Development of Low-Order Controllers for High-Order Reservoir Models and Smart Wells

Abstract

Abstract Recently, the oil industry has started instrumenting and deploying various controls for the enhancement of hydrocarbon extraction. However, due to system complexity, data acquisition and, in turn, decision making, are still issues to be resolved in large-scale reservoir management. A very challenging large-scale control problem that has emerged recently is the real-time control of smart wells. A fundamental reason for using feedback control in this setting is to achieve desired performance in the presence of external disturbances and model uncertainties. This work introduces iterative Krylov subspace projection (KSP) methods to generate low-order feedback controllers for smart wells employing high-order reservoir models. The main motivation for using these methods is to enable the efficient computation of low-order reservoir models derived from the highly sparse structure of fluxes and pressure coefficients after discretization. Compared to other model reduction methods, such as modal decomposition, balanced realization, subspace identification and the proper orthogonal decomposition (POD), the proposed approach is very efficient since it is fundamentally based on sparse matrix-vector products. For a problem size of n simulation gridblocks, the KSP method presents a complexity of O(n2k) (or even O(nk), by exploiting sparsity), where k is the number of iterations required to generate the low-order model. This is highly desirable for the design of timely low-controller mechanisms in smart wells. We illustrate the potential of the method by linearizing an oil-water model and showing how to establish stability and error bounds for production responses under certain perturbations. We also discuss how the method can be extended to cases of non-stationary and nonlinear fluid coefficients.