A Priori Fourier Analysis for 2.5D Finite Elements Simulations of Logging-While-Drilling (LWD) Resistivity Measurements

Triaxial induction measurements provided by LWD tools generate crucial petrophysical data to determine several quantities of interest around the drilled formation under exploration, such as a map of resistivities. However, the corresponding forward modeling requires the simulation of a large-scale three-dimensional computational problem for each tool position. When the material properties are assumed to be homogeneous in one spatial direction, the problem dimensionality can be reduced to a so called 2.5 dimensional (2.5D) formulation.In this paper, we propose an a priori adaptive algorithm for properly selecting and interpolating Fourier modes in 2.5D simulations in order to speed up computer simulations. The proposed method first considers an adequate range of Fourier modes, and it then determines a subset of those which need to be estimated via solution of a Partial Differential Equation (PDE), while the remaining ones are simply interpolated in a logarithmic scale, without the need of solving any additional PDE. Numerical results validate our selection of Fourier modes, delivering superb results in real simulations when solving via PDE only for a very limited number of Fourier modes (below 50%).