SIMULATION AND INTERPRETATION OF BOREHOLE GEOPHYSICAL MEASUREMENTS USING hp FINTE ELEMENTS

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Demkowicz, C. Torres-Verdín, V. M. Calo, M. Paszynski, and P. J.

1 SIMULATION AND INTERPRETATION OF BOREHOLE GEOPHYSICAL MEASUREMENTS USING hp FINTE ELEMENTS hp-fem team: D. Pardo, M. J. Nam, L. Demkowicz, C. Torres-Verdín, V. M. Calo, M. Paszynski, and P. J. Matuszyk 8th Annual Formation Evaluation Research Consortium Meeting August, 8

2 Overview.Main Lines of Research and Applications (D. Pardo) Previous work Main features of our technology. Application : Tri-Axial Induction Instruments (M. J. Nam). Application : Dual-Laterolog Instruments (M. J. Nam). Multi-Physics Inversion (D. Pardo). Sonic Instruments (L. Demkowicz) 8 th Annual Formation Evaluation Consortium Meeting, 8

3 Previous Work Type of Problems We Can Solve with our hp-fem software Applications Spatial Dimensions Well Type Logging Instruments Frequency Materials Physical Devices Sources Borehole Measurements D Vertical Well LWD/MWD Induction Isotropic Magnetic Buffers Casing Imperfections Finite Size Antennas Toroidal Antennas Marine Controlled Source EM Deviated Well Normal/Laterolog Dielectric Instruments ~ GHz Insulators Displacement Currents Dipoles in Any Direction Electrodes D Anisotropic Eccentered Tool Dual-Laterolog Cross-Well Casing Combination of All Solenoidal Antennas Combination of All Invasion Water Oil MOST (OIL-INDUSTRY) GEOPHYSICAL PROBLEMS etc. 8 th Annual Formation Evaluation Consortium Meeting, 8

4 Main Features of Our Technology. Self-Adaptive Goal-Oriented hp-refinements. Fourier Finite-Element Method. Parallel Implementation 8 th Annual Formation Evaluation Consortium Meeting, 8

Self-Adaptive Goal-Oriented hp-fem We vary locally the element size h and the polynomial order of approximation p throughout the grid. Optimal grids are automatically generated by the hp-algorithm.

5 Self-Adaptive Goal-Oriented hp-fem We vary locally the element size h and the polynomial order of approximation p throughout the grid. Optimal grids are automatically generated by the hp-algorithm. The self-adaptive goal-oriented hp-fem provides exponential convergence rates in terms of the CPU time vs. the error in a user prescribed quantity of Interest. 8 th Annual Formation Evaluation Consortium Meeting, 8

6 6 D Deviated Well Cartesian system of coordinates: (x, x, x ) New non-orthogonal system of coordinates: (ζ, ζ, ζ ) Subdomain Subdomain x = ζcosζ x = ζsinζ x = ζ x = ζcosζ x = ζsinζ ζ ρ x = ζ + tanθ ρ cosζ ρ ρ Subdomain x = ζcosζ x = ζsinζ x = ζ + ζtanθ cosζ 8 th Annual Formation Evaluation Consortium Meeting, 8

7 7 D Deviated Well Cartesian system of coordinates: (x, x, x ) New non-orthogonal system of coordinates: (ζ, ζ, ζ ) Constant material coefficients in the quasi-azimuthal direction ζ in the new non-orthogonal system of coordinates!!!! 8 th Annual Formation Evaluation Consortium Meeting, 8

8 8 D Deviated Well For each Fourier mode, we obtain a D problem. Each D problem couples with up to five different D problems corresponding to different Fourier modes, therefore, constituting the resulting D problem. When we use 9 Fourier Modes for the Solution: A, A, A, x b A, A, A, A, x b A, A, A, A, A, x b A, A, A, A, A,6 x b A, A, A, A,6 A,7 x = b A6, A6, A6,6 A6,7 A6,8 x6 b6 A7, A7,6 A7,7 A7,8 A 7,9 x 7 b 7 A8,6 A8,7 A8,8 A8,9 x8 b8 A9,7 A9,8 A 9,9 x 9 b 9 A i,j : represents a full D problem for each Fourier basis function 8 th Annual Formation Evaluation Consortium Meeting, 8

9 9 D Deviated Well For each Fourier mode, we obtain a D problem. Each D problem couples with up to five different D problems corresponding to different Fourier modes, therefore, constituting the resulting D problem. 8 th Annual Formation Evaluation Consortium Meeting, 8

