A Finite Difference Formulation of Biot's Equations for Vertically Heterogeneous Full Waveform Acoustic Logging Problems

In this paper we present a finite difference formulation for Biot's equations for wave propagation in saturated porous media which vary in range and depth. One objective of well logging petroleum exploration holes is to determine the permeability of a section. There are indications that the Stoneley wave in the full waveform acoustic logging tool is sensitive to permeability, but we need modeling techniques to fully understand the mechanism. One question to be addressed is how narrow horizontal fissures of varying permeability affect Stoneley or tube wave propagation in the borehole. Another question is the trade-off between attenuation due to viscous losses in the pore fluid and due to scattering. A technique for modeling acoustic logs in two-dimensionally varying Biot solids will give insight into these problems