Computational Geosciences

, Volume 22, Issue 4, pp 1093–1113 | Cite as

Machine-learning-based modeling of coarse-scale error, with application to uncertainty quantification

  • Sumeet TrehanEmail author
  • Louis J. Durlofsky
Original Paper


The use of upscaled models is attractive in many-query applications that require a large number of simulation runs, such as uncertainty quantification and optimization. Highly coarsened models often display error in output quantities of interest, e.g., phase production and injection rates, so the direct use of these results for quantitative evaluations and decision making may not be appropriate. In this work, we introduce a machine-learning-based post-processing framework for modeling the error in coarse-model results in the context of uncertainty quantification. Coarse-scale models are constructed using an accurate global single-phase transmissibility upscaling procedure. The framework entails the use of high-dimensional regression (random forest in this work) to model error based on a number of error indicators or features. Many of these features are derived from approximations of the subgrid effects neglected in the coarse-scale saturation equation. These features are identified through volume averaging, and they are generated by solving a fine-scale saturation equation with a constant-in-time velocity field. Our approach eliminates the need for the user to hand-design a small number of informative (relevant) features. The training step requires the simulation of some number of fine and coarse models (in this work we perform either 10 or 30 training simulations), followed by construction of a regression model for each well. Classification is also applied for production wells. The methodology then provides a correction at each time step, and for each well, in the phase production and injection rates. Results are presented for two- and three-dimensional oil–water systems. The corrected coarse-scale solutions show significantly better accuracy than the uncorrected solutions, both in terms of realization-by-realization predictions for oil and water production rates, and for statistical quantities important for uncertainty quantification, such as P10, P50, and P90 predictions.


Upscaling Machine learning Uncertainty quantification Reservoir simulation Classification Error modeling Random forest Surrogate model 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We thank the sponsors of the Stanford Smart Fields Consortium for financial support. We are grateful to Hai X. Vo for providing the channelized geological models used in this study, and to Olav Møyner for his support with MRST. We also thank Stanford’s Center for Computational Earth & Environmental Science for providing computational resources.


