Summary
Well-control management is nowadays frequently approached by means of mathematical optimization. However, in many practical situations the optimization algorithms used are still computationally expensive. In this paper, we present progressive optimization (PO), a simulator-nonintrusivefour-stage methodology to accelerate optimal search substantially in well-controlapplications.
The first stage of PO comprises a global exploration of the search space using design of experiments (DOEs). Thereafter, in the second stage, a fast-to-evaluate proxy model is constructed with the points considered in the experimental design. This proxy is based on generalized barycentric coordinates (GBCs), a generalization of the concept of barycentric coordinates used within a triangle. GBCs can be especially suited to problems in which nonlinearities are not strong, as is the case often for well-control optimization. This fact is supported by the good performance in these types of optimization problems of techniques that rely strongly on linearity assumptions, such as trajectory piecewise linearization, a procedure that is not always applicable due to its simulator-intrusive nature. In the third stage, the precision of the proxy model is iteratively improved and the enhanced surrogate model is reoptimized by means of manifold mapping (MM), a method that combines models with different levels of accuracy. MM has solid theoretical foundations and leads to efficient optimization schemes in multiple engineering disciplines. The final and fourth stage aims at additional improvement, resorting to direct optimization of the best solution from the previous stages. Nonlinear (operational) constraints are handled in PO with the filter method. The optimal search may be finalized earlier than at the fourth stage whenever the solution obtained is of satisfactory quality.
PO is tested on two waterflooding problems built upon a synthetic model previously studied in well-control optimization literature. In these problems, which have 120 and 40 well controls and include nonlinear constraints, we observe for PO reductions in computational cost, for solutions of comparable quality, of approximately 30% and 50% with respect to Hooke-Jeeves direct search (HJDS), which, in turn, outperforms particle swarm optimization (PSO). HJDS and PSO are simulator-nonintrusive algorithms that usually perform well in optimization for oilfield operations.
The novel concepts of GBC and MM within the framework of the PO paradigm can be extremely helpful for practitioners to efficiently deal with optimized well-control management. Savings of 50% in computing cost may be translated in practice into days of computations for just a single field and optimization run.
A Polygonal Finite Element Method for Stokes Equations
