Computational Geosciences

, Volume 17, Issue 5, pp 773–788 | Cite as

Controllability and observability in two-phase porous media flow

  • Jorn F. M. Van Doren
  • Paul M. J. Van den Hof
  • Okko H. Bosgra
  • Jan Dirk Jansen


Reservoir simulation models are frequently used to make decisions on well locations, recovery optimization strategies, etc. The success of these applications is, among other aspects, determined by the controllability and observability properties of the reservoir model. In this paper, it is shown how the controllability and observability of two-phase flow reservoir models can be analyzed and quantified with aid of generalized empirical Gramians. The empirical controllability Gramian can be interpreted as a spatial covariance of the states (pressures or saturations) in the reservoir resulting from input perturbations in the wells. The empirical observability Gramian can be interpreted as a spatial covariance of the measured bottom-hole pressures or well bore flow rates resulting from state perturbations. Based on examples in the form of simple homogeneous and heterogeneous reservoir models, we conclude that the position of the wells and of the front between reservoir fluids, and to a lesser extent the position and shape of permeability heterogeneities that impact the front, are the most important factors that determine the local controllability and observability properties of the reservoir.


Reservoir engineering Reservoir simulation Controllability Observability Two-phase flow Porous media Empirical Gramians Covariance matrix Proper orthogonal decomposition POD 


