Linear poroelasticity

For the linear poroelasticity model, the strain energy density is given by

\[ \psi = \frac{\firstlame_{u}}{2} (\trace \bm{\varepsilon})^2 + \secondlame_{d} \bm{\varepsilon} \tcolon \bm{\varepsilon} - \alpha \, M \, \trace \bm{\varepsilon} \, \zeta + \frac{1}{2} M \zeta^2, \]

where \(\firstlame_{u} = \firstlame_{d} + \alpha^2 M\) is undrained first Lamé parameter while \(\firstlame_{d}, \secondlame_{d}\) are Lamé parameters measured in drained condition, \(\alpha, M\) are the Biot effective stress coefficient and Biot modulus defined by

(102)\[ \begin{aligned} \alpha &= 1 - \frac{\bulk_{d}}{\bulk_s}, \\ \frac{1}{M} &= \frac{\phi}{\bulk_f} + \frac{\alpha - \phi}{\bulk_s}, \end{aligned} \]

with mixture bulk modulus \(\bulk_{d}\) measured in drained condition and solid and fluid bulk moduli \(\bulk_s, \bulk_f\), respectively.

To derive the constitutive law (stress-strain relationship) for the linear poroelasticity model we have

(103)\[ \begin{aligned} \bm\sigma(\bm{u}, p) &= \frac{\partial \psi}{\partial \bm{\varepsilon}} = \bm\sigma'(\bm{u}) - \alpha \, p \, \bm{I} = \firstlame_{d} \nabla\cdot \bm{u} \, \bm{I} + 2 \, \secondlame_{d} \, \bm{\varepsilon} - \alpha \, p \, \bm{I}, \\ p &= \frac{\partial \psi}{\partial \zeta} = M \left(\zeta - \alpha \nabla \cdot \bm{u} \right), \end{aligned} \]

where , \(\bm\sigma' = \firstlame_{d} (\trace \bm\varepsilon) \bm{I} + 2 \, \secondlame_{d} \, \bm{\varepsilon}\) is effective stress, \(p\) is pore pressure, and the variation of fluid content \(\zeta\) is

(104)\[ \zeta = \frac{p}{M} + \alpha \nabla\cdot \bm u. \]