Poroelasticity

Linear

The strong form of linear poroelasticity (mixed \(\bm u - p\) formulation) for constitutive equation (103) 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:

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

with \(\bm n\) be the unit normal on the boundary and in the conservation of mass (second equation), we replaced \(\zeta\) from constitutive equation (104). The weak form can be derived as:

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

where we have used Darcy’s law \(\dot{\bm w} = - \varkappa \left(\nabla p - \bm f^f \right)\) (equation (33) for static case).

Matrix-free implementation

The matrix-free formulation for poroelasticity is similar to incompressible elasticity explained in Matrix-free implementation (mixed fields). However, in equation (241) we have time derivative of displacement and pressure fields and for deriving jacobian we need to add \(\mathrm{shift_v}\) as shown in (145).

• For linear poroelasticity described in (241) 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 (145).