- Email: [email protected]
Studies of soil-water transport by MRI
Magnetic
ELSEVIER
l
Resonance Imaging, Vol. 14, Nos. 7/8, pp. 879-882, 1996 Copyright 0 1996 Elsevier Science Inc. Printed in the USA. All rights reserved 0730-725X/96 $15.00 + .OO
PI1 SO730-725X( 96) 00171-2
Shor ; Communication STUDIES
OF SOIL-WATER
M.H.G. AMIN,*
TRANSPORT
K.S. RICHARDS,* R.J. CHORLEY,* T.A. CARPENTER,? AND L.D. HALL?
BY MRI S.J. GIBBS,-~
*Department of Geography, University of Cambridge, Downing Place, Cambridge, CB2 3EN, UK, tHerche1 Smith Laboratory for Medicinal Chemistry, University of Cambridge School of Clinical Medicine, Robinson Way, Cambridge, CB2 2PZ, UK Sequential spin-echo spin-warp MRI pulse sequences have been used to study soil-water transport processes including infiltration, redistribution, and drainage of water in soil columns. Those images provide a means for monitoring and quantifying spatial and temporal changes of soil-water distributions and the movement of wetting fronts. In addition, temporal-geometric changes of unstable wetting fronts during water redistribution were estimated from 2D images and the temporal development of the longest length of finger was described by a fractal relation t - L1.38. Bulk dispersion-time-dependent displacement and velocity spectra, as well as 2D maps of flow velocities and dispersion coefficients in soil macropores during saturated steadystate flow, were reconstructed from data obtained using the alternating-pulsed-field-gradient (APFG) pulse sequences. Copyright 0 1996 Elsevier Science Inc. Keywords:
MRI; Soil-water
transport;
Unstable wetting front; Fractals.
INTRODUCTION
UK; (b) a sandy loam Cambisol soil from Korkusova Hut, Czech Republic; and (c) medium (0.21-0.36 mm equiv. diam.) and coarse (0.5-l mm equiv. diam.) quartz sand. The clay and sandy-loam soil samples were taken from the top lo-30 cm of the soil profile. The main difficulties and limitations encountered in soil-water MRI are image distortions, poor signal-tonoise ratio (S/N) and poor spatial resolution, all of which are mainly caused by background magnetic gradients produced by variations in the magnetic susceptibility within the soil pore system. These problems can be alleviated or overcome by stronger field gradients and appropriate imaging pulse sequences. This study employs prefocused selective pulses, 5 spin-echo (SE) spin-warp imaging pulse sequences and alternatingpulsed-field-gradient stimulated-echo pulse sequences. It is important to use as short as possible echo times (TE) in an SE sequence. In this work TE values of 3 ms were used. After optimising the experimental conditions, the successful techniques were applied to two-dimensional
Although the application of MRI to studies of soilwater transport can be dated back to 1979 when Gummerson et al.’ used MRI to study capillary absorption in several porous media, including a clay loam soil, few MRI studies of water transport in natural soils have been reported.2-4 Therefore, a wide range of MRI studies of soil-water transport still needs development. This article presents a brief overview of the authors’ work aimed at identifying opportunities for application of MRI to studies of soil-water transport and the difficulties and limitations of such applications. The MR equipment used in the present work was an Oxford Research Systems Biospec I spectrometer operating at 84 MHz for protons, connected to an Oxford Instruments 3 l-cm horizontal bore, 2 T superconducting magnet. The maximum gradient strengths used were -3.8 and 30.4 G/cm from a 20-cm and lo-cm gradient coil, respectively. Soil materials used in the studies reported here were: (a) a Hanslope clay soil from near Cambridge, Address correspondence to M.H.G. Amin, Department of Geography, University of Cambridge, Downing Place,
Cambridge, CB2 3EN, UK.
