solve?)

Time-Harmonic Maxwell’s Equations

∇ × H = (¯̄σ + jω¯̄ǫ)E + Jimp Ampere’s law

∇ × E = −jω ¯̄µH − Mimp Faraday’s law

∇ · (¯̄ǫE) = ρ Gauss’ law of Electricity

∇ · (¯̄µH) = 0 Gauss’ law of Magnetism

E-VARIATIONAL FORMULATION:

Find E ∈ ED + HD(curl; Ω) such that:∫

Ω

(¯̄µ−1∇ × E) · (∇ × F̄) dV −

∫

Ω

(¯̄k2E) · F̄ dV = −jω

∫

Ω

Jimp · F̄ dV

+jω

∫

ΓN

JimpΓN · F̄t dS −

∫

Ω

(¯̄µ−1Mimp) · (∇ × F̄) dV ∀ F ∈ HD(curl; Ω)





Flexibility (What Problems Can We Solve?)AXISYMMETRIC PROBLEMS

Eφ -Variational Formulation (Azimuthal)

Find Eφ ∈ Eφ,D + H̃1D(Ω) such that:∫

Ω

(¯̄µ−1ρ,z∇ × Eφ) · (∇ × F̄φ) dV −

∫

Ω

(¯̄k2φEφ) · F̄φ dV = −jω

∫

Ω

J impφ F̄φ dV

+jω

∫

ΓN

J impφ,ΓN F̄φ dS −

∫

Ω

(¯̄µ−1ρ,zMimpρ,z ) · F̄φ dV ∀ Fφ ∈ H̃

1D(Ω)

Eρ,z -Variational Formulation (Meridian)

Find (Eρ, Ez) ∈ ED + H̃D(curl; Ω) such that:∫

Ω

(¯̄µ−1φ ∇ × Eρ,z) · (∇ × F̄ρ,z) dV −

∫

Ω

(¯̄k2ρ,zEρ,z) · F̄ρ,z dV =

−jω

∫

Ω

J impρ F̄ρ + Jimpz F̄z dV + jω

∫

ΓN

J impρ,ΓN F̄ρ + Jimpz,ΓN

F̄z dS

−

∫

Ω

(¯̄µ−1φ Mimpφ ) · F̄ρ,z dV ∀ (Fρ, Fz) ∈ H̃D(curl; Ω)





Flexibility (What Problems Can We Solve?)

• Physical Devices: Casing, Casing Imperfections, Mandrel,Magnetic Buffers, Insulators, Displacement Currents,Combination of All, etc.

• Materials: Isotropic, Anisotropic*.

• Sources: Toroidal Antennas, Solenoidal Antennas, Dipolesin Any Direction, Electrodes, Finite Size Antennas,Combination of All, etc.

• Logging Instruments: Logging While Drilling (LWD),Laterolog, Normal, Induction, Dielectric Instruments,Cross-well, etc.

• Any Frequency (0-10 Ghz).

ALL AXISYMMETRIC RESISTIVITY LOGGING PROBLEMS





Reliability (Can We Trust the Solutions?)

• Comparison Against Analytical Results.

1. Exact solution in a homogeneous media.

2. Exact solution in a homogeneous media with a mandrel.

3. Exact solution in a homogeneous media with casing.

• Verification of Physical Properties.

1. Reciprocity principle (Gregory Itskovich).

2. Discrete divergence free approximation for edge elements.

• Numerical Verifications.

1. Different size of domain and antennas.

2. Comparison against other numerical software (Yang Wei).

3. Error control provided by the fine grid solution.

4. Comparison between continuous elements vs. edge elements.

HIGHLY RELIABLE SOFTWARE






Problem with casing at 10 kHz.

Continuous ElementsQuantity of Interest Real Part Imag Part

COARSE GRID 0.1516098429E-08 -0.1456374493E-08FINE GRID 0.1516094029E-08 -0.1456390824E-08

Edge Elements

Quantity of Interest Real Part Imag PartCOARSE GRID 0.1516060872E-08 -0.1456337248E-08

FINE GRID 0.1516093804E-08 -0.1456390864E-08

Error control provided by the fine grid solution.






Problem with casing at 10 kHz.

Continuous ElementsQuantity of Interest Real Part Imag Part

COARSE GRID 0.1516098429E-08 -0.1456374493E-08FINE GRID 0.1516094029E-08 -0.1456390824E-08

Edge Elements

Quantity of Interest Real Part Imag PartCOARSE GRID 0.1516060872E-08 -0.1456337248E-08

FINE GRID 0.1516093804E-08 -0.1456390864E-08

Comparison between continuous elements vs. edge elements.






