Imaging of Water Flow in Porous Media by Magnetic Resonance Imaging Microscopy
Markus Deurer, Iris Vogeler,* Alexander Khrapitchev, and Dave Scotter
ABSTRACT
pared the column-scale breakthrough of a chloride step-
pulse with a dispersion estimate based on an MRI mea-
Magnetic resonance imaging (MRI) was used to study the flow of
sured slice-scale velocity distribution.
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
THEORY
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
Nuclear Magnetic Resonance
proton density and velocities within a 2-mm slice were taken at a
We will give only some specific background on the theory
spatial resolution of 0.15 0.15 2 mm3. At a mean pore water
of MRI. A thorough introduction and more details are given
velocity of 1 mm s 1 we approximated hydrodynamic dispersion using
by Callaghan (1991).
NMR-measured velocity distributions in a 2-mm slice through the
sample. Additionally, we conducted a step pulse tracer experiment
Measurement of Volumetric Water Content
with chloride through the same column and at identical initial and
boundary conditions. We fitted the convection dispersion equation
The volumetric water content ( ) at one voxel is directly
to the breakthrough curve and compared the column scale dispersion
proportional to the number of resonating spinning water pro-
of the tracer experiment with the respective NMR estimate derived
tons in the voxel, the so-called spin density, . The precessing
at the slice scale.
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
t present it is possible to measure pore characteris- moments of the individual resonating spinning water protons,
and is therefore a measure of their density. Consider the spins
Atics of porous media three-dimensionally using,
at the spatial position r within the sample occupying an infini-
for example, a series of thin sections (Vogel and Kretz-
tesimal volume dV. The local spin density (r) will contain
schmar, 1996) or computer aided tomography (Perret
(r) dV spins. The spatial position of each volume element
et al., 1999). But with these techniques we cannot study
is labeled in three dimensions with magnetic field gradients
which part of all the potential flow pathways are realized
denoted as Gx, Gy, Gz (G), where the subscripts refer to the
under various initial and boundary conditions. Magnetic
x, y, and z coordinates. They impose on the received signal
resonance imaging (MRI) can be used to obtain infor-
an image position specific frequency (frequency gradient, e.g.,
mation on soil properties and flow at a high spatial reso-
Gx) and a phase shift (phase gradient, e.g., Gy). The signal
lution. arising from this infinitesimal volume is (Callaghan, 1991):
Magnetic resonance imaging techniques have been
dS(G,t) (r)dV exp[iw(r)t] [1]
used to spatially determine soil water content (Amin et
al., 1994; Hall et al., 1997), and to measure the local where i denotes the real part of the complex signal and w(r)
is the position specific precession frequency. The latter de-
velocity of water and paramagnetic tracers in porous
pends on the main magnetic field B0 (applied externally to
media (Seymour and Callaghan, 1997; Sederman et al.,
cause the precession of the water protons) and the gradient
1998) and soils (Pearl et al., 1993; Amin et al., 1994;
G. Therefore, Eq. [1] can be written as (Callaghan, 1991):
Hoffmann et al., 1996; Oswald et al., 1997).
We measured the velocities of moving water protons
dS(G,t) (r)dV exp[i( B0 Gr)t] [2]
directly with a dynamic imaging pulse sequence (see
1
where is the gyromagnetic ratio for H.
Callaghan, 1991). The study aimed to explore the feasi-
For convenience the received signal is mixed with a refer-
bility of measuring the volumetric water content ( ) and
ence oscillation, canceling out the precession contribution of
velocity in a porous medium at the pore scale. To do
the main magnetic field to the signal. The signal now oscillates
this we studied water flow in a column packed with glass
at Gr. Integrating over the volume of each voxel the signal
beads, 2 mm in diameter. The NMR images of the spin
becomes (Callaghan, 1991):
density (which is directly proportional to ) and velocity
S(t) [3]
(r)exp[i Grt]
distributions were taken within a 2-mm slice through
the column. The voxel size was 0.15 0.15 2 mm3.
