Fast inversion of 3D Borehole Resistivity Measurements using Model Reduction Techniques based on 1D Semi-Analytical Solutions.

Transcription

1 Fast inversion of 3D Borehole Resistivity Measurements using Model Reduction Techniques based on 1D Semi-Analytical Solutions. Aralar Erdozain Supervised by: Hélène Barucq, David Pardo and Victor Péron 1 / 30

2 MOTIVATION Main goal: To obtain a better characterization of the Earth s subsurface How: Recording borehole resistivity measurements Procedure: Well Logging Instrument Receivers Transmitter Transmitters Receivers 2 / 30

3 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... 3 / 30

4 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 the high conductivity and thinness of the casing 4 / 30

5 REALISTIC SCENARIO 14 cm 5 cm 1 m 2 m 5 m 5 m Scenarios: As we are considering axisymmetric scenarios, we can work with two dimensional scenarios 20 m 5 m 5 m 2 m Conductivity and casing width: { δ = 1.27e 2 m = 4.34e6 Ω 1 m 1 σ c 1.27cm, 2.3e 7 m 5 / 30

6 CONFIGURATIONS OF INTEREST Develop: Asymptotic method for avoiding the conflictive part of the domain (casing) Main idea: Equivalent conditions for substituting the casing Configuration A Configuration B C A S I N G ROCK B O R E H O L E C A S I N G C E M E N T ROCK 1 ROCK 2 6 / 30

7 REFERENCES A.A. Kaufman. The electrical field in a borehole with a casing. Geophysics, Vol.55, Issue 1, pp , D.Pardo, C.Torres-Verdín and Z.Zhang. Sensitivity study of borehole-to-surface and crosswell electromagnetic measurements acquired with energized steel casing to water displacement in hydrocarbon-bearing layers. Geophysics, 73 No.6, F261-F268, M. Duruflé, V. Péron and C. Poignard. Thin Layer Models for Electromagnetism.Communications in Computational Physics 16(1): , / 30

8 OUTLINE 1 Introduction 2 Equivalent Conditions 3 Numerical Results 4 Extra Configuration 5 Perspectives 8 / 30

9 OUTLINE 1 Introduction 2 Equivalent Conditions 3 Numerical Results 4 Extra Configuration 5 Perspectives 9 / 30

10 MODEL PROBLEM EQUATIONS FOR THE ELECTRIC POTENTIAL div [(σ iδω) u] = divj (ω = 0 first approach) e σ e u e = f in Ω e σ c u c = 0 in Ω c u e = u c on Γ σ c n u c = σ e n u e on Γ u c = 0 on Ω Ω = Ω e Ω c Γ Where the solution is expressed as { u e in Ω e u = u c in Ω c and σ e, σ c, f are known data 10 / 30

11 MAIN IDEA REFERENCE MODEL Solution: u ASYMPTOTIC MODEL Solution: u [n] Equivalent conditions e e Definition: Let u be the reference solution. We say an asymptotic model is of Order n+1, if its solution u [n] satisfies u u [n] L 2 Cδ n+1 11 / 30

12 METHODOLOGY Step1: Derive an Asymptotic Expansion for u when δ 0 In the casing: u c (t, s) = ( δ n Uc n t, s ) δ n N Outside the casing: u e (x, y) = n N δ n u n e (x, y) Step2: Obtain Equivalent Conditions of order k + 1 by identifying a simpler problem satisfied by the truncated expansion u k,δ := u 0 e + δu 1 e + δ 2 u 2 e δ k u k e 12 / 30

13 MULTISCALE EXPANSION In the Casing: σ c t 2 Uc n 2 + σ c s 2 Uc n = 0 1) s (0, σ c s Uc n = σ e n u n 1 e s = 0 Uc n = 0 s = 1 Outside the Casing: { σe u n e = fδ0 n in Ω e u n e = U n c on Γ e We collect the equations for n = 0, 1, 2 to derive the equivalent conditions 13 / 30

14 EQUIVALENT MODELS We identify simpler problems satisfied by truncated expansions outside the casing (up to residual terms) Order 1: Order 3: { σe u = f in Ω e u = 0 on Γ { σ e u = f in Ω e u + δ σe σ c n u = 0 on Γ Remark: Second model has already order 3 of convergence due to the flat configuration of the layer 14 / 30