10 D Parallelization Implementation Distributed Domain Decomposition Shared Domain Decomposition!! 8 th Annual Formation Evaluation Consortium Meeting, 8

11 SELF-ADAPTIVE hp FINITE-ELEMENT SIMULATION OF MULTI-COMPONENT INDUCTION MEASUREMENTS ACUIRED IN DIPPING, INVADED, AND ANISOTROPIC FORMATIONS M. J. Nam, D. Pardo, and C. Torres-Verdín, The University of Texas at Austin hp-fem team: D. Pardo, M. J. Nam, L. Demkowicz, C. Torres-Verdín, V. M. Calo, M. Paszynski, and P. J. Matuszik 8th Annual Formation Evaluation Research Consortium Meeting August, 8

12 Overview. Main Lines of Research and Applications (D. Pardo) Previous work Main features of our technology. Application : Tri-Axial Induction Instruments (M. J. Nam). Application : Dual-Laterolog Instruments (M. J. Nam). Multi-Physics Inversion (D. Pardo). Sonic Instruments (L. Demkowicz) 8 th Annual Formation Evaluation Consortium Meeting, 8

13 Outline Introduction to Tri-Axial Induction Method Numerical Results: Verification of D Method for Tri-Axial Induction Tool Dipping, Invaded, Anisotropic Formations Conclusions 8 th Annual Formation Evaluation Consortium Meeting, 8

14 Tri-Axial Induction Tool L =.6 m ( In.) Operating frequency: khz θ: dip angle H xy α: tool orientation angle 8 th Annual Formation Evaluation Consortium Meeting, 8

15 D Source Implementation. Solenoidal Coil (J φ ) for M z becoming a D source in (ρ, φ, z). Delta Function for D source M x or M y intensity f ( φ) = δφ ( φ) Delta function Nr. of modes = 7 Nr. of modes = Nr. of modes = Nr. of modes = M x : Delta function φ : the position of the center of the peak ( for M x ; 9 for M y ) Gibb s Phenomenon - - φ in degrees 8 th Annual Formation Evaluation Consortium Meeting, 8

16 D Source and Receiver (Delta Functions) 6 intensity Mx with modes Hx with modes Mx times Hx Source times Receiver intensity φ in degrees Mx with 9 modes Hx with 9 modes Mx times Hx Source times Receiver -. Coupling between source and receiver: less Gibb s phenomenon φ in degrees 8 th Annual Formation Evaluation Consortium Meeting, 8

17 7 Method Combination of:. A Self-Adaptive Goal-Oriented hp-fem for AC problems. A Fourier Series Expansion in a Non-Orthogonal System of Coordinates. Parallel Implementation 8 th Annual Formation Evaluation Consortium Meeting, 8

18 Verification of.d Simulation (H xx = H yy ) 8 Real part of Hxx at khz - Imaginary part of Hxx at khz emd hp with modes hp with modes Re(Hxx) field (A/m). emd hp with modes hp with modes -.. Im(Hxx) field (A/m) Converged solutions with Fourier modes emd: K. H. Lee 98, pers. comm. 8 th Annual Formation Evaluation Consortium Meeting, 8

19 9 Verification of.d Simulation (H zz ) Real part of Hzz at khz - Imaginary part of Hzz at khz emd hp with mode.. Re(Hzz) field (A/m). emd hp with mode Im(Hzz) field (A/m) The same solutions with Fourier mode 8 th Annual Formation Evaluation Consortium Meeting, 8

20 Verification of.d Simulation (H xy = H yx ) Real part of Hxy at khz - Imaginary part of Hxy at khz emd hp with modes hp with modes Re(Hxy) field (A/m) x -. emd hp with modes hp with modes Im(Hxy) field (A/m) x Converged solutions with Fourier modes 8 th Annual Formation Evaluation Consortium Meeting, 8

21 Verification of.d Simulation (H xz = H zx ) Real part of Hxz at khz - Imaginary part of Hxz at khz emd hp with mode Re(Hxz) field (A/m). emd hp with mode Im(Hxz) field (A/m) x - The same solutions with Fourier mode 8 th Annual Formation Evaluation Consortium Meeting, 8

22 Verification of D Simulation (H xx = H yy ) Real part of Hxx at khz Dip angle: 6 degrees Re(Hxx) field (A/m) emd -. hp with modes hp with 7 modes hp with 9 modes Distance between TX and RX (m) Im(Hxx) field (A/m) x - Imaginary part of Hxx at khz - - emd hp with modes - hp with 7 modes hp with 9 modes Distance between TX and RX (m) Converged solutions with 9 Fourier mode 8 th Annual Formation Evaluation Consortium Meeting, 8

