Efficient Closed-Loop Production Optimization Under Uncertainty
Authors: Denney Dennis 
Published: February 2015

This article, written by Technology Editor Dennis Denney, contains highlights of paper SPE 94241, “Efficient Closed-Loop Production Optimization Under Uncertainty,” by P. Sarma, L.J. Durlofsky, and K. Aziz, Stanford U., prepared for the 2005 SPE Europec/EAGE Annual Conference, Madrid, Spain, 13–16 June.
This closed-loop approach for real-time production optimization consists of three key elements—adjoint models for efficient parameter and control-gradient calculation, polynomial chaos expansions for efficient uncertainty propagation, and Karhunen-Loeve (K-L) expansions and Bayesian inversion theory for efficient real-time model updating (i.e., history matching). The closed-loop procedure provides a substantial improvement in net present value (NPV) over the base case, and the results seem to be very close to those obtained when the reservoir description is known a priori.
Introduction
A primary goal of reservoir modeling and management is to enable decisions that maximize the production potential of the reservoir. Real-time model-based reservoir management, also known as the closed-loop approach, has generated significant interest. This method uses model-based optimization of reservoir performance under geological uncertainty while incorporating dynamic information in real time, which acts to reduce model uncertainty. For such schemes to be practical, several algorithmic advances are required. The full-length paper details algorithms for efficient closed-loop reservoir management that are based on the application of optimal-control theory, Bayesian inversion theory, K-L expansions, and polynomial chaos expansions.
Closed-Loop Approach
Fig. 1 shows the “System” box that represents the real system over which some cost function (e.g., NPV or cumulative oil produced) is to be optimized for real-time model-based reservoir management. The system consists of the reservoir, wells, and surface facilities. Here, a set of controls [well rates and bottomhole pressures (BHPs)] are used to maximize or minimize NPV or cumulative oil produced. The low-order-model box represents the approximate model of the system, which, in this case, is the simulation model of the reservoir and facilities. Because knowledge of the reservoir generally is uncertain, the simulation model and its output also are uncertain. The basic idea of the closed-loop approach is to perform an optimization step on the approximate model, apply the obtained optimal controls on the real reservoir, gather new data as they become available, and assimilate these data to reduce model uncertainty. This loop can be repeated many times over the life of the reservoir.
The closed-loop approach for efficient real-time optimization consists of three key components: efficient optimization algorithms, efficient model-updating algorithms, and efficient algorithms for uncertainty propagation. In this work, an adjoint model is applied for the efficient calculation of gradients of the objective function with respect to the controls, which then are used by gradient-based optimization algorithms, such as sequential quadratic programming, for optimization. For the model-updating procedure, Bayesian inversion theory is used, and an efficient parameterization of the uncertain parameter field is applied using the K-L expansion. The K-L expansion essentially transforms the correlated input random field into a much smaller set of independent random variables, thereby enabling the use of standard gradient-based inversion algorithms and associated adjoints for very efficient model updating while maintaining the geological correlation.