SUMMARY We model seismic wave propagation in media with discrete distributions of fractures using the pseudospectral method. The implementation of fractures with a vanishing width in the 2‐D finite‐difference grids is done using an effective medium theory (that is, the Coates and Schoenberg method). Fractures are treated as highly compliant interfaces inside a solid rock mass. For the physical representation of the fractures the concept of linear slip deformation or the displacement discontinuity method is used. According to this model, the effective compliance of a rock mass with one or several fracture sets can be found as the sum of the compliances of the host (background) rock and those of all the fractures. To first order, the background rock and fracture parameters can be related to the effective anisotropic coefficients, which govern the influence of anisotropy on various seismic signatures. We test the validity of the method and examine the accuracy of the synthetic seismograms by a comparison with theoretical ray traveltimes. We present three numerical examples to show the effects of different fracture distributions. The first example shows that different spatial distributions of the same fractures produce different wavefield characteristics. The second example examines the effects of variation of fracture scale length (size) compared with the wavelength. The final example examines the case of fractures with a power‐law (fractal) distribution of sizes and shows how that affects the wavefield propagation in fractured rock. We conclude that characterization of fractured rock based on the concept of seismic anisotropy using effective medium theories must be used with caution. Scale length and the spatial distributions of fractures, which are not properly treated in such theories, have a strong influence on the characteristics of wave propagation.

S U M M A R YFluid flow in the Earth's crust plays an important role in a number of geological processes. In relatively tight rock formations such flow is usually controlled by open macrofractures, with significant implications for ground water flow and hydrocarbon reservoir management. The movement of fluids in the fractured media will result in changes in the pore pressure and consequently will cause changes to the effective stress, traction and elastic properties. The main purpose of this study is to numerically examine the effect of pore pressure changes on seismic wave propagation (i.e. the effects of pore pressures on amplitude, arrival time, frequency content). This is achieved by using dual simulations of fluid flow and seismic propagation in a common 2-D fracture network. Note that the dual simulations are performed separately as the coupled simulations of fluid flow and seismic wave propagations in such fracture network is not possible because the timescales of fluid flow and wave propagation are considerably different (typically, fluid flows in hours, whereas wave propagation in seconds). The flow simulation updates the pore pressure at consecutive time steps, and thus the elastic properties of the rock, for the seismic modelling. In other words, during each time step of the flow simulations, we compute the elastic response corresponding to the pore pressure distribution. The relationship between pore pressure and fractures is linked via an empirical relationship given by Schoenberg and the elastic response of fractures is computed using the equivalent medium theory described by Hudson and Liu. Therefore, we can evaluate the possibility of inferring the changes of fluid properties directly from seismic data. Our results indicate that P waves are not as sensitive to pore pressure changes as S and coda (or scattered) waves. The increase in pore pressure causes a shift of the energy towards lower frequencies, as shown from the spectrum (as a result of scattering attenuation). Another important observation is that the fluid effects on the wavefield vary significantly with the source-receiver direction, that is, the azimuth relative to the fracture orientation. These results have significant implications for the characterization of naturally fractured reservoirs using seismic methods, and may impact on experimental design to infer such attributes in a real reservoir situation, particularly in acquiring time-lapse seismic data.

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

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