Application of Koopman operator for model-based control of fracture propagation and proppant transport in hydraulic fracturing operation

Narasingam, Abhinav; Kwon, Joseph Sang-Il

doi:10.1016/j.jprocont.2020.05.003

Application of Koopman operator for model-based control of fracture propagation and proppant transport in hydraulic fracturing operation

Journal Article · Fri May 22 00:00:00 EDT 2020 · Journal of Process Control

DOI:https://doi.org/10.1016/j.jprocont.2020.05.003· OSTI ID:1799395

Narasingam, Abhinav ^[1]; Kwon, Joseph Sang-Il ^[2]

Texas A & M University, College Station, TX (United States); OSTI
Texas A & M University, College Station, TX (United States)

This work explores the application of the recently developed Koopman operator approach for model identification and feedback control of a hydraulic fracturing process. Controlling fracture propagation and proppant transport with precision is a challenge due in large part to the difficulty of constructing approximate models that accurately capture the characteristic moving boundary and highly-coupled dynamics exhibited by the process. Koopman operator theory is particularly attractive here as it offers a way to explicitly construct linear representations for even highly nonlinear dynamics. The method is data-driven and relies on lifting the states to an infinite-dimensional space of functions called observables where the dynamics are governed by a linear Koopman operator. Here this work considers two problems: (a) fracture geometry control, and (b) proppant concentration control. In both cases, an approximate linear model of the corresponding dynamics is constructed and used to design a model predictive controller (MPC). The manuscript shows that in the case of highly nonlinear dynamics, as observed in the proppant concentration, use of canonical functions in the observable basis fails. In such cases, a priori system knowledge can be leveraged to choose the required basis. The numerical experiments demonstrate that the Koopman linear model shows excellent agreement with the real system and successfully achieves the desired target values maximizing the oil and gas productivity. Additionally, due to its linear structure, the Koopman models allow convex MPC formulations that avoid any issues associated with nonlinear optimization.

View Accepted Manuscript (DOE)

Research Organization:: Texas A & M University, College Station, TX (United States)

Sponsoring Organization:: National Science Foundation (NSF); Texas A&M Energy Institute; USDOE; USDOE Office of Energy Efficiency and Renewable Energy (EERE)

Grant/Contract Number:: EE0007888

OSTI ID:: 1799395

Alternate ID(s):: OSTI ID: 1630472

Journal Information:: Journal of Process Control, Journal Name: Journal of Process Control Journal Issue: C Vol. 91; ISSN 0959-1524

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (43)

