SPE Journal
Volume 17, Number 2, June 2012, pp. 523-539

SPE-141473-PA

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

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DOI  More information 10.2118/141473-PA http://dx.doi.org/10.2118/141473-PA

Citation

  • Zhou, H. and Tchelepi, H.A. 2012. Two-Stage Algebraic Multiscale Linear Solver for Highly Heterogeneous Reservoir Models. SPE J.  17 (2): 523-539. SPE-141473-PA. http://dx.doi.org/10.2118/141473-PA.

Discipline Categories

  • 6.5 Reservoir Simulation
  • 6.5.1 Simulator Development

Keywords

  • Reservoir Simulation, Multiscale method, Large-scale simulation

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.

© 2012. Society of Petroleum Engineers

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History

  • Original manuscript received: 19 March 2011
  • Meeting paper published: 21 February 2011
  • Revised manuscript received: 19 August 2011
  • Manuscript approved: 23 August 2011
  • Published online: 6 February 2012
  • Version of record: 11 June 2012