Abstract
Macroscopic transport properties of natural porous media, such as the permeability tensor \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{K} }\), are the sum of uncountable microscopic events. Understanding how these microscopic events come together to yield a descriptive property, such as \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{K} }\), is facilitated if a set of porous descriptors P i i=1,N can be measured that synthesize all the critical microscopic properties of a real porous medium and that can be related to the computed \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{K} }\). The main difficulty lies in choosing the correct P i i=1,N. A description of the microgeometry of a porous medium by a set of P i i=1,N will be declared adequate if synthetic numerical porous media generated from the P i i=1,N possess the same \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{K} }\) as the real medium. This paper is another step for creating such synthetic porous media. The candidate geometrical descriptor P 1 considered herein is the autocorrelation function (ACF) measured directly on a sample of finite size v included in a porous medium of size V; V ≫ v. Capitalizing on the phase retrieval problem (retrieving an object from knowledge of its Fourier modulus) encountered in general imaging, it is shown that there exists a one-to-one relation between a digital thin section and its ACF. This is demonstrated using an iterative procedure, the Error Reduction/Hybrid Input Output algorithm, that allows one to recover uniquely, to within a pixel, a finite image from its ACF. A theoretical implication of this is that a direct measurement on a finite image cannot characterize the geometry of a porous medium. Yet, in stochastic modeling, quasi-infinite numerical porous media are commonly generated from acquired ACF. Such quasi-infinite stochastic porous media must include a structural noise as a practical consequence of the unicity of the relation between an image and its ACF. To correctly interpret the relation between \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{K} }\) and the microgeometry, it becomes necessary to verify that this nonimposed structural noise does not control the output \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{K} }\) of the numerical simulations.
Similar content being viewed by others
References
Adler, P. M., 1992, Porous media: Butterworth, Heinemann, London, 544p.
Adler, P. M., Jacquin, C. G., and Quiblier, J. A., 1990, Flow in simulated porous media: Int. J. Multiphase Flow, v.16, no.4, p. 691–712.
Anderson, T. B., and Jackson, R., 1967, A fluid mechanical description of fluidized beds: Ind. Eng. Chem. Fundam., v.6, p. 527–538.
Anguy, Y., Bernard, D., and Ehrlich, R., 1994, The local change of scale method for modeling flow in natural porous media (I): Numerical tools: Adv. Water Resour., v.17, p. 337–351.
Anguy, Y., Bernard, D. and Ehrlich, R., 1996, Towards realistic flow modeling. Creation and evaluation of two-dimensional simulated porous media: An image analysis approach: Surv. Geophys., v.17, p. 365–287.
Anguy, Y., Ehrlich, R., Prince, C. M., Riggert, V., and Bernard, D., 1994, The sample support problem for permeability assessment in sandstone reservoirs, in Yarus, J. M. and Chambers, R. L., eds., Stochastic modeling and geostatistics: Am. Assoc. Petroleum Geologists Publication, Tulsa, NE, p. 37–54.
Bachmat, Y., and Bear, J., 1986, Macroscopic modeling of transport phenomena in porous media 1. The continuum approach: Trans. Porous Media, v.1, p. 213–240.
Bakke, S., and Oren, P.-E., 1997, 3-D pore-scale modeling of sandstones and flow simulations in the pore networks: SPE J., v.2, p. 136–149.
Barrére, J., Gipouloux, O., and Whitaker, S., 1992, On the closure problem for Darcy's law: Transp. Porous Media, v.7, p. 209–222.
Barrett, J. F., and Coales, J. F., 1955, An introduction to the analysis of non-linear control systems with random inputs: Proc. IEEE Monogr. 154 M, p. 190–199.
Bear, J., 1972, Dynamics of fluids in porous media: Elsevier, New York, 796p.
Bensoussan, A., Lions, J. L., and Papanicolaou, G., 1978, Asymptotic analysis for periodic structures: North Holland, New York.
Berg, R. R., 1975, Capillary pressure in stratigraphic traps: Am. Assoc. Petroleum Geol. Bull., v.59, no.6, p. 939–956.
Brenner, H., 1980, Dispersion resulting from flow through spatially porous media: Trans. R. Soc. (Lond.), v.297, p. 81–133.
