Optimization of the well locations and completions for vertical and horizontal wells using a derivative-free optimization algorithm

We introduce a new methodology for the estimation of the location of horizontal and vertical wells that maximizes the life-cycle net-present-value (NPV) of production from a given reservoir. Because of the way wells are commonly modeled in reservoir simulators, the optimal well placement is usually formulated as a discrete optimization problem, where the center point of the well is moved from the center of one gridblock to the center of another gridblock at each iteration of whatever optimization algorithm is used. However, in reality, the optimal well placement is a continuous optimization problem as the center point of the well can be located at any point within the reservoir and it does not have to be at the center of a gridblock. Here, the well-placement problem is formulated in terms of four continuous variables, the xw, yw and zw coordinates of the center point of the well and the length, lw, of the well. A procedure is developed to modify well-productivity indices in the reservoir simulator to account for the location of the centerline of a well within gridblocks and to define the life-cycle NPV of production as a function of these four continuous well parameters, (xw,yw,zw,lw). This NPV functional is maximized using a derivative-free optimization (DFO) algorithm, BOBYQA. This algorithm can conceptually be applied to maximize any cost function subject to bound constraints on the independent variables. However, for the straightforward formulation of the optimal well-placement problem, BOBYQA performs relatively poorly; we show that the performance can be improved by a transformation of the control variables. Because a DFO algorithm is applied, the technology present here can easily be applied using any reservoir simulator; a code for calculating the gradient of NPV by the adjoint method is not needed. The applicability of our methodology is illustrated for a set of synthetic problems.