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Equivalent diffusion coefficient of clay-rich geological formations: comparison between numerical and analytical estimates

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Abstract

Laboratory experiments in rock samples collected from clay-rich formations indicate that the effective molecular diffusion coefficient (D) is a heterogeneous and anisotropic property. Since laboratory measurements of D are representative of a very small volume, upscaling is necessary in order to incorporate these data in large-scale numerical models of diffusive transport. In this work we address the problem of the estimating the equivalent diffusion coefficient (D eq ), in terms of total diffusive flux, in a three-dimensional domain characterized by a heterogeneous and anisotropic spatial distribution of D. D eq was estimated from the results of steady-state diffusive transport simulations through several realizations of the D field. The ensemble averages of D eq from fields with different degrees of heterogeneity and anisotropy were then compared with estimates from analytical upscaling expressions based on stochastic as well as power-averaging approaches. These expressions are largely based on similar expressions developed for calculating the effective hydraulic conductivity in heterogeneous and anisotropic domains. Comparisons showed that stochastic expressions provide accurate estimates of D eq only for fields characterized by low heterogeneity. Within the range of heterogeneity and anisotropy considered, our results showed that a power-averaging expression is very accurate in predicting D eq especially when the parameter p i is estimated through fitting of the numerical results. Nonetheless, the relationship between this parameter and the anisotropy ratio is linear.

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References

  • Ababou R (1991) Identification of effective conductivity tensor in randomly heterogeneous and stratified Aquifers. In: Bachu S (ed) Proceedings of the 5th Canadian/American conference on hydrogeology: parameter identification and estimation for aquifer and reservoir characterization. National Well Water Association, Dublin, pp 155–157

  • Aertsens M, Wemaere I, Wouters L (2004) Spatial variability of transport parameters in the Boom Clay. Appl Clay Sci 26:37–45

    Article  CAS  Google Scholar 

  • ANDRA (2005) Dossier 2005 Référentiel du site Meuse/Haute-Marne. Rapport Andra n° C RP ADS 04-0022

  • Appelo CAJ, Wersin P (2007) 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

    Article  CAS  Google Scholar 

  • Appelo CAJ, Van Loon LR, Wersin P (2010) 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

    Article  CAS  Google Scholar 

  • Bear J (1979) Hydraulics of groundwater. Dover Publications, Mineola

    Google Scholar 

  • Bianchi M, Zheng C, Wilson C, Tick GR, Liu G, Gorelick SM (2011) Spatial connectivity in a highly heterogeneous aquifer: From cores to preferential flow paths. Water Resour Res 47:W05524. doi:10.1029/2009WR008966

    Article  Google Scholar 

  • Bradbury KR, Gotkowitz MG, Cherry JA, Hart D J, Eaton TT, Parker BL, Borchardt MA (2007) Contaminant transport through aquitards: technical guidance for aquitard assessment. Awwa Research Foundation, report 91133B, Denver

  • Cardwell WT, Parsons RL (1945) Averaging permeability of heterogeneous oil sands. Trans Am Inst Min Metall Petroleum Eng 160:34–42

    Google Scholar 

  • Cherry JA, Parker BL, Bradbury KR, Eaton TT, Gotkowitz MG, Hart DJ, Borchardt MA (2007) contaminant transport through aquitards: a state of the science review. Awwa Research Foundation, Report 91133A. Denver

  • Churakov S, Gimmi T (2011) Up-scaling of molecular diffusion coefficients in clays: a two-step approach. J Phys Chem C 115(14):6703–6714

    Article  CAS  Google Scholar 

  • Cormenzana JL, García-Gutiérrez M, Missana T, Alonso Ú (2008) Modelling large-scale laboratory HTO and strontium diffusion experiments in Mont Terri and Bure clay rocks. Phys Chem Earth 33(14–16):949–956

    Google Scholar 

  • Dagan G (1984) Solute transport in heterogenous porous formations. J Fluid Mech 145:151–177

    Article  Google Scholar 

  • Dagan G (1988) Time-dependent macrodispersion for solute transport in anisotropic heterogeneous aquifers. Water Resour Res 24(9):1491–1500

    Article  CAS  Google Scholar 

  • Dai Z, Wolfsberg A, Lu Z, Reimus P (2007) Upscaling matrix diffusion coefficients for heterogeneous fractured rocks. Geophys Res Lett 34:L07408. doi:10.1029/2007GL029332

    Article  Google Scholar 

  • De Wit A (1995) Correlation structure dependence of the effective permeability of heterogeneous porous-media. Phys Fluids 7(11):2553–2562

    Article  Google Scholar 

  • Desbarats AJ (1992) Spatial averaging of hydraulic conductivity in three-dimensional heterogeneous porous media. Math Geol 24(3):249–267

    Article  Google Scholar 

  • Descostes M, Blin V, Bazer-Bachi F, Meier P, Grenut B, Radwan J, Schlegel ML, Buschaert S, Coelho D, Tevissen E (2008) Diffusion of anionic species in Callovo-Oxfordian argillites and Oxfordian limestones (Meuse/Haute–Marne, France). Appl Geochem 23:655–677