Brigham, E. O., 1974, The fast Fourier transform: Prentice-Hall, Englewood Cliffs, NJ, 252p.
Broste, N. A., 1970, Digital generation of random sequences with specified autocorrelation and probability density functions: Report no. RE-TR–70-5, Advanced Sensors Laboratory, Research and Engineering Directorate, U.S. Army Missile Command, Redstone Arsenal, AL.
Bryant, S. L., King, P. R., and Mellor, D. W., 1993, Network model evaluation of permeability and spatial correlation in a real random sphere packing: Transp. Porous Media, v.11, 53–70.
Carbonell, R. G., and Whitaker, S., 1984, Heat and mass transfer in porous media, in Bear, J., and Corapcioglu, M. Y., eds., Fundamentals of transport phenomena in porous media: Martinus Nijhoff, Dordrecht, The Netherlands, p. 121–198.
Cushman, J. H., 1990, An introduction to hierarchical porous media, in Cushman, J. H., ed., Dynamics of fluids in hierarchical porous media: Academic Press, New York, p. 1–6.
Dagan, G., 1979, The generalization of Darcy's law for non uniform flows: Water Resour. Res., v.15, p. 1–7.
Dagan, G., 1990, Flow and transport in porous formations: Springer-Verlag, New York, 465p.
Dullien, F. A. L., 1979, Porous media—Fluid transport and pore structure: Academic Press, New York, 396p.
Ehrlich, R., Crabtree, S. J., Horkowitz, K. O., and Horkowitz, J. P., 1991a, Petrography and reservoir physics I: Objective classification of reservoir porosity: Am. Assoc. Petroleum. Geol. Bull., v.75, no.10, p. 1547–1562.
Ehrlich, R., Etris, E. L., Brumfield, D., Yan, L. P. and Crabtree, S. J., 1991b, Petrography and reservoir physics III: Physical models for pemeability and formation factor: Am. Assoc. Petroleum Geol. Bull., v.75, no.10, p. 1579–1592.
Ferreol, B., and Rothman, D. H., 1995, Lattice–Boltzmann simulations of flow through Fontainebleau sandstone: Transp. Porous Media, v.20, p. 3–20.
Fienup, J. R., 1982, Phase retrieval algorithms: A comparison: Appl. Opt., v.21, p. 2758–2769.
Fienup, J. R., and Wackerman, C. C., 1986, Phase retrieval stagnation problems and solutions: J. Opt. Soc. Am. A, v.3, p. 1897–1907.
Garboczi, E. J., 1990, Permeability, diffusivity, and micro-structural parameters: A critical review: Cement Concrete Res., v.20, p. 591–601.
Gelhar, L. W., 1984, Stochastic analysis of flow in heterogeneous porous media, in Bear, J., and Corapcioglu, M. Y., eds., Fundamentals of transport phenomena in porous media: Martinus Nijhoff, Dordrecht, The Netherlands, p. 615–717.
Gerchberg, R. W., and Saxton, W. O., 1972, A practical algorithm for the determination of phase from image and diffraction plane pictures: Optik, v.35, 237–246.
Giona, M. and Adrover, A., 1996, Closed-form solution for the reconstruction problem in porous media: AIChE J., v.42, no.5, p. 1407–1415.
Graton, L. C., and Fraser, H. J., 1935, Systematic packing of spheres with particular relation to porosity and permeability: J. Geol., v.43, p. 785–909.
Gray, W. G., 1975, A derivation of the equations for multiphase transport: Chem. Eng. Sci., v.30, p. 229–233.
Gujar, U. G., and Kavanagh, R. J., 1968, Generation of random signals with specified probability density function and power density spectra: IEEE Trans. Automatic Control, p. 716–719.
Holiday, E. M., 1969, Transformation of a set of pseudo-random numbers into a set representing any desired probability and correlation: Report no. RE-TR–69-25, Advanced Sensors Laboratory, Research and Engineering Directorate, U.S. Army Missile Command, Redstone Arsenal, AL.
Ionnadis, M. A., Kwiecien M. J., and Chatzis, I., 1997, Electrical conductivity and percolation aspects of statistically homogeneous porous media: Transp. Porous Media, v.29, p. 61–83.