• Comparison Against Analytical Results.

1. Exact solution in a homogeneous media.

2. Exact solution in a homogeneous media with a mandrel.

3. Exact solution in a homogeneous media with casing.

• Verification of Physical Properties.

1. Reciprocity principle (Gregory Itskovich).

2. Discrete divergence free approximation for edge elements.

• Numerical Verifications.

1. Different size of domain and antennas.

2. Comparison against other numerical software (Yang Wei).

3. Error control provided by the fine grid.

4. Comparison between continuous elements vs. edge elements.

HIGHLY RELIABLE SOFTWARE





Performance (How Fast Can We Solve the Problems?)

80 Vert. Pos. 10−6Ω · m 10−5Ω · m

Toroid (10 Khz) 19’ 46” 16’ 28”

Ring of Vert. Dipoles (10 Khz) 22’ 47” 17’ 02”

Ring of Horiz. Dipoles (10 Khz) 19’ 25” 13’ 25”

Electrodes (0 Hz) 10’ 10” 8’ 35”

IBM Power 4 compiler 1.3 Ghz (4 years old)

Possible improvements in performance:

• To use a 3.4 Ghz processor.

• To execute the code in 8 processors (10 positions per processor).

• To improve implementation.

HIGH PERFORMANCE SOFTWARE




SIMULATION OF LOGGING INSTRUMENTS

List of Four Model Problems Solved with 2Dhp90 (v.7.2)

• Problem I: Through Casing Resistivity Tool ( TCRT). Study ofwater invasion in an oil-based formation.

• Problem II: Study of oil invasion on an anisotropic formationwith laminated sands.

• Problem III: Detection of oil-water contact below the positionof an induction logging instrument.

• Problem IV: Through Casing Resistivity Tool ( TCRT). Studyof anisotropy and water invasion effects on a modelformation typical from the Gulf of Mexico.





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1 Ohm−m

1 Ohm−m

5.5"6"

8"

20 fe

et30

00 fe

et30

00 fe

et

1500 feet

Cas

ing:

0.0

0001

Ohm

−m

4 fe

et1

foot

INV

AS

ION

ZO

NE

Cem

ent:

2 O

hm−

m

Bor

ehol

e:1

Ohm

−m

Target Zone200 Ohm−m

0.8

Ohm

−m




Objective: Study the effectof water invasion THROUGHCASING.





10−8

10−7

10−6

10−5

−5

−4

−3

−2

−1

0

1

2

3

4

5


Posit

ion

of th

e re

ceive

r ant

enna

in th

e z−

axis

(in m

)

NO WATER INVASION

1 Hz100 Hz10000 Hz

Variations due to frequency are small (below 5%)





10−8

10−7

10−6

10−5

−5

−4

−3

−2

−1

0

1

2

3

4

5


Po

sitio

n o

f th

e r

ece

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

1 Hz


Water invasion through casing can be accurately assessed





10−8

10−7

10−6

10−5

−5

−4

−3

−2

−1

0

1

2

3

4

5


Po

sitio

n o

f th

e r

ece

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

100 Hz







10−8

10−7

10−6

10−5

−5

−4

−3

−2

−1

0

1

2

3

4

5


Po

sitio

n o

f th

e r

ece

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

10000 Hz







102

103

104

−5

−4

−3

−2

−1

0

1

2

3

4

5


Pos

ition

of t

he re

ceiv

er a

nten

na in

the

z−ax

is (i

n m

)

NO WATER INVASION, TOOL DESIGN 1

1 Hz100 Hz10000 Hz

Mandrel Through Casing provides meaningless results





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5.5"

Bo

reh

ole

:

10 Ohm−m10 Ohm−m10 Ohm−m10 Ohm−m10 Ohm−m10 Ohm−m10 Ohm−m10 Ohm−m10 Ohm−m

1 Ohm−m

20

fe

et

1.8 feet

0.2 feet

10 Ohm−m3 Ohm−m3 Ohm−m3 Ohm−m3 Ohm−m3 Ohm−m3 Ohm−m3 Ohm−m3 Ohm−m3 Ohm−m3 Ohm−m

5.7"

6"

7 O

hm

−m

70

0 fe

et

70

0 fe

et

700 feet

48

"

1 Ohm−m (horizontal)5 Ohm−m (vertical)


Mu

d c

ake

: 1

5 O

hm

−m



Laminated sands.

