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Solute Concentration Statistics in Heterogeneous Aquifers for Finite Péclet Values

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Abstract

A Lagrangian framework is used for analysing the concentration fields associated with transport of nonreactive solutes in heterogeneous aquifers. This is related to two components: advection by the random velocity field v(x) and pore-scale dispersion, characterized by the dispersion tensor D d; the relative effect of the two components is quantified by the Péclet number. The principal aim of this paper is to define the probability density function (pdf) of a nonreactive solute concentration and its relevant moments >C< and σ2 c as sampled on finite detection volumes. This problem could be relevant in technical applications such as risk analysis, field monitoring and pollution control. A method to compute the concentration statistical moments and pdf is developed in the paper on the basis of the reverse formulation widely adopted to study solute dispersion in turbulent flows. The main advantages of this approach are: (i) a closed form solution for concentration mean and variance is attained, in case of small size of the sampling volume; (ii) a numerically efficient estimate of the concentration pdf can be derived. The relative effects of injection and sampling volume size and Péclet number on concentration statistics are assessed. The analysis points out that the concentration pdf can be reasonably fitted by the beta function. These results are suitable to be employed in practical applications, when the estimate of probability related to concentration thresholds is required.

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References

  • Andričcević, R.: 1998, Effects of local dispersion and sampling volume on the evolution of concentration fluctuations in aquifers, Water Resour. Res. 34(5), 1115–1129.

    Google Scholar 

  • Bedient, P. B., Rifai, H. S. and Newell, C. J.: 1994, Ground Water Contamination: Transport and Remediation, Prentice Hall, Englewood Cliffs, NJ.

    Google Scholar 

  • Bellin, A., Rubin, Y. and Rinaldo, A.: 1994, Eulerian–Lagrangian approach for modeling of flow and transport in heterogeneous geological formations, Water Resour. Res. 30(11), 2913–2924.

    Google Scholar 

  • Benjamin, J. R. and Cornell, C. A.: 1970, Probability, Statistics, and Decision for Civil Engineers, McGraw-Hill Book Company, New York.

    Google Scholar 

  • Burr, D. T., Sudicky, E. A. and Naff R. L.: 1994, Nonreactive and reactive solute transport in three-dimensional porous media: mean displacement, plume spreading, and uncertainty, Water Resour. Res. 30(3), 791–814.

    Google Scholar 

  • Caroni, E. and Fiorotto, V.: 2000, On the statistics of concentration fluctuations in aquifer, in: Tracers and Modelling in Hydrogeology, IAHS Publ. No 262, pp. 115–120.

  • Chatwin, P. C. and Sullivan, P. J.: 1990, A simple and unifying physical interpretation of scale fluctuation measurement from many turbulent shear flows, J. Fluid Mech. 212, 533–566.

    Google Scholar 

  • Dagan, G.: 1984, Solute transport in heterogeneous porous formation, J. Fluid Mech. 145, 152–157.

    Google Scholar 

  • Dagan, G.: 1986, Statistical theory of groundwater flow and transport: pore to laboratory, laboratory to formations and formations to regional scale, Water Resour. Res. 22(9), 120s–134s.

    Google Scholar 

  • Dagan, G.: 1989, Flow and Transport in Porous Formation, Springer-Verlag, New York.

    Google Scholar 

  • Dagan, G.: 1991, Dispersion of a passive solute in non-ergodic transport by steady velocity fields in heterogeneous formation, J. Fluid Mech. 233, 197–210.

    Google Scholar 

  • Dagan, G. and Fiori, A.: 1997, The influence of pore-scale dispersion on concentration statistical moments in transport through heterogeneous aquifers, Water Resour. Res. 33(7), 1595–1605.

    Google Scholar 

  • Egbert, G. D. and Baker, M. B.: 1984, Comments on paper ‘The effect of Gaussian particle-pair distribution functions in the statistical theory of concentration fluctuations in homogeneous turbulence’ and reply by B. L. Sawford, Quart. J. R. Met. Soc. 110, 1195–1200.

    Google Scholar 

  • Fiori, A.: 1996, Finite Peclet extension of Dagan's solutions to transport in anisotropic heterogeneous formations, Water Resour. Res. 32, 193–198.

    Google Scholar 

  • Fiori, A. and Dagan, G.: 1999, Concentration fluctuations in transport by groundwater: comparison between theory and field experiments, Water Resour. Res. 35(1), 105–112.

    Google Scholar 

  • Fiorotto, V.: 1992, Non linear effects in the velocity field related to the porous formations heterogen-eity (in italian), in: Proc. XXIII Convegno di Idraulica e Costruzioni Idrauliche, Vol. 1, Firenze, pp. B37–B48.

    Google Scholar 

  • Fitts, C. R.: 1996, Uncertainty in deterministic groundwater transport models due to the assumption of macrodispersive mixing: evidence from the Cape Cod (Massachusetts, U.S.A.) and Borden (Ontario, Canada) tracer tests, J. Contam. Hydrol. 23, 69–84.

    Google Scholar 

  • Freyberg, D. L.: 1986, A natural gradient experiment of solute transport in a sand aquifer, 2, spatial moment and the advection and dispersion of non-reactive tracers, Water Resour. Res. 22(13), 2031–2046.

    Google Scholar 

  • Garabedian, S. P., Leblanc, D. R., Gelhar, L. W. and Celia, M. A.: 1991, Large-scale natural gradient test in sand gravel, Cape Cod, Massachusetts: analysis of spatial moments for a non-reactive tracer, Water Resour. Res. 27(5), 911–924.

