Finite Difference Modelling Of Acoustic Logs In Vertically Heterogeneous Biot Solids

Abstract

This paper discusses the results of tests carried out on a finite difference formulation of Biot's equations for wave propagation in saturated porous media which vary in range and depth (Stephen, 1987). A technique for modeling acoustic logs in two dimensionally varying Biot solids will give insight into the behavior of tube waves at permeable fractures and fissures which intersect the borehole. The code agrees well with other finite difference codes and the discrete wavenumber code for small porosity in the elastic limit of Biot's equations. For large porosity (greater than one per cent) in the elastic limit or for the acoustic limit, good agreement is not obtained with the discrete wavenumber method for vertically homogeneous media. The agreement is worst for amplitudes of the pseudo-Rayleigh wave. The amplitude of the Stoneley wave and the phase velocities of both waves could be acceptable for some applications. An example is shown of propagation across a horizontal high porosity stringer in a Berea sandstone. Reflections from the stringer are observed but given the inaccuracies of the pseudo-Rayleigh waves for vertically heterogeneous media the amplitudes for the stringer model are questionable. We propose a three stage approach for further work: 1) Use the Virieux scheme instead of the Bhasavanija scheme for the finite difference template. The Virieux scheme has been shown in other studies to be more accurate for liquid-solid interfaces. 2) Run the present code for lower frequency sources to emphasize Stoneley waves and diminish pseudo-Rayleigh waves. Stoneley waves are most sensitive to permeability variations which are the primary objective of Biot wave studies. 3) Develop a finite difference code for Biot media with the fluid-solid boundary conditions specifically coded. This code would be suitable for studying constant radius boreholes in vertically varying Biot media.