Linearized reduced-order models for subsurface flow simulation

A trajectory piecewise linearization (TPWL) procedure for the reduced-order modeling of two-phase flow in subsurface formations is developed and applied. The method represents new pressure and saturation states using linear expansions around states previously simulated and saved during a series of preprocessing training runs. The linearized representation is projected into a low-dimensional space, with the projection matrix constructed through proper orthogonal decomposition of the states determined during the training runs. The TPWL model is applied to two example problems, containing 24,000 and 79,200 grid blocks, which are characterized by heterogeneous permeability descriptions. Extensive test simulations are performed for both models. It is shown that the TPWL model provides accurate results when the controls (bottom hole pressures of the production wells in this case) applied in test simulations are within the general range of the controls applied in the training runs, even though the well pressure schedules for the test runs can differ significantly from those of the training runs. This indicates that the TPWL model displays a reasonable degree of robustness. Runtime speedups using the procedure are very significant-a factor of 100-2000 (depending on model size and whether or not mass balance error is computed at every time step) for the cases considered. The preprocessing overhead required by the TPWL procedure is the equivalent of about four high-fidelity simulations. Finally, the TPWL procedure is applied to a computationally demanding multiobjective optimization problem, for which the Pareto front is determined. Limited high-fidelity simulations demonstrate the accuracy and applicability of TPWL for this optimization. Future work should focus on error estimation and on stabilizing the method for large models with significant density differences between phases.