    Article  CAS  Google Scholar 

  • Frippiat C, Holeyman AE (2008) A comparative review of upscaling methods for solute transport in heterogeneous porous media. J Hydrol 362(1–2):150–176

    Article  Google Scholar 

  • García-Gutiérrez M, Cormenzana JL, Missana T, Mingarro M, Martín PL (2006) Large-scale laboratory diffusion experiments in clay rocks. Phys Chem Earth 31:523–530

    Article  Google Scholar 

  • García-Gutiérrez M, Cormenzana JL, Missana T, Mingarro M, Alonso U, Samper J, Yang Q, Yi S (2008) Diffusion experiments in Callovo-Oxfordian clay from the Meuse/Haute-Marne URL, France. Experimental setup and data analyses. Phys Chem Earth Parts A/B/C 33(Supplement 1):S125–S130

    Google Scholar 

  • Gelhar L (1993) Stochastic subsurface hydrology. Prentice Hall, 390 pp

  • Gelhar LW, Axness CL (1983) Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resour Res 19(1):161–180

    Article  Google Scholar 

  • Gelhar LW, Gutjahr AL, Naff RL (1979) Stochastic analysis of macrodispersion in a stratified aquifer. Water Resour Res 15(6):1387–1397

    Article  Google Scholar 

  • Gomez-Hernandez JJ, Wen X-H (1998) To be or not to be multi-Gaussian? A reflection on stochastic hydrogeology. Adv Water Resour 21(1):47–61

    Article  Google Scholar 

  • Huysmans M, Dassargues A (2006) Stochastic analysis of the effect of spatial variability of diffusion parameters on radionuclide transport in a low permeability clay layer. Hydrogeol J 14:1094–1106

    Article  CAS  Google Scholar 

  • Huysmans M, Dassargues A (2007) Equivalent diffusion coefficient and equivalent diffusion accessible porosity of a stratified porous medium. Transp Porous Media 66(3):421–438

    Article  Google Scholar 

  • Indelman P, Abramovich B (1994) A higher-order approximation to effective conductivity in media of anisotropic random structure. Water Resour Res 30(6):1857–1864

    Article  Google Scholar 

  • Jougnot D, Revil A, Leroy P (2009) Diffusion of ionic tracers in the Callovo-Oxfordian clay-rock using the Donnan equilibrium model and the electrical formation factor. Geochim Cosmochim Acta 73:2712–2726

    Article  CAS  Google Scholar 

  • Journel AG, Deutsch CV, Desbarats AJ (1986) Power averaging for block effective permeability. Paper presented at 56th California Regional Meeting, Society of Petroleum Engineers, Oakland, CA

  • Matheron G (1967) Elements pour une Théorie des Miliéux Poruex. Masson, Paris

    Google Scholar 

  • Mazurek M, Alt-Epping P, Bath A, Gimmi T, Waber NH, Buschaert S, De Cannière P, De Craen M, Gautschi A, Savoye S, Vinsot A, Wemaere I, Wouters L (2011) Natural tracer profiles across argillaceous formations. Appl Geochem 6(7):1035–1064

    Article  Google Scholar 

  • NAGRA (2002) Project Opalinus Clay: safety report. Demonstration of disposal feasibility for spent fuel, vitrified high-level waste and long-lived intermediate level waste (Entsorgungsnachweis). Nagra technical report NTB 02-05, Wettingen

  • Neuman SP, Winter CL, Newman CM (1987) Stochastic theory of field-scale Fickian dispersion, in ansiotropic porous media. Water Resour Res 23(3):453–466

    Article  CAS  Google Scholar 

  • Ochs M, Lothenbach B, Shibata M, Sato H, Yui M (2003) Sensitivity analysis of radionuclide migration in compacted bentonite: a mechanistic model approach. J Contam Hydrol 61(1–4):313–328

    Article  CAS  Google Scholar 

  • Paleologos EK, Sarris T, Desbarats A (2000) Numerical estimation of effective hydraulic conductivity in leaky heterogeneous aquitards. GSA Special Volume: “Theory, Modeling and Field Investigation in Hydrogeology: A Special Volume in Honor of Shlomo P. Neuman’s 60th Birthday”: 119–127

  • Patriarche D, Michelet J-L, Ledoux E, Savoye S (2004) Diffusion as the main process for mass transport in very low water content argillites: 1. Chloride as a natural tracer for mass transport—diffusion coefficient and concentration measurements in interstitial water. Water Resour Res 40(1):W01517. doi:10.1029/2003WR002700

    Google Scholar 

  • Rehfeldt KR, Boggs JM, Gelhar LW (1992) Field study of dispersion in a heterogeneous aquifer: 3. Geostatistical analysis of hydraulic conductivity. Water Resour Res 28(12):3309–3324. doi:10.1029/92WR01758

    Article  Google Scholar 

  • Remy N, Boucher A, Wu J (2009) Applied geostatistics with SGeMS: a user’s guide. Cambridge University Press, New York

    Book  Google Scholar 

  • Renard P, de Marsily G (1997) Calculating equivalent permeability: a review. Adv Water Resour 20:253–278

