Transport in Porous Media

, Volume 21, Issue 2, pp 123–144 | Cite as

Impacts of local dispersion and first-order decay on solute transport in randomly heterogeneous porous media

  • Dongxiao Zhang
Article

Abstract

Stochastic subsurface transport theories either disregard local dispersion or take it to be constant. We offer an alternative Eulerian-Lagrangian formalism to account for both local dispersion and first-order mass removal (due to radioactive decay or biodegradation). It rests on a decomposition of the velocityv into a field-scale componentvω, which is defined on the scale of measurement supportω, and a zero mean sub-field-scale componentv s , which fluctuates randomly on scales smaller thanω. Without loss of generality, we work formally with unconditional statistics ofv s and conditional statistics ofvω. We then require that, within this (or other selected) working framework,v s andvω be mutually uncorrelated. This holds whenever the correlation scale ofvω is large in comparison to that ofv s . The formalism leads to an integro-differential equation for the conditional mean total concentration 〈cω which includes two dispersion terms, one field-scale and one sub-field-scale. It also leads to explicit expressions for conditional second moments of concentration 〈c′c′ω. We solve the former, and evaluate the latter, for mildly fluctuatingvω by means of an analytical-numerical method developed earlier by Zhang and Neuman. We present results in two-dimensional flow fields of unconditional (prior) mean uniformvω. These show that the relative effect of local dispersion on first and second moments of concentration dies out locally as the corresponding dispersion tensor tends to zero. The effect also diminishes with time and source size. Our results thus do not support claims in the literature that local dispersion must always be accounted for, no matter how small it is. First-order decay reduces dispersion. This effect increases with time. However, these concentration moments 〈cω and 〈c′c′ω of total concentrationc, which are associated with the scale belowω, cannot be used to estimate the field-scale concentrationcω directly. To do so, a spatial average over the field measurement scaleω is needed. Nevertheless, our numerical results show that differences between the ensemble moments ofcω and those ofc are negligible, especially for nonpoint sources, because the ensemble moments ofc are already smooth enough.

Key words

stochastic analysis conditional probability solute transport local dispersion first-order decay heterogeneous porous media 

