# A Scalable Multigrid Reduction Framework for Multiphase Poromechanics of Heterogeneous Media

## TL;DR

This paper introduces a scalable algebraic multigrid reduction framework tailored for efficiently solving the large, coupled linear systems arising in multiphase poromechanics simulations involving heterogeneous media.

## Contribution

It presents a novel algebraic multigrid reduction approach that effectively decouples complex PDE systems for improved solver performance in multiphase poromechanics.

## Key findings

- Framework is flexible for various scenarios.
- Achieves efficiency and scalability for large problems.
- Outperforms traditional black-box solvers.

## Abstract

Simulation of multiphase poromechanics involves solving a multi-physics problem in which multiphase flow and transport are tightly coupled with the porous medium deformation. To capture this dynamic interplay, fully implicit methods, also known as monolithic approaches, are usually preferred. The main bottleneck of a monolithic approach is that it requires solution of large linear systems that result from the discretization and linearization of the governing balance equations. Because such systems are non-symmetric, indefinite, and highly ill-conditioned, preconditioning is critical for fast convergence. Recently, most efforts in designing efficient preconditioners for multiphase poromechanics have been dominated by physics-based strategies. Current state-of-the-art "black-box" solvers such as algebraic multigrid (AMG) are ineffective because they cannot effectively capture the strong coupling between the mechanics and the flow sub-problems, as well as the coupling inherent in the multiphase flow and transport process. In this work, we develop an algebraic framework based on multigrid reduction (MGR) that is suited for tightly coupled systems of PDEs. Using this framework, the decoupling between the equations is done algebraically through defining appropriate interpolation and restriction operators. One can then employ existing solvers for each of the decoupled blocks or design a new solver based on knowledge of the physics. We demonstrate the applicability of our framework when used as a "black-box" solver for multiphase poromechanics. We show that the framework is flexible to accommodate a wide range of scenarios, as well as efficient and scalable for large problems.