Discussion

In the well placement optimization applications considered in this work, HPSDE algorithm yielded higher NPV than DE and PSO algorithms. While we note that these findings and results are interesting and potentially useful, we acknowledge that there are issues or limitations that need to be addressed. Chief among these limitations is the issue of control parameters tuning. For instance, in the second and third examples, there are instances where DE outperformed PSO, and vice versa. It is important to understand how these behaviors are influenced by relevant control parameters of the algorithm. We note that DE parameters (F = 0.5, CR = 0.1) used in this work were adapted from Storn and Price (1997), and the PSO parameters (c₁ c₂ 1.193,0.721) were adapted from Onwunalu and Durlofsky (2010). For reasons bothering on fair comparison of results, all three algorithms were used without parameter tuning of any kind. Although the population size and the maximum number of iteration are largely dependent on the complexity of the underlying well optimization problem, as well as the number of optimization variables; we believe that effective parameter tuning (which will be computationally expensive, as it will require extra function evaluations) would further enhance the performance of the algorithms. This is so because generalized adjustment of metaheuristic control parameters cannot be achieved from theoretical analyses on the algorithms alone. Thus, an effective mechanism for control parameter tuning would depend on the demands of the underlying optimization problem as well as the experience and background knowledge of the user.

A closely related limitation is the issue of usability in practical field optimization problem. Indeed, a hybridized metaheuristic optimization algorithm such as HPSDE is potentially a viable and promising alternative in reservoir engineering optimization problems; however, issues of usability have to be addressed before it can be deployed for practical use in the

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industry. To some degree, the usability limitation is intertwined with parameter tuning. We note that usability of metaheuristic algorithms would be greatly enhanced if the issue of parameter tuning is sorted out at the design-end, and not at the user-end of the algorithms. This is so because it is generally unrealistic for industrial end-users to waste expensive function evaluations in correcting the weakness of the design phase of an algorithm. We also note that the performance of HPSDE algorithm, and indeed other hybridized stochastic algorithms; could be further improved by incorporating into the algorithms, prior knowledge and relevant information about the optimization problem.

4.8 Summary

In this chapter, the importance of well optimization was highlighted; we showed that it is a field development decision input that can ultimately determine a reservoir‘s production profile, and therefore, the recoverability of the reservoir. For all intents and purposes, the recoverability is a direct measure of the economic value of the portfolio or NPV of the asset. In this work, we employed three metaheuristic algorithms in this problem domain; one of the algorithms (HPSDE) is a ‗hybrid‘ of the other two algorithms – DE and PSO. With NPV as performance measure, we considered three examples involving the placement of one, two and nine vertical wells in reservoir models of varying complexities.

Based on suggestions from Vasiljevic and Golobic (1996) and Ciaurri et al. (2011), five runs of each of the algorithms are performed in the first application; owing to the need to carry out a more detailed, reliable and systematic analyses of the algorithms, thirty optimization runs of the algorithms were performed in the second and third applications. In all cases, the results were averaged over the number of optimization runs so as to determine the relative strength of each algorithm. The HPSDE algorithm consistently outperformed DE and PSO algorithms. In two of the examples, we factored in geological uncertainty by addressing the discrepancies between physical reservoir and reservoir model. To this end, we performed a max-mean objective robust optimization of the performance measure; and HPSDE yielded better results than DE and PSO. In the third example, we compared the performance of the metaheuristic algorithms with the NPVs attained via different specific well pattern arrangements; and the stochastic algorithms yielded higher NPV than the specific well pattern arrangements. We also showed that the performance of DE and PSO was dependent on the total number of simulations – in other words, there was a variation in performance in the early, mid and later

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stages of simulation. DE attained higher NPV than PSO at very low and very high number of total simulation. However, in all examples considered, the overall performance of PSO was better than that of DE. More importantly, we note that HPSDE outperformed both algorithms in all cases. Besides, the performance of the HPSDE algorithm was compared with the performance achieved by more established optimization techniques such as LP and GA; and HPSDE outperformed both algorithms. Although the result attained by GA was comparable to that of PSO and DE, the result emanating from LP fell way behind those attained by the stochastic algorithms. Because HPSDE algorithm was created as a result of the hybridization of two global stochastic algorithms – DE and PSO; we compared its performance with results from another hybrid algorithm created by the hybridization of a global algorithm (PSO) and a local search algorithm (TS). In this regards, HPSDE was compared with PSOTS algorithm; and the result showed that the performance of PSOTS was 2.6% less than the performance of HPSDE. Interestingly, the PSOTS algorithm outperformed the remaining metaheuristic algorithms in the order DE, PSO and GA respectively.

Furthermore, with the aid of statistical tools, we used a collection of six benchmark tests of varying complexity to gain further insight into the relative performance of the algorithms. Based on the analyses of the results from the benchmark tests, the effectiveness of HPSDE algorithm over DE and PSO as demonstrated in the well placement optimization problems was reinforced. Also, due to the fact that computational complexity of population-based stochastic algorithms are critical to understanding relative algorithmic efficiency, the runtime and space complexity of the algorithms were analyzed by evaluating fundamental arithmetic and logical operations performed by the algorithms. By and large, these tools afforded us the ability to draw conclusions on the relative performance of the algorithms.

Despite the limitations that were highlighted in terms of usability of metaheuristic algorithms in the industry; this work demonstrates the potential benefit of hybridized metaheuristic algorithms over more established stochastic techniques in reservoir engineering applications. Besides the fact that these findings are promising, the applicability of HPSDE algorithm in well placement optimization problem shows that hybridization could be key to unlocking some of the challenging optimization problems in field development planning and reservoir engineering in general.

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