volume | PIER Journals

Abstract

We develop a new adaptive inversion procedure: Data-adaptive Resolution Inversion (DRI) method, which eliminates the need of selecting a parameterization prior to inversion. Instead, one performs a hierarchical search for the correct parameterization while solving a sequence of inverse problems with an increasing dimension of parameterization. A parsimonious approach to inverse problems usually involves the application of the same refinement consistently over the complete spatial domain. Such an approach may lead to over-parameterization, subsequently, to unrealistic conductivity estimates and excessive computational work. With DRI, the new parameterization at an arbitrary stage of inversion sequence is allocated such that new degrees of freedom are not necessarily introduced all over the spatial domain of the problem. The aim is to allocate new degrees of freedom only where it is warranted by the available data. Inversion results confirm that DRI is robust and efficient for multiparameter inversion of multicomponent borehole electromagnetic measurements.

1. Abubakar, A., T. M. Habashy, V. Druskin, S. Davydycheva, H. Wang, T. Barber, and L. Knizhnerman, "A three-dimensional parametric inversion of multi-component multi-spacing induction logging data," Society of Exploration Geophysicists International Exposition and 74th Annual Meeting, Denver, Colorado, 2004. Google Scholar

2. Alpak, F. O., T. M. Habashy, C. Torres-Verdín, and E. B. Dussan V., "Joint inversion of transient-pressure and time-lapse electromagnetic logging measurements," Petrophysics, Vol. 45, 251-267, 2004. Google Scholar

3. Anderson, B., T. Barber, and T. M. Habashy, "The interpretation and inversion of fully triaxial induction data," Transactions of 43rd Annual Logging Symposium: Society of Professional Well Logging Analysts, Paper O, Oiso, Japan, 2002. Google Scholar

4. Borcea, L., "Nonlinear multigrid for imaging electrical conductivity and permittivity at low frequency," Inverse Problems, Vol. 17, 329-359, 2001.
doi:10.1088/0266-5611/17/2/312 Google Scholar

5. Cheryauka, A. B. and M. S. Zhdanov, "Focusing inversion of tensor induction logging data in anisotropic formations and deviated well," Society of Exploration Geophysicists International Exposition and 71st Annual Meeting, San Antonio, Texas, 2001. Google Scholar

6. Druskin, V. and L. Knizhnerman, "Gaussian spectral rules for three-point second differences: I. A two-point positive definite problem in a semi-indefinite domain," SIAM J. Numer. Anal., Vol. 37, 403-422, 1999.
doi:10.1137/S0036142997330792 Google Scholar

7. George, B. K., C. Torres-Verdín, M. Delshad, R. Sigal, F. Zouioueche, and B. Anderson, "Assessment of in-situ hydrocarbon saturation in the presence of deep invasion and highly saline connate water," Petrophysics, Vol. 45, 141-156, 2003. Google Scholar

8. Goharian, M., M. Soleimani, and G. Moran, "A trust region subproblem for 3D electrical impedance tomography inverse problem using experimental data," Progress In Electromagnetics Research, Vol. 94, 19-32, 2009.
doi:10.2528/PIER09052003 Google Scholar

9. Grimstad, A.-A., T. Mannseth, G. Nævdal, and H. Urkedal, "Adaptive multiscale permeability estimation," Computational Geosciences, Vol. 7, 1-25, 2003.
doi:10.1023/A:1022417923824 Google Scholar

10. Hou, J., R. K. Mallan, and C. Torres-Verdín, "Finite-difference simulation of borehole EM measurements in 3D anisotropic media using coupled scalar-vector potentials," Geophysics, Vol. 71, G225-G233, 2006.
doi:10.1190/1.2245467 Google Scholar

11. Klein, J. D., P. R. Martin, and D. F. Allen, "The petrophysics of electrically anisotropic reservoirs," Transactions of 36th Annual Logging Symposium: Society of Professional Well Logging Analysts, Paper HH, Paris, France, 1995. Google Scholar

12. Kriegshäuser, B., O. Fanini, S. Forgang, G. Itskovich, M. Rabinovich, L. Tabarovsky, L. Yu, M. Epov, and J. van der Horst, "A new multicomponent induction logging tool to resolve anisotropic formations," Transactions of 40th Annual Logging Symposium: Society of Professional Well Logging Analysts, Paper D, Keystone, Colorado, 2000. Google Scholar

