Numerical simulation of borehole acoustic logging in the frequency and time domains with hp-adaptive finite elements

Accurate numerical simulation of borehole acoustic measurements is of great relevance to improving the efficacy of acoustic logging techniques and to computationally estimating elastic formation properties. Such simulations require sound physical modeling combined with accurate and efficient numerical discretization and solution techniques. The objective of this paper is to concomitantly model acoustic wave propagation in a fluid-filled borehole coupled with elastic wave propagation both in the probed rock formation and in the elastic logging tool. To ensure the accuracy and efficiency of our simulations, we use a self-adaptive finite-element discretization method enhanced with Perfectly-Matched-Layer spatial-domain truncation. This work constitutes the first application of automatic hp-adaptivity to a coupled multi-physics problem, which requires the non-trivial capability of propagating refinements between acoustics and elasticity subdomains through their common interface. Computations are carried out in the frequency domain. Subsequently, using an inverse Fourier transform, frequency-domain solutions are transformed into the time domain to obtain waveforms at the receiver positions. Numerical results are presented for monopole and dipole sources with and without the presence of the logging tool, and for a layered formation. To validate our method, we compare our results to published reference data and to results obtained using an in-house finite-difference code. Convergence to a user-specified tolerance for the discretization error confirms the accuracy delivered by our method in the presence of complex geometrical and physical conditions and indicates its potential for the simulation of borehole acoustic measurements.