  1. 1.
    Aliyev, E., Durlofsky, L.J.: Multilevel field development optimization under uncertainty using a sequence of upscaled models. Math. Geosci. 49(3), 307–339 (2017). CrossRefGoogle Scholar
  2. 2.
    Arnold, D., Demyanov, V., Christie, M., Bakay, A., Gopa, K.: Optimisation of decision making under uncertainty throughout field lifetime: A fractured reservoir example. Comput. Geosci. 95, 123–139 (2016)CrossRefGoogle Scholar
  3. 3.
    Bakay, A., Demyanov, V., Arnold, D.: Uncertainty quantification in fractured reservoirs based on outcrop modelling from northeast Brazil. In: 7th EAGE international conference and exhibition (2016)Google Scholar
  4. 4.
    Bardy, G., Biver, P.: Sorting reservoir models according to flow criteria: A methodology, using fast marching methods and multi-dimensional scaling. In: Mathematics of Planet Earth: Proceedings of the 15th Annual Conference of the International Association for Math. Geosci., pp. 643–646. Springer. (2014)
  5. 5.
    Breiman, L.: Random forests. Mach. Learn. 45(1), 5–32 (2001)CrossRefGoogle Scholar
  6. 6.
    Breiman, L., Cutler, A., Liaw, A., Wiener, M.: Package random forest version 4.6–12. (2015)
  7. 7.
    Cardoso, M., Durlofsky, L.J.: Linearized reduced-order models for subsurface flow simulation. J. Comput. Phys. 229(3), 681–700 (2010)CrossRefGoogle Scholar
  8. 8.
    Chen, Y., Durlofsky, L.J.: Ensemble-level upscaling for efficient estimation of fine-scale production statistics. SPE J. 13(4), 400–411 (2008)CrossRefGoogle Scholar
  9. 9.
    Chen, Y., Durlofsky, L.J., Gerritsen, M., Wen, X.H.: A coupled local–global upscaling approach for simulating flow in highly heterogeneous formations. Adv. Water Resour. 26(10), 1041–1060 (2003)CrossRefGoogle Scholar
  10. 10.
    Chen, Y., Mallison, B.T., Durlofsky, L.J.: Nonlinear two-point flux approximation for modeling full-tensor effects in subsurface flow simulations. Comput. Geosci. 12(3), 317–335 (2008)CrossRefGoogle Scholar
  11. 11.
    Chen, Y., Park, K., Durlofsky, L.J.: Statistical assignment of upscaled flow functions for an ensemble of geological models. Comput. Geosci. 15(1), 35–51 (2011)CrossRefGoogle Scholar
  12. 12.
    Drohmann, M., Carlberg, K.: The ROMES method for statistical modeling of reduced-order-model error. SIAM/ASA J. Uncertain. Quantif. 3(1), 116–145 (2015)CrossRefGoogle Scholar
  13. 13.
    Durlofsky, L.J.: Coarse scale models of two phase flow in heterogeneous reservoirs: Volume averaged equations and their relationship to existing upscaling techniques. Comput. Geosci. 2(2), 73–92 (1998)CrossRefGoogle Scholar
  14. 14.
    Durlofsky, L.J.: Upscaling and gridding of fine scale geological models for flow simulation. In: 8th International Forum on Reservoir Simulation (2005)Google Scholar
  15. 15.
    Durlofsky, L.J., Chen, Y.: Uncertainty quantification for subsurface flow problems using coarse-scale models. In: Numerical Analysis of Multiscale Problems, pp. 163–202. Springer (2012)Google Scholar
  16. 16.
    Efendiev, Y., Datta-Gupta, A., Ma, X., Mallick, B.: Efficient sampling techniques for uncertainty quantification in history matching using nonlinear error models and ensemble level upscaling techniques. Water Resources Research 45(11) (2009)Google Scholar
  17. 17.
    Efendiev, Y.R., Durlofsky, L.J.: A generalized convection-diffusion model for subgrid transport in porous media. Multiscale Model. Simul. 1(3), 504–526 (2003)CrossRefGoogle Scholar
  18. 18.
    Efendiev, Y.R., Durlofsky, L.J.: Accurate subgrid models for two-phase flow in heterogeneous reservoirs. SPE J. 9(2), 219–226 (2004)CrossRefGoogle Scholar
  19. 19.
    Floris, F., Bush, M., Cuypers, M., Roggero, F., Syversveen, A.R.: Methods for quantifying the uncertainty of production forecasts: A comparative study. Pet. Geosci. 7(S), S87—S96 (2001)CrossRefGoogle Scholar
  20. 20.
    Glimm, J., Hou, S., Lee, Y., Sharp, D., Ye, K.: Solution error models for uncertainty quantification. Contemp. Math. 327, 115–140 (2003)CrossRefGoogle Scholar
  21. 21.
    Guyon, I., Elisseeff, A.: An introduction to variable and feature selection. J. Mach. Learn. Res. 3, 1157–1182 (2003)Google Scholar
  22. 22.
    Hastie, T., Tibshirani, R., Friedman, J., Franklin, J.: The elements of statistical learning: Data mining, Inference, and prediction, vol. 27. Springer-Verlag, New York (2005)Google Scholar
  23. 23.
    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)CrossRefGoogle Scholar
  24. 24.
    James, G., Witten, D., Hastie, T., Tibshirani, R.: An introduction to statistical learning, vol. 6. Springer, New York (2013)CrossRefGoogle Scholar
  25. 25.
    Josset, L., Ginsbourger, D., Lunati, I.: Functional error modeling for uncertainty quantification in hydrogeology. Water Resour. Res. 51(2), 1050–1068 (2015)CrossRefGoogle Scholar
  26. 26.
    Khodabakhshi, M., Jafarpour, B., King, M.J.: Field applications of a multi-scale multi-physics history matching approach. In: SPE Reservoir Simulation Symposium, SPE 173239-MS (2015)Google Scholar
  27. 27.
    Kovscek, A., Wang, Y.: Geologic storage of carbon dioxide and enhanced oil recovery. I. Uncertainty quantification employing a streamline based proxy for reservoir flow simulation. Energy Convers. Manag. 46(11), 1920–1940 (2005)CrossRefGoogle Scholar
  28. 28.
