Abstract
By using nuclear magnetic resonance imaging (NMRI) we have been able to analyse dispersion at the microscopic scale during steady-state flow through water-saturated glass beads. The flow rate through the porous medium was chosen high enough in order to neglect the influence of molecular diffusion on dispersion. Velocity statistics were measured, by NMRI, within slices of increasing thickness perpendicular to the direction of flow. It took more than two bead diameters before a representative elementary volume (REV) for the mean velocity was reached. This was in a region in the middle of the column that was not influenced by the boundary conditions. There the velocity variance decreased exponentially as a function of the slice thickness, due we consider to the formation of an interconnecting streamline network. The exponential decrease in the velocity variance reflects the transition from a local pattern of stochastic–convective flow to a convective–dispersion regime at the scale of the REV. We found that the point-like preferential influx and efflux boundary condition increased velocity variances and thus enhanced longitudinal hydrodynamic dispersion. Using the transverse correlation length of longitudinal velocity variance, we derived a mean transverse dispersivity that agreed well with Saffman’s (1959) model. So we have been able to provide for the first time a direct observation verification of a part of Saffman’s (1959) conjectures. By NMRI we observed this value to be independent of the observation scale of the slice thickness.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aitcheson, J. and Brown, J. A. C.: 1976, The Lognormal Distribution, Cambridge University Press, Cambridge.
Amin, M. H. G., Chorley, R. J., Richards, K. S., Bache, B. W., Hall, L. D. and Carpenter, T. A.: 1993, Spatial and Temporal Mapping of Water in Soil by Magnetic Resonance Imaging, Hydrol. Processes 7, 279–286.
Bear, J.: 1972, Dynamics of Fluids in Porous Media, Elsevier, New York.
Beven, K. and Germann, P.: 1982, Macropores and water flow in soils, Water Resour. Res. 18, 1311–1325.
Blackwell, R. J.: 1963, Laboratory studies of microscopic dispersion phenomena, in: Miscible Processes, Vol. 8 Soc. of Petroleum Engineers of AIME, pp. 69–76.
Brusseau, M. L.: 1993, The influence of solute size, pore water velocity, and intraparticle porosity on solute dispersion and transport in soil, Water Resour. Res. 29, 1071–1080.
Callaghan, P. T.: 1991, Principles of Nuclear Magnetic Resonance Microscopy, Clarendon Press, Oxford.
Callaghan, P. T. and Xia, Y.: 1991, Velocity and diffusion imaging in dynamic NMR microscopy, J. Magn. Reson. 91, 326–352.
Chen, Q., Kinzelbach, W. and Oswald, S.: 2002, Nuclear magnetic resonance imaging for studies of flow and transport in porous media, J. Env. Qual. 31, 477–486.
Corapcioglu, M. Y. and Fedirchuk, P.: 1999, Glass bead micromodel study of solute transport. J. Cont. Hydrol. 36, 209–230.
Dagan, G.: 1989, Flow and Transport in Porous Formations, Springer, Berlin.
Deurer, M., Vogeler, I. and Khrapitchev, A.: 2002, Imaging of water flow in porous media by MRI microscopy, J. Env. Qual. 31, 487–493.
Dronfield, D. G. and Silliman, S. E.: 1993, Velocity dependence of dispersion for transport through a single fracture of variable roughness, Water Resour. Res. 29, 3477–3483.
Dullien, F. A. L.: 1992, Porous Media. Fluid Transport and Pore Structure, 2nd edn, Academic Press, New york.
Eccles, C. D., Clark, C. J., Codd, S. L. and Dykstra, R.: 1998, Construction of an MRI system for horticultural and material science research, In: Proceedings of the 5th Electronics NZ Conference.
Farrar, T. C.: 1987, An Introduction to Pulse NMR Spectroscopy, Farragut Press, Chicago.
Freeman, R.: 1988, A Handbook of Nuclear Magnetic Resonance, Longman, New York.
