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Simulations of 3D DC borehole resistivity measurements with a goal-oriented hp finite-element method. Part II: through-casing resistivity instruments

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Abstract

We simulate direct current (DC) borehole resistivity measurements acquired in steel-cased deviated wells for the assessment of rock formation properties. The assumed data acquisition configuration considers one current (emitter) and three voltage (collector) electrodes that are utilized to measure the second difference of the electric potential along the well trajectory. We assume a homogeneous, 1.27-cm-thick steel casing with resistivity equal to 10 − 5 Ω· m. Simulations are performed with two different numerical methodologies. The first one is based on transferring two-dimensional (2D) axisymmetric optimal grids to a three-dimensional (3D) simulation software. The second one automatically produces optimal 3D grids yielded by a 3D self-adaptive goal-oriented algorithm. Both methodologies utilize high-order finite elements (FE) that are specially well-suited for problems with high-contrast coefficients and rapid spatial variations of the electric field, as it occurs in simulations that involve steel-cased wells. The method based on transferring 2D-optimal grids is efficient in terms of CPU time (few seconds per logging position). Unfortunately, it may produce inaccurate 3D simulations in deviated wells, even though the error remains below 1% for the axisymmetric (vertical) well. The method based on optimal 3D grids, although less efficient in terms of CPU time (few hours per logging position), produces more accurate results that are validated by a built-in a posteriori error estimator. This paper provides the first existing simulations of through-casing resistivity measurements in deviated wells. Simulated resistivity measurements indicate that, for a 30° deviated well, measurements in conductive layers 0.01 Ω· m) are similar to those obtained in vertical wells. However, in resistive layers (10,000 Ω· m), we observe 100% larger readings in the 30° deviated well. This difference becomes 3,000% for the case of a 60° deviated well. For this highly-deviated well, readings corresponding to the conductive formation layer are about 30% smaller in magnitude than those in a vertical well. Shoulder effects significantly vary in deviated wells.

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References

  1. Demkowicz, L., Buffa, A.: H 1, H(curl), and H(div) conforming projection-based interpolation in three dimensions: quasi optimal p-interpolation estimates. Comput. Methods Appl. Mech. Eng. 194, 267–296 (2005)

    MATH  MathSciNet  Google Scholar 

  2. Druskin, V., Knizhnerman, L.: Gaussian spectral rules for the three-point second differences: I. A two-point postive definite problem in a semi-infinite domain. SIAM J. Numer. Anal. 37(2), 403–422 (1999)

    Article  MathSciNet  Google Scholar 

  3. Druskin, V., Knizhnerman, L.: Gaussian spectral rules for second order finite-difference schemes. Numer. Algorithms 25, 139–159 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kaufman, A.A.: The electrical field in a borehole with casing. Geophysics 55(1), 29–38 (1990)

    Article  Google Scholar 

  5. Kurtz, J., Demkowicz, L.: A fully automatic hp-adaptivity for elliptic PDEs in three dimensions. Comput. Methods Appl. Mech. Eng. 196(37–40), 3534–3545 (2007)

    Article  MathSciNet  Google Scholar 

  6. Pardo, D., Demkowicz, L.: Integration of hp-adaptivity with a two grid solver for elliptic problems. Comput. Methods Appl. Mech. Eng. 195(7–8), 674–710 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Pardo, D., Demkowicz, L., Torres-Verdin, C., Paszynski, M.: Simulation of resistivity logging-while-drilling (LWD) measurements using a self-adaptive goal-oriented hp-finite element method. SIAM J. Appl. Math. 66(6), 2085–2106 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Pardo, D., Demkowicz, L., Torres-Verdin, C., Tabarovsky, L.: A goal-oriented hp-adaptive finite element method with electromagnetic applications. Part I: electrostatics. Int. J. Numer. Methods Eng. 65(8), 1269–1309 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Pardo, D., Torres-Verdin, C., Demkowicz, L.: Simulation of multi-frequency borehole resistivity measurements through metal casing using a goal-oriented hp-finite element method. IEEE Trans. Geosci. Remote Sens. 44(8), 2125–2135 (2006)

    Article  MathSciNet  Google Scholar 

  10. Pardo, D., Torres-Verdin, C., Demkowicz, L.: Feasibility study for two-dimensional frequency dependent electromagnetic sensing through casing. Geophysics 72(3), F111–F118 (2007)

    Article  Google Scholar 

  11. Paszynski, M., Demkowicz, L., Pardo, D.: Verification of goal-oriented hp-adaptivity. Comput. Math. Appl. 50(8–9), 1395–1404 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Prudhomme, S., Oden, J.T.: On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comput. Methods Appl. Mech. Eng. 176(1–4), 313–331 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Schenkel, C.J., Morrison, H.F: Electrical resistivity measurement through metal casing. Geophysics 59(7), 1072–1082 (1994)

    Article  Google Scholar 

  14. Wu, X., Habashy, T.M.: Influence of steel casings on electromagnetic signals. Geophysics 59(3), 378–390 (1994)

    Article  Google Scholar 

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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, 83–89 (2008). https://doi.org/10.1007/s10596-007-9061-y

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  • DOI: https://doi.org/10.1007/s10596-007-9061-y

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