Mechanics of stratified anisotropic poroelastic media |

Throughout most of the paper, I will not introduce '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 (no summation over repeated indices) are strains in the directions. The are the corresponding stresses, assumed to be positive in tension. The fluid pressure is , which is positive in compression. The increment of fluid content is , and is often defined via:

where 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, is the corresponding pore volume, with being the fluid-saturated porosity of the same volume. is the volume occupied by the pore-fluid, so that before any new deformations begin. The '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 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 (

where is again the Reuss average modulus of the grains. The drained Reuss average bulk modulus is defined by

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

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, , where 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
,
,
and stresses
,
,
) 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 are the elements of the compliance
matrix
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
, 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
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 (
), 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

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

Furthermore, I have

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

Note that both (8) and (9) contain dependence on the distinct

Mechanics of stratified anisotropic poroelastic media |

2010-05-19