Abstract
In this article, we present fundamental transport property relations incorporating direct descriptors of the pore structure. The pore structure descriptors are defined from streamline decomposition of the numerical solutions of the transport equations. These descriptors have been introduced earlier, while the calculations are extended to voxel-based microstructures in this article. The pore structure descriptors for the respective transport equations are used in turn to obtain rigorous cross-property relations for porous media. We derive such cross-property relations exemplarily for computed tomography (CT) data and digital rock models of Fontainebleau sandstone, and CT data of two reservoir sandstone facies. Pore structure parameterizations of these porous media are given for electrical conductance and fluid permeability in the microstructure, yielding correlations for the transport property-dependent descriptors of effective porosity, tortuosity and constriction. These relations are shown to be well-correlated functions over the range of sample porosities for the Fontainebleau sandstone. Differences between the outcrop Fontainebleau sandstone and the reservoir samples are observed mainly in the derived hydraulic length descriptor. A quantitative treatment of the obtained cross-property functions is provided that could be applied for porous medium characterization. It is suggested that such cross-property investigation honoring the detailed microstructure will lead to more fundamental relations between porous medium properties, which could be exploited for example in rock typing or wire-line log interpretation.
Similar content being viewed by others
Notes
e-Core software version 1.5.2 from FEI (www.fei.com).
References
Adler, P.: Porous Media: Geometry and Transports. Butterworth-Heinemann, Oxford (1992)
Andrä, H., Combaret, N., Dvorkin, J., Glatt, E., Han, J., Kabel, M., Keehm, Y., Krzikalla, F., Lee, M., Madonna, C., et al.: Digital rock physics benchmarks–part ii: computing effective properties. Computers & Geosciences 50, 33–43 (2013)
Archie, G.: The electrical resistivity log as an aid in determining some reservoir characteristics. Trans. AIMe 146(99), 54–62 (1942)
Arns, C.H., Knackstedt, M.A., Martys, N.S.: Cross-property correlations and permeability estimation in sandstone. Phys. Rev. E 72(4), 046,304 (2005)
Arns, C.H., Knackstedt, M.A., Pinczewski, M.V., Lindquist, W.: Accurate estimation of transport properties from microtomographic images. Geophys. Res. Lett. 28(17), 3361–3364 (2001)
Auzerais, F., Dunsmuir, J., Ferreol, B., Martys, N., Olson, J., Ramakrishnan, T., Rothman, D., Schwartz, L.: Transport in sandstone: a study based on three dimensional microtomography. Geophys. Res. Lett. 23(7), 705–708 (1996)
Avellaneda, M., Torquato, S.: Rigorous link between fluid permeability, electrical conductivity, and relaxation times for transport in porous media. Phys. Fluids A Fluid Dyn. 3, 2529 (1991)
Bear, J.: Dynamics of Fluids in Porous Media. Dover publications, New York (1988)
Bear, J., Bachmat, Y.: A generalized theory on hydrodynamic dispersion in porous media. In: IASH Symposium on Artificial Recharge and Management of Aquifers, vol. 72, pp. 7–16 (1967)
Berg, C.F.: Re-examining Archie’s law: conductance description by tortuosity and constriction. Phys. Rev. E 86, 046,314 (2012)
Berg, C.F.: Permeability description by characteristic length, tortuosity, constriction and porosity. Transp. Porous Media 103(3), 381–400 (2014)
Berryman, J.G., Milton, G.W.: Normalization constraint for variational bounds on fluid permeability. J. Chem. Phys. 83(2), 754–760 (1985)
Biswal, B., Manwart, C., Hilfer, R., Bakke, S., Øren, P.: Quantitative analysis of experimental and synthetic microstructures for sedimentary rock. Phys. A Stat. Mech. Appl. 273(3), 452–475 (1999)
Brinkman, H.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1(1), 27–34 (1949)
Carcione, J.M., Ursin, B., Nordskag, J.I.: Cross-property relations between electrical conductivity and the seismic velocity of rocks. Geophysics 72(5), E193–E204 (2007)
Carman, P.: Fluid flow through granular beds. Trans. Inst. Chem. Eng. 15, 150–166 (1937)
