Numerical simulation of wave propagation in 2-D fractured media : scattering attenuation at different stages of the growth of a fracture population

摘要

In this paper, we systematically examine the multiple scattering process of seismic waves at consecutive stages of the evolution of 2-D fracture population. Synthetic seismograms are computed using the pseudo-spectral method for elastic wave propagation, where spatial derivations are computed using fast Fourier transforms and time derivatives are computed using second-order finite differences. The grid sizes are 2560 × 2560 with 1 m interval and a Ricker wavelet with a peak frequency of 30 Hz is used (or equivalently a wavelength of 10 m for the P-wave velocity of 3000 m s–1 used in our modelling). Fracture patterns are generated using a 2-D cellular automaton model of rupture with healing to account for clustering and anisotropy in the fracture growth process. The cellular automation model takes into account the discontinuous and segmented nature of a fracture population, and reproduces in the statistical sense the intermediate stages of fracture growths. To estimate the frequency-dependence of scattering attenuation (quantified by the inverse quality factor Q−1) at different stages of the fracture evolution, we use the spectral ratio method. Variations of Q−1 with frequency are then fitted to a polynomial of order up to 8 for each state of the fracture evolution as we do not want to make an assumption about how Q−1 should depend on frequency or scales. This allows us to determine the nature of the frequency-dependence of scattering attenuation as a function of fracture evolution. Our results confirm, as expected, the dependence of scattering attenuation on frequency, and the fifth-order polynomial seems to fit the measured attenuation from synthetic seismograms better. In addition, the inverse quality factor Q−1 is shown to be linearly dependent on fracture density, reaching a maximum when fracture density is the highest. In summary, our numerical results confirm that scattering attenuation has a complex dependence on frequency, and measurements of attenuations may be potentially used to characterize spatial distributions of fracture networks in particular, the scale distributions.