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Non-Fickian Transport in Transparent Replicas of Rough-Walled Rock Fractures

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Abstract

We present an experimental investigation and modeling analysis of tracer transport in two transparent fracture replicas. The original fractures used in this work are a Vosges sandstone sample with nominal dimensions approximately 26 cm long and 15 cm wide, and a granite sample with nominal dimensions approximately 33 cm long and 15.5 cm wide. The aperture map and physical characteristics of the fractures reveal that the aperture map of the granite fracture has a higher spatial variability than the Vosges sandstone one. A conservative methylene blue aqueous solution was injected uniformly along the fracture inlets, and exited through free outlet boundaries. A series of images was recorded at known time intervals during each experiment. Breakthrough curves were subsequently determined at the fracture outlets and at different distances, using an image processing based on the attenuation law of Beer–Lambert. These curves were then interpreted using a stratified medium model that incorporates a permeability distribution to account for the fracture heterogeneity, and a continuous time random walk (CTRW) model, as well as the classical advection–dispersion equation (ADE). The stratified model provides generally satisfactory matches to the data, while the CTRW model captures the full evolution of the long tailing displayed by the breakthrough curves. The transport behavior is found to be non-Fickian, so that the ADE is not applicable. In both stratified and CTRW models, parameter values related to the aperture field spatial variability indicate that the granite fracture is more heterogeneous than the Vosges sandstone fracture.

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Abbreviations

\(a\) :

Power law constant, Eq. (16)

\(\alpha \) :

Dispersivity (m)

\(b\) :

Power law exponent, Eq. (16)

\(\beta \) :

Exponent, Eq. (8)

\(C\) :

Concentration (g/l)

\(C^{*}\) :

Dimensionless concentration

\(C_{0}\) :

Injected concentration (g/l)

\(D\) :

Dispersion coefficient (\(\text{ m }^{2}\)/s)

\(D_{m}\) :

Molecular diffusion coefficient (\({\text{ m }}^{2}\)/s)

\(D_{\psi }\) :

Transport dispersion coefficient (\({\text{ m }}^{2}\)/s)

\(\varepsilon \) :

Solute absorptivity (\({\text{ m }}^{2}\)/g)

\(G(k)\) :

Probability distribution function of the permeability

\(\gamma \) :

(Semi)variogram (\({\text{ mm }}^{2}\))

\(H\) :

Heterogeneity factor

\(h\) :

Thickness or local aperture (mm)

\({\langle }h{\rangle }\) :

Mean aperture (mm)

\(I\) :

Intensity

\(I_{0}\) :

Intensity at \(C=0\)

\(k\) :

Permeability (\(k=h^{2}/12\)) (\(\text{ m }^{2}\))

\({\langle }k{\rangle }\) :

Mean permeability (\(\text{ m }^{2}\))

\(L_{x}\) :

Fractures length (m)

\(L_{y}\) :

Fractures wide (m)

\(M\) :

Memory function

\(\mu \) :

Dynamic viscosity (mPa s)

\(N\) :

Number of measured values, Eq. (15)

\(N_{r}\) :

Number of pairs, Eq. (11)

\(P\) :

Pressure (Pa)

\(p(s)\) :

Probability distribution of transition displacements

\(Pe\) :

Peclet number

\(Q\) :

Flow rate (ml/h)

\(RMSE\) :

Root-mean-square error

\(\rho \) :

Density (kg/m\(^{3}\))

\(\sigma \) :

Standard deviation

\(t\) :

Time (s)

\(t^{*}\) :

Dimensionless time

\(t_{1}\) :

Median transition time in \(\psi \) (s)

\(t_{2}\) :

Cutoff time in \(\psi \) (s)

\(U\) :

Average fluid velocity (m/s)

\(u_{\psi }\) :

Transport velocity (m/s)

\(V_{\mathrm{p}}\) :

Pore volume (ml)

\(w\) :

Laplace variable

\(x\) :

Location in space (m)

\(x^{*}\) :

Dimensionless distance

\(y\) :

Location in space (m)

\(\psi (t)\) :

Probability rate for a transition time \(t\)

References

  • Aris, R.: On the dispersion of a solute particle in a fluid moving through a tube. Proc. R. Soc. London. Ser. A. 235, 67–77 (1956)

    Google Scholar 

  • Bauget, F., Fourar, M.: Non-Fickian dispersion in a single fracture. J. Contam. Hydrol. 100(3–4), 137–148 (2008). doi:10.1016/j.jconhyd.2008.06.005