23 Verification of D Simulation (H zz ) Re(Hzz) field (A/m) Im(Hzz) field (A/m) Distance between TX and RX (m) x - Imaginary part of Hzz at khz Real part of Hzz at khz emd hp with modes hp with modes hp with 9 modes -8 emd hp with modes - hp with modes hp with 9 modes Distance between TX and RX (m) Dip angle: 6 degrees Converged solutions with Fourier mode 8 th Annual Formation Evaluation Consortium Meeting, 8

24 Description of the Tri-Axial Tool Operating frequency: khz ohm-m 8 th Annual Formation Evaluation Consortium Meeting, 8

25 Verification of.d Simulation (H xx ) Relative errors of tri-axial induction solutions with respect to the solution with 9 Fourier modes vertical well in a homogeneous formation Relative Error (in %) - real (without tool propt.) imaginary (without tool propt.) 7 9 Number of Fourier Modes 8 th Annual Formation Evaluation Consortium Meeting, 8

26 6 Verification of D Simulation (H xx ) θ = 6 degrees Relative errors of tri-axial Induction solutions with respect to the solution for the vertical well 6 degree deviated well in a homogeneous formation Relative Error (in %) - real (without tool propt.) imaginary (without tool propt.) 7 9 Number of Fourier Modes 8 th Annual Formation Evaluation Consortium Meeting, 8

27 7 Model for Numerical Experiments Five layers:,.,, and ohm-m from top to bottom Borehole:. m in radius ohm-m in resistivity Invasion in the third and fourth layers Anisotropy in the second and fourth layers θ =, and 6 degrees 8 th Annual Formation Evaluation Consortium Meeting, 8

28 8 Convergence History of H xx in Vertical Well Real part of Hxx at khz ohm-m hp with modes hp with modes hp with 7 modes hp with 9 modes Imaginary part of Hxx at khz ohm-m hp with modes hp with modes hp with 7 modes hp with 9 modes -. ohm-m -. ohm-m ohm-m - ohm-m ohm-m ohm-m ohm-m Re(Hxx) field (A/m) ohm-m -... Im(Hxx) field (A/m) Converged solutions with Fourier modes 8 th Annual Formation Evaluation Consortium Meeting, 8

29 Convergence History of H xx in Deviated Well 9 Real part of Hxx at khz ohm-m hp with modes hp with 7 modes hp with 9 modes hp with modes Imaginary part of Hxx at khz ohm-m hp with modes hp with 7 modes hp with 9 modes hp with modes θ = 6 degrees -. ohm-m -. ohm-m ohm-m - ohm-m ohm-m ohm-m Re(Hxx) field (A/m) ohm-m ohm-m Im(Hxx) field (A/m) Converged solutions with 9 Fourier modes 8 th Annual Formation Evaluation Consortium Meeting, 8

30 Deviated Wells (, & 6 degrees) Real part of Hzz at khz vertical degrees 6 degrees Real part of Hxx at khz vertical degrees 6 degrees Real part of Hyy at khz vertical degrees 6 degrees ohm-m ohm-m ohm-m -. ohm-m -. ohm-m -. ohm-m ohm-m - ohm-m - ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m Re(Hzz) field (A/m) x Dip angle has larger effects Re(Hxx) field (A/m) Re(Hyy) field (A/m) on tri-axial tools 8 th Annual Formation Evaluation Consortium Meeting, 8

31 Deviated Wells (, & 6 degrees) Imaginary part of Hzz at khz vertical degrees 6 degrees Imaginary part of Hxx at khz vertical degrees 6 degrees Imaginary part of Hyy at khz vertical degrees 6 degrees ohm-m ohm-m ohm-m -. ohm-m -. ohm-m -. ohm-m ohm-m - ohm-m - ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m Im(Hzz) field (A/m) x -... Im(Hxx) field (A/m) Dip angle has larger effects on tri-axial tools -... Im(Hyy) field (A/m) 8 th Annual Formation Evaluation Consortium Meeting, 8

32 H zz in Deviated Wells with Invasion (Re.) Real part of Hzz at khz No invasion With invasion ohm-m Real part of Hzz at khz No invasion With invasion ohm-m Shallow invasion with R =. m -. ohm-m -. ohm-m ohm-m ( ohm-m) - ohm-m ( ohm-m) ohm-m ( ohm-m) ohm-m ( ohm-m) ohm-m ohm-m Re(Hzz) field (A/m) x vertical Re(Hzz) field (A/m) x 6 degrees 8 th Annual Formation Evaluation Consortium Meeting, 8

