Mechanics of stratified anisotropic poroelastic media

The $\beta_i$ coefficients and effective stress

Making use of the previous definitions, it is easy to see that the coefficients $\beta_i$ are closely related to different sort of effective stress coefficient, for the individual principal strain coefficients:

$$e_{ii} = -\frac{1}{3\overline{K}_i^d}(p_c - D_ip_f), \qquad\hbox{for}\qquad i = 1,2,3,$$ (17)

where

$$D_i = 3\overline{K}_i^d\beta_i = 1 - \frac{\overline{K}_i^d}{\overline{K}_i^g}, \qquad\hbox{for}\qquad i = 1,2,3,$$ (18)

and $-p_c = \sigma_{11} = \sigma_{22} = \sigma_{33}$ in the case of uniform applied confining pressure $p_c$ . Then clearly, the $D_i$ 's are completely analogous to the usual Biot [or Biot and Willis (1957)] coefficient $\alpha_R = 1 - K_R^d/K_R^g$ , as commonly defined for isotropic poroelasticity.