    Article  Google Scholar 

  • Becker, M.W., Shapiro, A.M.: Tracer transport in fractured crystalline rock: evidence of nondiffusive breakthrough tailing. Water Resour. Res. 36(7), 1677–1686 (2000). doi:10.1029/2000WR900080

    Article  Google Scholar 

  • Berkowitz, B., Cortis, A., Dentz, M., Scher, H.: Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys. 44, RG2003 (2006). doi:10.1029/2005RG000178

    Article  Google Scholar 

  • Berkowitz, B., Kosakowski, G., Margolin, G., Scher, H.: Application of continuous time random walk theory to tracer test measurements in fractured and heterogeneous porous media. Ground Water 39(4), 593–604 (2001). doi:10.1111/j.1745-6584.2001.tb02347.x

    Article  Google Scholar 

  • Berkowitz, B., Scher, H.: Anomalous transport in random fracture networks. Phys. Rev. Lett. 79(20), 4038–4041 (1997). doi:10.1103/PhysRevLett.79.4038

    Article  Google Scholar 

  • Berkowitz, B., Scher, H.: Exploring the nature of non-Fickian transport in laboratory experiments. Adv. Water Res. 32(5), 750–755 (2008). doi:10.1016/j.advwatres.2008.05.004

    Article  Google Scholar 

  • Berkowitz, B., Scher, H., Silliman, S.E.: Anomalous transport in laboratory scale, heterogeneous porous media. Water Resour. Res. 36(1), 149–158 (2000). doi:10.1029/1999WR900295

    Article  Google Scholar 

  • Bolster, D., Valdés-Parada, F.J., LeBorgne, T., Dentz, M., Carrera, J.: Mixing in confined stratified aquifers. J. Contam. Hydrol. 120–120, 198–212 (2010). doi:10.1016/j.jconhyd.2010.02.003

  • Brown, S.R., Caprihan, A., Hardy, R.: Experimental observation of fluid flow channels in a single fracture. J. Geophys. Res. 103(B3), 5125–5132 (1998). doi:10.1029/97JB03542

    Article  Google Scholar 

  • Christakos, G., Bogaert, P., Serre, M.L.: Temporal GIS: Advanced Functions for Field-Based Applications. Springer, New York (2002)

    Google Scholar 

  • Communar, G.M.: A solute transport in stratified media. Transp. Porous Media 31(2), 133–134 (1998). doi:10.1023/A:1006557032656

    Article  Google Scholar 

  • Dentz, M., Cortis, A., Scher, H., Berkowitz, B.: Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport. Adv. Water Resour. 27(2), 155–173 (2003). doi:10.1016/j.advwatres.2003.11.002

    Article  Google Scholar 

  • Detwiler, R.L., Pringle, S.E., Glass, R.J.: Measurement of fracture aperture fields using transmitted light: an evaluation of measurement errors and their influence on simulations of flow and transport through a single fracture. Water Resour. Res. 35(9), 2605–2617 (1999). doi:10.1029/1999WR900164

    Article  Google Scholar 

  • Detwiler, R., Rajaram, H., Glass, R.J.: Solute transport in variable aperture fractures: An investigation of the relative importance of Taylor dispersion and macrodispersion. Water Resour. Res. 36(7), 1611–1625 (2000). doi:10.1029/2000WR900036

    Article  Google Scholar 

  • Didierjean, S., Amaral Souto, H.P., Christian, Moyne C.: Dispersion in periodic porous media. Experience versus theory for two-dimensional systems. Chem. Eng. Sci. 52(12), 1861–1874 (1997). doi:10.1016/S0009-2509(96)00518-0

    Article  Google Scholar 

  • Dronfield, D.G., Silliman, S.E.: Velocity dependence of dispersion for transport through a single fracture of variable roughness. Water Resour. Res. 29(10), 3477–3483 (1993). doi:10.1029/93WR01407

    Article  Google Scholar 

  • Fourar, M.: Characterization of heterogeneities at the core-scale using the equivalent stratified porous medium approach. In: Proceedings of the International Symposium of the Society of Core Analysts, Trondheim, 12–16 Sept 2006

  • Fourar M., Radilla G. (2009) Non-Fickian description of tracer transport through heterogeneous porous media. Transp. Porous Media (80)3:561–579. doi:10.1007/s11242-009-9380-7

