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 *p*_{f} in the fluid is the new field parameter that can be
controlled. Allowing sufficient time for global pressure equilibration will
permit us to consider *p*_{f} 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 *p*_{f}.
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:

(45) |

(46) |

(47) |

When the external stress is hydrostatic so , equation (51) telescopes down to

(48) |

(49) |

Comparing (51) and (54), we find easily that

(50) |

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

(51) |

(52) |

(53) |

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 *a*_{u}, *b*_{u}, *c*_{u}, and *f*_{u} for the
undrained system. The constants without *u* subscripts are assumed to
be drained, *i.e.*, *a* = *a*_{dr}, 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 *G*_{eff}^{(2)},
we find, after some rearrangement of terms, that

(54) |

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.

7/8/2003