Temporal clustering for order reduction of nonlinear parabolic PDE systems with time‐dependent spatial domains: Application to a hydraulic fracturing process Narasingam, Abhinav; Siddhamshetty, Prashanth; Sang‐Il Kwon, Joseph AIChE Journal, Vol. 63, Issue 9 https://doi.org/10.1002/aic.15733	journal	April 2017
Feedback control of proppant bank heights during hydraulic fracturing for enhanced productivity in shale formations Siddhamshetty, Prashanth; Kwon, Joseph Sang‐Il; Liu, Shuai AIChE Journal, Vol. 64, Issue 5 https://doi.org/10.1002/aic.16031	journal	December 2017
Koopman Lyapunov‐based model predictive control of nonlinear chemical process systems Narasingam, Abhinav; Kwon, Joseph Sang‐Il AIChE Journal, Vol. 65, Issue 11 https://doi.org/10.1002/aic.16743	journal	August 2019
A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition Williams, Matthew O.; Kevrekidis, Ioannis G.; Rowley, Clarence W. Journal of Nonlinear Science, Vol. 25, Issue 6 https://doi.org/10.1007/s00332-015-9258-5	journal	June 2015
On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator Korda, Milan; Mezić, Igor Journal of Nonlinear Science, Vol. 28, Issue 2 https://doi.org/10.1007/s00332-017-9423-0	journal	November 2017
Spectral Properties of Dynamical Systems, Model Reduction and Decompositions Mezić, Igor Nonlinear Dynamics, Vol. 41, Issue 1-3 https://doi.org/10.1007/s11071-005-2824-x	journal	August 2005
Optimal control formulation of pulse-based control using Koopman operator Sootla, Aivar; Mauroy, Alexandre; Ernst, Damien Automatica, Vol. 91 https://doi.org/10.1016/j.automatica.2018.01.036	journal	May 2018
Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control Korda, Milan; Mezić, Igor Automatica, Vol. 93 https://doi.org/10.1016/j.automatica.2018.03.046	journal	July 2018
Koopman operator-based model reduction for switched-system control of PDEs Peitz, Sebastian; Klus, Stefan Automatica, Vol. 106 https://doi.org/10.1016/j.automatica.2019.05.016	journal	August 2019
Development of local dynamic mode decomposition with control: Application to model predictive control of hydraulic fracturing Narasingam, Abhinav; Kwon, Joseph Sang-Il Computers & Chemical Engineering, Vol. 106 https://doi.org/10.1016/j.compchemeng.2017.07.002	journal	November 2017
Modeling of hydraulic fracturing and designing of online pumping schedules to achieve uniform proppant concentration in conventional oil reservoirs Siddhamshetty, Prashanth; Yang, Seeyub; Kwon, Joseph Sang-Il Computers & Chemical Engineering, Vol. 114 https://doi.org/10.1016/j.compchemeng.2017.10.032	journal	June 2018
Data-driven identification of interpretable reduced-order models using sparse regression Narasingam, Abhinav; Sang-Il Kwon, Joseph Computers & Chemical Engineering, Vol. 119 https://doi.org/10.1016/j.compchemeng.2018.08.010	journal	November 2018
Extending Data-Driven Koopman Analysis to Actuated Systems Williams, Matthew O.; Hemati, Maziar S.; Dawson, Scott T. M. IFAC-PapersOnLine, Vol. 49, Issue 18 https://doi.org/10.1016/j.ifacol.2016.10.248	journal	January 2016
Linear observer synthesis for nonlinear systems using Koopman Operator framework Surana, Amit; Banaszuk, Andrzej IFAC-PapersOnLine, Vol. 49, Issue 18 https://doi.org/10.1016/j.ifacol.2016.10.250	journal	January 2016
Computer simulation of hydraulic fractures Adachi, J.; Siebrits, E.; Peirce, A. International Journal of Rock Mechanics and Mining Sciences, Vol. 44, Issue 5 https://doi.org/10.1016/j.ijrmms.2006.11.006	journal	July 2007
Extracting spatial–temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition Brunton, Bingni W.; Johnson, Lise A.; Ojemann, Jeffrey G. Journal of Neuroscience Methods, Vol. 258 https://doi.org/10.1016/j.jneumeth.2015.10.010	journal	January 2016
Unified Fracture Design for very low permeability reservoirs Bhattacharya, S.; Nikolaou, M.; Economides, M. J. Journal of Natural Gas Science and Engineering, Vol. 9 https://doi.org/10.1016/j.jngse.2012.06.005	journal	November 2012
A tutorial review of economic model predictive control methods Ellis, Matthew; Durand, Helen; Christofides, Panagiotis D. Journal of Process Control, Vol. 24, Issue 8 https://doi.org/10.1016/j.jprocont.2014.03.010	journal	August 2014
Linearization in the large of nonlinear systems and Koopman operator spectrum Lan, Yueheng; Mezić, Igor Physica D: Nonlinear Phenomena, Vol. 242, Issue 1 https://doi.org/10.1016/j.physd.2012.08.017	journal	January 2013
Isostables, isochrons, and Koopman spectrum for the action–angle representation of stable fixed point dynamics Mauroy, A.; Mezić, I.; Moehlis, J. Physica D: Nonlinear Phenomena, Vol. 261 https://doi.org/10.1016/j.physd.2013.06.004	journal	October 2013
Dynamic mode decomposition of numerical and experimental data Schmid, Peter J. Journal of Fluid Mechanics, Vol. 656 https://doi.org/10.1017/S0022112010001217	journal	July 2010
Analysis of unsteady behaviour in shockwave turbulent boundary layer interaction Grilli, Muzio; Schmid, Peter J.; Hickel, Stefan Journal of Fluid Mechanics, Vol. 700 https://doi.org/10.1017/jfm.2012.37	journal	February 2012
