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Optimal control of polymer flooding based on maximum principle. (English) Zbl 1252.49032

Summary: Polymer flooding is one of the most important technologies for Enhanced Oil Recovery (EOR). In this paper, an optimal control model of Distributed Parameter Systems (DPSs) for polymer injection strategies is established, which involves the performance index as maximum of the profit, the governing equations as the fluid flow equations of polymer flooding and the inequality constraint as the polymer concentration limitation. To cope with the Optimal Control Problem (OCP) of this DPS, necessary conditions for optimality are obtained through application of the calculus of variations and Pontryagin’s weak maximum principle. A gradient method is proposed for the computation of optimal injection strategies. The numerical results of an example illustrate the effectiveness of the proposed method.

MSC:

49K20 Optimality conditions for problems involving partial differential equations
49J20 Existence theories for optimal control problems involving partial differential equations
49M30 Other numerical methods in calculus of variations (MSC2010)
90C30 Nonlinear programming
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[1] Y. Qing, D. Caili, W. Yefei, T. Engao, Y. Guang, and Z. Fulin, “A study on mass concentration determination and property variations of produced polyacrylamide in polymer flooding,” Petroleum Science and Technology, vol. 29, no. 3, pp. 227-235, 2011. · doi:10.1080/10916461003752496
[2] B. K. Maitin, “Performance analysis of several polyacrylamide floods in North German oil fields,” in Proceedings of the SPE/DOE Enhanced Oil Recovery Symposium, pp. 159-165, 1992.
[3] M. A. de Melo, C. R. C. de Holleben, I. P. G. da Silva et al., “Evaluation of polymer injection projects in Brazil,” in Proceedings of the SPE Latin American and Caribbean Petroleum Engineering Conference, pp. 1-17, 2005. · doi:10.1080/10916460903330098
[4] Q. Yu, H. Jiang, and C. Zhao, “Study of interfacial tension between oil and surfactant polymer flooding,” Petroleum Science and Technology, vol. 28, no. 18, pp. 1846-1854, 2010. · doi:10.1080/10916466.2010.506466
[5] H. Jiang, Q. Yu, and Z. Yi, “The influence of the combination of polymer and polymer-surfactant flooding on recovery,” Petroleum Science and Technology, vol. 29, no. 5, pp. 514-521, 2011. · doi:10.1080/10916466.2010.529551
[6] W. F. Ramirez, Z. Fathi, and J. L. Cagnol, “Optimal injection policies for enhanced oil recovery: part 1 theory and computational strategies,” Society of Petroleum Engineers Journal, vol. 24, no. 3, pp. 328-332, 1984.
[7] Z. Fathi and W. F. Ramirez, “Use of optimal control theory for computing optimal injection policies for enhanced oil recovery,” Automatica, vol. 22, no. 1, pp. 33-42, 1986. · Zbl 0586.93035 · doi:10.1016/0005-1098(86)90103-2
[8] W. Liu, W. F. Ramirez, and Y. F. Qi, “Optimal control of steamflooding,” SPE Advanced Technology Series, vol. 1, no. 2, pp. 73-82, 1993.
[9] J. Ye, Y. Qi, and Y. Fang, “Application of optimal control theory to making gas-cycling decision of condensate reservoir,” Chinese Journal of Computational Physics, vol. 15, no. 1, pp. 71-76, 1998.
[10] P. Daripa, J. Glimm, B. Lindquist, and O. McBryan, “Polymer floods: a case study of nonlinear wave analysis and of instability control in tertiary oil recovery,” SIAM Journal on Applied Mathematics, vol. 48, no. 2, pp. 353-373, 1988. · Zbl 0641.76093 · doi:10.1137/0148018
[11] P. Daripa and G. Pa\csa, “An optimal viscosity profile in enhanced oil recovery by polymer flooding,” International Journal of Engineering Science, vol. 42, no. 19-20, pp. 2029-2039, 2004. · Zbl 1211.86005 · doi:10.1016/j.ijengsci.2004.07.008
[12] P. Daripa and G. Pa\csa, “Stabilizing effect of diffusion in enhanced oil recovery and three-layer Hele-Shaw flows with viscosity gradient,” Transport in Porous Media, vol. 70, no. 1, pp. 11-23, 2007. · doi:10.1007/s11242-007-9122-7
[13] P. Daripa and G. Pasa, “On diffusive slowdown in three-layer Hele-Shaw flows,” Quarterly of Applied Mathematics, vol. 68, no. 3, pp. 591-606, 2010. · Zbl 1425.76076
[14] P. Daripa, “Studies on stability in three-layer Hele-Shaw flows,” Physics of Fluids, vol. 20, no. 11, pp. 1-11, 2008. · Zbl 1182.76180 · doi:10.1063/1.3021476
[15] P. Daripa, “Hydrodynamic stability of multi-layer Hele-Shaw flows,” Journal of Statistical Mechanics, vol. 12, pp. 1-32, 2008. · Zbl 1182.76180
[16] D. R. Brouwer and J. D. Jansen, “Dynamic optimization of water flooding with smart wells using optimal control theory,” in Proceedings of the SPE European Petroleum Conference, pp. 391-402, 2002.
[17] P. Sarma, K. Aziz, and L. J. Durlofsky, “Implementation of adjoint solution for optimal control of smart wells,” in Proceedings of the SPE Reservoir Simulation Symposium, pp. 1-17, 2005.
[18] L. L. Guo, S. R. Li, Y. B. Zhang, and Y. Lei, “Solution of optimal control of polymer flooding based on parallelization of iterative dynamic programming,” Journal of China University of Petroleum, vol. 33, no. 3, pp. 167-174, 2009, (Edition of Natural Science).
[19] S. R. Li, Y. Lei, X. D. Zhang, and Q. Zhang, “Optimal control solving of polymer flooding based on a hybrid genetic algorithm,” in Proceedings of the 29th Chinese Control Conference, pp. 5194-5198, 2010.
[20] Y. Lei, S. R. Li, and X. D. Zhang, “Optimal control solving of polymer flooding based on real-coded genetic algorithm,” in Proceedings of the 8th World Congress on Intelligent Control and Automation, pp. 5111-5114, 2010. · doi:10.1109/WCICA.2010.5554942
[21] K. Aziz and A. Settari, Fundamentals of Reservoir Simulation, Elsevier Applied Science, New York, NY, USA, 1986. · Zbl 0256.65020
[22] P. Sarma, W. H. Chen, L. J. Durlofsky, and K. Aziz, “Production optimization with adjoint models under nonlinear control-state path inequality constraints,” in Proceedings of the SPE Intelligent Energy Conference and Exhibition, pp. 1-19, 2006.
[23] J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, NY, USA, 2000. · Zbl 0930.65067
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