Modeling of hydraulic fracturing and designing of online pumping schedules to achieve uniform proppant concentration in conventional oil reservoirs

We present a novel control framework for the closed-loop operation of a hydraulic fracturing process. Initially, we focus on the development of a first-principle model of a hydraulic fracturing process. Second, a novel numerical scheme is developed to efficiently solve the coupled partial differential equations defined over a time-dependent spatial domain. Third, a reduced-order model is constructed, which is used to design a Kalman filter to accurately estimate unmeasurable states. Lastly, model predictive control theory is applied for the design of a feedback control system to achieve uniform proppant concentration across the fracture at the end of pumping by explicitly taking into account the desired fracture geometry, total amount of proppant injected, actuator limitations, and safety considerations. We demonstrate that the proposed control scheme is able to generate a spatial concentration profile which is uniform and close to the target concentration compared to that of the benchmark, Nolte's pumping schedule.