Inst. d'Electron. et de Microelectron. du Nord, UMR CNRS 9929, Dept. ISEN, 41 boulevard Vauban, 59046 Lille Cedex, France
In the context of petroleum acoustics, it is of great interest to modelize the radiation of the piezoelectric transducer in a borehole surrounded by an homogeneous isotropic elastic formation of infinite extent without restrictive assumptions on the geometry, radiation pattern, or types of waves. The finite element method is well suited to solve such problems if the troncature of the infinite formation is correctly treated. This troncature generates ingoing waves which normally do not exist in infinite domain. In this paper, classical finite elements (atila code) are used to model in steady state the transducer, the fluid-filled borehole, and part of the formation inside a spherical boundary. On this exterior boundary, impedance elements are used to take into account the infinite character of the formation. These elements are obtained by discretizing the mechanical impedance of outgoing spherical P and S waves. The method is validated by studying two configurations having analytical solutions: the pulsating sphere and the oscillating point. Results include displacement fields and radiation patterns for P or/and S waves. Finally, the radiation of a cylindrical piezoelectric transducer in an oil-filled borehole [S. Kostek et al., J. Acoust. Soc. Am. 95, 109 (1994)] is analyzed. P and S components of the displacement field, electrical admittance of the transducer, directivity patterns, and distribution of radiated energy are displayed.