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Modelling of block-scale macrodispersion as a random function

Published online by Cambridge University Press:  19 April 2011

FELIPE P. J. DE BARROS*
Affiliation:
Institute of Applied Analysis and Numerical Simulations/SimTech, University of Stuttgart, Pfaffenwaldring 7a, 705969 Stuttgart, Germany
YORAM RUBIN
Affiliation:
Department of Civil and Environmental Engineering, University of California, 627 Davis Hall, Berkeley, CA 94720-1710, USA
*
Email address for correspondence: felipe.debarros@simtech.uni-stuttgart.de

Abstract

Numerical modelling of solute dispersion in natural heterogeneous porous media is facing several challenges. Amongst these we highlight the challenge of accounting for high-frequency variability that is filtered out by homogenization at the subgrid scale and the uncertainty in the dispersive flux for transport under non-ergodic conditions. These two effects when combined lead to inaccurate representation of the dispersive fluxes. We propose to compensate for this deficiency by defining a block-scale dispersion tensor and modelling it as a random space function ℳij. The derived dispersion tensor is a function of several length scales and time. Grid blocks will be assigned dispersion coefficients generated from the ℳij distribution. We will show the dependence of ℳij on the spatial variability of the conductivity field, on the contaminant source size, on the travel time and on the grid-block scale. For an ergodic source, a statistically uniform conductivity field and very large grid blocks, ℳij is equal to the macrodispersion coefficients proposed by Dagan (J. Fluid Mech., vol. 145, 1984, p. 151) with zero variance. For an ergodic source and non-uniform conductivity field with a finite-size grid block, ℳij approaches the model proposed by Rubin et al. (J. Fluid Mech., vol. 395, 1999, p. 161). In both cases, ℳij is defined by its mean value with zero variance. ℳij is subject to uncertainty when the source is non-ergodic and when the grid block is defined by a finite scale. When the grid-block scale approaches zero, which means that the spatial variability is captured completely on the numerical grid, ℳij approaches zero with zero variance. In addition, we provide a complete statistical characterization of ℳij by invoking the concept of minimum relative entropy, thus providing upper bounds on the uncertainty associated with ℳij.

Type
Papers
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Ababou, R., McLaughlin, D., Gelhar, L. W. & Tompson, A. F. B. 1989 Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media. Transp. Porous Media 4 (6), 549565.Google Scholar
Andricevic, R. & Cvetkovic, V. 1998 Relative dispersion for solute flux in aquifers. J. Fluid Mech. 361, 145174.CrossRefGoogle Scholar
Attinger, S., Dentz, M., Kinzelbach, H. & Kinzelbach, W. 1999 Temporal behaviour of a solute cloud in a chemically heterogeneous porous medium. J. Fluid Mech. 386, 77104.Google Scholar
Attinger, S., Dentz, M. & Kinzelbach, W. 2004 Exact transverse macro dispersion coefficients for transport in heterogeneous porous media. Stochastic Environ. Res. Risk Assessment 18 (1), 915.Google Scholar
Auriault, J. L. & Adler, P. M. 1995 Taylor dispersion in porous media: analysis by multiple scale expansions. Adv. Water Resour. 18 (4), 217226.Google Scholar
Batchelor, G. K. 1952 Diffusion in a field of homogeneous turbulence. II. The relative motion of particles. Proc. Camb. Phil. Soc. 48, 345362.Google Scholar
Beckie, R., Aldama, A. A. & Wood, E. F. 1996 Modeling the large-scale dynamics of saturated groundwater flow using spatial-filtering theory. 1. Theoretical development. Water Resour. Res. 32 (5), 12691280.Google Scholar
