Journal of Environmental Quality 31:487-493 (2002)
© 2002 American Society of Agronomy, Crop Science Society of America, and Soil Science Society of America
Article
SYMPOSIUM PAPERS
Imaging of Water Flow in Porous Media by Magnetic Resonance Imaging Microscopy
Markus Deurera,
Iris Vogeler*,a,
Alexander Khrapitchevb and
Dave Scottera
a Environment and Risk Management Group, Hort Research Institute, Private Bag 11-030, Palmerston North, New Zealand
b Institute of Fundamental Sciences, Massey University, Private Bag 11-222, Palmerston North, New Zealand
* Corresponding author (ivogeler{at}hort.cri.nz)
Received for publication June 6, 2001.
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ABSTRACT
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Magnetic resonance imaging (MRI) was used to study the flow of water in a column 14 mm in diameter packed with glass beads. The sample was fully saturated and water was pumped through the column using a peristaltic pump, at flow rates of 125 and 250 mL h-1. This corresponds to mean velocities of 0.5 and 1 mm s-1, given a porosity of 0.46 m3 m-3. Nuclear magnetic resonance (NMR) images of the proton density and velocities within a 2-mm slice were taken at a spatial resolution of 0.15 x 0.15 x 2 mm3. At a mean pore water velocity of 1 mm s-1 we approximated hydrodynamic dispersion using NMR-measured velocity distributions in a 2-mm slice through the sample. Additionally, we conducted a step pulse tracer experiment with chloride through the same column and at identical initial and boundary conditions. We fitted the convection–dispersion equation to the breakthrough curve and compared the column scale dispersion of the tracer experiment with the respective NMR estimate derived at the slice scale.
Abbreviations: CDE, convection–dispersion equation • CLT, convective–lognormal transfer • MRI, magnetic resonance imaging • NMR, nuclear magnetic resonance
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INTRODUCTION
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AT PRESENT IT IS POSSIBLE to measure pore characteristics of porous media three-dimensionally using, for example, a series of thin sections (Vogel and Kretzschmar, 1996) or computer aided tomography (Perret et al., 1999). But with these techniques we cannot study which part of all the potential flow pathways are realized under various initial and boundary conditions. Magnetic resonance imaging (MRI) can be used to obtain information on soil properties and flow at a high spatial resolution.
Magnetic resonance imaging techniques have been used to spatially determine soil water content (Amin et al., 1994; Hall et al., 1997), and to measure the local velocity of water and paramagnetic tracers in porous media (Seymour and Callaghan, 1997; Sederman et al., 1998) and soils (Pearl et al., 1993; Amin et al., 1994; Hoffmann et al., 1996; Oswald et al., 1997).
We measured the velocities of moving water protons directly with a dynamic imaging pulse sequence (see Callaghan, 1991). The study aimed to explore the feasibility of measuring the volumetric water content (
) and velocity in a porous medium at the pore scale. To do this we studied water flow in a column packed with glass beads, 2 mm in diameter. The NMR images of the spin density (which is directly proportional to ) and velocity distributions were taken within a 2-mm slice through the column. The voxel size was 0.15 x 0.15 x 2 mm3.
A second aim was to document the potential of MRI to analyze the spatial scale dependence of processes such as hydrodynamic dispersion. To do this we compared the column-scale breakthrough of a chloride step-pulse with a dispersion estimate based on an MRI measured slice-scale velocity distribution.
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THEORY
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Nuclear Magnetic Resonance
We will give only some specific background on the theory of MRI. A thorough introduction and more details are given by Callaghan (1991).
Measurement of Volumetric Water Content
The volumetric water content () at one voxel is directly proportional to the number of resonating spinning water protons in the voxel, the so-called spin density, 
. The precessing macroscopic magnetization at each voxel induces the electric signal we measure with NMR. This macroscopic magnetization is caused by the ensemble of precessing microscopic magnetic moments of the individual resonating spinning water protons, and is therefore a measure of their density. Consider the spins at the spatial position r within the sample occupying an infinitesimal volume dV. The local spin density (r) will contain (r) dV spins. The spatial position of each volume element is labeled in three dimensions with magnetic field gradients denoted as Gx, Gy, Gz (G), where the subscripts refer to the x, y, and z coordinates. They impose on the received signal an image position specific frequency (frequency gradient, e.g., Gx) and a phase shift (phase gradient, e.g., Gy). The signal arising from this infinitesimal volume is (Callaghan, 1991):
 | [1] |
where i denotes the real part of the complex signal and w(r) is the position specific precession frequency. The latter depends on the main magnetic field B0 (applied externally to cause the precession of the water protons) and the gradient G. Therefore, Eq. [1] can be written as (Callaghan, 1991):
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where 
is the gyromagnetic ratio for 1H.