15 OUTLINE 1 Introduction 2 Equivalent Conditions 3 Numerical Results 4 Extra Configuration 5 Perspectives 15 / 30

16 NUMERICAL DISCRETIZATION FINITE ELEMENT METHOD (Matlab Code) Straight triangular elements Lagrange shape functions of any degree Domain Mesh Reference Solution Σc 0.5 Σ e 2 16 / 30

17 NUMERICAL SOLUTIONS Reference Model (in Ω e ) Order 1 Model Order 3 Model Definition: We define the relative error between the reference solution u and the asymptotic solution u [n], as u u [n] L 2 u L 2 17 / 30

18 CONVERGENCE RATES ERROR DEPENDING ON EPSILON IN LOGARITHMIC COORDINATES 10 2 L2 Relative Error (%) order 1 order Casing Thickness (m) Casing Thickness Order 1 Slopes Order 3 Slopes / 30

19 INTERPOLATION DEGREE Relative error between a solution of degree 10 and solutions of lower degrees 10 2 REFERENCE MODEL 10 2 ORDER 3 MODEL L2 Relative Error L2 Relative Error Interpolation Degree Interpolation Degree CONCLUSION: Error analysis is not relevant once we reach a relative error of / 30

20 OUTLINE 1 Introduction 2 Equivalent Conditions 3 Numerical Results 4 Extra Configuration 5 Perspectives 20 / 30

21 MAIN IDEA REFERENCE MODEL Solution: u ASYMPTOTIC MODEL Solution: u [n] b c b c cem c cem e2 b c b Equivalent Conditions cem c cem e2 e1 e1 Definition: We define the jump and mean value of the solution u across the casing as [u] = u Γ b c u Γ c cem {u} = 1 ) (u 2 Γ + u bc Γ ccem 21 / 30

22 MODEL PROBLEM b c cem cem e2 cem e1 e2 e2 e1 σ i u i = f i in Ω i σ c u c = 0 in Ω c u i = u j on Γ j i σ i n u i = σ j n u j on Γ j i u = 0 on Ω i, j = b, c, cem, e1, e2 c b c cem e1 Where σ i and f i are known data 22 / 30

23 EQUIVALENT CONDITIONS Using a similar procedure than for the previous configuration we obtain equivalent conditions Order 1: { [u] = 0 [σ n u] = 0 Order 3: { [u] [σ n u] = ɛ σ c {σ n u} = ɛσ c x 2 {u} Remark: Second model has already order 3 of convergence due to the flat configuration of the layer 23 / 30

24 NUMERICAL SOLUTIONS Domain Mesh Σ e1 5 Σb 2 Σc 0.5 Σcem 3 Σ e1 4 Reference Solution outside Ω c Order 1 Model Order 3 Model 24 / 30

25 CONVERGENCE RATES ERROR DEPENDING ON EPSILON IN LOGARITHMIC COORDINATES 10 2 L2 Relative Error (%) order 1 order Casing Thickness (m) Casing Thickness Order 1 Slopes Order 3 Slopes / 30

26 OUTLINE 1 Introduction 2 Equivalent Conditions 3 Numerical Results 4 Extra Configuration 5 Perspectives 26 / 30

27 REALISTIC SCENARIO 1 m 2 m 14 cm 5 m 5 cm 5 m Conductivity and casing width: { δ = 1.27e 2 m = 4.34e6 Ω 1 m 1 σ c 20 m 5 m 1.27cm, 2.3e 7 m 5 m 2 m σ c δ 3 First approach: σ c = α α R Case to be studied: σ c = αδ 3 α R 27 / 30

28 SECOND DIFFERENCE OF POTENTIAL 1.27cm, 2.3e 7 ohm m y y 3 y y 2 Receivers y y 1 Equation: div [(σ iδω) u] = f Right hand side: { 1 In the transmitter f = 0 Outside the transmitter y y 0 Transmitter Objective: Measure the second difference of potential on the Receivers U 2 = u(y 1 ) 2u(y 2 ) + u(y 3 ) Expected Result: The second difference of potential proportional to the rock resistivity U 2 = α ρ rock α R 28 / 30

29 Perspectives Short Term: Asymptotic models with σ c = 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 29 / 30

30 THANK YOU FOR YOUR ATTENTION 30 / 30