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 high efficient simulations, are very useful for optimizing problems. In this paper we describe a recently developed reduced-order modeling technique which has been used in other application areas, 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 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 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 450.
We assess the TPWL representation for simulations of waterflood in a heterogeneous 3D model containing over 20,000 grid blocks and six wells. The high degree of accuracy of the TPWL model is first demonstrated for several testing simulations in which producer will settings differ from those used in the preprocessing runs. The TPWL representations are then used in several optimizations involving the determination of optimal bottomhole pressures in four producers at six different times (24 control variables). Both gradient-based and generalized pattern search optimization algorithms are considered. Results for optimized net present value using TPWL are shown to be in consistently close agreement with those computed using high-fidelity simulations. Most significantly, when the optimal well settings obtained using TPWL are applied in high-fidelity models, the resulting net present values are within 0.8% of the values determined using the high-fidelity simulations. Our overall conclusion is that the TPWL representation is very well suited for use in production optimization problems.
