A VEM-fully discrete polytopal scheme with bubble stabilisation for mixed-dimensional models with frictional contact at matrix--fracture interfaces
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The objective of this work 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. An extension of the first order Virtual Element Method (VEM) to contact-mechanics is proposed based on a mixed formulation combining a VEM discretisation of the displacement field with a face-wise constant approximation of the Lagrange multiplier accounting for the surface tractions along the fracture/fault network. The required stabilisation of such mixed formulation is obtained by bubble enrichment of the VEM displacement space at one side of each fracture face. This P1-bubble VEM - P0 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.
