Schoenberg's angle on fractures and anisotropy: A study in orthotropy

VERTICAL FRACTURES IN VTI EARTH

Another model in a very similar context that has been discussed frequently by Schoenberg and Helbig (1997), Bakulin et al. (2000), and others is the model of a VTI earth system (where the background elastic medium is transversely isotropic with vertical axis of system, as it would be in a layered earth model having isotropic layers [Backus (1962)]) with superposed vertical fractures. A result that is often quoted in this context concerns a condition that is necessarily satisfied by the elastic stiffness matrix elements for such a system:

$$c_{13}(c_{22}+c_{12}) = c_{23}(c_{11}+c_{12}).$$ (10)

Using the same ideas applied here already, we can reduce this equation to a simple statement about the system compliances. The resulting statement is the formula:

$$S_{13} = S_{23},$$ (11)

by which we mean to say that the only requirement imposed on the compliances after the introduction of the vertical fractures to the VTI earth background is that the new overall system compliance must satisfy the conditions in (11) after the changes due to the fractures are included in the values of these two compliance components. No other special constraints appear.

To see that (11) is the correct condition, note that

$$\begin{array}{c} S_{13} = (c_{12}c_{23} - c_{13}c_{22})/\det... ... S_{23} = (c_{12}c_{13} - c_{23}c_{11})/\det{(C)}, \end{array}$$ (12)

where $\det{(C)}$ is the determinant of the upper left $3\times3$ sub-matrix of the orthotropic stiffness matrix $C$. Equating these two expressions from (12) and rearranging the final result gives a formula that is precisely the same condition (10) given by Schoenberg and Helbig (1997). What this condition implies for the physical system -- since the background earth medium is assumed to be VTI with vertical axis of symmetry and also since $S_{13} = S_{23}$ for the background medium itself (before the fractures are added to it) -- is that our final result must be the condition:

$$\Delta S_{13} = \Delta S_{23}.$$ (13)

This very simple equality means the changes (i.e., increases) in these off-diagonal compliances -- when due to the addition of the vertical fractures to this model -- have just one constraint, and that single constraint is that changes in these two off-diagonal compliances $S_{13}$ and $S_{23}$ must occur in unison. This result is also seen as a limiting case found in Table 1, when $A = B$, which occurs only when $\phi = 0^o$. Since $\phi = 0^o$ means that all the vertical fractures are parallel, and therefore being all aligned fractures, I therefore then have exactly the case studied explicitly by Schoenberg and Helbig (1997).

Schoenberg's angle on fractures and anisotropy: A study in orthotropy