Variational inequalities for modeling flow in heterogeneous porous media
One of the driving forces in porous media flow is the capillary pressure. In standard models, it is given depending on the saturation. Recently, this relationship was enhanced by a dynamic retardation term which leads to a dependency on the saturation and its time-derivative. The situation becomes even complexer when heterogeneous porous media is considered. Here, the continuity condition for the capillary pressure does not guarantee that the saturation has to be continuous at the material interfaces. Moreover, to model capillary barriers, an entry pressure is often included into the capillary pressure relationship which has to be treated correctly in the numerical simulation.
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.