Gradual Deformation and Iterative Calibration of Gaussian-Related Stochastic Models
Tóm tắt
This paper describes a new method for gradually deforming realizations of Gaussian-related stochastic models while preserving their spatial variability. This method consists in building a stochastic process whose state space is the ensemble of the realizations of a spatial stochastic model. In particular, a stochastic process, built by combining independent Gaussian random functions, is proposed to perform the gradual deformation of realizations. Then, the gradual deformation algorithm is coupled with an optimization algorithm to calibrate realizations of stochastic models to nonlinear data. The method is applied to calibrate a continuous and a discrete synthetic permeability fields to well-test pressure data. The examples illustrate the efficiency of the proposed method. Furthermore, we present some extensions of this method (multidimensional gradual deformation, gradual deformation with respect to structural parameters, and local gradual deformation) that are useful in practice. Although the method described in this paper is operational only in the Gaussian framework (e.g., lognormal model, truncated Gaussian model, etc.), the idea of gradually deforming realizations through a stochastic process remains general and therefore promising even for calibrating non-Gaussian models.
Tài liệu tham khảo
Blanc, G., Guérillot, D., Rahon, D., and Roggero, F., 1996, Building geostatistical models constrained by dynamic data—a posteriori constraints, Paper SPE 35478: Proc. NPF/SPE European 3D Reservoir Modelling Conference, Stavanger, Norway, p. 19–33.
Deutsch, C., 1993, Conditioning reservoir models to well test information, in Soares, A., ed., Geostatistics Toria '92, vol. 1: Kluwer Acad. Publ., Dordrecht, The Netherlands, p. 505–518.
Feller, W., 1971, An introduction to probability theory and its applications, vol. II: John Wiley & Sons, New York, 669 p.
Fletcher, R., 1987, Practical methods of optimization: John Wiley & Sons, Chichester, 436 p.
Freulon, X., and de Fouquet, C., 1993, Conditioning a Gaussian model with inequalities, in Soares, A., ed., Geostatistics Toria'92, vol. 1: Kluwer Acad. Publ., Dordrecht, The Netherlands, p. 201–212.
Geman, S., and Geman, D., 1984, Stochastic relaxation, Gibbs distribution and Bayesian restoration of images: I.E.E.E. Transactions on Pattern Analysis and Machine Intelligence, no. 6, p. 721–741.
Hegstad, B. K., Omre, H., Tjelmeland, H., and Tyler, K., 1994, Stochastic simulation and conditioning by annealing in reservoir description, in Armstrong, M., and Dowd, P., eds., Geostatical simulations: Kluwer Acad. Publ., Dordrecht, The Netherlands, p. 43–55.
Hu, L.Y., and Blanc, G., 1988, Constraining a reservoir facies model to dynamic data using a gradual deformation method: Proc. 6th European Conference on Mathematics of Oil Recovery(ECMORVI), Peebles, Scotland.
Journel, A. G., and Huijbregts, C. J., 1978, Mining geostatistics: Academic Press, London, 600 p.
Lantuéjoul, C., 1994, Nonconditional simulation of stationary isotropic multigaussian random functions, in Armstrong, M., and Dowd, P., eds., Geostatistical simulations: Kluwer Acad. Publ., Dordrecht, The Netherlands, p. 147–177.
Lantuéjoul, C., and Rivoirard, J., 1984, Une méthode de détermination d'anamorphose, Note du Centre de Géostatistique, no. 916: Ecole des Mines de Paris, Fontainebleau, France.
Le Loc'h, G., and Galli, A., 1997, Truncated plurigaussian method: Theoretical and practical points of view, in Baafi, E. and others, eds., Geostatistics Wollongong 96, vol. 1: Kluwer Acad. Publ., Dordrecht, The Netherlands, p. 211–223.
Marsily de, G., Lavedan, G., Boucher, M., and Fasanino, G., 1984, Interpretation of interference tests in a well field using geostatistical techniques to fit the permeability distribution in a reservoir model, in Verly, G. and others, eds., Geostatistics for natural resources characterization, Part 2: D. Reidel Publ. Co., Dordrecht, The Netherlands, p. 831–849.
Matheron, G., 1971, The theory of regionalized variables and its applications, fascicule 5: Les Cahiers du Centre de Morphologie Mathématique, Ecole des Mines de Paris, Fontainebleau, France, 212 p.
Matheron, G., Beucher, H., de Fouquet, C., Galli, A., and Ravenne, C., 1987, Conditional simulation of the geometry of fluvio-deltaic reservoirs, Paper SPE 16753: Proc. SPE Annual Technical Conference and Exhibition, Las Vegas, NV.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E., 1953, Equation of state calculations by fast computing machines: J. Chem. Phys., v. 21, p. 1087–1091.
Oliver, D. S., 1995, Moving averages for Gaussian simulation in two and three dimensions: Math. Geology, v. 27, no. 8, p. 939–960.
Oliver, D. S., Cunha, L. B., and Reynolds, A. C., 1997, Markov chain Monte Carlo methods for conditioning a permeability field to pressure data: Math. Geology, v. 29, no. 1, p. 61–91.
RamaRao, B. S., LaVenue, A. M., Marsily de, G., and Marietta, M. G., 1995, Pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields, 1. Theory and computational experiments: Water Resour. Res., v. 31, no. 3, p. 475–493.
Roggero, F., and Hu, L. Y., 1998, Gradual deformation of continuous geostatistical models for history matching, Paper SPE 49004: Proc. SPE Annual Technical Conference and Exhibition, New Orleans, LA.
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, Paper SPE 24754: Proc. SPE Annual Technical Conference and Exhibition, Washington, DC.
Srivastava, R. M., 1994, The interactive visualization of spatial uncertainty, Paper SPE 27965: Proc. University of Tulsa Centennial Petroleum Engineering Symposium, Tulsa, OK.
Sun, N. Z., 1994, Inverse problems in groundwater modeling: Kluwer Acad. Publ., Dordrecht, The Netherlands, 337 p.
Tarantola, A., 1987, Inverse problem theory—Methods for data fitting and model parameter estimation: Elsevier, Amsterdam, 613 p.