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A Simple Hysteretic Constitutive Model for Unsaturated Flow

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Abstract

In this paper, we present a constitutive model to describe unsaturated flow that considers the hysteresis phenomena. This constitutive model provides simple mathematical expressions for both saturation and hydraulic conductivity curves, and a relationship between permeability and porosity. The model is based on the assumption that the porous media can be represented by a bundle of capillary tubes with throats or “ink bottles” and a fractal pore size distribution. Under these hypotheses, hysteretic curves are obtained for saturation and relative hydraulic conductivity in terms of pressure head. However, a non-hysteretic relationship is obtained when relative hydraulic conductivity is expressed as a function of saturation. The proposed relationship between permeability and porosity is similar to the well-known Kozeny–Carman equation but depends on the fractal dimension. The performance of the constitutive model is tested against different sets of experimental data and previous models. In all of the cases, the proposed expressions fit fairly well the experimental data and predicts values of permeability and hydraulic conductivity better than others models.

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References

  • Assouline, S., Tessier, D., Bruand, A.: A conceptual model of the soil water retention curve. Water Resour. Res. 34(2), 223–231 (1998)

    Article  Google Scholar 

  • Assouline, S.: A model for soil relative hydraulic conductivity based on the water retention characteristic curve. Water Resour. Res. 37(2), 265–271 (2001)

    Article  Google Scholar 

  • Assouline, S.: On the relationships between the pore size distribution index and characteristics of the soil hydraulic functions. Water Resour. Res. 41(7) (2005). doi:10.1029/2004WR003511

  • Bear, J.: Dynamics of Fluids in Porous Media. Dover Publications Inc., Mineola (1998)

    Google Scholar 

  • Beliaev, A.Y., Hassanizadeh, S.M.: A theoretical model of hysteresis and dynamic effects in the capillary relation for twophase flow in porous media. Transp. Porous Media 43, 487–510 (2001)

    Article  Google Scholar 

  • Blunt, M.J., Jackson, M.D., Piri, M., Valvatne, P.H.: Detailed physics, predictive capabilities and macroscopic consequences for pore-network models of multiphase flow. Adv. Water Resour. 25(8), 1069–1089 (2002)

    Article  Google Scholar 

  • Bodurtha, P.: Novel Techniques for Investigating the Permeation Properties of Environmentally Friendly Paper Coatings: The Influence of Structural Anisotropy on Fluid Permeation in Porous Media. University of Plymouth, Plymouth (2003)

    Google Scholar 

  • Bousfield, D.W., Karles, G.: Penetration into three-dimensional complex porous structures. J. Colloid Interface Sci. 270(2), 396–405 (2004)

    Article  Google Scholar 

  • Brooks, R.H., Corey, A.T.: Hydraulic properties of porous media and their relation to drainage design. Trans. ASAE 7(1), 0026–0028 (1964). doi:10.13031/2013.40684

    Article  Google Scholar 

  • Buckingham, E.: Studies on the Movement of Soil Moisture. Bulletin 38. US Gov. Print Office, Washington, DC (1907)

    Google Scholar 

  • Burdine, N.: Relative permeability calculations from pore size distribution data. J. Pet. Technol. 5(03), 71–78 (1953)

    Article  Google Scholar 

  • Carman, P.C.: Fluid flow through granular beds. Trans. Inst. Chem. Eng. 15, 150–166 (1937)

    Google Scholar 

  • Carsel, R.F., Parrish, R.S.: Developing joint probability distributions of soil water retention characteristics. Water Resour. Res. 24(5), 755–769 (1988)

    Article  Google Scholar 

  • Chilindar, G.V.: Relationship Between Porosity, Permeability and Grain Size Distribution of Sands and Sandstones, in Deltaic and Shallow Marine Deposits, vol. I, pp. 71–75. Elsevier, New York (1964)

    Google Scholar 

  • Darcy, H.P.G.: Exposition et application des principes à suivre et des formules à employer dans les questions de distribution d’eau. In: Dalmont, V. (ed.) Les fontaines publiques de la ville de Dijon. Victor Dalmont, Paris (1856)

  • Dentz, M., Le Borgne, T., Englert, A., Bijeljic, B.: Mixing, spreading and reaction in heterogeneous media: a brief review. J. Contam. Hydrol. 120, 1–17 (2011)

    Article  Google Scholar 

  • Dong, H., Blunt, M.J.: Pore-network extraction from micro-computerized-tomography images. Phys. Rev. E 80(3), 036307 (2009)

    Article  Google Scholar 

  • Feng, M., Fredlund, D.G.: Hysteretic influence associated with thermal conductivity sensor measurements. In: Proceedings from Theory to the Practice of Unsaturated Soil Mechanics, 52nd Canadian Geotechnical Conference and the Unsaturated Soil Group, Regina, 23–24 October, vol. 14, p. 2 (1999)

