Abstract
A method of 3-D stochastic reconstruction of porous media based on statistical information extracted from 2-D sections is evaluated with reference to the steady transport of electric current. Model microstructures conforming to measured and simulated pore space autocorrelation functions are generated and the formation factor is systematically determined by random walk simulation as a function of porosity and correlation length. Computed formation factors are found to depend on correlation length only for small values of this parameter. This finding is explained by considering the general percolation behavior of a statistically homogeneous system. For porosities lower than about 0.2, the dependence of formation factor on porosity shows marked deviations from Archie's law. This behavior results from the relatively high pore space percolation threshold (∼0.09) of the simulated media and suggests a limitation to the applicability of the method to low porosity media. It is additionally demonstrated that the distribution of secondary porosity at a larger scale can be simulated using stochastic methods. Computations of the formation factor are performed for model media with a matrix-vuggy structure as a function of the amount and spatial distribution of vuggy porosity and matrix conductivity. These results are shown to be consistent with limited available experimental data for carbonate rocks.
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Adler, P. M., Jacquin, C. G. and Quiblier, J. A.: 1990, Flow in simulated porous media, Int. J. Multiphase Flow 16, 691-712.
Adler, P. M., Jacquin, C. G. and Thovert, J. F.: 1992, The formation factor of reconstructed porous media, Water Resour. Res. 28, 1571-1576.
Adler, P. M.: 1992, Porous Media: Geometry and Transports, Butterworth-Heinemann, Stoneham.
Archie, G. E.: 1942, Electrical resistivity log as an aid in determining some reservoir characteristics, Trans. AIME 146, 54-67.
Bentz, D. P. and Martys, N. S.: 1994, Hydraulic radius and transport in reconstructed model threedimensional porous media, Transport in Porous Media 17, 221-238.
Berryman, J. G.: 1985, Measurement of spatial correlation functions using image processing techniques, J. Appl. Phys. 57, 2374- 2384.
Borai, A. M.: 1987, A new correlation for the cementation factor in low-porosity carbonates, SPE Formation Evaluation 2, 495-499.
Bryant, S., Cade, C. and Mellor, D.: 1993, Permeability prediction from geologic models, AAPG Bull. 77, 1338-1350.
Carmichael, R. S.: 1982, Handbook of Physical Properies of Rocks, Vol. 1, Chemical Rubber Company, Boca Raton, FL.
Crossley, P. A., Schwartz, L. M. and Banavar, J. R.: 1991, Image-based models of porous media: application to Vycor glass and carbonate rocks, Appl. Phys. Lett. 59, 3553-3555.
Debye, P., Anderson, H. R. and Brumberger, H.: 1957, Scattering by an inhomogeneous solid. II. the correlation function and its application, J. Appl. Phys. 28, 679-683.
Doyen, P. M.: 1988, Permeability, conductivity and pore geometry of sandstone, J. Geophys. Res. 93, 7729-7740.
Dullien, F. A. L.: 1992, Porous Media: Fluid Transport and Pore Structure, 2nd edn, Academic Press, San Diego, CA.
Focke, J. W. and Munn, D.: 1987, Cementation exponents in middle eastern carbonate reservoirs, SPE Formation Evaluation 2, 155-167.
de Gennes, P. G.: 1976, La percolation: un concept unificateur, La Recherche 7, 919-927.
Gutjahr, A. L.: 1989, Fast Fourier transform for random field generation, Project Report, New Mexico Institute of Mining and Technology, Contact No. 4-R58-2690R.
Hearst, J. R. and Nelson, P. H.: 1985, Well Logging for Physical Properties, McGraw-Hill, New York.
Hilfer, R.: 1991, Geometric and dielectric characterization of porous media, Phys. Rev. B 44, 60-75.
Hoshen, J. and Kopelman, R.: 1976, Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm, Phys. Rev. B. 14, 3438-3445.
Ioannidis, M. A. and Chatzis, I.: 1993, Network modelling of pore structure and transport properties of porous media, Chem. Eng. Sci. 48, 951-972.
