This paper proposes two algorithms to impose seepage boundary conditions in the context of Richards' equation for groundwater flows in unsaturated media. Seepage conditions are non-linear boundary conditions, that can be formulated as a set of unilateral constraints on both the pressure head and the water flux at the ground surface, together with a complementarity condition: these conditions in practice require switching between Neumann and Dirichlet boundary conditions on unknown portions on the boundary. Upon realizing the similarities of these conditions with unilateral contact problems in mechanics, we take inspiration from that literature to propose two approaches: the first method relies on a strongly consistent penalization term, whereas the second one is obtained by an hybridization approach, in which the value of the pressure on the surface is treated as a separate set of unknowns. The flow problem is discretized in mixed form with div-conforming elements so that the water mass is preserved. Numerical experiments show the validity of the proposed strategy in handling the seepage boundary conditions on geometries with increasing complexity.

翻译：本文针对非饱和介质中地下水流动的Richards方程，提出了两种施加渗流边界条件的算法。渗流条件是非线性边界条件，可表述为地表处压力水头和水分通量的一组单边约束，并附带互补条件：这些条件在实际应用中需要在边界未知区域上切换Neumann和Dirichlet边界条件。通过认识到这些条件与力学中单边接触问题的相似性，我们借鉴相关文献提出了两种方法：第一种方法依赖于强相容的惩罚项，而第二种方法通过杂交法获得，其中地表压力值被作为独立的未知量组处理。流动问题采用满足散度协调的混合元进行离散化，从而保持水体质量守恒。数值实验表明，所提策略在处理几何复杂度递增情况下的渗流边界条件时具有有效性。