Handling Spatial Heterogeneity in Reservoir Parameters Using Proper Orthogonal Decomposition Based Ensemble Kalman Filter for Model-Based Feedback Control of Hydraulic Fracturing Narasingam, Abhinav; Siddhamshetty, Prashanth; Kwon, Joseph Sang-Il Industrial & Engineering Chemistry Research, Vol. 57, Issue 11 https://doi.org/10.1021/acs.iecr.7b04927	journal	February 2018
Enlarging the Domain of Attraction of the Local Dynamic Mode Decomposition with Control Technique: Application to Hydraulic Fracturing Bangi, Mohammed Saad Faizan; Narasingam, Abhinav; Siddhamshetty, Prashanth Industrial & Engineering Chemistry Research, Vol. 58, Issue 14 https://doi.org/10.1021/acs.iecr.8b05995	journal	March 2019
Model-Based Closed-Loop Control of the Hydraulic Fracturing Process Gu, Qiuying; Hoo, Karlene A. Industrial & Engineering Chemistry Research, Vol. 54, Issue 5 https://doi.org/10.1021/ie5024782	journal	January 2015
On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics Mauroy, A.; Mezić, I. Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 22, Issue 3 https://doi.org/10.1063/1.4736859	journal	September 2012
Hamiltonian Systems and Transformation in Hilbert Space Koopman, B. O. Proceedings of the National Academy of Sciences, Vol. 17, Issue 5 https://doi.org/10.1073/pnas.17.5.315	journal	May 1931
Estimation of perturbations in robotic behavior using dynamic mode decomposition Berger, Erik; Sastuba, Mark; Vogt, David Advanced Robotics, Vol. 29, Issue 5 https://doi.org/10.1080/01691864.2014.981292	journal	March 2015
Discovering dynamic patterns from infectious disease data using dynamic mode decomposition Proctor, Joshua L.; Eckhoff, Philip A. International Health, Vol. 7, Issue 2 https://doi.org/10.1093/inthealth/ihv009	journal	February 2015
Koopman operator based observer synthesis for control-affine nonlinear systems Surana, Amit 2016 IEEE 55th Conference on Decision and Control (CDC) https://doi.org/10.1109/CDC.2016.7799268	conference	December 2016
Koopman operator framework for constrained state estimation Surana, Amit; Williams, Matthew O.; Morari, Manfred 2017 IEEE 56th Annual Conference on Decision and Control (CDC) https://doi.org/10.1109/CDC.2017.8263649	conference	December 2017
A Data-Driven Koopman Model Predictive Control Framework for Nonlinear Partial Differential Equations Arbabi, Hassan; Korda, Milan; Mezic, Igor 2018 IEEE Conference on Decision and Control (CDC) https://doi.org/10.1109/CDC.2018.8619720	conference	December 2018
Feedback Stabilization Using Koopman Operator Huang, Bowen; Ma, Xu; Vaidya, Umesh 2018 IEEE Conference on Decision and Control (CDC) https://doi.org/10.1109/CDC.2018.8619727	conference	December 2018
Global Stability Analysis Using the Eigenfunctions of the Koopman Operator Mauroy, Alexandre; Mezic, Igor IEEE Transactions on Automatic Control, Vol. 61, Issue 11 https://doi.org/10.1109/TAC.2016.2518918	journal	November 2016
Generalizing Koopman Theory to Allow for Inputs and Control Proctor, Joshua L.; Brunton, Steven L.; Kutz, J. Nathan SIAM Journal on Applied Dynamical Systems, Vol. 17, Issue 1 https://doi.org/10.1137/16M1062296	journal	January 2018
Determination of Proppant and Fluid Schedules From Fracturing-Pressure Decline Nolte, K. G. SPE Production Engineering, Vol. 1, Issue 04 https://doi.org/10.2118/13278-PA	journal	July 1986
Experimental and Numerical Modeling of Convective Proppant Transport(includes associated papers 31036 and 31068 ) Barree, R. D.; Conway, M. W. Journal of Petroleum Technology, Vol. 47, Issue 03 https://doi.org/10.2118/28564-PA	journal	March 1995
Propagation of a Vertical Hydraulic Fracture Nordgren, R. P. Society of Petroleum Engineers Journal, Vol. 12, Issue 04 https://doi.org/10.2118/3009-PA	journal	August 1972
Numerical Solution of Sand Transport in Hydraulic Fracturing Daneshy, A. A. Journal of Petroleum Technology, Vol. 30, Issue 01 https://doi.org/10.2118/5636-PA	journal	January 1978
New Pump Schedule Generator for Hydraulic Fracturing Treatment Design Gu, H.; Desroches, J. SPE Latin American and Caribbean Petroleum Engineering Conference, All Days https://doi.org/10.2118/81152-MS	conference	April 2003
Widths of Hydraulic Fractures Perkins, T. K.; Kern, L. R. Journal of Petroleum Technology, Vol. 13, Issue 09 https://doi.org/10.2118/89-PA	journal	September 1961
Sample Complexity for Nonlinear Stochastic Dynamics Chen, Yongxin; Vaidya, Umesh 2019 American Control Conference (ACC) https://doi.org/10.23919/ACC.2019.8815138	conference	July 2019
On dynamic mode decomposition: Theory and applications Kutz, J. Nathan; Brunton, Steven L.; Luchtenburg, Dirk M. Journal of Computational Dynamics, Vol. 1, Issue 2 https://doi.org/10.3934/jcd.2014.1.391	journal	December 2014

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Application of Koopman operator for model-based control of fracture propagation and proppant transport in hydraulic fracturing operation

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