Fourier series expansion in a non-orthogonal system of coordinates for the simulation of 3D alternating current borehole resistivity measurements

We describe a method to simulate 3D resistivity borehole measurements acquired with alternating current (AC) logging instruments. It combines the use of a Fourier series expansion in a non-orthogonal system of coordinates with an existing 2D goal-oriented higher-order self-adaptive hp-finite element (FE) algorithm. The method is used to simulate measurements acquired with both wireline and logging-while-drilling (LWD) borehole logging instruments in deviated wells. It enables a considerable reduction of the computational complexity with respect to available 3D simulators, since the number of Fourier modes (basis functions) needed to solve practical applications is limited (typically, below 13). The fast convergence of the method is studied via numerical experimentation by simulating two wireline and one LWD instruments in a deviated well across a possibly invaded formation. Numerical results confirm the efficiency and reliability of the method for simulating challenging 3D AC resistivity borehole problems in deviated wells, while avoiding the expensive construction of optimal 3D grids. We accurately simulate challenging electrodynamic problems within a few seconds (or minutes) of CPU time per logging position. The method is especially well-suited for inversion of triaxial electromagnetic (EM) measurements, since we demonstrate that the number of Fourier modes needed for the exact representation of the material function is limited to only one (the central mode) for the case of borehole measurements acquired in deviated wells.