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Flow-based dissimilarity measures for reservoir models: a spatial-temporal tensor approach


In reservoir engineering, it is attractive to characterize the difference between reservoir models in metrics that relate to the economic performance of the reservoir as well as to the underlying geological structure. In this paper, we develop a dissimilarity measure that is based on reservoir flow patterns under a particular operational strategy. To this end, a spatial-temporal tensor representation of the reservoir flow patterns is used, while retaining the spatial structure of the flow variables. This allows reduced-order tensor representations of the dominating patterns and simple computation of a flow-induced dissimilarity measure between models. The developed tensor techniques are applied to cluster model realizations in an ensemble, based on similarity of flow characteristics.


  1. Afra, S., Gildin, E.: Permeability parametrization using higher order singular value decomposition (hosvd). In: 2013 12th International Conference on Machine Learning and Applications (ICMLA), vol. 2, pp. 188–193. IEEE (2013). doi:10.1109/icmla.2013.121

  2. Afra, S., Gildin, E., Tarrahi, M.: Heterogeneous reservoir characterization ussing efficient parameterization through higher order svd (hosvd). In: American Control Conference, pp. 147–152, Portland, Oregon (2014). doi:10.1109/acc.2014.6859,246

  3. Aloise, D., Deshpande, A., Hansen, P., Popat, P.: Np-hardness of euclidean sum-of-squares clustering. Mach. Learn. 75(2), 245–248 (2009). doi:10.1007/s10,994-009-5103-0

    Article  Google Scholar 

  4. Aziz, K., Settari, A.: Petroleum reservoir simulation, vol. 476 Applied Science Publishers London (1979)

  5. Bader, B.W., Kolda, T.G., et al.: Matlab tensor toolbox version 2.6. Available online (2015)

  6. Barros, E.G.D., Van den Hof, P.M.J., Jansen, J.D.: Value of information in closed-loop reservoir management. Comput. Geosci. 20(3), 737–749 (2016). doi:10.1007/s10,596-015-9509-4

    Article  Google Scholar 

  7. Borg, I., Groenen, P.J.F.: Modern multidimensional scaling: Theory and applications. Springer. doi:10.4324/9780203767719 (2005)

  8. Caers, J., Park, K., Scheidt, C.: Modeling uncertainty of complex earth systems in metric space. In: Handbook of Geomathematics, pp. 865–889. Springer (2010). doi:10.1007/978-3-642-01,546-5-29

  9. Cardoso, M.A., Durlofsky, L.J., Sarma, P.: Development and application of reduced-order modeling procedures for subsurface flow simulation. Int. J. Numer. Methods Eng. 77(9), 1322–1350 (2009). doi:10.1002/nme.2453

    Article  Google Scholar 

  10. Chen, Y., Oliver, D.S., Zhang, D., et al: Efficient ensemble-based closed-loop production optimization. SPE J. 14(04), 634–645 (2009). doi:10.2118/112,873-pa

    Article  Google Scholar 

  11. De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000). doi:10.1137/s0895479896305,696

    Article  Google Scholar 

  12. De Lathauwer, L., De Moor, B., Vandewalle, J.: On the best rank-1 and rank- (r 1,r 2, ...,r n ) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 21 (4), 1324–1342 (2000). doi:10.1137/s0895479898346,995

    Article  Google Scholar 

  13. Durlofsky, L.J.: Upscaling and gridding of fine scale geological models for flow simulation. In: 8th International Forum on Reservoir Simulation Iles Borromees, Stresa, Italy, pp. 20–24 (2005).

  14. Gildin, E., Afra, S.: Efficient inference of reservoir parameter distribution utilizing higher order svd reparameterization. In: ECMOR XIV-14th European conference on the mathematics of oil recovery. Catania, Italy (2014). doi:10.3997/2214-4609.20141826

  15. Golub, G.H., Van Loan, C.F.: Matrix computations, vol. 3. JHU Press. doi:10.1137/1028073 (2012)

  16. Insuasty, E., Van den Hof, P.M.J., Weiland, S., Jansen, J.D.: Tensor-based reduced order modeling in reservoir engineering: An application to production optimization. IFAC-PapersOnLine 48(6), 254–259 (2015). doi:10.1016/j.ifacol.2015.08.040

    Article  Google Scholar 

  17. Jansen, J.D.: A systems description of flow through porous media. Springer Briefs in Earth Sciences, Springer. doi:10.1007/978-3-319-00260-6 (2013)

  18. Jansen, J.D., Bosgra, O.H., Van den Hof, P.M.J.: Model-based control of multiphase flow in subsurface oil reservoirs. J. Process Control 18(9), 846–855 (2008). doi:10.1016/j.jprocont.2008.06.011

    Article  Google Scholar 

  19. Jansen, J.D., Fonseca, R.M., Kahrobaei, S., Siraj, M.M., Van Essen, G.M., Van den Hof, P.M.J.: The egg model–a geological ensemble for reservoir simulation. Geosci. Data J. 1(2), 192–195 (2014). doi:10.1002/gdj3.21

    Article  Google Scholar 

  20. Jegelka, S., Sra, S., Banerjee, A.: Approximation algorithms for tensor clustering. In: Algorithmic learning theory, pp. 368–383. Springer (2009). doi:10.1007/978-3-642-04,414-4-30

  21. Ketchen, D.J., Shook, C.L.: The application of cluster analysis in strategic management research: an analysis and critique. Strateg. Manag. J. 17(6), 441–458 (1996). doi:10.1002/(SICI)1097-0266(199,606)17:6<441::AID-SMJ819>3.0.CO;2-G

