Finite-element solution to multidimensional multisource electromagnetic problems in the frequency domain using non-conforming meshes

We propose an approach to solving multisource induction logging problems in multidimensional media. According to the type of induction logging tools, the measurements are performed in the frequency range of 10 kHz to 14 MHz, transmitter-receiver offsets vary in the range of 0.5-8 m or more, and the trajectory length is up to 1 km. For calculating the total field, the primary-secondary field approach is used. The secondary field is calculated with the use of the finite-element method (FEM), irregular non-conforming meshes with local refinements and a direct solver. The approach to constructing basis functions with the continuous tangential components (from H-curl(Omega)) on the non-conforming meshes from the standard shape vector functions is developed. On the basis of this method, the algorithm of generating global matrices and a vector of the finite-element equation system is proposed. We also propose the method of grouping the logging tool positions, which makes it possible to significantly increase the computational effectiveness. This is achieved due to the compromise between the possibility of using the 1-D background medium, which is very similar to the investigated multidimensional medium for a small group, and the decrease in the number of the finite-element matrix factorizations with the increasing number of tool positions in one group. For calculating the primary field, we propose the method based on the use of FEM. This method is highly effective when the 1-D field is required to be calculated at a great number of points. The use of this method significantly increases the effectiveness of the primary-secondary field approach. The proposed approach makes it possible to perform modelling both in the 2.5-D case (i.e. without taking into account a borehole and/or invasion zone effect) and the 3-D case (i.e. for models with a borehole and invasion zone). The accuracy of numerical results obtained with the use of the proposed approach is compared with the one obtained by other codes for 1-D and 3-D anisotropic models. The results of this comparison lend support to the validity of our code. We also present the numerical results proving greater effectiveness of the finite-element approach proposed for calculating the 1-D field in comparison with the known codes implementing the semi-analytical methods for the case in which the field is calculated at a large number of points. Additionally, we present the numerical results which confirm the accuracy advantages of the automatic choice of a background medium for calculating the 1-D field as well as the results of 2.5-D modelling for a geoelectrical model with anisotropic layers, a fault and long tool-movement trajectory with the varying dip angle.