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**Derivative-free trust region optimization for robust well control under geological uncertainty.**
*(English)*
Zbl 1484.86004

Summary: A Derivative-Free Trust-Region (DFTR) algorithm is proposed to solve the robust well control optimization problem under geological uncertainty. Derivative-Free (DF) methods are often a practical alternative when gradients are not available or are unreliable due to cost function discontinuities, e.g., caused by enforcement of simulation-based constraints. However, the effectiveness of DF methods for solving realistic cases is heavily dependent on an efficient sampling strategy since cost function calculations often involve time-consuming reservoir simulations. The DFTR algorithm samples the cost function space around an incumbent solution and builds a quadratic polynomial model, valid within a bounded region (the trust-region). A minimization of the quadratic model guides the method in its search for descent. Because of the curvature information provided by the model-based routine, the trust-region approach is able to conduct a more efficient search compared to other sampling methods, e.g., direct-search approaches. DFTR is implemented within FieldOpt, an open-source framework for field development optimization, and is tested in the Olympus benchmark against two other types of methods commonly applied to production optimization: a direct-search (Asynchronous Parallel Pattern Search) and a population-based (Particle Swarm Optimization). Current results show that DFTR has improved performance compared to the model-free approaches. In particular, the method presented improved convergence, being capable to reach solutions with higher NPV requiring comparatively fewer iterations. This feature can be particularly attractive for practitioners who seek ways to improve production strategies while using an ensemble of full-fledged models, where good convergence properties are even more relevant.

### MSC:

86-08 | Computational methods for problems pertaining to geophysics |

90C90 | Applications of mathematical programming |

86A20 | Potentials, prospecting |

86A32 | Geostatistics |

### Keywords:

derivative-free trust-region algorithm; well control optimization; robust optimization under geological uncertainty
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\textit{T. L. Silva} et al., Comput. Geosci. 26, No. 2, 329--349 (2022; Zbl 1484.86004)

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### References:

[1] | Baumann, EJ; Dale, SI; Bellout, MC, FieldOpt: A powerful and effective programming framework tailored for field development optimization, Comput. Geosci., 135, 104, 379 (2020) |

[2] | Brouwer, DR; Jansen, JD, Dynamic optimization of waterflooding with smart wells using optimal control theory, SPE J., 9, 4, 391-402 (2004) |

[3] | Bukshtynov, V.; Volkov, O.; Durlofsky, LJ; Aziz, K., Comprehensive framework for gradient-based optimization in closed-loop reservoir management, Comput. Geosci., 19, 4, 877-897 (2015) · Zbl 1392.86006 |

[4] | Capolei, A.; Suwartadi, E.; Foss, B.; Jørgensen, JB, Waterflooding optimization in uncertain geological scenarios, Comput. Geosci., 17, 6, 991-1013 (2013) · Zbl 1393.86014 |

[5] | Chen, C., Li, G., Reynolds, A.: Robust constrained optimization of short- and long-term net present value for closed-loop reservoir management. SPE J. 17(3) (2012) |

[6] | Codas, A.; Foss, B.; Camponogara, E., Output-constraint handling and parallelization for oil-reservoir control optimization by means of multiple shooting, SPE J., 20, 4, 856-871 (2015) |

[7] | Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust Region Methods. Society for Industrial and Applied Mathematics (SIAM) and Mathematical Programming Society (MPS) (2000) |

[8] | Conn, AR; Scheinberg, K.; Toint, PL, Recent progress in unconstrained nonlinear optimization without derivatives, Math. Program., 79, 1-3, 397-414 (1997) · Zbl 0887.90154 |

[9] | Conn, AR; Scheinberg, K.; Vicente, LN, Geometry of interpolation sets in derivative free optimization, Math. Program., 111, 1-2, 141-172 (2008) · Zbl 1163.90022 |

[10] | Conn, A.R., Scheinberg, K., Vicente, L.N.: Introduction to Derivative-Free Optimization. Society for Industrial and Applied Mathematics (SIAM) and Mathematical Programming Society (MPS) (2009) · Zbl 1163.49001 |

[11] | Datta-Gupta, A.; Alhuthali, AHH; Yuen, B.; Fontanilla, J., Field applications of waterflood optimization via optimal rate control with smart wells, SPE Reserv. Eval. Eng., 13, 3, 406-422 (2010) |

[12] | Dehdari, V.; Oliver, DS, Sequential quadratic programming for solving constrained production optimization-case study from brugge field, SPE J., 17, 3, 874-884 (2012) |

[13] | Echeverría Ciaurri, D.; Isebor, O.; Durlofsky, L., Application of derivative-free methodologies to generally constrained oil production optimization problems, Procedia Comput. Sci., 1, 1, 1301-1310 (2010) |

[14] | van Essen, G.; Van den Hof, P.; Jansen, JD, Hierarchical long-term and short-term production optimization, SPE J., 16, 1, 191-199 (2011) |

[15] | Fonseca, R., Rossa, E.D., Emerick, A., Hanea, R., Jansen, J.: Overview of the olympus field development optimization challenge. In: ECMOR XVI - 16th European Conference on the Mathematics of Oil Recovery, EAGE, pp 1-10 (2018) |