    Krogstad, S., Lie, K.A., Møyner, O., Nilsen, H.M., Raynaud, X., Skaflestad, B.: MRST-AD–An open-source framework for rapid prototyping and evaluation of reservoir simulation problems. In: SPE Reservoir Simulation Symposium, SPE 173317-MS (2015)Google Scholar
  29. 29.
    Krogstad, S., Raynaud, X., Nilsen, H.M.: Reservoir management optimization using well-specific upscaling and control switching. Comput. Geosci. 20(3), 695–706 (2016)CrossRefGoogle Scholar
  30. 30.
    Li, H., Durlofsky, L.J.: Ensemble level upscaling for compositional flow simulation. Comput. Geosci. 20 (3), 525–540 (2016)CrossRefGoogle Scholar
  31. 31.
    Li, H., Durlofsky, L.J.: Local–global upscaling for compositional subsurface flow simulation. Transp. Porous Media 111(3), 701–730 (2016)CrossRefGoogle Scholar
  32. 32.
    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. Comput. Geosci. 16(2), 297–322 (2012)CrossRefGoogle Scholar
  33. 33.
    Lødøen, O.P.: Bayesian calibration of reservoir models using a coarse-scale reservoir simulator in the prior specification. In: EAGE Conference on Petroleum Geostatistics (2007)Google Scholar
  34. 34.
    Lødøen, O.P., Omre, H.: Scale-corrected ensemble Kalman filtering applied to production-history conditioning in reservoir evaluation. SPE J. 13(2), 177–194 (2008)CrossRefGoogle Scholar
  35. 35.
    Lødøen, O.P., Omre, H., Durlofsky, L.J., Chen, Y.: Assessment of uncertainty in reservoir production forecasts using upscaled flow models. In: Geostatistics Banff, pp. 713–722. Springer (2005)Google Scholar
  36. 36.
    Ma, X., Al-Harbi, M., Datta-Gupta, A., Efendiev, Y.: An efficient two-stage sampling method for uncertainty quantification in history matching geological models. SPE J. 13(1), 77–87 (2008)CrossRefGoogle Scholar
  37. 37.
    Møyner, O., Krogstad, S., Lie, K.A.: The application of flow diagnostics for reservoir management. SPE J. 20(2), 306–323 (2015)CrossRefGoogle Scholar
  38. 38.
    Ng, L.W.T., Eldred, M.: Multifidelity uncertainty quantification using non-intrusive polynomial chaos and stochastic collocation. In: 14th AIAA Non-Deterministic Approaches Conference, vol. 43 (2012)Google Scholar
  39. 39.
    Omre, H., Lødøen, O.P.: Improved production forecasts and history matching using approximate fluid-flow simulators. SPE J. 9(3), 339–351 (2004)CrossRefGoogle Scholar
  40. 40.
    Peaceman, D.W.: Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability. SPE J. 23(3), 531–543 (1983)CrossRefGoogle Scholar
  41. 41.
    Remy, N., Boucher, A., Wu, J.: Applied geostatistics with SGeMS: A user’s guide. Cambridge University Press, New York (2009)CrossRefGoogle Scholar
  42. 42.
    Salehi, A., Voskov, D., Tchelepi, H.: Thermodynamically consistent transport coefficients for upscaling of compositional processes. In: SPE Reservoir Simulation Symposium, SPE 163576-MS (2013)Google Scholar
  43. 43.
    Scheidt, C., Caers, J.: Representing spatial uncertainty using distances and kernels. Math. Geosci. 41(4), 397–419 (2009)CrossRefGoogle Scholar
  44. 44.
    Scheidt, C., Caers, J.: Uncertainty quantification in reservoir performance using distances and kernel methods–application to a west Africa deepwater turbidite reservoir. SPE J. 14(4), 680–692 (2009)CrossRefGoogle Scholar
  45. 45.
    Scheidt, C., Caers, J., Chen, Y., Durlofsky, L.J.: A multi-resolution workflow to generate high-resolution models constrained to dynamic data. Comput. Geosci. 15(3), 545–563 (2011)CrossRefGoogle Scholar
  46. 46.
    Shahvali, M., Mallison, B., Wei, K., Gross, H.: An alternative to streamlines for flow diagnostics on structured and unstructured grids. SPE J. 17(3), 768–778 (2012)CrossRefGoogle Scholar
  47. 47.
    Shirangi, M.G., Durlofsky, L.J.: A general method to select representative models for decision making and optimization under uncertainty. Comput. Geosci. 96, 109–123 (2016)CrossRefGoogle Scholar
  48. 48.
    Shook, G.M., Mitchell, K.M.: A robust measure of heterogeneity for ranking earth models: The F-PHI curve and dynamic Lorenz coefficient. In: SPE Annual Technical Conference and Exhibition, SPE 124625-MS (2009)Google Scholar
  49. 49.
    Strebelle, S.: Conditional simulation of complex geological structures using multiple-point statistics. Math. Geol. 34(1), 1–21 (2002)CrossRefGoogle Scholar
  50. 50.
    Suzuki, S., Caers, J.K.: History matching with an uncertain geological scenario. In: SPE Annual Technical Conference and Exhibition, SPE 102154-MS (2006)Google Scholar
  51. 51.
    Trehan, S.: Surrogate modeling for subsurface flow: A new reduced-order model and error estimation procedures. Ph.D. thesis, Stanford University (2016)Google Scholar
  52. 52.
    Trehan, S., Carlberg, K.T., Durlofsky, L.J.: Error modeling for surrogates of dynamical systems using machine learning. Int. J. Numer. Methods Eng. 112(12), 1801–1827 (2017). CrossRefGoogle Scholar
  53. 53.
    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)CrossRefGoogle Scholar
  54. 54.
    Vo, H.X., Durlofsky, L.J.: Data assimilation and uncertainty assessment for complex geological models using a new PCA-based parameterization. Comput. Geosci. 19(4), 747–767 (2015)CrossRefGoogle Scholar
  55. 55.
    Zhang, P., Pickup, G.E., Christie, M.A.: A new practical method for upscaling in highly heterogeneous reservoir models. SPE J. 13(1), 68–76 (2008)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Energy Resources EngineeringStanford UniversityStanfordUSA

Personalised recommendations