(118b) An Operator Theoretic Framework for Data-Driven Identification and Control of a Hydraulic Fracturing Process

A need for delivering well-informed decisions in the context of production optimization and optimal control, especially due to lack of simple or often completely unknown physics-based models, has fueled data-driven learning in solving complex oil and gas engineering problems. Within this context, an operator theoretic perspective of dynamical systems offers an intuitive approach in the quest for developing accurate practical models. In 1931, Koopman first showed that a principled global linear representation of nonlinear dynamical systems is possible using an infinite dimensional linear operator by shifting focus from the traditional state space to the space of functions called observables which can simply be system outputs [1]. Expressing the system in a function (output) space makes sense especially in the era of big data. This means that the spectral properties of the linear operator (i.e., eigenvalues and eigenfunctions) encode global information that allows future state prediction and scalable reconstruction of the underlying dynamics from measurement data [2].

The goal is to determine a (nearly) global linear representation of the nonlinear dynamics such that the use of established linear control design methodologies becomes readily applicable in a larger domain. In this work, we tackle the problem of identification and control of proppant concentration in a hydraulic fracturing process. Specifically, we consider simultaneous fracture propagation and multi-phase proppant flow within a single fracture based on the widely used PKN model [3]. The physics-based model for such a flow can be obtained by combining lubrication theory, elasticity equations and conservation laws which results in a complex set of partial differential equations, thereby making real-time analysis and control computationally intractable. Moreover, the validity of these high-fidelity models is directly dependent on heterogeneous rock mechanical properties which are very difficult to characterize. In contrast, we use the Koopman operator theory to develop linear predictors in a completely data-driven route using eigenfunctions of the operator. In order to achieve this, the first step is to learn the eigenvalues and eigenfunctions of the Koopman operator from data using a dictionary of observable functions [4], for example, polynomial functions of proppant concentration, or in an unsupervised fashion by using deep autoencoder networks from machine learning. In the next step, the system is transformed from the state space to the space of functions whose basis is given by the eigenfunctions of the operator. A large number of eigenfunctions must be constructed such that the state and any other observable quantities of interest lie in the span of these eigenfunctions and hence can be predicted in a linear fashion [5]. In the final step, the control part is incorporated by solving a multi-step error minimization problem which is simply a least-squares regression. The final model so identified will be in the form of a linear time invariant system which results in a standard convex optimization problem in the model predictive control (MPC) formulation allowing it to be solved faster than in the case of nonlinear MPC problems. Moreover, we can integrate these linear predictors within a stabilizing control framework and show that the stability properties of the linear system (in the observable space) are inherited by the original nonlinear system under certain assumptions. Based on numerical simulations we show that the developed linear predictors perform favorably in terms of prediction accuracy compared to several standard techniques. We also show that the Koopman MPC framework successfully achieves the control objective and has superior performance compared to feedback strategies based on local linearization.

Literature cited:

[1] Koopman, B.O. Hamiltonian systems and transformation in Hilbert space. Proceedings of National Academy of Sciences USA, 17(5):315, 1931.

[2] Mezic, I. Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dynamics, 41(1-3): 309-325, 2005.

[3] Economides, M.J. and Nolte, K.G. Reservoir Stimulation. John Wiley & Sons, Chichester, 2000.

[4] Williams, M.O., Rowley, C.W. and Kevrekidis, I.G. A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition. Journal of Nonlinear Science, 25(6):1307–1346, 2015.

[5] Korda, M. and Mezic, I. On convergence of extended dynamic mode decomposition to the Koopman operator. Journal of Nonlinear Science, 28(2):687–710, 2018.