  • Gao, G., Zhan, H., Feng, S., Huang, G., Mao, X.: Comparison of alternative models for simulating anomalous solute transport in a large heterogeneous soil column. J. Hydrol. 377(3–4), 391–404 (2009). doi:10.1016/j.jhydrol.2009.08.036

    Article  Google Scholar 

  • Güven, O., Molz, F.J., Melville, J.G.: An analysis of macrodispersion in a stratified aquifer. Water Resour. Res. 20(10), 1337–1354 (1984). doi:10.1029/WR020i010p01337

    Article  Google Scholar 

  • Hakami, E., Larsson, E.: Aperture measurements and flow experiments on a single natural fracture. Int. J. Rock Mech. Min. Sci. Geomech. Abstr 33(4), 395–404 (1996). doi:10.1016/0148-9062(95)00070-4

    Article  Google Scholar 

  • Ippolito, I., Hinch, E.J., Daccord, G., Hulin, J.P.: Tracer dispersion in 2-D fractures with flat and rough walls in a radial flow geometry. Phys. Fluids A 5(8), 1952–1962 (1993). doi:10.1063/1.858822

  • Isakov, E., Ogilvie, S.R., Taylor, C.W., Glover, P.W.J.: Fluid flow through rough fractures in rocks I: high resolution aperture determinations. Earth Planet. Sci. Lett. 191(3–4), 267–282 (2001). doi:10.1016/S0012-821X(01)00424-1

    Article  Google Scholar 

  • Iwano, M., Einstein, H.H.: Stochastic analysis of surface roughness, aperture and flow in a single fracture. In: Proceedings of the International Symposium EUROCK ’93, pp. 135–141, Lisbon (1993)

  • Jiménez-Hornero, F.J., Giráldez, J.V., Laguna, A., Pachepsky, Y.: Continuous time random walks for analyzing the transport of a passive tracer in a single fissure. Water Resour. Res. 41, W04009 (2005). doi:10.1029/2004WR003852

    Article  Google Scholar 

  • Johns, R.A., Steude, J.S., Castanier, L.M., Roberts, P.V.: Nondestructive measurements of fracture aperture in crystalline rock cores using X-ray computed tomography. J. Geophys. Res. 98(B2), 1889–1900 (1993). doi:10.1029/92JB02298

    Article  Google Scholar 

  • Kosakowski, G., Berkowitz, B., Scher, H.: Analysis of field observations of tracer transport in a fractured till. J. Contam. Hydrol. 47(1), 29–51 (2001). doi:10.1016/S0169-7722(00)00140-6

    Article  Google Scholar 

  • Kuntz, B.W., Rubin, S., Berkowitz, B., Singha, K.: Quantifying solute transport at the Shale Hills Critical Zone Observatory. Vadose Zone J. 10(3), 843–857 (2011). doi:10.2136/vzj2010.0130

    Article  Google Scholar 

  • Levy, M., Berkowitz, B.: Measurement and analysis of non-Fickian dispersion in heterogeneous porous media. J. Contam. Hydrol. 64(3–4), 203–226 (2003). doi:10.1016/S0169-7722(02)00204-8

    Article  Google Scholar 

  • Margolin, G., Berkowitz, B.: Application of continuous time random walks to transport in porous media. J. Phys. Chem. B104(16), 3942–3947 (2000). doi:10.1021/jp993721x

    Google Scholar 

  • Moreno, L., Neretnieks, I., Eriksen, T.: Analysis of some laboratory tracer runs in natural fissures. Water Resour. Res. 21(7), 951–958 (1985). doi:10.1029/WR021i007p00951

    Article  Google Scholar 

  • Moreno, L., Tsang, Y.W., Tsang, C.F., Hale, F.V., Neretnieks, I.: Flow and tracer transport in a single fracture: a stochastic model and its relation to some field observations. Water Resour. Res. 24(12), 2033–2048 (1988). doi:10.1029/WR024i012p02033

    Article  Google Scholar 

  • Neretnieks, I., Eriksen, T., Tahtinen, P.: Tracer movement in a single fissure in granitic rock: some experimental results and their interpretation. Water Resour. Res. 18(4), 849–858 (1982). doi:10.1029/WR018i004p00849