    Google Scholar 

  • Graham, W. D. and McLaughlin: 1989a, Stochastic analysis of nonstationarity subsurface solute transport, 1, unconditional moments, Water Resour. Res. 25(2), 215–232.

    Google Scholar 

  • Graham, W. D. and McLaughlin, 1989b, Stochastic analysis of nonstationarity subsurface solute transport, 2, conditional moments, Water Resour. Res. 25(2), 215–232.

    Google Scholar 

  • Hinze, J. O.: 1975, Turbulence, 2nd edn., reissued 1987, McGraw Hill, New York.

    Google Scholar 

  • Hsu, K. C., Zhang, D. and Neuman, S. P.: 1996, Higher-order effects on flow and transport in randomly heterogeneous porous media, Water Resour. Res. 32(3), 571–582.

    Google Scholar 

  • Kapoor, V. and Gelhar, L. W.: 1994a, Transport in three-dimensionally heterogeneous aquifer 1: dynamics of concentration fluctuations, Water Resour. Res. 30(6), 1775–1788.

    Google Scholar 

  • Kapoor, V. and Gelhar, L. W.: 1994b, Transport in three-dimensionally heterogeneous aquifer 2: prediction and observations of concentration fluctuations, Water Resour. Res. 30(6), 1789–1801.

    Google Scholar 

  • Kapoor, V. and Kitanidis, P. K.: 1997, Advection-diffusion in spatially random flows: formulation of concentration covariance, Stochast. Hydrol. Hydr. 11(5), 397–422.

    Google Scholar 

  • Kapoor, V. and Kitanidis P. K.: 1998, Concentration fluctuations and dilution in aquifers, Wat er Resour. Res. 34(5), 1181–1193.

    Google Scholar 

  • LeBlanc, D. R., Garabedian, S. P., Hess, K. M., Gelhar L. W., Quadri, R. D., Stollenwerk, K. G. and Wood, W. W.: 1991, Large-scale natural gradient tracer test in sand and gravel, Cape Cod, Massachusetts, 1, experimental design and observed tracer movement, Water Resour. Res. 27(5), 895–910.

    Google Scholar 

  • Lundgren, T. S.: 1995, Turbulent pair dispersion and scalar diffusion, J. Fluid Mech. 111, 27–57, 1981.

    Google Scholar 

  • NAG: Fortran Library, Nag. Ltd, Oxford, U.K.

  • Rubin, Y.: 1991a, Prediction of tracer plume migration in disordered porous media by the method of conditional probabilities, Water Resour. Res. 27(6), 1291–1308.

    Google Scholar 

  • Rubin, Y.: 1991b, Transport in heterogeneous porous media: prediction and uncertainty, Water Resour. Res. 27(7), 1723–1738.

    Google Scholar 

  • Rubin, Y., Cushey, M. A. and Bellin, A.: 1994, Modeling of transport in groundwater for environmental risk assessment, Stochast. Hydrol. Hydr. 8, 57–77.

    Google Scholar 

  • Salandin, P. and Fiorotto V.: 1998, Solute transport in highly heterogenous aquifers, Water Resour. Res. 34(5), 949–961.

    Google Scholar 

  • Salandin, P. and Fiorotto, V.: 2000, Dispersion tensor evaluation in heterogeneous media for finite Peclet number, Water Resour. Res. 36(6), 1449–1458.

    Google Scholar 

  • Sawford, B. L.: 1983, The effect of Gaussian particle-pair distribution functions in the statistical theory of concentration fluctuations in homogeneous turbulence, Quart. J. R. Met. Soc. 109, 339–354.

    Google Scholar 

  • Sudicky, E. A.: 1986, A natural gradient experiment on solute transport in a sand aquifer: spatial variability of hydraulic conductivity and its role in the dispersion process, Water Resour. Res. 22(13), 2069–2082.

    Google Scholar 

  • Taylor, G. I.: 1921, Diffusion by continuous movements, Proc. London Math. Soc. Ser. 2, 20, pp.196–212.

    Google Scholar 

  • Thomson, D. J.: 1986, On the relative dispersion of two particles in homogeneous stationarity turbu-lence and implications for size of concentration fluctuations at large times, Quart. J. R. Met. Soc. 112, 890–894.

    Google Scholar 

  • Thomson, D. J.: 1990, A stochastic model for the motion of particle pairs in isotropic high-Reynolds turbulence, and its application to the problem of concentration variance, J. Fluid Mech. 111, 113–153.

    Google Scholar 

  • Van Lent, T. and Kitanidis, P. K.: 1989, A numerical spectral approach for the derivation of piezometric head covariance functions, Water Resour. Res. 25(11), 2287–2298.

    Google Scholar 

  • Vomvoris, E. G. and Gelhar, L. W.: 1990, Stochastic analysis of the concentration variability in a three-dimensional heterogeneous aquifer, Water Resour. Res. 26(10), 2591–2602.

    Google Scholar 

  • Wise, D.L.and Trantolo, D.J.:Remediation of Hazardous Waste Contaminated Soils, Marcel Dekker, New York.

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Authors and Affiliations

  1. Dipartimento di Ingegneria Civile, Università di Trieste, Piazzale Europa, 1, 34127, Trieste, Italy

    Virgilio Fiorotto & Elpidio Caroni

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  1. Virgilio Fiorotto
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  2. Elpidio Caroni
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Fiorotto, V., Caroni, E. Solute Concentration Statistics in Heterogeneous Aquifers for Finite Péclet Values. Transport in Porous Media 48, 331–351 (2002). https://doi.org/10.1023/A:1015744421033

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