Lubrication boundary condition for a crack in a Darcy medium (${\partial }_x(h^3{p}_x/12)+k[p_y{]}_{-}^{+}=0$)

In a thin fluid-filled crack, lubrication theory gives flux per unit depth $q=-h^3{p}_x/(12\mu )$. The surrounding Darcy flow has normal velocity $-k{p}_y/\mu$. Conservation in the crack gives $q_x=-[u_y{]}_{-}^{+}$, yielding the displayed condition. Pressure is continuous across the crack. This reduced model neglects transverse pressure variation and assumes wall slip lengths are small compared with $h$.