Freeze, R. A. and Cherry, J. A.: 1979, Groundwater, Prentice-Hall, Englewood Cliffs, NJ.
Frissel, M. J. and Poelstra, P.: 1967, Chromatographic transport through soils. I. Theoretical evaluations, Plant and Soil 26, 285–302.
Gadian, D. G.: 1995, NMR and Its Application to Living Systems, Oxford University Press, Oxford.
Goodknight, R. C., Klikoff, W. A. and Fatt, I.: 1960, Non-steady-state fluid flow and diffusion in porous media containing dead-end pore volume, J. Phys. Chem. 64, 1162–1168.
Greiner, A., Schreiber, W., Brix, G. and Kinzelbach, W.: 1997, Magnetic resonance imaging of paramagnetic tracers in porous media: quantification of flow and transport parameters, Water Resour. Res. 33, 1461–1473.
Hall, L. D., Amin, M. H. G., Dougherty, E., Sanda, M., Votrubova, J., Richards, K. S., Chorley, R. J. and Cislerova, M.: 1997, MR properties of water in saturated soils and resulting loss of MRI signal in water content detection at 2 tesla, Geoderma 80, 431–448.
Herrmann, K. H., Pohlmeier, A., Wiese, S., Shah, N. J., Nitzsche, O. and Vereecken, H.: 2002, Three-dimensional nickel ion transport through porous media using magnetic resonance imaging, J. Env. Qual. 31, 506–514.
Horvath, C. and Lin, H.: 1976, Movement and band spreading of unsorbed solutes in liquid chromatography, J. Chromatography 126, 401–420.
Jensen, J. R.: 1983, Chloride dispersion in packed columns during saturated steady flow, J. Soil Sci. 34, 249–262.
Journel, A. G. and Huijbregts, C. J.: 1978, Mining Geostatistics, Academic Press, New York.
Jury, W. A.: 1982, Simulation of solute transport with a transfer function model, Water Resour. Res. 18, 363–368.
Jury, W. A. and Roth, K.: 1990, Transfer Functions and Solute Movement through Soil: Theory and Applications, Birkhäuser, Basel.
Jury, W. A., Focht, D. D., and Farmer, W. J.: 1987, Evaluation of pesticide ground water pollution potential from standard indices of soil-chemical adsorption and biodegradation, J. Env. Qual. 16, 422–428.
Klotz, D., Seiler, K. P., Moser, H. and Neumaier, F.: 1980, Dispersivity and velocity relationships from laboratory and field experiments, J. Hydrol. 45, 169–184.
Koch, D. L. and Brady, J. F.: 1987, A non-local description of advection–diffusion with application to dispersion in porous media, J. Fluid Mech. 180, 387–403.
Kung, K.-J. S.: 1990a, Preferential flow in a sandy vadosze zone. 1. Field Observation, Geoderma 46, 51–58.
Kung, K.-J. S.: 1990b, Preferential flow in a sandy vadosze zone: 2. Mechanism and Implications, Geoderma 46, 59–71.
Kutsovsky, Y. E., Alvarado, V., Davis, H. T., Scriven, L. E. and Hammer, B. E.: 1996, Dispersion of paramagnetic tracers in bead packs by T 1 mapping: experiments and simulations, Magn. Reson. Im. 14(7/8), 833–839.
List, E. J. and Brooks, N. H.: 1967, Lateral dispersion in saturated porous media, J. Geophys. Res. 72, 2531–2541.
Lumley, J. L. and Panofsky, A.: 1964, The Structure of Atmospheric Turbulence, Wiley, New York.
Ma, L. and Selim, H. M.: 1996, Solute transport in soils under conditions of variable flow velocities, Water Resour. Res. 32, 3277–3283.
Maier, R. S., Kroll, D. M., Kutsovsky, Y. E., Davis, H. T. and Bernard, R. S.: 1998, Simulation of flow through bead packs using the lattice Boltzmann method, Phys. Fluids 10, 60–74.
