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Model Reduction of Parametrized Systems pp 471–490Cite as

Model Reduction for Coupled Near-Well and Reservoir Models Using Multiple Space-Time Discretizations

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Part of the book series: MS&A ((MS&A,volume 17))

Abstract

In reservoir simulations, fine fully-resolved grids deliver accurate model representations, but lead to large systems of nonlinear equations to solve every time step. Numerous techniques are applied in porous media flow simulations to reduce the computational effort associated with solving the underlying coupled nonlinear partial differential equations. Many models treat the reservoir as a whole. In other cases, the near-well accuracy is important as it controls the production rate. Near-well modeling requires finer space and time resolution compared with the remaining of the reservoir domain. To address these needs, we combine Model Order Reduction (MOR) with local grid refinement and local time stepping for reservoir simulations in highly heterogeneous porous media. We present a domain decomposition algorithm for a gas flow model in porous media coupling near-well regions, which are locally well-resolved in space and time with a coarser reservoir discretization. We use a full resolution for the near-well regions and apply MOR in the remainder of the domain. We illustrate our findings with numerical results on a gas flow model through porous media in a heterogeneous reservoir.

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References

  1. Afra, S., Gildin, E., Tarrahi, M.: Heterogeneous reservoir characterization using efficient parameterization through higher order svd (hosvd). In: American Control Conference. IEEE, Portland, OR (2014)

    Book  Google Scholar 

  2. Antil, H., Heinkenschloss, M., Hoppe, R.H.W., Sorensen, D.C.: Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables. Comput. Vis. Sci. 13(6), 249–264 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Antoulas, A., Sorensen, D., Gugercin, S.: A survey of model reduction methods for large-scale systems. Contemp. Math. Numer. Algorithms 280, 193–220 (2001)

    MathSciNet  MATH  Google Scholar 

  4. Baiges, J., Codina, R., Idelsohn, S.: A domain decomposition strategy for reduced order models. Application to the Incompressible Navier-Stokes Equations. Comput. Methods Appl. Mech. Eng. 267, 23–42 (2013)

    Article  MATH  Google Scholar 

  5. Buffoni M., Telib, H., Lollo, A.: Iterative methods for model reduction by domain decomposition. Comput. Fluids 38(6), 1160–1167 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cardoso, M., Durlofsky, L.: Use of reduced-order modeling procedures for production optimization. SPE J. 15(2), 426–435 (2010)

    Article  Google Scholar 

  7. Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010). doi:10.1137/090766498

    Article  MathSciNet  MATH  Google Scholar 

  8. Corigliano, A., Dossi, M., Mariani, S.: Domain decomposition and model order reduction methods applied to the simulation of multi-physics problems in MEMS. Comput. Struct. 122, 113–127 (2013)

    Article  Google Scholar 

  9. Corigliano, A., Dossi, M., Mariani, S.: Model order reduction and domain decomposition strategies for the solution of the dynamic elastic-plastic structural problem. Comput. Methods Appl. Mech. Eng. 290, 127–155 (2015)

    Article  MathSciNet  Google Scholar 

  10. Doren, J., Markovinovic R., Jansen, J.-D.: Reduced-order optimal control of water flooding using proper orthogonal decomposition. Comput. Geosci. 10(1), 137–158 (2006). doi:10.1007/s10596-005-9014-2. http://dx.doi.org/10.1007/s10596-005-9014-2

    Article  MathSciNet  MATH  Google Scholar 

  11. Doren, J.V., Markovinovic, R., Cansen, J.: Reduced-order optimal control of waterflooding using pod. In: 9th European Conference of the Mathematics of Oil Recovery. EAGE, Cannes (2004)

    Google Scholar 

  12. Efendiev, Y., Romanovskay, A., Gildin, E., Ghasemi, M.: Nonlinear complexity reduction for fast simulation of flow in heterogeneous porous media. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers, The Woodlands, TX. SPE 163618-MS (2013). http://dx.doi.org/10.2118/163618-MS

  13. Ewing, R.E., Lazarov, R.D., Vassilevski, P.S.: Finite difference schemes on grids with local refinement on time and space for parabolic problems. Derivation, stability and error analysis. Computing 45, 193–215 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. Handb. Numer. Anal. 7, 713–1018 (2000)

    MathSciNet  MATH  Google Scholar 

  15. Gander, M.J.: Optimized Schwarz methods. SIAM J. Numer. Anal. 44, 699–731 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ghasemi, M., Zhao, S., Insperger, T., Kalmar-Nagy, T.: Act-and-wait control of discrete systems with random delays. In: American Control Conference (ACC), pp. 5440–5443. IEEE, Montreal (2012). http://dx.doi.org/10.1109/ACC.2012.6315674

  17. Ghasemi, M., Ashraf, I., Gildin, E.: Reduced order modeling in reservoir simulation using the bilinear approximaion techniques. In: SPE Latin American and Caribbean Petroleum Engineering Conference. Society of Petroleum Engineers, Maracaibo. SPE 169357-MS (2014). http://dx.doi.org/10.2118/169357-MS

