Uncertainty Assessment of Well-Placement Optimization

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Summary Determining the best location for new wells is a complex problem that depends on reservoir and fluid properties, well and surface-equipment specifications, and economic criteria. Numerical simulation is often the most appropriate tool to evaluate the feasibility of well configurations. However, because the data used to establish numerical models have uncertainty, so do the model forecasts. The uncertainties in the model reflect themselves in the uncertainties of the outcomes of well-configuration decisions. We never possess the true and deterministic information about the reservoir, but we may have geostatistical realizations of the truth constructed from the information available. An approach that can translate the uncertainty in the data to uncertainty in the well-placement decision in terms of monetary value was developed in this study. The uncertainties associated with well placement were addressed within the utility-theory framework using numerical simulation as the evaluation tool. The methodology was evaluated by use of the Production forecasting with UNcertainty Quantification (PUNQ)-S3 model, which is a standard test case that was based on a real field. Experiments were carried out on 23 history-matched realizations, and a truth case was also available. The results were verified by comparison to exhaustive simulations. Utility theory not only offered the framework to quantify the influence of uncertainties in the reservoir description in terms of monetary value, but it also provided the tools to quantify the otherwise arbitrary notion of the risk attitude of the decision maker. A hybrid genetic algorithm (HGA) was used for optimization. In addition, a computationally cheaper alternative was also investigated. The well-placement problem was formulated as the optimization of a random function. The genetic algorithm (GA) was used as the optimization tool. Each time a well configuration was to be evaluated, a different realization of the reservoir properties was selected randomly from the set of realizations, all of which honored the geologic and dynamic data available from the reservoir. Numerical simulation was then carried out with this randomly selected realization to calculate the objective function value. This approach has the potential to incorporate the risk attitudes of the decision maker and was observed to be approximate but computationally feasible. Introduction Optimization of well placement is a complex problem that has been investigated in previous studies.1-9 Some of these studies looked into the assessment of uncertainty. However, none has suggested a robust way to assess the uncertainty of well placement within a direct-optimization context using the numerical simulator as the evaluation tool. Quantification of the qualitative notion of risk attitude also has not been addressed. Direct optimization constitutes the coupling of an optimization tool with the numerical model that, although accurate, is in most cases computationally infeasible. A hybrid approach used by Güyagüler et al.9 was able to reduce the computational burden of making numerous simulations and was applied with good results to the Gulf of Mexico Pompano field. This hybrid approach is referred to as the HGA because it makes use of GAs, the polytope method, and the proxy approach using the ordinary kriging algorithm. The HGA as proposed by Güyagüler et al.9 was based on the algorithm developed by Bittencourt and Horne,4 who hybridized the polytope method with the GA and the proxy approach that was proposed by Pan and Horne.5 The HGA was also used in this study for the PUNQ-S3 reservoir. The HGA is based on the GA in that a population is kept and modified with GA operators such as mutation and crossover. Greedy local search is also carried out with the polytope algorithm. In addition, a proxy to numerical simulation is constructed and updated at each iteration and is used to guide the overall search toward the feasible areas of the search space. It is realized that the numerical model, on which we chose to base the well-placement decision, relies on data that are uncertain. This uncertainty in the data translates into uncertainty in the numerical-simulation forecasts. Optimization in such an uncertain case has many additional complications. A deterministic global solution is not available in the case of uncertainty. Given the data available, what we hope to achieve is to estimate the expected outcome of any proposed decision as well as the risks associated with it. The established framework of decision and utility theory enables us to manage uncertainty.10 Realizing the fact that every decision maker would act differently given options with probabilistic outcomes, the framework also provides the tools necessary to quantify the risk attitudes of the decision maker.11 Decision-theory framework has been used extensively and successfully in a wide range of industries,12 including the petroleum industry.13-17 However, it was observed that in the petroleum industry, decision-analysis tools are generally used during exploration and initial development stages.15 Application of decision-analysis tools to the reservoir-development process has been limited owing to the exponentially increasing number of options with added uncertainty in the decision trees. Because of this exponential growth property of the decision trees, decision makers have been forced to use approximate evaluation tools. In this study, we insist on using full numerical simulation as the evaluation tool. The use of numerical simulation was rendered computationally feasible by transforming the problem into a deterministic problem through utility functions that quantify risk attitudes and by using the HGA for optimization. The utility framework requires the outcome probabilities for proposed well configurations. In some cases, the determination of outcome probabilities might be computationally infeasible, particularly for very large numerical models. A second approach is presented in which the well-placement problem has been formulated as the optimization of a random function that does not require the prior knowledge of outcome probabilities. The GA was used for optimization. Utility Framework The problem of well placement can be studied within the decision- analysis framework because the problem consists of the decision of the location to drill the new well and the probable events thereafter.

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