The Parameter Averaging Technique in Finite-Difference Modeling of Elastic Waves in Combined Structures with Solid, Fluid and Porous Subregions

To finite-difference model elastic wave propagation in a combined structure with solid, fluid and porous subregions, a set of modified Biot's equations are used, which can be reduced to the governing equations in solids, fluids as well as fluid-saturated porous media. Based on the modified Biot's equations, the field quantities are finite-difference discretized into unified forms in the whole structure, including those on any interface between the solid, fluid and porous subregions. For the discrete equations on interfaces, however, the harmonic mean of shear modulus and the arithmetic mean of the other parameters on both sides of the interfaces are used. These parameter averaging equations are validated by deriving from the continuity conditions on the interfaces. As an example of using the parameter averaging technique, a 2-D finite-difference scheme with a velocity-stress staggered grid in cylindrical coordinates is implemented to simulate the acoustic logs in porous formations. The finite-difference simulations of the acoustic logging in a homogeneous formation agree well with those obtained by the analytical method. The acoustic logs with mud cakes clinging to the borehole well are simulated for investigating the effect of mud cake on the acoustic logs. The acoustic logs with a varying radius borehole embedded in a horizontally stratified formation are also simulated by using the proposed finite-difference scheme.