[16] | G., Kt; Michael, LR; Virginia, T., Optimization by direct search: New perspectives on some classical and modern methods, SIAM Rev., 45, 3, 385-482 (2003) · Zbl 1059.90146 |

[17] | GeoQuest, S.: Eclipse reservoir simulator. Man. Tech. Descr. Houston TX (2014) |

[18] | Giuliani, C.M.: Contributions to derivative-free optimization: an exact-penalty method and decompositions for distributed control. Ph.D. thesis, Universidade Federal de Santa Catarina. https://bu.ufsc.br/teses/PEAS0331-T.pdf(2019) |

[19] | Giuliani, C.M., Camponogara, E., Conn, A.R.: A derivative-free exact penalty algorithm: Basic ideas, convergence theory and computational studies. To appear in Computational and Applied Mathematics (2021) · Zbl 07490224 |

[20] | Hasan, A., Gunnerud, V., Foss, B., Teixeira, A.F., Krogstad, S.: Decision analysis for long-term and short-term production optimization applied to the voador field. In: Proc. of SPE Reservoir Characterization and Simulation Conference and Exhibition. Society of Petroleum Engineers, pp 16-18 (2013) |

[21] | Hough, PD; Kolda, TG; Torczon, VJ, Asynchronous parallel pattern search for nonlinear optimization, SIAM J. Sci. Comput., 23, 1, 134-156 (2001) · Zbl 0990.65067 |

[22] | Isebor, O.J., Echeverría Ciaurri, D., Durlofsky, L.J.: Generalized field-development optimization with derivative-free procedures. SPE J. 19(5) (2014) |

[23] | Jansen, JD, Adjoint-based optimization of multi-phase flow through porous media - a review, Comput. Fluids, 46, 1, 40-51 (2011) · Zbl 1305.76107 |

[24] | Jansen, J.D., Brouwer, R., Douma, S.G.: Closed loop reservoir management. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2009) |

[25] | Kourounis, D.; Durlofsky, LJ; Jansen, JD; Aziz, K., Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow, Comput. Geosci., 18, 2, 117-137 (2014) · Zbl 1393.76122 |

[26] | Kraaijevanger, J.F.B.M., Egberts, P.J.P., Valstar, J.R., Buurman, H.W.: Optimal waterflood design using the adjoint method. SPE Reservoir Simulation Symposium p. 15. SPE-105764-MS (2007) |

[27] | Nocedal, J.; Wright, S., Numerical Optimization (2006), New York: Springer, New York · Zbl 1104.65059 |

[28] | Nwankwor, E.; Nagar, AK; Reid, D., Hybrid differential evolution and particle swarm optimization for optimal well placement, Comput. Geosci., 17, 2, 249-268 (2013) · Zbl 1382.90051 |

[29] | Pardalos, PM; Vavasis, SA, Quadratic programming with one negative eigenvalue is NP-hard, J. Glob. Optim., 1, 1, 15-22 (1991) · Zbl 0755.90065 |

[30] | Powell, MJD, Least frobenius norm updating of quadratic models that satisfy interpolation conditions, Math. Program., 100, 1, 183-215 (2004) · Zbl 1146.90526 |

[31] | Sampaio, PR; Toint, PL, Numerical experience with a derivative-free trust-funnel method for nonlinear optimization problems with general nonlinear constraints, Optim. Methods Softw., 31, 3, 511-534 (2016) · Zbl 1369.90138 |

[32] | Sarma, P.; Durlofsky, LJ; Aziz, K.; Chen, WH, Efficient real-time reservoir management using adjoint-based optimal control and model updating, Comput. Geosci., 10, 1, 3-36 (2006) · Zbl 1161.86303 |

[33] | Scheinberg, K.; Toint, PL, Self-correcting geometry in model-based algorithms for derivative-free unconstrained optimization, SIAM J. Optim., 20, 6, 3512-3532 (2010) · Zbl 1209.65017 |

[34] | Silva, TL; Codas, A.; Stanko, M.; Camponogara, E.; Foss, B., Network-constrained production optimization by means of multiple shooting, SPE Reserv. Eval. Eng., 22, 2, 709-733 (2019) |

[35] | Suwartadi, E.; Krogstad, S.; Foss, B., Nonlinear output constraints handling for production optimization of oil reservoirs, Comput. Geosci., 16, 2, 499-517 (2011) · Zbl 1253.90105 |

[36] | Volkov, O.; Bellout, MC, Gradient-based production optimization with simulation-based economic constraints, Comput. Geosci., 21, 5, 1385-1402 (2017) · Zbl 1403.90297 |

[37] | Volkov, O.; Voskov, DV, Effect of time stepping strategy on adjoint-based production optimization, Comput. Geosci., 20, 3, 707-722 (2016) · Zbl 1392.86047 |

[38] | Wang, C.; Li, G.; Reynolds, AC, Production optimization in closed-loop reservoir management, SPE J., 14, 3, 506-523 (2010) |

[39] | Yan, X., Reynolds, A.C.: Optimization algorithms based on combining FD approximations and stochastic gradients compared with methods based only on a stochastic gradient. SPE J. 19(5) (2014) |

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