For convenience the received signal is mixed with a reference oscillation, canceling out the precession contribution of the main magnetic field to the signal. The signal now oscillates at Gr. Integrating over the volume of each voxel the signal becomes (Callaghan, 1991):
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Using the following substitution (the so-called reciprocal space vector K):
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the received signal S and the spin density are mutually conjugate Fourier pairs (Callaghan, 1991):
 | [5] |
Therefore, after Fourier transformation, we can directly derive the spin density as the spectrum of the Fourier-transformed signal.
Spatial Measurements of Velocities
We want to characterize the bulk motion of the spins in the direction of flow (in our case z). At the time t = 0 the spins are at position z and move with a longitudinal velocity vz. To label and analyze their motion we apply an additional (independent of the gradients for spatial encoding) pulsed gradient gz in the direction of flow. For a very short time 
the spins at z will therefore precess faster. They dephase by an angle 
1 (C.D. Eccles, personal communication, 2000):
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where is the duration gz is applied. After the time 
, gz is reapplied. Due to the movement along the z axis the protons are now at position z = vz. Consequently, they will rephase by an angle 2:
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The resultant phase shift is then:
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Note that this resulting phase shift is not position-dependent but only velocity-dependent. We repeat this basic experiment several times varying either gz, , or . Let us assume that we vary . We then obtain a signal amplitude that varies sinusoidally with (C.D. Eccles, personal communication, 2000):
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where the angular frequency term in the waveform of Eq. [9] is proportional to the velocity of the spins.
The concept of measuring velocities with MRI (phase shift, see Eq. [8]) puts the movement of water in the Lagrangian framework (Seymour and Callaghan, 1997). The measured voxel-scale vz is not space-fixed.
Solute Transport
We assume that the one-dimensional convection–dispersion equation (CDE) is appropriate for our experimental conditions: horizontal flow through a homogeneous medium with spatially and temporally constant mean longitudinal pore water velocities v in the direction of flow (z axis). The CDE is given by:
 | [10] |
where Cf is the tracer flux concentration and D the longitudinal dispersion coefficient. For negligible molecular diffusion D can be approximated by:
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where 
is the longitudinal dispersivity.
The column is initially free of chemicals and at time t = 0 a tracer with the concentration C0 is applied as a step pulse to the inlet end. Thus, the boundary conditions are:
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For the lower boundary condition it was assumed that the column was part of an effectively semi-infinite system.
The solution for the flux concentration is given by (Kirkham and Powers, 1971, p. 379–420):
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We also predicted the outflow concentration Cf through the entire sample using the NMR-measured slice-scale voxel velocities. One approach to link measurements of water and/or solute transport from different scales, in our case the nuclear MRI measurements in the 2-mm slice and the column scale tracer experiment, is the use of transfer functions (Jury, 1982; Jury and Roth, 1990). This approach models hydrodynamic dispersion either with a convective–lognormal transfer (CLT) function (Jury et al., 1987), or with the convection–dispersion equation (CDE). In the first case the dispersivity grows linearly with the travel distance and has no asymptotic limit. In the second case it is constant. The CLT is considered to represent the short distance limit and the CDE the long-distance limit of solute transport (Sposito et al., 1986). Physically, water and/or solute transport is probably best represented by a transition from a flow in isolated streamtubes (CLT) to one in a fully interconnected network of flow pathways (CDE). There are no theoretical concepts to predict the spatial or temporal scales for such a transition. Only little experimental evidence exists for such a transition (Butters and Jury, 1989).
At the observation scale of the 2-mm slice imaged with NMR the flow is restricted to isolated areas between the glass beads, as they have a diameter of 2 mm. Thus, on this scale hydrodynamic dispersion is truly stochastic–convective and takes place in isolated stream tubes.