Using the following substitution (the so-called reciprocal space
A second aim was to document the potential of MRI
vector K):
to analyze the spatial scale dependence of processes
1
such as hydrodynamic dispersion. To do this we com-
K Gt [4]
2
M. Deurer, I. Vogeler, and D. Scotter, Environment and Risk Manage-
the received signal S and the spin density are mutually conju-
ment Group, Hort Research Institute, Private Bag 11-030, Palmerston
North, New Zealand. A. Khrapitchev, Institute of Fundamental Sci- gate Fourier pairs (Callaghan, 1991):
ences, Massey University, Private Bag 11-222, Palmerston North, New
Zealand. Received 6 June 2001. *Corresponding author (ivogeler@
Abbreviations: CDE, convection dispersion equation; CLT, convec-
hort.cri.nz).
tive lognormal transfer; MRI, magnetic resonance imaging; NMR,
Published in J. Environ. Qual. 31:487 493 (2002). nuclear magnetic resonance.
487
488 J. ENVIRON. QUAL., VOL. 31, MARCH APRIL 2002
For the lower boundary condition it was assumed that the
S(K) (r)exp[i2 Kr]dr
column was part of an effectively semi-infinite system.
The solution for the flux concentration is given by (Kirkham
(r) S(K)exp[ i2 Kr]dK [5]
and Powers, 1971, p. 379 420):
Therefore, after Fourier transformation, we can directly
derive the spin density as the spectrum of the Fourier-trans-
1 z vt v z z vt
Cf(z,t) erfc. exp erfc.
formed signal.
2 D
2"Dt 2"Dt
Spatial Measurements of Velocities
[13]
We want to characterize the bulk motion of the spins in
We also predicted the outflow concentration Cf through
the direction of flow (in our case z). At the time t 0 the
the entire sample using the NMR-measured slice-scale voxel
spins are at position z and move with a longitudinal velocity
velocities. One approach to link measurements of water and/
vz. To label and analyze their motion we apply an additional
or solute transport from different scales, in our case the nuclear
(independent of the gradients for spatial encoding) pulsed
MRI measurements in the 2-mm slice and the column scale
gradient gz in the direction of flow. For a very short time
tracer experiment, is the use of transfer functions (Jury, 1982;
the spins at z will therefore precess faster. They dephase by
Jury and Roth, 1990). This approach models hydrodynamic
an angle 1 (C.D. Eccles, personal communication, 2000):
dispersion either with a convective lognormal transfer (CLT)
function (Jury et al., 1987), or with the convection dispersion
1 w1t1 gzz [6]
equation (CDE). In the first case the dispersivity grows lin-
where is the duration gz is applied. After the time , gz is
early with the travel distance and has no asymptotic limit. In
reapplied. Due to the movement along the z axis the protons
the second case it is constant. The CLT is considered to repre-
are now at position z vz . Consequently, they will rephase
sent the short distance limit and the CDE the long-distance
by an angle 2:
limit of solute transport (Sposito et al., 1986). Physically, water
and/or solute transport is probably best represented by a tran-
2 w2t2 gz(z vz ) [7]
sition from a flow in isolated streamtubes (CLT) to one in a
The resultant phase shift is then:
fully interconnected network of flow pathways (CDE). There
are no theoretical concepts to predict the spatial or temporal
2 1 gzvz [8]
scales for such a transition. Only little experimental evidence
Note that this resulting phase shift is not position-depend-
exists for such a transition (Butters and Jury, 1989).
ent but only velocity-dependent. We repeat this basic experi-
At the observation scale of the 2-mm slice imaged with
ment several times varying either gz, , or . Let us assume
NMR the flow is restricted to isolated areas between the glass
that we vary . We then obtain a signal amplitude that varies
beads, as they have a diameter of 2 mm. Thus, on this scale
sinusoidally with (C.D. Eccles, personal communication,
hydrodynamic dispersion is truly stochastic convective and
2000):
takes place in isolated stream tubes.
