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：本文旨在解决断裂/断层孔隙力学模型的三维多面体网格离散化问题，以应对断层地质模型的几何复杂性。针对接触力学问题，本文提出了一种基于混合形式的多面体格式，该格式将位移场的全离散空间与合适的重构算子相结合，并采用面片常数近似方法处理沿断裂/断层网络的拉格朗日乘子（表征表面牵引力）。为确保混合形式的inf-sup稳定性，在位移离散空间中引入气泡型自由度（并纳入重构算子考量）。理论证明，该位移全离散格式等价于低阶虚拟单元方法（Virtual Element Method, VEM）的气泡富集格式。这种$\mathbb{P}^1$气泡VEM-$\mathbb{P}^0$混合离散格式与达西流动的混合有限体积法相结合。综合而言，所提方法适用于含平面断层/断裂网络（包括角点、尖端和交叉点）的复杂几何构型，可构建接触力学的高效半光滑牛顿求解器，并保持全耦合模型的耗散特性。本文通过2D和3D测试案例（采用解析解或数值参考解）分别验证了纯静态接触力学模型及全耦合孔隙力学模型的收敛性和鲁棒性。