1. Alexandrov Dmitriy, Saint-Petersburg State University Numerical modeling: Tube-wave reflections in cased borehole

2. Outline  Modeling approaches: 1D effective wavenumber approach finite-difference  Wave field in cased borehole wave field in isotropic homogeneous fluid wave field in isotropic homogeneous elastic media  Reflection from geological interfaces behind casing;  Reflection from corroded section of the casing;  Response of perforation in cased borehole: Idealized disk-shaped perforation Idealized zero-length disk-shaped perforation  1D approach limitations;  Conclusions. Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Outline Limitations Model 1 Model 2 Model 3 Conclusions 1D effective wavenumber approach Modeling approaches

3. Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Introduction Outline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

4.  Finite-difference (FD) code flexible little analytical insight  1D effective wavenumber approach Attractive for analysis Approximate Validity for cased borehole is unknown  Validate 1D approach using FD code Outline Limitations Model 1 Model 2 Model 3 Conclusions 1D effective wavenumber approach Modeling approaches Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.

5. 1D effective wavenumber approach  Helmholtz equations: Solution form: Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Outline Limitations Wavefield in cased borehole Results Conclusions Modeling approaches 1D effective wavenumber approach

6. Boundary conditions: Boundary conditions:  continuity of pressure:  continuity of fluid flow: Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Outline Limitations Wavefield in cased borehole Results Conclusions Modeling approaches 1D effective wavenumber approach

7.  Multilayered model Boundary conditions: continuity of pressure: continuity of fluid flow: Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Outline Limitations Wavefield in cased borehole Results Conclusions Modeling approaches 1D effective wavenumber approach

8. Motion equation: Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Wave field in isotropic homogeneous fluid Outline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

9. Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Outline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches Wave field in isotropic homogeneous fluid

10. Wave field in isotropic homogeneous elastic media Motion equation: Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Outline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

11. Boundary conditions     Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Outline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

12. Reflection from geological interfaces behind casing Reflection coefficient for tube wave Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Outline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

13. Reflection from corroded section of the casing Reflection of tube wave from three different types of corroded section. Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Outline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

14. Idealized perforation in cased borehole Considered models: Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.  Finite-length perforation (10 cm)  Zero-length perforation (break in casing) Outline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

15. Reflection of the tube wave from perforation with 10 cm length. Reflection of the tube wave from zero-length perforation. Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Idealized perforation in cased borehole Outline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

16. Limitations Low frequency approximation for tube-wave slowness (White J.E. 1984): Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Outline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

17. Relative error defined as: Relative error of 1D approach Considered model: Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Limitations Outline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

18. Finite-difference code 1D approach Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Reflection coefficientsOutline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

19. Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Finite-difference code 1D approach Reflection coefficientsOutline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

20. Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Finite-difference code 1D approach Reflection coefficientsOutline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

21.  Validated 1D approach for multi-layered media (cased boreholes) inhomogeneous borehole casing idealized perforations in cased borehole  Defined the limitations for 1D approach Conclusions Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Outline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

22. Thank you for attention! Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia. Outline Limitations Wavefield in cased borehole Results Conclusions 1D effective wavenumber approach Modeling approaches

23. References Outline Limitations Model 1 Model 2 Model 3 Conclusions 1D effective wavenumber approach Modeling approaches Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.  References  Bakulin, A., Gurevich, B., Ciz, R., and Ziatdinov S., 2005, Tube-wave reflection from a porous permeable layer with an idealized perforation: 75th Annual Meeting, Society of Exploration Geophysicists, Expanded Abstract, 332-335.  Krauklis, P. V., and A. P. Krauklis, 2005, Tube Wave Reflection and Transmission on the Fracture: 67th Meeting, EAGE, Expanded Abstracts, P217.  Medlin, W.L., Schmitt, D.P., 1994, Fracture diagnostics with tube-wave reflections logs: Journal of Petroleum Technology, March, 239-248.  Paige, R.W., L.R. Murray, and J.D.M. Roberts, 1995, Field applications of hydraulic impedance testing for fracture measurements: SPE Production and Facilities, February, 7-12.  Tang, X. M., and C. H. Cheng, 1993, Borehole Stoneley waves propagation across permeable structures: Geophysical Prospecting, 41, 165-187.  Tezuka, K., C.H. Cheng, and X.M. Tang, 1997, Modeling of low-frequency Stoneley- wave propagation in an irregular borehole: Geophysics, 62, 1047-1058.  White, J. E., 1983, Underground sound, Elsevier.  Winkler, K. W., H. Liu, and D.L. Johnson, 1989, Permeability and borehole Stoneley waves: Comparison between experiment and theory: Geophysics, 54, 66–75.

24. Formation parameters Longitudinal velocity (m/s) Shear velocity (m/s) Density (kg/m 3 ) Elastic half- spaces 350025003400 Fluid1500-1000 Casing 1 (steel) 600030007000 Casing 2 (plastic) 284014801200 Layer 1310018002600 Layer 2370024003000 Corroded section 1 12006001400 Corroded section 2 300015003500 Corroded section 3 420021004900 Outline Limitations Model 1 Model 2 Model 3 Conclusions 1D effective wavenumber approach Modeling approaches Tube-wave reflections in cased borehole AlexandrovDmitriy, StPSU, Saint-Petersburg, Russia.