Effective and undrained shear moduli G_eff and G_u

Four shear moduli are easily and unambiguously defined for the anisotropic system under study. Furthermore, since we are treating only soft anisotropy, all of these moduli are the same, i.e., G_i = G_dr for $i=1,\ldots,4$. These are all related to the four shear eigenvectors of the systems, and they do not couple to the pore-fluid mechanics. But, the eigenvectors in the reduced $2\times 2$ system studied here are usually mixed in character, being quasi-compressional or quasi-shear modes. It is therefore somewhat problematic to find a proper definition for a fifth shear modulus. The author has analyzed this problem previously (Berryman, 2004b), and concluded that a sensible (though approximate) definition can be made using G₅ = G_eff. There are several different ways of arriving at the same result, but for the present analysis the most useful of these is to express G_eff in terms of the product $\Lambda_+\Lambda_-$ (the eigenvalue product, which is also the determinant of the $2\times 2$ compliance system). The result, which will be quoted here without further discussion [see Berryman (2004b) for details], is

$$\begin{eqnarray} {{1}\over{3K_u}}\cdot{{1}\over{2G_{eff}}} \equiv \Lambda_+\Lambda_- = 18\left[A^*_{11}A^*_{33} - (A^*_{13})^2\right], \end{eqnarray}$$ (31)

which we take as the definition of G_eff here. And, since A^*₁₁ = 1/9K_u, we have

$$\begin{eqnarray} {{1}\over{G_{eff}}} = 12\left[A^*_{33} - (A^*_{13})^2/A^*_{11}\right]. \end{eqnarray}$$ (32)

To obtain an isotropic average overall undrained shear modulus, we next take the arithmetic mean of these five shear compliances:

$$\begin{eqnarray} {{1}\over{G_u}} \equiv {{1}\over{5}}\sum_{i=1}^5 {{1}\over{G_i}}. \end{eqnarray}$$ (33)

Combining these definitions and results gives:

$$\begin{eqnarray} {{1}\over{G_u}} - {{1}\over{G_{dr}}} = - {{4}\over{15}}{{(\beta... ...{1-\alpha B}} \left[{{1}\over{K_u}} - {{1}\over{K_{dr}}}\right], \end{eqnarray}$$ (34)

where the $\beta'$s are defined by $\beta'_i = \beta_iK_{dr}/\alpha$.The final equality is presented to emphasize the similarity of the present results to those of both Mavko and Jizba (1991) and Berryman et al. (2002b). Setting $\beta'_1 = 0$, $\beta'_3 = 1$, B = 1, and $\alpha \simeq 0$ recovers the form of Mavko and Jizba (1991) for the case of a very dilute system of flat cracks.

TABLE. Elastic and poroelastic parameters of the three rock samples considered in the text. Bulk and shear moduli of the grains K_m and G_m, bulk and shear moduli of the drained porous frame K_dr and G_dr, the effective and undrained shear moduli G_eff and G_u, and the Biot-Willis parameter $\alpha= 1 - K_{dr}/K_m$.The porosity is $\phi$.

Elastic/Poroelastic Sierra White Schuler-Cotton Valley Spirit River

Parameters Granite Sandstone Sandstone

G_m (GPa) 31.7 36.7 69.0 

G_u (GPa) 28.3 17.7 12.41

G_dr (GPa) 26.4 15.7 11.33

G_eff (GPa) 39.8 35.8 20.11

K_m (GPa) 57.7 41.8 30.0 

K_dr (GPa) 38.3 13.1  7.04

$$\alpha$$ 0.336 0.687 0.765

$$\phi$$ 0.008 0.033 0.052