Flexural waves in pressurized borehole: A finite‐difference approach

Abstract

The propagation of elastic waves in a fluid-filled borehole is adequately described by the linear equations of elasticity. However, when the borehole fluid is pressurized either due to the hydrostatic head at a given depth or with the aid of packers at the wellhead, both the fluid and the surrounding formation are subjected to biasing stresses. In this situation, wave propagation along the borehole is described by the equations of motion for small dynamic fields superposed on a static bias. The effective elastic coefficients in these equations account for the radial decay of biasing stresses in the surrounding formation. We present a velocity-stress finite-difference formulation that can readily account for such radially varying coefficients in the equations of motion. Computational results have been obtained for the synthetic dipole waveforms at an array of receivers in a fluid-filled borehole before and after pressurization. The corresponding flexural dispersions are obtained by processing the waveforms with the Prony’s method. The processed dispersion from the waveforms obtained before pressurization is compared with the flexural dispersion obtained from a modal search routine. The flexural dispersion in the presence of borehole pressurization is compared with the results obtained by a previously reported perturbation method. Good agreement has been obtained in both cases. This study shows that while the borehole pressure-induced changes in the flexural dispersions are negligibly small at low frequencies, they can be significantly large at higher frequencies when the flexural wave becomes more confined to the borehole. This change in the flexural dispersion due to an increase in borehole pressurization is a measure of the acoustoelastic coefficient of the formation that can provide additional information about the mechanical properties of the formation.