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An efficient preconditioner for mixed-dimensional contact poromechanics based on the fixed stress splitting scheme

Yury Zabegaev, Inga Berre, Eirik Keilegavlen, Kundan Kumar

TL;DR

The paper addresses the challenging linear solves arising from fully implicit simulations of fracture contact poromechanics by introducing a robust, scalable preconditioner built on a linearly transformed Jacobian and three nested Schur complements to decouple contact mechanics, momentum balance, and interface flow. The method extends the fixed-stress decoupling to both matrix and fracture subdomains and uses AMG/ILU subsolvers to maintain efficiency on large-scale problems. A contraction-based analysis under simplifying assumptions guarantees convergence of the sequential fixed-stress scheme, providing a theoretical foundation for the preconditioner. Numerical experiments in 2D and 3D demonstrate strong robustness and scalability across parameter variations, grid refinements, and complex fracture networks, indicating practical utility for subsurface applications such as CO2 storage, geothermal energy, and underground gas storage. Overall, the work delivers a principled, scalable solver for coupled poromechanics with frictional fracture contact that can be integrated with existing discretizations that employ Lagrange multipliers.

Abstract

Numerical simulation of fracture contact poromechanics is essential for various applications, including CO2 sequestration, geothermal energy production and underground gas storage. Modeling this problem accurately presents significant challenges due to the complex physics involved in strongly coupled poromechanics and frictional contact mechanics of fractures. The robustness and efficiency of the simulation heavily depends on a preconditioner for the linear solver, which addresses the Jacobian matrices arising from Newton's method in fully implicit time-stepping schemes. Developing an effective preconditioner is difficult because it must decouple three interdependent subproblems: momentum balance, fluid mass balance, and contact mechanics. The challenge is further compounded by the saddle-point structure of the contact mechanics problem, a result of the Augmented Lagrange formulation, which hinders the direct application of the well-established fixed stress approximation to decouple the poromechanics subproblem. In this work, we propose a preconditioner hat combines nested Schur complement approximations with a linear transformation, which addresses the singular nature of the contact mechanics subproblem. This approach extends the fixed stress scheme to both the matrix and fracture subdomains. We investigate analytically how the contact mechanics subproblem affects the convergence of the proposed fixed stress-based iterative scheme and demonstrate how it can be translated into a practical preconditioner. The scalability and robustness of the method are validated through a series of numerical experiments.

An efficient preconditioner for mixed-dimensional contact poromechanics based on the fixed stress splitting scheme

TL;DR

The paper addresses the challenging linear solves arising from fully implicit simulations of fracture contact poromechanics by introducing a robust, scalable preconditioner built on a linearly transformed Jacobian and three nested Schur complements to decouple contact mechanics, momentum balance, and interface flow. The method extends the fixed-stress decoupling to both matrix and fracture subdomains and uses AMG/ILU subsolvers to maintain efficiency on large-scale problems. A contraction-based analysis under simplifying assumptions guarantees convergence of the sequential fixed-stress scheme, providing a theoretical foundation for the preconditioner. Numerical experiments in 2D and 3D demonstrate strong robustness and scalability across parameter variations, grid refinements, and complex fracture networks, indicating practical utility for subsurface applications such as CO2 storage, geothermal energy, and underground gas storage. Overall, the work delivers a principled, scalable solver for coupled poromechanics with frictional fracture contact that can be integrated with existing discretizations that employ Lagrange multipliers.

Abstract

Numerical simulation of fracture contact poromechanics is essential for various applications, including CO2 sequestration, geothermal energy production and underground gas storage. Modeling this problem accurately presents significant challenges due to the complex physics involved in strongly coupled poromechanics and frictional contact mechanics of fractures. The robustness and efficiency of the simulation heavily depends on a preconditioner for the linear solver, which addresses the Jacobian matrices arising from Newton's method in fully implicit time-stepping schemes. Developing an effective preconditioner is difficult because it must decouple three interdependent subproblems: momentum balance, fluid mass balance, and contact mechanics. The challenge is further compounded by the saddle-point structure of the contact mechanics problem, a result of the Augmented Lagrange formulation, which hinders the direct application of the well-established fixed stress approximation to decouple the poromechanics subproblem. In this work, we propose a preconditioner hat combines nested Schur complement approximations with a linear transformation, which addresses the singular nature of the contact mechanics subproblem. This approach extends the fixed stress scheme to both the matrix and fracture subdomains. We investigate analytically how the contact mechanics subproblem affects the convergence of the proposed fixed stress-based iterative scheme and demonstrate how it can be translated into a practical preconditioner. The scalability and robustness of the method are validated through a series of numerical experiments.
Paper Structure (24 sections, 4 theorems, 72 equations, 3 figures, 4 tables, 2 algorithms)

This paper contains 24 sections, 4 theorems, 72 equations, 3 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model description
  3. Mixed-dimensional geometry
  4. Conservation laws
  5. Contact mechanics relations
  6. Constitutive relations
  7. Initial and boundary conditions
  8. Discretization
  9. Preconditioner for mixed-dimensional contact poromechanics
  10. Preprocessing
  11. First-level Schur complement: Eliminating contact mechanics
  12. Second-level Schur complement: Eliminating the momentum balance equations
  13. Third-level Schur complement: Eliminating the interface flow
  14. Preconditioner algorithm summary
  15. Analysis of fixed stress stabilization for mixed-dimensional contact poromechanics
  16. ...and 9 more sections

Key Result

Lemma 1

Under the assumptions assumption:1 - assumption:coercivity, the following holds:

Figures (3)

  • Figure 1: The block structure of the matrices on different stages of the preconditioner algorithm. White cells represent empty submatrices, light-gray cells represent non-empty submatrices of the original Jacobian. Dark-gray cells represent modified submatrices. The "S" letter corresponds to the singular submatrix. From left to right: 1 -- the original Jacobian, 2 -- the Jacobian after the linear transformation (see \ref{['sec:linear_transformation']}), 3 -- the first-level Schur complement (see \ref{['sec:eliminating_contact_mechanics']}), 4 -- the second-level Schur complement (see \ref{['sec:eliminating_force_balance']}), 5 -- the third-level Schur complement (see \ref{['sec:eliminating_intf_flow']}).
  • Figure 2: Material parameters used in the numerical experiments. $K_m$ denotes matrix isotropic permeability. The other symbols are defined in \ref{['sec:model_description']}.
  • Figure 3: State (sticking, sliding or open) of fracture cells in the 2D experiment with multiple fractures (left). Shaded and white background colors distinguish different time steps. Geometry of the fractures in the same experiment (right).

Theorems & Definitions (7)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Corollary 1