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.
Well-Placement Optimization in an Enhanced Geothermal System Based on the Fracture Continuum Method and 0-1 Programming