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  1. 1.
    Zandvliet, M.J., Van Doren, J.F.M., Bosgra, O.H., Jansen, J.D., Van den Hof, P.M.J.: Controllability, observability and identifiability in single-phase porous media flow. Comput. Geosci. 12(4), 605–622 (2008)CrossRefGoogle Scholar
  2. 2.
    Sudaryanto, B., Yortsos, Y.C.: Optimization of fluid front dynamics in porous media using rate control. Phys. Fluids 12(7), 1656–70 (2000)CrossRefGoogle Scholar
  3. 3.
    Fyrozjaee, M.H., Yortsos, Y.C.: Control of a displacement front in potential flow using flow-rate partition. Paper SPE 99524, presented at the SPE intelligent energy conference, Amsterdam, The NetherlandsGoogle Scholar
  4. 4.
    Ramakrishnan, T.S.: On reservoir fluid-flow control with smart completions. SPE Prod. Oper. 22(1), 4–12 (2007)Google Scholar
  5. 5.
    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, 846–855 (2008)CrossRefGoogle Scholar
  6. 6.
    Jansen, J.D., Van Doren, J.F.M., Heidary-Fyrozjaee, M., Yortsos, Y.C.: Front controllability in two-phase porous media flow. In: Van den Hof, P.M.J., Scherer, C., Heuberger, P.S.C. (eds.) Model- based Control—Bridging Rigorous Theory and Advanced Control, pp 203–219. Springer, New York (2009)CrossRefGoogle Scholar
  7. 7.
    Jansen, J.D.: Adjoint-based optimization of multiphase flow through porous media—a review. Comput. Fluids 46(1), 40–51 (2011)CrossRefGoogle Scholar
  8. 8.
    Watson, A.T., Gavalas, G.R., Seinfeld, J.H.: Identifiability of estimates of two-phase reservoir properties in history matching. SPE J. 24(6), 697–706 (1984)Google Scholar
  9. 9.
    Datta-Gupta, A., Vasco, D.W., Long, J.C.S.: On the sensitivity and spatial resolution of transient pressure and tracer data for heterogeneity characterization. SPE Form. Eval. 12(2), 137–144 (1997)Google Scholar
  10. 10.
    Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse theory for petroleum reservoir characterization and history matching. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  11. 11.
    Van Doren, J.F.M., Van den Hof, P.M.J., Jansen, J.D., Bosgra, O.H.: Determining identifiable parameterizations for large-scale physical models in reservoir engineering. In: Chung, M., Misra, P., Shim, H. (eds.) Proceedings 17th International Federation for Automatic Control (IFAC) World Congress, pp. 11421–11426. Seoul, 6–11 July 2008Google Scholar
  12. 12.
    Van Doren, J.F.M., Douma, S.G., Van den Hof, P.M.J., Jansen, J.D., Bosgra, O.H.: Identifiability: from qualitative analysis to model structure approximation. In: Proceedings 15th IFAC Symposium on System Identification (SYSID), St. Malo, 6–8 July 2009Google Scholar
  13. 13.
    Oliver, D.S., Chen, Y.: Recent progress on reservoir history matching: a review. Comput. Geosci. 15(1), 185–221 (2011)CrossRefGoogle Scholar
  14. 14.
    Vakili-Ghahani, S.A., Jansen, J.D.: Control-relevant upscaling. SPE J. 15(2), 471–479 (2010)Google Scholar
  15. 15.
    Vakili-Ghahani, S.A., Jansen, J.D.: A system-theoretical approach to selective grid coarsening of reservoir models. Comput. Geosci. 16(1), 159–176 (2012)CrossRefGoogle Scholar
  16. 16.
    Heijn, T., Markovinovi, R., Jansen, J.D.: Generation of low-order reservoir models using system-theoretical concepts. SPE J. 9(2), 202–218 (2004)Google Scholar
  17. 17.
    Van Doren, J.F.M., Markovinovi´c, R., Jansen, J.D.: Reduced-order optimal control of waterflooding using POD. Comput. Geosci. 10(1), 137–158 (2006)CrossRefGoogle Scholar
  18. 18.
    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)CrossRefGoogle Scholar
  19. 19.
    Wu, Z., Reynolds, A.C., Oliver, D.S.: Conditioning geostatistical models to two-phase production data. SPE J. 4(2), 144–152 (1999)Google Scholar
  20. 20.
    Rodrigues, J.R.P.: Calculating derivatives for automatic history matching. Comput. Geosci. 10(1), 119–136 (2006)CrossRefGoogle Scholar
  21. 21.
    Tavakoli, R., Reynolds, A.C.: History matching with parameterization based on the SVD of a dimensionless sensitivity matrix. SPE J. 15(2), 495–508 (2010)Google Scholar
  22. 22.
    Nijmeijer, H., Van der Schaft, A.: Nonlinear Dynamical Control Systems. Springer, New York (1996)Google Scholar
  23. 23.
    Hermann, R., Krener, A.J.: Nonlinear controllability and observability. IEEE Trans. Autom. Control 22(5), 728 (1997)CrossRefGoogle Scholar
  24. 24.
    Isidori, A.: Nonlinear Control Systems. Springer, New York (1995)CrossRefGoogle Scholar
  25. 25.
    Lall, S., Marsden, J.E., Glavaski, S.A.: A subspace approach to balanced truncation for model reduction of nonlinear control systems. Int. J. Robust Nonlinear Control 12(5), 519–535 (2002)CrossRefGoogle Scholar
  26. 26.
    Hahn, J., Edgar, T.F., Marquardt, W.: Controllability and observability covariance matrices for the analysis and order reduction of stable nonlinear systems. J. Process Control 13(2), 115–127 (2003)CrossRefGoogle Scholar
  27. 27.
    Kailath, T.: Linear Systems. Prentice-Hall, Englewood Cliffs (1980)Google Scholar
  28. 28.
    Willcox, K., Peraire, J.: Balanced model reduction via the proper orthogonal decomposition. J. Am. Inst. Aeronaut. Astronaut. 40(11), 2323–2330 (2002)CrossRefGoogle Scholar
  29. 29.
    Rowley, C.W.: Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifurcation Chaos 15(3), 997–1013 (2005)CrossRefGoogle Scholar
  30. 30.
    Moore, B.C.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Automat. Control 26(1), 17–32 (1981)CrossRefGoogle Scholar
  31. 31.
    Antoulas, A.C.: Approximation of Large-scale Dynamical Systems. SIAM, Philadelphia (2005)CrossRefGoogle Scholar
  32. 32.
    Verriest, E.I., Kailath, T.: On generalized balanced realizations. IEEE Trans. Automat. Control 28(8), 833–844 (1983)CrossRefGoogle Scholar
  33. 33.
    Van Doren, J.F.M.: Model structure analysis for model-based operation of petroleum reservoirs. PhD thesis, Delft University of Technology (2010)Google Scholar
  34. 34.
    Jansen, J.D.: A systems description of flow through porous media. Springer Briefs in Earth Sciences (2013)Google Scholar
  35. 35.
    Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Science Publishers, London (1979)Google Scholar
  36. 36.
    Astrom, K.J., Wittenmark, B., 2nd ed.: Computer Controlled Systems. Prentice Hall, Englewood Cliffs (1990)Google Scholar

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Jorn F. M. Van Doren
    • 1
  • Paul M. J. Van den Hof
    • 2
  • Okko H. Bosgra
    • 1
  • Jan Dirk Jansen
    • 3
  1. 1.Delft Center for Systems and ControlDelft University of TechnologyDelftThe Netherlands
  2. 2.Department of Electrical EngineeringEindhoven University of TechnologyEindhovenThe Netherlands
  3. 3.Department of Geoscience and EngineeringDelft University of TechnologyDelftThe Netherlands

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