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Numerical Simulation of a Multi-Frequency Resistivity Logging-While-Drilling Tool Using a Highly Accurate and Adaptive Higher-Order Finite Element Method

  • Zhonghua Ma (a1) (a2), Dejun Liu (a1), Hui Li (a1) and Xinsheng Gao (a1)


A novel, highly efficient and accurate adaptive higher-order finite element method (hp-FEM) is used to simulate a multi-frequency resistivity logging-while-drilling (LWD) tool response in a borehole environment. Presented in this study are the vector expression of Maxwell’s equations, three kinds of boundary conditions, stability weak formulation of Maxwell’s equations, and automatic hp-adaptivity strategy. The new hp-FEM can select optimal refinement and calculation strategies based on the practical formation model and error estimation. Numerical experiments show that the new hp-FEM has an exponential convergence rate in terms of relative error in a user-prescribed quantity of interest against the degrees of freedom, which provides more accurate results than those obtained using the adaptive h-FEM. The numerical results illustrate the high efficiency and accuracy of the method at a given LWD tool structure and parameters in different physical models, which further confirm the accuracy of the results using the Hermes library ( with a multi-frequency resistivity LWD tool response in a borehole environment.


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[1]Chimedsurong, Z. and Wang, H., Forward modeling of induction well logging tools in dipping boreholes and their response, Chinese. J. Comput. Phys., 20 (2003), pp. 161168.
[2]Chen, X., Liu, D. and Ma, Z., Numerical simulation of electric field in resistivity LWD using High accuracy self-adaptive hp-FEM, Chinese. J. Comput. Phys., 28 (2011), pp. 5056.
[3]Pardo, D., Two-dimensional high accuracy simulation of resistivity logging while drilling (LWD) measurements using a self adaptive goal oriented finite element method, SIAM. J. Appl. Math., 66 (2006), pp. 20852106.
[4]Lee, H. O., Cylindrical FDTD Analysis of LWD Tools Through Anisotropic Dipping Layered Earth Media, M.S. Thesis, The Ohio State University, 2005.
[5]Tan, M., Gao, J., Wang, X. and Zhang, S., Numerical simulation of the dual laterolog for carbonate cave reservoirs and response characteristics, Appl. Geophys., 8 (2011), pp. 7985.
[6]Lovell, J. R., Finite Element Methods in Resistivity Logging, Ph.D Thesis, Delft University of Technology, 1993.
[7]Chen, Q., Pardo, D., Li, H. and Wang, F., New post-processing method for interpretation of through casing resistivity (TCR) measurements, J. Appl. Geophys., 74 (2011), pp. 1925.
[8]Dubcova, L., Solin, P., Cerveny, J. and Kus, P., Space and time adaptive two-mesh hp-finite element method for transient microwave heating problems, Electromagnetics., 30 (2010), pp. 2340.
[9]Vejchodsky, T., Solin, P. and Zitka, M., Modular hp-FEM system HERMES and its application to the Maxwell’s equations, Math. Comput. Simul., 76 (2007), pp. 223228.
[10]Solin, P., Segeth, K. and Dolezel, I., Higher-Order Finite Element Methods, Chapman & Hall/CRC Press, Philadelphia, 2002.
[11]Demkowicz, L., Computing with hp-Adaptive Finite Elements: One and Two Dimensional Elliptic and Maxwell Problems, Chapman & Hall/CRC Press, Boca Raton, 2006.
[12]Solin, P., Cerveny, J. and Dolezel, I., Arbitrary-level hanging nodes and automatic adap-tively in the hp-FEM, Math. Comput. Simul., 77 (2008), pp. 117132.
[13]Pardo, D., Demkowicz, L., Torres-Verdin, C. and Paszynski, M., A self-adaptive goal-oriented hp finite element method with electromagnetic applications, part II: electrodynamics, Comput. Methods. Appl. Mech. Eng., 196 (2007), pp. 35853597.
[14]Solin, P., Cerveny, J., Dubcova, L. and Andrs, D., Monolithic discretization of linear ther-moelasticity problems via adaptive multimesh hp-FEM, J. Comput. Appl. Math., 234 (2010), pp. 23502357.
[15]Solin, P., Dubcova, L., Cerveny, J. and Dolezel, I., Adaptive hp-FEM with arbitrary-level hanging nodes for Maxwell’s equations, Adv. Appl. Math. Mech., 2 (2010), pp. 518532.



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