Summary Determining the optimal location of wells with the aid of an automated search method can significantly increase a project's net present value (NPV) as modeled in a reservoir simulator. This paper has two main contributions: first, to determine the effect of production constraints on optimal well locations and, second, to determine optimal well locations using a gradient-based optimization method. Our approach is based on the concept of surrounding the wells whose locations have to be optimized by so-called pseudowells. These pseudowells produce or inject at a very low rate and, thus, have a negligible influence on the overall flow throughout the reservoir. The gradients of NPV over the lifespan of the reservoir with respect to flow rates in the pseudowells are computed using an adjoint method. These gradients are used subsequently to approximate improving directions (i.e., directions to move the wells to achieve an increase in NPV), on the basis of which improving well locations can be determined. The main advantage over previous approaches such as finite-difference or stochastic-perturbation methods is that the method computes improving directions for all wells in only one forward (reservoir) and one backward (adjoint) simulation. The process is repeated until no further improvements are obtained. The method is applied to three waterflooding examples. Introduction Determining the location of wells is a crucial decision during a field-development plan because it can affect a project's NPV significantly. Well placement is often posed as a discrete optimization problem (Yeten 2003) (i.e., involving integers as decision variables). Solving such problems is an arduous task; therefore, well locations often are determined manually. However, several automated well-placement optimization methods are available in the literature. They can be classified broadly into two categories. The first category consists of local methods such as finite-difference-gradient (FDG) (Bangerth et al. 2006), simultaneous-perturbation-stochastic-approximation (Bangerth et al. 2003, Spall 2003), and Nelder-Mead simplex (Spall 2003) methods. The second category consists of global methods such as simulated annealing (Beckner and Song 1995), genetic algorithms (Montes et al. 2001, Güyagüler et al. 2002, Yeten et al. 2003), and neural networks (Centilmen et al. 1999). The first category is generally very efficient, requires only a few forward reservoir simulations, and increases NPV at each iteration. However, these methods can get stuck in a local optimal solution. The second category can, in theory, avoid this problem but has the disadvantages of not increasing NPV at each iteration and requiring many forward reservoir simulations. A rather different approach is proposed by Lui and Jalali (2006), where standard reservoir models are transformed to maps of production potential to screen regions that are most favorable for well placement. In this paper, we present a gradient-based method that is distinct from those previously mentioned. The adjoint method used in optimal-control theory has been used previously for optimization of injection and production rates in a fixed-well configuration (Ramirez 1987, Asheim 1988, Sudaryanto and Yortsos 2001, Zakirov et al. 1996, Virnovsky 1991, Brouwer and Jansen 2004, Sarma et al. 2005, Kraaijevanger et al. 2007). In these applications, the parameters to be optimized are usually well-flow rates, bottomhole pressures (BHPs), or choke-valve settings. Because these are not mixed-integer problems, gradient-based methods are used commonly to solve them and the adjoint method efficiently generates the required gradients. We propose to use the adjoint method for well-placement optimization. An example of well-placement optimization using optimal control theory has been proposed previously by Virnovsky and Kleppe (1995). Our approach, however, is significantly different. Moreover, two further applications of adjoint-based well-placement optimization were published recently (Wang et al. 2007, Sarma and Chen 2007.) The outline of our paper is as follows: First, the effect of production constraints on optimal well locations is investigated. Then, an adjoint-based well-placement-optimization method is presented. Finally, the benefits of this method are demonstrated by three waterflooding examples.