  • Guarracino, L.: A fractal constitutive model for unsaturated flow in fractured hard rocks. J. Hydrol. 324(1), 154–162 (2006)

    Article  Google Scholar 

  • Guarracino, L.: Estimation of saturated hydraulic conductivity \(K_S\) from the van Genuchten shape parameter \(\alpha \). Water Resour. Res. 43, W11502 (2007). doi:10.1029/2006WR005766

    Article  Google Scholar 

  • Guarracino, L., Rötting, T., Carrera, J.: A fractal model to describe evolution of multiphase flow properties during mineral dissolution. Adv. Water Resour. 67, 78–86 (2014)

    Article  Google Scholar 

  • Ghanbarian-Alavijeh, B., Millán, H., Huang, G.: A review of fractal, prefractal and pore-solid-fractal models for parameterizing the soil water retention curve. Can. J. Soil Sci. 91(1), 1–14 (2011)

    Article  Google Scholar 

  • Hirst, J.P.P., Davis, N., Palmer, A.F., Achache, D., Riddiford, F.A.: The tight gas challenge: appraisal results from the Devonian of Algeria. Pet. Geosci. 7, 13–21 (2001)

    Article  Google Scholar 

  • Jerauld, G.R., Salter, S.J.: The effect of pore-structure on hysteresis in relative permeability and capillary pressure: pore-level modeling. Transp. Porous Med. 5(2), 103–151 (1990)

    Article  Google Scholar 

  • Jougnot, D., Linde, N., Revil, A., Doussan, C.: Derivation of soil-specific streaming potential electrical parameters from hydrodynamic characteristics of partially saturated soils. Vadose Zone J. 11(1) (2012). doi:10.2136/vzj2011.0086

  • Jurin, J.: An account of some experiments shown before the royal society; with an enquiry into the cause of the ascent and suspension of water in capillary tubes. By James Jurin, MD and R. Soc. S. Philos. Trans. 30(351–363), 739–747 (1717)

    Article  Google Scholar 

  • Jury, W.A., Gardner, W.R., Gardner, W.H.: Soil Physics. John Wiley, New York (1991)

    Google Scholar 

  • Karube, D., Kawai, K.: The role of pore water in the mechanical behavior of unsaturated soils. Geotech. Geol. Eng. 19(3–4), 211–241 (2001)

    Article  Google Scholar 

  • Klausner, Y.: Fundamentals of Continuum Mechanics of Soils. Springer, New York (1991)

    Book  Google Scholar 

  • Kozeny, J.: ijber kapillare Leitung des Wassers im Boden. Sitzungsber. Kais. Akad. Wiss. Wien. 136, 271–306 (1927)

    Google Scholar 

  • Lindquist, W.B., Venkatarangan, A., Dunsmuir, J., Wong, T.F.: Pore and throat size distributions measured from synchrotron X-ray tomographic images of Fontainebleau sandstones. J. Geophys. Res. Solid Earth 105(B9), 21509–21527 (2000)

    Article  Google Scholar 

  • Luffel, D.L., Howard, W.E., Hunt, E.R.: Travis Peak core permeability and porosity relationships at reservoir stress. Soc. Pet. Eng. Form. Eval. 6(3), 310–318 (1991)

    Google Scholar 

  • Monachesi, L.B., Guarracino, L.: A fractal model for predicting water and air permeabilities of unsaturated fractured rocks. Transp. Porous Med. 90(3), 779–789 (2011)

    Article  Google Scholar 

  • Mualem, Y.: Modified approach to capillary hysteresis based on a similarity hypothesis. Water Resour. Res. 9(5), 1324–1331 (1973)

    Article  Google Scholar 

  • Mualem, Y.: A Catalogue of the Hydraulic Properties of Unsaturated Soils. Technion-Israel Institute of Technology, Haifa (1974)

    Google Scholar 

  • Mualem, Y.: A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12(3), 513–522 (1976)

    Article  Google Scholar 

  • Mualem, Y.: Extension of the similarity hypothesis used for modeling the soil water characteristics. Water Resour. Res. 13(4), 773–780 (1977)

    Article  Google Scholar 

  • Mualem, Y.: Hydraulic conductivity of unsaturated soils: prediction and formulas. In: Klute, A. (ed.) Methods of Soil Analysis. Part 1. Physical and Mineralogical Methods, vol. 9, pp. 799–823. Soil Science Society of America, American Society of Agronomy (1986)

  • Néel, L.: Théories des lois daimantation de Lord Rayleigh. Cah. Phys. 12, 1–20 (1942)

