The Parallel Subdomain-Levelset Deflation Method in Reservoir Simulation

TL;DR

This paper introduces a parallel subdomain-levelset deflation method for reservoir simulation that effectively improves linear solver convergence by addressing harmful eigenvalues caused by permeability heterogeneity, with strong potential for commercial applications.

Contribution

The paper proposes a novel physics-based, parallelizable subdomain-levelset deflation method that constructs deflation vectors a priori, offering a scalable alternative to traditional eigenvector deflation in reservoir simulation.

Findings

Both deflation methods improve solver performance.

Subdomain-levelset deflation is highly suitable for parallel implementation.

Parallel physics-based deflation shows potential for handling high permeability jumps.

Abstract

Extreme and isolated eigenvalues are known to be harmful to the convergence of an iterative solver. These eigenvalues can be produced by strong heterogeneity in the underlying physics. We can improve the quality of the spectrum by "deflating" the harmful eigenvalues. In this work, deflation is applied to linear systems in reservoir simulation. In particular, large, sudden differences in the permeability produce extreme eigenvalues. The number and magnitude of these eigenvalues is linked to the number and magnitude of the permeability jumps. Two deflation methods are discussed. Firstly, we state that harmonic Ritz eigenvector deflation, which computes the deflation vectors from the information produced by the linear solver, is unfeasible in modern reservoir simulation due to high costs and lack of parallelism. Secondly, we test a physics-based subdomain-levelset deflation algorithm that constructs the deflation vectors a priori. Numerical experiments show that both methods can improve the performance of the linear solver. We highlight the fact that subdomain-levelset deflation is particularly suitable for a parallel implementation. For cases with well-defined permeability jumps of a factor 104 or higher, parallel physics-based deflation has potential in commercial applications. In particular, the good scalability of parallel subdomain-levelset deflation combined with the robust parallel preconditioner for deflated system suggests the use of this method as an alternative for AMG.