Nitsche's and Mixed formulations of frictionless contact-mechanics for mixed-dimensional poro-mechanical models

  • Beaude, Laurence (BRGM)
  • Chouly, Franz ( Université de Bourgogne Franche-Comté Institut de Mathématiques de Bourgogne, CNRS)
  • Laaziri, Mohamed (Université Cote d'Azur, CNRS, Inria, LJAD)
  • Masson, Roland (Université Cote d'Azur, CNRS, Inria, LJAD)

Please login to view abstract download link

The problem of coupled fluid flow and rock mechanics is encountered in many areas of geoscience. Changes in the pore pressure of a geological formation due to injection or removal of fluid can lead to rock deformation. In applications such as CO2 sequestration, models that couple flow and mechanics are used to determine the safe injection pressures to avoid compromising the integrity of the caprock. In this work,we address the discretization of single-phase Darcy flows in a fractured and deformable porous medium, including frictional contact between the matrix-fracture interfaces. Fractures are described as a network of planar surfaces leading to the so-called mixed or hybrid-dimensional models. Small displacements and a linear elastic behavior are considered for the matrix. To simulate the coupled model, we employ a Hybrid Finite Volume scheme for the flow and for the contact mechanics we investigate two different formulations, a second-order $\left(\mathbb{P}_2\right)$ finite elements for the mechanical displacement coupled with face-wise constant $\left(\mathbb{P}_0\right)$ Lagrange multipliers on fractures, representing normal and tangential stresses, to discretize the frictional contact conditions [Bonaldi et al, JCP 2022] and the Nitsche's method which has been originally proposed by Nitsche in 1971, to treat boundary or interface conditions in a weak sense, with appropriate consistent terms that involve only the primary variables. Furthermore, no additional unknowns (Lagrange multiplier) are required and, therefore, no discrete inf-sup conditions need to be satisfied, unlike the mixed methods. Then, the Nitsche's formulation is compared with the mixed formulation. A variant of The Nistche's method is also investigated both theoretically and numerically which is shown to match the mixed formulation with piecewise constant Lagrange multipliers at the limit of large stabilization parameters. We present several numerical test cases to compare both formulations in terms of accuracy and robustness.