Previous research on multiscale methods for subsurface flow aims to obtain an efficient multiscale solution to the fine-scale problems. Such multiscale solution is usually a good approximation to the fine-scale problem. However, it has been reported that the multiscale solution deteriorates for high aspect ratios and channelized structures of permeability. Moreover, the multiscale solution does not converge to the fine-scale solution unless some special techniques are used.

In this paper, we propose an efficient two-stage algebraic multiscale (TAMS) method that converges to the fine-scale solution. The TAMS method consists of two stages, namely global and local stages. In the global stage, a multiscale solution is obtained purely algebraically from the fine-scale matrix. In the local stage, a local solution is constructed from a local preconditioner such as Block ILU(0) (BILU) or Additive Schwarz (AS) method. Spectral analysis shows that the multiscale solution step captures the low-frequency spectra in the original matrix very well and when combined with a local preconditioner that represents the high-frequency spectra, the full spectra can be well approximated. Thus the TAMS is guaranteed to converge to the fine-scale solution. Moreover, the spectra of the TAMS method tend to cluster together, which is a favorable property for Krylov subspace methods to converge fast. We also show that local mass conservation can be preserved if a multiscale solution step with the finite-volume restriction operator is applied before the iterative procedure exits. This allows for the TAMS method to be used to build efficient approximate solutions for multiphase flow problems that need to solve transport equations.

We test the numerical performance of the TAMS method using several challenging large-scale problems (fine-scale grid in the magnitude of one million) with complex heterogeneous structures and high aspect ratios. Different choices in the TAMS algorithm are employed, including the Galerkin or finite-volume type of restriction operator, BILU or AS preconditioner for the second stage, and the size of blocks for BILU and AS. The performance of TAMS is comparable or superior to the state-of-the-art algebraic multigrid (AMG) preconditioner when some optimal choice in the TAMS method is adopted. Moreover, the convergence of the TAMS method is insensitive to problem sizes, and the CPU time is almost linear to problem sizes. These indicate the TAMS method is efficient and robust for large-scale problems.

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