T-matrix approach to shale acoustics

Abstract

The T-matrix approach of quantum scattering theory is used here to place many long-wavelength equivalent-medium approximations for porous composites, polycrystals and cracked media on a common footing and to indicate their limitations, but also to derive some new results based on two-point statistics. In this way, we have obtained an insight into the difficult problem of elastic inclusions at finite concentrations, which is of foremost relevance when estimating the effective material parameters of porous/cracked shales, involving stacks of more or less horizontally aligned clay platelets, mixed together with more rounded silt minerals, and with fluid filling the spaces. A rather involved perturbative analysis of the effects of interactions between (or structural correlations among) the various inclusions (minerals and cavities) making up a real shale of hexagonal symmetry was performed in an attempt to obtain a better match between theoretical predictions (based on a combination of coherent and optical potential approximations) and experimental results (recovered from ultrasonic wave speeds) for the effective elastic stiffness tensor. For the particular data set considered in this study, the T-matrix approach was able to match the data better than the approach of Hornby et al., but the match was not completely satisfactory. Further progress in theoretical shale modelling may come from a better knowledge of the elastic properties of pure clay minerals, a more detailed knowledge of the microstructure of shales, the incorporation of constraints obtained from comparisons between theoretical predictions and experimental results, as well as a continuing development of the T-matrix approach. Numerical results (also for the effect of bedding parallel microcracks on the elasticity of such a real shale) have value in illustrating the importance of taking into account the effects of spatial distribution when trying to deal with non-dilute mixtures of highly contrasting material properties.