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

Abstract

International audienceIn 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 distribution

Fundamental controls on fluid flow in carbonates: current workflows to emerging technologies

“…Tillotson et al (2012) reported experiments on synthetic rock samples to support theoretical predictions that shear-wave splitting can be used as a good estimate of fracture density. Further research has highlighted the frequency-and scale-dependent nature of fracture compliance (Hobday & Worthington 2012), the use of scattered seismic energy to constrain fracture spacing and intensities (Vlastos et al 2007), and advanced numerical methods to simulate seismic-wave propagation in discrete representations of fracture populations (Hall & Wang 2012). Although by no means mature, improved approaches to predict anisotropic permeability have been explored through the analysis of frequencydependent seismic amplitude v. angle and azimuth data (Ali & Jakobsen 2013).…”

Seismic attenuation in fractured media

“…So compared to Vlastos et al (2007), Wang (2012), the crack density is replaced with porosity within the fracture, and two constants representing a compact cluster of dry cracks are replaced with N C and T C , to represent fluid filled fractures.…”

“…But at low frequencies, the elastic properties of fractured media satisfy anisotropic Gassmann (or Brown-Korringa) theory applied to a porous material with linear-slip interfaces. Vlastos et al (2007) showed that scattering attenuation is not only frequency dependent but also related to the direction of propagation and the orientation of the fractures. That is the attenuation anisotropy.…”

Seismic response of fractures by numerical simulation

“…They indicate that the size of the fractures strongly affects the resultant waveform, with fractures shorter than wavelength acting more as scatterers and longer ones more as reflectors. A fracture system can evolve with time, and thus so will the resultant wavefield (Vlastos et al 2007). Strong differences were also observed in wavefields produced by different clustering of random distributions.…”