Numerical simulation and interpretation of sonic arrival times in high-angle wells using the eikonal equation

Borehole sonic measurements acquired in high-angle wells in general do not exhibit axial symmetry in the vicinity of bed boundaries and thin layers, while sonic waveforms remain strongly affected by the corresponding contrast in elastic properties across bed boundaries. The latter conditions often demand sophisticated and time-consuming numerical modeling to reliably interpret borehole sonic measurements into rock elastic properties. We circumvent this problem by implementing the eikonal equation based on the fast-marching method to (a) calculate first-arrival times of borehole acoustic waveforms, and (b) trace ray paths between sonic transmitters and receivers in high-angle wells. Furthermore, first-arrival times of compressional and shear waves are calculated at different azimuthal receivers included in wireline borehole sonic instruments and are verified against waveforms obtained via three-dimensional (3D) finite-difference time-domain simulations (3D-FDTD). Calculations of travel times, wavefronts, and ray paths for challenging synthetic examples with effects due to formation anisotropy and different inclination angles show a transition from a head wave to a boundary-induced refracted wave as the borehole sonic instrument moves across bed boundaries. Apparent slownesses obtained from first-arrival times at receivers can be faster or slower than the actual slownesses of rock formations surrounding the borehole, depending on formation dip, azimuth, anisotropy, and bed boundaries. Differences in apparent acoustic slownesses measured by adjacent azimuthal receivers reflect the behavior of wave propagation within the borehole and across bed boundaries and can be used to estimate bed-boundary orientation and anisotropy. The high-frequency approximation of travel times obtained with the eikonal equation saves more than 99% of calculation time with acceptable numerical errors, with respect to rigorous time-domain numerical simulation of the wave equation, and is therefore amenable to inversion-based measurement interpretation. Apparent slownesses extracted from acoustic arrival times suggest a potential method for estimating formation elastic properties and inferring boundary geometries