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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Agkun, M., Haftka, R.T., and Wu, K.C. 1999. Sensitivity of LumpedConstraints Using the Adjoint Method. 40th AIAA/ASME/ASCE/AHS/ASC Structures,Structural Dynamics and Materials Conference, St. Louis, Missouri, 12-15April.
Asheim, H. 1988. Maximizationof Water Sweep Efficiency by Controlling Production and Injection Rates.Paper SPE 18365 presented at the SPE European Petroleum Conference, London,16-19 October. DOI: 10.2118/18365-MS.
Brouwer, D.R. and Jansen, J.-D. 2004. Dynamic Optimization of WaterFlooding With Smart Wells Using Optimal Control Theory. SPEJ9 (4): 391-402. SPE-78278-PA. DOI: 10.2118/78278-PA.
Brouwer, D.R., Nævdal, G., Jansen, J.D., Vefring, E.H., and van Kruijsdiik,C.P.J.W. 2004. Improved ReservoirManagement Through Optimal Control and Continuous Model Updating. Paper SPE90149 presented at the SPE Annual Technical Conference and Exhibition, Houston,26-29 September. DOI: 10.2118/90149-MS.
Bryson, A.E. and Ho, Y.C. 1975. Applied Optimal Control: Optimization,Estimation, & Control. New York City: Taylor & Francis.
Bryson, A.E., Denham, W.F. and Dreyfus, S.E. 1963. Optimal ProgrammingProblems with Inequality Constraints I: Necessary Conditions for ExtremalSolutions. AIAA Journal 1 (11): 2544-2550.
Cao, H. 2002. Development of Techniques for General Purpose Simulators. PhDdissertation. Stanford, California: Stanford University.
Chavent, G., Dupuy, M., and Lemonnier, P. 1975. History Matching by Use of OptimalControl Theory. SPEJ 15 (1): 74-86; Trans., AIME,259. SPE-4627-PA. DOI: 10.2118/4627-PA.
Chen, W.H., Gavalas, G.R., Seinfeld, J.H., and Wasserman, M.L. 1974. A New Algorithm for Automatic HistoryMatching. SPEJ 14 (6): 593-608; Trans., AIME,257. SPE-4545-PA. DOI: 10.2118/4545-PA.
Denham, W.F. and Bryson, A.E. 1964. Optimal Programming Problems WithInequality Constraints II: Solution by Steepest-Descent. AIAA Journal2 (11): 25-34.
ECLIPSE 100 Technical Description 2001A. 2001. Abingdon, UK:Schlumberger.
Feehery, W.F. 1998. Dynamic Optimization with Path Constraints. PhDdissertation, Cambridge, Massachusetts: Massachusetts Institute ofTechnology.
Fisher, M.E. and Jennings, L.S. 1992. Discrete-Time Optimal ControlProblems With General Constraints. ACM Transactions on MathematicalSoftware 18 (4): 401-413. DOI: 10.1145/138351.138356.
Jacobson, D. and Lele, M. 1969. A Transformation Techniquefor Optimal Control Problems With a State Variable Inequality Constraint.IEEE Transactions on Automatic Control 14 (5): 457-464. DOI:10.1109/TAC.1969.1099283.
Jansen, J.D., Brouwer, D.R., Naevdal, G., and van Kruijsdiik, C.P.J.W. 2005.Closed-Loop Reservoir Management. First Break 23: 43-48.
Kelley, C.T. 1999. Iterative Methods for Optimization. Philadelphia,Pennsylvania: Society for Industrial and Applied Mathematics.
Li, R., Reynolds, A.C., and Oliver, D.S. 2001. History Matching of Three-Phase FlowProduction Data. Paper SPE 66351 presented at the SPE Reservoir SimulationSymposium, Houston, 11-14 February. DOI: 10.2118/66351-MS.
Liu, S.-Q., Shi, J., Dong, J., and Wang, S. 2004. A Modified PenaltyFunction Method for Inequality Constraints Minimization. http://www.mmm.muroran-it.ac.jp/~shi/LiuShiDongWang.pdf.
Mehra, R.K. and Davis, R.E. 1972. A Generalized Gradient Methodfor Optimal Control Problems With Inequality Constraints and Singular Arcs.IEEE Transactions on Automatic Control 17 (1): 69-79. DOI:10.1109/TAC.1972.1099881.
Rao, S.S. 1978. Optimization: Theory and Applications. New York City:John Wiley and Sons Ltd.
Sarma, P., Aziz, K., and Durlofsky, L.J. 2005a. Implementation of Adjoint Solutionfor Optimal Control of Smart Wells. Paper SPE 92864 presented at the SPEReservoir Simulation Symposium, The Woodlands, Texas, 31 January-2 February.DOI: 10.2118/92864-MS.
Sarma, P., Chen, W.H., Durlofsky L.J., and Aziz, K. 2006a. Production Optimization With AdjointModels Under Nonlinear Control-State Path Inequality Constraints. Paper SPE99959 presented at the SPE Intelligent Energy Conference and Exhibition,Amsterdam, 11-13 April. DOI: 10.2118/99959-MS.
Sarma, P., Durlofsky L.J., and Aziz, K. 2005b. Efficient Closed-Loop ProductionOptimization Under Uncertainty. Paper SPE 94241 presented at the SPEEuropec/EAGE Annual Conference, Madrid, Spain, 13-16 June. DOI:10.2118/94241-MS.
Sarma, P., Durlofsky, L.J., Aziz, K., and Chen, W.H. 2006b. Efficient Real-TimeReservoir Management Using Adjoint-Based Optimal Control and ModelUpdating. Computational Geosciences 10 (1): 3-36. DOI:10.1007/s10596-005-9009-z.
Sengul, M., Yeten, B., and Kuchuk, F. 2004. The CompletionChallenge—Modeling Potential Well Solutions. Middle East and Asia ReservoirReview 5: 4-17.
Stengel, R.F. 1985. Optimal Control and Estimation. New York City:Dover Books on Advanced Mathematics.
Vironovsky, G.A. 1991. Waterflooding Strategy Design Using Optimal ControlTheory. 6th European IOR Symposium, Stavanger, 21-23 May.
Wu, Z., Reynolds, A.C., and Oliver, D.S. 1999. Conditioning Geostatistical Models toTwo-Phase Production Data. SPEJ 4 (2): 142-155. SPE-56855-PA.DOI: 10.2118/56855-PA.
Yeten, B. 2003. Optimum Deployment of Nonconventional Wells. PhDdissertation, Stanford, California: Stanford University.
Zakirov, I.S., Aanonsen, S.I., Zakirov, E.S., and Palatnik, B.M. 1996.Optimization of Reservoir Performance by Automatic Allocation of Well Rates.5th European Conference on the Mathematics of Oil Recovery, Leoben, Austria,3-6 September.
Zhang, F., Skjervheim, J.A., Reynolds, A.C., and Oliver, D.S. 2005. Automatic History Matching in aBayesian Framework, Example Applications. SPEREE 8 (3):214-223. SPE-84461-PA. DOI: 10.2118/84461-PA.