The noninteraction approximation (NIA) is commonly used for prediction of the anisotropic elastic stiffnesses of cracked rocks. At large crack density, the NIA has a desirable nonlinear stiffness behavior; however, this is inconsistent with the dilute crack assumption. The nonlinear behavior of stiffness predicted by the NIA at high crack density is produced by defining compliance to be a linear function of crack density and then inverting the compliance tensor to stiffness. The linear behavior of compliance is strictly valid only when there is no crack interaction (at low crack density), so the resulting nonlinear stiffnesses at high crack density are unconstrained extrapolations. Comparison of results from the NIA method, an effective stiffness (T-matrix) method, and numerical modeling shows that: (1) first-order compliance methods are not better than first-order stiffness methods; they are equivalent and valid only under the noninteraction (or dilute crack) assumption; and (2) it is misleading to use the NIA at high crack density; crack interactions (shielding and amplification) should not be assumed to cancel for all crack distributions, so they require explicit consideration, for example, with a high-order T-matrix formulation. Examples of unreliable predictions of the NIA at high crack density include nonzero stiffnesses at 100% porosity for a model with dry cracks, and errors in relative stiffnesses in the isotropic limit.
Parameters associated with the presence of inclusions (cracks or vugs) significantly influence seismic responses. The T-matrix method is used to define approximate anisotropic effective elastic stiffness tensors for media with inclusions. This allows 3D, three-component, eighth-order staggered-grid finite-difference modeling to simulate the seismic responses of anisotropic media with high inclusion density, various aspect ratios, spatial distributions and orientations of inclusions, and fluid content. The model parameters are chosen to represent aligned inclusions ranging from vertical cracks to vugs in a carbonate reservoir encased in clastics. The magnitude of anisotropy increases with increasing inclusion density for P- and S-waves. Cracks in a high-velocity carbonate reduce the stiffnesses and correspondingvelocities; this results in smaller contrasts with the surrounding (lower-stiffness) clastics and, hence, smaller reflection coefficients. S-wave splitting and the anisotropy of [Formula: see text] are clearly visible.The aspect ratio of the spatial distribution of the cracks potentially produces larger anisotropy than the crack aspect ratio, especially at large crack density. The crack distribution has little effect on stiffnesses parallel to the cracks but a large effect perpendicular to the cracks. As the crack orientation moves farther from vertical, changes in the resulting seismograms are more systematic in the direction parallel to the crack strike than perpendicular to it. The seismic signatures resulting from variations of the inclusion parameters are significant and easily visible in the data. This is a computational basis for obtaining more accurate, complete, and quantitative characterizations of inclusions.
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