Bellin, A., Lawrence, A. E. & Rubin, Y. 2004 Models of sub-grid variability in numerical simulations of solute transport in heterogeneous porous formations: three-dimensional flow and effect of pore-scale dispersion. Stochastic Environ. Res. Risk Assessment 18 (1), 3138.Google Scholar
Bellin, A., Salandin, P. & Rinaldo, A. 1992 Simulation of dispersion in heterogeneous porous formations: statistics, first-order theories, convergence of computations. Water Resour. Res. 28 (9), 22112227.CrossRefGoogle Scholar
Bolster, D., Benson, D. A., Le Borgne, T. & Dentz, M. 2010 Anomalous mixing and reaction induced by superdiffusive nonlocal transport. Phys. Rev. E 82 (2), 21119.CrossRefGoogle ScholarPubMed
Bolster, D., Dentz, M. & Le Borgne, T. 2009 Solute dispersion in channels with periodically varying apertures. Phys. Fluids 21, 056601.Google Scholar
Bolster, D., Valdés-Parada, F. J., LeBorgne, T., Dentz, M. & Carrera, J. 2011 Mixing in confined stratified aquifers. J. Contam. Hydrol. 120–121, 198212.CrossRefGoogle ScholarPubMed
Bras, R. L. & Rodríguez-Iturbe, I. 1994 Random Functions and Hydrology. Dover.Google Scholar
Chatwin, P. C. & Allen, C. M. 1985 Mathematical models of dispersion in rivers and estuaries. Annu. Rev. Fluid Mech. 17, 119149.CrossRefGoogle Scholar
Chatwin, P. C. & Sullivan, P. J. 1979 The relative diffusion of a cloud of passive contaminant in incompressible turbulent flow. J. Fluid Mech. 91 (2), 337355.CrossRefGoogle Scholar
Cirpka, O. A. & Kitanidis, P. K. 2000 Characterization of mixing and dilution in heterogeneous aquifers by means of local temporal moments. Water Resour. Res. 36 (5), 1221–1136.Google Scholar
Cushman, J. H. & Moroni, M. 2001 Statistical mechanics with three-dimensional particle tracking velocimetry experiments in the study of anomalous dispersion. I. Theory. Phys. Fluids 13, 75.CrossRefGoogle Scholar
Cvetkovic, V. & Dagan, G. 1994 Transport of kinetically sorbing solute by steady random velocity in heterogenous porous formations. J. Fluid Mech. 265, 189215.CrossRefGoogle Scholar
Cvetkovic, V., Dagan, G. & Cheng, H. 1998 Contaminant transport in aquifers with spatially variable hydraulic and sorption properties. Proc. Math. Phys. Engng Sci. 454 (1976), 21732207.Google Scholar
Cvetkovic, V. & Shapiro, A. M. 1989 Solute advection in stratified formations. Water Resour. Res. 25 (6), 12831289.Google Scholar
Dagan, G. 1984 Solute transport in heterogeneous porous formations. J. Fluid Mech. 145, 151177.CrossRefGoogle Scholar
Dagan, G. 1987 Theory of solute transport by groundwater. Annu. Rev. Fluid Mech. 19, 183215.Google Scholar
Dagan, G. 1989 Flow and Transport in Porous Formations. Springer.Google Scholar
Dagan, G. 1990 Transport in heterogeneous porous formations: spatial moments, ergodicity & effective dispersion. Water Resour. Res. 26 (6), 12811290.CrossRefGoogle Scholar
Dagan, G. 1991 Dispersion of a passive solute in non-ergodic transport by steady velocity fields in heterogeneous formations. J. Fluid Mech. 233, 197210.CrossRefGoogle Scholar
Dagan, G. 1994 Transport by two-dimensional random velocity fields: effective dispersion coefficients of a finite plume. Stochastic Meth. Hydrol. Environ. Engng 2, 113126.CrossRefGoogle Scholar
Dagan, G. & Fiori, A. 1997 The influence of pore-scale dispersion on concentration statistical moments in transport through heterogeneous aquifers. Water Resour. Res. 33 (7), 15951605.Google Scholar
Dagan, G., Fiori, A. & Jankovic, I. 2003 Flow and transport in highly heterogeneous formations. Part 1. Conceptual framework and validity of first-order approximations. Water Resour. Res. 39 (9), 1268.Google Scholar
Dentz, M. & Carrera, J. 2007 Mixing and spreading in stratified flow. Phys. Fluids 19, 017107.Google Scholar
Dentz, M., Kinzelbach, H., Attinger, S. & Kinzelbach, W. 2000 a Temporal behaviour of a solute cloud in a heterogeneous porous medium. 1. Point-like injection. Water Resour. Res. 36 (12), 35913604.Google Scholar
