Optimal well placement using an adjoint gradient

In the process of reservoir development, one wishes to drill wells at optimal locations so that more hydrocarbons can be extracted at a lower cost. Because well locations in a reservoir simulator are commonly treated as discrete variables (well gridblock indices), standard implementations of gradient-based optimization are not applicable for optimal well placement, so optimization is commonly done with a non-gradient-based method such as the genetic algorithm. This paper presents a novel idea to convert the discrete optimization problem into an optimization problem with continuous variables. The idea is to initialize the problem by putting a well in every gridblock and maximize the net present value (NPV) with respect to the rates of the hypothesized wells. The NPV includes an additional term to account for the cost of “drilling a well." As the cost of “drilling a well” detracts from the NPV, when NPV is maximized with respect to the well rates, the rates of some wells will be driven to zero and hence these wells are eliminated from the system. As the control variables (well rates) are continuous, we are able to use adjoint gradient for the optimization process and the gradient projection method to ensure that the constraints are satisfied. For the synthetic homogeneous and heterogeneous reservoir examples where the problem is to determine the optimal location of water injection wells, this problem formulation yields good results after the optimization process.