Dynamic optimization of waterflooding using optimal control theory has significant potential to increase ultimate recovery, as has been shown in various studies. However, optimal control strategies often lack robustness to geological uncertainties. We present an approach to reduce the effect of geological uncertainties in the field-development phase known as robust optimization (RO). RO uses a set of realizations that reflect the range of possible geological structures honoring the statistics of the geological uncertainties. In our study, we used 100 realizations of a 3D reservoir in a fluvial depositional environment with known main-flow direction. We optimized the rates of the eight injection and four production wells over the life of the reservoir, with the objective to maximize the average net present value (NPV). We used a gradient-based optimization method in which the gradients are obtained with an adjoint formulation. We compared the results of the RO procedure to two alternative approaches: a nominal-optimization (NO) and a reactive-control approach. In the reactive approach, each production well is shut in when production is no longer profitable. The NO procedure is based on a single realization. In our study, the NO procedure is performed on each of the 100 realizations in the set individually, resulting in 100 different NO-production strategies. The control strategies were applied to each realization, from which the average NPVs, the standard deviation, the cumulative-distribution functions, and the probability-density functions were determined. The RO results displayed a much smaller variance than the alternatives, indicating an increased robustness to geological uncertainty. Moreover, the RO procedure significantly improved the expected NPV compared to the alternative methods (on average 9.5% higher than using reactive-control and 5.9% higher than the average of the NO strategies).

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Dynamic optimization of waterflooding using optimal control theory has a significant potential to increase ultimate recovery, as has been shown in various studies. However, optimal control strategies often lack robustness to geological uncertainties. We present an approach to reduce the impact of geological uncertainties in the field development phase known as a robust optimization (RO). RO uses a set of realizations that reflect the range of possible geological structures honoring the statistics of the geological uncertainties. In our study we used 100 realizations of a 3-dimensional reservoir in a fluvial depositional environment with known main flow direction. We optimized the rates of the 8 injection and 4 production wells over the life of the reservoir, with the objective to maximize the average net present value (NPV). We used a gradient-based optimization method where the gradients are obtained with an adjoint formulation. We compared the results of the RO procedure to two alternative approaches: a nominal optimization and a reactive control approach. The nominal optimization is based on a single realization, while in the reactive approach each production well is shut in when production is no longer profitable. The three control strategies were applied to each realization, from which the average NPV's and three cumulative distribution functions were determined. The RO results displayed a much smaller variance than the alternatives, indicating the increased robustness to geological uncertainty. Moreover, the RO procedure significantly improved the average NPV, compared to the alternative methods: on average 9% higher than using reactive control and 3% higher than for the nominal case. Introduction Within this paper, we consider the secondary recovery phase of a petroleum reservoir using waterflooding. In this case a number of injection and production wells are drilled to preserve a steady reservoir pressure and sweep the reservoir. The use of smart wells expands the possibilities to manipulate and control fluid flow paths through the oil reservoir. The ability to manipulate (to some degree) the progression of the oil-water front provides the possibility to search for a control strategy that will result in maximization of ultimate oil recovery. Dynamic optimization of waterflooding using optimal control theory has a significant potential to increase ultimate recovery by delaying water breakthrough and increasing sweep, as has been shown in various studies1. However, optimal control strategies often lack robustness to geological uncertainties. By discarding these uncertainties, bounding the sensitivity to a possibly large system-model mismatch is not taken into account within the optimization procedure. As a result, the optimal control strategy may seize to be optimal or may even result into very poor performance. Dealing with uncertainty is a topic encountered in many fields related to modeling and control. It can essentially be divided into two different strategies, which are not mutually exclusive: reducing the uncertainty itself using measurements, i.e. history matching2,3 and reducing the sensitivity to the uncertainty. However, within this paper, we consider a situation in which no production data is assumed available, which rules out any history matching approach to reduce the geological uncertainty. Our study forms part of a larger research project to enable closed-loop model-based reservoir management4. A suggested approach from the downstream process industry, to optimization problems which suffer from vast uncertainty and limited measurement information, is the use of a so-called robust optimization technique5,6,7. In robust optimization, the optimization procedure is carried out over a set of realizations, in this way actively accounting for the influence of the uncertainty. The implementation of multiple realizations within the optimization process has been addressed by Yeten8, however this study deviates in way they are incorporated in the objective function and in the number of realizations. The goal of this paper is to present a robust optimization procedure based on a set of 100 realizations of a 3-dimensional oil-water reservoir, which leads to a control strategy that explicitly accounts for geological uncertainty.

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