Dentz, M., Kinzelbach, H., Attinger, S. & Kinzelbach, W. 2000 b Temporal behaviour of a solute cloud in a heterogeneous porous medium. 2. Spatially extended injection. Water Resour. Res. 36 (12), 36053614.CrossRefGoogle Scholar
Dentz, M. & Tartakovsky, D. M. 2008 Self-consistent four-point closure for transport in steady random flows. Phys. Rev. E 77 (6), 66307.Google Scholar
de Dreuzy, J. R., Beaudoin, A. & Erhel, J. 2007 Asymptotic dispersion in 2D heterogeneous porous media determined by parallel numerical simulations. Water Resour. Res. 43, W10439.CrossRefGoogle Scholar
Eberhard, J. 2004 Approximations for transport parameters and self-averaging properties for point-like injections in heterogeneous media. J. Phys. A: Math. Gen. 37, 25492571.CrossRefGoogle Scholar
Eberhard, J. 2005 Upscaling for stationary transport in heterogeneous porous media. Multiscale Model. Simulation (SIAM) 3 (4), 957976.Google Scholar
Elder, J. W. 1959 The dispersion of marked fluid in turbulent shear flow. J. Fluid Mech. 5 (4), 544560.CrossRefGoogle Scholar
Fiori, A. 1996 Finite Peclet extensions of Dagan's solutions to transport in anisotropic heterogeneous formations. Water Resour. Res. 32 (1), 193198.CrossRefGoogle Scholar
Fiori, A. 1998 On the influence of pore-scale dispersion in nonergodic transport in heterogeneous formations. Transp. Porous Media 30 (1), 5773.Google Scholar
Fiori, A., Boso, F., de Barros, F. P. J., De Bartolo, S., Frampton, A., Severino, G., Suweis, S. & Dagan, G. 2010 An indirect assessment on the impact of connectivity of conductivity classes upon longitudinal asymptotic macrodispersivity. Water Resour. Res. 46 (8), W08601.CrossRefGoogle Scholar
Fiori, A. & Dagan, G. 2000 Concentration fluctuations in aquifer transport: a rigorous first-order solution and applications. J. Contam. Hydrol. 45 (1–2), 139163.Google Scholar
Fiori, A. & Dagan, G. 2002 Transport of a passive scalar in a stratified porous medium. Transp. Porous Media 47 (1), 8198.CrossRefGoogle Scholar
Fischer, H. B. 1973 Longitudinal dispersion and turbulent mixing in open-channel flow. Annu. Rev. Fluid Mech. 5 (1), 5978.CrossRefGoogle Scholar
Fischer, H. B., List, J. E., Koh, R. C. Y., Imberger, J. & Brooks, N. H. 1979 Mixing in Inland and Coastal Waters. Academic.Google Scholar
Gelhar, L. W. & Axness, C. L. 1983 Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resour. Res. 19 (1), 161180.CrossRefGoogle Scholar
Gotovac, H., Cvetkovic, V. & Andricevic, R. 2009 Flow and travel time statistics in highly heterogeneous porous media. Water Resour. Res. 45 (7), W07402.CrossRefGoogle Scholar
Herrera, P. & Beckie, R. 2009 A new approach to model solute mixing in porous media. Eos Trans. AGU Fall Meet. Suppl., Abstract H23D-0990 90 (52).Google Scholar
Hou, Z. & Rubin, Y. 2005 On minimum relative entropy concepts and prior compatibility issues in vadose zone inverse and forward modeling. Water Resour. Res. 41 W12425.Google Scholar
Hsu, K. C., Zhang, D. & Neuman, S. P. 1996 Higher-order effects on flow and transport in randomly heterogeneous porous media. Water Resour. Res. 32 (3), 571582.Google Scholar
Indelman, P. & Dagan, G. 1999 Solute transport in divergent radial flow through heterogeneous porous media. J. Fluid Mech. 384, 159182.CrossRefGoogle Scholar
Jaynes, E. T. 1957 Information theory and statistical mechanics. Phys. Rev. 108 (2), 171190.Google Scholar
Kitanidis, P. K. 1988 Prediction by the method of moments of transport in a heterogeneous formation. J. Hydrol. 102 (1–4), 453473.Google Scholar
Koch, D. L. & Brady, J. F. 1988 Anomalous diffusion in heterogeneous porous media. Phys. Fluids 31, 965.Google Scholar
Kraichnan, R. H. 1970 Diffusion by a random velocity field. Phys. Fluids 13, 22.Google Scholar
Kramer, P. R., Majda, A. J. & Vanden-Eijnden, E. 2003 Closure approximations for passive scalar turbulence: a comparative study on an exactly solvable model with complex features. J. Stat. Phys. 111 (3), 565679.Google Scholar