    Article  Google Scholar 

  • Revil A, Linde N (2006) Chemico-electromechanical coupling in microporous media. J Colloid Interface Sci 302:682–694

    Article  CAS  Google Scholar 

  • Rubin Y (1990) Stochastic modeling of macrodispersion in heterogeneous porous media. Water Resour Res 26(1):133–141

    Article  CAS  Google Scholar 

  • Rubin Y, Dagan G (1988) Stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers, 1: constant head boundary. Water Resour Res 24(10):1689–1697

    Article  Google Scholar 

  • Sammartino S, Bouchet A, Prêt D, Parneix JC, Tevissen E (2003) Spatial distribution of porosity and minerals in clay rocks from the Callovo-Oxfordian formation (Meuse/Haute-Marne, Eastern France)—implications on ionic species diffusion and rock sorption capability. Appl Clay Sci 23:157–166

    Article  CAS  Google Scholar 

  • Samper J, Yang Q, Yi S, García-Gutiérrez M, Missana T, Mingarro M, Alonso Ú, Cormenzana JL (2008) Numerical modelling of large-scale solid-source diffusion experiment in Callovo-Oxfordian clay. Phys Chem Earth 33(Suppl. 1):S208–S215

    Google Scholar 

  • Sanchez-Vila X, Guadagnini A, Carrera J (2006) Representative hydraulic conductivities in saturated groundwater flow. Rev Geophys 44:RG3002. doi:10.1029/2005RG000169

    Article  Google Scholar 

  • Sarris T, Paleologos EK (2004) Numerical investigation of the anisotropic hydraulic conductivity behavior in heterogeneous porous media. Stoch Environ Res Risk Assess 18:188–197

    Article  Google Scholar 

  • Sato H, Yui M, Yoshikawa H (1994) Diffusion behavior for Se and Zr in Sodium-Bentonite. In: MRS Proceedings 353, pp 269–353

  • Van Loon LR, Soler JM, Muller W, Bradbury MH (2004a) Anisotropic diffusion in layered argillaceous rocks: a case study with opalinus clay. Environ Sci Technol 38:5721–5728

    Article  Google Scholar 

  • Van Loon LR, Wersin P, Soler JM, Eikenberg J, Gimmi T, Hernan P, Dewonck S, Savoye S (2004b) In situ diffusion of HTO, 22Na+, Cs+ and I− in Opalinus Clay at the Mont Terri underground rock laboratory. Radiochim Acta 92:757–763

    Article  Google Scholar 

  • Van Loon LR, Baeyens B, Bradbury MH (2005) 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 20:2351–2363

    Article  Google Scholar 

  • Van Marcke Ph, Laenen B (2005) The Ypresian clays as possible host rock for radioactive waste disposal: an evaluation. ONDRAF/NIRAS, Brussel

  • Wen X-H, Gómez-Hernández J (1996) Upscaling hydraulic conductivities in heterogeneous media: an overview. J Hydrol 183(1–2):ix–xxxii

    Article  CAS  Google Scholar 

  • Zhang D, Neuman SP (1996) Effect of local dispersion on solute transport in randomly heterogeneous media. Water Resour Res 32(9):2715–2723

    Article  Google Scholar 

  • Zhang Y, Gable CW, Sheets B (2010) Equivalent hydraulic conductivity of three-dimensional heterogeneous porous media: an upscaling study based on an experimental stratigraphy. J Hydrol 388(3–4):304–320

    Article  Google Scholar 

  • Zheng C (2010) MT3DMS v5.3 a modular three-dimensional multispecies transport model for simulation of advection, dispersion and chemical reactions of contaminants in groundwater systems. Supplemental user’s guide, technical report to the U.S. Army Engineer Research and Development Center, Department of Geological Sciences, University of Alabama

  • Zhou L, Selim HM (2003) Scale-dependent dispersion in soils: an overview. Adv Agron 80:223–263

    Article  Google Scholar 

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Acknowledgments

Funding for this work was provided by the Used Fuel Disposition Campaign, Office of Nuclear Energy, of the U.S. Department of Energy under Contract Number DE-AC02-05CH11231 with the Lawrence Berkeley National Lab. We thank the Associate Editor and two anonymous referees for their careful review of this manuscript.

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Authors and Affiliations

  1. Earth Sciences Division, Lawrence Berkeley National Laboratory, One Cyclotron Road, Blgd. 74, Berkeley, CA, 94720, USA

    Marco Bianchi & Jens T. Birkholzer

  2. Earth Sciences Division, Lawrence Berkeley National Laboratory, One Cyclotron Road, Blgd. 84, Berkeley, CA, 94720, USA

    Hui-Hai Liu

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  1. Marco Bianchi
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Correspondence to Marco Bianchi.

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Bianchi, M., Liu, HH. & Birkholzer, J.T. Equivalent diffusion coefficient of clay-rich geological formations: comparison between numerical and analytical estimates. Stoch Environ Res Risk Assess 27, 1081–1091 (2013). https://doi.org/10.1007/s00477-012-0646-1

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