A stochastic–convective transport of water and/or solutes is like a superposition of piston flow processes within individual subcolumns (Jury and Roth, 1990). Each voxel within the slice is hypothesized to be a representative section of one of those subcolumns. We then use the slice-scale results and the CLT to predict the column-scale breakthrough from our NMR measurements. According to Bear (1972)(p. 608–609), at a mean flow rate of 1 mm-1 the effect of diffusion on dispersion is negligible, as our experiment would be classified to belong to his Zone IV, a region of dominant mechanical dispersion. Hydrodynamic dispersion is then entirely driven by the variability of velocities. Knowing the length l of the sample and the voxel velocities, solutes in each of these subcolumns have a characteristic travel time from the inlet to the outlet, tl(s):
 | [14] |
where s denotes the spatial coordinates (x,y) of the center of each subcolumn. We assume equal probabilities for solute molecules to move into each of the subcolumns. The result is a probability density function of the travel times of the solute molecules to travel from the inlet to the outlet port, ff (l,t). The respective flux concentration at the outlet l and at time t is then given by (Jury and Roth, 1990):
 | [15] |
where Pf (l,t) is the cumulative travel time distribution function. After normalization of Cf by C0, Eq. [15] can be directly compared with Eq. [13].
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METHODS AND MATERIALS
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Flow Experiments
A column with a 14.1-mm inner diameter and length l of 46 mm was used for the experiment. The column was filled with glass beads 2 mm in diameter giving a porosity of 0.45 m3 m-3. The inlet and outlet ports occupied only a fraction of the proximal and distal surfaces, thereby representing a point-like influx and efflux. These ports were connected to a silicon tube with a 1.73-mm diameter. For the flow experiments tap water was pumped through the glass beads with a Pharmacia double syringe pump at rates of 125 and 250 mL h-1. This corresponds to mean longitudinal velocities through the porous medium of v = 0.5 and 1 mm s-1.
For the tracer experiment, a step pulse with C0 = 0.001 M KCl was fed directly to the inlet port of the column at a flow rate of 250 mL h-1. The concentrations of chloride were analyzed with high performance liquid chromatography (HPLC) (Dionex [Sunnyvale, CA] 500).
Magnetic Resonance Imaging Methods
We used a horizontal wide-bore MRI system. It is based around an Oxford Instruments 4.7 Tesla superconducting magnet and consists of a number of commercial and homebuilt components (C.D. Eccles, personal communication, 1998).
For the measurement of spin densities we used a two-dimensional spin–echo pulse sequence within a slice of 2 mm. A field of view of 20 x 20 mm was resolved with 128 phase steps. The sequence consisted of a 4-µs 90° hard pulse and a 180° soft pulse. We used 10 averages, an echo time of 8.4 ms, and a repetition time of 1 s.
For the velocity we used a dynamic imaging sequence with two different parameter sets ("slow" and "fast"). The respective times between the two velocity gradient pulses were 7.9 and 6.5 ms, respectively. For the analysis we combined the results of both sequences. From the "slow" sequence we inferred velocities up to 2.4 mm s-1 and from the "fast" sequence velocities that were higher than 2.4 mm s-1.
Calibration Measurements
Transformation of Spin Densities into Volumetric Water Contents
We assumed that the voxels with the highest 5% of spin densities contain only water. We used the median of this population for normalization to get the fractions. For this calculation we excluded voxels with a zero spin density.
Velocity Calibration
Figure 1
shows the velocity distribution of three of the velocity measurements through the capillary tube, at 9, 90, and 210 mL h-1. This corresponds to mean velocities of 0.6, 6, and 14 mm s-1. All three images show a parabolic velocity distribution, as expected. At low velocities some distortion occurs, which is probably due to the shaking of the magnet. Figure 2
shows the relationship between the actual applied flow rates and those measured by NMR. The relationship is linear. Different NMR parameter settings had to be used for low ("slow" sequence) and high velocities ("fast" sequence). These velocity measurements in capillary tubes provided us with a NMR sequence, which we could use to measure velocity spectra in porous media.