Ionnadis, M. A., and Lange, E., 1998, Micro-geometry and topology of statistically homogeneous porous media, in Burganos, V. N., Karatzas, G. P., Payatakes, A. C., Brebbia, C. A., Gray, W. G., and Pinder, G. F., eds., Proceedings of computational methods in water resources XII, v.1: Computational Mechanics Southampton, UK, p. 223–230.
Joshi, M. Y., 1974, A class of stochastic models for porous pedia: PhD Dissertation, University of Kansas, 154p.
Kim, W., and Hayes, M. H., 1990, Phase retrieval using two Fourier-transform intensities: J. Opt. Soc. Am. A, v.7, no.3, p. 441–449.
Kwiecien, M. J., Macdonald, I. F. and Dullien, F. A. L., 1990, Three-dimensional reconstruction of porous media from serial section data: J. Microsc., v.159, no.3, p. 343–359.
Lane, R. G., 1987, Recovery of complex images from Fourier magnitudes: Optics commun., v.63, no.1, p. 6–10.
Marle, C. M., 1967, Ecoulements monophasiques en milieux poreux: Revue Institut Français du Pétrole, v.22, p. 1471–1509.
Matheron, G., 1965, Les variables régionalisées et leur estimation: Masson & Cie, Paris, 305p.
Matheron, G., 1970, La théorie des variables régionalisées, et ses applications: Les Cahiers du Centre de Morphologie de Fontainebleau, v.5, Ecole des Mines de Paris, eds., Paris, 212p.
McCreesh, C. A., Ehrlich, R. and Crabtree, S. J., 1991, Petrography and reservoir physics II: Relating thin section porosity to capillary pressure curves, the association between pore types and throat size: Am. Assoc. Petroleum Geol. Bull., v.75, no.10, p. 1563–1578.
Millane, R., 1990, Phase retrieval in crystallography and optics: J. Opt. Soc. Am. A, v.7, no.3, p. 394–411.
Nieto-Vesperinas, M., 1993, A Fortran routine to estimate a function of two variables form its autocorrelation: Comput. Phys. Commun., v.78, p. 211–217.
Nozad, I., Carbonell, R. G., and Whitaker, S., 1985, Heat conduction in multiphase systems—1: Theory and experiment for two-phase systems: Chem. Eng. Sci., v.40, no.5, p. 843–855.
Pérez-Ilzarbe, M. J., 1992, Phase retrieval from the power spectrum of a periodic objet: J. Opt. Soc. Am. A, v.9, no.12, p. 2138–2148.
Pérez-Ilzarbe, M. J., Nieto-Vesperinas, N., and Navarro, R., 1990, Phase retrieval from experimental far-field intensity data: J. Opt. Soc. Am. A, v.7, no.3, p. 434–440.
Pilotti, M., 1998, Generation of realistic porous media by grains sedimentation: Transp. Porous Media, v.33, p. 257–278.
Prat, M., 1990, Modelling of heat transfer by conduction in a transition region between a porous medium and an external fluid: Transp. Porous Media, v.5, p. 71–95.
Prince, C. M., and Ehrlich, R., 1990, Analysis of spatial order in sandstones I: Basic principles: Math. Geol., v.22, no.3, p. 333–359.
Prince, C. M., Ehrlich, R. and Anguy, Y., 1995, Analysis of spatial order in sandstones II: Grain clusters, packing flaws and the small-scale structure of sandstones: J. Sedim Res., v.A65, p. 13–28.
Quiblier, J. A., 1984, A new three-dimensional modeling technique for studying porous media: J. Colloid Interface Sci., v.98, no.1, p. 84–101.
Quintard, M., and Whitaker, S., 1987, Ecoulement monophasique en milieu poreux: effets des hétérogéités locales: J. de Mécanique Théorique et Appliquée, v.6, no.5, p. 691–726.
Quintard, M., and Whitaker, S., 1990, Two-phase flow in heterogeneous porous media I: The influence of large spatial and temporal gradients: Transp. Porous Media, v.5, p. 341–379.
Quintard, M., and Whitaker, S., 1993, Transport in ordered and disordered porous media: Volume-averaged equations, closure problems, and comparison with experiment: Chem. Eng. Sci., v.48, no.14, p. 2537–2564.