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References

  1. Batchelor, G. K.: 1952, Diffusion in a field of homogeneous turbulence: II. The relative motion of particles,Proc. Cambridge Philos. Soc. 48, 345–363.Google Scholar
  2. Baveye, P. and Sposito, G.: 1984, The operational significance of the continuum hypothesis in the theory of water movement through soils and aquifers,Water Resour. Res. 20(5), 521–530.Google Scholar
  3. Baveye, P. and Sposito, G.: 1985, Macroscopic balance equations in soils and aquifers: The case of space and time-dependent instrumental response,Water Resour. Res. 21(8), 1116–1120.Google Scholar
  4. Beckie, R., Aldama, A. A. and Wood, E. F.: 1994, The universal structure of the groundwater flow equations,Water Resour. Res. 30(5), 1407–1419.Google Scholar
  5. Bear, J.: 1972,Dynamics of Fluids in Porous Media, Dover Publ., New York.Google Scholar
  6. Brusseau, M. L., Rao, P. S. C. and Bellin, C. A.: 1992, Modeling coupled processes in porous media: Sorption, transformation, and transport of organic solutes, in R. J. Wagenet, P. Baveye and B. A. Stewart (eds),Interacting Processes in Soil Science, Lewis Publ., Ann Arbor, pp. 147–184.Google Scholar
  7. Cushman, J. H.: 1984, On unifying the concept of scale, instrumentation, and stochastics in the development of multiphase transport theory,Water Resour. Res. 20(11), 1668–1678.Google Scholar
  8. Cushman, J. H. and Ginn, T. R.: 1993, Nonlocal dispersion in media with continuous evolving scales of heterogeneity,Transport in Porous Media 13, 123–138.Google Scholar
  9. Dagan, G.: 1984, Solute transport in heterogeneous porous formations,J. Fluid Mech. 145, 151–177.Google Scholar
  10. Dagan, G.: 1989,Flow and Transport in Porous Formations, Springer-Verlag, New York.Google Scholar
  11. Dagan, G., Cvetkovic, V. and Shapiro, A. M.: 1992, A solute flux approach to transport in heterogeneous formations: 1. The general framework,Water Resour. Res. 28(5), 1369–1376.Google Scholar
  12. Deng, F. -W., Cushman, J. H. and Delleur, J. W.: 1993, A fast Fourier transform stochastic analysis of the contaminant transport problem,Water Resour. Res. 29(9), 3241–3247.Google Scholar
  13. Dullien, F. A. L.: 1992,Porous Media: Fluid Transport and Pore Structure, 2nd edn, Academic Press, New York.Google Scholar
  14. Gelhar, L. W. and Axness, C. L.: 1983, Three-dimensional stochastic analysis of macrodispersion in aquifers,Water Resour. Res. 19(1), 161–180.Google Scholar
  15. Graham, W. and McLaughlin, D.: 1989a, Stochastic analysis of nonstationary subsurface solute transport: 1. Unconditional moments,Water Resour. Res. 25(2), 215–232.Google Scholar
  16. Graham, W. and McLaughlin, D.: 1989b, Stochastic analysis of nonstationary subsurface solute transport: 2. Conditional moments,Water Resour. Res. 25(11), 2331–2355.Google Scholar
  17. Kapoor, V. and Gelhar, L. W.: 1994a, Transport in three-dimensional heterogeneous aquifers: 1. Dynamics of concentration fluctuations,Water Resour Res. 30(6), 1775–1788.Google Scholar
  18. Kapoor, V. and Gelhar, L. W., 1994b, Transport in three-dimensional heterogeneous aquifers: 2. Predictions and observations of concentration fluctuations,Water Resour. Res. 30(6), 1789–1801.Google Scholar
  19. Kim, J. -H., Ochoa, J. A. and Whitaker, S.: 1987, Diffusion in anisotropic porous media,Transport in Porous Media 2, 327–356.Google Scholar
  20. Koch, D. L. and Brady, J. F.: 1988, Anomalous diffusion in heterogeneous porous media,Phys. Fluids 31(5), 965–973.Google Scholar
  21. Naff, R. L.: 1992, Arrival times and temporal moments of breakthrough curves for an imperfectly stratified aquifer,Water Resour. Res. 28(1), 53–68.Google Scholar
  22. Naff, R. L.: 1994, An Eulerian scheme for the second-order approximation of subsurface transport moments,Water Resour. Res. 30(5), 1439–1455.Google Scholar
  23. Neuman, S. P.: 1990, Universal scaling of hydraulic conductivities and dispersivities in geologic media,Water Resour. Res. 26(8), 1749–1758.Google Scholar
  24. Neuman, S. P.: 1993, Eulerian-Lagrangian theory of transport in space-time nonstationary velocity fields: Exact nonlocal formalism by conditional moments and weak approximations,Water Resour. Res. 29(3), 633–645.Google Scholar
  25. Neuman, S. P., Winter, C. L. and Newman, C. M.: 1987, Stochastic theory of field-scale Fickian dispersion in anisotropic porous media,Water Resour. Res. 23(3), 453–466.Google Scholar
  26. Neuman, S. P. and 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), 887–902.Google Scholar
  27. Rajaram, H. and Gelhar, L. W.: 1993, Plume scale-dependent dispersion in heterogeneous aquifers: 2. Eulerian analysis and three-dimensional aquifers,Water Resour. Res. 29(9), 3249–3260.Google Scholar
  28. Rubin, Y.: 1990, Stochastic modeling of macrodispersion in heterogeneous porous media,Water Water Resour. Res. 26(1), 133–141.Google Scholar
  29. Quintard, M.: 1993, Diffusion in isotropic and anisotropic porous systems: Three-dimensional calculations,Transport in Porous Media 11, 187–199.Google Scholar
  30. Scow, K. M., 1993, Effect of sorption-desorption and diffusion processes on the kinetics of biodegradation of organic chemicals in soil, inSorption and Degradation of Pesticides and Organic Chemicals in Soil, SSSA Special Publication No. 32, pp. 73–114.Google Scholar
  31. Shvidler, M. I.: 1993, Correlation model of transport in random fields,Water Resour. Res. 29(9), 3189–3199.Google Scholar
  32. Simmons, C. S.: 1982, A stochastic-convective transport representation of dispersion in one-dimensional porous media,Water Resour. Res. 18(4), 1193–1214.Google Scholar
  33. Sposito, G. and Barry, D. A.: 1987, On the Dagan model of solute transport in groundwater: Foundational aspects,Water Resour. Res. 23(10), 1867–1875.Google Scholar
  34. Winter, C. L., Newman, C. M. and Neuman, S. P.: 1984, A perturbation expansion for diffusion in a random velocity field,SIAM J. Appl. Math. 44(2), 411–424.Google Scholar
  35. Zhang, D.: 1992, Some aspects of stochastic flow and transport in complex geologic media, M.S. thesis, Univ. of Ariz., Tucson.Google Scholar
  36. Zhang, D.: 1993, Conditional stochastic analysis of solute transport in heterogeneous geologic media, Ph.D. thesis, Univ. of Ariz., Tucson.Google Scholar
  37. Zhang, D. and Neuman, S. P.: 1995a, Eulerian-Lagrangian analysis of transport conditioned on hydraulic data: 1. Analytical-Numerical approach,Water Resour. Res. 31(1), 39–51.Google Scholar
  38. Zhang, D. and Neuman, S. P.: 1995b, Eulerian-Lagrangian analysis of transport conditioned on hydraulic data: 2. Effects of log transmissivity and hydraulic head measurements,Water Resour. Res. 31(1), 53–63.Google Scholar
  39. Zhang, D. and Neuman, S. P.: 1995c, Eulerian-Lagrangian analysis of transport conditioned on hydraulic data: 3. Spatial moments, travel time distribution, mass flow rate and cumulative release across a compliance surface,Water Resour. Res. 31(1), 65–75.Google Scholar
  40. Zhang, D. and Neuman, S. P.: 1995d, Eulerian-Lagrangian analysis of transport conditioned on hydraulic data: 4. Uncertain initial plume state and non-Gaussian velocity,Water Resour. Res. 31(1), 77–88.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Dongxiao Zhang
    • 1
  1. 1.6020 Academy NE, Suite 100Daniel B. Stephens & Associates, Inc.AlbuquerqueUSA

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