Single-well imaging (SWI) is a borehole measurement technique aimed at detecting geologic interfaces or structures tens of meters away from the borehole. In SWI, acoustic signals generated by a monopole or a dipole source in the borehole are reflected by geologic interfaces and received by arrays of receivers of the same type. Current SWI cannot determine the azimuth angle of the geologic interfaces uniquely. We have developed a new SWI system with combined dipole and monopole receivers for a dipole source. This system can be used to determine the azimuth angle uniquely. The other difficult issue is to quickly simulate the wavefield of SWI to meet the needs of real-time data processing in field logging. We derive asymptotic solutions of the horizontal displacement and the fluid pressure for the reflected wave from the geologic interfaces in this nonaxisymmetric system. The radiation fields of a monopole and a dipole are computed using the steepest-descent integration method. The wave reflected from a reflector outside the borehole is proved to be equivalent to the radiation of a virtual source outside the borehole. We use the reciprocity relation between the virtual source and a monopole or a dipole source to obtain the asymptotic solutions of fluid pressure or horizontal displacement for the reflected wave. The asymptotic results agree well with those from 3D finite-difference simulation. The asymptotic solution demonstrates the polarity of fluid pressure changes when the azimuth angle changes by 180°. Based on the characteristics of asymptotic solutions, we evaluate a scheme to determine the azimuth uniquely.

SUMMARY: We formulate and implement a spectral element method to simulate elastic wave propagation in an arbitrary anisotropic background with discrete anisotropic fractures. The approach uses a general linear-slip condition to incorporate the anisotropic fractures into SEM, which allows for discontinuities of displacement fields across the surfaces of fractures. We treat the extremely thin fractures as geometry interfaces instead of meshing them, thus reducing computational cost. The results obtained by the proposed method agree well with the reference solutions for both a single horizontal anisotropic planar fracture and a tilted fracture. Based on the numerical simulation, we analyse the effects of anisotropic fracture on the wavefields. We find that due to the presence of off-diagonal element in the anisotropic fracture stiffness matrix, the wavefields generated by the anisotropic fracture are different from the isotropic fracture. The method can also simulate the wavefields in the anisotropic background including the multiple fractures or the intersecting fractures.

We propose a hybrid method based on the finite-difference time-domain (FDTD) algorithm and analytical methods to solve the elastic scattering of 3-D objects outside a borehole. In the proposed method, the FDTD algorithm containing the incident waves generated by a borehole acoustic source is first used to calculate the wave fields inside a rectangular volume enclosing the object. Then, the reciprocity theorem for the wave fields in the fluid/solid configurations is employed to transform the near fields around the object into the far fields inside the borehole. This method uses the regional domain which tightly encloses the object, thus avoiding the simulation for the whole domain and leading to high efficiency for the large-scale and multiscale problems containing both the borehole and far-field objects. The validations with both analytical solutions and independent numerical methods demonstrate the accuracy and efficiency of the hybrid method.

Single-well imaging (SWI) is a borehole measurement technique for detecting geologic structures outside a borehole by using arrays of receivers to record the waves reflected from these structures. The asymptotic solutions of P-P, SV-SV, and SH-SH waves in SWI have been obtained, but the P-SV and SV-P waves have been ignored in previous studies. It is necessary to know when these conversion waves are large and to estimate these waves when they are not ignorable. The analytical solutions of P-SV and SV-P waves are first derived using reciprocity relations between virtual concentrated forces and acoustic sources. The analytical results agree well with finite-difference solutions for monopole and dipole sources. We have found that, for a given source-receiver offset, the detected converted waves first increase and then decrease with the increase of the source-reflector distance. The relative amplitudes of converted waves to reflected waves are larger than 20% and cannot be ignored when the ratio of source-reflector distance to source-receiver offset is smaller than four. However, when the source-reflector distance is 10 or more times the source-receiver offset and the dip angle of the reflector is smaller than 10°, the amplitudes of the converted waves are smaller than 5% of the reflected waves. Furthermore, the relative amplitudes of converted waves increase with the dip angle of the reflector. The analytical solutions of converted waves are useful for simulating SWI wavefields and checking the inversion results in field logging.

We have used the spectral-element method (SEM) to simulate 3D acoustic wavefields in the fluid-filled borehole embedded in 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 compared with meshing each fracture directly. For the fracture-free case, we validate SEM through the comparison with the real-axis integration method for 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 the finite-difference method. Good agreement is achieved between the results from our 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 fractures. Through analyzing the synthetic data in the time and frequency domains, we discuss how to evaluate the properties of fractures intersected by a borehole.