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Fig. 1. Spatial distribution of velocities inside a capillary tube with a diameter of 2.3 mm. The mean velocities were (a) 0.6, (b) 6, and (c) 14 mm s-1.
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Fig. 2. Relationship between the actual applied flow rates and those measured by nuclear magnetic resonance (NMR).
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RESULTS
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Nuclear Magnetic Resonance Images of Water Content
Figure 3
shows an image of the volumetric water content from a 2-mm slice in the column packed with glass beads. The regions where the glass beads are and the regions between them are clearly visible. The mean value of all these normalized voxels was 0.44 m3 m-3. This compares well with the gravimetrically determined bulk porosity of 0.45 m3 m-3.
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Fig. 3. Nuclear magnetic resonance (NMR) image of the volumetric water content from a 2-mm slice in the center of the column packed with glass beads.
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Velocity Measurements through Beads of Glass
Figure 4
shows the spatial distribution of velocities between glass beads at the set flow rates of 125 and 250 mL h-1, corresponding to mean flow rates of 0.5 and 1 mm s-1. For both flow rates two images were taken, with different settings (according to the calibration measurements). These two images were then overlaid. The regions with high velocities are clustered and occur close to the wall where the packing is more regular. The mean velocities calculated from these NMR images agree well with the actual applied velocities (see Table 1). The flow rates obtained by multiplying these mean velocities with the NMR measured , also agree well with the actual ones (see Table 1, bulk characteristics). However, when the voxel-based velocities were multiplied by the voxel-based , the flow rates obtained were only about 50% of those applied. We think this discrepancy between flux based on voxels and flux measured is mainly due to errors in the spin density measurements. The glass beads themselves impose small local magnetic fields. Therefore, in the vicinity of the surface of the glass beads the magnetic susceptibility of water changes. This inhomogeneity of the magnetic susceptibility leads to an additional transverse relaxation and consequential signal attenuation (Callaghan, 1990). This effect is used to analyze, for example, the pore size distributions of porous materials (Davies et al., 1991; Allen et al., 1998). In the study of Hall et al. (1997) using a standard spin–echo sequence, only 0.2 to 57% of the bulk water was detected for soils of varying types including sands. Currently, we try to use different NMR protocols such as an inversion–recovery sequence (Callaghan, 1991) to overcome this problem. For the calibration we omitted voxel values that had zero spin density. These are voxels with very low water contents. Voxels with small water contents contain either small pore(s) or mainly glass bead material. The effects of a changing magnetic susceptibility are strongest close to the surface of the glass beads. Therefore, voxels with small water contents are likely to show the resulting effect of signal attenuation most pronounced. In those voxels the underestimation of the spin density, and thus the water content, would be highest. If the respective voxel velocities are simultaneously high then their contribution to the overall flux will be considerable. But generally, all voxels and their water contents that are influenced by the inhomogeneity of magnetic susceptibility due to the glass bead material are affected.
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Fig. 4. Spatial distribution of velocities in the glass bead mean flow rates of (a) 0.5 and (b) 1 mm s-1.
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Table 1. Comparison between applied flow rates (Q) and velocities (v) and those obtained by nuclear magnetic resonance (NMR).
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Tracer Experiment
The breakthrough curve (BTC) of chloride is shown in Fig. 5
. Also shown as the dotted line is the prediction of the convection–dispersion equation (CDE) using a dispersivity () of 2 mm (theoretical value according the diameter of the glass beads; Dullien, 1992), and assuming that all the water is mobile (m). However, the chloride appears in the leachate much earlier than predicted. Next, the CDE was fitted to the data, giving a of 3 mm and a m of 0.33 (solid line). The dispersivity as a correlation length of the pore space is larger than the glass-bead diameter, and the respective transport volume is smaller than the porosity. This might be due to our experimental setup, with a point-source entry of water and/or solutes as an upper boundary condition. Existing macroscopic experiments measuring dispersion in glass bead porous media show that dispersivity on the column scale was about the bead diameter (Pfannkuch, 1963). But all of them used a uniform Neumann-type boundary condition (Neumann et al., 1974). In many situations we can encounter a similar boundary condition rather than a homogeneous entry of water and/or solutes into the soil (e.g., drip irrigation systems, preferential or unstable flow). This shows the dependence of transport parameter values on the type of boundary condition.