Coker, D.A., Torquato, S., Dunsmuir, J.H.: Morphology and physical properties of fontainebleau sandstone via a tomographic analysis. J. Geophys. Res. Solid Earth (1978–2012) 101(B8), 17,497–17,506 (1996)
Cushman, J.H., et al.: Dynamics of Fluids in Hierarchical Porous Media. Academic Press Inc.(London) Ltd., London (1990)
Duda, A., Koza, Z., Matyka, M.: Hydraulic tortuosity in arbitrary porous media flow. Phys. Rev. E 84(3), 036,319 (2011)
Dullien, F.: Porous Media: Fluid Transport and Pore Structure, vol. 26. Academic press, London (1992)
Dunn, K.J., LaTorraca, G.A., Bergman, D.J.: Permeability relation with other petrophysical parameters for periodic porous media. Geophysics 64(2), 470–478 (1999)
Fredrich, J.T., Lakshtanov, D., Lane, N., Liu, E.B., Natarajan, C., Ni, D.M., Toms, J., et al.: Digital rocks: developing an emerging technology through to a proven capability deployed in the business. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2014)
Gomari, K.A.R., Berg, C.F., Mock, A., Øren, P.E,. Petersen Jr., E., Rustad, A., Lopez, O.: Electrical and petrophysical properties of siliciclastic reservoir rocks from pore-scale modeling. In: SCA2011-20 Presented at the 2011 SCA International Symposium, Austin, Texas
Gomez, C.T., Dvorkin, J., Vanorio, T.: Laboratory measurements of porosity, permeability, resistivity, and velocity on fontainebleau sandstones. Geophysics 75(6), E191–E204 (2010)
Hasimoto, H.: On the periodic fundamental solutions of the stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5(02), 317–328 (1959)
Herrick, D.C., Kennedy, W.D.: Electrical efficiency-a pore geometric theory for interpreting the electrical properties of reservoir rocks. Geophysics 59(6), 918–927 (1994)
Hilfer, R.: Local-porosity theory for flow in porous media. Phys. Rev. B 45(13), 7115 (1992)
Hilfer, R.: Local porosity theory for electrical and hydrodynamical transport through porous media. Phys. A Stat. Mech. Appl. 194(1), 406–414 (1993)
Hilfer, R.: Review on scale dependent characterization of the microstructure of porous media. Transp. Porous Media 46(2–3), 373–390 (2002)
Hilfer, R., Manwart, C.: Permeability and conductivity for reconstruction models of porous media. Phys. Rev. E 64(2), 021,304 (2001)
Holzer, L., Wiedenmann, D., Münch, B., Keller, L., Prestat, M., Gasser, P., Robertson, I., Grobéty, B.: The influence of constrictivity on the effective transport properties of porous layers in electrolysis and fuel cells. J. Mater. Sci. 48(7), 2934–2952 (2013)
Hoogschagen, J.: Diffusion in porous catalysts and adsorbents. Ind. Eng. Chem. 47(5), 906–912 (1955)
Jacquin, C.: Corrélation entre la perméabilité et les caractéristiques géométriques du grès de fontainebleau. Rev. Inst. Fr. Pet. 19, 921–937 (1964)
Johnson, D.L., Koplik, J., Schwartz, L.M.: New pore-size parameter characterizing transport in porous media. Phys. Rev. Lett. 57(20), 2564 (1986)
Katz, A., Thompson, A.: Quantitative prediction of permeability in porous rock. Phys. Rev. B 34(11), 8179 (1986)
Katz, A., Thompson, A.: Prediction of rock electrical conductivity from mercury injection measurements. J. Geophys. Res. Solid Earth (1978–2012) 92(B1), 599–607 (1987)
Kenyon, W.: Nuclear magnetic resonance as a petrophysical measurement. The international journal of radiation applications and instrumentation. Part E. Nucl. Geophys. 6(2), 153–171 (1992)
Kenyon, W., Day, P., Straley, C., Willemsen, J.: A three-part study of nmr longitudinal relaxation properties of water-saturated sandstones. SPE Form. Eval. 3(3), 622–636 (1988)
Koplik, J.: Creeping flow in two-dimensional networks. J. Fluid Mech. 119, 219–247 (1982)
Koponen, A., Kataja, M., Timonen, J.: Permeability and effective porosity of porous media. Phys. Rev. E 56(3), 3319 (1997)
Kozeny, J.: Ueber kapillare leitung des wassers im boden. Wien Akad. Wiss 136(2a), 271 (1927)
Manwart, C., Aaltosalmi, U., Koponen, A., Hilfer, R., Timonen, J.: Lattice-boltzmann and finite-difference simulations for the permeability for three-dimensional porous media. Phys. Rev. E 66(1), 016,702 (2002)
Martys, N.S., Torquato, S., Bentz, D.: Universal scaling of fluid permeability for sphere packings. Phys. Rev. E 50(1), 403 (1994)
Milton, G.: Correlation of the electromagnetic and elastic properties of composites and microgeometries corresponding with effective medium approximations. In: Physics and Chemistry of Porous Media, vol. 107, pp. 66–77. AIP Publishing (1984)