A stochastic convective transport of water and/or solutes is
S( ) S(t 0)exp(i[ gzvz ] ) [9]
like a superposition of piston flow processes within individual
where the angular frequency term in the waveform of Eq. [9]
subcolumns (Jury and Roth, 1990). Each voxel within the slice
is proportional to the velocity of the spins.
is hypothesized to be a representative section of one of those
The concept of measuring velocities with MRI (phase shift,
subcolumns. We then use the slice-scale results and the CLT
see Eq. [8]) puts the movement of water in the Lagrangian
to predict the column-scale breakthrough from our NMR mea-
framework (Seymour and Callaghan, 1997). The measured
surements. According to Bear (1972, p. 608 609), at a mean
voxel-scale vz is not space-fixed.
flow rate of 1 mm 1 the effect of diffusion on dispersion is
negligible, as our experiment would be classified to belong to
Solute Transport
his Zone IV, a region of dominant mechanical dispersion.
Hydrodynamic dispersion is then entirely driven by the vari-
We assume that the one-dimensional convection dispersion
equation (CDE) is appropriate for our experimental condi- ability of velocities. Knowing the length l of the sample and
the voxel velocities, solutes in each of these subcolumns have
tions: horizontal flow through a homogeneous medium with
spatially and temporally constant mean longitudinal pore wa- a characteristic travel time from the inlet to the outlet, tl(s):
ter velocities v in the direction of flow (z axis). The CDE is
l
given by:
tl(s) [14]
vz(s)
Cf 2Cf Cf
D v [10]
where s denotes the spatial coordinates (x,y) of the center of
t z2 z
each subcolumn. We assume equal probabilities for solute
where Cf is the tracer flux concentration and D the longitudinal
molecules to move into each of the subcolumns. The result is
dispersion coefficient. For negligible molecular diffusion D
a probability density function of the travel times of the solute
can be approximated by:
molecules to travel from the inlet to the outlet port, ff (l,t).
The respective flux concentration at the outlet l and at time
D v [11]
t is then given by (Jury and Roth, 1990):
where is the longitudinal dispersivity.
t
The column is initially free of chemicals and at time t 0
Cf(l,t) C0 ff(l,t )dt C0 Pf(l,t) [15]
a tracer with the concentration C0 is applied as a step pulse
0
to the inlet end. Thus, the boundary conditions are:
where Pf (l,t) is the cumulative travel time distribution func-
Cf 0 for z 0 and t 0
tion. After normalization of Cf by C0, Eq. [15] can be directly
Cf C0 for z 0 and t 0 [12] compared with Eq. [13].
DEURER ET AL.: IMAGING OF WATER FLOW IN POROUS MEDIA BY MRI MICROSCOPY 489
m3 m 3. The inlet and outlet ports occupied only a fraction
METHODS AND MATERIALS
of the proximal and distal surfaces, thereby representing a
Flow Experiments
point-like influx and efflux. These ports were connected to a
silicon tube with a 1.73-mm diameter. For the flow experi-
A column with a 14.1-mm inner diameter and length l of
ments tap water was pumped through the glass beads with a
46 mm was used for the experiment. The column was filled
Pharmacia double syringe pump at rates of 125 and 250 mL
with glass beads 2 mm in diameter giving a porosity of 0.45
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 ana-
lyzed 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 mag-
net 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-dimen-
sional spin echo pulse sequence within a slice of 2 mm. A
field of view of 20 20 mm was resolved with 128 phase
steps. The sequence consisted of a 4- s 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 respec-
tive 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 se-
quence 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 popu-
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 Fig. 2. Relationship between the actual applied flow rates and those
(c ) 14 mms 1. measured by nuclear magnetic resonance (NMR).
490 J. ENVIRON. QUAL., VOL. 31, MARCH APRIL 2002
lation for normalization to get the fractions. For this calcula-
RESULTS
tion we excluded voxels with a zero spin density.
Nuclear Magnetic Resonance Images
of Water Content
Velocity Calibration
Figure 3 shows an image of the volumetric water
Figure 1 shows the velocity distribution of three of the
content from a 2-mm slice in the column packed with
velocity measurements through the capillary tube, at 9, 90,
glass beads. The regions where the glass beads are and
and 210 mL h 1. This corresponds to mean velocities of 0.6,
the regions between them are clearly visible. The mean
6, and 14 mm s 1. All three images show a parabolic velocity
value of all these normalized voxels was 0.44 m3 m 3.
distribution, as expected. At low velocities some distortion
occurs, which is probably due to the shaking of the magnet. This compares well with the gravimetrically determined
Figure 2 shows the relationship between the actual applied
bulk porosity of 0.45 m3 m 3.
flow rates and those measured by NMR. The relationship is
linear. Different NMR parameter settings had to be used for
Velocity Measurements through Beads of Glass
low ( slow sequence) and high velocities ( fast sequence).
