Efficient Closed-loop Production Optimization Under Uncertainty

Abstract

Abstract This work discusses a closed-loop approach for efficient real-time production optimization that 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 (history-matching). The control gradients provided by the adjoint solution are used by a gradient-based optimization algorithm to determine optimal control settings, while the parameter gradients are used for model updating. Polynomial chaos expansions provide optimal encapsulation of information contained in the input random fields and output random variables. This approach allows the forward model to be used as a black box but is much faster than standard Monte Carlo techniques. The K-L representation allows for the direct application of adjoint techniques for history matching while assuring that the two-point geostatistics of the reservoir description are maintained. The benefits and efficiency of the overall closed-loop approach are demonstrated through real-time optimizations of net present value (NPV) for synthetic reservoirs under waterflood subject to production constraints and uncertain reservoir description. The closed-loop procedure is shown to provide a substantial improvement in NPV over the base case, and the results are seen to be very close to those obtained when the reservoir description is known apriori. Introduction One of the primary goals of the reservoir modeling and management process is to enable decisions that maximize the production potential of the reservoir. Among the various existing approaches to accomplish this, real-time model-based reservoir management, also known as the "closed-loop" approach, has recently generated significant interest. This methodology entails model-based optimization of reservoir performance under geological uncertainty, while also incorporating dynamic information in real-time, which acts to reduce model uncertainty. For such schemes to be practically applicable, a number of algorithmic advances are required. In this paper, we describe and apply algorithms for efficient closed-loop reservoir management based on the application of optimal control theory, Bayesian inversion theory, Karhunen-Loeve expansions and polynomial chaos expansions. Real-time model-based reservoir management can be explained with reference to Fig. 1. In the figure, the "System" box represents the real system over which some cost function (e.g., net present value or cumulative oil produced), designated J(u), is to be optimized. The system consists of the reservoir, wells and surface facilities. Here is a set of controls (well rates, bottom hole pressures), which can be controlled in order to maximize or minimize J(u). The "Low-order model" box represents the approximate model of the system, which in our case is the simulation model of the reservoir and facilities. Since our knowledge of the reservoir is generally uncertain, the simulation model and its output are also uncertain. The basic idea behind the closed-loop approach is to perform an optimization step on the approximate model, apply the optimal controls thus obtained on the real reservoir, gather new data as it becomes available, and assimilate this data to reduce model uncertainty. This loop can thus be repeated over the life of the reservoir [1,2].