Adjoint-Based Well-Placement Optimization Under Production Constraints

Abstract

Determining the optimal location of wells with the aid of an automated search algorithm 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 pseudo-wells. These pseudo-wells 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 pseudo-wells are computed using an adjoint model. These are subsequently used to approximate ‘improving directions’, i.e. directions in which to move the wells to achieve an increased NPV, based on which improving well positions 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 and one backward (adjoint) simulation. The process is repeated until no further improvements are obtained. The method is illustrated by two waterflooding examples. In the first the location of a single injector is optimized to maximize NPV. Starting from four different initial injector locations the algorithm converges to four similar local optima. The second example involves optimization of the locations of 9 producers and 4 injectors. Starting from two different initial well configurations we obtain nearly the same (local) optimum. 1 Introduction Determining the location of wells is a crucial decision to be made during a field development plan, because it can significantly affect a project's Net Present Value (NPV). Well placement optimization is often posed as a so-called mixed integer problem.1 Because these types of problems involve a mixture of continuous and discrete variables solving them is an arduous task, which is why well locations are often determined manually. However, there are several automated well placement optimization algorithms available in the literature. They can broadly be classified into two categories. The first category concerns gradient-based algorithms such as Simultaneous Perturbation Stochastic Approximation2,3 and Finite Difference Gradient (FDG)2. The second category concerns stochastic algorithms such as the Nelder-Mead simplex algorithm 2,3, Simulated Annealing4, Genetic Algorithms5,6, and Artificial Neural Networks7. The first category is generally very efficient, requires only a few forward reservoir simulations and increases NPV at each iteration. However, these algorithms 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 in Ref. 8, where standard reservoir models are transformed to maps of production potential in order to screen regions that are most favorable for well placement. 1.1 Well placement optimization with adjoint models In this paper we present a gradient-based algorithm that is distinct from those mentioned above. The adjoint method used in optimal control theory has been used previously for optimization of injection and production rates in a fixed well configuration 9–15. In particular, the method is currently receiving considerable attention in the field of closed-loop reservoir management 16. In these applications, the parameters to be optimized are often flow rates, bottom-hole pressures (BHP's) and / or valve settings. Since these are not mixed integer problems, gradient-based algorithms are commonly used to solve them, and the adjoint method efficiently generates the required gradients.

Using Evolution Strategy with Meta-models for Well Placement Optimization

“…The use of gradient based methods (e.g., Handels et al, 2007 andSarma &Chen, 2008) was motivated by the fact that it is by far less reservoir simulations consuming compared to stochastic methods. However, these deterministic methods find a difficulty in general when dealing with multi-modal, non-smooth and non-convex problems such as the well placement problem.…”

Joint optimization of oil well placement and controls

“…wells operated at a negligible rate) that have been placed around the current well location (Zandvliet, 2008). The main advantage of this approach is that only one (forward) reservoir simulation, and then a (backward) adjoint simulation, of comparable computational cost, are required to compute improving directions for all wells Handels et al, 2007). Vlemmix et al (2009) later applied this approach to horizontal well trajectories.…”

Data Integration for the Generation of High Resolution Reservoir Models

“…Gradient based optimization algorithms, with the gradient of a functional or objective function to be optimized most commonly computed by the adjoint (optimal control) method, have been used in both automatic history matching (Chen et al (1974) However, to the best of our knowledge, the first few papers that use the gradient directly to solve the optimal well placement problem are Handels et al (2007), Wang et al (2007a) and Sarma et al (2008).…”