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Over the past few decades, several data‐driven methods have been developed for identifying a model that accurately describes the process dynamics. Lately, sparse identification of nonlinear dynamics (SINDy) has delivered promising results for various nonlinear processes. However, at any instance of plant‐model mismatch or process upset, retraining the model using SINDy is computationally expensive and cannot guarantee to catch up with rapidly changing dynamics. Hence, we propose operable adaptive sparse identification of systems (OASIS) framework that extends the capabilities of SINDy for accurate, automatic, and adaptive approximation of process models. First, we use SINDy to obtain multiple models from historical data for varying input settings. Next, using these models and their training data, we build a deep neural network that is incorporated in a model predictive control framework for closed‐loop operation. We demonstrate the OASIS methodology on the identification and control of a continuous stirred tank reactor.

 

In this work, we propose the integration of Koopman operator methodology with Lyapunov‐based model predictive control (LMPC) for stabilization of nonlinear systems. The Koopman operator enables global linear representations of nonlinear dynamical systems. The basic idea is to transform the nonlinear dynamics into a higher dimensional space using a set of observable functions whose evolution is governed by the linear but infinite dimensional Koopman operator. In practice, it is numerically approximated and therefore the tightness of these linear representations cannot be guaranteed which may lead to unstable closed‐loop designs. To address this issue, we integrate the Koopman linear predictors in an LMPC framework which guarantees controller feasibility and closed‐loop stability. Moreover, the proposed design results in a standard convex optimization problem which is computationally attractive compared to a nonconvex problem encountered when the original nonlinear model is used. We illustrate the application of this methodology on a chemical process example.

 

Summary Slickwater fracturing has become one of the most leveraging completion technologies in unlocking hydrocarbon in unconventional reservoirs. In slickwater treatments, proppant transport becomes a big concern because of the inefficiency of low-viscosity fluids to suspend the particles. Many studies have been devoted to proppant transport experimentally and numerically. However, only a few focused on the proppant pumping schedules in slickwater fracturing. The impact of proppant schedules on well production remains unclear. The goal of our work is to simulate the proppant transport under real pumping schedules (multisize proppants and varying concentration) at the field scale and quantitatively evaluate the effects of proppant schedules on well production for slickwater fracturing. The workflow consists of three steps. First, a validated 3D multiphase particle-in-cell (MP-PIC) model has been used to simulate the proppant transport at real pumping schedules in a field-scale fracture (180-m length, 30-m height). Second, we applied a propped fracture conductivity model to calculate the distribution of propped fracture width, permeability, and fracture conductivity. In the last step, we incorporated the fracture geometry, propped fracture conductivity, and the estimated unpropped fracture conductivity into a reservoir simulation model to predict gas production. Based on the field designs of pumping schedules in slickwater treatments, we have generated four proppant schedules, in which 100-mesh and 40/70-mesh proppants were loaded successively with stair-stepped and incremental stages. The first three were used to study the effects of the mass percentages of the multisize proppants. From Schedules 1 through 3, the mass percentage of 100-mesh proppants is 30, 50, and 70%, respectively. Schedule 4 has the same proppant percentage as Schedule 2 but has a flush stage after slurry injection. The comparison between Schedules 2 and 4 enables us to evaluate the effect of the flush stage on well production. The results indicate that the proppant schedule has a significant influence on treatment performance. The schedule with a higher percentage of 100-mesh proppants has a longer proppant transport distance, a larger propped fracture area, but a lower propped fracture conductivity. Then, the reservoir simulation results show that both the small and large percentages of 100-mesh proppants cannot maximize well production because of the corresponding small propped area and low propped fracture conductivity. Schedule 2, with a median percentage (50%) of 100-mesh proppants, has the highest 1,000-day cumulative gas production. For Schedule 4, the flush stage significantly benefits the gas production by 8.2% because of a longer and more uniform proppant bed along the fracture. In this paper, for the first time, we provide both the qualitative explanation and quantitative evaluation for the impact of proppant pumping schedules on the performance of slickwater treatments at the field scale by using an integrated numerical simulation workflow, providing crucial insights for the design of proppant schedules in the field slickwater treatments.