The workflow for modeling sonic logging response of boreholes in anisotropic formations subject to the formation stress is presented. It enables computing the dispersion curves of any borehole mode, including the flexural and quadrupole ones. The key steps are determining static reference stress-induced state of the formation, finding effective elastic moduli tensor in the framework of the third order elasticity theory, and computing the dispersion curves of borehole modes using semi-analytical technique. The workflow is exemplified by computing the effect of the formation stress on the dispersion curves of the flexural modes of boreholes for Berea sandstone. Developed methodology can be used to gain new insights into formation stress estimation from sonic logging measurement. In addition, it presents opportunities for developing advanced computationally efficient processing algorithms.
The anisotropy measurement from the dispersion analysis of sonic logging data has proven to be a vital source of information for the subsurface stress field (Donald et al., 2013). It allows estimating both the direction and the magnitude of the stresses. To analyze the possibilities and limitations of this measurement, an efficient numerical procedure is required. Such modeling involves solving both the static mechanical problem to determine the prestressed state of the formation and finding the characteristics of elastic wave propagation in the borehole, for example, flexural waves. Direct 3D simulation, although possible, is quite computationally demanding. The situation is further complicated by the fact that the subsurface stress results into inhomogeneous distribution of the effective elastic moduli tensor and lowers its symmetry. For example, its crystal system can become orthorhombic or even a less symmetric one. The alternative procedure employs the semi-analytical finite element method (SAFE) to compute the spectrum of the borehole modes (Ellefsen et al., 1991; Zharnikov and Syresin 2015; Fang et al., 2015). Using SAFE method is the key step, which results in significant speed up and improved accuracy. Among its advantages is the ability to handle reliably and in uniform manner arbitrary inhomogeneities in the plane orthogonal to the borehole axis and arbitrary anisotropy. All steps of the procedure can be implemented using standard algorithms (either open-source or commercial ones). Separate steps of this workflow were already reported. For example, the methodology to compute the reference static stressed state was demonstrated by (Gaede et al., 2012); the technique to compute the dispersion curves of boreholes modes in anisotropic medium was proposed by (Ellefsen et al., 1991); Fang et al. modeled acoustic response of the borehole under stress adopting Mavko's approach to the computation of the deformed reference state (Fang et al., 2015). One of the important differences of our work from that of Fang et al. is in using laboratory measured thirdorder elastic constants. Present work describes the approach and the workflow and exemplifies its capabilities by modeling and analyzing the results of anisotropy measurements for several cases.