Objective: Study the effects ofinvasion and anisotropy.





10−1

100

101

102

−5

−4

−3

−2

−1

0

1

2

3

4

5


Po

sitio

n o

f th

e r

ece

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

NO WATER INVASION1 Hz1 Hz (ANISOTROPY)2 MHz2 MHz (ANISOTROPY)

−100 0 100−5

−4

−3

−2

−1

0

1

2

3

4

5

Phase (degrees)

Po

sitio

n o

f th

e r

ece

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

NO WATER INVASION

1 Hz1 Hz (ANISOTROPY)2 MHz2 MHz (ANISOTROPY)

Anisotropy effects are significant. Frequency variations are below 10%





10−1

100

101

102

−5

−4

−3

−2

−1

0

1

2

3

4

5


Po

sitio

n o

f th

e r

ece

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

2 MHz


−100 0 100−5

−4

−3

−2

−1

0

1

2

3

4

5

Phase (degrees)

Po

sitio

n o

f th

e r

ece

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

2 MHz


Accurate software is needed for water invasion assessment





z=1m

z=−1m

z=0m


D=2,5,10,15z=−D m

0.0

00

00

1 O

hm

−m


3.375"3.75"

TX

RX




Objective: Study the effect ofinvasion.





10−4

10−2

100

102

104

106

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Frequency (kHz)

Am

plitu

de

of

Eφ (

V/m

)

NO ANISOTROPY

2m5m10m15m





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1500 feet

Ca

sin

g: 0

.00

00

1 O

hm

−m

4 fe

et

1 fo

ot

Ce

me

nt: 2

Oh

m−

m

Bo

reh

ole

:3

Oh

m−

m

30

00

fe

et

Shale: 1 Ohm−m (horizontal)10 Ohm−m (vertical)

30

00

fe

et

1 O

hm

−m

Oil: 60 Ohm−m


2 O

hm

−m

Oil−Water: 10 Ohm−m

10 Ohm−m (vertical)Shale: 1 Ohm−m (horizontal)

0.5

Oh

m−

m


20

fe

et

25

fe

et

30

fe

et

15 feet

12 feet

6"

Water: 0.3 Ohm−m

4.5"5"

7"

z=0


Seven different materials withhigh contrast on resistivity.


Objective: Study the effectof invasion and anisotropyTHROUGH CASING.





10−9

10−8

10−7

10−6

10−5

−5

0

5

10

15

20

25

30

35

40


Po

sitio

n o

f th

e r

ece

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

NO INVASION. ANISOTROPY STUDY

1 Hz10000 Hz1 Hz (ANISOTROPY)10000 Hz (ANISOTROPY)

Study of anisotropy and frequency effects require from high accuracy simulati ons





10−9

10−8

10−7

10−6

10−5

−5

0

5

10

15

20

25

30

35

40


Po

sitio

n o

f th

e r

ece

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

ANISOTROPIC

1 Hz (NO INVASION)1 Hz (6" INVASION)10000 Hz (NO INVASION)10000 Hz (6" INVASION)

Variations due to invasion are below 20%.




CONCLUSIONS AND FUTURE WORK

• The self-adaptive goal-oriented hp-adaptive strategyconverges exponentially in terms of a user-prescribedquantity of interest vs. the CPU time.

• It is possible to simulate a variety of EM logging instrumentsby using the self-adaptive goal-oriented hp-FEM.

– The software can be utilized to simulate ALL axisymmetric resistivi tylogging instruments in possibly cased wells.

– Furthermore, by using 2Dhp90, we can accurately describe the eff ectof water/oil-based mud invasion, anisotropy, magnetic buffers, etc.

Department of Petroleum and Geosystems Engineering, andInstitute for Computational Engineering and Sciences (ICES)




FUTURE WORK

Simulation of 3D Resistivity Logging Problems

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• PROJECT I: Simulate 3D DC andAC Resistivity Logging Problems.

– Main challenge: To Perform FastLarge Computations.

– Expected results: Similar results asin 2D.

• PROJECT II: Invert 2D Multi-Physic Problems.

– Main challenge: To deal withdifferent physics.

– Expected results: Similar results asin 2D.




ACKNOWLEDGMENTS

THANKS!!!



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