We use the spectral element method (SEM) to simulate 3D acoustic wavefields in the fluid-filled borehole embedded in the fractured media. The fractures are characterized by the linear-slip model (LSM), which is incorporated into the surface integral of the SEM weak form, avoiding meshing individual fractures, thus reducing the degrees of freedom of the fractures comparing with meshing each fracture directly. For the fracture-free case, we validate SEM through the comparison with the real-axis integration (RAI) method for both monopole and dipole sources. For the case with a fracture, we compare the SEM-LSM solutions with the reference numerical solutions of a thin layer model using finite-difference method. Good agreement is achieved between the results from the proposed method and the reference finite-difference solutions. We find that the acoustic wavefields excited by a dipole source are more sensitive to the fractures than those by a monopole source. To show the ability of the approach to handle complex problems, we simulate the cases with a tilted fracture and multiple fractures. Based on the simulated results, we investigate the influence of the fracture parameters (e.g., stiffness, tilt angle, azimuth, thickness, number and spatial intervals of fractures) on the scattered wavefields. We find that the tilt angle has an obvious influence on the scattered waveforms and amplitudes. The results also demonstrate that the wavefields are quite sensitive to the number of fractures. The magnitudes of the horizontal-components transmitted wavefields decrease linearly with the number of the fractures. Through analyzing the synthetic data in time and frequency domains, we discuss how to evaluate the properties of fractures intersected by a borehole.
Mesh Update Techniques for Free-Surface Flow Solvers Using Spectral Element Method