33 H zz in Deviated Wells with Invasion (Im.) Imaginary part of Hzz at khz No invasion With invasion ohm-m Imaginary part of Hzz at khz No invasion With invasion ohm-m Shallow invasion with R =. m -. ohm-m -. ohm-m ohm-m ( ohm-m) - ohm-m ( ohm-m) ohm-m ( ohm-m) ohm-m ( ohm-m) ohm-m Im(Hzz) field (A/m) x vertical ohm-m Im(Hzz) field (A/m) x 6 degrees Almost no effects of invasion regardless of the dip angle 8 th Annual Formation Evaluation Consortium Meeting, 8

34 H xx in Deviated Wells with Invasion (Re.) Real part of Hxx at khz No invasion with invasion ohm-m Real part of Hxx at khz No invasion with invasion ohm-m Shallow invasion with R =. m -. ohm-m -. ohm-m ohm-m ( ohm-m) - ohm-m ( ohm-m) ohm-m ( ohm-m) ohm-m ( ohm-m) ohm-m ohm-m Re(Hxx) field (A/m) Re(Hxx) field (A/m) vertical 6 degrees 8 th Annual Formation Evaluation Consortium Meeting, 8

35 H xx in Deviated Wells with Invasion (Im.) Imaginary part of Hxx at khz No invasion with invasion ohm-m Imaginary part of Hxx at khz No invasion with invasion ohm-m Shallow invasion with R =. m -. ohm-m -. ohm-m ohm-m ( ohm-m) - ohm-m ( ohm-m) ohm-m ( ohm-m) ohm-m ( ohm-m) ohm-m -... Im(Hxx) field (A/m) vertical ohm-m -... Im(Hxx) field (A/m) 6 degrees Small effects of invasion 8 th Annual Formation Evaluation Consortium Meeting, 8

36 6 H yy in Deviated Wells with Invasion (Re.) Real part of Hyy at khz No invasion with invasion ohm-m Real part of Hyy at khz No invasion with invasion ohm-m Shallow invasion with R =. m -. ohm-m -. ohm-m ohm-m ( ohm-m) - ohm-m ( ohm-m) ohm-m ( ohm-m) ohm-m ( ohm-m) ohm-m ohm-m Re(Hyy) field (A/m) Re(Hyy) field (A/m) vertical 6 degrees 8 th Annual Formation Evaluation Consortium Meeting, 8

37 7 H yy in Deviated Wells with Invasion (Im.) Imaginary part of Hyy at khz No invasion with invasion ohm-m Imaginary part of Hyy at khz No invasion with invasion ohm-m Shallow invasion with R =. m -. ohm-m -. ohm-m ohm-m ( ohm-m) - ohm-m ( ohm-m) ohm-m ( ohm-m) ohm-m ( ohm-m) ohm-m -... Im(Hyy) field (A/m) vertical ohm-m -... Im(Hyy) field (A/m) 6 degrees Small effects of invasion 8 th Annual Formation Evaluation Consortium Meeting, 8

38 H zz in Deviated Wells with Anisotropy (Re.) 8 Real part of Hzz at khz Isotropy TI anisotropy ohm-m Real part of Hzz at khz Isotropy TI anisotropy ohm-m Real part of Hzz at khz Isotropy θ = 6, & degrees TI anisotropy ohm-m - -. ohm-m. ohm-m - -. ohm-m. ohm-m - -. ohm-m. ohm-m - ohm-m - ohm-m - ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m Re(Hzz) field (A/m) x Effects of anisotropy increase with increasing dip angle Re(Hzz) field (A/m) x Re(Hzz) field (A/m) x vertical degrees 6 degrees 8 th Annual Formation Evaluation Consortium Meeting, 8

39 H zz in Deviated Wells with Anisotropy (Im.) ohm-m Imaginary part of Hzz at khz. ohm-m Isotropy TI anisotropy ohm-m - -. ohm-m Imaginary part of Hzz at khz. ohm-m Isotropy TI anisotropy ohm-m - -. ohm-m Imaginary part of Hzz at khz. ohm-m Isotropy TI anisotropy ohm-m - ohm-m - ohm-m - ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m Im(Hzz) field (A/m) x Effects of anisotropy increase with increasing dip angle Im(Hzz) field (A/m) x Im(Hzz) field (A/m) x vertical degrees 6 degrees 8 th Annual Formation Evaluation Consortium Meeting, 8

