An Improved High-Order Rotated Staggered Finite-Difference Algorithm for Simulating Elastic Waves in Heterogeneous Viscoelastic/Anisotropic Media

Abstract

Standard and rotated staggered grid algorithms, to eighth order in space and second order in time, have been developed for simulating seismic wave propagation in heterogeneous viscoelastic/anisotropic media. These algorithms have been validated by comparing their numerical results with analytical solutions for homogeneous models. Numerical simulation results show that high-order staggered finite-difference methods are better than staggered pseudo-spectral methods in modelling strong heterogeneous viscoelastic/anisotropic media. For strong velocity or density contrast interfaces, such as fluid-solid interfaces, staggered pseudo-spectral methods generate apparent artefacts although they give minimum spatial dispersion; while high-order staggered finite-difference methods, whether rotated-staggered or standard-staggered, give reasonable modelling results. This is especially true for fracture modelling, if the fracture is simulated with zero compressional and shear velocities and a very low density such as less than 10-8 kg.m-3.Based on the proposed algorithms, a uniform random function is used to establish random models with two kinds of components, i.e., shales or clays and pure sandstones. These two media consist of various synthetic shale sand formations to simulate random formations with various shale contents. Elastic wave velocities and attenuations for various shale contents are simulated properly at a frequency range around 10 kHz to 1 MHz. The numerical results show that our algorithms can be used for modelling composite media to study elastic wave attenuation and velocity variation for various rocks, including fracture modelling. The modelling analyses demonstrate that velocity models with shale contents from 30% to 60% generate strong back wave scattering for a wave length around 10 times the minimum size of the synthetic shale lumps. The velocity variations with respect to the shale contents evaluated from synthetic waveforms are in good agreement with the results predicted by Wyllie’s time-average equation at high frequencies.