For the discretization, we use a mortar method on non-matching meshes. More precisely, the flux is introduced as new variable at the interfaces playing the role of a Lagrange multiplier. This method can be applied to both the standard and the enhanced capillary model. To correctly model the penetration process into porous media with entry pressure, we introduce an inequality constraint. The weak formulation of which can be written as a variational inequality. As non-linear solver, we use a primal-dual active-set strategy which can be reformulated as semi-smooth Newton method. Several numerical examples demonstrate the efficiency and flexibility of the new algorithm.