Modeling of Acoustic Attenuation in Porous Media

Overview & Synopsis

In recent years [1-3], the authors have been trying to develop a quantitative understanding of what seismic attenuation in rocks and sediments is primarily due to. Although many attenuation mechanisms have been proposed in the literature, only a few more recent models have both begun to predict attenuation levels that are consistent with field measurements in the seismic band (10 Hz to a few kHz) and been based on a realistic physical description of rocks. These models assume that rock contains " mesoscopic-scale " heterogeneity, which is defined as heterogeneity over scales larger than the grains, but smaller than the seismic wavelengths. When rocks containing mesoscopic structure are stressed, the more compliant parts of the rock respond with a larger fluid pressure change than do the stiffer parts, which results in fluid flow and wave attenuation. One approach we have taken to better understand the mechanism of seismic attenuation in the presence of mesoscopic heterogeneity is to perform numerical simulations [2-3]. Computer-generated synthetic rock samples are created that have spatially variable poroelastic moduli and porous-continuum properties (e.g., porosity and permeability) distributed over the pixels. The numerical experiments consist of applying a time-varying stress to the sample surface, and measuring the resultant sample strain (defined as the average local strain throughout a sample). The Fourier transform of the stress and strain determine the sample's complex frequency-dependent moduli, while the ratios of the imaginary and real parts of these moduli define the inverse quality factor 1/Q for the compressional and shear modes. The local response within such synthetic samples is obtained using finite-difference approximations of Biot's poroelasticity equations. The finite-difference algorithm we employ [4-5] is an explicit time-stepping scheme on a staggered grid that can allow for the full-range of dynamics contained in Biot's equations including the onset of viscous-boundary layers in the pores at high-enough frequency. Using both numerical and analytical modeling, the level of attenuation in samples containing mesoscopic-scale heterogeneity is shown to be proportional to the square of the contrast in elastic moduli present. Stated differently, in materials possessing random fluctuations, the attenuation is proportional to the variance of the probability distribution used to randomly distribute the local elastic moduli. Further, when the heterogeneity is smoothly varying (i.e., not placed in the form of " patches " that are characterized by step function changes), the high frequency limit of seismic Q-1 (a measure of attenuation) is not the often assumed (and modeled) f-1/2 dependence, but is instead f-1 where f is wave frequency. Further, when the poroelastic moduli are distributed as self-affine fractals having a Hurst exponent H, both numerical results and analytical arguments demonstrate the scaling law Q(f) = f H. Last, in pure shear, it is demonstrated that contrasts in the bulk modulus can be responsible for local fluid pressure changes (some positive and some negative) that average to zero in pure shear, but that are responsible for a considerable amount of mesoscopic-scale flow and associated shear attenuation. In the laboratory, we have been measuring the mesoscale fluctuation in the elastic properties of actual rocks using a novel servo-controlled indentor. The device measures the fluctuations of the " surface " or " indention " modulus on the surface of a planar rock slab. We will present several maps of the