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 and 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 .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 )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 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:
When the external stress is hydrostatic so , equation (51) telescopes down to
Comparing (51) and (54), we find easily that
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 in the layer, , and .So the undrained bulk modulus of the layer is
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 appeared explicitly before, now 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 or .
Carrying through this analysis for our main estimate Geff(2), we find, after some rearrangement of terms, that
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.