The Parameter Averaging Technique in Finite-Difference Modeling of Elastic Waves in Combined Structures with Solid, Fluid and Porous Subregions (Communications in Computational Physics, 2011)
To finite-difference model elastic wave propagation in a combined structurewith 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-saturatedporous media. Based on the modified Biot's equations, the field quantitiesare finite-difference discretized into unified forms in the whole structure, includingthose on any interface between the solid, fluid and porous subregions. For the discreteequations on interfaces, however, the harmonic mean of shear modulus and thearithmetic mean of the other parameters on both sides of the interfaces are used. Theseparameter averaging equations are validated by deriving from the continuity conditionson the interfaces. As an example of using the parameter averaging technique, a2-D finite-difference scheme with a velocity-stress staggered grid in cylindrical coordinatesis implemented to simulate the acoustic logs in porous formations. The finite differencesimulations of the acoustic logging in a homogeneous formation agree wellwith those obtained by the analytical method. The acoustic logs with mud cakes clingingto the borehole well are simulated for investigating the effect of mud cake on theacoustic logs. The acoustic logs with a varying radius borehole embedded in a horizontallystratified formation are also simulated by using the proposed finite-differencescheme.