Pareto-based robust optimization of water-flooding using multiple realizations

highlights
• Two multi-objective robust optimization problems have been investigated.
• They have been considered to reduce the sensitivity to the geological uncertainty.
• Robust optimization has been done based on sample set of realizations.
• The comparative test studies demonstrated the superiority of the proposed methodology.
• Mean and variance (robustness) of NPVs, were selected as objective functions.

Robust optimization (RO) approach is inherently a multi-objective paradigm. The proposed multi-objective optimization formulation would attempt to find the optimum – yet robust – water injection policies. Two multi-objective, Pareto-based robust optimization scenarios have been investigated to encounter the permeability uncertainties. These multi-objective RO scenarios have been done based on a small representative set of realizations but they have introduced optimum points that could be reliable for the original set of realizations either. In both scenarios, the desired objective functions are expected value and variance of Net Present Value (NPV). The underlying RO scenarios have been done without any observation/measurement of pressures or well flows. Therefore, an ensemble of equally probable realizations has been used and ranked using Monte Carlo simulation technique. The Non-dominated Sorting Genetic Algorithm second version (NSGA-II) has been used as the optimization algorithm. The multi-objective robust optimization scheme has been applied for both scenarios via a twin setup of 100 realizations, one for investigation and the other one for validation purposes. The test studies demonstrated the superiority of the proposed methodology to give a robust optimal Pareto-based solution(s) (injection policies) under permeability uncertainties that could be reliable for the original set of realizations. Probability distribution functions (PDFs) of the original and small set of realizations have been depicted for comparison. Both optimization scenarios introduced optimum and robust injection policies that lead to higher expected value of NPV and lower variance, besides preserving the first and second moments of the original population of the original set of realizations.