Ioannidis, M. A., Chatzis, I. and Sudicky, E. A.: 1993, The effect of spatial correlations on the accessibility characteristics of three-dimensional cubic networks as related to drainage displacements in porous media, Water Resour. Res. 29, 1777-1785.
Ioannidis, M. A., Kwiecien, M. J. and Chatzis, I.: 1995, Computer generation and application of 3-D model porous media: from pore-level geostatistics to the estimation of formation factor, SPE Petroleum Computer Conf., paper 30201, 11-14 June Houston, TX.
Ioannidis, M. A., Kwiecien, M. J. and Chatzis, I.: 1996, Statistical analysis of the porous microstructure as a method for estimating reservoir permeability, J. Petrol. Sci. Eng. 16, 251-261.
Joshi, M.: 1974, A class of stochastic models for porous media, PhD Dissertation, University of Kansas.
Journel, A. B. and Huijbregts, C. J.: 1978, Mining Geostatistics, Academic Press, San Diego, CA.
Kim, I. C. and Torquatto, S.: 1990, Determination of the effective conductivity of heterogeneous media by Brownian motion simulation, J. Appl. Phys. 68, 3892-3903.
Kong, T. Y.: 1989, Digital topology: introduction and survey, Comput. Vision Graphics Image Process. 48, 357-374.
Kwiecien, M. J.: 1994, On the comprehensive characterization of porous media via computer reconstruction and stochastic modeling, PhD Dissertation, University of Waterloo.
Lucia, F. J.: 1983, Petrophysical parameters estimated from visual descriptions of carbonate rocks, J. Petrol. Technol. 35, 629-637.
Martys, N. and Garboczi, E. J.: 1992, Length scales relating the fluid permeability and electrical conductivity in random two-dimensional model porous media, Phys. Rev. B 46, 6080.
Mendelson, K. S. and Cohen, M. H.: 1982, The effect of grain anisotropy on the electrical properties of sedimentary rocks, Geophysics 47, 257-263.
Quiblier, J. A.: 1984, A new three-dimensional modeling technique for studying porous media, J. Colloid Interface Sci. 98, 84-102.
Renault, P.: 1991, The effect of spatially correlated blocking-up of some bonds or nodes of a network on the percolation threshold, Transport in Porous Media 6, 451-468.
Roberts, J. N. and Schwartz, L. M.: 1985, Grain consolidation and electrical conductivity in porous media, Phys. Rev. B 31, 5990-5997.
Sahimi, M. and Stauffer, D.: 1991, Efficient simulation of flow and transport in porous media, Chem. Eng. Sci. 46, 2225-2233.
Schwartz, L.M. and Kimminau, S.: 1987, Analysis of electrical conduction in the grain consolidation model, Geophysics 52, 1402-1411.
Sen, P. N., Scala, C. and Cohen, M. H.: 1981, A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads, Geophysics 46, 781-795.
Spanne, P., Thovert, J. F., Jacquin, C. J., Lindquist, W. B., Jones, K. W. and Adler, P. M.: 1994, Synchrotron computed microtomography of porous media: topology and transports, Phys. Rev. Lett. 73, 2001-2004.
Stauffer, D.: 1985, Introduction to Percolation Theory, Taylor and Francis, London.
Thovert, J. F., Salles, J. M. and Adler, P. M.: 1993, Computerized characterization of the geometry of real porous media: their discretization, analysis and interpretation, J. Microscopy 170, 65-79.
Wong, P., Koplik, J. and Tomanic, J. P.: 1984, Conductivity and permeability of rocks, Phys. Rev. B 30, 6606-6614.
Yao, J., Frykman, P., Kalaydjian, F., Thovert, J. F. and Adler, P. M.: 1993, High-order moments of the phase function for real and reconstructed porous media: a comparison, J. Colloid Interface Sci. 156, 478-490.
Ziman, J. M.: 1979, Models of Disorder, Cambridge University Press, New York.
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Ioannidis, M.A., Kwiecien, M.J. & Chatzis, I. Electrical Conductivity and Percolation Aspects of Statistically Homogeneous Porous Media. Transport in Porous Media 29, 61–83 (1997). https://doi.org/10.1023/A:1006557614527
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DOI: https://doi.org/10.1023/A:1006557614527