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

Abstract

Unlike the common situation for which vertical wells penetrate horizontal layers, the trajectory of high-angle wells is usually not aligned with the principal axes of elastic rock properties. Borehole sonic measurements acquired in high-angle wells in general do not exhibit axial symmetry in the vicinity of bed boundaries and thin layers, and 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. The problem is circumvented by implementing the eikonal equation based on the fast marching method to (1) calculate first-arrival times of borehole acoustic waveforms and (2) trace raypaths between sonic transmitters and receivers in high-angle wells. Furthermore, first-arrival times of P and S waves are calculated at different azimuthal receivers included in wireline borehole sonic instruments and are verified against waveforms obtained via 3D finite-difference time-domain simulations. Calculations of traveltimes, wavefronts, and raypaths for challenging synthetic examples with effects due to formation anisotropy and different inclination angles indicate 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 traveltimes 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.