On the parallel solution of hydro-mechanical problems with fracture networks and contact conditions

TL;DR

The paper tackles hydro-mechanical coupling in fractured rock by adopting a discrete fracture-matrix (DFM) framework that represents the rock mass and fracture network with coupled poroelastic equations. It introduces a stable finite element discretization using P1 displacements, RT0 velocities, and P0 pressures, and solves the resulting nonlinear system via a robust fixed-stress iterative splitting that decouples the flow and mechanical subproblems. The mechanical component is formulated as a quadratic programming (QP) problem to enforce non-penetration constraints on fractures and is solved efficiently through a dual MPGP-based approach within an in-house library; the flow subproblem remains linear and is solved with parallel solvers. The method is demonstrated on 2D validation and a 3D tunnel-excavation benchmark with hundreds of fractures, showing numerical scalability and the ability to handle large DFN-driven hydro-mechanical responses in near-realistic settings. The work advances practical large-scale simulations of fractured media by integrating DFM poroelasticity, robust iterative coupling, and efficient QP solvers, with potential extensions to friction laws and domain-decomposition techniques for even larger problems.

Abstract

The paper presents a numerical method for simulating flow and mechanics in fractured rock. The governing equations that couple the effects in the rock mass and in the fractures are obtained using the discrete fracture-matrix approach. The fracture flow is driven by the cubic law, and the contact conditions prevent fractures from self-penetration. A stable finite element discretization is proposed for the displacement-pressure-flux formulation. The resulting nonlinear algebraic system of equations and inequalities is decoupled using a robust iterative splitting into the linearized flow subproblem, and the quadratic programming problem for the mechanical part. The non-penetration conditions are solved by means of dualization and an optimal quadratic programming algorithm. The capability of the numerical scheme is demonstrated on a benchmark problem for tunnel excavation with hundreds of fractures in 3D. The paper's novelty consists in a combination of three crucial ingredients: (i) application of discrete fracture-matrix approach to poroelasticity, (ii) robust iterative splitting of resulting nonlinear algebraic system working for real-world 3D problems, and (iii) efficient solution of its mechanical quadratic programming part with a large number of fractures in mutual contact by means of own solvers implemented into an in-house software library.

Figures (12)

• Figure 1: Illustration of a domain $\Omega_m$ representing the rock matrix with discrete fractures $\Omega_f$ and the boundaries, where Dirichlet and Neumann conditions are prescribed for the mechanics. Circles denote the boundary of $\Omega_f$. The flow boundary conditions can be prescribed on different parts of the boundary.
• Figure 2: Scheme of contact forces acting on the matrix-fracture interface.
• Figure 3: Scheme of spacial discretization in 2D.
• Figure 4: Geometry, initial and boundary conditions of the fluid injection test.
• Figure 5: Fluid injection test: Comparison of fracture apertures obtained by numerical solution to the semi-analytical results at times 500 s and 2000 s.