Numerical analysis of a mixed-dimensional poromechanical model with frictionless contact at matrix-fracture interfaces

TL;DR

This paper tackles the numerical analysis of a coupled flow–mechanics model in fractured porous media, where fractures are treated as mixed-dimensional interfaces and contact is frictionless. The authors develop a Gradient Discretization Method (GDM) framework that unifies a broad class of discretizations for flow (including Hybrid Finite Volume) with a mixed mechanical formulation using Lagrange multipliers for contact. They prove convergence of discrete solutions to a weak solution of the continuous problem, deriving robust energy estimates and compactness results that also imply the existence of a weak solution. A 2D numerical experiment using HFV for flow and a mixed P2–P0 discretization for contact demonstrates the method’s convergence behavior and captures fracture-induced phenomena such as pressure transfer and aperture evolution, highlighting the approach’s practical viability for complex fracture networks.

Abstract

We present a complete numerical analysis for a general discretization of a coupled flow-mechanics model in fractured porous media, considering single-phase flows and including frictionless contact at matrix-fracture interfaces, as well as nonlinear poromechanical coupling. Fractures are described as planar surfaces, yielding the so-called mixed- or hybrid-dimensional models. Small displacements and a linear elastic behavior are considered for the matrix. The model accounts for discontinuous fluid pressures at matrix-fracture interfaces in order to cover a wide range of normal fracture conductivities. The numerical analysis is carried out in the Gradient Discretization framework, encompassing a large family of conforming and nonconforming discretizations. The convergence result also yields, as a by-product, the existence of a weak solution to the continuous model. A numerical experiment in 2D is presented to support the obtained result, employing a Hybrid Finite Volume scheme for the flow and second-order finite elements ($\mathbb P_2$) for the mechanical displacement coupled with face-wise constant ($\mathbb P_0$) Lagrange multipliers on fractures, representing normal stresses, to discretize the contact conditions.

Figures (9)

• Figure 1: Illustration of the dimension reduction in the fracture aperture for a 2D domain $\Omega$ with three intersecting fractures $\Gamma_i$, $i\in\{1,2,3\}$. Equi-dimensional geometry on the left, mixed-dimensional geometry on the right.
• Figure 2: Conceptual fracture model with contact at asperities, $d_0$ being the fracture aperture at contact state.
• Figure 3: Example of triangular mesh with three fracture edges in bold. The discrete unknowns are presented for the mixed-dimensional HFV discretization of the flow model in (a), and for the $\mathbb P_2$--$\mathbb P_0$ mixed discretization of the contact mechanics in (b). The discontinuities of the pressures and of the displacement are captured at matrix--fracture interfaces. Note that a nodal fracture pressure unknown is directly eliminated (hence not represented) when shared by two fracture edges, while it is kept as an additional discrete unknown when shared by three or more fracture edges.
• Figure 4: Two-dimensional, $2\times1$ m domain containing six fractures. Fracture 1 comprises two sub-fractures making a corner, and fracture 5 has a tip on the boundary. The contact state along each fracture over time is also shown.
• Figure 5: Variation of the matrix and fracture pressures (with respect to the initial value $p_m^0=p_f^0=~10^5$ Pa) at $t=T/4$ and $t=T$.

Theorems & Definitions (19)

• Definition 3.1: Weak solution
• Remark 3.2: Alternate formulation for the mechanical equations
• Lemma 4.1: Local contact conditions
• Theorem 5.1: Convergence of the gradient scheme
• Remark 5.2: GDM properties and existence of a solution to the weak formulation
• Theorem 5.3: Energy estimates for \ref{['eq:GS']}
• Remark 5.4: Assumptions on fracture width and porosity
• Remark 5.5: Time-dependent source term
• Theorem 5.6: Existence of a discrete solution
• Proposition 5.7: Estimates on the time translates of $d_{f,\mathcal{D}_\mathbf{u}}$ and $\phi_\mathcal{D}$