Figure 4 shows the spatial distribution of velocities
These velocity measurements in capillary tubes provided us
between glass beads at the set flow rates of 125 and 250
with a NMR sequence, which we could use to measure velocity
spectra in porous media. mL h 1, corresponding to mean flow rates of 0.5 and
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.
DEURER ET AL.: IMAGING OF WATER FLOW IN POROUS MEDIA BY MRI MICROSCOPY 491
1 mms 1. For both flow rates two images were taken, to the wall where the packing is more regular. The mean
with different settings (according to the calibration mea- velocities calculated from these NMR images agree well
surements). These two images were then overlaid. The with the actual applied velocities (see Table 1). The
regions with high velocities are clustered and occur close flow rates obtained by multiplying these mean velocities
Fig. 4. Spatial distribution of velocities in the glass bead mean flow rates of (a ) 0.5 and (b ) 1 mms 1.
492 J. ENVIRON. QUAL., VOL. 31, MARCH APRIL 2002
Table 1. Comparison between applied flow rates (Q) and veloci-
ties (v) and those obtained by nuclear magnetic resonance
(NMR).
Qv
mL h 1 mm s 1
Applied 125 250 0.5 1
NMR, bulk characteristics 137 249 0.55 1
NMR, voxel-based 66 115
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 be-
tween 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 Fig. 5. Measured and predicted breakthrough curve (BTC) of chlo-
ride using the convection dispersion equation (CDE) with theoret-
porous materials (Davies et al., 1991; Allen et al., 1998).
ical parameter values (dotted line), and with fitted parameter values
In the study of Hall et al. (1997) using a standard spin
(solid line). Also shown is the prediction based on the velocity
echo sequence, only 0.2 to 57% of the bulk water was
distribution measured by nuclear magnetic resonance (NMR) (bro-
detected for soils of varying types including sands. Cur-
ken line).
rently, we try to use different NMR protocols such as
an inversion recovery sequence (Callaghan, 1991) to
column scale was about the bead diameter (Pfannkuch,
overcome this problem. For the calibration we omitted
1963). But all of them used a uniform Neumann-type
voxel values that had zero spin density. These are voxels
boundary condition (Neumann et al., 1974). In many
with very low water contents. Voxels with small water
situations we can encounter a similar boundary condi-
contents contain either small pore(s) or mainly glass
tion rather than a homogeneous entry of water and/
bead material. The effects of a changing magnetic sus-
or solutes into the soil (e.g., drip irrigation systems,
ceptibility are strongest close to the surface of the glass
preferential or unstable flow). This shows the depen-
beads. Therefore, voxels with small water contents are
dence of transport parameter values on the type of
likely to show the resulting effect of signal attenuation
boundary condition.
most pronounced. In those voxels the underestimation
Using only the voxel-based slice-scale velocities for
of the spin density, and thus the water content, would
an estimate of the normalized outflow concentration
be highest. If the respective voxel velocities are simulta-
(see broken line in Fig. 5), the breakthrough starts much
neously high then their contribution to the overall flux
earlier and shows pronounced tailing. The dispersivity
will be considerable. But generally, all voxels and their
from this curve is estimated as 63 mm and the respective
water contents that are influenced by the inhomogeneity
fractional transport volume as only 0.129. The extrapo-
of magnetic susceptibility due to the glass bead material
lation of the slice measurement neglects the formation
are affected.
of a network of flow paths. This is indirect microscopic
experimental proof for a scale-dependent transition
Tracer Experiment
from a stochastic convective to a convective dispersive
The breakthrough curve (BTC) of chloride is shown
transport process. The variability of the velocity distri-
in Fig. 5. Also shown as the dotted line is the prediction
bution itself is obviously scale dependent and a result of
of the convection dispersion equation (CDE) using a
the flow network configuration. Our findings emphasize
dispersivity ( ) of 2 mm (theoretical value according
that we need more research in this particular area con-
the diameter of the glass beads; Dullien, 1992), and
cerning the scales of the transition from a stochastic
assuming that all the water is mobile ( m). However,
convective to a convective dispersive dispersion regime.
