Computational Geosciences

, 15:691

Application of the two-stage Markov chain Monte Carlo method for characterization of fractured reservoirs using a surrogate flow model

  • Victor Ginting
  • Felipe Pereira
  • Michael Presho
  • Shaochang Wo
Original Paper

Abstract

In this paper, we develop a procedure for subsurface characterization of a fractured porous medium. The characterization involves sampling from a representation of a fracture’s permeability that has been suitably adjusted to the dynamic tracer cut measurement data. We propose to use a type of dual-porosity, dual-permeability model for tracer flow. This model is built into the Markov chain Monte Carlo (MCMC) method in which the permeability is sampled. The Bayesian statistical framework is used to set the acceptance criteria of these samples and is enforced through sampling from the posterior distribution of the permeability fields conditioned to dynamic tracer cut data. In order to get a sample from the distribution, we must solve a series of problems which requires a fine-scale solution of the dual model. As direct MCMC is a costly method with the possibility of a low acceptance rate, we introduce a two-stage MCMC alternative which requires a suitable coarse-scale solution method of the dual model. With this filtering process, we are able to decrease our computational time as well as increase the proposal acceptance rate. A number of numerical examples are presented to illustrate the performance of the method.

Keywords

Dual porosity Dual permeability Markov chain Monte Carlo method 

References

  1. 1.
    Al-Kobaisi, M., Kazemi, J., Ramirez, B., Ozkan, E., Atan, S.: A critical review for proper use of water/oil/gas transfer functions in dual-porosity naturally fractured reservoirs: part II. SPE 124213, 211–217 (2009)Google Scholar
  2. 2.
    Arbogast, T.: Analysis of the simulation of single phase flow through a naturally fractured reservoir. SIAM J. Math. Anal. 26, 12–29 (1989)MathSciNetMATHGoogle Scholar
  3. 3.
    Arbogast, T., Douglas, J., Hornung, U.: Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal. 21(4), 823–836 (1990)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Balogun, A., Kazemi, H., Ozkan, E., Al-Kobaisi, M., Ramirez, B.: Verification and proper use of water-oil transfer function for dual-porosity and dual-permeability reservoirs. SPE Reserv. Evalu. Eng. 104580, 189–199 (2009)Google Scholar
  5. 5.
    Barenblatt, G., Zheltov, I., Kochina, I.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]. SMM 24(5), 852–864 (1960)Google Scholar
  6. 6.
    Chen, Z., Douglas, J.: Modelling of compositional flow in naturally fractured reservoirs. In: IMA Volumes in Mathematics and its Applications, vol. 79, pp. 65–96. Springer, New York (1996)Google Scholar
  7. 7.
    Choi, E., Cheema, T., Islam, M.: A new dual-porosity/dual-permeability model with non-Darcian flow through fractures. J. Petrol. Sci. Eng. 17, 331–344 (1977)CrossRefGoogle Scholar
  8. 8.
    Diaconis, P.: The Markov chain Monte Carlo revolution. Bull. Am. Math. Soc. 46, 179–205 (2009)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Ding, Y., Li, T., Zhang, D., Zhang, P.: Adaptive stroud stochastic collocation method for flow in random porous media via Karhunen–Loève expansion. Commun. Comput. Phys. 4(1), 102–123 (2008)MathSciNetGoogle Scholar
  10. 10.
    Douglas, J., Arbogast, T.: Dual-porosity models for flow in naturally fractured reservoirs. In: Dynamics of Fluids in Heirarchical Porous Media, pp. 177–221. Academic, London (1990)Google Scholar
  11. 11.
    Efendiev, Y., Datta-Gupta, A., Ginting, V., Ma, X., Mallick, B.: An efficient two-stage Markov chain Monte Carlo method for dynamic data integration. Water Resour. Res. 41, W12423 (2005). doi:10.1029/2004WR003764 CrossRefGoogle Scholar
  12. 12.
    Efendiev, Y., Hou, T., Luo, W.: Preconditioning Markov chain Monte Carlo simulations using coarse-scale models. SIAM J. Sci. Comput. 28(2), 776–803 (2006)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Gamerman, D.: Markov Chain Monte Carlo. Stochastic simulation for Bayesian inference. Chapman & Hall, Boca Raton (1997)MATHGoogle Scholar
