Hybrid FDG optimization method and kriging interpolator to optimize well locations

Hybrid FDG optimization method and kriging interpolator to optimize well locations Gholamreza Khademi 0 Paknoosh Karimaghaee 0 0 School of Electrical and Computer Engineering, Shiraz University , Shiraz , Iran As the number of new significant oilfields discoveries are reduced and as production operations become more challenging and expensive, the efficient development of oil reservoirs in order to satisfy increasing worldwide demand for oil and gas becomes crucial. A key decision engineers must make is where to drill wells in the reservoir to maximize net present value or some other objectives. Since the number of possible solutions that depend on the size of reservoir can be very large, the use of an optimization algorithm is necessary. Optimization methods are divided into two main categories: non-gradient-based and gradient-based algorithms. In the former, the search strategy is to find global optimum while they need a great number of reservoir simulation runs. On the other hand, gradient-based optimization algorithms search locally but require fewer reservoir simulations. The computational cost of optimization method in the optimal well placement problem is substantial. Thus, in practical problems with large models, implying the gradient-based method is preferable. In the present paper, finite difference gradient (FDG) algorithm as one of the easy implemented gradientbased family is used. The main disadvantage of the mentioned technique is its dependency on the number of decision variables. The major contribution of this paper is to hybrid the FDG method and kriging interpolator. This interpolator is used as a proxy to decrease the required number of function evaluations and estimate the direction of movements in the FDG algorithm. Moreover, the idea of local grid refinement is proposed to eliminate the mixed Optimal well placement; FDG optimization method; Kriging interpolator; Reservoir simulation - & Gholamreza Khademi integer problem of well placement. Then, the method is applied to some sample reservoirs and the simulation results verify the performance of the proposed method. In the context of oilfield development, production optimization has been an attractive research area in recent years. The motivation of such a considerable focus of attention is the need for producing limited existing fields as efficiently as possible, while decreasing economical and operating costs at the same time. Production optimization is often divided into well placement and well control optimization problems. In the well placement problems, the purpose is to drill wells at optimal locations so that more oil and gas can be extracted, while in well control problems, the well parameters such as producer or injector flow rates or bottom hole pressures (BHPs) are optimized. The focus of this paper is on the literature review of optimal well placement procedure. Well placement is a challenging problem due to the existence of different decision variables e.g., well types and the presence of geological uncertainty which leads to multiple realization of reservoir. Thus, different possible solutions and scenarios exist, and only trusting to experienced reservoir engineers may lead to insufficient solution far from the optimal one. Consequently, the need for a systematic optimization method is obvious. The literature relevant to optimal well placement is extensive. Numerous optimization methods to find optimal well place have been introduced over the past few years. These methods fall into two wide categories: gradient-free and gradient-based algorithms. Gradient-free methods are also categorized into stochastic algorithms, global search, and deterministic algorithms, local search strategy. Both stochastic and deterministic algorithms do not need the derivatives of objective function respect to the decision variables. However, the stochastic approach requires a large number of objective function evaluations. Simulated annealing (Beckner and Song 1995), genetic algorithm (Yeten et al. 2003), particle swarm optimization (Onwunalu and Durlofsky 2010), the covariance matrix adaptation evolution strategy (CMAES) (Ding 2008) are popular stochastic optimization algorithms applied to the well placement problem. In Onwunalu (2010), standard PSO method is applied as an alternative to GA for well placement and showed that PSO resulted in better performance than GA. In addition, GPS (Isebor 2009), HJDS (Hooke and Jeeves 1961), polytope (Guyaguler 2002), and MADS (Ciaurri et al. 2011) are examples of deterministic local search methods. For detailed information on the implementation of the mentioned methods refer to (Mathworks 2009). Briefly, it can be concluded from the literature that stochastic methods are well suited for the well placement problem since normally the problem type is discrete. However, the disadvantage of this kind of optimization strategy covers its benefit such as global search optimum. The defect is the need for many forward reservoir simulation runs and also disability to improve objective function monotonically. On the other hand, in gradient-based methods, the gradient of objective function subject to optimization variables is needed. This kind of optimization method is computationally efficient because it requires fewer function evaluations though it is capable of getting stuck in a local optimum. Gradient-based optimization methods are divided into two main groups in terms of calculating gradients. They include approximation and adjoint-based gradient algorithms. Well-known examples of gradient approximation method are simultaneous perturbation stochastic approximation (SPSA) and FDG. In Bangerth et al. (2006), the performance of FDG and SPSA methods is compared to the very fast simulated annealing (VFSA). They concluded that both FDG and SPSA algorithms are more efficient than gradient-free VSFA method. The main drawback of FDG and SPSA methods is that the step size along the search direction has to be chosen such that each function evaluation point corresponds to the lattice points in the simulation grid. Thus, a treatment to resolve the problem is to change the discrete optimization problem into continuous one. As a result, in Sarma and Chen (2008), adjoint method is suggested where the derivative of objective function is computed using the concept of optimal control theory and production optimization. In Zhang et al. (2010), indirect optimal well placement based on the use of adjoint model and optimal well control is applied. The complexity of adjoint-based algorithms in optimal well placement is similar to solving reservoir dynamic equations, which is the major drawback of the method. Since the problem is too complicated to compute gradients analytically, the simplest approach is to approximate gradients numerically using FDG or SPSA methods. In fact, it is easy to implement the method and the reservoir model is considered as a black box. In the present paper, (...truncated)