Efficient closed-loop optimal control of petroleum reservoirs under uncertainty

This work discusses a closed-loop approach for efficient realtime production optimization of petroleum reservoirs that consists of three key elements---adjoint models for efficient parameter and control gradient calculation, polynomial chaos expansions for efficient uncertainty propagation, and Karhunen-Loeve (K-L) expansions and Bayesian inversion theory for efficient realtime model updating (history matching). The control gradients provided by the adjoint solution are used by a gradient-based optimization algorithm to determine optimal control settings, while the parameter gradients are used for model updating. We also investigate an adjoint construction procedure that makes it relatively easy to create the adjoint and is applicable to any level of implicitness of the forward model. Polynomial chaos expansions provide optimal encapsulation of information contained in the input random fields and output random variables. This approach allows the forward model to be used as a black box but is much faster than standard Monte Carlo techniques. The K-L representation of input random fields allows for the direct application of adjoint techniques for history matching and uncertainty propagation algorithms while assuring that the two-point geostatistics of the reservoir description are maintained.;We further extend the basic closed-loop algorithms discussed above to address two important issues. The first concerns handling non-linear path inequality constraints during optimization, which are quite difficult to maintain with existing optimal control algorithms. We propose an approximate feasible direction algorithm combined with a feasible line-search to satisfy such constraints efficiently. The second issue concerns the Karhunen-Loeve expansion. It is computationally very expensive and impractical for large-scale simulation models, and since it only preserves two-point statistics of the input random field, it may not always be suitable for arbitrary non-Gaussian random fields. We use Kernel Principal Component Analysis (PCA) to address these issues efficiently. This approach is much more efficient, preserves high-order statistics of the random field, and is differentiable, meaning that gradient-based methods (and adjoints) can still be utilized with this representation.;The benefits and efficiency of the overall closed-loop approach are demonstrated through realtime optimizations of net present value (NPV) for synthetic and real reservoirs under waterflood subject to production constraints and uncertain reservoir description.