# 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

$$
\begin{aligned}
  \alpha &= 1 - \frac{\bulk_{d}}{\bulk_s}, \\
  \frac{1}{M} &= \frac{\phi}{\bulk_f} + \frac{\alpha - \phi}{\bulk_s},
\end{aligned}
$$ (biot-coefficient-modulus)

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

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


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

$$
  \zeta = \frac{p}{M} + \alpha \nabla\cdot \bm u.
$$ (fluid-variation2)