Finite Difference Synthetic Acoustic Logs for Boreholes with Sharp, Rough Interfaces

Previous finite difference codes developed for this consortium were unstable at rough, sharp liquid-solid interfaces. In this paper we present results from a method developed by Nicoletis (1981) and Bhasavanija (1983) which is stable at rough, sharp liquid-solid interfaces. The method gives acceptable accuracy for the vertically homogeneous case when compared with discrete wavenumber results. Examples are also given for waShouts and horizontal fissures with sharp liquid-solid boundaries. Head waves tend to follow the shape of the borehole wall in the washout examples, but are greatly attenuated at fissures. In the fissure case, pseudoRayleigh waves are reflected from the fissure, but Stoneley waves are almost unaffected. INTRODUCTION This paper is the third in· a· series on finite difference synthetic acoustic logs. The first paper (Stephen et al., 1983) introduced a finite difference code for the acoustic logging problem and the second paper (Stephen and Pardo-Casas, 1984) demonstrated the applicability of the method to acoustic logging problems with vertically varying velocity and density profiles. The code used in these papers was stable and accurate at flat (vertical), sharp interfaces and at two-dimensionally varying interfaces where the transition in elastic parameters was smooth. In the first case boundary conditions were specifically coded (the boundary condition method) and in the second case solutions were obtained directly from the elastic wave equation for heterogeneous media (the Stephen method). It is too inconvenient to specifically code boundary conditions for non-planar, or even piecewise planar, interfaces. At sharp interfaces, where the elastic parameters varied from liquid to solid over a few grid points, the finite difference formulation of the wave equation for heterogeneous media was unstable. In this report we show the results of a scheme developed by Nicoletis (1981) and Bhasavanija (1983) which is stable for rough, sharp interfaces and has acceptable accuracy. Bhasavanija (1983) applied the method to the acoustic logging problem by considering examples consisting of homogeneous blocks. He applied his code for heterogeneous media to the edges and corners of the blocks, and used the traditional code for homogeneous media inside the blocks. (The Bhasavanija code reduces to the traditional code in homogeneous media). At the borehole wall he used a code which specifically included the liquid-solid boundary conditions to first order in 101