Production Optimization With Adjoint Models Under Nonlinear Control-State Path Inequality Constraints
- Pallav Sarma (Chevron Corp.) | Wen H. Chen (Chevron Corp.) | Louis J. Durlofsky (Stanford University) | Khalid Aziz (Stanford University)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- April 2008
- Document Type
- Journal Paper
- 326 - 339
- 2008. Society of Petroleum Engineers
- 2.3 Completion Monitoring Systems/Intelligent Wells, 5.4.1 Waterflooding, 2.2.2 Perforating, 5.2.1 Phase Behavior and PVT Measurements, 4.1.5 Processing Equipment, 5.5.8 History Matching, 2.3.4 Real-time Optimization, 4.3.4 Scale, 6.5.2 Water use, produced water discharge and disposal, 5.8.7 Carbonate Reservoir, 5.5 Reservoir Simulation, 5.1.5 Geologic Modeling, 4.1.2 Separation and Treating
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The general petroleum-production optimization problem falls into the category of optimal control problems with nonlinear control-state path inequality constraints (i.e., constraints that must be satisfied at every time step), and it is acknowledged that such path constraints involving state variables can be difficult to handle. Currently, one category of methods implicitly incorporates the constraints into the forward and adjoint equations to address this issue. However, these methods either are impractical for the production optimization problem or require complicated modifications to the forward-model equations (the simulator). Therefore, the usual approach is to formulate this problem as a constrained nonlinear-programming (NLP) problem in which the constraints are calculated explicitly after the dynamic system is solved. The most popular of this category of methods for optimal control problems has been the penalty-function method and its variants, which are, however, extremely inefficient. All other constrained NLP algorithms require a gradient for each constraint, which is impractical for an optimal control problem with path constraints because one adjoint must be solved for each constraint at each time step in every iteration.
The authors propose an approximate feasible-direction NLP algorithm based on the objective-function gradient and a combined gradient for the active constraints. This approximate feasible direction is then converted into a true feasible direction by projecting it onto the active constraints and solving the constraints during the forward-model evaluation itself. The approach has various advantages. First, only two adjoint evaluations are required in each iteration. Second, the solutions obtained are feasible (within a specified tolerance) because feasibility is maintained by the forward model itself, implying that any solution can be considered a useful solution. Third, large step sizes are possible during the line search, which may lead to significant reductions in the number of forward- and adjoint-model evaluations and large reductions in the magnitude of the objective function. Through two examples, the authors demonstrate that this algorithm provides a practical and efficient strategy for production optimization with nonlinear path constraints.
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. Some earlier papers by the authors (Sarma et al. 2006b; Sarma et al. 2005b) and also papers by other authors such as Brouwer et al. (2004) have discussed efficient algorithms for such closed-loop production optimization problems.
This paper, however, focuses only on the optimization component of the closed-loop process, which is essentially a large-scale optimal control problem. A large variety of methods for solving discrete-time optimal control problems now exist in the control-theory literature, including dynamic programming, neighboring extremal methods, and gradient-based nonlinear-programming (NLP) methods. These are discussed in detail in Stengel (1985) and Bryson and Ho (1975). Of these approaches, the NLP method combined with the Maximum Principle (Bryson and Ho 1975) (adjoint models) generates a class of NLP methods in which only the control variables are the decision variables and the state variables are obtained from the dynamic equations. These algorithms are generally considered more efficient compared to the other methods. Furthermore, within this class of NLP methods, there are many existing techniques available for handling nonlinear control-state path inequality constraints (Bryson and Ho 1975; Mehra and Davis 1972; Feehery 1998; Fisher and Jennings 1992). However, as will be discussed later, these techniques are either impractical for the production-optimization problem or difficult to implement with existing reservoir simulator codes.
In the petroleum-engineering literature, papers by various authors such as Asheim (1988), Vironovsky (1991), Brouwer and Jansen (2004) have discussed in significant detail the application of adjoint models and gradient techniques to the production-optimization problem. However, an important element that is missing from most of these papers is an effective treatment of nonlinear control-state path inequality constraints (for example, a maximum water-injection-rate constraint). Such constraints are always present in practical production-optimization problems, and therefore appropriate treatments are essential for such algorithms to be useful. In an earlier paper by the authors (Sarma et al. 2005a), two methods to handle such constraints were discussed; however, they either do not satisfy the constraints exactly or are applicable only for small problems. Zakirov et al. (1996) also discussed an approach to implementing path constraints; there are, however, certain issues with this approach, as discussed in a later section.
It should be noted that adjoints and gradient methods have also been applied to the history-matching problem. Such approaches were pioneered by Chen et al. (1974) and Chavent et al. (1975) and have more recently been applied by Wu et al. (1999), Li et al. (2001), and Zhang et al. (2005), among others. However, the problem of nonlinear path constraints usually does not appear in the history-matching problem.
In this paper, an approximate feasible-direction optimization algorithm is proposed, suitable for large-scale optimal control problems, that is able to handle nonlinear inequality path constraints effectively while maintaining feasibility within a specified tolerance. Other advantages of this approach are that only two adjoint simulations are required for each iteration and that large step sizes are possible during the line search in each iteration, potentially leading to large reductions in the magnitude of the objective function. This method belongs to the class of NLP methods combined with the Maximum Principle (adjoint models) discussed previously. Although the algorithmic components implemented here have been applied previously in various contexts, to the authors' knowledge this is the first integration of a feasible-direction algorithm, constraint lumping (with the particular lumping function used), and a feasible-line search algorithm. Thus the resulting feasible-direction optimization algorithm can be considered to be a new treatment for an important problem.
This paper proceeds with a brief description of the mathematical formulation of the problem and the application of adjoint models for efficient calculation of objective-function gradients with respect to the controls. This is followed by a discussion of existing methods for handling nonlinear path constraints for optimal control problems. The next section discusses the traditional feasible-direction optimization algorithm, which is the basis of the proposed algorithm. This is followed by detailed discussions of the proposed approximate feasible-direction and feasible-line search algorithms. The validity and effectiveness of the approach for handling nonlinear path inequality constraints is demonstrated through two examples, one with a maximum water-injection constraint, and the other with a maximum liquid-production constraint (both of these are nonlinear with respect to the BHP controls).
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