    Article  Google Scholar 

  22. Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009). doi:10.1137/07070,111x

    Article  Google Scholar 

  23. Krogstad, S.: A sparse basis pod for model reduction of multiphase compressible flow. In: SPE Reservoir Simulation Symposium. The Woodlands, Texas. doi:10.2118/141973-ms (2011)

  24. 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). doi:10.1007/s10,596-011-9244-4

    Article  Google Scholar 

  25. Lloyd, S.: Least squares quantization in pcm. IEEE Trans. Inf. Theory 28(2), 129–137 (1982). doi:10.1109/TIT.1982.1056,489

    Article  Google Scholar 

  26. Markovinovic, R., Jansen, J.D.: Accelerating iterative solution methods using reduced-order models as solution predictors. Int. J. Numer. Methods Eng. 68(5), 525–541 (2006). doi:10.1002/nme.1721

    Article  Google Scholar 

  27. Park, K., Caers, J.: History matching in low-dimensional connectivity-vector space. In: EAGE Petroleum Geostatistics. Cascais, Portugal. doi:10.3997/2214-4609.201403075 (2007)

  28. Sarma, P., Chen, W., Xie, J.: Selecting representative models from a large set of models. In: SPE Reservoir Simulation Symposium. The Woodlands, Texas. doi:10.2118/163671-MS (2013)

  29. Sarma, P., Durlofsky, L.J., Aziz, K.: Computational techniques for closed–loop reservoir modeling with application to a realistic reservoir. Pet. Sci. Technol. 26(10–11), 1120–1140 (2008). doi:10.1080/10916460701829,580

    Article  Google Scholar 

  30. Scheidt, C., Caers, J.: Representing spatial uncertainty using distances and kernels. Math. Geosci. 41(4), 397–419 (2009). doi:10.1007/s11,004-008-9186-0

    Article  Google Scholar 

  31. Scheidt, C., Caers, J., Chen, Y., Durlofsky, L.: A multi-resolution workflow to generate high-resolution models constrained to dynamic data. Comput. Geosci. 15(3), 545–563 (2011). doi:10.1007/s10,596-011-9223-9

    Article  Google Scholar 

  32. Scheidt, C., Caers, J., et al: Uncertainty quantification in reservoir performance using distances and kernel methods–application to a west africa deepwater turbidite reservoir. SPE J. 14(04), 680–692 (2009). doi:10.2118/118,740-PA

    Article  Google Scholar 

  33. Shekhawat, H.S., Weiland, S.: On the problem of low rank approximation of tensors. In: 21st International Symposium on Mathematical Theory of Networks and Systems. Groningen, The Netherlands. (2014)

  34. Suzuki, S., Caers, J.: A distance based prior model parameterization for constraining solution of spatial inverse problems. Math. Geosci. 40(4), 445-469 (2008). doi:10.1007/s11,004-008-9154-8

    Article  Google Scholar 

  35. Suzuki, S., Caumon, G., Caers, J.: Dynamic data integration for structural modeling: model screening approach using a distance-based model parameterization. Comput. Geosci. 12(1), 105-119 (2008). doi:10.1007/s10,596-007-9063-9

    Article  Google Scholar 

  36. Van Doren, J.F.M., Van den Hof, P.M.J., Bosgra, O.H., Jansen, J.D.: Controllability and observability in two-phase porous media flow. Comput. Geosci. 17(5), 773-788 (2013). doi:10.1007/s10,596-013-9355-1

    Article  Google Scholar 

  37. Van Essen, G.M., Zandvliet, M.J., Van den Hof, P.M.J., Bosgra, O.H., Jansen, J.D.: Robust waterflooding optimization of multiple geological scenarios. SPE J. 14(01), 202-210 (2009). doi:10.2118/102,913-ms

    Article  Google Scholar 

  38. Vervliet, N., Debals, O., Sorber, L., Barel, M.V., Lathauwer, L.D.: Tensorlab v3.0. Available online (2016)

  39. 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). doi:10.1007/s10,596-015-9483-x

    Article  Google Scholar 

  40. Weiland, S., Van Belzen, F.: Singular value decompositions and low rank approximation of tensors. IEEE Trans. Signal Process. 58(3), 1171-1182 (2010). doi:10.1109/tsp.2009.2034,308

    Article  Google Scholar 

  41. Yeh, T.H., Jimenez, E., Van Essen, G., Chen, C., Jin, L., Girardi, A., Gelderblom, P., Horesh, L., Conn, A.R., et al: Reservoir uncertainty quantification using probabilistic history matching workflow. In: SPE Annual Technical Conference and Exhibition. Amsterdam, The Netherlands. doi:10.2118/170893-ms (2014)

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We acknowledge the discussions with Dr.Tzu-hao Yeh from the Quantitative Reservoir Management group at Shell for his views on the potential application of the techniques presented in this paper on field cases. The authors acknowledge financial support from the Recovery Factory program sponsored by Shell Global Solutions International.

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Correspondence to Paul M. J. Van den Hof.

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Insuasty, E., Van den Hof, P.M.J., Weiland, S. et al. Flow-based dissimilarity measures for reservoir models: a spatial-temporal tensor approach. Comput Geosci 21, 645–663 (2017).

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  • Reduced-order modeling
  • Tensor decompositions
  • Tensor algebra
  • Flow characterization