Abstract
The electrostatic properties of clay mineral surfaces play a significant role in their diffusion properties. The negative electrostatic potential field at clay mineral surfaces results in the presence of a diffuse layer that balances the mineral surface charge. The diffusion properties of the porosity fraction that is affected by this phenomenon are different from the diffusion properties of electroneutral bulk water. These properties have attracted growing interest from diverse communities in the past years, especially in the field of study of radioactive waste disposal. The influence of the diffuse layer can be described at the continuum scale by a set of equations that are formulated in terms of the Nernst-Planck equation. The number of codes that can handle the coupling between transport properties in clay affected by the presence of a diffuse layer in the porosity and chemical reactions is very limited, and no benchmark exercises have been published yet that make it possible to validate the numerical implementation of these equations in reactive transport codes. The present study proposes a set of benchmark exercises of increasing complexity that highlight caveats related to the finite difference (volume) treatment of the Nernst-Planck equation in the presence of a diffuse layer in heterogeneous systems. Once these problems are identified and solved, the codes PHREEQC, CrunchClay, and a new Fortran routine written for this study gave results in very good agreement for most of the benchmark exercises. When present, the differences in results were directly traceable to the differences in averaging methods at grid cell boundaries, and to the consideration or the omission of the activity gradient term in the Nernst-Planck equation.
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Altmann, S.: Geo’chemical research: a key building block for nuclear waste disposal safety cases. J. Contam. Hydrol. 102, 174–179 (2008)
Altmann, S., Tournassat, C., Goutelard, F., Parneix, J. -C., Gimmi, T., Maes, N.: Diffusion-driven transport in clayrock formations. Appl. Geochem. 27, 463–478 (2012)
Glaus, M. A., Birgersson, M., Karnland, O., Van Loon, L. R.: Seeming steady-state uphill diffusion of 22na+ in compacted montmorillonite. Environ. Sci. Technol. 47, 11522–11527 (2013)
Grangeon, S., Vinsot, A., Tournassat, C., Lerouge, C., Giffaut, E., Heck, S., Groschopf, N., Denecke, M. A., Wechner, S., Schäfer, T.: The influence of natural trace element distribution on the mobility of radionuclides. The exemple of nickel in a clay-rock. Appl. Geochem. 52, 155–173 (2015)
Jacquier, P., Hainos, D., Robinet, J. -C., Herbette, M., Grenut, B., Bouchet, A., Ferry, C.: The influence of mineral variability of Callovo-Oxfordian clay rocks on radionuclide transfer properties. Appl. Clay Sci. 83, 129–136 (2013)
Robinet, J.-C., Sardini, P., Coelho, D., Parneix, J.-C., Prêt, D., Sammartino, S., Boller, E., Altmann, S.: Effects of mineral distribution at mesoscopic scale on solute diffusion in a clay-rich rock: example of the Callovo-Oxfordian mudstone (Bure, France). Water Resources Research, 48, W05554 (2012)
Tournassat, C., Gaboreau, S., Robinet, J. -C., Bourg, I. C., Steefel, C. I.: Impact of microstructure on anion exclusion in compacted clay media. CMS Workshop Lect Ser. 21, 137–149 (2016)
Appelo, C. A. J.: Multicomponent diffusion modeling in clay systems with application to the diffusion of tritium, iodide, and sodium in Opalinus clay. Supporting information (2007)
Appelo, C. A. J., Van Loon, L. R., Wersin, P.: Multicomponent diffusion of a suite of tracers (HTO, Cl, Br, I, Na, Sr, Cs) in a single sample of Opalinus clay. Geochim. Cosmochim. Acta 74, 1201–1219 (2010)
Bourg, I. C., Bourg, A. C. M., Sposito, G.: Modeling diffusion and adsorption in compacted bentonite: a critical review. J. Contam. Hydrol. 61, 293–302 (2003)
Bourg, I. C., Sposito, G.: Connecting the molecular scale to the continuum scale for diffusion processes in smectite-rich porous media. Environ. Sci. Technol. 44, 2085–2091 (2010)
Tournassat, C., Steefel, C. I.: Ionic transport in nano-porous clays with consideration of electrostatic effects. Rev. Mineral. Geochem. 80, 287–330 (2015)