Monteagudo, J.E., Lage, P.L.: Cross-properties relations in 3d percolation networks: I. Network characteristic length determination. Transp. Porous Media 61(2), 143–156 (2005a)
Monteagudo, J.E., Lage, P.L.: Cross-properties relations in 3d percolation networks: Ii. Network permeability. Transp. Porous Media 61(3), 259–274 (2005b)
Mukerji, T., Mavko, G., Gomez, C.: Cross-property rock physics relations for estimating low-frequency seismic impedance trends from electromagnetic resistivity data. Lead. Edge 28(1), 94–97 (2009)
Øren, P., Antonsen, F., Rueslåtten, H., Bakke, S., et al.: Numerical simulations of nmr responses for improved interpretations of nmr measurements in reservoir rocks. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2002)
Øren, P.E., Bakke, S.: Process based reconstruction of sandstones and prediction of transport properties. Transp. Porous Media 46(2–3), 311–343 (2002)
Øren, P.E., Bakke, S., Held, R.: Direct pore-scale computation of material and transport properties for north sea reservoir rocks. Water Resour. Res. 43(12), 1–11 (2007)
Owen, J.: The resistivity of a fluid filled porous body. J. Pet. Technol. 4, 169–176 (1952)
Paterson, M.: The equivalent channel model for permeability and resistivity in fluid-saturated rock–a re-appraisal. Mech. Mater. 2(4), 345–352 (1983)
Pilotti, M.: Reconstruction of clastic porous media. Transp. Porous Media 41(3), 359–364 (2000)
Pollock, D.W.: Semianalytical computation of path lines for finite-difference models. Gr. Water 26(6), 743–750 (1988)
Prager, S.: Viscous flow through porous media. Phys. Fluids (1958–1988) 4(12), 1477–1482 (1961)
Schwartz, L., Martys, N., Bentz, D., Garboczi, E., Torquato, S.: Cross-property relations and permeability estimation in model porous media. Phys. Rev. E 48(6), 4584 (1993)
Sevostianov, I., Shrestha, M.: Cross-property connections between overall electric conductivity and fluid permeability of a random porous media with conducting sceleton. Int. J. Eng. Sci. 48(12), 1702–1708 (2010)
Spanne, P., Thovert, J., Jacquin, C., Lindquist, W., Jones, K., Adler, P.: Synchrotron computed microtomography of porous media: topology and transports. Phys. Rev. Lett. 73(14), 2001 (1994)
Thovert, J.F., Yousefian, F., Spanne, P., Jacquin, C., Adler, P.: Grain reconstruction of porous media: application to a low-porosity fontainebleau sandstone. Phys. Rev. E 63(6), 061,307 (2001)
Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties, vol. 16. Springer, Berlin (2002)
Torquato, S., Beasley, J.: Bounds on the permeability of a random array of partially penetrable spheres. Phys. Fluids (1958–1988) 30(3), 633–641 (1987)
Walderhaug, O., Eliassen, A., Aase, N.E.: Prediction of permeability in quartz-rich sandstones: examples from the norwegian continental shelf and the fontainebleau sandstone. J. Sediment. Res. 82(12), 899–912 (2012)
Widjajakusuma, J., Manwart, C., Biswal, B., Hilfer, R.: Exact and approximate calculations for the conductivity of sandstones. Phys. A Stat. Mech. Appl. 270(1), 325–331 (1999)
Wilkinson, D.: Modified drag theory of permeability. Phys. Fluids (1958–1988) 28(4), 1015–1022 (1985)
Wong, Pz, Koplik, J., Tomanic, J.: Conductivity and permeability of rocks. Phys. Rev. B 30(11), 6606 (1984)
Wyllie, M., Rose, W.D., et al.: Some theoretical considerations related to the quantitative evaluation of the physical characteristics of reservoir rock from electrical log data. J. Pet. Technol. 2(04), 105–118 (1950)
Zhan, X., Schwartz, L.M., Toksöz, M.N., Smith, W.C., Morgan, F.D.: Pore-scale modeling of electrical and fluid transport in berea sandstone. Geophysics 75(5), F135–F142 (2010)
Zhang, X., Knackstedt, M.A.: Direct simulation of electrical and hydraulic tortuosity in porous solids. Geophys. Res. Lett. 22(17), 2333–2336 (1995)
Zick, A., Homsy, G.: Stokes flow through periodic arrays of spheres. J. Fluid Mech. 115, 13–26 (1982)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Berg, C.F., Held, R. Fundamental Transport Property Relations in Porous Media Incorporating Detailed Pore Structure Description. Transp Porous Med 112, 467–487 (2016). https://doi.org/10.1007/s11242-016-0661-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-016-0661-7