P-wave attenuation and dispersion in a fluid-saturated rock with aligned rectangular cracks

It is well known that mesoscopic-scale wave-induced fluid flow (WIFF) between cracks and surrounding porous matrix is one of significant mechanisms in causing seismic dispersion and attenuation. Previous theoretical models that were used to interpret the interaction between cracks and passing waves assumed the cracks to be circular or slit-like. However, in many actual cases, the cracks can have a shape more close to a rectangle. In this paper, we develop an effective medium model to estimate the P-wave attenuation and velocity dispersion in a saturated porous rock containing a random distribution of permeable, aligned, rectangular cracks of infinitesimal thickness. This is done by combining the far-field displacement representation of the solution of a normally incident P wave scattered by a single crack and Foldy's scattering theorem. The effective low-frequency P-wave velocity predicted by the present model is asymptotically consistent with the Gassmann's theory. It is shown that thorough knowledge of the crack shape is important for understanding seismic signatures. Specifically, the WIFF relaxation frequency is mainly determined by the crack intermediate dimension, while it is insensitive to the crack largest dimension. Moreover, P-wave velocity of the rectangular crack model can be quite different from those of the circular crack model and the slit crack model, although the overall frequency-dependent trends of the velocities for different models are very similar. Analogously to the circular crack model and the slit crack model, at the normal incidence of a fast-P wave, the WIFF is found to dominate the wave attenuation and velocity dispersion. These results are helpful to estimate formation parameters from different frequency components of observed seismic data.