POD-Based Deflation Method For Reservoir Simulation

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Simulation of flow through highly heterogeneous porous media results in large ill-conditioned systems of equations. In particular, solving the linearized pressure system can be especially time-consuming. Therefore, extensive efforts to find ways to address this issue effectively are required. In this work, we introduce a POD-based deflation method that combines the advantages of two state of the art techniques: Proper Orthogonal Decomposition (POD) and the deflation method. The dominant features of the system are captured in a set of POD basis vectors, used later to accelerate the solution of linear systems with a deflation procedure.
If all of the system information is contained in the POD basis, the deflation method converges in one iteration. This behavior was compared with the usual choices of deflation vectors, which require more than 18 iterations for the same number of deflation vectors. If only part of this information is obtained, the POD-based deflation method gives a good initial solution, after one iteration the error of the solution is of order 10^{-4}. The applicability of the POD-based deflation method does not depend on the test case. It is implemented for reservoir simulation problems, but it can be implemented for any time-varying problem. Furthermore, we study its applicability for various 2L-PCG methods, but it can also be implemented together with many other linear solvers, e.g., multigrid, multilevel, and domain decomposition techniques. The implementation can also be extended to include various preconditioners.