fast inversion of 3d borehole resistivity measurements ......aralar erdozain supervised by: h el ene...
TRANSCRIPT
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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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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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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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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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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OUTLINE
1 Introduction
2 Equivalent Conditions
3 Numerical Results
4 Extra Configuration
5 Perspectives
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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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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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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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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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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
)
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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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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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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THANK YOU FOR
YOUR ATTENTION
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