fast inversion of 3d borehole resistivity measurements ......aralar erdozain supervised by: h el ene...

Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

Fast inversion of 3D Borehole ResistivityMeasurements using Model Reduction Techniques

based on 1D Semi-Analytical Solutions.

Aralar Erdozain

Supervised by:

Helene Barucq, David Pardo and Victor Peron

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

MOTIVATION

Main goal: To obtain a better characterization of the Earth’ssubsurface

How: Recording borehole resistivity measurements

Procedure:

Well

Logging Instrument

Transmitters

Receivers

Receivers

Transmitter

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

MOTIVATION

Practical Difficulties:

It is not easy to drill a borehole

It may collapse

Practical Solutions:

Use a metallic casing

Surround with a cement layer

Problem solved, but...

Numerical problems due to thehigh conductivity and thinness of the casing

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

MOTIVATION

Practical Difficulties:

It is not easy to drill a borehole

It may collapse

Practical Solutions:

Use a metallic casing

Surround with a cement layer

Problem solved, but... Numerical problems due to thehigh conductivity and thinness of the casing

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

REALISTIC SCENARIO1

Wm

2W

m

5 Wm

5 Wm

20 Wm

5m

5m

2m

5 cm14 cm

1.27cm, 2.3e-7 Wm

Scenarios: As we are consideringaxisymmetric scenarios, we can workwith two dimensional scenarios

Conductivity and casing width:δ = 1.27e− 2 mσc = 4.34e6 Ω−1m−1

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

CONFIGURATIONS OF INTEREST

Develop: Asymptotic method for avoiding the conflictive partof the domain (casing)

Main idea: Equivalent conditions for substituting the casing

Configuration A Configuration B

ROCK

C

A

S

I

N

G

ROCK 1

ROCK 2

C

A

S

I

N

G

B

O

R

E

H

O

L

E

C

E

M

E

N

T

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

REFERENCES

A.A. Kaufman. The electrical field in a borehole with a casing.Geophysics, Vol.55, Issue 1, pp. 29-38, 1990.

D.Pardo, C.Torres-Verdın and Z.Zhang. Sensitivity study ofborehole-to-surface and crosswell electromagnetic measurementsacquired with energized steel casing to water displacement inhydrocarbon-bearing layers. Geophysics, 73 No.6, F261-F268, 2008.

M. Durufle, V. Peron and C. Poignard. Thin Layer Models forElectromagnetism.Communications in Computational Physics16(1):213-238, 2014.

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

OUTLINE

1 Introduction

2 Equivalent Conditions

3 Numerical Results

4 Extra Configuration

5 Perspectives

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

OUTLINE

1 Introduction

2 Equivalent Conditions

3 Numerical Results

4 Extra Configuration

5 Perspectives

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

MODEL PROBLEM

EQUATIONS FOR THE ELECTRIC POTENTIAL

div [(σ − iδω)∇u] = −divj (ω = 0 first approach)

We

Wc

G∆

Ω = Ωe ∪ Ωc ∪ Γ

σe∆ue = f in Ωe

σc∆uc = 0 in Ωc

ue = uc on Γσc∂nuc = σe∂nue on Γ

uc = 0 on ∂Ω

Where the solution is expressed as

u =

ue in Ωe

uc in Ωc

and σe, σc, f are known data

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

MAIN IDEA

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REFERENCE MODELSolution: u

We

Wc

G∆

ASYMPTOTIC MODELSolution: u[n]

Equivalent conditions

We

G

−→

Definition: Let u be the reference solution. We say an asymptoticmodel is of Order n+1, if its solution u[n] satisfies

||u− u[n]||L2 ≤ Cδn+1

Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

METHODOLOGY

Step1: Derive an Asymptotic Expansion for u when δ −→ 0

In the casing: uc(t, s) =∑n∈N

δnUnc

(t,s

δ

)Outside the casing: ue(x, y) =

∑n∈N

δnune (x, y)

Step2: Obtain Equivalent Conditions of order k + 1 byidentifying a simpler problem satisfied by the truncatedexpansion

uk,δ := u0e + δu1

e + δ2u2e + . . .+ δkuke

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

MULTISCALE EXPANSION

In the Casing:σc∂

2t U

n−2c + σc∂

2sU

nc = 0 s ∈ (0, 1)

σc∂sUnc = σe∂nu

n−1e s = 0

Unc = 0 s = 1

Outside the Casing:σe∆u

ne = fδn0 in Ωe

une = Unc on Γe

We collect the equations for n = 0, 1, 2 to derive the equivalentconditions

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

EQUIVALENT MODELS

We identify simpler problems satisfied by truncated expansionsoutside the casing (up to residual terms)

Order 1: −→σe∆u = f in Ωe

u = 0 on Γ

Order 3: −→

σe∆u = f in Ωe

u+ δ σeσc ∂nu = 0 on Γ

Remark: Second model has already order 3 of convergence due tothe flat configuration of the layer

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

OUTLINE

1 Introduction

2 Equivalent Conditions

3 Numerical Results

4 Extra Configuration

5 Perspectives

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

NUMERICAL DISCRETIZATION

FINITE ELEMENT METHOD (Matlab Code)

