A surrogate-based approach to waterflood optimisation under uncertainty

The Markowitz classical theory has been applied in the robust optimisation of petroleum engineering operations by many researchers. It involves the computation of the means and standard deviation of a specified reservoir performance measure(s), and the creation of an efficient frontier which qualifies the relationship between the optimised mean and standard deviation. However, the optimisation routine is computationally expensive as numerous simulations are required for calculating the means and standard deviations. Also, to simplify the optimisation problem many significant uncertainties are not considered in the optimisation routine. Also, previous researches have used a limited number of reservoir-model sample points of the uncertain variable(s) to calculate the means and standard deviations values. For example, if the uncertain parameter is uniformly distributed, three equiprobable (the low, median and high values) are used to correlate the uncertainty. However, this approach leads to erroneous calculations of the means and standard deviations because the actual distribution of the uncertainty is ignored. In this study, we apply the Markowitz classical robust optimisation routine to a validated approximation model of the cumulative oil production of a case study reservoir to optimise oil recovery after waterflooding. Using this approach, we can reduce computational costs and for the first time, consider up to four geological uncertain variables in reservoir optimisation under uncertainty. We show that at least 100 sample points (realisations) of the uncertain geological parameters are required to obtain accurate computations of the means (reward) and standard deviations (risk). This allows for adequate sampling of the distribution of the uncertain parameters. We then construct an efficient frontier of the optimal solutions for various risk-aversion factors and compare the results to that obtained from a deterministic optimisation routine. This approach was applied for the first time to optimisation under uncertainty. The result indicates that considering geological uncertainties while solving to the optimisation problem results in more realistic optimal solutions when compared to the deterministic optimisation case. This is because engineering control variables that lead to a riskquantified strategy for the waterflooding operation are obtained.