A two-stage well placement optimization method based on adjoint gradient

Abstract

Abstract As well locations are represented as integer well gridblock indices in reservoir simulators, determining optimal well locations is most naturally treated as a discrete optimization problem. A few recent papers in the literature have attempted to convert this discrete optimization problem into a continuous optimization problem, but most of these papers require that one specify a priori both the number of wells to be drilled and the operating rates (or bottomhole pressures) for the specified operational life of the reservoir. Therefore, the final optimal well locations may not be optimal when different well rates (or bottomhole pressure) and/or reservoir production life are specified. The methodology in this paper attempts to mitigate the effects of an ad hoc specification of both operating well rates (or pressures) and operational reservoir life by using an initialization step to determine an appropriate total reservoir water injection rate and/or total reservoir production rate for the specified operational life. After the initialization stage a second stage is performed in which we estimate (i) the optimal number of water injection wells and/or producing wells; (ii) the optimal location of these wells and (iii) the optimal rates at these wells. Both stages use a gradient based optimization algorithm with the relevant gradient computed by a combination of the adjoint method and analytical methods for linear constraints which are imposed using a gradient projection method. A practical method is proposed for imposing nonlinear bound constraints on the bottomhole pressure of each well.