Skip to main content
SpringerLink
Log in

Trajectory-based DEIM (TDEIM) model reduction applied to reservoir simulation

  • Original Paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

Two well-known model reduction methods, namely the trajectory piecewise linearization (TPWL) approximation and the discrete empirical interpolation method (DEIM), are combined to take advantage of their benefits and avoid their shortcomings to generate reduced-order models for reservoir simulation. To this end, we use the trajectory-based DEIM (TDEIM) to approximate the nonlinear terms in the simulation. Explicitly, we express such quantities in the test simulation as the sum of their equivalents evaluated at the closest available training point from the high-fidelity training trajectory and a perturbed contribution defined as the difference between the test and the training runs. We only interpolate this difference term in the reduced space of DEIM instead of the original one, resulting in computational savings and improvement in accuracy. TDEIM is further combined with the proper orthogonal decomposition (POD) method to provide an efficient POD-TDEIM framework. We test our new methodology on three examples, involving two-phase (water-oil) heterogeneous reservoir models. First, the performance of POD-TDEIM is compared with POD-TPWL and POD-DEIM on a 2D reservoir model. For the same set of high-fidelity training runs, POD-TDEIM outperforms the other two methods. We further propose an extended TDEIM in which the nonlinear term is expanded along the training trajectory to include an additional higher order derivative. An example of a 3D reservoir model is then presented to show the capability of the extended TDEIM to improve the accuracy of the reduced model further when handling significant discrepancies between the training and test boundary controls. We also present another 3D example to demonstrate the superior correctness and numerical stability of using Petrov-Galerkin projection with TDEIM over the Galerkin projection.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, R., Fournier, J.: Sobolev Spaces. Academic Press, Oxford (1975)

    Google Scholar 

  2. Amsallem, D., Zahr, M., Choi, Y., Farhat, C.: Design optimization using hyper-reduced-order models. Struct. Multidiscip. Optim. 51(4), 919–940 (2015)

    Article  Google Scholar 

  3. Aziz, K., Settari, A.: Petroleum reservoir simulation. Elsevier Applied Science Publishers, New York (1986)

    Google Scholar 

  4. Cardoso, M., Durlofsky, L.J.: Linearized reduced-order models for subsurface flow simulation. J. Comput. Phys. 229(3), 681–700 (2010)

    Article  Google Scholar 

  5. Cardoso, M.A., Durlofsky, L.J., et al.: Use of reduced-order modeling procedures for production optimization. SPE J. 15(02), 426–435 (2010)

    Article  Google Scholar 

  6. Carlberg, K., Bou-Mosleh, C., Farhat, C.: Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations. Int. J. Numer. Methods Eng. 86(2), 155–181 (2011)

    Article  Google Scholar 

  7. Carlberg, K., Farhat, C., Cortial, J., Amsallem, D.: The gnat method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. J. Comput. Phys. 242, 623–647 (2013)

    Article  Google Scholar 

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

    Article  Google Scholar 

  9. Chaturantabut, S., Sorensen, D.C.: Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media. Math. Comput. Model. Dyn. Syst. 17(4), 337–353 (2011)

    Article  Google Scholar 

  10. Christie, M., Blunt, M., et al.: Tenth spe comparative solution project: a comparison of upscaling techniques. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2001)

  11. van Doren, J.F., Markovinović, R., Jansen, J.D.: Reduced-order optimal control of water flooding using proper orthogonal decomposition. Comput. Geosci. 10(1), 137–158 (2006)

  12. van Essen, G., Van den Hof, P., Jansen, J.D., et al.: A two-level strategy to realize life-cycle production optimization in an operational setting. SPE J. 18(06), 1–057 (2013)

  13. Florez, H.: Domain decomposition methods for geomechanics. Ph.D. thesis The University of Texas at Austin (2012)

  14. Florez, H., Argáez, M.: Applications and comparison of model-order reduction methods based on wavelets and POD. In: Fuzzy Information Processing Society (NAFIPS), 2016 Annual Conference of the North American, pp. 1–8. IEEE (2016)

  15. Florez, H., Argáez, M.: A model-order reduction method based on wavelets and POD to solve nonlinear transient and steady-state continuation problems. Appl. Math. Model. 53, 12–31 (2018)

    Article  Google Scholar 

  16. Florez, H., Argaez, M.: A reduced order Gauss-Newton method for nonlinear problems based on compressed sensing for PDE applications, chap. 6, pp. 107–128. In Nonlinear Systems - Modeling, Estimation, and Stability. InTech Open. https://doi.org/10.5772/intechopen.74439. ISBN 978-1-78923-405-3 (2018)