Tournassat, C., Bourg, I. C., Steefel, C. I., Bergaya, F.: Chapter 1 - surface properties of clay minerals. In: Tournassat, C., Steefel, C. I., Bourg, I. C., Bergaya, F. (eds.) Natural and Engineered Clay Barriers, vol. 6, pp 5–31. Developments in Clay Science; Elsevier (2015)
Tachi, Y., Yotsuji, K., Suyama, T., Ochs, M.: Diffusion model for bentonite Integrated sorption Part 2: porewater chemistry, sorption and diffusion modeling in compacted systems. J. Nucl. Sci. Technol. 51, 1–14 (2014)
Tinnacher, R. M., Holmboe, M., Tournassat, C., Bourg, I. C., Davis, J. A.: Ion adsorption and diffusion in smectite: molecular, pore, and continuum scale views. Geochim. Cosmochim. Acta 177, 130–149 (2016)
Glaus, M., Aertsens, M., Appelo, C., Kupcik, T., Maes, N., Van Laer, L., Van Loon, L.: Cation diffusion in the electrical double layer enhances the mass transfer rates for Sr2 + , Co2 + and Zn2 + in compacted illite. Geochim. Cosmochim. Acta 165, 376–388 (2015)
Bestel, M., Glaus, M. A., Frick, S., Gimmi, T., Juranyi, F., Van Loon, L. R., Diamond, L. W.: Combined tracer through-diffusion of HTO and 22Na through Na-montmorillonite with different bulk dry densities. Appl. Geochem. 93, 158–166 (2018)
Parkhurst, D. L., Appelo, C. A. J.: Description of input and examples for PHREEQC Version 3– a computer program for speciation,batch-reaction, one-dimensional transport, and inverse geochemical calculations; U.S. Geological Survey Techniques and Methods, book 6, chap. A43, 497 p., available at http://pubs.usgs.gov/tm/06/a43/ (2013)
Steefel, C. I., Appelo, C. A. J., Arora, B., Jacques, D., Kalbacher, T., Kolditz, O., Lagneau, V., Lichtner, P. C., Mayer, K. U., Meeussen, J. C. L., Molins, S., Moulton, D., Shao, H., Šimunek, J., Spycher, N., Yabusaki, S. B., Yeh, G. T.: Reactive transport codes for subsurface environmental simulation. Comput. Geosci. 19, 445–478 (2015)
Gimmi, T., Alt-Epping, P.: Simulating Donnan equilibria based on the Nernst-Planck equation. Geochim. Cosmochim. Acta 232, 1–13 (2018)
Alt-Epping, P., Gimmi, T., Wersin, P., Jenni, A.: Incorporating electrical double layers into reactive-transport simulations of processes in clays by using the Nernst-Planck equation: a benchmark revisited. Appl. Geochem. 89, 1–10 (2018)
Appelo, C. A. J., Wersin, P.: Multicomponent diffusion modeling in clay systems with application to the diffusion of tritium, iodide, and sodium in Opalinus clay. Environ. Sci. Technol. 41, 5002–5007 (2007)
Tournassat, C., Bourg, I. C., Holmboe, M., Sposito, G., Steefel, C. I.: Molecular dynamics simulations of anion exclusion in clay interlayer nanopores. Clays Clay Miner. 64, 374–388 (2016)
Crank, J.: The Mathematics of Diffusion. Oxford University Press, Oxford (1975)
Glaus, M., Baeyens, B., Bradbury, M. H., Jakob, A., Van Loon, L. R., Yaroshchuk, A.: Diffusion of 22na and 85sr in montmorillonite: evidence of interlayer diffusion being the dominant pathway at high compaction. Environ. Sci. Tech. 41, 478–485 (2007)
Tachi, Y., Yotsuji, K.: Sorption of cs+, na+ Diffusion i− and HTO in compacted sodium montmorillonite as a function of porewater salinity: integrated sorption and diffusion model. Geochim. Cosmochim. Acta 132, 75–93 (2014)
Van Loon, L. R., Soler, J. M., Jakob, A., Bradbury, M. H.: Effect of confining pressure on the diffusion of HTO, 36Cl- and 125I- in a layered argillaceous rock (Opalinus Clay): diffusion perpendicular to the fabric. Appl. Geochem. 18, 1653–1662 (2003)
Acknowledgments
This work was supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy (BES-DOE) under Contract No. DE-AC02-05CH11231; the French National Radioactive Waste Management Agency (Andra, project CTEC, P.I. J-C. Robinet and Mélanie Lundy) (Andra, project CTEC, P.I. J-C. Robinet); and the CNRS Défi NEEDS (project MIPOR-TRANSREAC). Carl I. Steefel acknowledges funding from L’Institut Carnot for his visit to the BRGM.
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Tournassat, C., Steefel, C.I. Modeling diffusion processes in the presence of a diffuse layer at charged mineral surfaces: a benchmark exercise. Comput Geosci 25, 1319–1336 (2021). https://doi.org/10.1007/s10596-019-09845-4
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DOI: https://doi.org/10.1007/s10596-019-09845-4