Constraint reduction procedures for reduced‐order subsurface flow models based on POD–TPWL

International Journal for Numerical Methods in Engineering - Tập 103 Số 1 - Trang 1-30 - 2015
Junjing He1, Louis J. Durlofsky1
1Department of Energy Resources Engineering, Stanford University, Stanford, CA, 94305, USA

Tóm tắt

SummaryThe properties and numerical performance of reduced‐order models based on trajectory piecewise linearization (TPWL) and proper orthogonal decomposition (POD) are assessed. The target application is subsurface flow modeling, although our findings should be applicable to a range of problems. The errors arising at each step in the POD–TPWL procedure are described. The impact of constraint reduction on accuracy and stability is considered in detail. Constraint reduction entails projection of the overdetermined system into a low‐dimensional subspace, in which the system is solvable. Optimality conditions for constraint reduction, in terms of error minimization, are derived. Galerkin and Petrov–Galerkin projections are shown to correspond to optimality in norms that involve weighting with the Jacobian matrix. Two new treatments, inverse projection and weighted inverse projection, are suggested. These methods minimize error in appropriate norms, although they require substantial preprocessing computations. Numerical results are presented for oil reservoir simulation problems. Galerkin projection provides reasonable accuracy for simpler oil–water systems, although it becomes unstable in more challenging cases. Petrov–Galerkin projection is observed to behave stably in all cases considered. Weighted inverse projection also behaves stably, and it provides the highest accuracy. Runtime speedups of 150–400 are achieved using these POD–TPWL models. Copyright © 2015 John Wiley & Sons, Ltd.

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International Journal for Numerical Methods in Engineering

Tập 103 Số 1

1-30

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