[Paper Review] Multigrid reduction preconditioning framework for coupled processes in porous and fractured media

Quan M. Bui, François P. Hamon+4 more|arXiv (Cornell University)|Jan 27, 2021

Hydraulic Fracturing and Reservoir Analysis64 references27 citations

TL;DR

This paper proposes a physics-informed, algebraic multigrid reduction (MGR) preconditioning framework for solving large-scale, coupled linear systems arising in subsurface simulations involving flow, mechanics, and fracturing. By leveraging block structure and physics-based coarsening, MGR enables scalable, efficient solution of fully-implicit time-stepping schemes across diverse discretizations—demonstrated on single-phase flow, compositional flow with wells, and hydraulic fracturing—achieving strong and weak scalability on field-scale problems.

ABSTRACT

Many subsurface engineering applications involve tight-coupling between fluid flow, solid deformation, fracturing, and similar processes. To better understand the complex interplay of different governing equations, and therefore design efficient and safe operations, numerical simulations are widely used. Given the relatively long time-scales of interest, fully-implicit time-stepping schemes are often necessary to avoid time-step stability restrictions. A major computational bottleneck for these methods, however, is the linear solver. These systems are extremely large and ill-conditioned. Because of the wide range of processes and couplings that may be involved--e.g. formation and propagation of fractures, deformation of the solid porous medium, viscous flow of one or more fluids in the pores and fractures, complicated well sources and sinks, etc.--it is difficult to develop general-purpose but scalable linear solver frameworks. This challenge is further aggravated by the range of different discretization schemes that may be adopted, which have a direct impact on the linear system structure. To address this obstacle, we describe a flexible framework based on multigrid reduction that can produce purely algebraic preconditioners for a wide spectrum of relevant physics and discretizations. We demonstrate its broad applicability by constructing scalable preconditioners for several problems, notably: a hybrid discretization of single-phase flow, compositional multiphase flow with complex wells, and hydraulic fracturing simulations. Extension to other systems can be handled quite naturally. We demonstrate the efficiency and scalability of the resulting solvers through numerical examples of difficult, field-scale problems.

Motivation & Objective

• Address the computational bottleneck of solving large, ill-conditioned linear systems arising from fully-implicit time discretization in coupled subsurface processes.
• Develop a general-purpose, scalable linear solver framework applicable to diverse physics (flow, mechanics, fracturing) and discretization schemes (e.g., mimetic finite differences, finite volumes).
• Overcome limitations of existing block preconditioners that are highly specialized, require manual tuning, or lack scalability for complex, large-scale problems.
• Enable efficient and portable solution strategies for high-fidelity simulations in carbon sequestration, geothermal energy, and unconventional hydrocarbon recovery.
• Demonstrate the framework’s robustness and scalability on realistic, field-scale problems involving complex couplings and heterogeneous media.

Proposed method

• Formulate the coupled system as a B×B block linear system (Ax = b), where blocks represent physical fields (e.g., pressure, saturation, displacement).
• Apply multigrid reduction (MGR) as a purely algebraic framework to construct preconditioners without geometric assumptions, enabling application to unstructured and complex grids.
• Use physics-based coarsening and smoother selection to preserve the structure of the system, particularly the coupling between flow, mechanics, and transport.
• Integrate MGR with Newton’s method for nonlinear time integration, solving the resulting Jacobian systems efficiently at each Newton step.
• Leverage existing solver libraries (e.g., PETSc, Trilinos) to enable modular, extensible, and portable implementation across simulators.
• Design the framework to allow rapid exploration of solver parameters (e.g., reduction order, smoother types, coarse grid operators) without re-implementation.

Experimental results

Research questions

• RQ1 Can a general-purpose, algebraic multigrid reduction (MGR) framework effectively precondition large, block-structured linear systems arising from coupled flow-geomechanics simulations?
• RQ2 How does the MGR framework perform in terms of scalability and convergence for complex, field-scale problems involving single-phase flow, compositional flow with wells, and hydraulic fracturing?
• RQ3 To what extent can physics-informed coarsening and block structure be leveraged within an algebraic MGR framework to maintain robustness across diverse discretization schemes?
• RQ4 Can MGR outperform traditional block preconditioners (e.g., CPR, fixed-stress) in terms of convergence rate and parallel scalability for complex, tightly coupled systems?
• RQ5 How flexible is the MGR framework in handling different types of well models and nonlinearities (e.g., phase transitions, fracture propagation) without reconfiguration?

Key findings

• The MGR preconditioner achieves robust convergence for all three test cases: hybrid mimetic finite difference single-phase flow, compositional multiphase flow with complex wells, and hydraulic fracturing simulations.
• The framework demonstrates strong and weak scalability on large-scale, field-relevant problems, with near-linear speedup on thousands of processors.
• For hydraulic fracturing, the MGR solver successfully handles the strongly nonlinear coupling between fracture aperture, fluid pressure, and solid deformation, even during dynamic propagation.
• The method maintains efficiency across different discretization schemes—including mimetic finite differences and finite volumes—demonstrating its generality and adaptability.
• The MGR framework enables efficient solution of fully-implicit systems without requiring problem-specific tuning, significantly reducing solver setup time and computational cost.
• Numerical results confirm that MGR outperforms standard black-box preconditioners in convergence rate and scalability, especially for ill-conditioned, large-scale systems with complex couplings.