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General analysis for layered poroelastic systems

I will next formulate the layered porelastic earth problem in a way so that both of these standard boundary conditions can be imposed, as needed in any particular modeling problem.

Now I assume throughout the rest of the paper that the porous layers are stacked vertically ($ 3$ - or $ z$ -axis) and for this geometry it is easy to see that the three horizontal strains $ e_{11}$ , $ e_{22}$ , and $ e_{12}$ must be continuous if the layers are in solid-welded contact. Furthermore, the vertical stress $ \sigma_{33}$ , and rotational stresses involving the vertical direction $ \sigma_{13}$ and $ \sigma_{23}$ must also be continuous. These conditions are equivalent to an assumption of welded contact between layers. If contact is not welded, then the system can have more complicated behaviors than those being considering here.

Appendix A summarizes the Backus (1962) and/or Schoenberg and Muir (1989) approach to elastic layer averaging. The method I present here provides a small generalization of this approach, taking the presence of the pore-fluid into account. For the drained situation, the influence of the fluid on the system mechanics is minimal (as will be shown). But I should nevertheless have this result available to compare it with the more interesting case of the undrained layers.

Although the shear moduli normally associated with the twisting shear components $ e_{23}$ , $ e_{31}$ , and $ e_{12}$ usually do not interact with the pore-fluid itself in systems as symmetric or more symmetric than orthotropic, I nevertheless need to carry these terms along in the poroelastic formulation for layered systems because of possible boundary effects due to welded contact at interfaces. To accomplish this goal, I will generalize the form of equation (42) from Appendix A. In compliance form, the equations will relate the strains

$\displaystyle E_T \equiv \left(\begin{array}{c} e_{11} \cr e_{22} \cr e_{12} \e...
... \equiv \left(\begin{array}{c} e_{33} \cr e_{32} \cr e_{31} \end{array}\right),$ (27)

and fluid increment $ \zeta$ to the stresses

$\displaystyle \Pi_T \equiv \left(\begin{array}{c} \sigma_{11} \cr \sigma_{22} \...
...begin{array}{c} \sigma_{33} \cr \sigma_{32} \cr \sigma_{31} \end{array}\right),$ (28)

and the fluid pressure change $ p_f$ .

The required general relationship is:

$\displaystyle \left(\begin{array}{c} E_T \cr -\zeta \cr E_N \end{array}\right) ...
...rray}\right) \left(\begin{array}{c} \Pi_T \cr -p_f \cr \Pi_N\end{array}\right),$ (29)

where, for example, in the orthotropic media considered here I have:

$\displaystyle {\bf S}_{TT} \equiv \left(\begin{array}{ccc} s_{11} & s_{12} & s_...
...ccc} s_{11} & s_{12} & \cr s_{21} & s_{22} & \cr & & s_{66} \end{array}\right),$ (30)

$\displaystyle {\bf S}_{NN} \equiv \left(\begin{array}{ccc} s_{33} & s_{34} & s_...
...\begin{array}{ccc} s_{33} & & \cr & s_{44} & \cr & & s_{55} \end{array}\right),$ (31)


$\displaystyle {\bf S}_{NT} \equiv \left(\begin{array}{ccc} s_{31} & s_{32} & s_...
...ft(\begin{array}{ccc} s_{31} & s_{32} & \cr & 0 & \cr & & 0 \end{array}\right),$ (32)

with $ {\bf S}_{TN} = {\bf S}_{NT}^T$ (the $ T$ superscript indicates the matrix transpose). Here all these expressions for elastic compliance refer specifically to drained compliances $ s_{ij} = s^d_{ij}$ , for all $ i,j = 1,\ldots,6$ within each poroelastic anisotropic layer.

All the poroelastic contributions to (29) are determined by $ \gamma$ , $ {\bf g}_{12}$ , and $ {\bf g}_3$ . The scalar $ \gamma$ within the $ 7\times7$ matrix in (29) was defined earlier in (8), and is the only term in the $ 7\times7$ matrix that includes fluid effects directly through $ K_f$ . The remaining pair of vectors contained within the $ 7\times7$ matrix in (29) is defined by:

$\displaystyle {\bf g}_{12}^T = (\beta_1, \beta_2, 0)$ (33)


$\displaystyle {\bf g}_{3}^T = (\beta_3, 0, 0),$ (34)

where the $ \beta$ 's were defined previously following (1).

I now consider two examples of special uses of the general equation (29) for different choices of boundary conditions. These two physical circumstances covered in the cases considered are distinct end-members. For relatively high-frequency wave propagation, it is appropriate to consider that the fluids do not have time to equilibrate, and therefore fluid pressures can be different in distinct layers, while the fluid particles do not have time to move very far during wave passage time, so the fluid increment $ \zeta = 0$ essentially everywhere. This situation is called the ``undrained'' condition. An alternative condition is the fully drained condition, in which the fluid particles have as much time as they need to achieve fluid-pressure equilibration, so that $ p_f =$ constant. These two limiting situations are clearly connected physically via Darcy's law, which provides the mechanism to move fluid particles, and ultimately to guarantee that the fluid pressure reaches an equilibrium state. Bringing Darcy's law actively into oplay in the equations would result in Biot-style equations which are beyond my current scope. So I consider only the end-member conditions for the present contribution.

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