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Markov chain Monte Carlo methods for conditioning a permeability field to pressure data

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Abstract

Generating one realization of a random permeability field that is consistent with observed pressure data and a known variogram model is not a difficult problem. If, however, one wants to investigate the uncertainty of reservior behavior, one must generate a large number of realizations and ensure that the distribution of realizations properly reflects the uncertainty in reservoir properties. The most widely used method for conditioning permeability fields to production data has been the method of simulated annealing, in which practitioners attempt to minimize the difference between the ’ ’true and simulated production data, and “true” and simulated variograms. Unfortunately, the meaning of the resulting realization is not clear and the method can be extremely slow. In this paper, we present an alternative approach to generating realizations that are conditional to pressure data, focusing on the distribution of realizations and on the efficiency of the method. Under certain conditions that can be verified easily, the Markov chain Monte Carlo method is known to produce states whose frequencies of appearance correspond to a given probability distribution, so we use this method to generate the realizations. To make the method more efficient, we perturb the states in such a way that the variogram is satisfied automatically and the pressure data are approximately matched at every step. These perturbations make use of sensitivity coefficients calculated from the reservoir simulator.

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References

  • Alabert, F., 1987, The practice of fast conditional simulations through the LU decomposition of the covariance matrix: Math. Geology, v. 19, no. 5, p. 369–386.

    Article  Google Scholar 

  • Chu, L., Reynolds, A. C., and Oliver, D. S., 1995, Computation of sensitivity coefficients for conditioning the permeability field to well-test pressure data: In Situ, v. 19, no. 2, p. 179–223.

    Google Scholar 

  • Davis, M., 1987a, Generating large stochastic simulations—the matrix polynomial approximation method: Math. Geology, v. 19, no. 2, p. 99–107.

    Google Scholar 

  • Davis, M., 1987b, Production of conditional simulations via the LU decomposition of the covariance matrix: Math. Geology, v. 19, no. 2, p. 91–98.

    Google Scholar 

  • Deutsch, C. V., 1992, Annealing techniques applied to reservoir modeling and the integration of geological and engineering (well test) data: unpubl. doctoral dissertation, thesis, Stanford University, 325 p.

  • Deutsch, C. V., and Journel, A. G., 1992, GSLIB: Geostatistical Software Library and user’s guide: Oxford Univ. Press, New York, 340 p.

    Google Scholar 

  • Dietrich, C. R., 1993, Computationally efficient generation of Gaussian conditional simulations over regular sample grids: Math. Geology, v. 25, no. 4, p. 439–451.

    Article  Google Scholar 

  • Dietrich, C. R., and Newsam, G. N., 1995, Efficient generation of conditional simulations by Chebyshev matrix polynomial approximations to the symmetric square root of the covariance matrix: Math. Geology, v. 27, no. 2, p. 207–228.

    Article  Google Scholar 

  • Farmer, C. L., 1992, Numerical rocks,in King, P. R., ed., Mathematics of oil recovery: Clarendon Press, Oxford, p. 437–447.

    Google Scholar 

  • Feller, W., 1968, An introduction to probability theory and its applications, v. I (3rd ed.): John Wiley & Sons, New York, 509 p.

    Google Scholar 

  • Geman, S., and Geman, D., 1984, Stochastic relaxation, Gibbs distributions, and Bayesian restoration of images: IEEE Trans. Pattern Analysis and Machine Intelligence, v. PAMI-6, no. 6, p. 721–741.

    Article  Google Scholar 

  • Gupta, A. D., 1992, Stochastic heterogeneity, dispersion, and field tracer response: unpubl. doctoral dissertation, Univ. Texas, Austin, 248 p.

    Google Scholar 

  • Hammersley, J. M., and Handscomb, D. C., 1964, Monte Carlo methods: John Wiley & Sons, New York, 178 p.

    Google Scholar 

  • Hastings, W. K., 1970, Monte Carlo sampling methods using Markov chains and their applications: Biometrika, v. 57, no. 1, p. 97–109.

    Article  Google Scholar 

  • Hegstad, B. K., Omre, H., Tjelmeland, H., and Tyler, K., 1993, Stochastic simulation and conditioning by annealing in reservoir description,in Armstrong, M., and Dowd, P. A., eds., Geostatistical simulation: Kluwer Acad. Publ., Dordrecht, The Netherlands, p. 43–55.

