(results from the DuMuX fracture exercise)
(results from the DuMuX two-phase discrete fracture model)
Consider the model equation $\nabla \cdot \left( - \mathbf{K} \nabla p \right) = q$
\begin{equation} \begin{aligned} \\ \mathbf{u}_i &= - \mathbf{K}_i \nabla p_i \\ \nabla \cdot \mathbf{u}_i &= q_i &&\mathrm{in} \, \Omega_i, \\ \mathbf{U}_f &=- a \mathbf{K}_{f, \tau} \nabla_\tau P_f \\ \nabla_\tau \cdot\mathbf{U}_f &= q_f + \left( \mathbf{u}_1 \cdot \mathbf{n}_1 + \mathbf{u}_2 \cdot \mathbf{n}_2 \right) \\ \mathbf{u}_i \cdot \mathbf{n}_i &= - \frac{2 k_\eta}{a} ( P_f - p_i ) &&\mathrm{in} \, \gamma. \end{aligned} \end{equation}
Consider the model equation $\nabla \cdot \left( - \boldsymbol{\Lambda} \nabla u \right) = q$
\begin{equation} \begin{aligned} \\ \mathbf{u}_i &= - \mathbf{K}_i \nabla p_i \\ \nabla \cdot \mathbf{u}_i &= q_i &&\mathrm{in} \, \Omega_i, \\ \mathbf{U}_f &=- a \mathbf{K}_{f, \tau} \nabla_\tau P_f \\ \nabla_\tau \cdot\mathbf{U}_f &= q_f + \left( \mathbf{u}_1 \cdot \mathbf{n}_1 + \mathbf{u}_2 \cdot \mathbf{n}_2 \right) \\ P_f &= p_i &&\mathrm{in} \, \gamma. \end{aligned} \end{equation}