Bounds of elastic parameters characterizing transversely isotropic media: Application to shales

Abstract

Several rocks, and in particular shales, are often described as transversely isotropic (TI) materials. Geophysical data coverage does not always permit reliable determination of all five elastic parameters, neither in seismic and sonic data from the field nor in laboratory measurements. Data may, however, be constrained by the existence of bounds on elastic moduli, derived from the fundamental requirement of positive elastic energy. Conditioned bounds are described for engineering parameters such as Poisson’s ratios as well as anisotropy coefficients such as the moveout parameter δδ and the anellipticity parameter ηη. “Conditioned bounds” means bounds that in general depend on some of the other elastic moduli: The bounds we have evaluated are controlled primarily by P- and S-wave moduli obtained from wave propagation along a symmetry axis and to some extent by P- and S-wave anisotropies. Such data may be acquired more easily from geophysical measurements. We have inspected the laboratory data obtained with various types of shales under different testing conditions, and none of them failed to adapt to the bounds. The data indicate, for instance, clear distinctions between how the proximity to bounds is driven by stress changes for saturated versus nonsaturated shales.