    Article  Google Scholar 

  • Nordqvist, A.W., Tsang, Y.W., Tsang, C.F., Dverstorp, B., Andersson, J.: Effects of high variance of fracture transmissivity on transport and sorption at different scales in a discrete model for fractured rocks. J. Contam. Hydrol. 22(1–2), 39–66 (1996). doi:10.1016/0169-7722(95)00064-X

    Article  Google Scholar 

  • Ogata, A., Banks, R.B.: A solution of the differential equation of longitudinal dispersion in porous media. U.S. Geol. Survey, Professional Papers, 411-A, pp. A1–A7 (1961)

  • Oron, A.P., Berkowitz, B.: Flow in rock fractures: The local cubic law assumption re-examined. Water Resour. Res. 34(11), 2811–2825 (1998). doi:10.1029/98WR02285

    Article  Google Scholar 

  • Pyrak-Nolte, L.J., Montemagno, C.D., Nolte, D.D.: Volumetric imaging of aperture distributions in connected fracture networks. Geophy. Res. Lett. 24(18), 2343–2346 (1997). doi:10.1029/97GL02057

    Article  Google Scholar 

  • Radilla, G., Sausse, J., Sanjuan, B., Fourar, M.: Interpreting tracer tests in the enhanced geothermal system (EGS) of Soultz-sous-Forêts using the equivalent stratified medium approach. Geothermics (2012). doi:10.1016/j.geothermics.2012.07.001

    Google Scholar 

  • Sharifzadeh, M., Mitani, Y., Esaki, T., Urakawa, F.: An investigation of joint aperture distribution using precise surface asperities measurement and GIS data processing. In: Ohnishi Y., Aoki K. (eds.) Proceedings of the 3rd International Asian Rock Mechanics Symposium, pp. 165–171, Kyoto (2004)

  • Taylor, G.I.: Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. London. Ser. A. 219, 186–203 (1953)

    Article  Google Scholar 

  • Tsang, C.F., Tsang, Y.W., Hale, F.V.: Tracer transport in fractures: analysis of field data based on a variable-aperture channel model. Water Resour. Res. 27(12), 3095–310 (1991). doi:10.1029/91WR02270

    Article  Google Scholar 

  • Wan, J., Tokunaga, T.K., Orr, T.R., O’Neill, J., Connors, R.W.: Glass casts of rock fracture surfaces: a new tool for studying flow and transport. Water Resour. Res. 36(1), 355–360 (2000). doi:10.1029/1999WR900289

    Article  Google Scholar 

  • Xiong, Y., Huang, G., Huang, Q.: Modeling solute transport in one-dimensional homogeneous and heterogeneous soil columns with continuous time random walk. J. Contam. Hydrol. 86(3–4), 163–175 (2006). doi:10.1016/j.jconhyd.2006.03.001

    Article  Google Scholar 

  • Yeo, I.W.: Effect of fracture roughness on solute transport. Geosciences J. 5(2), 145–151 (2001). doi:10.1007/BF02910419

    Article  Google Scholar 

  • Zavala-Sanchez, V., Dentz, M., Sanchez-Vila, X.: Characterization of mixing and spreading in a bounded stratified medium. Adv. Water Resour. 32(5), 635–648 (2009). doi:10.1016/j.advwatres.2008.05.003

    Article  Google Scholar 

  • Zimmerman, R.W., Bodvarsson, G.S.: Hydraulic conductivity of rock fractures. Transp. Porous Media 23(1), 1–30 (1996). doi:10.1007/BF00145263

    Article  Google Scholar 

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Authors and Affiliations

  1. LEMTA, Nancy-University, CNRS, 2 avenue de la Forêt de Haye, BP 160, 54504 , VANDOEUVRE CEDEX, France

    Ali Nowamooz, Giovanni Radilla & Mostafa Fourar

  2. Arts et Métiers ParisTech, rue St Dominique, BP 508, 51006, Châlons-en-Champagne, France

    Ali Nowamooz & Giovanni Radilla

  3. Department of Environmental Sciences and Energy Research, Weizmann Institute of Science, 76100 , Rehovot, Israel

    Brian Berkowitz

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  1. Ali Nowamooz
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  4. Brian Berkowitz
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Correspondence to Ali Nowamooz.

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Nowamooz, A., Radilla, G., Fourar, M. et al. Non-Fickian Transport in Transparent Replicas of Rough-Walled Rock Fractures. Transp Porous Med 98, 651–682 (2013). https://doi.org/10.1007/s11242-013-0165-7

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