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Now we want to broaden our outlook and suppose that the materials composing the layers are not homogeneous isotropic elastic materials, but rather poroelastic materials containing voids or pores. The pores may be air-filled, or alternatively they may be partially or fully saturated with a liquid, some gas, or a fluid mixture. For simplicity, we will suppose here that the pores are either air-filled or they are fully saturated with some other homogeneous fluid. When the porous layers are air-filled, it is generally adequate to assume that the analysis of the preceding section holds, but with the new interpretation that the Lamé parameters are those for the porous elastic medium in the absence of saturating liquids. The resulting effective constants $\lambda_{dr}$ and $\mu_{dr}$are then said to be those for ``dry'' -- or somewhat more accurately ``drained'' -- conditions. These constants are also sometimes called the ``frame'' constants, to distinguish them from the constants associated with the solid materials composing the frame, which are often called the ``grain'' or ``mineral'' constants.

One simplification that arises immediately is due to the fact, according to a quasi-static analysis of Gassmann (1951), that the presence of pore fluids has no mechanical effect on the layer shear moduli, so $\mu_{dr} = \mu$.There may be other effects on the layer shear moduli due to the presence of pore fluids, such as softening of cementing materials or expansion of interstitial clays, which we will term ``chemical'' effects to distinguish them from the purely mechanical effects to be considered here. We neglect all such chemical effects in the following analysis. This means that the lamination analysis for the four effective shear moduli (l, l, m, m) associated with eigenvectors (since they are uncoupled from the analysis involving $\lambda$)does not change in the presence of fluids. Thus, equations (15) and (16) continue to apply for the poroelastic problem, and we can therefore simplify our system of equations in order to focus on the parts of the analysis that do change in the presence of fluids.

The presence of a saturating pore fluid introduces the possibility of an additional control field and an additional type of strain variable. The pressure pf in the fluid is the new field parameter that can be controlled. Allowing sufficient time for global pressure equilibration will permit us to consider pf to be a constant throughout the percolating (connected) pore fluid, while restricting the analysis to quasistatic processes. The change $\zeta$ in the amount of fluid mass contained in the pores [see Berryman and Thigpen (1985)] is the new type of strain variable, measuring how much of the original fluid in the pores is squeezed out during the compression of the pore volume while including the effects of compression or expansion of the pore fluid itself due to changes in pf. It is most convenient to write the resulting equations in terms of compliances rather than stiffnesses, so the basic equation for an individual layer in the stack of layers to be considered takes the form:
{c} e_{11} e_{22} e_{33} -\zeta\\ end{arra...
 ... \sigma_{11} \sigma_{22} \sigma_{33} - p_f \\ end{array}\right).
 \end{eqnarray} (45)
The constants appearing in the matrix on the right hand side will be defined in the following two paragraphs. It is important to write the equations this way rather than using the inverse relation in terms of the stiffnesses, because the compliances Sij appearing in (51) are simply related to the drained constants $\lambda_{dr}$ and $\mu_{dr}$ in the same way they are related in normal elasticity, whereas the individual stiffnesses obtained by inverting the equation in (51) must contain coupling terms through the parameters $\beta$ and $\gamma$ that depend on the pore and fluid compliances. Thus, we find easily that
S_{11} = {{1}\over{E_{dr}}} =
S_{12} = - {{\nu_{dr}}\over{E_{dr}}},
 \end{eqnarray} (46)
where the drained Young's modulus Edr is defined by the second equality of (52) and the drained Poisson's ratio is determined by
\nu_{dr} = {{\lambda_{dr}}\over{2(\lambda_{dr}+\mu)}}.
 \end{eqnarray} (47)

When the external stress is hydrostatic so $\sigma= \sigma_{11} = \sigma_{22} =
\sigma_{33}$, equation (51) telescopes down to
e -\zeta\end{array}\right) =
{cc} \sigma-p_f \\ end{array}\right),
 \end{eqnarray} (48)
where e = e11 + e22 + e33, $K_{dr} = \lambda_{dr} +
{2\over3}\mu$ is the drained bulk modulus, $\alpha= 1 - K_{dr}/K_m$ is the Biot-Willis parameter (Biot and Willis, 1957) with Km being the bulk modulus of the solid minerals present, and Skempton's pore-pressure buildup parameter B (Skempton, 1954) is given by
B = {{1}\over{1 + K_p(1/K_f - 1/K_m)}}.
 \end{eqnarray} (49)
New parameters appearing in (55) are the bulk modulus of the pore fluid Kf and the pore modulus $K_p^{-1} = \alpha/\phi K_{dr}$ where $\phi$ is the porosity. The expressions for $\alpha$ and B can be generalized slightly by supposing that the solid frame is composed of more than one constituent, in which case the Km appearing in the definition of $\alpha$is replaced by Ks and the Km appearing explicitly in (55) is replaced by $K_{\phi}$ [see Brown and Korringa (1975), Rice and Cleary (1976), Berryman and Milton (1991), Berryman and Wang (1995)]. This is an important additional complication (Berge and Berryman, 1995), but one that we choose not to pursue here.

