RELATIONS FOR ANISOTROPY IN POROELASTIC MATERIALS

Gassmann (1951), Brown and Korringa (1975), and others have considered the problem of obtaining effective constants for anisotropic poroelastic materials when the pore fluid is confined within the pores. The confinement condition amounts to a constraint that the increment of fluid content $\zeta= 0$, while the external loading $\sigma$ is changed and the pore-fluid pressure p_f is allowed to respond as necessary and thus equilibrate.

To provide an elementary derivation of the Gassmann equation for anisotropic materials, we consider the anisotropic generalization of (1)

$$\begin{eqnarray} \left( \begin{array} {c} e_{11} \\ e_{22} \\ e_{33} \\ -\zet... ... \\ \sigma_{22} \\ \sigma_{33} \\ -p_f \\ \end{array}\right). \end{eqnarray}$$ (9)

Three shear contributions have been immediately excluded from consideration since they can easily be shown not to interact mechanically with the fluid effects. This form is not completely general in that it includes orthorhombic, cubic, hexagonal, and all isotropic systems, but excludes triclinic, monoclinic, trigonal, and some tetragonal systems that would have some nonzero off-diagonal terms in the full elastic matrix. Also, we have assumed that the material axes are aligned with the spatial axes. But this latter assumption is not significant for the derivation that follows. Such an assumption is important when properties of laminated materials having arbitrary orientation relative to the spatial axes need to be considered, but we do not treat this more general problem here.

If the fluid is confined (or undrained on the time scales of interest), then $\zeta\equiv 0$ in (9) and p_f becomes a linear function of $\sigma_{11}$, $\sigma_{22}$,$\sigma_{33}$. Eliminating p_f from the resulting equations, we obtain the general expression for the strain dependence on external stress under such confined conditions:

$$\begin{eqnarray} \left( \begin{array} {c} e_{11} \\ e_{22} \\ e_{33} \\ \end{... ...igma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \end{array}\right). \end{eqnarray}$$ (10)

The s_ij's are fluid-drained constants, while the s^*_ij's are the fluid-undrained (or fluid-confined) constants.

The fundamental result (10) was obtained earlier by both Gassmann (1951) and Brown and Korringa (1975), and may be written simply as

$$\begin{eqnarray} s^*_{ij} = s_{ij} - {{\beta_i\beta_j}\over{\gamma}},\quad\hbox{for}\quad i,j = 1,2,3. \end{eqnarray}$$ (11)

This expression is just the anisotropic generalization of the well-known Gassmann equation for isotropic, microhomogeneous porous media.