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Nonlinear State Constraints Handling in Waterflooding Optimization Through Reduced Order Models
- Publisher: European Association of Geoscientists & Engineers
- Source: Conference Proceedings, ECMOR XVII, Sep 2020, Volume 2020, p.1 - 19
Abstract
Summary
This study addresses strategies to efficiently impose nonlinear state constraints using reduced order models. Nonlinear constraints imposed on state variables are of practical interest in optimizing reservoir production performance (NPV or oil production), but they are difficult to handle numerically. Constraints involve bounds on control themselves (e.g. rates, BHP or valve openings), linear functions involving the design variables, but oftentimes nonlinear constraints involving state variables must also be imposed. Examples are minimum (maximum) BHP’s at producer (injector) wells subject to rate controls, or vice versa. Enforcement of these constraints involves repeated computation of state variables, and possibly their derivatives, not only at the ends of control steps but at numerous intermediate times. Both computations are time consuming and, thus, it is proposed to make use of reduced order methods to decrease the numerical effort. The contributions of this paper are twofold: (1) we propose correction points based on a time series within the control cycle to impose state constraints thus reducing the computational effort; (2) we are coupling the optimizer with physics-based and data-driven reduced-order models to enforce state complexity reduction.
Here, two strategies are compared: Proper Orthogonal Decomposition / Trajectory Piecewise Linearization (POD/TPWL) and Dynamic Mode Decomposition (DMD). Both methods are snapshot-based linearizations but are implemented differently. TPWL/POD technique reduces the complexity of the problem by linearizing the governing equations around converged and stored states during a training simulation, and reduction is obtained by projecting states onto smaller subspaces by POD. This method requires access to the simulator code and, thus, is an intrusive method. DMD also rely on state snapshots that are used to generate a small set of optimal basis vectors called modes. The snapshot data also permits extraction of a coherent dynamic structure of the problem through the assumption that there exists a linear mapping connecting temporal evolution of the state system. This evolution can be computed without further simulation runs. DMD does not require access to the simulator code and therefore is nonintrusive. The reduced-order techniques are compared in the optimization of a BHP controlled synthetic reservoir where the objective function is maximization of oil production subject to field water production rate constraints. We will demonstrate the handling of non-linear constraints and the resulting computational savings using the MATLAB Reservoir Simulation Toolbox (MRST). We performed modifications to some of its routines to store Jacobian matrices and also snapshots, both used at POD/TPWL and DMD.
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