Abstract
We consider problems where it is desirable to maximize multiple objective functions, but it is impossible to find a single design vector (vector of optimization variables) which maximizes all objective functions. In this case, the solution of the multi-objective optimization problem is defined as the Pareto front. The defining characteristic of the Pareto front is that, given any specific point on the Pareto front, it is impossible to find another point on the Pareto front or another feasible point which yields a greater value of all objective functions. The focus of this work is on the generation of the Pareto front for bi-objective optimization problems with specific applications to waterflooding optimization.
The most straightforward way to obtain the Pareto front is by application of the weighted sum method. We provide a procedure for scaling the optimization problem which makes it more straightforward to obtain points which are approximately uniformly distributed on the Pareto front when applying the weighted sum method. We also compare the performance of implementations of the weighted sum and normal boundary intersection (NBI) procedures where, with both methodologies, a gradient-based algorithm is used for optimization.
The vector of objective functions maps the set of feasible design vectors onto a set Z, and it is well known that all points on the Pareto front are on the boundary of Z. The weighted sum method cannot find points which are on the concave part of the boundary of Z, whereas the NBI method can be used to find all points on the boundary of Z, even though all points on this boundary may not correspond to Pareto optimal points. We develop and implement an NBI algorithm based on the augmented Lagrange method where the maximization of the augumented Lagrangian in the inner loop of the augmented Lagrange method is accomplished by a gradient-based optimization algorithm with the necessary gradients computed by the adjoint method.
Two waterflooding optimization problems are considered where we wish to optimize (maximize) two conflicting objectives. In the first, the two objectives are to maximize the life-cycle net present value (NPV) of production and to maximize the short-term NPV of production. In the second application, given an uncertain reservoir description, we wish to maximize the expected value of the NPV of life-cycle production and minimize the standard deviation of NPV over the ensemble of geological realizations.
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References
Aannonsen, S.I., Cominelli, A., Gosselin, O., Aavatsmark, I., Barkve, T.: Integration of 4D data in the history match loop by investigating scale dependent correlations in the acoustic impedance cube. In: Proceedings of 8th European conference on the mathmatics of oil recovery, pp 1–8 (2002)
Bailey, W.J., Couët, B., Wilkenson, D.: Framework for field optimization to maximize asset value. SPE Reserv. Eval. Eng. 8(1), 7–21 (2005)
Chen, C.: Adjoint-gradient-based production optimization with the augmented Lagrangian method. Ph.D. thesis, University of Tulsa (2011)
Chen, C., Li, G., Reynolds, A.C.: Robust constrained optimization of short- and long-term net present value for closed-loop reservoir management. SPE J. 17, 849–864 (2012)
Chen, Y., Oliver, D.S., Zhang, D.: Efficient ensemble-based closed-loop production optimization. SPE J. 14(4), 634–645 (2009)
Conn, A.R., Gould, N.I.M., Toint, P.L.: A globally convergent augumented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Numer. Anal. 28(2), 545–572 (1991)
Das, I., Dennis, J.E.: Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J. Optim. 8, 631–657 (1998)
Emerick, A.A., Reynolds, A.C.: History-matching production and seismic data in a real field case using the ensemble smoother with multiple data assimilation. In: Proccedings of the SPE reservoir simulation symposium, The Woodlands, Texas, USA, 18-20 February, SPE-163645 (2013)
Hajizadeh, Y., Christie, M., Demyanov, V.: Towards multiobjective history matching: faster convergence and uncertainty quantification. SPE Reserv. Simul. Symp. (2011)
Isebor, O.J., Durlofsky, L.J.: Biobjective optimization for general oil field development. J. Pet. Sci. Eng. (2014)
Li, R., Reynolds, A.C., Oliver, D.S.: History matching of three-phase flow production data. SPE J. 8 (4), 328–340 (2003)
Liu, X., Reynolds, A.C.: Multi-objective optimization for maximizing expectation and minimizing uncertainty or risk with application to optimal well control. In: Proceedings of SPE reservoir simulation symposium, SPE 173216 (2015)
Miettinen, K.: Nonlinear multi-objective optimization. Kluwer Academic Publishers, New York (1999)
Mohamed, L., Christie, M., Demyanov, V.: History matching and uncertainty quantification: multiobjective particle swarm optimization approach. In: SPE EUROPEC/EAGE annual conference and exhibition (2011)
Nocedal, J., Wright, S.J.: Numerical optimization. Springer, New York (2006)
Park, H.-Y., Datta-Gupta, A., King, M.J.: Handling conflicting multiple objectives using Pareto-based evolutionary algorithm during history matching of reservoir performance. SPE Reserv. Simul. Symp. (2013)
Rahuraman, B., Couët, B., Savundararaj, P., Bailey, W.J., Wilkenson, D.J.: Framework for field optimization to maximize asset value. SPE Reserv. Eval. Eng. 6(6), 307–315 (2003)
Reyes-Sierra, M., Coello, C.A.: Multi-objective partical swarm optimizers: a survey of the state-of-the-art. Int. J. Comput. Intell. Res. 2, 287–308 (2006)
Reynolds, A.C., He, N., Oliver, D.S.: Reducing uncertainty in geostatistical description with well testing pressure data. In: Schatzinger, R.A., Jordan, J.F. (eds.) Reservoir characterization–recent advances, pp 149–162, American Association of Petroleum Geologists (1999)
Sayyafzadeh, M., Haghighi, M.: Regularization in history matching using multi-objective genetic algorithm and Bayesian framework. In: EAGE annual conferance & exhibition incorporating SPE Europec (2012)
Schulze-Riegert, R., Krosche, M., Fahimuddin, A., Ghedan, S.: Multiobjective optimization with application to model validation and uncertainty quantification. In: 15th SPE Middle East oil & gas show and conference (2007)
Toint, P.L.: Non-monotone trust-region algorithms for nonlinear optimization subject to convex constraints. Math. Program. 77, 69–94 (1997)
van Essen, G., den Hof, P.V., Jansen, J.: Hierarchical long-term and short-term production optimization. SPE J. 16(1), 191–199 (2011)
van Essen, G., Zandvliet, M., den Hof, P.V., Bosgra, O., Jansen, J.: Robust waterflooding optimization of multiple geological scenarios, SPE 102913. SPE J. 14(1), 202–210 (2009)
Wang, S.: Second-order necessary, sufficient conditions in multi-objective programming. Numer. Funct. Anal. Optim. 12, 237–252 (1991)
Wu, Z., Reynolds, A.C., Oliver, D.S.: Conditioning geostatistical models to two-phase production data. SPE J. 3(2), 142–155 (1999)
Yasari, E., Pishvaie, M.R., Khorasheh, F., Salarhshoor, K.: Application of multi-criterion robust optimization in water-flooding of oil reservoir. J. Pet. Sci. Eng. 109(0), 1–J11 (2013)
Zhao, Y., Li, G., Reynolds, A.C.: Characterization of the measurement error in time-lapse seismic data and production data with an EM algorithm, oil & gas science and technology. Rev. IFP 62(2), 6181–193 (2007)
Zitzler, E., Deb, K., Thiele, L.: Comparison of multi-objective evolutionary algorithm. Evol. Comput. 8, 173–195 (2000)
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Liu, X., Reynolds, A.C. Gradient-based multi-objective optimization with applications to waterflooding optimization. Comput Geosci 20, 677–693 (2016). https://doi.org/10.1007/s10596-015-9523-6
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DOI: https://doi.org/10.1007/s10596-015-9523-6