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Fig. 5. Measured and predicted breakthrough curve (BTC) of chloride using the convection–dispersion equation (CDE) with theoretical parameter values (dotted line), and with fitted parameter values (solid line). Also shown is the prediction based on the velocity distribution measured by nuclear magnetic resonance (NMR) (broken line).
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Using only the voxel-based slice-scale velocities for an estimate of the normalized outflow concentration (see broken line in Fig. 5), the breakthrough starts much earlier and shows pronounced tailing. The dispersivity from this curve is estimated as 63 mm and the respective fractional transport volume as only 0.129. The extrapolation of the slice measurement neglects the formation of a network of flow paths. This is indirect microscopic experimental proof for a scale-dependent transition from a stochastic–convective to a convective–dispersive transport process. The variability of the velocity distribution itself is obviously scale dependent and a result of the flow network configuration. Our findings emphasize that we need more research in this particular area concerning the scales of the transition from a stochastic–convective to a convective–dispersive dispersion regime. We think that NMR has the potential to solve these questions in the future. It also shows that what we interpret as the mobile (transport volume) water fraction might be rather a function of the flow network, the observation scale, and the type of boundary condition.
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OUTLOOK
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In a few experiments NMR has been used to study the transport of water and/or solutes in soils (Pearl et al., 1993; Hoffman et al., 1996; Amin et al., 1997; Hemminga and Buurman, 1997; Cislerová et al., 1999).
Natural soils regularly contain ferromagnetic or paramagnetic elements, for example, free iron oxides or copper sulfates, respectively. These lead to faster spin–spin and spin–lattice relaxation times. Consequently, when the signal is measured a considerable part of it has already been attenuated and the results are accordingly biased. The study of Greiner et al. (1997) took advantage of this effect using copper sulfate as a paramagnetic tracer. Other soil physical and chemical properties, such as high clay contents (which result in pronounced inhomogeneities in the magnetic susceptibility), organic matter, and even exchangeable cations such as calcium and potassium can also reduce the relaxation times (Hall et al., 1997). However, advances in NMR technology and the development of new adapted pulse sequences might solve these problems.
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CONCLUSIONS
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The study has described the use of NMR to observe water movement in porous media, here in columns packed with glass beads. The volumetric water content inferred from the NMR image agreed well with the one determined gravimetrically. The mean velocities and flow rates obtained by NMR also agreed well with the applied flow rate when based on bulk characteristics. However, when the flow rate was calculated on a voxel basis only about 50% of the actual applied flow rate was obtained. This was probably due to a spin density distribution that was biased toward voxels with small volumetric water contents: spin–spin relaxation times will be rapid and due to technical limitations no signal can be detected. We intend to further improve our NMR measuring technique.
We compared the results of a conventional breakthrough experiment (column-scale) with the prediction based on the slice-scale velocity distribution obtained by MRI. We showed that solute transport parameters such as the dispersivity and the transport volume depend on the network of flow pathways, the type of boundary condition, and the observation scale. Thus, MRI can give valuable information concerning transport processes in porous media, which at present can only be studied on a macroscopic scale.
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ACKNOWLEDGMENTS
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The study was funded by the Royal Society through a Marsden Fund, Contract HRT 805. We would like to thank Paul Callaghan and Sarah Codd for many helpful discussions and Brent Clothier for the encouragement to do this work.
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REFERENCES
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- Allen, S., M. Mallet, M.E. Smith, and J.H. Strange. 1998. Susceptibility effects in unsaturated porous silica. Magn. Reson. Imag. 16:597– 600.[Medline]
- Amin, M.H.G., R.J. Chorley, K.S. Richards, L.D. Hall, T.A. Carpenter, M. Cislerová, and T. Vogel. 1997. Study of infiltration into a heterogeneous soil using magnetic resonance imaging. Hydrol. Processes 11:471–483.
- Amin, M.H.G., L.D. Hall, R.J. Chorley, T.A. Carpenter, K.S. Richards, and B.W. Bache. 1994. Magnetic resonance imaging of soil-water phenomena. Magn. Reson. Imag. 12:319–321.[Web of Science][Medline]
- Bear, J. 1972. Dynamics of fluids in porous media. Elsevier, New York.
