A model of groundwater flow in porous media influenced by wells (boreholes, channels) is considered. The model is motivated by the reduced dimension approach which suits fractured porous media problems, especially in granite rocks. The wells are modeled as lower dimensional 1d objects intersecting the surrounding bulk rock domain and causing singularities in the solution. The domains are discretized independently resulting in a mesh of mixed dimensions.

The extended finite element methods (XFEM) with a proper enrichment is applied to couple the flow between the wells and the bulk rock and to better approximate the singularities.

In this contribution, we derive and solve a mixed model for pressure and velocity. We suggest a new enrichment (similar to Stable generalized FEM) for velocity on top of standard zero order Raviart–Thomas finite elements. The properties of the suggested XFEM are validated on a set of numerical tests and the optimal convergence rate is demonstrated. Test cases both in 2d and 3d are presented.

考虑了受井（钻孔，通道）影响的多孔介质中地下水的流动模型。该模型是由减小尺寸的方法驱动的，该方法适合破裂的多孔介质问题，尤其是在花岗岩岩石中。井被建模为与周围的块状岩石区域相交并在解决方案中引起奇点的低维一维对象。域被独立离散，从而产生混合尺寸的网格。

具有适当富集的扩展有限元方法（XFEM）用于耦合井与大块岩石之间的流动并更好地近似奇点。

在此贡献中，我们推导并求解了压力和速度的混合模型。我们建议在标准零阶Raviart–Thomas有限元之上对速度进行新的富集（类似于稳定广义FEM）。建议的XFEM的特性在一组数值测试中得到了验证，并证明了最佳收敛速度。介绍了2d和3d中的测试用例。