Extracting shear and normal compliances of crack-like defects from pressure dependences of elastic-wave velocities

We present a differential formulation of effective-medium model in which the normal and shear compliances of the high-compliance porosity are explicitly decoupled. This feature of the decoupled-compliance model (DC model) is in contrast to conventional models in which such defect's properties are implicitly assumed and are subject to strong limitations defined by the used particular crack model. Comparison with the DC model makes it possible to reveal such implicit assumptions in the conventional models. Furthermore, for the conventional cracks, our approach gives the same results as the conventional models. The ability of the DC model to incorporate arbitrary defect properties in terms of their normal-to-shear compliance ratio (q-ratio) is used to formulate an analogue of Hashin-Shtrikman constraints on the range of feasible crack-induced variations in the moduli. Comparison of the DC model with experimental pressure dependences of elastic-wave velocities in rocks makes it possible to extract the q-ratio for real crack-like defects. These results demonstrate that properties of real cracks usually noticeably differ from those of popular crack models such as cracks with free faces (e.g., penny-shape) or pure shear cracks. We discuss an example of sandstone with pronouncedly negative Poisson's ratio that is due to the fact that the ratio of normal-to-shear compliances of voids in this rock (q~7-8) is significantly higher than for the conventional cracks (q~2). Ability of the DC model to accurately extrapolate pressure dependences of the moduli from relatively low pressures to several times greater is demonstrated, including the cases, for which the conventional models give huge errors. The introduced parameter q - the ratio of normal-to-shear compliances of voids - provides additional insight into properties of real crack-like defect in rocks.