  18. Ghasemi M., Yang, Y., Gildin, E., Efendiev Y., Calo, V.: Fast multiscale reservoir simulations using POD-DEIM model reduction. In: SPE Reservoir Simulation Symposium. Houston, TX, pp. 23–25 (2015)

    Google Scholar 

  19. Ghommem, M., Calo, V.M., Efendiev, Y., Gildin, E.: Complexity reduction of multi-phase flows in heterogeneous porous media. In: SPE Kuwait Oil and Gas Show and Conference. SPE, Kuwait City. SPE 167295 (2013)

    Google Scholar 

  20. Gildin, E., Ghasemi, M.: A new model reduction technique applied to reservoir simulation. In: 14th European conference on the mathematics of oil recovery. European Association of Geoscientists and Engineers, Sicily (2014). http://dx.doi.org/10.3997/2214-4609.20141820

  21. Gildin, E., Lopez, T.J.: Closed-loop reservoir management: do we need complex models. In: SPE Digital Energy Conference and Exhibition. The Woodlands, TX (2011)

    Google Scholar 

  22. Heijn, T., Markovinovic, R., Jansen, J.: Generation of low-order reservoir models using system-theoretical concepts. SPE J. 9(2) (2004)

    Google Scholar 

  23. Jafarpour, B., Tarrahi, M.: Assessing the performance of the ensemble kalman filter for subsurface flow data integration under variogram uncertainty. Water Resour. Res. 47(5) (2011)

    Google Scholar 

  24. Kheriji, W., Masson, R., Moncorgé, A.: Nearwell local space and time refinement in reservoir simulation. Math. Comput. Simul. 118, 273–292 (2015)

    Article  MathSciNet  Google Scholar 

  25. Lerlertpakdee, P., Jafarpour, B., Gildin, E.: Efficient production optimization with flow-network models. SPE J. 19, 1–83 (2014)

    Article  Google Scholar 

  26. Mlacnik, M.J.: Using well windows in full field reservoir simulations. Ph.D. Thesis, University of Leoben (2002)

    Google Scholar 

  27. Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse Theory for Petroleum Reservoir Characterization and History Matching, vol. 1. Cambridge University Press, Cambridge (2008)

    Book  Google Scholar 

  28. Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulations. Elsevier, Amsterdam (1977)

    Google Scholar 

  29. Queipo, N.V., Pintos, S., Rincón, N., Contreras, N., Colmenares, J.: Surrogate modeling-based optimization for the integration of static and dynamic data into a reservoir description. J. Pet. Sci. Eng. 35(3), 167–181 (2002)

    Article  Google Scholar 

  30. Sun, K., Glowinski, R., Heinkenschloss, M., Sorensen, D.C.: Domain decomposition and model reduction of systems with local nonlinearities. In: Proceedings of ENUMATH 2007. The 7th European Conference on Numerical Mathematics and Advanced Applications, Graz, pp. 389–396 (2008)

    Google Scholar 

  31. Voneiff, G., Sadeghi, S., Bastian, P., Wolters, B., Jochen, J., Chow, B., Gatens, M.: Probabilistic forecasting of horizontal well performance in unconventional reservoirs using publicly-available completion data. In: SPE Unconventional Resources Conference. Society of Petroleum Engineers, The Woodlands, TX (2014)

    Book  Google Scholar 

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Acknowledgements

This publication was made possible by NPRP award [NPRP 7-1482-1-278] from the Qatar National Research Fund (a member of The Qatar Foundation). Additionally, this project was partially supported by the European Union’s Horizon 2020, research and innovation programme under the Marie Sklodowska-Curie grant agreement N 644202.

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Authors and Affiliations

  1. Texas A&M University at Qatar, PO Box 23874, Education City Doha, Qatar

    Walid Kheriji

  2. Texas A&M University, College Station, TX, 77843, USA

    Yalchin Efendiev & Eduardo Gildin

  3. Mineral Resources, CSIRO, Curtin University, Kensington, WA, Australia

    Victor Manuel Calo

Authors
  1. Walid Kheriji
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  2. Yalchin Efendiev
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  3. Victor Manuel Calo
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  4. Eduardo Gildin
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Corresponding author

Correspondence to Walid Kheriji .

Editor information

Editors and Affiliations

  1. Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

    Peter Benner

  2. Institute for Computational and Applied Mathematics, University of Münster , Münster, Germany

    Mario Ohlberger

  3. Massachusetts Institute of Technology, MIT , Cambridge, Massachusetts, USA

    Anthony Patera

  4. SISSA MathLab, International School for Adv. Studies , Trieste, Italy

    Gianluigi Rozza

  5. Institute of Numerical Mathematics, Ulm University , Ulm, Germany

    Karsten Urban

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Kheriji, W., Efendiev, Y., Manuel Calo, V., Gildin, E. (2017). Model Reduction for Coupled Near-Well and Reservoir Models Using Multiple Space-Time Discretizations. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds) Model Reduction of Parametrized Systems. MS&A, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-58786-8_29

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