    Google Scholar 

  • Journel, A. G., and Huijbregts, C. J., 1978, Mining geostatistics: Academic Press, New York, 600 p.

    Google Scholar 

  • Koren, Z., Mosegaard, K., Landa, E., Thore, P., and Tarantula, A., 1991, Monte Carlo estimation and resolution analysis of seismic background velocities: Jour. Geophys. Res., v. 96, no. B12, p. 20,289–20,299.

    Google Scholar 

  • Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E., 1953, Equations of state calculations by fast computing machines: Jour. Chem. Physics, v. 21, no. 6, p. 1087–1092.

    Article  Google Scholar 

  • Neal, R. M., 1993, Probabilistic inference using Markov chain Monte Carlo methods: Tech. Rept. CRG-TR-93-1, Dept. Computer Science, Univ. Toronto, 140 p.

  • Oliver, D. S., 1994, The incorporation of transient pressure data into reservoir characterization: In Situ, v. 18, no. 3, p. 243–275.

    Google Scholar 

  • Oliver, D. S., 1995, Moving averages for Gaussian simulation in two and three dimensions: Math. Geology, v. 27, no. 8, p. 939–960.

    Article  Google Scholar 

  • Oliver, D. S., 1996, Multiple realizations of the permeability field from well test data: Soc. Petrol. Eng. Jour., v. 1, no. 2, p. 145–154.

    Google Scholar 

  • Ouenes, A., 1992, Application of simulated annealing to reservoir characterization and petrophysics inverse problems: unpubl. doctoral dissertation, New Mexico Tech., 205 p.

  • Ouenes, A., Bréfort, B., Muenier, G., and Dupéré, S., 1993, A new algorithm for automatic history matching: Application of simulated annealing method (SAM) to reservoir inverse modeling: unsolicited manuscript SPE 26297, 30 p.

  • Pérez, G., 1991, Stochastic conditional simulation for description of reservoir properties: unpubl. doctoral dissertation, Univ. Tulsa, 245 p.

  • Rao, C. R., 1965. Linear statistical inference and its applications. John Wiley & Sons, New York, 522 p.

    Google Scholar 

  • Ripley, B. D., 1981, Spatial statistics: John Wiley & Sons, New York, 252 p.

    Google Scholar 

  • Ripley, B. D., 1987, Stochastic simulation: John Wiley & Sons, New York, 237 p.

    Google Scholar 

  • Sagar, R. K., Kelkar, B. G., and Thompson, L. G., 1993, Reservoir description by integration of well test data and spatial statistics,in Proc. 68th Ann. Tech. Conf. Exhibition: Soc. Petrol. Eng., p. 475–489.

  • Scheuer, E. M., and Stoller, D. S., 1962, On the generation of normal random vectors: Technometrics, v. 4, no. 2, p. 278–281.

    Article  Google Scholar 

  • Sen, M. K., Gupta, A. D., Stoffa, P. L., Lake, L. W., and Pope, G. A., 1992, Stochastic reservoir modeling using simulated annealing and genetic algorithm,in Proc. 67th Ann. Tech. Conf. Exhibition: Soc. Petrol. Eng., p. 939–950.

  • Srivastava, R. M., 1994, The interactive visualization of spatial uncertainty,in Proc. Univ. Tulsa Centennial Petroleum Engineering Symposium: Soc. Petrol. Eng., p. 87–95.

  • Tarantola, A., and Valette, B., 1982, Inverse problems = quest for information: Jour. Geophysics, v. 50, no. 3, p. 159–170.

    Google Scholar 

  • Tierney, L., 1994, Markov chains for exploiting posterior distributions: Tech. Rept. 560 (revised), School of Statistics, Univ. Minnesota, 25 p.

  • Tjelmeland, H., Omre, H., and Hegstad, B. K., 1994, Sampling from Bayesian models in reservoir characterization: Tech. Rept. Statistics No. 2/1994, Univ. Trondheim, 15 p.

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Correspondence to Dean S. Oliver.

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Oliver, D.S., Cunha, L.B. & Reynolds, A.C. Markov chain Monte Carlo methods for conditioning a permeability field to pressure data. Math Geol 29, 61–91 (1997). https://doi.org/10.1007/BF02769620

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  • DOI: https://doi.org/10.1007/BF02769620

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