Mechanics of stratified anisotropic poroelastic media

Drained scenario ( $p_f \equiv 0$ )

Now, recall that, in the drained scenario, changes in pore-fluid pressure are assumed to be zero (or at least negligibly small), so $p_f \equiv 0$ in these equations. Accounting for this condition, the results should (and do) recover the Backus (1962) and Schoenberg and Muir (1989) results for the elastic parts of the system (found in Appendix A) exactly. Also, I find the additional (expected) result for the poroelastic case that the average fluid increment is:

$$\left<\zeta\right> = \left<\beta_1\sigma_{11}\right> + \left<\beta_2\sigma_{22}\right> + \left<\beta_3\right>\sigma_{33},$$ (35)

if $\sigma_{33}$ is nearly constant. Or, if $\sigma_{33}$ is not uniform from one layer to the next (as might happen due to weight of solid overburden pressure), then the third expression in (35) should be modified, by moving $\sigma_{33}$ inside the averaging operator. So then I have

$$\left<\zeta\right> = \left<\beta_1\sigma_{11}\right> + \left<\beta_2\sigma_{22}\right> + \left<\beta_3\sigma_{33}\right>,$$ (36)

whenever $\sigma_{33}$ taken constant is a poor approximation. The results shown in (35) and (36) are easy to reconcile with the definitions of the $\beta$ 's, and the meaning of averaging operator $\left<\cdot\right>$ across all layers. When $p_f$ vanishes everywhere, the final results for the averaging and the various stresses and strains are identical to the results in Appendix A. For the drained scenario, the only difference is the addition of equations (35) or (36).

Mechanics of stratified anisotropic poroelastic media