Abstract
A new two-scale computational model is proposed to construct the constitutive law of the swelling pressure which appears in the modified form of the macroscopic effective stress principle for expansive clays saturated by an aqueous electrolyte solution containing multivalent ionic species. The microscopic non-local nanoscale model is constructed based on a coupled Poisson-Fredholm integral equation arising from the thermodynamics of inhomogeneous fluids in nanopores (Density Functional Theory), which governs the local electric double layer potential profile coupled with the ion-particle correlation function in an electrolytic solution of finite size ions. The local problem is discretized by invoking the eigenvalue expansion of the convolution kernel in conjunction with the Galerkin method for the Gauss-Poisson equation. The discretization of the Fredholm equation is accomplished by a collocation scheme employing eigenfunction basis. Numerical simulations of the local ionic profiles in rectangular cell geometries are obtained showing considerable discrepancies with those computed with Poisson-Boltzmann based models for point charges, particularly for divalent ions in calcium montmorillonite. The constitutive law for the disjoining pressure is reconstructed numerically by invoking the contact theorem within a post-processing approach. The resultant computational model is capable of capturing ranges of particle attraction characterized by negative values of the disjoining pressure overlooked by the classical electric double layer theory. Such results provide further insight in the role the swelling pressure plays in the modified macroscopic effective stress principle for expansive porous media.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Attard, P.: Thermodynamics and Statistical Mechanics: Equilibrium by Entropy Maximisation. Academic Press, London (2002)
Biot, M.A.: General Theory of Three-dimensional Consolidation. J. Appl. Phys. 12(2), 155–164 (1941)
Carnie, S., Chan, Y.: The statistical mechanics of the electrical double layer: stress tensor and contact conditions. J. Chem. Phys. 74(2), 1293–1297 (1981)
Ebeling, D., van den Ende, D., Mugele, F.: Electrostatic interaction forces in aqueous salt solutions of variable concentration and valency. Nanotechnology 22(30), 305706 (2011)
Derjaguin, B.V., Churaev, N., Muller, V.: Surface Forces. Plenum Press, New York (1987)
Dormieux, L., Lemarchand, E., Coussy, O.: Macroscopic and micromechanical approaches to the modelling of the osmotic swelling in clays. Transp. Porous Media 50, 75–91 (2003)
Evans, R.: The nature of the Liquid-Vapor interface and other topics in the statistical mechanics of Non-Uniform, kclassical fluids. Adv. Phys. 28(2), 143–200 (1979)
Hackbusch, W.: Integral Equations. Theory and Numerical Treatment, Birkhäuser, Basel (1995)
Hansen, J.-P., McDonald, I.R.: Theory of Simple Liquids, Third Edition, Elsevier (2006)
Hill, T.L.: Statistical Mechanics: Principles and Selected Applications. McGraw-Hill Book Company, Inc, New York 1956 Reprinted by Dover Publications (1987)
Irving, J.H., Kirkwood, J.: The statistical mechanical theory of transport processes. IV. The equation of hydrodynamics. J. Chem. Phys. 18(6), 817–829 (1950)
Kjellander, R., Marčelja, S., Pashley, R.M., Quirk, J.P.: Double layer ion correlation forces restrict calcium-clay swelling. J. Phys. Chem. 92, 6489–6492 (1988)
Kjellander, R., Pashley, R.M., Quirk, J.P., Theoretical, A.: Experimental study of forces between charged Mica surfaces in aqueous CaCl2 solutions. J. Chem Phys. 92(7), 4399–4407 (1990)
Lai, W.M., Hou, J.S., Mow, V.C.: A triphasic theory for the swelling and deformation behaviors of articular cartilage. ASME J. Biomech. Eng. 113, 245–258 (1991)
Le, T.D., Moyne, C., Murad, M.A., Lima, S.A.: A two-scale non-local model of swelling porous media incorporating ion size correlation effects. J. Mech. Phys. Solids 61(12), 2493–2521 (2013)
Le, T.D., Moyne, C., Murad, M.A.: A three-scale model for ionic solute transport in swelling clays incorporating ion-ion correlation effects. Adv. Water Resour. 75, 31–52 (2015)
Lyklema, J.: Fundamentals of Colloid and Interface Science. Academic, London (1993)
Looker, J.R., Carnie, S.L.: Homogenization of the ionic transport equations in periodic porous media. Transp. Porous Media 65, 107–131 (2006)
Lozada-Cassou, M.: The force between two planar electrical double layers. J. Chem. Phys. 80(7), 3344–3349 (1984)