879
880
Magnetic ResonanceImaging0 Volume 14, Numbers 7/8, 1996
(2D) imaging of water movement in unsaturated soil materials.6’7 The results showed that MRI can be used to observe water transport in a wide range of model and natural soils, especially for soils with an iron content of <3% and a water content of >20%. The images obtained revealed different dynamic behaviour in medium sand, coarse sand and packed clay soil columns (54 mm diam. X 60 mm height) during the wetting, redistribution, and drainage processes. Wetting front instability was observed in the coarse sand during the wetting and water redistribution periods.6 Temporal geometrical changes of an unstable wetting front (“fingering”) during water redistribution in a packed coarse sand column (50 X 51 mm) were monitored by sequential 2D imaging. Twelve 2D images obtained at time (t) after a step-feed of 20 mL water (t = 0) are shown in Fig. 1. The experimental details are given in the figure caption. Fingering of the wetting front has features similar to viscous fingering developed in a linear Hele-Shaw cell.g*9 A critical wavelength expression was derived for wetting front fingering, which is the same as that for viscous fingerindos” This suggests that the geometry of fingering should also possess similar fractal features I2 to those of viscous fingering for which case the fractal properties are well documented (e.g., as reviewed by Feder 13). As wetting-front fingering develops with time it should develop a multifractal character. To test this, perimeters, areas, and the longest finger length at different times were measured in 128 x 128, 256 x 256, and 5 12 X 5 12 image matrix sizes. Fractal dimensions (D) l2 of perimeters vary with the passage of time (D = 1.02, 1.15, 1.14, 1.38, 1.28, 1.27, 1.24,
Fig. 1. Vertical (2D, 128 x 128) images of water redistribution in a packed coarse quartz sand column (54 X 5 1 mm) after a step-feed of 20 mL water at t = 0. A spin-echo spinwarp pulse sequence was used with repetition time of 0.2 s and echo time 2.97 ms. The slice thickness was 12.5 mm, read gradient 10.4 kHz/cm and field of view 91 mm. A single signal average was obtained and the acquisition time for each image was 45 s.
0
1
2 In L (mm)
3
4
0
1
2
3
4
In L (mm)
Fig. 2. Plots of the longest length of the finger (L) vs: (a) time (t) and b) total pixel number (N) of the area of fingering.
1.34, 1.42, 1.31, 1.30, and 1.81 for each time step in Fig. 1 ), which agrees well with the trend and range in the values of D obtained by Chang et all2 the last D in the sequence is very large due to the significant split in the finger. Relationships between the longest length of the finger (L) and time (t) , and between L and total pixel number (N) of the wetted area can be expressed as - LD’ ; and N - LD. Figure 2 shows plots of these two relations, and gives D 1 of 1.38 and D of 1.18, which are smaller than the corresponding D values ( 1.6- 1.7) for viscous fingers obtained from a radial Hele-Shaw cell. Thus, the fractal features of wettingfront fingering and relations among those fractal dimensions need further investigation. The conventional method of investigating wetting-front fingers uses a large size Hele-Shaw cell, which gives 2D infiltration by restricting the third dimension effectively to the narrow gap between the cell walls. It has already been shown that MRI provides nondestructive visualisation of 2D viscous fingering.14 In the present work, the wetting-front finger that developed in a 3D cylindrical space was observed by 2D imaging. Investigations of such 3D phenomena using 3D imaging are unfortunately difficult due to the short time of the infiltration process relative to image acquisition time and poor S/ N. Consequently, future studies in this area should be based on fast data acquisition MRI protocols. In addition to these experiments, one-dimensional ( 1D) and 2D imaging techniques were employed to detect unsaturated flow patterns, to track the movement of wetting fronts, to obtain water content profiles, and to estimate sorptivities and steady-state infiltration rates in an undisturbed core (46 millimeters X 80 mm) of the sandy-loam soil during infiltration7 The results showed preferential flow phenomena, as well as air entrapment in the soil. It is important to contrast the strengths and weaknesses of 1D and 2D MRI methods. Clearly, the 2D method has the advantage of enhanced spatial coverage, but
t
Studies of soil-water transport by MRI
