This paper extends and applies novel computational procedures for the efficient closed-loop optimal control of petroleum reservoirs under uncertainty. It addresses two important issues that were present in our earlier implementation [2] that limited the application of the procedure to practical problems. <br><br>Specifically, the previous approach encountered difficulties in handling nonlinear path constraints (constraints that must be satisfied at every time-step of the forward model) during optimization. Such constraints (e.g., maximum liquid production rate) are frequently present in practical problems. To address this issue, an approximate feasible direction optimization algorithm was proposed. The algorithm uses the objective function gradient and a combined gradient of the active constraints [3], both of which can be obtained efficiently with adjoint models. The second limitation of the implementation in [2] was the use of the standard Karhunen-Loeve (K-L) expansion for parameterization of the input random fields of the simulation model. This parameterization is computationally expensive and preserves only two-point statistics of the random field. It is thus not suitable for large simulation models or for complex geological scenarios, such as channelized systems. In another paper [4], a nonlinear form of the K-L expansion, referred to as kernel PCA, is applied for parameterizing arbitrary random fields. Kernel PCA successfully addresses the limitations of the K-L expansion, and is differentiable, meaning that gradient-based methods can be utilized in conjunction with this parameterization within the closed-loop.<br><br>An example based on a Gulf of Mexico reservoir model is considered. For this case it is demonstrated that the proposed algorithms indeed provide a viable real-time closed-loop optimization framework. Application of the closed-loop methodology is shown to result in a 25% increase in NPV over the base case. This is almost the same improvement achieved using an open-loop approach, which is an idealized formulation in which the geological model is assumed to be known.


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