Effects of Tool Positions on Borehole Acoustic Measurements: a Stretched Grid Finite Difference Approach

Abstract

This dissertation made three contributions to numerical simulation and borehole acoustic logging. The first one is a novel finite difference time domain algorithm that features non- uniform grid, wavelet-based difference operator and anisotropic perfectly matched layer. This algorithm reduces numerical reflections and wave distortions introduced by grid change to a minimum by sampling the physical space with gradually varying mesh. By coordinate stretching, the algorithm discretizes the physical space with variable grid, while solving the wave equation on a uniform mesh. That approach helps retain the advantages pertaining to uniform mesh. Further improvement in efficiency is achieved without losing accuracy by the development of a wavelet-based difference operator. By using a family of compactly supported wavelet function, the wavelet- based finite difference time domain algorithm allows less grid point per wavelength. Coordinate stretching is also employed in deriving an anisotropic perfectly matched layer, superior to currently available perfectly matched layer formulation which requires field splitting, a process that results in more computer memory requirement for the storage of extra variables. Validations of the algorithm include comparison with analytical solutions, uniform grid FDTD solutions and discrete wavenumber results. The second contribution is a time domain investigation of wave propagations in the logging while drilling situation. Logging while drilling is an emerging downhole acoustic acquisition method. The investigation is focused on soft formations where formation shear velocity is slower than borehole fluid velocity, because shear velocity measurement, one of the key measurements that acoustic logging is designed to acquire, is the most problematic in soft formations. Special attention is paid to mode excitations, with respect to frequencies, tool positions and source types, in the hope to shed some light on some highly debated questions regarding tool design and data interpretation. The stretched grid finite difference algorithm is applied. The third contribution is the development of an inversion method to estimate stress magnitudes and directions from borehole acoustic measurements. It is predicted in theory that a crossover in flexural dispersion is an indicator of stress-induced anisotropy dominating over other sources of intrinsic anisotropy. The prediction is subsequently verified in a scaled-borehole experiment. We are the first ones that observe flexural dispersion crossover in field data. Using the flexural crossover as a stress signature on the borehole acoustic data, we are able to isolate stressed zones. The maximum horizontal stress direction coincides with the polarization direction of far field fast shear. The stress magnitude is related to velocity changes in the stressed state from the zero stress or hydrostatically balanced state, through a perturbation theory developed in the late 1990's. Stress directions estimated in this dissertation are consistent with focal mechanism and borehole breakout data present in the world stress map database.