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IAMCS Workshop in Large-Scale Inverse Problems and Uncertainty Quantification

Stein Krogstad, SINTEF
Local POD-Based Multiscale Mixed FEMs for Model Reduction of Multiphase Compressible Flow


Model-based decision support for reservoir management can involve a high number of reservoir simulations, e.g., in optimization loops and uncertainty sampling. From one simulation to the next, changes in model parameters and input are typically limited, and accordingly the potential to "reuse" computations through model reduction is large. In this talk we present a local basis model-order reduction technique for approximation of flux/pressure fields based on local proper orthogonal decompositions (PODs) "glued" together using a Multiscale Mixed Finite Element Method (MsMFEM) framework on a coarse grid. Based on snapshots from one or more simulation runs, we perform SVDs for the flux distribution over coarse grid interfaces and use the singular vectors corresponding to the largest singular values as boundary conditions for the multiscale flux basis functions. The span of these basis functions matches (to prescribed accuracy) the span of the snapshots over coarse grid faces. Accordingly, the complementary span (what's left) can be approximated by local PODs on each coarse block giving a second set of local/sparse basis functions.

One natural property of such a model reduction technique is that it should reproduce the tuning simulations to given accuracy, and in addition, to keep the storage requirements low, we wish to build a reduced basis for the velocity (or flux) only, and use piecewise constant basis functions for pressure (as in the original MsMFEM) when solving the coarse systems. We show that this is straight forward for incompressible flow, while when compressibility is present, a difficulty arises due to certain orthogonality requirements between the velocity-basis source-functions and the fine-scale pressure variations. We discuss and compare a few formulations to get around this problem, and present results for two-phase test problems including compressibility and gravity.

The numerical experiments suggest that the localized basis approach gives good results "further away" from the tuning run(s) than the global version. Another advantage of the local version is that local changes in the model can be accounted for by adding a few extra local basis functions.