Lozada-Cassou, M., Díaz-herrera, E.: Three point extension for the hypernetted chain and other integral equation theories. J. Chem. Phys. 92(2), 1194–1210 (1990)
McQuarrie, D.A.: Statistical Mechanics, University Science Books (2000)
Mier-y-Teran, L., Suh, S.H., White, S., Davis, H.T.: A non-local free-energy density-functional approximation for the electrical double layer. J. Chem. Phys. 92(8), 5087–5098 (1990)
Mitchell, J.K.: Fundamentals of Soil Behaviour, John Wiley & Sons Inc. (1993)
Mody, F.K., Hale, A.H.: A borehole stability model to couple the mechanics and chemistry of drilling fluid shale interaction, Paper SPE/IADC 25728 Society of Petroleum Enginneers, 473–489 (1993)
Moyne, C., Murad, M.A.: Electro-chemo-mechanical couplings in swelling clays derived from Micro/Macro homogenization procedure. Int. J. Solids Struct. 39, 6159–6190 (2002)
Moyne, C., Murad, M.A.: Macroscopic behaviour of swelling porous media derived from micromechanical analysis. Transp. Porous Media 50, 127–151 (2003)
Moyne, C., Murad, M.A.: A two-scale model for coupled electro-chemo-mechanical phenomena and onsager’s reciprocity relations in expansive clays: i homogenization analysis. Transp. Porous Media 62, 333–380 (2006)
Moyne, C., Murad, M.A.: A two-scale model for coupled electro-chemo-mechanical phenomena and onsager’s reciprocity relations in expansive clays: II computational validation. Transp. Porous Media 63, 13–56 (2006)
Murad, M.A., Moyne, C.: Micromechanical computational modeling of expansive porous media. C. R. Mecanique 330, 865–870 (2002)
Murad, M.A., Moyne, C.: A dual-porosity model for ionic solute transport in expansive clays. Comput. Geosci. 12, 47–82 (2008)
Nelson, J.D., Miller, D.J.: Expansive soils: Problem and Practice in Foundation and Pavement Engineering, John Wiley & Sons Inc. (1992)
Nickell, R.E., Gartling, D.K., Strang, G.: Spectral decomposition in advection-diffusion analysis by finite element methods. Comput. Methods Appl. Mech. Eng. 17-18, 561–580 (1979)
Oliveira, S.P., Azevedo, J.S.: Spectral element approximation of Fredholm integral eigenvalue problems. J. Comput. Appl. Math. 257, 46–56 (2014)
Ort, V.: On the physical and chemical stability of shales. J. Pet. Sci. Eng. 38(3–4), 213–235 (2003)
Ponce, R.V., Murad, M.A., Lima, S.: A two-scale computational model of pH sensitive expansive porous media. J. Appl. Mech. 80(2), 0209031–2090314 (2013)
Quesada-Pérez, M., González-Tovar, E., Martín-molina, A., Lozada-Cassou, M., Hidalgo-Álvarez, R.: Overcharging in colloids: beyond the Poisson-Boltzmann approach. ChemPhysChem 4(3), 234–248 (2003)
Jellander, R., Marčelja, S., Quirk, J.P.: Attractive double-layer interactions between calcium clay particles. J. Colloid Interface Sci. 126(01), 194–211 (1988)
Rowe, R.K.: Long-term performance of contaminant barrier systems. Geotechinique 35(09), 631–678 (2005)
Ruhl, J.L., Daniel, D.E.: Geosynthetic clay liners permeated with chemical solutions and leachates. J. Geotech. Geoenviron. Eng. 123, 369–381 (1997)
Segad, M., Jönsson, B., Åkesson, T., Cabane, B.: Ca/Na montmorillonite: structure, forces and swelling properties. Langmuir 26(08), 5782–5790 (2010)
Zixiang, T., Mier-y-Teran, L., Davis, H.T., Scriven, L.E., White, H.S.: Non-local free-energy density-functional theory applied to the electrical double layer. Part I: Symmetrical electrolytes. Mol. Phys. 71(2), 369–392 (1990)
Zixiang, T., Scriven, L.E., Davis, H.T.: Interactions between primitive electrical double layers. J. Chem. Phys. 97(12), 9258–9266 (1992)
Thiele, E.: Equation of state of hard spheres. J. Chem. Phys. 39(2), 474–479 (1963)
Van Olphen, H.: An Introduction to Clay Colloid Chemistry: For Clay Technologists, Geologists, and Soil Scientists. Wiley, New York (1977)
Waisman, E., Lebowitz, J.L.: Mean spherical model integral equation for charged hard spheres. I. Method of solution. J. Chem. Phys. 56(6), 3086–3093 (1972)
Waisman, E., Lebowitz, J.L.: Mean spherical model integral equation for charged hard spheres. II. Results. J. Chem. Phys. 56(6), 3093–3099 (1972)
Wertheim, M.S.: Exact solution of the Percus-Yevick integral equation for hard spheres. Phys. Rev. Lett. 10(8), 321–323 (1963)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rocha, A.C., Murad, M.A., Moyne, C. et al. A new methodology for computing ionic profiles and disjoining pressure in swelling porous media. Comput Geosci 20, 975–996 (2016). https://doi.org/10.1007/s10596-016-9572-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-016-9572-5