Mansfield, P. and Issa, B.: 1994, Studies of fluid transport in porous rocks by echo-planar MRI, Magn. Reson. Im. 12(2), 275–278.
Mansfield, P. and Issa, B.: 1996, Fluid transport in porous rocks. I. EPI studies and a stochastic model of flow, J. Magn. Reson. A 122, 137–148.
Neumann, S. P., Feddes, R. A. and Bresler, E.: 1974, Finite element simulation of flow in saturated–unsaturated soils considering water uptake by plants, Third Annual Report, Project No. A10-SWC-77, Hydraulic Engineering Lab., Technion, Haifa.
Perkins, T. K. and Johnston, O. C.: 1963, A review of diffusion and dispersion in porous media, In: Miscible Processes, Vol. 8., Soc. of Petroleum Engineers of AIME, pp. 77–91.
Pfannkuch, H. O.: 1963, Contribution a l'etude des deplacements de fluides miscible dans un milieu poreux, Revue de l'Institute Francais du Petrole 18, 215–270.
Pope, J.M. and Yao, S.: 1993, Quantitative NMR imaging of flow, Concepts Magn. Reso. 5, 281–302.
Price, W. S.: 1998, NMR imaging, Annual Reports on NMR Spectroscopy 35, 281–302.
Ritsema, C. J., Dekker, L. W., Hendrickx, J. M. H. and Hamminga, W.: 1993, Preferential flow mechanism in a water repellent sandy soil, Water Resour. Res. 29, 2183–2193.
Roth, K.: 1995, Steady state flow in an unsaturated, two-dimensional, macroscopically homogeneous, Miller-similar medium, Water Resour. Res. 31, 2127–2140.
Roth, K.: 1996, Lecture Notes in Soil Physics, Version 3.2, Institute of Soil Science, University of Hohenheim.
Saffman, P. G.: 1959, A theory of dispersion in a porous medium, J. Fluid Mech. 3, 321–349.
Saffman, P. G.: 1960, Dispersion due to molecular diffusion and macroscopic mixing in flow through a network of capillaries, J. Fluid Mech. 2, 194–208.
Scheidegger, A. E.: 1972, The Physics of Flow through Porous Media, 3rd edn, University of Toronto Press, Toronto, Ont., Canada.
Sederman, A. J., Johns, M. L., Alexander, P. and Gladden, L. F.: 1998, Visualisation of structure and flow in packed beds, Magn. Reson. Im. 16, 497–500.
Seymour, J. D. and Callaghan, P. T.: 1997, Generalized approach to NMR analysis of flow and dispersion in porous media, AICHE J. 43, 2096–2111.
Simmons, C. S.: 1982, A stochastic–convective transport representation of dispersion in one dimensional porous media systems, Water Resour. Res. 18, 1193–1214.
Sposito, G., Jury, W. A. and Gupta, V.: 1986, Fundamental problems in the stochastic convection– dispersion model of solute transport in aquifers and field soils, Water Resour. Res. 22, 77–88.
van Genuchten, M. T. and Wierenga, P. J.: 1976, Mass transfer studies in sorbing porous media. I. Analytical solutions, Soil Sci. Soc. Am. J. 40, 473–480.
Wang, Z., Feyen, J. and Ritsema, C. J.: 1998, Susceptibility and predictability of conditions for preferential flow, Water Resour. Res. 34, 2169–2182.
Whitaker, S.: 1967, Diffusion and dispersion in porous media, J. Am. Inst. Chem. Eng. 3, 420–427.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Deurer, M., Vogeler, I., Clothier, B.E. et al. Magnetic Resonance Imaging of Hydrodynamic Dispersion in a Saturated Porous Medium. Transport in Porous Media 54, 145–166 (2004). https://doi.org/10.1023/A:1026358431442
Issue Date:
DOI: https://doi.org/10.1023/A:1026358431442