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.