## ABSTRACT

The problem of theoretical prediction of the elastic stiffness parameter $C13$ is important, as well as proper establishment of its theoretical bounds, which is extremely useful in the reliable $C13$ estimation. We revise a formula for the lower bound $C13_min$, 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., $ν13>ν12$. In doing this, we consider a theoretical counterexample derived on theoretical grounds by J. Sarout leading to the opposite inequality $ν13<ν12$. 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 $ν13<ν12$, and group 2 by $ν13>ν12$. In the framework of the linear slip (LS) model, we found another restriction on two Poisson’s ratios $ν13=ν12$, which must be satisfied in the LS model. In addition, we found that the theoretical formula for the $C13LS$ in the LS model and the bound $C13_min$ should be equal: $C13_min≡C13LS$. Also, we found that $C13$ depends on the normal fracture compliance $ZN$. We postulate that there is a critical compliance $ZNLS$ predicted by the LS model that marks the transition between two inequalities of Poisson’s ratios: $ν13<ν12$, and $ν13>ν12$. These inequalities become a part of a broader interpretation, in which the compliance $ZNLS$ acts as a transitional intermediate response (i.e., $ν13=ν12$) between the two shale groups 1 and 2. Each shale group (i.e., $ν13<ν12$, and $ν13>ν12$) is assigned to different $ZN$ (i.e., $ZN and $ZN>ZNLS$), which imply differentiation of rock properties in two groups, such as different fluid saturation, as an example.