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The moving boundary nature of hydraulic fracturing process makes it extremely difficult to approximate using local models (i.e., approximate models whose validity is limited by the training data). In this work, we implement the Koopman operator methodology for closed-loop control of fracture propagation. Koopman theory models nonlinear dynamical systems as linear systems by lifting the states to an infinite-dimensional space of functions called observables. It is particularly attractive because of its ability to provide (nearly) global linearization valid in a larger domain (in some cases the entire basin of attraction) compared to local models. Additionally, due to its linear structure, it allows convex predictive control formulations that avoid any issues associated with nonlinear optimization. The numerical results show that the Koopman linear model shows excellent agreement with the real system and successfully achieves the desired fracture geometry. |
