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Two-Stage Algebraic Multiscale Linear Solver for Highly Heterogeneous Reservoir Models

Authors
Hui Zhou (ConocoPhillips Subsurface Technology) | Hamdi A. Tchelepi (Stanford University)
DOI
https://doi.org/10.2118/141473-PA
Document ID
SPE-141473-PA
Publisher
Society of Petroleum Engineers
Source
SPE Journal
Volume
17
Issue
02
Publication Date
June 2012
Document Type
Journal Paper
Pages
523 - 539
Language
English
ISSN
1086-055X
Copyright
2012. Society of Petroleum Engineers
Disciplines
4.1.2 Separation and Treating, 4.3.4 Scale
Keywords
Large-scale simulation, Reservoir Simulation, Multiscale method
Downloads
1 in the last 30 days
456 since 2007
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SPE Non-Member Price: USD 35.00

Summary

An efficient two-stage algebraic multiscale solver (TAMS) that converges to the fine-scale solution is described. The first (global) stage is a multiscale solution obtained algebraically for the given fine-scale problem. In the second stage, a local preconditioner, such as the Block ILU (BILU) or the Additive Schwarz (AS) method, is used. Spectral analysis shows that the multiscale solution step captures the low-frequency parts of the error spectrum quite well, while the local preconditioner represents the high-frequency components accurately. Combining the two stages in an iterative scheme results in efficient treatment of all the error components associated with the fine-scale problem. TAMS is shown to converge to the reference fine-scale solution. Moreover, the eigenvalues of the TAMS iteration matrix show significant clustering, which is favorable for Krylov-based methods. Accurate solution of the nonlinear saturation equations (i.e., transport problem) requires having locally conservative velocity fields. TAMS guarantees local mass conservation by concluding the iterations with a multiscale finite-volume step. We demonstrate the performance of TAMS using several test cases with strong permeability heterogeneity and large-grid aspect ratios. Different choices in the TAMS algorithm are investigated, including the Galerkin and finite-volume restriction operators, as well as the BILU and AS preconditioners for the second stage. TAMS for the elliptic flow problem is comparable to state-of-the-art algebraic multigrid methods, which are in wide use. Moreover, the computational time of TAMS grows nearly linearly with problem size.

File Size  4 MBNumber of Pages   17

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