(672g) Big Data for Enhanced Reduced Order Modeling: Application to Hydraulic Fracturing

Many chemical processes undergo non-linear dynamics which are usually described by high-dimensional high-fidelity models whose complexities limit their usage in the design of model-based feedback control systems. These hindrances have paved the way for the use of reduced-order models (ROMs) in the design of controllers. Among the plethora of available techniques, Dynamic Mode Decomposition (DMD) [1], has gained a lot of attention as it accurately captures coherent structures that are dynamically relevant and contribute towards the long term dynamics of the system [2]. The primary assumption of the DMD technique is that the non-linear dynamics of a complex system can be represented in a linear form. But the dynamics of many chemical processes are influenced by the external inputs applied on them and therefore, Dynamic Mode Decomposition with Control (DMDc) technique [3] was developed to include the effect of the external inputs on the dynamics of the system. However, for a highly non-linear chemical process, a global linear representation may not accurately capture its dynamics considering that DMDc has very limited degrees of freedom. To overcome this limitation of DMDc, Local Dynamic Mode Decomposition with Control (Local DMDc) technique [4] was developed which utilizes unsupervised learning to cluster the sequential data, use DMDc to build local ROMs which individually capture the local dynamics accurately and together represent the dynamical system.

However, the domain of attraction (DOA) of Local DMDc is narrow; meaning, it will be less accurate when used for prediction purposes under conditions outside the training data. In this work, we overcome the problem of limited DOA of Local DMDc by using multiple training data sets obtained under distinct training conditions. Specifically, training data is divided into clusters such that each cluster represents certain unique, local dynamics of the system and a DMDc-based local ROM is built for each cluster. Also, a unique identity is built for each cluster by transforming the data through normalization, applying weights, and Principal Component Analysis (PCA). The identity for each cluster is the average of the PCA scores of all the data points it contains. During prediction, given a state and input, we utilize the k-nearest neighbors (kNN) technique to select the appropriate local ROM in order to calculate the future state of the system. We demonstrate the performance of our proposed algorithm by applying it to hydraulic fracturing, a highly non-linear and complex process represented by a system of nonlinear highly-coupled PDEs with time-dependent spatial domain [5]. We utilize its high-fidelity model to generate training data obtained by using distinct training inputs, build multiple local ROMs based on DMDc, and utilize kNN for the purposes of model selection. We demonstrate the enlarged DOA through increased applicability of the LDMDc models obtained using the proposed algorithm for a wide range of operating conditions

Literature cited:

[1] Schmid, P.J., 2010. Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.

[2] Ghommem, M., Carlo, V. M., Efendiev, Y., 2014. Mode decomposition methods for flows in high-contrast porous media. A Global approach. J. Comput. Phys. 257, 400-413.

[3] Proctor, J. L., Brunton, S. L., Kutz, J. N., 2016. Dynamic mode decomposition with control. SIAM J. Appl. Dyn. Syst. 15, 142-161.

[4] Narasingam, A., Kwon, J. S., 2017. Development of local dynamic mode decomposition with control: Application to model predictive control of hydraulic fracturing. Comput. Chem. Eng. 106, 501-511.

[5] Economides, M.J., Nolte, K.G., 2000. Reservoir stimulation. Chichester, Wiley.