Poroelastic measurements resulting in complete data sets for granular and other anisotropic porous media

ANISOTROPIC POROELASTICITY

If the overall porous medium is anisotropic due either to some preferential alignment of the constituent particles or to externally imposed stress (such as a gravity field and weight of overburden, for example), I consider the orthorhombic anisotropic version of the poroelastic equations:

$$\left(\begin{array}{c} e_{11} \ e_{22} \ e_{33} \ -\zeta\end{a... ...array}{c} \sigma_{11} \ \sigma_{22} \ \sigma_{33} \ -p_f \end{array}\right).$$ (22)

From here on throughout the paper, I will drop the $\delta$'s from the stresses and strains, as this extra notation is truly redundant when they are all being treated as small (and therefore resulting in linear effects), as I do here.

The $e_{ii}$ are strains in the $i = 1,2,3$ directions. The $\sigma_{ii}$ are the corresponding stresses. The fluid pressure is $p_f$. The increment of fluid content is $\zeta$. The drained compliances are $s_{ij} = s_{ij}^d$. Undrained compliances (not yet shown) are symbolized by $s_{ij}^u$. Coefficients $\beta_i = s_{i1} + s_{i2} + s_{i3} - 1/3K_R^g$, where $K_R^g$ is again the Reuss average modulus of the grains. The drained Reuss average bulk modulus is defined by

$$\frac{1}{K_R^d} = \sum_{ij=1,2,3} s_{ij}^d.$$ (23)

For the Reuss average undrained bulk modulus $K_R^u$, I have drained compliances replaced by undrained compliances. A similar definition of $K_R^g$, with drained compliances replaced by grain compliances has already been introduced earlier in the discussion. The alternative Voigt (1928) average [also see Hill (1952)] of the stiffnesses will play no role in the present work. And, finally, $\gamma = \sum_{i=1-3}\beta_i/BK_R^d$, where $B$ is the second Skempton (1954) coefficient, which will be defined carefully again later in our discussion.

The shear terms due to twisting motions (i.e., strains $e_{23}$, $e_{31}$, $e_{12}$ and stresses $\sigma_{23}$, $\sigma_{31}$, $\sigma_{12}$) are excluded from this discussion since they typically do not couple to the modes of interest for anisotropic systems having orthotropic symmetry, or any more symmetric system such as transversely isotropic or isotropic. I have also assumed that the true axes of symmetry are known, and make use of them in my formulation of the problem. Note that the $s_{ij}$'s are the elements of the compliance matrix ${\bf S}$ and are all independent of the fluid, and therefore would be the same if the medium were treated as elastic (i.e., by ignoring the fluid pressure, or assuming that the fluid saturant is air). In keeping with the earlier discussions, I typically call these compliances the drained compliances and the corresponding matrix the drained compliance matrix ${\bf S}^d$, since the fluids do not contribute to the stored mechanical energy if they are free to drain into a surrounding reservoir containing the same type of fluid. In contrast, the undrained compliance matrix ${\bf S}^u$ presupposes that the fluid is trapped (unable to drain from the system into an adjacent reservoir) and therefore contributes in a significant and measureable way to the compliance and stiffness ( ${\bf C}^u = \left[{\bf S}^u\right]^{-1}$), and also therefore to the stored mechanical energy of the undrained system.

Although the significance of the formula is somewhat different now, I find again that

$$\beta_1+\beta_2+\beta_3 = \frac{1}{K_R^d} - \frac{1}{K_R^g} = \frac{\alpha_R}{K_R^d},$$ (24)

if I also define (as I did for the isotropic case) a Reuss effective stress coefficient:

$$\alpha_R \equiv 1 - K_R^d/K_R^g.$$ (25)

Furthermore, I have

$$\gamma = \frac{\beta_1+\beta_2+\beta_3}{B} = \frac{\alpha_R}{K_R^d} + \phi\left(\frac{1}{K_f} - \frac{1}{K_R^\phi}\right),$$ (26)

since I have the rigorous result in this notation (Berryman, 1997) that Skempton's $B$ coefficient is given by

$$B \equiv \frac{1-K_R^d/K_R^u}{1-K_R^d/K_R^g} = \frac{\alpha_R/K_R^d}{\alpha_R/K_R^d + \phi(1/K_f - 1/K_R^\phi)}.$$ (27)

Note that both (26) and (27) contain dependence on the pore bulk modulus $K_R^\phi$ that comes into play when the pores are heterogeneous, regardless of whether the system is isotropic or anisotropic. I want to emphasize that all these formulas are rigorous statements based on the earlier anisotropic analysis. The appearance of $K_R^d$ and $\alpha_R$ is not an approximation, but merely a useful choice of notation made here because it will make clear the similarity between the rigorous anisotropic formulas and the isotropic ones.

Poroelastic measurements resulting in complete data sets for granular and other anisotropic porous media