Kullback, S. 1997 Information Theory and Statistics. Dover.Google Scholar
Lawrence, A. E. & Rubin, Y. 2007 Block-effective macrodispersion for numerical simulations of sorbing solute transport in heterogeneous porous formations. Adv. Water Resour. 30 (5), 12721285.Google Scholar
Le Borgne, T., Dentz, M. & Carrera, J. 2008 Lagrangian statistical model for transport in highly heterogeneous velocity fields. Phys. Rev. Lett. 101 (9), 090601.Google Scholar
Majda, A. J. & Kramer, P. R. 1999 Simplified models for turbulent diffusion: theory, numerical modelling & physical phenomena. Phys. Rep. 314 (4–5), 237574.CrossRefGoogle Scholar
Matheron, G. & De Marsily, G. 1980 Is transport in porous media always diffusive: a counterexample. Water Resour. Res. 16 (5), 901917.Google Scholar
McComb, W. D. 1990 The Physics of Fluid Turbulence. Clarendon Press.Google Scholar
Mole, N., Schopflocher, T. P. & Sullivan, P. J. 2008 High concentrations of a passive scalar in turbulent dispersion. J. Fluid Mech. 604, 447474.Google Scholar
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics, vol. 2. MIT Press.Google Scholar
Munro, R. J., Chatwin, P. C. & Mole, N. 2003 A concentration PDF for the relative dispersion of a contaminant plume in the atmosphere. Boundary-Layer Meteorol. 106 (3), 411436.Google Scholar
Neuman, S. P. 1990 Universal scaling of hydraulic conductivities and dispersivities in geologic media. Water Resour. Res. 26 (8), 17491758.Google Scholar
Neuman, S. P. 2006 Blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition. Water Resour. Res. 42 (6), W06D04.CrossRefGoogle Scholar
Neuman, S. P. & Zhang, Y. K. 1990 A quasi-linear theory of non-Fickian and Fickian subsurface dispersion. 1. Theoretical analysis with application to isotropic media. Water Resour. Res. 26 (5), 887902.Google Scholar
Phythian, R. & Curtis, W. D. 1978 The effective long-time diffusivity for a passive scalar in a Gaussian model fluid flow. J. Fluid Mech. 89 (02), 241250.Google Scholar
Rajaram, H. & Gelhar, L. W. 1993 a Plume scale-dependent dispersion in heterogeneous aquifers 1. Lagrangian analysis in a stratified aquifer. Water Resour. Res. 29 (9), 32493260.Google Scholar
Rajaram, H. & Gelhar, L. W. 1993 b Plume scale-dependent dispersion in heterogeneous aquifers 2. Eulerian analysis and three-dimensional aquifers. Water Resour. Res. 29 (9), 32613276.CrossRefGoogle Scholar
Risken, H. 1989 The Fokker-Planck Equation: Methods of Solution and Applications, 2nd edn. Springer.Google Scholar
Rubin, Y. 1990 Stochastic modeling of macrodispersion in heterogeneous porous media. Water Resour. Res. 26 (1), 133141.Google Scholar
Rubin, Y. 2003 Applied Stochastic Hydrogeology. Oxford University Press.CrossRefGoogle Scholar
Rubin, Y., Bellin, A. & Lawrence, A. E. 2003 On the use of block-effective macrodispersion for numerical simulations of transport in heterogeneous formations. Water Resour. Res. 39 (9), 12421252.Google Scholar
Rubin, Y., Sun, A., Maxwell, R. & Bellin, A. 1999 The concept of block-effective macrodispersivity and a unified approach for grid-scale- and plume-scale-dependent transport. J. Fluid Mech. 395, 161180.CrossRefGoogle Scholar
Sawford, B. 2001 Turbulent relative dispersion. Annu. Rev. Fluid Mech. 33 (1), 289317.Google Scholar
Valocchi, A. J. 1990 Numerical simulation of the transport of adsorbing solutes in heterogeneous aquifers. In Computational Methods in Subsurface Hydrology (ed. Gambolati, G. et al. ), chap. 9, pp. 373382. Computational Mechanics.Google Scholar
Woodbury, A. D. & Ulrych, T. J. 1993 Minimum relative entropy: forward probabilistic modeling. Water Resour. Res. 29 (8), 28472860.Google Scholar
Zavala-Sanchez, V., Dentz, M. & Sanchez-Vila, X. 2009 Characterization of mixing and spreading in a bounded stratified medium. Adv. Water Resour. 32 (5), 635648.CrossRefGoogle Scholar