Abstract

The surrogate model of choice in this investigation uses Trajectory Piecewise Linearization as numerical complexity reduction technique and Proper Orthogonal Decomposition for dimension reduction. The stability of the model is assured through the use of Petrov-Galerkin left projector in finding the reduced space solution of the linearized problem. The TPWL/POD approximation is further refined by the addition of a kriging correction model. The high fidelity model is a two-phase, 3D, fully implicit simulator that uses mass fraction-based formulation. The approximate model is used to expedite a waterflooding optimization problem where design variables are BHP controls to maximize the lifecycle Net Present Value. The proposed optimization strategy is based on a trust region framework. It decomposes the original problem into a sequence of local problems performed on the surrogate model bounded by a trust region whose extent is adaptively managed by the strategy during the optimization process depending on surrogate accuracy. Should surrogate accuracy deteriorate as a result of changes in well controls during the optimization process TPWL/POD model must be retrained to incorporate new state snapshots corresponding to those controls. This work proposes TPWL/POD retraining criteria based on the trust region accuracy parameter and an error indicator that represents the average distance between stored snapshots and the corresponding simulation states. The proposed optimization strategy is applied to a 24000 cell reservoir based on SPE-10 problem with two injector and four producer wells considering four control cycles. Differences in fluid densities and fluid compressibilities are take into account increasing problem nonlinearity. Different trends are used for the correction model. A parameter study is conducted to fine tune the proposed correction criteria. Excellent results are obtained with NPV values exceeding those obtained by coupling the SQP optimizer directly to the simulator as well as the TPWL/POD approximation with no correction or retraining. The strategy proved to be very effective because of the reuse of previous computation through stored Jacobians and continuous refinement of the kriging correction model. Unnecessary retraining simulations were avoided by the criteria which can be used by other surrogate based strategies making use of TPWL/POD approximation.

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