摘要
A novel,highly efficient and accurate adaptive higher-order finite element method(hp-FEM)is used to simulate a multi-frequency resistivity loggingwhile-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 hpadaptivity 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(http://hpfem.org/hermes)with a multi-frequency resistivity LWD tool response in a borehole environment.
基金
The work for this paper was supported by the National Natural Science Foundation of China under Projects No.41074099。
