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.
Similar content being viewed by others
References
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)
Arbogast, T.: Analysis of the simulation of single phase flow through a naturally fractured reservoir. SIAM J. Math. Anal. 26, 12–29 (1989)
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)
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)
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)
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)
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)
Diaconis, P.: The Markov chain Monte Carlo revolution. Bull. Am. Math. Soc. 46, 179–205 (2009)
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)
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)
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
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)
Gamerman, D.: Markov Chain Monte Carlo. Stochastic simulation for Bayesian inference. Chapman & Hall, Boca Raton (1997)
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)
Greenbaum, A.: Iterative Methods for Solving Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia (1997)
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)
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
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)
Kazemi, H., Merrill, L., Porterfield, K., Zeman, P.: Numerical simulation of water-oil flow in naturally fractured reservoirs. SPE 5719, 317–326 (1976)
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)
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)
Le Maître, O., Knio, O.: Spectral methods for uncertainty quantification. With applications to computational fluid dynamics. Springer, Dordrecht (2010)
LeVeque, R.: Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge (2002)
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
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)
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)
Robert, C., Casella, G.: Monte Carlo Statistical Methods. Springer, New York (1999)
Stuart, A., Voss, J., Wiberg., P.: Conditional path sampling of SDEs and the langevin MCMC method. Commun. Math. Sci. 2(4), 685–697 (2004)
Warren, J., Root, P.: The behavior of naturally fractured reservoirs. SPE 426, 245–255 (1963)
Wong, E.: Stochastic processes in information and dynamical systems. McGraw-Hill, New York (1971)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ginting, V., Pereira, F., Presho, M. et al. Application of the two-stage Markov chain Monte Carlo method for characterization of fractured reservoirs using a surrogate flow model. Comput Geosci 15, 691–707 (2011). https://doi.org/10.1007/s10596-011-9236-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-011-9236-4