An Efficient Multiscale Multigrid Preconditioner for Darcy Flow in High-Contrast Fractured Media

Core Concepts

The authors develop a multigrid preconditioner to efficiently solve Darcy flow problems in highly heterogeneous and fractured porous media. The key component is constructing a sequence of nested subspaces using spectral problems defined on local coarse and coarse-coarse elements.

Abstract

The paper presents a novel multiscale multigrid preconditioner for solving Darcy flow problems in high-contrast and fractured porous media. The key aspects are:

Constructing a hierarchy of nested subspaces WL ⊂ WL-1 ⊂ ... ⊂ W1 = Wh by solving local spectral problems on coarse and coarse-coarse elements.
Applying this spectral multigrid preconditioner to solve a positive semi-definite linear system resulting from discretizing the Darcy flow equation using the lowest-order Raviart-Thomas spaces.
Providing theoretical analysis on the condition number of the preconditioned operator based on the convergence theory of inexact two-grid methods.
Extensive numerical experiments demonstrating the strong and weak scalability, as well as the robustness of the preconditioner against high-contrast and fractured media.
Demonstrating an application of the preconditioner for solving a two-phase flow benchmark problem.

The authors show that the proposed preconditioner can efficiently handle problems with resolutions up to 10243 and is robust against high-contrast and fractured media.

Stats

The permeability field K exhibits high heterogeneity, with the ratio of maximum to minimum eigenvalues being much greater than 1.
The finest mesh resolution considered is up to 10243.
The number of MPI processes (proc) ranges from 33 to 123.

Quotes

"Simulating fluid flow in porous media is critical for many practical applications, such as reservoir simulation, CO2 sequestration, nuclear water storage, and underground water contamination."
"Although the aforementioned multiscale approaches have been applied successfully to solve a wide range of multiscale problems, multiscale solutions can deteriorate with increasing contrast in permeability and correlation length in certain scenarios."

Key Insights Distilled From

An efficient multiscale multigrid preconditioner for Darcy flow in high-contrast media

by Changqing Ye... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2403.19342.pdf

Deeper Inquiries

How can the proposed preconditioner be extended to handle more general anisotropic permeability fields beyond the orthotropic case considered in the paper

To extend the proposed preconditioner to handle more general anisotropic permeability fields beyond the orthotropic case considered in the paper, we can introduce a more flexible spectral problem-based coarse space construction method. One approach is to solve separate spectral problems for each component of the anisotropic permeability tensor. This means defining spectral problems for each direction of the permeability field and constructing corresponding coarse spaces based on the eigenvectors obtained from these spectral problems. By incorporating the anisotropy of the permeability field into the construction of the coarse spaces, the preconditioner can effectively handle a wider range of anisotropic permeability fields.

What are the potential limitations of the spectral problem-based coarse space construction, and how can they be addressed to further improve the scalability and robustness of the method

The potential limitations of the spectral problem-based coarse space construction method include the computational cost associated with solving multiple spectral problems, especially for highly heterogeneous media with complex geometries. To address these limitations and further improve the scalability and robustness of the method, several strategies can be implemented:

Adaptive Coarse Space Construction: Implement adaptive strategies to dynamically adjust the dimensions of the coarse spaces based on the local heterogeneity of the permeability field. This adaptive approach can optimize the balance between accuracy and computational efficiency.

Parallelization and Load Balancing: Enhance the parallelization of the spectral problem-solving process to efficiently distribute the computational workload across multiple processors. Implement load balancing techniques to ensure an even distribution of computational tasks and minimize idle time.

Reduced Order Models: Explore the use of reduced order models or model order reduction techniques to approximate the spectral problems and construct coarse spaces more efficiently. These techniques can help reduce the computational complexity while maintaining the accuracy of the preconditioner.

Hybrid Methods: Combine the spectral problem-based coarse space construction with other preconditioning techniques, such as algebraic multigrid methods or domain decomposition methods, to leverage the strengths of each approach and enhance the overall scalability and robustness of the method.

Given the success of the preconditioner in solving Darcy flow problems, how could it be adapted to handle more complex multiphysics problems involving coupled flow and transport processes in fractured porous media

To adapt the preconditioner for more complex multiphysics problems involving coupled flow and transport processes in fractured porous media, several modifications and enhancements can be made:

Incorporation of Additional Physics: Extend the preconditioner to handle additional physics equations, such as transport equations for species transport or heat transfer, by incorporating the corresponding variables and coupling terms into the preconditioning process.

Fracture Network Modeling: Develop specialized coarse space construction techniques to capture the behavior of fractures in the porous media, considering the connectivity and permeability of the fracture network. This can involve defining separate spectral problems for fractures and integrating them into the preconditioner.

Coupling Strategies: Implement robust coupling strategies to ensure the effective interaction between different physics components, such as flow and transport, within the preconditioner. This may involve iterative coupling schemes or implicit coupling methods to handle the multiphysics interactions accurately.

Adaptive Mesh Refinement: Utilize adaptive mesh refinement techniques to dynamically adjust the mesh resolution in regions of interest, such as fractures, to capture the complex flow and transport phenomena more accurately. This adaptive approach can enhance the preconditioner's performance in modeling multiphysics processes in fractured porous media.