An efficient semianalytical finite element scheme developed in the Hamiltonian system and its applications in well loggings

Abstract

An efficient semianalytical finite element method is proposed to simulate electromagnetic wave propagation in layered media. A piecewise homogeneous structure is divided into several layers. Each layer is uniform in the longitudinal direction, and the distributions of geometry and material can be arbitrary on the transverse plane, or cross-section of the layer. To develop this semianalytical finite element scheme, the standard functional corresponding to the vector wave equation is cast to a new form in the Hamiltonian system based on dual variables, which are transverse components of both electric and magnetic fields on the cross-section of the layer. 2D finite elements are employed to discretize the cross-section, and a high precision integration scheme based on the Riccati equations is used to exploit the longitudinal homogeneity in the layer [1]. By transforming a 3D layered problem into a series of 2D problems, this semianalytical finite element method can save a great amount of computational costs and meanwhile achieve a much higher level of accuracy compared with conventional finite element schemes. The flexibility of this semianalytical method can be greatly increased by hybridization with conventional finite elements, and this strategy works well for layered structures with local inhomogeneities such as borehole washouts. Examples will be given to show the applications of this method in borehole resistivity well logging as well as electromagnetic telemetry.