Abstract
Over the past few years, more and more systems and control concepts have been applied in reservoir engineering, such as optimal control, Kalman filtering, and model reduction. The success of these applications is determined by the controllability, observability, and identifiability properties of the reservoir at hand. The first contribution of this paper is to analyze and interpret the controllability and observability of single-phase flow reservoir models and to investigate how these are affected by well locations, heterogeneity, and fluid properties. The second contribution of this paper is to show how to compute an upper bound on the number of identifiable parameters when history matching production data and to present a new method to regularize the history matching problem using a reservoir’s controllability and observability properties.
References
Abacioglu, Y., Oliver, D., Reynolds, A.: Efficient reservoir history matching using subspace vectors. Comput. Geosci. 5, 151–172 (2001)
Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia (2005)
Astrom, K.J., Wittenmark, B.: Computer Controlled Systems, 2nd edn. Prentice Hall, Englewood Cliffs (1990)
Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Science, Barking (1979)
Bellman, R., Astrom, K.J.: On structural identifiability. Math. Biosci. 7, 329–339 (1970)
Berre, I., Lien, M., Mannseth, T.: A level-set corrector to an adaptive multiscale permeability prediction. Comput. Geosci. 11, 27–42 (2007)
Bissell, R.C.: Calculating optimal parameters for history matching. In: 4th European Conference on the Mathematics of Oil Recovery, Roros, 7–10 June 1994
Bissell, R.C., Sharma, Y., Killough, J.E.: History matching using the method of gradients: two case studies. In: SPE Annual Conference and Exhibition (SPE 28590-MS), New Orleans, 25–28 September 1994
Brun, B., Gosselin, O., Barker, J.W.: Use of prior information in gradient-based history matching. SPE J. (SPE 87680-PA) 9(1), 67–78 (2004)
Carter, R.D., Kemp, L.F., Pierce, A.C., Williams, D.L.: Performance matching with constraints. SPE J. (SPE 4260) 14, 187–196 (1974)
Chavent, G.: History matching by use of optimal theory. SPE J. (SPE 4627) 1, 74–86 (1975)
Chen, C.T.: Linear System Theory and Design, 2nd edn. Holt, Rinehart and Winston, New York (1984)
Denham, M.J.: Canonical forms for the identification of multiviarable linear systems. IEEE Trans. Automat. Contr. 19(6), 646–656 (1974)
Gavalas, G.R., Shah, P.C., Seinfeld, J.H.: Reservoir history matching by Bayesian estimation. SPE J. (SPE 5740-PA) 16(6), 337–350 (1976)
Gildin, E., Klie, H., Rodriguez, A., Wheeler, M.F.: Development of low-order controllers for high-order reservoir models and smart wells. In: SPE Annual Techical Conference and Exhibition (SPE 102214-MS), San Antonio, 24–27 September 2006
Glover, K.: All optimal hankel-norm approximations of linear multivariable systems and their \(\mathcal{L}^{\infty}\text{-error}\) bounds. Int. J. Control 39(6), 1115–1193 (1984)
Glover, K., Willems, J.C.: Parameterizations of linear dynamical systems: Canonical forms and identifiability. IEEE Trans. Automat. Contr. 19(6), 640–646 (1974)
Grimstad, A., Mannseth, T., Naevdal, G.: Adaptive multiscale permeability estimation. Comput. Geosci. 7(1), 1–25 (2003)
Heijn, T., Markovinovic, R., Jansen, J.D.: Generation of low-order reservoir models using system-theoretical concepts. SPE J. (SPE 88361) 9(2), 202–218 (2004)
Jacquard, P., Jain, C.: Permeability distribution from field pressure data. SPE J. (SPE 1307) 5, 281–294, (1965)
Jafarpour, B., McLaughlin, D.B.: Efficient permeability parameterization with the discrete cosine transform. In: SPE Reservoir Simulation Symposium (SPE 106453), Houston, February 2007
Jafarpour, B., McLaughlin, D.B.: History matching with an ensemble Kalman filter and discrete cosine parameterization. In: SPE Annual Technical Conference and Exhibition (SPE 108761), Anaheim, November 2007
Jahns, H.O.: A rapid method for obtaining a two-dimensional reservoir description from well pressure response data. SPE J. (SPE 1473-PA) 6(4), 315–327 (1966)
Kalman, R.E.: Mathematical description of linear dynamical systems. SIAM J. Control 1(2), 152–192 (1963)
Li, R., Reynolds, A.C., Oliver, D.S.: History matching of three-phase flow production data. SPE J. (SPE 87336) 8(4), 328–340 (2003)
Ljung, L.: System Identification—Theory for the User, 2nd edn. Prentice Hall, Englewood Cliffs (1999)
Luenberger, D.G.: Canonical forms for linear multivariable systems. IEEE Trans. Automat. Contr. 12, 290–293 (1967)
Markovinovic, R., Geurtsen, E.L., Heijn, T., Jansen, J.D.: Generation of low-order reservoir models using POD, empirical Gramians and subspace identification. In: 8th European Conference on the Mathematics of Oil Recovery, Freiberg, 3–6 September 2002
Moore, B.C.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Automat. Contr. 26(1), 17–32 (1981)
Reynolds, A.C., He, N., Chu, L., Oliver, D.S.: Reparameterization techniques for generating reservoir descriptions conditioned to variograms and well-test pressure data. SPE J. (SPE 30588-PA) 1(4), 413–426 (1996)
Sarma, P., Durlofsky, L.J., Aziz, K., Chen, W.H.: A new approach to automatic history matching using kernel PCA. In: SPE Reservoir Simulation Symposium (SPE 106176), Houston, February 2007
Shah, P.C., Gavalas, G.R., Seinfeld, J.H.: Error analysis in history matching: the optimum level of parameterization. SPE J. (SPE 6508-PA) 18(3), 219–228 (1978)
Tarantola, A.: Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM, Philadelphia (2005)
Tavassoli, Z., Carter, J.N., King, P.R.: Errors in history matching. SPE J. (SPE 86883) 9(3), 352–361 (2004)
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: Proceedings of the 17th IFAC World Congress, Seoul, 6–11 July 2008
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Zandvliet, M.J., Van Doren, J.F.M., Bosgra, O.H. et al. Controllability, observability and identifiability in single-phase porous media flow. Comput Geosci 12, 605–622 (2008). https://doi.org/10.1007/s10596-008-9100-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-008-9100-3
Keywords
- Reservoir engineering
- Controllability
- Observability
- Identifiability