40 H xx in Deviated Wells with Anisotropy (Re.) Real part of Hxx at khz Isotropy TI anisotropy ohm-m Real part of Hxx at khz Isotropy TI anisotropy ohm-m Real part of Hxx at khz Isotropy TI anisotropy ohm-m - -. ohm-m. ohm-m - -. ohm-m. ohm-m - -. ohm-m. ohm-m - ohm-m - ohm-m - ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m Effects of anisotropy decrease Re(Hxx) field (A/m) Re(Hxx) field (A/m) with increasing dip angle ohm-m Re(Hxx) field (A/m) vertical degrees 6 degrees 8 th Annual Formation Evaluation Consortium Meeting, 8

41 H xx in Deviated Wells with Anisotropy (Im.) Imaginary part of Hxx at khz Isotropy TI anisotropy ohm-m Imaginary part of Hxx at khz Isotropy TI anisotropy ohm-m Imaginary part of Hxx at khz Isotropy TI anisotropy ohm-m - -. ohm-m. ohm-m - -. ohm-m. ohm-m - -. ohm-m. ohm-m - ohm-m - ohm-m - ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m -... Im(Hxx) field (A/m) ohm-m ohm-m Effects of anisotropy decrease -... with increasing Im(Hxx) field (A/m) dip angle -... Im(Hxx) field (A/m) vertical degrees 6 degrees 8 th Annual Formation Evaluation Consortium Meeting, 8

42 H yy in Deviated Wells with Anisotropy (Re.) Real part of Hyy at khz Isotropy TI anisotropy ohm-m Real part of Hyy at khz Isotropy TI anisotropy ohm-m Real part of Hyy at khz Isotropy TI anisotropy ohm-m - -. ohm-m. ohm-m - -. ohm-m. ohm-m - -. ohm-m. ohm-m - ohm-m - ohm-m - ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m Effects of anisotropy decrease Re(Hyy) field (A/m) Re(Hyy) field (A/m) with increasing dip angle Re(Hyy) field (A/m) vertical degrees 6 degrees 8 th Annual Formation Evaluation Consortium Meeting, 8

43 H yy in Deviated Wells with Anisotropy (Im.) Imaginary part of Hyy at khz Isotropy TI anisotropy ohm-m Imaginary part of Hyy at khz Isotropy TI anisotropy ohm-m Imaginary part of Hyy at khz Isotropy TI anisotropy ohm-m - -. ohm-m. ohm-m - -. ohm-m. ohm-m - -. ohm-m. ohm-m - ohm-m - ohm-m - ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m ohm-m -... Im(Hyy) field (A/m) ohm-m Effects of anisotropy decrease -... Im(Hyy) field (A/m) with increasing dip angle ohm-m -... Im(Hyy) field (A/m) vertical degrees 6 degrees 8 th Annual Formation Evaluation Consortium Meeting, 8

44 H xx at KHz and MHz in Vertical Well Real part of Hxx at khz and MHz khz MHz ohm-m Imaginary part of Hxx at khz and MHz khz MHz ohm-m -. ohm-m -. ohm-m ohm-m - ohm-m ohm-m ohm-m ohm-m Re(Hxx) field (A/m) ohm-m Im(Hxx) field (A/m) Larger variations at MHz than at khz 8 th Annual Formation Evaluation Consortium Meeting, 8

45 Conclusions We successfully simulated D tri-axial induction measurements by combining the use of a Fourier series expansion in a non-orthogonal system of coordinates with a D high-order, self-adaptive hp finite-element method. Dip angle effects on tri-axial tools are larger than on more traditional induction logging instruments. Anisotropy effects on H xx and H yy decrease with increasing dip angle, while those on H zz increase. H xx at khz exhibits smaller variations than at MHz. 8 th Annual Formation Evaluation Consortium Meeting, 8

46 6 Acknowledgements Sponsors of UT Austin s consortium on Formation Evaluation: 8 th Annual Formation Evaluation Consortium Meeting, 8

A Multiphysics Framework Using hp-finite Elements for Electromagnetics Applications

Univ. Carlos III of Madrid A Multiphysics Framework Using hp-finite Elements for Electromagnetics Applications D. Pardo, L. E. García-Castillo, M. J. Nam, C. Torres-Verdín Team: D. Pardo, M. J. Nam, V.