Numerical Simulation of a Multi-Frequency Resistivity Logging-While-Drilling Tool Using a Highly Accurate and Adaptive Higher-Order Finite Element Method (Advances in Applied Mathematics and Mechanics, 2012)
A novel, highly efficient and accurate adaptive higher-order finite elementmethod ($hp$-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 expressionof Maxwell's equations, three kinds of boundary conditions, stability weak formulation ofMaxwell's equations, and automatic $hp$-adaptivity strategy. The new $hp$-FEM can selectoptimal refinement and calculation strategies based on the practical formation model anderror estimation. Numerical experiments show that the new $hp$-FEM has an exponentialconvergence rate in terms of relative error in a user-prescribed quantity of interestagainst the degrees of freedom, which provides more accurate results than those obtainedusing the adaptive $h$-FEM. The numerical results illustrate the high efficiency andaccuracy of the method at a given LWD tool structure and parameters in different physicalmodels, which further confirm the accuracy of the results using the Hermeslibrary (http://hpfem.org/hermes) with a multi-frequency resistivity LWD toolresponse in a borehole environment.