the chloride appears in the leachate much earlier than
We think that NMR has the potential to solve these
predicted. Next, the CDE was fitted to the data, giving
questions in the future. It also shows that what we inter-
a of 3 mm and a m of 0.33 (solid line). The dispersivity
pret as the mobile (transport volume) water fraction
as a correlation length of the pore space is larger than
might be rather a function of the flow network, the
the glass-bead diameter, and the respective transport
observation scale, and the type of boundary condition.
volume is smaller than the porosity. This might be due
to our experimental setup, with a point-source entry of
OUTLOOK
water and/or solutes as an upper boundary condition.
Existing macroscopic experiments measuring dispersion In a few experiments NMR has been used to study
in glass bead porous media show that dispersivity on the the transport of water and/or solutes in soils (Pearl et
DEURER ET AL.: IMAGING OF WATER FLOW IN POROUS MEDIA BY MRI MICROSCOPY 493
ards, and B.W. Bache. 1994. Magnetic resonance imaging of soil-
al., 1993; Hoffman et al., 1996; Amin et al., 1997; Hem-
water phenomena. Magn. Reson. Imag. 12:319 321.
minga and Buurman, 1997; Cislerová et al., 1999).
Bear, J. 1972. Dynamics of fluids in porous media. Elsevier, New York.
Natural soils regularly contain ferromagnetic or para-
Butters, G.L., and W.A. Jury. 1989. Field scale solute transport of
magnetic elements, for example, free iron oxides or
bromide in an unsaturated soil. 2. Dispersion modeling. Water
copper sulfates, respectively. These lead to faster spin Resour. Res. 25:1583 1589.
Callaghan, P.T. 1990. Susceptibility-limited resolution in nuclear mag-
spin and spin lattice relaxation times. Consequently,
netic resonance microscopy. J. Magn. Reson. 87:304 318.
when the signal is measured a considerable part of it has
Callaghan, P.T. 1991. Principles of nuclear magnetic resonance micros-
already been attenuated and the results are accordingly
copy. Clarendon Press, Oxford.
biased. The study of Greiner et al. (1997) took advan-
Cislerová, M., J. Votrubová, T. Vogel, M.H.G. Amin, and L.D. Hall.
tage of this effect using copper sulfate as a paramagnetic 1999. Magnetic resonance imaging and preferential flow in soils.
p. 397 411. In Proc. of the Int. Workshop Characterization and
tracer. Other soil physical and chemical properties, such
Measurement of the Hydraulic Properties of Unsaturated Porous
as high clay contents (which result in pronounced inho-
Media, Riverside, CA. 22 24 Oct. 1999. Part I. USDA, River-
mogeneities in the magnetic susceptibility), organic mat-
side, CA.
ter, and even exchangeable cations such as calcium and
Davies, S., K.J. Packer, D.R. Roberts, and F.O. Zelanya. 1991. Pore
potassium can also reduce the relaxation times (Hall et size distributions from NMR spin lattice relaxation data. Water
Resour. Res. 9:681 685.
al., 1997). However, advances in NMR technology and
Dullien, F.A.L. 1992. Porous media. Fluid transport and pore struc-
the development of new adapted pulse sequences might
ture. 2nd ed. Academic Press, San Diego, CA.
solve these problems.
Greiner, A., W. Schreiber, G. Brix, and W. Kinzelbach. 1997. Magnetic
resonance imaging of paramagnetic tracers in porous media: Quan-
tification of flow and transport parameters. Water Resour. Res.
CONCLUSIONS
33:1461 1473.
Hall, L.D., M.H.G. Amin, E. Dougherty, M. Sanda, J. Votrubova,
The study has described the use of NMR to observe
K.S. Richards, R.J. Chorley, and M. Cislerova. 1997. MR properties
water movement in porous media, here in columns
of water in saturated soils and resulting loss of MRI signal in water
packed with glass beads. The volumetric water content
content detection at 2 Tesla. Geoderma 80:431 448.
inferred from the NMR image agreed well with the one
Hemminga, M.A., and P. Buurman. 1997. NMR in soil science. Ge-
oderma 80:221 224.
determined gravimetrically. The mean velocities and
Hoffmann, F., D. Ronen, and Z. Pearl. 1996. Evaluation of flow
flow rates obtained by NMR also agreed well with the
characteristics of a sand column using magnetic resonance imaging.
applied flow rate when based on bulk characteristics.
