Physical constraints on C13 for transversely isotropic shales and their applications

Abstract

The problem of theoretical prediction of the elastic stiffness parameter [Formula: see text] is important, as well as proper establishment of its theoretical bounds, which is extremely useful in the reliable [Formula: see text] estimation. We revise a formula for the lower bound [Formula: see text], empirically derived by F. Yan with coauthors, based on their empirical inequality between two Poisson’s ratios in a transversely isotropic elastic medium; i.e., [Formula: see text]. In doing this, we consider a theoretical counterexample derived on theoretical grounds by J. Sarout leading to the opposite inequality [Formula: see text]. Our analysis on shale anisotropy (using published data on various shale samples) proves both inequalities to be true. Therefore, we can divide shale data into two groups: 1 and 2; i.e., group 1 we define by the inequality [Formula: see text], and group 2 by [Formula: see text]. In the framework of the linear slip (LS) model, we found another restriction on two Poisson’s ratios [Formula: see text], which must be satisfied in the LS model. In addition, we found that the theoretical formula for the [Formula: see text] in the LS model and the bound [Formula: see text] should be equal: [Formula: see text]. Also, we found that [Formula: see text] depends on the normal fracture compliance [Formula: see text]. We postulate that there is a critical compliance [Formula: see text] predicted by the LS model that marks the transition between two inequalities of Poisson’s ratios: [Formula: see text], and [Formula: see text]. These inequalities become a part of a broader interpretation, in which the compliance [Formula: see text] acts as a transitional intermediate response (i.e., [Formula: see text]) between the two shale groups 1 and 2. Each shale group (i.e., [Formula: see text], and [Formula: see text]) is assigned to different [Formula: see text] (i.e., [Formula: see text] and [Formula: see text]), which imply differentiation of rock properties in two groups, such as different fluid saturation, as an example.