- Butters, G.L., and W.A. Jury. 1989. Field scale solute transport of bromide in an unsaturated soil. 2. Dispersion modeling. Water Resour. Res. 25:1583–1589.
- Callaghan, P.T. 1990. Susceptibility-limited resolution in nuclear magnetic resonance microscopy. J. Magn. Reson. 87:304–318.
- Callaghan, P.T. 1991. Principles of nuclear magnetic resonance microscopy. Clarendon Press, Oxford.
- Cislerová, M., J. Votrubová, T. Vogel, M.H.G. Amin, and L.D. Hall. 1999. Magnetic resonance imaging and preferential flow in soils. p. 397–411. In Proc. of the Int. Workshop Characterization and Measurement of the Hydraulic Properties of Unsaturated Porous Media, Riverside, CA. 22–24 Oct. 1999. Part I. USDA, Riverside, CA.
- Davies, S., K.J. Packer, D.R. Roberts, and F.O. Zelanya. 1991. Pore size distributions from NMR spin–lattice relaxation data. Water Resour. Res. 9:681–685.
- Dullien, F.A.L. 1992. Porous media. Fluid transport and pore structure. 2nd ed. Academic Press, San Diego, CA.
- Greiner, A., W. Schreiber, G. Brix, and W. Kinzelbach. 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., M.H.G. Amin, E. Dougherty, M. Sanda, J. Votrubova, K.S. Richards, R.J. Chorley, and M. Cislerova. 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.
- Hemminga, M.A., and P. Buurman. 1997. NMR in soil science. Geoderma 80:221–224.
- Hoffmann, F., D. Ronen, and Z. Pearl. 1996. Evaluation of flow characteristics of a sand column using magnetic resonance imaging. J. Contam. Hydrol. 22:95–107.
- Jury, W.A. 1982. Simulation of solute transport using a transfer function model. Water Resour. Res. 18:363–368.
- Jury, W.A., D.D. Focht, and W.J. Farmer. 1987. Evaluation of pesticide ground water pollution potential from standard indices of soil–chemical adsorption and biodegradation. J. Environ. Qual. 16: 422–428.[Abstract/Free Full Text]
- Jury, W.A., and K. Roth. 1990. Transfer functions and solute movement through soil. Theory and application. Birkhäuser Verlag, Basel, Switzerland.
- Kirkham, D., and W.L. Powers. 1971. Advanced soil physics. Wiley Interscience, New York.
- Neumann, S.P., R.A. Feddes, and E. Bresler. 1974. Finite element simulation of flow in saturated–unsaturated soils considering water uptake by plants. 3rd Annu. Rep., Project A10-SWC-77. Hydraulic Engineering Lab., Technion, Haifa, Israel.
- Oswald, S., W. Kinzelbach, A. Greiner, and G. Brix. 1997. Observing of flow and transport processes in artificial porous media via magnetic resonance imaging in three dimensions. Geoderma 80:417– 429.
- Pearl, Z., M. Magaritz, and P. Bendel. 1993. Nuclear magnetic resonance imaging of miscible fingering in porous media. Transp. Porous Media 12:107–123.
- Perret, J., S.O. Prasher, A. Kartzas, and C. Langford. 1999. Three-dimensional quantification of macropore networks in undisturbed soil cores. Soil Sci. Soc. Am. J. 63:1530–1543.[Abstract/Free Full Text]
- Pfannkuch, 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.
- Sederman, A.J., M.L. Johns, P. Alexander, and L.F. Gladden. 1998. Visualization of structure and flow in packed beads. Magn. Reson. Imag. 16:497–500.[Medline]
- Seymour, J.D., and P.T. Callaghan. 1997. Generalized approach to NMR analysis of flow and dispersion in porous media. AICHE J. 43:2096–2111.
- Sposito, G., W.A. Jury, and V. Gupta. 1986. Fundamental problems in the stochastic convection–dispersion model of solute transport in aquifers and field soils. Water Resour. Res. 22:77–88.
- Vogel, H.J., and A. Kretzschmar. 1996. Topological characterization of pore space in soil-sample preparation and digital image-processing. Geoderma 73:23–38.
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ABSTRACT
INTRODUCTION
