Scattering attenuation in short wavelengths has long been interesting to geophysicists.Ultrasonic coda waves,observed as the tail portion of ultrasonic wavetrains in laboratory ultrasonic measurements,are important for such studies where ultrasonic waves interact with smallscale random heterogeneities on a scale of micrometers,but often ignored as noises because of the contamination of boundary reflections from the side ends of a sample core.Numerical simulations with accurate absorbing boundary can provide insight into the effect of boundary reflections on coda waves in laboratory experiments.The simulation of wave propagation in digital and heterogeneous porous cores really challenges numerical techniques by digital image of poroelastic properties,numerical dispersion at high frequency and strong heterogeneity,and accurate absorbing boundary schemes at grazing incidence.To overcome these difficulties,we present a staggered-grid high-order finite-difference(FD) method of Biot's poroelastic equations,with an arbitrary even-order(2L) accuracy to simulate ultrasonic wave propagation in digital porous cores with strong heterogeneity.An unsplit convolutional perfectly matched layer(CPML) absorbing boundary,which improves conventional PML methods at grazing incidence with less memory and bettercomputational efficiency,is employed in the simulation to investigate the influence of boundary reflections on ultrasonic coda waves.Numerical experiments with saturated poroelastic media demonstrate that the 2L FD scheme with the CPML for ultrasonic wave propagation significantly improves stability conditions at strong heterogeneity and absorbing performance at grazing incidence.The boundary reflections from the artificial boundary surrounding the digital core decay fast with the increase of CPML thicknesses,almost disappearing at the CPML thickness of 15grids.Comparisons of the resulting ultrasonic coda Qsc values between the numerical and experimental ultrasonic S waveforms for a cylindrical rock sample demonstrate that the boundary reflection may contribute around one-third of the ultrasonic coda attenuation observed in laboratory experiments.
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