Abstract
The effective diffusivity is a key parameter in the diffusive transport calculations, thus decisive for predicting the radionuclide migration in low-permeable clay-rich formations. Potential host rocks such as the Opalinus clay exhibit pore network heterogeneities, critically modified due to compositional variability in the sandy facies and owing to diagenetic minerals. Meaningful estimation of the effective diffusivity requires an understanding of transport mechanisms at the nanometer-scale as a starting point and a combination with upscaling strategies for considering compositional heterogeneities at the micrometer-scale. In this study, we propose an upscaling workflow that integrates transport simulations at both the nanometer-scale and the micrometer-scale to predict the effective diffusivities of radionuclides in the sandy facies of the Opalinus clay. The respective synthetic digital rocks provide conceptually two types of materials at the pore scale, in which the pore space and pore network in the clay matrix at the nanometer scale and mineral complexity in shales at the micrometer scale are considered. The numerical approach using the introduced digital rocks is validated with published experimental data that confirm the general applicability of the models. Sensitivity studies reveal the increase of effective diffusivity of shales as a function of increased pore space, reduced tortuosity, and an increased sheet silicate concentration compared to other rock components. Thus, such spatial variabilities at the pore scale of more complex sedimentary rocks are now addressed in the proposed approach and available for studying heterogeneous diffusion patterns compared to commonly assumed homogeneous behavior. Finally, and as a starting point for further upscaling strategies, we investigate anisotropic diffusion by studying the effect of lamination of the shales toward enhanced predictability of radionuclide migration.
Article Highlights
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Upscaling workflow to predict effective diffusivity of radionuclides (RN) diffusion in shales showing compositional heterogeneity (sandy facies of Opalinus clay)
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Sensitivity studies demonstrate and quantify the effect of the pore network geometry in the clay matrix as well as the effect of clay mineral concentration variability in shales on RN diffusion
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Prediction of the heterogeneity of diffusivity based on multiple types of imaging data including compositional rock data
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Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
References
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(4), 1201–1219 (2010). https://doi.org/10.1016/j.gca.2009.11.013
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)
Bossart, P., Thurry, M.: Mont Terri Rock Laboratory. Project, programme 1996 to 2007 and results. In: no. 3. Swiss Geological Survey, Wabern (2008)
Boving, T.B., Grathwohl, P.: Tracer diffusion coefficients in sedimentary rocks: correlation to porosity and hydraulic conductivity. J. Contam. Hydrol. 53, 85–100 (2001)
Chen, L., Kang, Q., Dai, Z., Viswanathan, H.S., Tao, W.: Permeability prediction of shale matrix reconstructed using the elementary building block model. Fuel 160, 346–356 (2015). https://doi.org/10.1016/j.fuel.2015.07.070
FASP User Guide. http://www.multigrid.org/fasp (2018). Accessed 20 December 2018
Germanou, L., Ho, M.T., Zhang, Y., Wu, L.: Intrinsic and apparent gas permeability of heterogeneous and anisotropic ultra-tight porous media. J. Nat. Gas Sci. Eng. 60, 271–283 (2018). https://doi.org/10.1016/j.jngse.2018.10.003
Gimmi, T., Kosakowski, G.: How mobile are sorbed cations in clays and clay rocks? Environ. Sci. Technol. 45(4), 1443–1449 (2011). https://doi.org/10.1021/es1027794
Grathoff, G.H., Peltz, M., Enzmann, F., Kaufhold, S.: Porosity and permeability determination of organic-rich Posidonia shales based on 3-D analyses by FIB-SEM microscopy. Solid Earth 7(4), 1145–1156 (2016). https://doi.org/10.5194/se-7-1145-2016
Grathwohl, P.: Diffusion in natural porous media. Springer Science+Business Media, (1998)
