Skip to main content
SpringerLink
Log in

A quadrature-free method for simulation and inversion of 1.5D direct current (DC) borehole measurements

  • Original Paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

Resistivity inverse problems are routinely solved in order to characterize hydrocarbon bearing formations. They often require a large number of forward problems simulations. When considering a one dimensional (1D) planarly layered media, semi-analytical methods can be employed in order to solve a single forward problem in a fraction of a second. However, in some situations, a large number of (over one million) simulations is required, preventing this method to be used as a real time (logging) alternative. In this paper, we propose a novel semi-analytical method that dramatically reduces the total computational time, so it can be employed for real time inversion. In our proposed method, we select an ad hoc basis representation for the spectral solution such that its inverse Hankel transform can be computed analytically. The proposed method requires a pre-process that is expensive when compared with a single evaluation in classical semi-analytical methods. However, subsequent evaluations can be rapidly obtained, decreasing thus the total computational time by orders of magnitude when the number of required forward simulations is large.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Avdeev, D.B., Kuvshinov, A.V., Pankratov, O.V., Newman, G.A.: Three-dimensional induction logging problems, part I: an integral equation solution and model comparisons. Geophysics 67(2), 413–426 (2002)

    Article  Google Scholar 

  2. Chew, W.C., Chen, S.Y.: Response of a point source embedded in a layered medium. IEEE Antennas Wirel. Propag. Lett. 2(1), 254–258 (2003)

    Article  Google Scholar 

  3. Davydycheva, S., Druskin, V., Habashy, T., et al.: Finite-difference scheme for electromagnetic logging in 3D anisotropic media. In: 2004 SEG Annual Meeting. Society of Exploration Geophysicists (2004)

  4. Druskin, V.L., Knizhnerman, L., Lee, P.: New spectral Lanczos decomposition method for induction modeling in arbitrary 3-D geometry. Geophysics 64(3), 701–706 (1999)

    Article  Google Scholar 

  5. Dyatlov, G., Onegova, E., Dashevsky, Y.: Efficient 2.5 D electromagnetic modeling using boundary integral equations. Geophysics 80(3), E163–E173 (2015)

    Article  Google Scholar 

  6. Gajda-Zagórska, E., Schaefer, R., Smołka, M., Paszyński, M., Pardo, D.: A hybrid method for inversion of 3D DC resistivity logging measurements. Nat. Comput., 1–20 (2014)

  7. Ijasan, O., Torres-Verdín, C., Preeg, W. E.: Inversion-based petrophysical interpretation of logging-while-drilling nuclear and resistivity measurements. Geophysics 78(6), D473–D489 (2013)

    Article  Google Scholar 

  8. Kong, J.: Electromagnetic fields due to dipole antennas over stratified anisotropic media. Geophysics 37(6), 985–996 (1972)

    Article  Google Scholar 

  9. Løseth, L., Ursin, B.: Electromagnetic fields in planarly layered anisotropic media. Geophys. J. Int. 170 (1), 44–80 (2007)

    Article  Google Scholar 

  10. Merchant, G., Strickland, R.W., Jackson, C., et al.: Enhanced resolution LWD resistivity logs using a new inversion technique. In: SPWLA 37th Annual Logging Symposium. Society of Petrophysicists and Well-Log Analysts (1996)

  11. Meyer, W.H., et al.: Inversion of 2 MHz propagation resistivity logs in dipping thin beds. In: SPWLA 34th Annual Logging Symposium. Society of Petrophysicists and Well-Log Analysts (1993)

  12. Nam, M.J., Pardo, D., Torres-Verdín, C.: Simulation of dual-laterolog measurements in dipping, invaded and anisotropic formations using a Fourier series expansion in a non-orthogonal system of coordinates and a self-adaptive hp-finite element method. Geophysics 74(1) (2008)

  13. Nam, M.J., Pardo, D., Torres-Verdín, C.: Assessment of Delaware and Groningen effects on dual-laterolog measurements with a self-adaptive hp finite-element method. Geophysics 75(6), F143–F149 (2010)

    Article  Google Scholar 

  14. Newman, G.A., Alumbaugh, D.L.: Three-dimensional induction logging problems, Part 2: a finite-difference solution. Geophysics 67(2), 484–491 (2002)