limits the time resolution per image slice. In this research, the temporal resolution was enhanced to 2 s by reducing the spatial resolution to 1D. Clearly, reduced image dimensionality could lead to ambiguities if applied to heterogenous samples. Nevertheless, 2D images using a short repetition time and prefocused rf pulses in a SE sequence are feasible. In this work, the alternating-pulsed-field-gradient MR technique has been used to obtain displacement and velocity spectra of steady-state saturated flow through a column packed with glass beads.” The displacement spectra obtained by PFG MR correspond to travel-distance probability-density functions (pdf) for initial conditions of a concentration impulse in a column with zero concentration. These spectra show strong dispersion-time dependence and differ from Gaussian-shaped pdfs for short dispersion times. These data provide estimates of the dispersion-time dependence of transverse and longitudinal dispersion coefficients. The longitudinal dispersion coefficient reaches its long-term behaviour slower than the transverse one. Long-term values obtained from MR data fall between the values calculated using the existing empirical relationships of Gunn16 and of Wakao and Funazkri.17 A model based on three components of apparent velocity and dispersion coefficient has been developed for the description of dispersion-time dependent displacement spectra for water flow through the bead column.‘5 The short-distance component corresponds to convectiondispersion-diffusion within necks between particles, whereas the long-distance component represents the macroscopic convection-dispersion process. This research shows that PFG MR flow spectroscopy is a simple but potentially useful method for study of flow and hydrodynamic dispersion in porous media, especially for investigations of time-dependent phenomena. A similar technique was also used to obtain maps of macropore flow velocities and dispersion coefficients in cross-sections of steady-state saturated flow through a packed bead column and a clay soil column. The following conclusions can be drawn from this work. First, MRI is a potentially useful tool for a wide range of studies of soil-water transport, especially with respect to preferential flow in soils. Secondly, soils with an iron content of ~3% and a water content of >20% can be studied using MRI. The results reported here suggest that MRI can achieve measurements and allow inferences that would be difficult to obtain by any other technique. Because the spatial and temporal resolutions of MRI are restricted by the dynamic features of the transport process under investigation and by the MR properties of the soil water, it is clear that it is now important to investigate roles for faster MRI
l
M.H.G. AMIN ETAL.
881
protocols. Such methods will facilitate higher spatial and temporal resolution. It is obvious that advances in MRI of other porous media, especially of rocks, provide a further source of important methodology that should now be applied to soil. Acknowledgments-This
work was made possible by a munificent
benefaction to LDH andTAC fromDr. HerchelSmith,andpartially supported by NERC Research Grant GR3/9369 and a PGRA Grant to MHGA. Thanksarealsodueto Drs.M. Cislerova andT. Vogel
for helpfuldiscussions andprovidingthe sandy-loam soil sample, andto Dr. N. Herrod,Mr. J.A. Derbyshire, andMr. D. Xing for providing software for MR data processing and analysis.
REFERENCES 1. Gummerson,R.J.; Hall, C.; Hoff, W.D.; Hawkes,R.; Holland, G.N.; Moore, W.S. Unsaturatedwater flow within porousmaterialsobservedby NMR imaging.Nature 281:56-57; 1979. 2. Bottomley, P.A.; Rogers,H.H.; Foster, T.H. NMR imaging showswater distribution and transport in plant root systemsin situ. Proc. Natl. Acad. Sci. USA. 83:8789; 1986. 3. Liu, I.-W.Y.; Wong, S.T.S.; Waldron, L.J. The application of nuclear magneticresonanceimaging to study preferential water flow through root channels.In: S.H. Anderson;J.W. Hopmans(Eds.) Tomographyof SoilWater Root Processes.Madison, WI: ASA, Inc. and SSSA, Inc.; 1994:pp. 135-148. 4. Amin, M.H.G.; Hall, L.D.; Chorley, R.J.; Carpenter, T.A.; Richards,KS.; Bathe, B.W. Magnetic resonance imaging of soil-water phenomena.Magn. Reson.Imaging 12:319-321; 1994. 5. Roberts,T.P.; Carpenter,T.A.; Hall, L.D. Designand applicationof prefocusedpulsesby simulatedannealing. J. Magn. Reson.89:595-604; 1990. 6. Amin, M.H.G.; Hall, L.D.; Chorley, R.J.; Carpenter, T.A.; Richards,K.S.; Bathe, B.W. Visualisationof static and dynamic water phenomenain soil using magnetic resonanceimaging. In: V.P. Singh; B. Kumar (Eds). Subsurface-WaterHydrology. Dordrecht: Kluwer Academic Publishers,1996:pp. 3- 16. 7. Amin, M.H.G.; Chorley, R.J.; Richards, K.S.; Hall, L.D.; Carpenter,T.A.; Cislerova, M.; Vogel, T. Study of infiltration into a heterogeneoussoil using nuclear magneticresonanceimaging.Hydro. Proc. (in press). 8. Saffman,P.G.; Taylor, G. The penetrationof a fluid into a porousmediumof Hele-Shawcell containing a more viscousliquid. Proc. R. Sot. Lond. [A] 245:312-331; 1958. 9. Hill, D.; Parlange,J.Y. Wetting front instability in layeredsoils.Soil Sci. Sot. Am. Proc. 36:697-702; 1972. 10. Philip, J.R. Stability analysisof infiltration. Soil Sci. Sot. Am. Proc. 39:1042-1049; 1975. 11. Chuoke,R.L.; van Meurs, P.; vander Poel,C. The instability of slow, immiscible, viscous liquid-liquid dis-
882
Magnetic ResonanceImaging l Volume 14, Numbers 7/8, 1996
placements in permeable media. Trans. Am. Inst. Min. Metall. Pet. Eng. 216:188-194; 1959. 12. Chang, W.-L.; Biggar, J.W.; Nielsen, D.R. Fractal description of wetting front instability in layered soils. Water Resources Res. 30: 125- 132; 1994. 13. Feder, J. Fractals. New York: Plenum Press; 1988. 14. Davies, ES.; Roberts, T.P.L.; Carpenter, T.A.; Hall, L.D.; Hall, C. Visualization of viscous fingering by nuclear magnetic resonance imaging. J. Magn. Reson. 96:210-214; 1992.
