The Parallel Subdomain-Levelset Deflation Method in Reservoir Simulation

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.