A near-optimal quadrature for 2.5D EM logging-while-drilling tool modeling

In this paper, we present a novel near-optimal quadrature (NOQ) for 2.5-dimensional (2.5D) electromagnetic (EM) logging-while-drilling (LWD) tool modeling. We first briefly introduce the 2.5D finite difference scheme in anisotropic media. This leads to the inverse Fourier transform (IFT), i.e., integral along the k(y) plane, to derive the electric components. Then, we propose the near-optimal quadrature derived from an optimized integration path in the complex plane to implement the IFT procedure. We have proved that the selection of the integration nodes and weights is equivalent to finding the rational approximation of 1/root s, which is defined from i/k(y). Both analytic and 3D solutions have verified the effectiveness and efficiency of the proposed algorithm. Specifically, we show that the number of quadrature points is significantly reduced using the proposed method. Compared to the Gauss-Hermite quadrature, the NOQ can save 75% of the computation time. Therefore, the efficiency of the 2.5D modeling is significantly improved.