(582a) Centroidal Voronoi Tesselation Based Model Order Reduction for a Moving Boundary Problem: Application to a Hydraulic Fracturing System

More often than not a chemical process can be accurately described using a mathematical model which is generally comprised of nonlinear partial differential equations (PDE). One arrives at these equations by applying conservation laws and/or transport equations. Many chemical engineering processes are, owing to their nature, characterized by a moving boundary problem. Two such examples would be the propagation of fractures during a hydraulic fracturing operation [1] and the evolution of interface boundaries during a reaction-diffusion process [2, 3].

In order to accurately capture the dynamics of systems described by PDEs, a large number of state variables are required and this makes it computationally difficult to design online control systems. Although a model reduction technique, Proper orthogonal decomposition (POD), has proved to perform effectively, it often fails to capture the process dynamics in nonlinear systems, since it assumes that data belong to a linear space and therefore relies on the Euclidean distance as the metric to minimize. Hence, we can deal with the above mentioned problem by applying POD locally (with respect to time or spatial coordinates) to clusters instead of applying it globally because each cluster contains snapshots that show relatively close-in-distance behavior within itself, and considerably far with respect to other clusters. Also, the dominant behavior of each cluster can be captured by using less empirical Eigenfunctions as compared to that of global POD. Sahyoun and Djouadi [4] introduced different clustering schemes such as time snapshots clustering (TSC) and space vectors clustering (SVC), and applied them to a nonlinear convective PDE system governed by the Burgersâ?? equation for fluid flows over 1D and 2D domains. Following these contributions, the idea here is to develop an optimal clustering technique using Centroidal Voronoi Tesselation (CVT) scheme and apply POD locally to these clusters which extracts a relevant set of basis vectors for each cluster.

 In this work, we first transformed the PDE with a time-dependent spatial domain into the one with an appropriate time-invariant spatial coordinate, and a representative ensemble of solutions is constructed by solving a high-order discretization of the PDE. Then we divide the ensemble into clusters using CVT and apply POD locally to these clusters to derive a small set of empirical eigenfunctions (dominant spatial patterns) that capture almost all the energy of the ensemble of solutions. The empirical eigenfunctions are subsequently used as basis functions within the Galerkinâ??s model reduction framework to derive low-order ODE systems that accurately describe the dominant dynamics of the PDE system. These ODE systems are used as a basis for the synthesis of a low-dimensional nonlinear controller based on model predictive control theory. We applied the framework to the hydraulic fracturing problem and a reaction-diffusion process.

References:

[1] Qiuying Gu and Karlene A. Hoo, Evaluating the Performance of a Fracturing Treatment Design, Ind. Eng. Chem. Res., 2014, 53 (25) : 10491â??10503

[2] A. Armaou and P. D. Christofides, Nonlinear Feedback Control of Parabolic Partial Differential Equation Systems with Time-dependent Spatial Domains, J. Math. Anal. Appl., 1999, 239:124-157.

[3] M. Izadi and S. Dubljevic, Backstepping output-feedback control of moving boundary parabolic PDEs, European J. of Control, 2015, 21:27-35.

[4] S. Sahyoun and S. Djouadi, Time, Space, and Space-Time Hybrid Clustering POD with Application to the Burgersâ?? Equation, 53rd IEEE Conference on Decision and Control December 15-17, 2014. Los Angeles, California, USA