In this dissertation, we develop time-domain numerical algorithms to model the electromagnetic responses of logging-while-drilling (LWD) tools in complex Earth formations. The increasing need to model complex features of realistic formations calls for the development of increasingly sophisticated and flexible numerical algorithms. The new algorithms proposed in this dissertation are based upon the finite-difference time-domain (FDTD) framework. FDTD is highly suited for the purposes because it is matrix-free and discretizes Maxwell's equations directly on a discrete grid of points, providing unparalleled flexibility in handing complex Earth media.
In this dissertation, we employ FDTD directly in three-dimensional cylindrical coordinates, which suppress the staircasing error incurred when discretizing the cylindrical tool geometry, while keeping the method matrix-free. Another limitation of FDTD is its conditionally stability, which limits the maximum time increment that can be used on the marching-on-time algorithm. This maximum time increment is set up the Courant criterion and is proportional to the spatial cell size used. The Courant criteria leads to oversampling in time whenever a very fine spatial discretization is required. This implies excessive computation times. As an alternative to FDTD, the alternating direction implicit (ADI)-FDTD method offers unconditionally stability with little extra computation effort, which includes the need to solve a tridiagonal system at each time-step.
In order to properly truncate the computational domain in the modeling of open-space problems, an absorbing boundary condition is also needed. Here, a convolutional perfectly-matched-layer (CPML) absorbing boundary condition based on a complex frequency shifted (CFS) stretching and recursive convolution is developed and implemented in the 3-D cylindrical ADI-FDTD algorithm. Because the time step size in the ADI-FDTD simulations cannot be increased arbitrarily due to numerical errors such as splitting errors and numerical (grid) dispersion, a complex-envelope (CE) technique is further incorporated into the ADI-FDTD algorithm.
The cylindrical CE-ADI-FDTD makes it possible to reduce the overall computation time while maintaining the dispersion error at reasonable levels.
In order to further reduce the computational time, this dissertation also develops a heterodyne approach for time-domain simulations in highly refined grids.
The proposed approach is based on a complex-envelope algorithm with a carrier frequency and a complex-valued FDTD algorithm with a shifted spectrum centered at a higher simulation frequency. This corresponds to a shorter period and faster convergence for narrowband simulations. In practice, the logging tool axis is frequently misaligned with the borehole axis due to gravitational pull effects and/or mechanical vibrations.
In addition, Earth media exhibits anisotropy which is represented by a full conductivity tensor during deviated drilling. The study of anisotropy and eccentricity is important for correct interpretation of logging data. A new locally-conformal (LC) FDTD based on deriving effective anisotropic tensor conductivities for partially-filled grid cells, composed of an isotropic conductive region representing the borehole and an anisotropic conductive region representing the Earth formation, is developed using a quasi-static approximation. The new LC-FDTD algorithm allows for employing coarser grids than conventional FDTD, and hence for obtaining faster simulation times, for a given required accuracy level.