Hemes, S., Desbois, G., Urai, J.L., Schröppel, B., Schwarz, J.-O.: Multi-scale characterization of porosity in Boom Clay (HADES-level, Mol, Belgium) using a combination of X-ray μ-CT, 2D BIB-SEM and FIB-SEM tomography. Microporous Mesoporous Mater. 208, 1–20 (2015). https://doi.org/10.1016/j.micromeso.2015.01.022
Houben, M.E., Desbois, G., Urai, J.L.: Pore morphology and distribution in the Shaly facies of Opalinus Clay (Mont Terri, Switzerland): Insights from representative 2D BIB–SEM investigations on mm to nm scale. Appl. Clay Sci. 71, 82–97 (2013). https://doi.org/10.1016/j.clay.2012.11.006
Keller, L.M., Hilger, A., Manke, I.: Impact of sand content on solute diffusion in Opalinus Clay. Appl. Clay Sci. 112–113, 134–142 (2015). https://doi.org/10.1016/j.clay.2015.04.009
Keller, L.M., Holzer, L.: Image-Based Upscaling of Permeability in Opalinus Clay. J. Geophys. Res. Solid Earth 123(1), 285–295 (2018). https://doi.org/10.1002/2017jb014717
Keller, L.M., Holzer, L., Schuetz, P., Gasser, P.: Pore space relevant for gas permeability in Opalinus clay: statistical analysis of homogeneity, percolation, and representative volume element. J. Geophys. Res. Solid Earth 118(6), 2799–2812 (2013). https://doi.org/10.1002/jgrb.50228
Kulenkampff, J., Gründig, M., Zakhnini, A., Gerasch, R., Lippmann-Pipke, J.: Process tomography of diffusion, using PET, to evaluate anisotropy and heterogeneity. Clay Miner. 50(3), 369–375 (2015). https://doi.org/10.1180/claymin.2015.050.3.09
Kulenkampff, J., Zakhnini, A., Gründig, M., Lippmann-Pipke, J.: Quantitative experimental monitoring of molecular diffusion in clay with positron emission tomography. Solid Earth 7(4), 1207–1215 (2016). https://doi.org/10.5194/se-7-1207-2016
Leupin, O.X., Van Loon, L.R., Gimmi, T., Wersin, P., Soler, J.M.: Exploring diffusion and sorption processes at the Mont Terri rock laboratory (Switzerland): lessons learned from 20 years of field research. Swiss J. Geosci. 110(1), 391–403 (2017). https://doi.org/10.1007/s00015-016-0254-z
Lippmann-Pipke, J., Gerasch, R., Schikora, J., Kulenkampff, J.: Benchmarking PET for geoscientific applications: 3D quantitative diffusion coefficient determination in clay rock. Comput. Geosci. 101, 21–27 (2017). https://doi.org/10.1016/j.cageo.2017.01.002
Nagra: Opalinus Clay Project: Demonstration of feasibility of disposal for spent fuel, vitrified high-level waste and long-lived intermediate-level waste. In. Wettingen, Switzerland (2002)
Navarre-Sitchler, A., Steefel, C.I., Yang, L., Tomutsa, L., Brantley, S.L.: Evolution of porosity and diffusivity associated with chemical weathering of a basalt clast. J. Geophys. Res. 114(F2) (2009). https://doi.org/10.1029/2008jf001060
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 calculation. In: Techniques and Methods. Reston, VA, USA (2013)
Pearson, F.J., Arcos, D., Bath, A., Boisson, J.Y., Fernández, A.M., Gäbler, H.E., Gaucher, E., Gautschi, A., Griffault, L., Hernán, P., Waber, H.N.: Mont Terri Project—Geochemistry of Water in the Opalinus Clay Formation at the Mont Terri Rock Laboratory. In: Reports of Federal Office for Water and Geology. vol. 5, p. s. Bern, Switzerland (2003)
Philipp, T., Amann-Hildenbrand, A., Laurich, B., Desbois, G., Littke, R., Urai, J.L.: The effect of microstructural heterogeneity on pore size distribution and permeability in Opalinus Clay (Mont Terri, Switzerland): insights from an integrated study of laboratory fluid flow and pore morphology from BIB-SEM images. Geol. Soc. Lond. Special Pub. 454(1), 85–106 (2017). https://doi.org/10.1144/sp454.3
Steefel, C.I., Beckingham, L.E., Landrot, G.: Micro-continuum approaches for modeling pore-scale geochemical processes. Rev. Mineral. Geochem. 80(1), 217–246 (2015). https://doi.org/10.2138/rmg.2015.80.07
Thoenen, T., Hummel, W., Berner, U., Curti, E.: The PSI/Nagra Chemical Thermodynamic Database 12/07. In. Villigen PSI, Switzerland (2014)
Tournassat, C., Appelo, C.A.J.: Modelling approaches for anion-exclusion in compacted Na-bentonite. Geochim. Cosmochim. Acta 75(13), 3698–3710 (2011). https://doi.org/10.1016/j.gca.2011.04.001
Tournassat, C., Steefel, C.I.: Ionic transport in nano-porous clays with consideration of electrostatic effects. Rev. Mineral. Geochem. 80(1), 287–329 (2015). https://doi.org/10.2138/rmg.2015.80.09
