(Poroelasticity)=
# Poroelasticity

## Linear

The strong form of linear poroelasticity (mixed $\bm u - p$ formulation) for constitutive equation {eq}`constitutive-linear-poroelastic` based on conservation of momentum (for static case) and mass may be stated as follows: Given body forces $\bm f^s, \bm f^f$ and volumetric injected fluid rate $\dot{\gamma}$, Dirichlet boundaries $\bar{\bm u}, \bar{p}$, applied traction $\bar{\bm t}$ and fluid flux $\bar{s}$ and initial conditions $\bm u_0, p_0$, find the displacement and pressure variables $(\bm u, p) \in \mathcal{V} \times \mathcal{Q} $ (here $\mathcal{V} = H^1(\Omega), \mathcal{Q} = H^1(\Omega) $ ), such that:

$$
  \begin{aligned}
    -\nabla \cdot \bm \sigma - \bm f^s   &= 0, \qquad \text{in $\Omega$} \\
    \frac{\dot{p}}{M} + \alpha \nabla \cdot \dot{\bm u} + \nabla\cdot \dot{\bm{w}} - \dot{\gamma} &= 0, \qquad \text{in $\Omega$} \\
    \bm{u} &= \bar{\bm u}, \quad    \text{on $\partial \Omega^{D}$} \\
    p &= \bar{p}, \quad    \text{on $\partial \Omega^{D}$} \\
    \bm \sigma \cdot \bm n &= \bar{\bm t}, \qquad \text{on $\partial \Omega^{N}$} \\
    \dot{\bm{w}} \cdot \bm n &= \bar{s}, \qquad \text{on $\partial \Omega^{N}$} \\
    \bm u &= \bm u_0, \qquad \text{in $\Omega$} \\
    p &= p_0, \qquad \text{in $\Omega$}
  \end{aligned}
$$ (poro-linear-strong-form)

with $\bm n$ be the unit normal on the boundary and in the conservation of mass (second equation), we replaced $\zeta$ from constitutive equation {eq}`fluid-variation2`. The weak form can be derived as:

$$
  \begin{aligned}
    \int_{\Omega}{ \nabla \bm{v} \tcolon \bm{\sigma}} \, dv
    - \int_{\partial \Omega}{\bm{v} \cdot \bar{\bm t}} \, da
    - \int_{\Omega}{\bm{v} \cdot \bm f^s} \, dv
    &= 0, \quad \forall \bm v \in \mathcal V \\
    \int_{\Omega} q \left(\frac{\dot{p}}{M} + \alpha \nabla \cdot \dot{\bm u} - \dot{\gamma} \right) \, dv + \int_{\Omega} \nabla q \cdot \left( \varkappa \nabla p - \varkappa \bm f^f \right) + \int_{\partial \Omega}{ q \, \bar{s}} \, da & = 0. \quad \forall q \in \mathcal Q
  \end{aligned}
$$ (poro-linear-weak-form)

where we have used Darcy's law $\dot{\bm w} = - \varkappa \left(\nabla p - \bm f^f \right)$ (equation {eq}`generalized-darcy` for static case).

## Matrix-free implementation

The matrix-free formulation for poroelasticity is similar to incompressible elasticity explained in {ref}`matrix-free-mixed-fields`. However, in equation {eq}`poro-linear-weak-form` we have time derivative of displacement and pressure fields and for deriving jacobian we need to add $\mathrm{shift_v}$ as shown in {eq}`time-dependent-jacobian`.

* For linear poroelasticity described in {eq}`poro-linear-weak-form` we have

$$
  \begin{aligned}
    \bm f_0 &= -\bm f^s, \quad \bm f_1 = \bm{\sigma}(\bm u, p) = \firstlame_{d} \nabla\cdot \bm{u} \, \bm{I} + 2 \, \secondlame_{d} \, \bm{\varepsilon} - \alpha \, p \, \bm{I}, \\
    \bm g_0 &=\frac{\dot{p}}{M} + \alpha \nabla \cdot \dot{\bm u} - \dot{\gamma}, \quad \bm g_1 = \varkappa \nabla p - \varkappa \bm f^f, \\
		\diff \bm f_1 &= \bm f_1(\diff \bm{u}, \diff p), \quad \diff \bm g_0 = \mathrm{shift_v} \left( \frac{\diff p}{M} + \alpha \nabla \cdot \diff \bm{u} \right), \quad \diff \bm g_1 = \varkappa \nabla \diff p.
  \end{aligned}
$$

where $\mathrm{shift_v}$ evaluated at time  $\mathrm{shift_v} = \frac{\partial\dot{\phi}}{\partial\phi}|_{\phi_n}$ and we have used the linearization of time dependent variables explained {eq}`time-dependent-jacobian`.