In this paper, we consider unsaturated filtration in heterogeneous porous media with rough surface topography. The surface topography plays an important role in determining the flow process and includes multiscale features. The mathematical model is based on the Richards’ equation with three different types of boundary conditions on the surface: Dirichlet, Neumann, and Robin boundary conditions. For coarse-grid discretization, the Generalized Multiscale Finite Element Method (GMsFEM) is used. Multiscale basis functions that incorporate small scale heterogeneities into the basis functions are constructed. To treat rough boundaries, we construct additional basis functions to take into account the influence of boundary conditions on rough surfaces. We present numerical results for two-dimensional and three-dimensional model problems. To verify the obtained results, we calculate relative errors between the multiscale and reference (fine-grid) solutions for different numbers of multiscale basis functions. We obtain a good agreement between fine-grid and coarse-grid solutions.

中文翻译：

---

具有粗糙表面形貌的非均质多孔介质中不饱和流动问题的多尺度模型还原

在本文中，我们考虑了具有粗糙表面形貌的非均质多孔介质中的不饱和过滤。表面形貌在确定流动过程中起着重要作用，并且包括多尺度特征。数学模型基于Richards方程，表面上具有三种不同类型的边界条件：Dirichlet，Neumann和Robin边界条件。对于粗网格离散化，使用广义多尺度有限元方法（GMsFEM）。构建将小规模异质性纳入基本函数的多尺度基本函数。为了处理粗糙边界，我们构造了附加的基函数来考虑边界条件对粗糙表面的影响。我们提出了二维和三维模型问题的数值结果。为了验证所获得的结果，我们针对不同数量的多尺度基函数计算了多尺度和参考（细网格）解决方案之间的相对误差。我们在细网格解决方案和粗网格解决方案之间达成了良好的协议。