Massachusetts Institute of Technology. Earth Resources Laboratory
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.Massachusetts Institute of Technology. Full Waveform Acoustic Logging Consortiu