15. Amin, M.H.G.; Gibbs, S.J.; Hall, L.D.; Chorley, R.J.; Richards, K.S.; Carpenter, T.A. Study of flow and hydrodynamic dispersion in a porous medium using PFG MR. Proc. R. Sot. Lond. [A] (in press). 16. Gunn, D.J. Axial and radial dispersion in fixed beds. Chem. Eng. Sci. 42:363-373; 1987. 17. Wakao, N.; Funazkri, T. Effect of fluid dispersion coefficients on particle-to-fluid mass transfer coefficients in packed beds. Chem. Eng. Sci. 33:13751384; 1978.
ELSEVIER
l
Resonance Imaging, Vol. 14, Nos. 7/8, pp. 879-882, 1996 Copyright 0 1996 Elsevier Science Inc. Printed in the USA. All rights reserved 0730-725X/96 $15.00 + .OO
PI1 SO730-725X( 96) 00171-2
Shor ; Communication STUDIES
OF SOIL-WATER
M.H.G. AMIN,*
TRANSPORT
K.S. RICHARDS,* R.J. CHORLEY,* T.A. CARPENTER,? AND L.D. HALL?
BY MRI S.J. GIBBS,-~
*Department of Geography, University of Cambridge, Downing Place, Cambridge, CB2 3EN, UK, tHerche1 Smith Laboratory for Medicinal Chemistry, University of Cambridge School of Clinical Medicine, Robinson Way, Cambridge, CB2 2PZ, UK Sequential spin-echo spin-warp MRI pulse sequences have been used to study soil-water transport processes including infiltration, redistribution, and drainage of water in soil columns. Those images provide a means for monitoring and quantifying spatial and temporal changes of soil-water distributions and the movement of wetting fronts. In addition, temporal-geometric changes of unstable wetting fronts during water redistribution were estimated from 2D images and the temporal development of the longest length of finger was described by a fractal relation t - L1.38. Bulk dispersion-time-dependent displacement and velocity spectra, as well as 2D maps of flow velocities and dispersion coefficients in soil macropores during saturated steadystate flow, were reconstructed from data obtained using the alternating-pulsed-field-gradient (APFG) pulse sequences. Copyright 0 1996 Elsevier Science Inc. Keywords:
MRI; Soil-water
transport;
Unstable wetting front; Fractals.