  14. 14.
    Granet, S., Fabrie, P., Lemonnier P., Quintard M.: A two-phase flow simulation of a fractured reservoir using a new fissure element method. J. Petrol. Sci. Eng. 32(1), 35–52 (2001)CrossRefGoogle Scholar
  15. 15.
    Greenbaum, A.: Iterative Methods for Solving Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia (1997)MATHGoogle Scholar
  16. 16.
    Guo, G., George, S., Lindsey, R.: Statistical analysis of surface lineaments and fractures for characterizing naturally fractured reservoirs. In: Schatzinger, R., Jordan, J. (eds.) Reservoir characterization—recent Advances, AAPG Memoir, vol. 71, pp. 221–250 (1999)Google Scholar
  17. 17.
    Hoteit, H., Firoozabadi, A.: Multicomponent fluid flow by discontinuous Galerkin and mixed methods in unfractured and fractured media. Water Resour. Res. 41, W11412 (2005). doi:10.1029/2005WR004339 CrossRefGoogle Scholar
  18. 18.
    Hou, T., Wu, X.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134, 169–189 (1997)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Kazemi, H., Merrill, L., Porterfield, K., Zeman, P.: Numerical simulation of water-oil flow in naturally fractured reservoirs. SPE 5719, 317–326 (1976)Google Scholar
  20. 20.
    Kirby, M., Sirovich, L.: Application of the Karhunen–Loève procedure for the characterization of human faces. IEEE T. Pattern Anal. 12(1), 103–108 (1990)CrossRefGoogle Scholar
  21. 21.
    Lange, A., Bouzian, J., Bourbiaux, B.: Tracer-test simulation on discrete fracture network models for the characterization of fractured reservoirs. SPE 94344, 1–10 (2005)Google Scholar
  22. 22.
    Le Maître, O., Knio, O.: Spectral methods for uncertainty quantification. With applications to computational fluid dynamics. Springer, Dordrecht (2010)MATHCrossRefGoogle Scholar
  23. 23.
    LeVeque, R.: Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge (2002)MATHCrossRefGoogle Scholar
  24. 24.
    Monteagudo, J., Firoozabadi, A.: Control-volume method for numerical simulation of two-phase immiscible flow in two- and three-dimensional discrete-fracture media. Water Resour. Res. 40, W07405 (2004). doi:10.1029/2003WR002996 CrossRefGoogle Scholar
  25. 25.
    Ramirez, B., Kazemi, H., Al-Kobaisi, M., Ozkan, M., Atan, S.: A critical review for proper use of water/oil/gas transfer functions in dual-porosity naturally fractured reservoirs: part I. SPE 109821, 200–210 (2009)Google Scholar
  26. 26.
    Refunjol, B., Lake, L.: Reservoir characterization based on tracer response and rank analysis of production and injection rates. In: Schatzinger, R., Jordan, J. (eds.) Reservoir characterization—recent advances, AAPG Memoir, vol. 71, pp. 209–218 (1999)Google Scholar
  27. 27.
    Robert, C., Casella, G.: Monte Carlo Statistical Methods. Springer, New York (1999)MATHGoogle Scholar
  28. 28.
    Stuart, A., Voss, J., Wiberg., P.: Conditional path sampling of SDEs and the langevin MCMC method. Commun. Math. Sci. 2(4), 685–697 (2004)MathSciNetMATHGoogle Scholar
  29. 29.
    Warren, J., Root, P.: The behavior of naturally fractured reservoirs. SPE 426, 245–255 (1963)Google Scholar
  30. 30.
    Wong, E.: Stochastic processes in information and dynamical systems. McGraw-Hill, New York (1971)MATHGoogle Scholar
  31. 31.
    Zhang, D., Lu, Z.: An efficient, high-order perturbation approach for flow in random porous media via Karhunen–Loève and polynomial expansions. J. Comput. Phys. 194, 773–794 (2004)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Victor Ginting
    • 1
  • Felipe Pereira
    • 2
  • Michael Presho
    • 3
  • Shaochang Wo
    • 4
  1. 1.Department of MathematicsUniversity of WyomingLaramieUSA
  2. 2.Department of Mathematics and School of Energy ResourcesUniversity of WyomingLaramieUSA
  3. 3.Department of MathematicsColorado State UniversityFt. CollinsUSA
  4. 4.Enhanced Oil Recovery InstituteUniversity of WyomingLaramieUSA

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