  17. Ghasemi, M., Gildin, E.: Localized model reduction in porous media flow. IFAC-PapersOnLine 48(6), 242–247 (2015)

    Article  Google Scholar 

  18. Ghasemi, M., Ibrahim, A., Gildin, E., et al.: Reduced order modeling in reservoir simulation using the bilinear approximation techniques. In: SPE Latin America and Caribbean Petroleum Engineering Conference. Society of Petroleum Engineers (2014)

  19. Gildin, E., Ghasemi, M.: A new model reduction technique applied to reservoir simulation. In: ECMOR XIV-14Th European Conference on the Mathematics of Oil Recovery (2014)

  20. Gildin, E., Ghasemi, M., Romanovskay, A., Efendiev, Y.: Nonlinear complexity reduction for fast simulation of flow in heterogeneous porous media. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2013)

  21. He, J., Durlofsky, L.J.: Reduced-order modeling for compositional simulation by use of trajectory piecewise linearization. SPE J. 19(05), 858–872 (2014)

    Article  Google Scholar 

  22. He, J., Durlofsky, L.J.: Constraint reduction procedures for reduced-order subsurface flow models based on POD–TPWL. Int. J. Numer. Methods Eng. 103(1), 1–30 (2015)

    Article  Google Scholar 

  23. He, J., Sætrom, J., Durlofsky, L.J.: Enhanced linearized reduced-order models for subsurface flow simulation. Journal of Computational Physics 230(23), 8313–8341 (2011)

    Article  Google Scholar 

  24. Hinze, M., Volkwein, S.: Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control. In: Dimension Reduction of Large-Scale Systems, Pp. 261–306. Springer (2005)

  25. Jafarpour, B., Tarrahi, M.: Assessing the performance of the ensemble Kalman filter for subsurface flow data integration under variogram uncertainty. Water Resour. Res. 47, W05537 (2011). https://doi.org/10.1029/2010WR009090

    Article  Google Scholar 

  26. Lie, K.A., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H.M., Skaflestad, B.: Open-source matlab implementation of consistent discretisations on complex grids, vol. 16. https://doi.org/10.1007/s10596-011-9244-4(2012)

  27. Ljung, L.: System ddentification. In: Signal Analysis and Prediction, pp. 163–173. Springer (1998)

  28. Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse theory for petroleum reservoir characterization and history matching. Cambridge University Press, Cambridge (2008)

    Book  Google Scholar 

  29. Peaceman, D.W., et al.: Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability. Soc. Pet. Eng. J. 23(03), 531–543 (1983)

    Article  Google Scholar 

  30. Peherstorfer, B., Butnaru, D., Willcox, K., Bungartz, H.J.: Localized discrete empirical interpolation method. SIAM J. Sci. Comput. 36(1), A168–A192 (2014)

    Article  Google Scholar 

  31. Shirangi, M.G., Emerick, A.A.: An improved tsvd-based Levenberg–Marquardt algorithm for history matching and comparison with Gauss–Newton. J. Pet. Sci. Eng. 143, 258–271 (2016)

    Article  Google Scholar 

  32. Trehan, S., Durlofsky, L.J.: Trajectory piecewise quadratic reduced-order model for subsurface flow, with application to pde-constrained optimization. J. Comput. Phys. 326, 446–473 (2016)

    Article  Google Scholar 

  33. Yang, Y., Ghasemi, M., Gildin, E., Efendiev, Y., Calo, V., et al.: Fast multiscale reservoir simulations with pod-deim model reduction. SPE J. 21(06), 2–141 (2016)

    Article  Google Scholar 

  34. Yoon, S., Alghareeb, Z.M., Williams, J.R., et al.: Hyper-reduced-order models for subsurface flow simulation. SPE J. 21(06), 2–128 (2016)

    Article  Google Scholar 

Download references

Acknowledgments

We thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions. We also appreciate Dr. Lou Durlofsky for his valuable ideas to improve this paper.

Funding

The authors received financial support from ExxonMobil Upstream Research Company, and permission to publish this paper.

Author information

Authors and Affiliations

  1. Texas A&M University, College Station, TX, USA

    Xiaosi Tan, Eduardo Gildin & Horacio Florez

  2. ExxonMobil Upstream Research Company, Houston, TX, USA

    Sumeet Trehan, Yahan Yang & Nazish Hoda

Authors
  1. Xiaosi Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eduardo Gildin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Horacio Florez
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sumeet Trehan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yahan Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Nazish Hoda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eduardo Gildin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tan, X., Gildin, E., Florez, H. et al. Trajectory-based DEIM (TDEIM) model reduction applied to reservoir simulation. Comput Geosci 23, 35–53 (2019). https://doi.org/10.1007/s10596-018-9782-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-018-9782-0

Keywords

Search

Navigation