(613h) Global Optimization of Constrained Grey-Box Models for Well Injection and Production

Crude oil development and production using secondary enhanced oil recovery techniques increase oil production by 20 to 40 percent by injecting water to oil fields to displace and drive oil to production wellbore [1-2]. Secondary recovery is a water-intensive method and the future availability of water resources is greatly threatened due poor water management in oil production fields (roughly 210 million barrels of water produced for every 75 million barrels of oil) [3]. This creates a huge demand to design and operate, environmentally sustainable and optimal well-control methods in waterflooding of oil reservoirs. However, waterflooding well-control technology in well injection and production greatly suffers from the curse of dimensionality. Thus, the objective of this paper is to introduce a sustainable innovative water-control technology based on a new optimization framework that significantly reduces the computational effort for waterflooding well-control optimization problems by (a) reducing the dimensionality of the optimization formulation, and (b) using a constrained grey-box optimization algorithm, which couples deterministic global optimization with surrogate modeling.

The first step of this workflow is based on parameterization of the well-control variable domain through a set of functional relationships, which is denoted as Functional Well-Control Method (FCM). The functional forms can range from Polynomial to Exponential or Hybrid Control Methods (PCM, ECM, HCM). Through this approach, we transform the optimization search space from the traditional pressure-based or rate-based control to a reduced space formed by the coefficients of the selected functional method. These new formulations are then optimized by the ARGONAUT algorithm [4-5], which has been developed for constrained derivative-free optimization problems. This optimizer is comprised of several mixed-integer and/or nonlinear optimization sub-problems for (a) sampling selection, (b) surrogate model identification and parameter estimation and, (c) global optimization of the formulated constrained surrogate formulations. The resulted non-linear constrained optimization problems are solved to global optimality using deterministic global optimization solver ANTIGONE [6-9], which is a state-of-the-art global solver for mixed-integer nonlinear optimization, used in a wide range of applications.

We test the efficiency of the entire framework, with and without constraints, on a realistic three-dimensional model (the UNSIM-I-D benchmark) [10]. Our results demonstrate significant computational savings due to the coupling of ARGONAUT [4-5] and the FCM formulation. The FCM leads to a substantial reduction in the number of control parameters as we seek the optimal function coefficients to describe the control trajectories as opposed to directly searching for the optimal control values at each time interval. In addition, we compare our results with other gradient-free and gradient-based algorithms which have been traditionally used in the literature (such as NOMAD [11-12] and EGO [13-14]) and we demonstrate that our framework leads to improved solutions with higher consistency and with reduced sample-calls to the reservoir simulation.

References:

1. CA Floudas, AM Niziolek, O Onel, LR Matthews, Multi-Scale Systems Engineering for Energy and the Environment: Challenges and Opportunities. AIChE Journal, 62(3), 2016, 602-623.
2. United States Department of Energy, Office of Fossil Energy: Enhanced Oil Recovery, (n.d.). http://energy.gov/fe/science-innovation/oil-gas-research/enhanced-oil-re....
3. B Bailey, M Crabtree, J Tyrie, J Elphick, F Kuchuk, C Romano, et al., Water control, Oilfield Review,12, 2000, 30-51.
4. F Boukouvala, MMF Hasan, CA Floudas, Global Optimization of General Constrained Grey-Box Models: New Method and Its Application to Constrained PDEs for Pressure Swing Adsorption, Journal of Global Optimization, 2016, DOI: 10.1007/s10898-015-0376-2.
5. F Boukouvala, CA Floudas, ARGONAUT: Algorithms for Global Optimization of Constrained Grey-Box Computational Problems, Optimization Letters, 2016, DOI: 10.1007/s11590-016-1028-2.
6. R Misener, CA Floudas, GloMIQO: Global Mixed-Integer Quadratic Optimizer. Journal of Global Optimization, 57(1), 2013, 3-50.
7. R Misener, CA Floudas, A Framework for Globally Optimizing Mixed-Integer Signomial Programs, Journal of Optimization Theory and Applications, 161(3), 2014, 905-932.
8. R Misener, CA Floudas, ANTIGONE: Algorithms for coNTinuous Integer Global Optimization of Nonlinear Equations, Journal of Global Optimization, 59(2â??3), 2014, 503â??526.
9. F Boukouvala, R Misener, CA Floudas, Global Optimization Advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO, European Journal of Operational Research, 252(3), 2016, 701â??727.
10. AT Gaspar, GD Avansi, AA dos Santos, JC von Hohendorff Filho, DJ Schiozer. UNISIM-ID: Benchmark Studies for Oil Field Development and Production Strategy Selection. International Journal of Modeling and Simulation for the Petroleum Industry, 9(1), 2015, 47-55.
11. MA Abramson, C Audet, G Couture, JE Dennis, C Tribes, The NOMAD Project (2009). https://www.gerad.ca/nomad 52.
12. S Le Digabel: Algorithm 909: NOMAD: Nonlinear Optimization with the MADS Algorithm. ACM Trans. Math. Softw. (TOMS) 37(4), 2011, 1â??15.
13. DR Jones, M Schonlau and WJ Welch: Efficient Global Optimization of Expensive Black-Box Functions, Journal of Global Optimization, 13, 1998, 455-492.
14. MJ Sasena, P Papalambros, P Goovaerts, Exploration of Metamodeling Sampling Criteria for Constrained Global Optimization, Engineering Optimization, 34(3), 2002, 263-78.