13. Kriegshäuser, B., S. McWilliams, O. Fanini, and L. Yu, "An efficient and accurate pseudo 2-D inversion scheme for multicomponent induction log data," Society of Exploration Geophysicists International Exposition and 71st Annual Meeting, San Antonio, Texas, 2001. Google Scholar

14. Kunz, K. S. and J. H. Moran, "Some effects of formation anisotropy on resistivity measurements in boreholes," Geophysics, Vol. 23, 770-794, 1958.
doi:10.1190/1.1438527 Google Scholar

15. Lu, X. and D. Alumbaugh, "One-dimensional inversion of three-component induction logging in anisotropic media," Society of Exploration Geophysicists International Exposition and 71st Annual Meeting, San Antonio, Texas, 2001. Google Scholar

16. Rosthal, R., T. Barber, S. Bonner, K. Chen, S. Davydycheva, G. Hazen, D. Homan, C. Kibbe, G. Minerbo, R. Schlein, L. Villegas, H. Wang, and F. Zhou, "Field test results of an experimental fully-triaxial induction logging tool," Society of Exploration Geophysicists International Exposition and 73rd Annual Meeting, Dallas, Texas, 2003. Google Scholar

17. Schlumberger, C., M. Schlumberger, and E. G. Leonardon, "Some observations concerning electrical measurements in anisotropic media and their interpretation," Transactions of the American Institute of Mining Engineers, Vol. 100, 159-182, 1934. Google Scholar

18. Tarantola, A., Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation, Elsevier, 1987.

19. Tikhonov, A. N. and V. Y. Arsenin, Solution of Ill-posed Problems, John Wiley, 1977.

20. Tompkins, M. J. and D. L. Alumbaugh, "A transversely anisotropic 1-D electromagnetic inversion scheme requiring minimal a priori information," Society of Exploration Geophysicists International Exposition and 72nd Annual Meeting, Salt Lake City, Utah, 2002. Google Scholar

21. Torres-Verdín, C., V. L. Druskin, S. Fang, L. A. Knizhnerman, and A. Malinverno, "A dual-grid nonlinear inversion technique with applications to the interpretation of dc resistivity data," Geophysics, Vol. 65, 1733-1745, 2000.
doi:10.1190/1.1444858 Google Scholar

22. Wang, H., T. Barber, R. Rosthal, J. Tabanou, B. Anderson, and T. Habashy, "Fast and rigorous inversion of triaxial induction logging data to determine formation resistivity anisotropy, bed boundary position, relative dip, and azimuth angles," Society of Exploration Geophysicists International Exposition and 73rd Annual Meeting, Dallas, Texas, 2003. Google Scholar

23. Wang, T. and S. Fang, "3-D electromagnetic anisotropy modeling using finite differences," Geophysics, Vol. 66, 1386-1398, 2001.
doi:10.1190/1.1486779 Google Scholar

24. Yee, K. S., "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Transactions on Antennas and Propagation, Vol. 14, 302-307, 1966.
doi:10.1109/TAP.1966.1138693 Google Scholar

25. Yu, L., B. Kriegshäuser, O. Fanini, and J. Xiao, "A fast inversion method for multicomponent induction log data," Society of Exploration Geophysicists International Exposition and 71st Annual Meeting, San Antonio, Texas, 2001. Google Scholar

26. Zhang, Z., L. Yu, B. Kriegshäuser, and R. Chunduru, "Simultaneous determination of relative angles and anisotropic resistivity using multicomponent induction logging data," Transactions of 42nd Annual Logging Symposium: Society of Professional Well Logging Analysts, Houston, Texas, 2001. Google Scholar

27. Zhang, Z. and A. Mezzatesta, "2D anisotropic inversion of multicomponent induction logging data," Society of Exploration Geophysicists International Exposition and 71st Annual Meeting, San Antonio, Texas, 2001. Google Scholar

28. Zhang, Z., L. Yu, B. Kriegshäuser, and L. Tabarovsky, "Determination of relative angles and anisotropic resistivity using multicomponent induction logging data," Geophysics, Vol. 69, 898-908, 2004.
doi:10.1190/1.1778233 Google Scholar

Dr. Faruk Omer Alpak

Dr. Carlos Torres-Verdin