The objective of this article is to address the discretisation of fractured/faulted poromechanical models using 3D polyhedral meshes in order to cope with the geometrical complexity of faulted geological models. A polytopal scheme is proposed for contact-mechanics, based on a mixed formulation combining a fully discrete space and suitable reconstruction operators for the displacement field with a face-wise constant approximation of the Lagrange multiplier accounting for the surface tractions along the fracture/fault network. To ensure the inf--sup stability of the mixed formulation, a bubble-like degree of freedom is included in the discrete space of displacements (and taken into account in the reconstruction operators). It is proved that this fully discrete scheme for the displacement is equivalent to a low-order Virtual Element scheme, with a bubble enrichment of the VEM space. This $\mathbb{P}^1$-bubble VEM--$\mathbb{P}^0$ mixed discretization is combined with an Hybrid Finite Volume scheme for the Darcy flow. All together, the proposed approach is adapted to complex geometry accounting for network of planar faults/fractures including corners, tips and intersections; it leads to efficient semi-smooth Newton solvers for the contact-mechanics and preserve the dissipative properties of the fully coupled model. Our approach is investigated in terms of convergence and robustness on several 2D and 3D test cases using either analytical or numerical reference solutions both for the stand alone static contact mechanical model and the fully coupled poromechanical model.

翻译：本文旨在解决使用三维多面体网格对含裂缝/断层的多孔弹性力学模型进行离散化的问题，以应对断层地质模型中几何复杂性带来的挑战。针对接触力学问题，本文提出了一种基于混合格式的多面体方法，该格式将全离散空间与位移场的适当重构算子相结合，并采用面片常数近似来处理沿裂缝/断层网络的面力拉格朗日乘子。为确保混合格式的inf-sup稳定性，在位移的离散空间中引入了气泡自由度（并纳入重构算子中）。证明了该全离散位移格式等价于具有气泡（bubble）富集VEM空间的一个低阶虚拟元格式。将该$\mathbb{P}^1$-bubble VEM--$\mathbb{P}^0$混合离散与用于达西流的混合有限体积格式相结合。综上，所提方法适用于包含具有角点、尖端和交叉点的平面断层/裂缝网络的复杂几何结构；它为接触力学问题提供了高效的半光滑牛顿求解器，并保留了全耦合模型的耗散特性。通过若干二维和三维算例，分别针对纯静态接触力学模型和全耦合多孔弹性力学模型，基于解析解或数值参考解，对所提方法的收敛性和鲁棒性进行了研究。