A bubble VEM-fully discrete polytopal scheme for mixed-dimensional poromechanics with frictional contact at matrix fracture interfaces

TL;DR

The paper addresses the challenge of discretizing mixed-dimensional poromechanical models with Coulomb friction at matrix-fracture interfaces on polyhedral meshes. It introduces a fully discrete scheme that combines a $\mathbb{P}^1$-bubble Virtual Element discretisation for the displacement with a $P^0$ Lagrange multiplier for fracture tractions, stabilized to satisfy the inf-sup condition, and couples this with Hybrid Finite Volume for the Darcy flow. The authors prove that this discretisation is equivalent to a low-order bubble-VEM formulation and establish discrete energy estimates, while demonstrating robustness and convergence through extensive 2D and 3D numerical experiments on complex fracture networks including corners and intersections. The approach enables energy-dissipative, mesh-flexible simulations of fractured poromechanics applicable to CO$_2$ storage and geothermal contexts, and lays the groundwork for further theoretical analysis (Korn and inf-sup proofs) and extensions to Corner Point Geometries.

Abstract

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.

Figures (22)

• Figure 1: Mixed-dimensional geometry with the fracture network $\Gamma$ and the matrix domain $\Omega\setminus\overline\Gamma$. The poromechanical unknowns are defined by the matrix pressure $p_m$, the fracture pressure $p_f$ and the displacement vector field $\mathbf{u}$ in the matrix domain.
• Figure 2: Conceptual fracture model with contact at asperities. ${\rm d}_{f}^{c}$ is the fracture aperture at contact state.
• Figure 3: Degrees of freedom $\mathbf{v}_{\mathcal{K}s}$, ${\mathcal{K}s}\in \overline{\mathcal{M}}_s$, at a given node $s \in \mathcal{V}_\Gamma$ with three intersecting fractures. $\mathbf{v}_{\mathcal{K}s}$ corresponds to the nodal unknown at node s located on the side $K$ of the fractures. Here, $\mathbf{u}$ is a fictive continuous function that $\mathbf{v}$ could interpolate, and is used to give a clearer meaning to the degrees of freedom on each side of the fracture.
• Figure 4: $\mathbb P^1$-bubble VEM
• Figure 5: Example of randomly perturbated Cartesian cell with non planar faces cut into two triangles.

Theorems & Definitions (8)

• Remark 3.1: Two-sided bubbles vs. one-sided bubbles
• Remark 3.2: Discrete jump
• Proposition 3.3: Discrete energy estimate
• proof
• Lemma 3.4: Link between discrete reconstructions and elliptic projectors
• proof
• Proposition 3.5
• proof