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Compliance formulation

If we define the effective compliance matrix for the system as S* having the matrix elements given in (11), then the bulk modulus for this system is defined in terms of v1 by
{{1}\over{K_u}} = v_1^T S^* v_1 = {{1}\over{K_{dr}}} -
\gamma^{-1}\left(2\beta_1 + \beta_3\right)^2,
 \end{eqnarray} (13)
where the T superscript indicates the transpose, and $1/K_{dr} \equiv
\sum_{i,j = 1}^3 s_{ij}$.This is the result usually quoted as Gassmann's equation for the bulk modulus of the undrained (or confined) anisotropic (VTI) system. Also, note that in general
\sum_{i=1}^3 \beta_i = 2\beta_1 + \beta_3 = \alpha/K_{dr}.
 \end{eqnarray} (14)
Thus, even though v1 is not an eigenvector of this system, it nevertheless plays a fundamental role in the mechanics. Furthermore, this role is quite well-understood. What is perhaps not so well-understood then, especially for poroelastic systems, is the role of v3. Understanding this role will become our main focus for the remainder of this discussion.

The true eigenvectors of the $2\times 2$ subproblem of interest (i.e., in the space orthogonal to the four pure shear eigenvectors already discussed) are necessarily linear combinations of v1 and v3. We can construct the relevant contracted operator for the $2\times 2$ subsystem by considering:
{c} v_1^T \\  v_3^T\end{array}\right) S^*
 ...*_{11} & 18A^*_{13} \\ 18A^*_{13} & 36A^*_{33}\end{array}\right)
 \end{eqnarray} (15)
(in all cases the * superscripts indicate that the pore-fluid effects are included) and the reduced matrix
\Sigma^* = A^*_{11}v_1v_1^T + A^*_{13}(v_1v_3^T + v_3v_1^T)
+ A^*_{33} v_3v_3^T,
 \end{eqnarray} (16)
A^*_{11} = [2(s^*_{11}+s^*_{12}+2s^*_{13})+s^*_{33}]/9, \nonumb...
 ...A^*_{33} = (s^*_{11}+s^*_{12}-4s^*_{13}+2s^*_{33})/18. \nonumber
Providing some understanding of these connections and the implications for shear modulus dependence on fluid content is one of our goals.

First we remark that A*11 = 1/9Ku, where Ku is again the undrained (or Gassmann) bulk modulus for the system in (13). Therefore, A*11 is proportional to the undrained bulk compliance of this system. The other two matrix elements cannot be given such simple interpretations in general. To simplify the analysis we note that, at least for purposes of modeling, anisotropy of the compliances sij and the poroelastic coefficients $\beta_i$ can be treated independently. Anisotropy displayed in the sij's corresponds mostly to the anisotropy in the solid elastic components of the system, while anisotropy in the $\beta_i$'s corresponds mostly to anisotropy in the shapes and spatial distribution of the porosity. We will therefore distinguish these contributions by calling anisotropy appearing in the sij's the ``hard anisotropy,'' and the anisotropy in the $\beta_i$'s will in contrast be called the ``soft anisotropy.''

Now, it is clear (also see the discussion in the Appendix for more details) that the eigenvectors having unit magnitude $f(\theta)$ for this problem (i.e., for the reduced operator $\Sigma^*$) necessarily take the form
f(\theta) = \overline{v}_1 \cos\theta + \overline{v}_3\sin\theta,
 \end{eqnarray} (18)
where $\overline{v}_1 = v_1/\sqrt{3}$ and $\overline{v}_3 = v_3/\sqrt{6}$ are the normalized eigenvectors. Two solutions for the rotation angle are: $\theta_-$and $\theta_+ = \theta_- + {{\pi}\over{2}}$, guaranteeing that the two solutions (the eigenvectors) are orthogonal. It is easily seen that the eigenvalues are given by
\Lambda^*_\pm = 3\left[A^*_{33} + A^*_{11}/2\pm
 \sqrt{(A^*_{33}-A^*_{11}/2)^2 + 2(A^*_{13})^2}\right]
 \end{eqnarray} (19)
and the rotation angles are determined by
\tan\theta^*_\pm = {{\Lambda^*_\pm/3 - A^*_{11}}\over{\sqrt{2}A...
 ...A^*_{33}-A^*_{11}/2)^2 + 2(A^*_{13})^2}\right]/\sqrt{2}A^*_{13}.
 \end{eqnarray} (20)
One part of the rotation angle is due to the drained (fluid free) ``hard anisotropic'' nature of the rock frame material. We will call this part $\bar{\theta}$. The remainder is due to the presence of the fluid in the pores, and we will call this part $\delta\theta \equiv \theta^* - \bar{\theta}$ for the ``soft anisotropy.'' Using a standard formula for tangents, we have
\delta\theta_\pm = \tan^{-1}\left[{{\tan\theta^*_\pm
 ..._\pm}\over{1 +
\tan\theta^*_\pm \tan{\bar{\theta}}_\pm}}\right].
 \end{eqnarray} (21)
Furthermore, definite formulas for $\bar{\theta}_\pm$ are found from (20) by taking $\gamma \to \infty$ (corresponding to air saturation of the pores).

\tan\theta^*_+\cdot\tan\theta^*_- = -1,
 \end{eqnarray} (22)
it is sufficient to consider just one of the signs in front of the radical in (20). The most convenient choice for analytical purposes turns out to be the minus sign (which corresponds to the eigenvector with the larger component of pure compression). Furthermore, it is also clear from the form of (20) that often the behavior of most interest to us here occurs for cases when $A^*_{13} \ne 0$.

In the limit of a nearly isotropic solid frame (so the ``hard anisotropy'' vanishes and thus we will also call this the ``quasi-isotropic'' limit), it is not hard to see that
A^*_{33} \simeq {{1}\over{12G_{dr}}} - {{(\beta_1 -
 \end{eqnarray} (23)
where Gdr is the drained shear modulus of the quasi-isotropic solid frame. Similarly, the remaining coefficient
A^*_{13} \simeq - 
{{(\beta_1 - \beta_3)(2\beta_1+\beta_3)}\over{9\gamma}},
 \end{eqnarray} (24)
since all the solid contributions approximately cancel in this limit.

To clarify the situation further, we will enumerate three cases: