Reduced-order models (ROM) are considered powerful techniques to address computational challenges associated with reser-voir management decision-making. In this sense, they represent perfect alternatives that trade off accuracy for speed in a controllable manner. In this paper, we describe a model-order reduction technique that entails the use of proper orthogonal decomposition (POD), truncated balanced realization (TBR) and discrete empirical interpolation (DEIM) to accurately re-produce the full-order model (FOM) input/output behavior. POD allows for a concise representation of the FOM in terms of relatively small variables while TBR improves the overall stability and accuracy. DEIM improves the shortcomings of POD and TBR in the case of nonlinear PDEs, i.e.; saturation equation, by retaining nonlinearities at lower dimensional space. In this work, the use of ROM reduced the computational time by O(100) while providing good overall agreement with FOM. The use of large reservoir simulation models is expected to add additional speedup factors.
ROMs are potentially perfect alternatives to FOMs in reservoir management intensive studies such as production optimization. However, ROMs presented in this paper and the overall physics-based ROMs have the tendency to perform well within a restricted zone. This zone is generally dictated by the training simulations used to build the ROM. Therefore, special care is considered when implementing these training runs. To mitigate the heuristic process of implementing the training runs, we apply a ROM based trust-region method that provides an adaptive framework to systemically retrain ROM during the optimization run.
The ROM approach with trust-region methodology is applied to a heterogeneous model containing 13,200 grid blocks and five wells. The accuracy of the ROM is first demonstrated for several testing simulations in which the injection and production rates for each well differ from those used to build the ROM. A waterflood optimization case is then considered to determine the optimum injection and production rates for four producers and one injector at five different times (total of 25 control variables). Results for optimized net present value using ROM based trust-region is shown to be very close to those achieved using the full order model with a difference of only 0.2%. The runtime speedup factor for this case was about 31. The ROM approach thus appears to be well suited for use in applications in which many simulations must be performed such as production optimization and uncertainty assessment.