    Google Scholar 

  • Parker, J.C., Lenhard, R.J.: A model for hysteretic constitutive relations governing multiphase flow 1. Saturation pressure relations. Water Resour. Res. 23(4), 618–624 (1987)

    Article  Google Scholar 

  • Pham, H.Q., Fredlund, D.G., Barbour, S.L.: A study of hysteresis models for soil–water characteristic curves. Can. Geotech. J. 42, 1548–1568 (2005). doi:10.1139/T05-071

    Article  Google Scholar 

  • Pham, H.Q., Fredlund, D.G., Barbour, S.L.: A practical model for the soil–water characteristic curve for soils with negligible volume change. Gotechnique 53(2), 293–298 (2003)

    Article  Google Scholar 

  • Poulovassilis, A., Tzimas, E.: The hysteresis in the relationship between hydraulic conductivity and soil water content. Soil Sci. 120(5), 327–331 (1975)

    Article  Google Scholar 

  • Richards, L.A.: Capillary conduction of liquids through porous mediums. J. Appl. Phys. 1(5), 318–333 (1931)

    Google Scholar 

  • Rubin, J.: Numerical method for analyzing hysteresis-affected, post-infiltration redistribution of soil moisture. Soil Sci. Soc. Am. J. 31(1), 13–20 (1967)

    Article  Google Scholar 

  • Rubino, J.G., Guarracino, L., Müller, T.M., Holliger, K.: Do seismic waves sense fracture connectivity? Geophys. Res. Lett. 40(4), 692–696 (2013)

    Article  Google Scholar 

  • Spiteri, E.J., Juanes, R., Blunt, M.J., Orr, F.M.: A new model of trapping and relative permeability hysteresis for all wettability characteristics. Spe J. 13(03), 277–288 (2008)

    Article  Google Scholar 

  • Topp, G.C., Miller, E.E.: Hysteretic moisture characteristics and hydraulic conductivities for glass-bead media. Soil Sci. Soc. Am. J. 30(2), 156–162 (1966)

    Article  Google Scholar 

  • Topp, G.C.: Soil-water hysteresis: the domain theory extended to pore interaction conditions. Soil Sci. Soc. Am. J. 35(2), 219–225 (1971)

    Article  Google Scholar 

  • Tyler, S.W., Wheatcraft, S.W.: Fractal process in soil water retention. Water Resour. Res. 26(5), 1047–1054 (1990)

    Article  Google Scholar 

  • van Genuchten, M.T.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44, 892–898 (1980)

    Article  Google Scholar 

  • Wang, S., Wu, T., Qi, H., Zheng, Q., Zheng, Q.: A permeability model for power-law fluids in fractal porous media composed of arbitrary cross-section capillaries. Physica A 437, 12–20 (2015)

    Article  Google Scholar 

  • Wildenschild, D., Vaz, C.M.P., Rivers, M.L., Rikard, D., Christensen, B.S.B.: Using X-ray computed tomography in hydrology: systems, resolutions, and limitations. J. Hydrol. 267(3), 285–297 (2002)

    Article  Google Scholar 

  • Xu, C., Torres-Verdín, C.: Pore system characterization and petrophysical rock classification using a bimodal Gaussian density function. Math. Geosci. 45(6), 753–771 (2013)

    Article  Google Scholar 

  • Yu, B.: Analysis of flow in fractal porous media. Appl. Mech. Rev. 61(5), 050801 (2008). doi:10.1115/1.2955849

    Article  Google Scholar 

  • Yu, B., Li, J.: Some fractal characters of porous media. Fractals 9(03), 365–372 (2001)

    Article  Google Scholar 

  • Yu, B., Li, J., Li, Z., Zou, M.: Permeabilities of unsaturated fractal porous media. Int. J. Multiphas. Flow 29(10), 1625–1642 (2003)

    Article  Google Scholar 

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Acknowledgements

The authors thank the editor and two anonymous reviewers for their careful assessment of our work and the valuable comments and suggestions that helped to greatly improve the manuscript.

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Authors and Affiliations

  1. Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Consejo Nacional de Investigaciones Científicas y Técnicas, La Plata, Argentina

    Mariangeles Soldi & Luis Guarracino

  2. UPMC Univ Paris 06, CNRS, EPHE, UMR 7619 METIS, Sorbonne Universites, Paris, France

    Damien Jougnot

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  1. Mariangeles Soldi
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  2. Luis Guarracino
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Correspondence to Mariangeles Soldi.

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Soldi, M., Guarracino, L. & Jougnot, D. A Simple Hysteretic Constitutive Model for Unsaturated Flow. Transp Porous Med 120, 271–285 (2017). https://doi.org/10.1007/s11242-017-0920-2

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