A Numerical Mode-Matching Method Based on Multiple-Knot Cubic B-Splines Applied to the Analysis of Well-Logging Tools

José Ricardo Bergmann, Maiquel S. Canabarro, Fernando L. Teixeira

Abstract


The numerical mode-matching (NMM) technique is a very efficient approach to analyze well-logging tools used for hydrocarbon exploration. This problem is typically modeled as a cylindrically layered medium (subterranean formation) including both horizontal and vertical stratifications. In this paper, we present a new NMM formulation based on the use of B-splines expansion functions with variable knot multiplicity to represent the dependency of fields on the vertical direction, and cylindrical Bessel functions to represent the fields along the radial (horizontal) direction. We show that this type of expansion allows to better capture the spatial variation of the electromagnetic field near vertical layer interfaces. Illustrative examples elucidate the advantage of knot multiplicity in the modeling of earth formations with anisotropic and lossy layers. A case study on compensated dual resistivity (CDR) well-logging tool sensors is also presented based on the proposed method.


Keywords


Numerical mode-matching, cubic B-splines, well-logging tools, compensated dual resistivity

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References


REFERENCES

S. Y. Chen, W. C. Chew, V. R. N. Santos, K. Sainath, and F. L. Teixeira, "Electromagnetic subsurface remote sensing,`` in: Wiley Encyclopedia of Electrical and Electronics Engineering, John Wiley & Sons, New York, 2016.

Y.-K. Hue, F. L. Teixeira, L. San Martin, and M. Bittar, "Three-dimensional simulation of eccentric LWD tool response in boreholes through dipping formations,' IEEE Trans. Geosci. Remote Sens., vol. 43, no. 2, pp. 257-268, 2005.

H.O. Lee and F. L. Teixeira, "Cylindrical FDTD analysis of LWD tools through anisotropic dipping layered earth media,' IEEE Trans. Geosci. Remote Sens., vol. 45, no. 2, pp. 383-388, 2007.

D. Wu, C. R. Liu, and J. Chen, "An efficient FDTD method for axially

symmetric LWD environments,`` IEEE Trans. Geosci. Remote Sens.,

vol. 46, no. 6, pp. 1652–1656, 2008.

H. O. Lee, F. L. Teixeira, L. E. San Martin, and M. S. Bittar, `Numerical modeling of eccentered LWD borehole sensors in dipping and fully anisotropic Earth formations,' IEEE Trans. Geosci. Remote Sens., vol. 50, no. 3, pp. 727-735, 2012.

M. S. Novo, L. C. Silva, and F. L. Teixeira, `Three-dimensional finite-volume analysis of directional resistivity logging sensors,' IEEE Trans. Geosci. Remote Sens., vol. 48, no. 3, pp. 1151-1158, 2010.

M. S. Novo, L. C. da Silva and F. L. Teixeira, "Finite volume modeling of borehole electromagnetic logging in 3-D anisotropic formations using coupled scalar-vector potentials,`` IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 549-552, 2007.

J. R. Lovell and W. C. Chew, "Response of a point-source in a multicylindrically layered medium,`` IEEE Trans. Geosci. Remote Sens., vol. 25, no. 6, pp. 850–858, 1987.

J. R. Lovell and W. C. Chew, "Effect of tool eccentricity on some electrical well-logging tools,`` IEEE Trans. Geosci. Remote Sens., vol. 28, no. 1, pp. 127–136, 1990.

Z.-Q. Zhang and Q.-H. Liu, "Applications of the BCGS-FFT method to

-D induction well-logging problems,`` IEEE Trans. Geosci. Remote

Sens., vol. 41, no. 5, pp. 998–1004, 2003.

Y.-K. Hue and F. L. Teixeira, "Analysis of tilted-coil eccentric borehole antennas in cylindrical multilayered formations for well-logging applications,' IEEE Trans. Antennas Propagat., vol. 54, no. 4, pp. 1058-1064, 2006.

G.-S. Liu, F. L. Teixeira, and G.-J. Zhang, `Analysis of directional logging tools in anisotropic and multieccentric cylindrically-layered Earth formations,' IEEE Trans. Antennas Propagat., vol. 60, no. 1, pp. 318-327, 2012.

H. Moon, B. Donderici, and F. L. Teixeira, `Stable evaluation of Green's functions in cylindrically stratified regions with uniaxial anisotropic layers,' J. Comp. Phys., vol. 325, pp. 174-200, 2016.

G.-X. Fan, Q. H. Liu, and S. P. Blanchard, "3-D numerical mode-matching (NMM) method for resistivity well-logging tools,`` IEEE Trans. Geosci. Remote Sens., vol. 38, no. 10, pp. 1544–1552, 2000.

Y.-K. Hue and F. L. Teixeira, 'Numerical mode-matching method for tilted coil antennas in cylindrically layered anisotropic media with multiple horizontal beds,' IEEE Trans. Geosci. Remote Sens., vol. 45, no. 8, pp. 2451-2462, 2007.

W. C. Chew, Waves and Fields in Inhomogeneous Media. New York, NY, USA: John Wiley & Sons, 1995.

J. Li and L. C. Shen, "Vertical eigenstate method for simulation of induction and MWD resistivity sensors,`` IEEE Trans. Geosci. Remote Sens., vol. 31, no. 2, pp. 399-406, Mar. 1993.

E. Cavanagh and B. Cook, "Numerical evaluation of Hankel transforms via Gaussian-Laguerre polynomial expansions,`` IEEE Trans. Acoustics, Speech, Signal Proc., vol. 27, no. 4, pp. 361-366, Aug. 1979.

D. M. Pai, "Induction log modeling using vertical eigenstates,`` IEEE Trans. Geosci. Remote Sens., vol. 29, no. 2, pp. 209-213, Mar. 1991.

H. Yang, W. Yue, Y. He, H. Huang, and H. Xia, "The deduction of coefficient matrix for cubic non-uniform B-Spline curves,`` in 1st Int. Workshop Edu. Technol. Comp. Sci., Hubei, China, pp. 607-609, 2009.

R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York, NY, USA: McGraw-Hill, 1961.

B. Clark., M. G. Luling, J. Jundt, M. O. Ross, and D. L. Best, "A dual depth resistivity measurement for FEWD,`` in SPWLA 29th Annual Logging Symp., paper SPWLA-1988-A, San Antonio, TX, USA, 5-8 June, 1988.

B. Clark., D. F. Allen, D. L. Best, S. D. Bonner, J. Jundt, M. G. Luling, and M. O. Ross, "Electromagnetic propagation logging while drilling: Theory and experiment,`` SPE Format. Eval., vol. 5, no. 3, pp. 263-271, 1990.

F. L. Teixeira and W. C. Chew, "Finite-difference computation of transient electromagnetic waves for cylindrical geometries in complex media,`` IEEE Trans. Geosci. Remote Sens., vol. 38, no. 4, pp. 1530-1543, 2000.




DOI: http://dx.doi.org/10.1590/2179-10742017v16i3895

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