Efficient Well Placement Optimization with Gradient-based Algorithms and Adjoint Models

Abstract

A key reservoir management decision taken throughout the life of a reservoir is the determination of optimal well locations that maximizes asset value (such as Net Present Value, NPV). Because this well placement optimization problem is a discrete-parameter problem (well locations are discrete parameters in the simulation model), gradients of the objective function (NPV) with respect to these parameters are not defined. Thus, gradient-based methods have not found much applicability to this problem, and most existing algorithms applied to this problem are stochastic in nature, such as genetic algorithms, simulated annealing, and stochastic perturbation methods. These methods are usually quite inefficient requiring hundreds of simulations and thus may have limited application to large-scale simulation models with many wells. We propose a novel, continuous approximation to the original discrete-parameter well placement problem such that gradients can be calculated on the approximate problem, and gradient-based algorithms can then be employed for efficiently determining the optimal well locations. The basic idea is to first replace the discrete parameters (i, j well location indices) with their continuous counterparts in the spatial domain (x, y well locations) and then obtain a continuous functional relationship between the objective function and these continuous parameters. Such a functional relationship is obtained by replacing the discontinuous Dirac-delta functions (defining wells as point sources) in the underlying governing PDE with continuous functions (which in the limit tends to the Dirac-delta function, such as the bivariate Gaussian function). Numerical discretization of the modified PDE leads to well terms in the mass balance equations that are continuous functions of the continuous well location variables. As a result of this continuous functional relationship, adjoints and gradient-based optimizations algorithms can now be applied to obtain the optimal well locations. The efficiency and practical applicability of the approach is demonstrated on a few synthetic waterflood optimization problems. Introduction A key reservoir management decision necessary throughout the life of an oilfield is the determination of optimal well locations that maximize asset value. The current industry practice to do so is usually through manual approaches wherein the engineer essentially uses engineering judgment and numerical simulation to determine such locations. Although such an approach may be viable for small reservoirs with a small number of wells, it is unlikely that such approaches will be applicable when dealing with large reservoirs (with large-scale simulation models) and a large number of wells. Recently however, there has been an increasing interest in solving this problem more efficiently with automatic optimization algorithms. This optimal well placement problem is usually formulated as a discrete parameter optimization problem, because the well location variables are discrete variables (i, j indices of grid blocks where wells are located). Because of the discrete nature of the problem, gradients of the objective function (NPV for example) with respect to these discrete variables do not exist. As a result, gradient-based optimization algorithms have not found much applicability to this problem, and most existing algorithms applied to this problem are stochastic gradient-free algorithms, such as genetic algorithms (Montes and Bartolome, 2001, Yeten, 2003), simulated annealing (Beckner and Song, 1995), and stochastic perturbation methods (Spall, 2003, Bangerth et al., 2006). Although these algorithms are easy to apply and are supposedly global in nature, they are usually quite inefficient requiring hundreds of simulations and thus may have limited application to large-scale simulation models with many wells. Furthermore, they do not guarantee a monotonic increase in the objective function with successive iterations, implying that increasing the computational effort may not necessarily provide a better optimum.

Joint optimization of oil well placement and controls

“…Derivative-free and stochastic optimization approaches ordinarily require parallel computing implementations for efficiency. We note, however, that gradient-based techniques have been applied for well placement (e.g., Sarma and Chen, 2008;Zandvliet et al, 2008), and stochastic search has been used for well control (e.g., Echeverría Ciaurri et al, 2011a), so our observations here should not be viewed as absolute.…”

“…Therefore, the well placement optimization problem does not appear to be as amenable to solution using gradient-based methods because these approaches can get trapped in local minima. There have, however, been procedures presented for (2.8) that use gradients (see e.g., Chen, 2008, andZandvliet et al, 2008). These methods replace the problem with a related (though not necessarily equivalent) problem that has continuous variables.…”

Recent Developments in Closed-Loop Approaches for Real-Time Mining and Petroleum Extraction

“…Although some of them are adjoint-based (Wang et al 2007;Zandvliet et al 2008a;Sarma and Chen 2008;Vlemmix et al 2009), the most effective techniques use 'non-classical' methods such as genetic algorithms, particle swarm optimization or evolutionary strategies (e.g. Güyagüler et al 2002;Yeten et al 2003;Bangerth, 2006;Owunalu and Durlofsky, 2010, Bouzarkouna et al 2012Forouzanfar et al 2013b, Jesmani et al 2016.…”

Optimal Well Positioning under Geological Uncertainty by Equalizing the Arrival Time

“…The process is repeated until there is no change on injection well location. Sarma and Chen (2008) converted Handels et al (2008)'s discrete optimization problem into a continuous optimization problem. Therefore, the search direction is not limited to the eight directions of an initial well position guessed.…”