Straight triangular elements

Lagrange shape functions of any degree

Domain Mesh Reference Solution

Σe=2

Σc=

0.5

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

NUMERICAL SOLUTIONS

Reference Model (in Ωe) Order 1 Model Order 3 Model

Definition: We define the relative error between the referencesolution u and the asymptotic solution u[n], as

||u− u[n]||L2

||u||L2

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

CONVERGENCE RATES

0.001 0.003 0.010 0.032 0.10010

−6

10−4

10−2

100

102

Casing Thickness (m)

L2 R

elat

ive

Err

or (

%)

ERROR DEPENDING ON EPSILON IN LOGARITHMIC COORDINATES

order 1

order 3

Casing Thickness 0.001 0.002 0.004 0.007 0.013 0.024 0.043 0.078

Order 1 Slopes 0.975 0.956 0.925 0.873 0.792 0.677 0.534 0.382

Order 3 Slopes 2.990 2.074 1.468 1.440 2.188 2.362 2.148 1.872

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

INTERPOLATION DEGREE

Relative error between a solution of degree 10 and solutions oflower degrees

0 2 4 6 8 10

10−2

100

102

L2 R

elat

ive

Err

or

Interpolation Degree

REFERENCE MODEL

0 2 4 6 8 10

10−2

100

102

L2 R

elat

ive

Err

or

Interpolation Degree

ORDER 3 MODEL

CONCLUSION: Error analysis is not relevant once we reach arelative error of 10−2

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

OUTLINE

1 Introduction

2 Equivalent Conditions

3 Numerical Results

4 Extra Configuration

5 Perspectives

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

MAIN IDEA

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REFERENCE MODELSolution: u

Wc

We1

We2Wc

Wb

Wce

m

∆

GcemcGc

b

ASYMPTOTIC MODELSolution: u[n]

We1

We2Wb

Wce

m

∆

Eq

uiv

alen

tC

on

dit

ion

s

GcemcGc

b−→

Definition: We define the jump and mean value of the solution uacross the casing as

[u] = u|Γbc− u|Γc

cem

u =1

2

(u|Γb

c+ u|Γc

cem

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

MODEL PROBLEM

We2

We1

Wc

Wb

Wce

m

G

Gc

b Gcemc

Ge2cem

Ge1cem

Ge1e2

σi∆ui = fi in Ωi

σc∆uc = 0 in Ωc

ui = uj on Γjiσi∂nui = σj∂nuj on Γji

u = 0 on ∂Ω

i, j = b, c, cem, e1, e2

Where σi and fi are known data

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

EQUIVALENT CONDITIONS

Using a similar procedure than for the previous configuration weobtain equivalent conditions

Order 1: −→

[u] = 0[σ∂nu] = 0

Order 3: −→

[u] = εσcσ∂nu

[σ∂nu] = −εσc∂2x u

Remark: Second model has already order 3 of convergence due tothe flat configuration of the layer

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

NUMERICAL SOLUTIONS

Domain

Σe1=5

Σe1=4Σ

c=

0.5

Σb

=2

Σce

m=

3

Mesh

Reference Solutionoutside Ωc

Order 1 Model Order 3 Model

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

CONVERGENCE RATES

0.001 0.003 0.010 0.032 0.10010

−6

10−4

10−2

100

102

Casing Thickness (m)

L2 R

elat

ive

Err

or (

%)

ERROR DEPENDING ON EPSILON IN LOGARITHMIC COORDINATES

order 1

order 3

Casing Thickness 0.001 0.002 0.004 0.007 0.013 0.024 0.043 0.078

Order 1 Slopes 0.967 0.944 0.904 0.843 0.751 0.631 0.493 0.365

Order 3 Slopes 4.764 1.637 1.187 1.291 1.116 1.742 1.911 1.846

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

OUTLINE

1 Introduction

2 Equivalent Conditions

3 Numerical Results

4 Extra Configuration

5 Perspectives

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

REALISTIC SCENARIO1

Wm

2W

m

5 Wm

5 Wm

20 Wm5

m5

m2

m

5 cm14 cm

1.27cm, 2.3e-7 Wm

Conductivity and casing width:δ = 1.27e− 2 mσc = 4.34e6 Ω−1m−1

⇒ σc ≈ δ−3

First approach:

σc = α α ∈ R

Case to be studied:

σc = αδ−3 α ∈ R

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

SECOND DIFFERENCE OF POTENTIAL

1.27cm, 2.3e-7 ohm-m

Receivers

Transmittery=y0

y=y1

y=y2

y=y3

Equation:

div [(σ − iδω)∇u] = f

Right hand side:

f =

1 In the transmitter

0 Outside the transmitter

Objective: Measure the seconddifference of potential on theReceivers

U2 = u(y1)− 2u(y2) + u(y3)

Expected Result: The second difference of potential proportional tothe rock resistivity

U2 = α · ρrock α ∈ R28 / 30

Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

Perspectives

Short Term:

Asymptotic models with σc = αδ−3 α ∈ R

Measure the second difference of potential on the receivers

Long Term:

Consider physically more realistic scenarios

Develop 3D electromagnetic models

Study highly deviated boreholes

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Introduction Equivalent Conditions Numerical Results Extra Configuration Perspectives

THANK YOU FOR

YOUR ATTENTION

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