Comparing (51) and (54), we find easily that
\beta= {{\alpha}\over{3K_{dr}}}\qquad\hbox{and}\qquad \gamma= {{\alpha}\over{BK_{dr}}}.
 \end{eqnarray} (50)
All the constants are defined now in terms of ``easily'' measureable quantities.

When the mechanical changes (i.e., applied stress or strain increments) in the system happen much more rapidly than the fluid motion can respond, the system is ``undrained'' -- a situation commonly modeled by taking $\zeta = 0$ in the layer, $p_f = B\sigma$, and $e = (1-\alpha B)\sigma/K_{dr}$.So the undrained bulk modulus of the layer is
K_{u} \equiv {{K_{dr}}\over{(1-\alpha B)}}.
 \end{eqnarray} (51)
Then, it is not hard to show that
{c} \sigma_{11} \sigma_{22} \sigma_{33} \\...
{c} e_{11} e_{22} e_{33} \\ end{array}\right),
 \end{eqnarray} (52)
where the undrained Lamé constant is given by
\lambda_u = K_u - {{2\mu}\over{3}}.
 \end{eqnarray} (53)
For our purposes, the language used here to describe drained/undrained layers is intended to convey the same meaning as opened/closed pores at the boundaries separating the layers. If pores are open, fluid can move between layers, so pressure can equilibrate over long periods of time. If the pores are closed at these interfaces, pressure equilibration can occur within each layer, but not between layers -- no matter how long the observation time.

With (58) and (59), we can now repeat the Backus analysis for a layered system in which each layer is undrained. The only difference is that everywhere $\lambda$ appeared explicitly before, now $\lambda_u$ is substituted. Furthermore, the constants resulting from the Backus lamination analysis can now be distinguished as au, bu, cu, and fu for the undrained system. The constants without u subscripts are assumed to be drained, i.e., a = adr, etc. The constants l and m are the same in both drained and undrained systems, since they do not depend on either $\lambda$ or $\lambda_u$.

Carrying through this analysis for our main estimate Geff(2), we find, after some rearrangement of terms, that
G_{eff}^{(2)} = m - {{4c_u}\over{3}}\left[
- \left<{{\mu}\over{\lambda_u + 2\mu}}\right\gt^2\right].
 \end{eqnarray} (54)
Equation (60) is the main result of this paper. It incorporates all the earlier work by making use of the effective shear modulus Geff(2) selected in the end by using energy estimates. Then it is evaluated explicitly here for the simple layered system. The term in square brackets is in Cauchy-Schwartz form (i.e., $\left<\alpha^2\right\gt\left<\beta^2\right\gt \ge \left<\alpha\beta\right\gt^2$), and thus this term is always nonnegative. The contribution of this term is therefore a nonpositive correction to the leading term m. Further, it is clear that the correction is second order in the fluctuations in the shear modulus throughout the layered material. This fact means that fluctuations must be fairly large before any effect can be observed, since there are second order subtractions but no first order corrections at all. It is also easy to show from this formula that the range of possible values for this shear modulus is $l \le G_{eff}^{(2)} \le m$, because as Ku in the layers ranges from zero to infinity the corrections to m range from l-m to . Finally, we note that the effect of an increase in the undrained constant $\lambda_u$ is to reduce the magnitude of these correction terms. Thus, since a reduction in a negative contribution leads to a positive contribution, it is also clear that an increase in Skempton's coefficient B always leads to an increase in the effective shear modulus Geff(2), as we expected.

Since each layer in the stack is isotropic, the result (60) may appear to contradict the earlier results of Berryman and Wang (2001), showing that shear modulus dependence on fluid properties does not and cannot occur in microhomogeneous isotropic media. But these layered media are not isotropic everywhere. In particular, if we consider a point right on the boundary between any two layers, the elastic properties look very anisotropic at these points. So it is exactly at, and near the vicinity of, these boundary points where deviations from Gassmann's results can and do arise, leading to the result (60).

We will check these ideas by showing some numerical examples in the next section.

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