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Orthotropic poroelasticity

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

$\displaystyle \left(\begin{array}{c} e_{11}  e_{22}  e_{33}  -\zeta\end{a...
...array}{c} \sigma_{11}  \sigma_{22}  \sigma_{33}  -p_f \end{array}\right).$ (1)

Throughout most of the paper, I will not introduce $ \delta$ 's preceding the stresses and strains, as is sometimes done to emphasize their smallness, since this extra notation is truly redundant when they are all being treated as quantities pertinent to seismic wave propagation (and therefore resulting in linear effects) as I do here, for very small deviations from an initial rest state.

The $ e_{ii}$ (no summation over repeated indices) are strains in the $ i = 1,2,3$ directions. The $ \sigma_{ii}$ are the corresponding stresses, assumed to be positive in tension. The fluid pressure is $ p_f$ , which is positive in compression. The increment of fluid content is $ \zeta$ , and is often defined via:

$\displaystyle \zeta \equiv \frac{\delta(\phi V) - \delta V_f}{V} \simeq \phi\left(\frac{\delta V_\phi}{V_\phi} - \frac{\delta V_f}{V_f}\right),$ (2)

where $ V = V_{\phi}/\phi \simeq V_f/\phi$ is the pertinent local volume (within a layer in present circumstances) of the initially fully fluid-saturated porous layer at the first instant of consideration, $ V_\phi = \phi V$ is the corresponding pore volume, with $ \phi$ being the fluid-saturated porosity of the same volume. $ V_f$ is the volume occupied by the pore-fluid, so that $ V_f = \phi V$ before any new deformations begin. The $ \delta$ 's here do indicate small changes in the quantities immediately following them. For ``drained'' systems, there would ideally be a reservoir of the same fluid just outside the volume $ V$ that can either supply more fluid or absorb any excreted fluid as needed during the nonstationary phase of the poroelastic process. The amount of pore fluid (i.e., the number of fluid molecules) can therefore either increase or decrease from that of the initial amount of pore fluid; at the same time, the pore volume can also be changing, but -- in general -- not necessarily at exactly the same rate as the pore fluid itself. The one exception to these statements is when the surface pores of the layer volume $ V$ are sealed, in which case the layer is ``undrained'' and $ \zeta \equiv 0$ , identically. In such circumstances, it is still possible that both $ V_f$ and $ V_\phi = \phi V$ are changing; but, because of the imposed undrained boundary conditions, they are necessarily changing at the same rate. The drained compliances are $ s_{ij} = s_{ij}^d$ , with or without the $ d$ superscript. Undrained compliances (not yet shown) are symbolized by $ s_{ij}^u$ . Coefficients

$\displaystyle \beta_i = s_{i1} + s_{i2} + s_{i3} - 1/3K_R^g,$ (3)

where $ K_R^g$ is again the Reuss average modulus of the grains. The drained Reuss average bulk modulus is defined by

$\displaystyle \frac{1}{K_R^d} = \sum_{ij=1,2,3} s_{ij}^d.$ (4)

For the Reuss (1929) average undrained bulk modulus $ K_R^u$ , undrained compliances have replaced drained compliances in a formula analogous to (4). A similar definition of the effective grain modulus $ K_R^g$ is:

$\displaystyle \frac{1}{K_R^g} = \sum_{i,j=1,2,3} s_{ij}^g.$ (5)

with grain compliances replacing drained compliances as discussed earlier by Berryman (2010). 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=1,2,3}\beta_i/BK_R^d$ , where $ B$ is the second Skempton (1954) coefficient, which will be defined carefully later in my 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 poroelastic 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 those being either 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 - or vacuum). 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 measurable 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

$\displaystyle \beta_1+\beta_2+\beta_3 = \frac{1}{K_R^d} - \frac{1}{K_R^g} = \frac{\alpha_R}{K_R^d},$ (6)

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

$\displaystyle \alpha_R \equiv 1 - K_R^d/K_R^g.$ (7)

Furthermore, I have

$\displaystyle \gamma = \frac{\beta_1+\beta_2+\beta_3}{B} = \frac{\alpha_R}{K_R^d} + \phi\left(\frac{1}{K_f} - \frac{1}{K_R^\phi}\right),$ (8)

since I have the rigorous result in this notation that Skempton's $ B$ coefficient is given by

$\displaystyle B \equiv \frac{1-K_R^d/K_R^u}{1-K_R^d/K_R^g} = \frac{\alpha_R/K_R^d}{\alpha_R/K_R^d + \phi(1/K_f - 1/K_R^\phi)}.$ (9)

Note that both (8) and (9) contain dependence on the distinct pore bulk modulus $ K_R^\phi$ that comes into play when the pores are heterogeneous (Brown and Korringa, 1975), regardless of whether the system is isotropic or anisotropic. I emphasize that all these formulas are rigorous statements based on the earlier anisotropic analyses. The appearance of both the Reuss average quantities $ K_R^d$ and $ \alpha_R$ is not an approximation, but merely a useful choice of notation.

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