    Article  Google Scholar 

  15. Papoulis, A.: Systems and Transforms with Applications in Optics. McGraw-Hill Series in System Science, vol. 1, p 1968. Krieger, Malabar (1968)

  16. Pardo, D., Paszynski, M., Collier, N., Alvarez, J., Dalcin, L., Calo, V.M.: A survey on direct solvers for galerkin methods. SeMA Journal 57(1), 107–134 (2012)

    Article  Google Scholar 

  17. Pardo, D., Paszynski, M., Torres-Verdín, C., Demkowicz, L.: Simulations of 3D DC borehole resistivity measurements with a goal-oriented hp finite-element method. Part I: laterolog and LWD. Journal of the Serbian Society for Computational Mechanics 1, 62–73 (2007)

    Google Scholar 

  18. Pardo, D., Torres-Verdín, C.: Fast 1D inversion of logging-while-drilling resistivity measurements for improved estimation of formation resistivity in high-angle and horizontal wells. Geophysics 80(2), E111–E124 (2015)

    Article  Google Scholar 

  19. Pardo, D., Torres-verdín, C., Nam, M., Paszynski, M., Calo, V.: Fourier series expansion in a non-orthogonal system of coordinates for the simulation of 3D alternating current borehole resistivity measurements. Comput. Methods Appl. Mech. Eng. 197(45), 3836–3849 (2008)

    Article  Google Scholar 

  20. Pardo, D., Torres-Verdín, C., Paszynski, M.: Simulations of 3D DC borehole resistivity measurements with a goal-oriented hp finite-element method. Part II: through-casing resistivity instruments. Comput. Geosci. 12 (1), 83–89 (2008)

    Article  Google Scholar 

  21. Sjödahl, P., Dahlin, T., Zhou, B.: 2.5 D resistivity modeling of embankment dams to assess influence from geometry and material properties. Geophysics 71(3), G107–G114 (2006)

    Article  Google Scholar 

  22. Streich, R., Becken, M.: Sensitivity of controlled-source electromagnetic fields in planarly layered media. Geophys. J. Int. 187(2), 705–728 (2011)

    Article  Google Scholar 

  23. Tai, C.T.: Dyadic Green functions in electromagnetic theory. Institute of Electrical & Electronics Engineers (IEEE) (1994)

  24. Wait, J.R.: The magnetic dipole over the horizontally stratified Earth. Can. J. Phys. 29(6), 577–592 (1951)

    Article  Google Scholar 

  25. Wang, T., Fang, S.: 3-D electromagnetic anisotropy modeling using finite differences. Geophysics 66(5), 1386–1398 (2001)

    Article  Google Scholar 

  26. Wang, T., Signorelli, J.: Finite-difference modeling of electromagnetic tool response for logging while drilling. Geophysics 69(1), 152–160 (2004)

    Article  Google Scholar 

  27. Zhang, J., Mackie, R.L., Madden, T.R.: 3-D resistivity forward modeling and inversion using conjugate gradients. Geophysics 60(5), 1313–1325 (1995)

    Article  Google Scholar 

  28. Zhong, L., Li, J., Bhardwaj, A., Shen, L.C., Liu, R.C.: Computation of triaxial induction logging tools in layered anisotropic dipping formations. IEEE Trans. Geosci. Remote Sens. 46(4), 1148–1163 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Facultad de Ingeniería, Pontificia Universidad Católica de Chile, Santiago, Chile

    Sergio Rojas

  2. Centro de Análisis Estocástico y Aplicaciones (ANESTOC), Pontificia Universidad Católica de Chile, Santiago, Chile

    Sergio Rojas

  3. Instituto de Matemáticas, Pontificia Universidad Catótica de Valparaíso, Valparaíso, Chile

    Ignacio Muga

  4. University of the Basque Country (UPV/ EHU), Leioa, Spain

    David Pardo

  5. Basque Center for Applied Mathematics (BCAM), Bilbao, Spain

    David Pardo

  6. IKERBASQUE, Bilbao, Spain

    David Pardo

Authors
  1. Sergio Rojas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ignacio Muga
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Pardo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergio Rojas.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rojas, S., Muga, I. & Pardo, D. A quadrature-free method for simulation and inversion of 1.5D direct current (DC) borehole measurements. Comput Geosci 20, 1301–1318 (2016). https://doi.org/10.1007/s10596-016-9592-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-016-9592-1

Keywords

Search

Navigation