J. Contam. Hydrol. 22:95 107.
However, when the flow rate was calculated on a voxel
Jury, W.A. 1982. Simulation of solute transport using a transfer func-
basis only about 50% of the actual applied flow rate tion model. Water Resour. Res. 18:363 368.
Jury, W.A., D.D. Focht, and W.J. Farmer. 1987. Evaluation of pesti-
was obtained. This was probably due to a spin density
cide ground water pollution potential from standard indices of
distribution that was biased toward voxels with small
soil chemical adsorption and biodegradation. J. Environ. Qual. 16:
volumetric water contents: spin spin relaxation times
422 428.
will be rapid and due to technical limitations no signal
Jury, W.A., and K. Roth. 1990. Transfer functions and solute move-
ment through soil. Theory and application. Birkhäuser Verlag,
can be detected. We intend to further improve our NMR
Basel, Switzerland.
measuring technique.
Kirkham, D., and W.L. Powers. 1971. Advanced soil physics. Wiley
We compared the results of a conventional break-
Interscience, New York.
through experiment (column-scale) with the prediction
Neumann, S.P., R.A. Feddes, and E. Bresler. 1974. Finite element
based on the slice-scale velocity distribution obtained simulation of flow in saturated unsaturated soils considering water
uptake by plants. 3rd Annu. Rep., Project A10-SWC-77. Hydraulic
by MRI. We showed that solute transport parameters
Engineering Lab., Technion, Haifa, Israel.
such as the dispersivity and the transport volume depend
Oswald, S., W. Kinzelbach, A. Greiner, and G. Brix. 1997. Observing
on the network of flow pathways, the type of boundary
of flow and transport processes in artificial porous media via mag-
condition, and the observation scale. Thus, MRI can
netic resonance imaging in three dimensions. Geoderma 80:417
give valuable information concerning transport pro- 429.
Pearl, Z., M. Magaritz, and P. Bendel. 1993. Nuclear magnetic reso-
cesses in porous media, which at present can only be
nance imaging of miscible fingering in porous media. Transp. Po-
studied on a macroscopic scale.
rous Media 12:107 123.
Perret, J., S.O. Prasher, A. Kartzas, and C. Langford. 1999. Three-
ACKNOWLEDGMENTS
dimensional quantification of macropore networks in undisturbed
soil cores. Soil Sci. Soc. Am. J. 63:1530 1543.
The study was funded by the Royal Society through a Mars-
Pfannkuch, O. 1963. Contribution a l etude des deplacements de flu-
den Fund, Contract HRT 805. We would like to thank Paul
ides miscible dans un milieu poreux. Revue de l Institute Francais
Callaghan and Sarah Codd for many helpful discussions and
du Petrole 18:215 270.
Brent Clothier for the encouragement to do this work. 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.
REFERENCES
Seymour, J.D., and P.T. Callaghan. 1997. Generalized approach to
Allen, S., M. Mallet, M.E. Smith, and J.H. Strange. 1998. Susceptibility NMR analysis of flow and dispersion in porous media. AICHE J.
effects in unsaturated porous silica. Magn. Reson. Imag. 16:597 43:2096 2111.
600. Sposito, G., W.A. Jury, and V. Gupta. 1986. Fundamental problems
Amin, M.H.G., R.J. Chorley, K.S. Richards, L.D. Hall, T.A. Carpen- in the stochastic convection dispersion model of solute transport
ter, M. Cislerová, and T. Vogel. 1997. Study of infiltration into in aquifers and field soils. Water Resour. Res. 22:77 88.
Vogel, H.J., and A. Kretzschmar. 1996. Topological characterization
a heterogeneous soil using magnetic resonance imaging. Hydrol.
of pore space in soil-sample preparation and digital image-pro-
Processes 11:471 483.
Amin, M.H.G., L.D. Hall, R.J. Chorley, T.A. Carpenter, K.S. Rich- cessing. Geoderma 73:23 38.
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