Numerical Modeling of Seismic Wave Propagation at the Micro-Scale in Digitized Sandstone

Abstract

In this paper, we first examine the relationship between the relative particle motions of fluids and solids and the seismic signal received when compressional waves propagate through saturated porous materials. We use a rotated-staggered-grid finite difference modeling scheme to simulate elastic wave phenomena in a digitized 2D structural model obtained from micrographs of a loose beach sand. When considering ultrasonic wave propagation wave in models with explicit inclusion of granular structure, the heterogeneities of quartz and pores in size and shape lead to frequency-dependent seismic phenomena. By comparing the numerical results from models where three sources with different frequencies were used, we saw that (1) strong particle motions concentrate mostly in fluid; (2) significant variations in pressure are observed in the fluid; (3) during the dynamic process of wave propagation, relative particle motion of the fluid and solid phases induces stress concentration on the sharp tips and corner of grains; (4) coherent particle motion is generated by sources with low frequency content, while sources with higher frequencies induce disordered particle motions. Corresponding to these particle motions, less scattered energy is observed in cases with more coherent particle motion, and strong scattering is generated by disordered particle motion. Then we extend our work to a 3D digitized Fontainebleau sandstone sample. Though the size of the sample is small, we still consider a relative broad source frequency band (100 kHz – 20 MHz) so as to study the frequency dependent behavior of this sample. We notice a velocity minimum occurring at some “critical frequency” (750 kHz). Above this “critical frequency”, the velocity increases with frequency; while below this frequency, velocity goes to a low frequency value – effective medium value. The transition from low frequency to high frequency behavior can be viewed as going from wave-like to ray-like propagation. We then study the fluid effect by saturating the pores with non-viscous and viscous brine and oil. The velocities for samples saturated with fluids are generally larger than those of dry sample at frequencies below the critical one, which shows the significant effect of the compressibility of fluids. While the velocities become smaller than those of dry sample at frequencies above the critical one, which shows that the density of fluids comes into play a significant role. We see little effect of viscosity of fluids on velocity. To investigate the scale effect, we first compare the result from dynamic modeling for case with source frequency of 100 kHz to that from static modeling by using finite element method on a sub-cube selected from the original sample. Then we elongate the 3D sample in one direction by repeating the original sample five times, and compare the result from this elongated one to that from the original one at source frequency of 100 kHz. Velocities for these three cases are close to each other. Smaller velocity from static modeling might be due to the higher porosity of the selected sub-cube sample.