Tournassat, C., Steefel, C.I.: Modeling diffusion processes in the presence of a diffuse layer at charged mineral surfaces: a benchmark exercise. Comput. Geosci. (2019). https://doi.org/10.1007/s10596-019-09845-4
Van Loon, L., Soler, J.M., Müller, W., Bradbury, M.H.: Anisotropic diffusion in layered argillaceous rocks: a case study with opalinus clay. Environ. Sci. Technol. 38(21), 5721–5728 (2004a)
Van Loon, L.R., Baeyens, B., Bradbury, M.H.: Diffusion and retention of sodium and strontium in Opalinus clay: Comparison of sorption data from diffusion and batch sorption measurements, and geochemical calculations. Appl. Geochem. 20(12), 2351–2363 (2005). https://doi.org/10.1016/j.apgeochem.2005.08.008
Van Loon, L.R., Soler, J.M., Bradbury, M.H.: Diffusion of HTO, 36Cl− and 125I− in opalinus clay samples from mont terri. J. Contam. Hydrol. 61(1–4), 73–83 (2003). https://doi.org/10.1016/s0169-7722(02)00114-6
Van Loon, L.R., Wersin, P., Soler, J.M., Eikenberg, J., Gimmi, T., Hernan, P., Dewonck, S., Savoye, S.: In-situ diffusion of HTO, 22Na+, Cs+ and I− in Opalinus Clay at the Mont Terri underground rock laboratory. Radiochim. Acta 92, 757–763 (2004b)
Wang, M., Wang, J., Pan, N., Chen, S.: Mesoscopic predictions of the effective thermal conductivity for microscale random porous media. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(3 Pt 2), 036702 (2007). https://doi.org/10.1103/PhysRevE.75.036702
Wersin, P., Soler, J.M., Van Loon, L., Eikenberg, J., Baeyens, B., Grolimund, D., Gimmi, T., Dewonck, S.: Diffusion of HTO, Br−, I−, Cs+, 85Sr2+ and 60Co2+ in a clay formation: results and modelling from an in situ experiment in Opalinus Clay. Appl. Geochem. 23(4), 678–691 (2008). https://doi.org/10.1016/j.apgeochem.2007.11.004
Wersin, P., Van Loon, L.R., Soler, J.M., Yllera, A., Eikenberg, J., Gimmi, T., Hernán, P., Boisson, J.Y.: Long-term diffusion experiment at Mont Terri: first results from field and laboratory data. Appl. Clay Sci. 26(1–4), 123–135 (2004). https://doi.org/10.1016/j.clay.2003.09.007
Xiong, Q., Jivkov, A.P.: Anion diffusion in clay-rich sedimentary rocks – A pore network modelling. Appl. Clay Sci. 161, 374–384 (2018). https://doi.org/10.1016/j.clay.2018.05.010
Yuan, T., Qin, G.: Numerical investigation of wormhole formation during matrix acidizing of carbonate rocks by coupling stokes-brinkman equation with reactive transport model under radial flow conditions. In: Paper presented at the SPE International Conference and Exhibition on Formation Damage Control, Lafayette, Louisiana, USA, 19–21 Feb 2020
Yuan, T., Wei, C., Zhang, C.-S., Qin, G.: A numerical simulator for modeling the coupling processes of subsurface fluid flow and reactive transport processes in fractured carbonate rocks. Water 11(10) (2019). https://doi.org/10.3390/w11101957
Zhao, T., Zhao, H., Ning, Z., Li, X., Wang, Q.: Permeability prediction of numerical reconstructed multiscale tight porous media using the representative elementary volume scale lattice Boltzmann method. Int. J. Heat Mass Transf. 118, 368–377 (2018). https://doi.org/10.1016/j.ijheatmasstransfer.2017.11.004
Acknowledgements
We thank our colleagues J. Kulenkampff, J. Schabernack, and T. Bollermann (HZDR) for fruitful discussions. We gratefully acknowledge funding by the German Federal Ministry of Education and Research (BMBF), grant 02NUK053B and the Helmholtz Association, grant SO-093 (iCross).
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German Federal Ministry of Education and Research (BMBF), grant 02NUK053B and the Helmholtz Association, grant SO-093 (iCross).
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The study was designed by CF and TY. TY performed numerical modeling and simulation. TY and CF performed data interpretation and discussed the results. TY drafted the manuscript and CF contributed to writing the paper.
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Yuan, T., Fischer, C. Effective Diffusivity Prediction of Radionuclides in Clay Formations Using an Integrated Upscaling Workflow. Transp Porous Med 138, 245–264 (2021). https://doi.org/10.1007/s11242-021-01596-0
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DOI: https://doi.org/10.1007/s11242-021-01596-0