\[ \small \begin{aligned} \mathbf{u}_i &= - \mathbf{K}_i \nabla p_i, \\ \nabla \cdot \mathbf{u}_i &= q_i, &&\mathrm{in} \, \Omega_i, \\ \mathbf{U}_f &=- a \mathbf{K}_{f, \tau} \nabla_\tau P_f, \\ \nabla_\tau \cdot\mathbf{U}_f &= q_f + \left( \mathbf{u}_1 \cdot \mathbf{n}_1 + \mathbf{u}_2 \cdot \mathbf{n}_2 \right), &&\mathrm{in} \, \Omega_f, \\ \mathbf{u}_i \cdot \mathbf{n}_i &= - \frac{2 k_\eta}{a} ( P_f - p_i ), &&\mathrm{in} \, \gamma_N, \\ P_f &= p_i, &&\mathrm{in} \, \gamma_D. \end{aligned} \]