Abstract
The determination of optimal well settings is very demanding computationally, as the simulation model must be run many times during the course of the optimization. For this reason, reduced-order modeling procedures, which are a family of techniques that enable highly efficient simulations, may be very useful for optimization problems. In this paper we describe a recently developed reduced-order modeling technique which has been used in other application areas, the trajectory piecewise linear (TPWL) procedure, and incorporate it in production optimization computations. The TPWL methodology represents solutions encountered during the optimization runs in terms of Taylor series expansions around previously simulated states. This requires a small number of preprocessing (training) simulations using the full (high-fidelity) model, during which pressure and saturation states and Jacobian matrices are saved. These states and matrices are then projected into a low-dimensional space using proper orthogonal decomposition. Simulations in this reduced space can be performed very efficiently; in this work we observe runtime speedups of a factor of 1000. Overall speedups are, however, less due to the preprocessing overhead.
We assess the TPWL representation for simulations of waterflood in a heterogeneous 3D model containing 36,000 grid blocks and six wells. The high degree of accuracy of the TPWL model is first demonstrated for several testing simulations in which producer and injector well settings differ from those used in the training runs. The TPWL representations are then used in 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. Our overall conclusion is that the TPWL representation may be quite useful in production optimization problems.
