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Summary We used optimal control theory as an optimization algorithm for the valve settings in smart wells. We focused on their use in injectors and producers for the waterflooding of heterogeneous reservoirs. As a followup to an earlier intuitive optimization approach, a systematic dynamic optimization approach based on optimal control theory was developed. The objective was to maximize recovery or net present value of the waterflooding process over a given time period. We investigated the scope for optimization under purely pressure- and purely rate-constrained operating conditions, and concluded that:for wells operating on bottomhole-pressure constraints, the benefit of using smart wells is mainly reduced water production rather than increased oil production, andfor wells operating on rate constraints, there is generally a large scope for accelerating production and increasing recovery, in combination with a drastic reduction in water production. Introduction The development of wells containing permanent downhole measurement and control equipment (in the remainder of this paper referred to as smart wells) potentially enables significant improvement of the oil production process. In an earlier study, we investigated the static optimization of waterflooding with smart wells, using heuristic algorithms.1 Static implies that the injection and production rates of the inflow control valves (ICVs) in the wells were kept constant during the displacement process, until water breakthrough at the producers occurred. Significant improvements were realized for simple reservoir models. Results suggested that more improvement could be expected by dynamically optimizing injection and production rates. Later, we therefore addressed the same problem using an optimization technique known as optimal control theory.2 In addition to being more systematic, enabling optimization also for more complex heterogeneities, this technique allowed for dynamic waterflooding control. Optimal control theory has been used before in reservoir engineering. Fathi and Ramirez used it to optimize surfactant flooding processes,3,4 Mehos to optimize CO2 flooding, 5 Liu and Ramirez to optimize steamflooding,6 and Zakirov et al. to optimize production from a thin oil rim.7 Furthermore, the same technique has been used in history matching reservoir models with production data. Optimization of waterflooding using optimal control theory has been studied before by Asheim,8 Virnovski,9 Sudaryanto and Yortsos,10-12 and Dolle et al.2 The optimization objective was either to maximize water breakthrough time at given field rates, or to maximize cumulative oil production or net present value (NPV) within a given time. In all these waterflood-optimization cases, the flow in the reservoir was controlled with wells that operated at constant field injection and production rates (i.e., the total injection and production rates were kept constant but the distribution over the wells was changed over time). In real life, however, a production strategy with constant field rates will often not be feasible, because it may require unrealistic bottomhole pressures, and associated sandface pressures. These could be too low pressures at the producers, resulting in lift die-out, or too high at the injectors, exceeding maximum allowable formation or equipment pressures. In the present study, we therefore investigated the scope for dynamic optimization for two extreme well-operating conditions. The first is completely rate-constrained injection and production, in which field injection and production rates are always at the maximum. The second is completely pressure-constrained injection and production, in which the injectors always inject at the maximum allowable injection pressure and the producers always produce at the minimum allowable well-flowing pressure. Problem Formulation Dynamic System Model. We considered a heterogeneous, horizontal, 2D, two-phase (oil/water) reservoir with two horizontal smart wells, an injector and a producer, at opposite sides (see Fig. 1). The reservoir has no-flow boundaries at all sides. Each well is divided into segments with ICVs, allowing for individual inflow control of the segments. Alternatively, the two horizontal wells can be interpreted as rows of vertical injectors and producers. We used a conventional finite-difference approximation to describe the reservoir, with details as given in Appendix A. The resulting numerical model can be represented as a discrete-time dynamic system model: Equation 1 where g is a nonlinear vector function, x is the vector of state variables with elements corresponding to the oil pressures and water saturations in each gridblock, k=0, . . ., K is the timestep, and u is the vector of the input variables or control variables, with elements that correspond to the water injection or liquid production rates in those gridblocks that are penetrated by a well.

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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.