Low Frequency Modelling of Layered Media for Logging While Drilling Applications Using FDTD

摘要

In a typical Logging While Drilling (LWD) ap- plication, several coils operating in the frequency range of a few KHz to MHz are used as transmitters and receivers to appropriately characterize the earth formation. Electromagnetic modelling of such a low frequency system poses serious computational challenges. In the Method of Moment (MoM) formulation, contribution of vector potential to the total field becomes several orders of magnitude smaller than that of the scalar potential, thus making the resultant matrix highly ill-conditioned. Finite Difference Time Domain (FDTD) method, on the other hand, requires enormous number of time steps to capture the low frequency information. In this paper, we consider a layered-earth model and compute the electromagnetic field due to electric and magnetic dipoles embedded in the formation. To address the low frequency problem in FDTD, we consider the source and the receiver dipoles to be infinitesimally small and aligned with the computational grid, and we modify the update equations accordingly. This approach reduces the time convergence of FDTD by two-to-three orders of magnitude, and also reduces the memory requirements by the same factor. Numerical results for the fields reflected from the layered interfaces and the corresponding voltages induced in the receive coils are presented for multiple scenarios involving shale and sand zones.