Quintard, M., and Whitaker, S., 1994a, Transport in ordered and disordered porous media I: The cellular average and the use of weighting functions: Transp. Porous Media, v.14, p. 163–177.
Quintard, M., and Whitaker, S., 1994b, Transport in ordered and disordered porous media II: Generalized volume-averaging: Transp. Porous Media, v.14, p. 179–206.
Quintard, M., and Whitaker, S., 1994c, Transport in ordered and disordered porous media III: Closure and comparison between theory and experiment: Transp. Porous Media, v.15, p. 31–49.
Quintard, M., and Whitaker, S., 1994d, Transport in ordered and disordered porous media IV: Computer generated porous media for three-dimensional systems: Transp. Porous Media, v.15, p. 51–70.
Quintard, M. and Whitaker, S., 1994e, Transport in ordered and disordered porous media V: Geometrical results for two-dimensional systems: Transp. Porous Media, v.15, p. 183–196.
Renard, P., and de Marsily, G., 1997, Calculating equivalent permeability: A review: Adv. Water Resour., v.20, no.5–6, p. 253–278.
Roddier, F., 1985, Distributions et transformation de Fourier: McGraw-Hill, Paris, 323p.
Rubinstein, J., and Torquato, S., 1989, Flow in random porous media: Mathematical formulation; variational principles and rigorous bounds: J. Fluid Mech., v.206, p. 3–25.
Sallés, J., Thovert, J. F., and Adler, P. M., 1994, Transport in reconstructed porous media, in Rouquerol, J., and others, eds., Studies in surface science and catalysis, v.87: Elsevier, Amsterdam, p. 211.
Sanchez-Palencia, E., 1980, Non homogeneous media and vibrational theory: Springer-Verlag, Berlin.
Sasaki, O., and Yamagami, T., 1987, Phase retrieval algorithms for nonnegative and finite-extent objects: J. Opt. Soc. Am. A., v.4, no.4, p. 720–726.
Sault, R. J., 1984, Two procedures for phase estimation form visibility magnitudes: Aust. J. Phys., v.37, p. 209–229.
Saxton, W. O., 1978, computer techniques for image processing in electron microscopy: Advances in Electronics and Electron Physics, Suppl. v.10, Academic Press, New York.
Schultz, T. J., and Snyder, D. L., 1992, Image recovery from correlations: J. Opt. Soc. Am. A., v.9, no.8, p. 1266–1272.
Slattery, J. C., 1967, Flow of viscoelastic fluids through porous media: AIChE, v.13, p. 1066–1077.
Tartar, L., 1980, Incompressible fluid flow in a porous medium: Convergence of the homogeneization process, in Lecture notes in physics 127, Appendix 2: Springer-Verlag, New York.
Thovert, J. F., Sallés, J., and Adler, P. M., 1992, Computerized characterization of the geometry of real porous media: Their discretization, analysis and interpretation: J. Microsc., v.170, p. 65–79.
van Brakel, J., 1975, Pore space models for transport phenomena in porous media. Review and evaluation with special emphasis to capillary liquid transport: Powder Technol., v.11, p. s205–236.
Ventzel, H., 1973, Théorie des probabilités: MIR, Moscou, 584p.
Weathcraft, S. W., Sharp, G. A., and Tyler, S. W., 1990, Fluid flow and solute transport in fractal heterogeneous porous media, in Cushman, J. H., ed., Dynamics of fluids in hierarchical porous media: Academic Press, San Diego, CA, p. 305–326.
Whitaker, S., 1967, Diffusion and dispersion in porous media: AIChE, v.13, p. 420–427.
Whitaker, S., 1969, Advances in the theory of fluid motion in porous media: Ind. Eng. Chem, v.61, p. 14–28.
Whitaker, S., 1986, Flow in porous media I: A theoretical derivation of Darcy's law: Transp. Porous Media, v.1, p. 3–25.
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., v.146, p. 478.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Anguy, Y., Ehrlich, R. & Mercet, C. Is It Possible to Characterize the Geometry of a Real Porous Medium by a Direct Measurement on a Finite Section? 1: The Phase-Retrieval Problem. Mathematical Geology 35, 763–788 (2003). https://doi.org/10.1023/B:MATG.0000007778.94825.ed
Issue Date:
DOI: https://doi.org/10.1023/B:MATG.0000007778.94825.ed