INTRODUCTION
UK; (b) a sandy loam Cambisol soil from Korkusova Hut, Czech Republic; and (c) medium (0.21-0.36 mm equiv. diam.) and coarse (0.5-l mm equiv. diam.) quartz sand. The clay and sandy-loam soil samples were taken from the top lo-30 cm of the soil profile. The main difficulties and limitations encountered in soil-water MRI are image distortions, poor signal-tonoise ratio (S/N) and poor spatial resolution, all of which are mainly caused by background magnetic gradients produced by variations in the magnetic susceptibility within the soil pore system. These problems can be alleviated or overcome by stronger field gradients and appropriate imaging pulse sequences. This study employs prefocused selective pulses, 5 spin-echo (SE) spin-warp imaging pulse sequences and alternatingpulsed-field-gradient stimulated-echo pulse sequences. It is important to use as short as possible echo times (TE) in an SE sequence. In this work TE values of 3 ms were used. After optimising the experimental conditions, the successful techniques were applied to two-dimensional
Although the application of MRI to studies of soilwater transport can be dated back to 1979 when Gummerson et al.’ used MRI to study capillary absorption in several porous media, including a clay loam soil, few MRI studies of water transport in natural soils have been reported.2-4 Therefore, a wide range of MRI studies of soil-water transport still needs development. This article presents a brief overview of the authors’ work aimed at identifying opportunities for application of MRI to studies of soil-water transport and the difficulties and limitations of such applications. The MR equipment used in the present work was an Oxford Research Systems Biospec I spectrometer operating at 84 MHz for protons, connected to an Oxford Instruments 3 l-cm horizontal bore, 2 T superconducting magnet. The maximum gradient strengths used were -3.8 and 30.4 G/cm from a 20-cm and lo-cm gradient coil, respectively. Soil materials used in the studies reported here were: (a) a Hanslope clay soil from near Cambridge, Address correspondence to M.H.G. Amin, Department of Geography, University of Cambridge, Downing Place,
Cambridge, CB2 3EN, UK.
879
880
Magnetic ResonanceImaging0 Volume 14, Numbers 7/8, 1996
(2D) imaging of water movement in unsaturated soil materials.6’7 The results showed that MRI can be used to observe water transport in a wide range of model and natural soils, especially for soils with an iron content of <3% and a water content of >20%. The images obtained revealed different dynamic behaviour in medium sand, coarse sand and packed clay soil columns (54 mm diam. X 60 mm height) during the wetting, redistribution, and drainage processes. Wetting front instability was observed in the coarse sand during the wetting and water redistribution periods.6 Temporal geometrical changes of an unstable wetting front (“fingering”) during water redistribution in a packed coarse sand column (50 X 51 mm) were monitored by sequential 2D imaging. Twelve 2D images obtained at time (t) after a step-feed of 20 mL water (t = 0) are shown in Fig. 1. The experimental details are given in the figure caption. Fingering of the wetting front has features similar to viscous fingering developed in a linear Hele-Shaw cell.g*9 A critical wavelength expression was derived for wetting front fingering, which is the same as that for viscous fingerindos” This suggests that the geometry of fingering should also possess similar fractal features I2 to those of viscous fingering for which case the fractal properties are well documented (e.g., as reviewed by Feder 13). As wetting-front fingering develops with time it should develop a multifractal character. To test this, perimeters, areas, and the longest finger length at different times were measured in 128 x 128, 256 x 256, and 5 12 X 5 12 image matrix sizes. Fractal dimensions (D) l2 of perimeters vary with the passage of time (D = 1.02, 1.15, 1.14, 1.38, 1.28, 1.27, 1.24,
Fig. 1. Vertical (2D, 128 x 128) images of water redistribution in a packed coarse quartz sand column (54 X 5 1 mm) after a step-feed of 20 mL water at t = 0. A spin-echo spinwarp pulse sequence was used with repetition time of 0.2 s and echo time 2.97 ms. The slice thickness was 12.5 mm, read gradient 10.4 kHz/cm and field of view 91 mm. A single signal average was obtained and the acquisition time for each image was 45 s.
0
1
2 In L (mm)
3
4
0
1
2
3
4
In L (mm)
Fig. 2. Plots of the longest length of the finger (L) vs: (a) time (t) and b) total pixel number (N) of the area of fingering.
1.34, 1.42, 1.31, 1.30, and 1.81 for each time step in Fig. 1 ), which agrees well with the trend and range in the values of D obtained by Chang et all2 the last D in the sequence is very large due to the significant split in the finger. Relationships between the longest length of the finger (L) and time (t) , and between L and total pixel number (N) of the wetted area can be expressed as - LD’ ; and N - LD. Figure 2 shows plots of these two relations, and gives D 1 of 1.38 and D of 1.18, which are smaller than the corresponding D values ( 1.6- 1.7) for viscous fingers obtained from a radial Hele-Shaw cell. Thus, the fractal features of wettingfront fingering and relations among those fractal dimensions need further investigation. The conventional method of investigating wetting-front fingers uses a large size Hele-Shaw cell, which gives 2D infiltration by restricting the third dimension effectively to the narrow gap between the cell walls. It has already been shown that MRI provides nondestructive visualisation of 2D viscous fingering.14 In the present work, the wetting-front finger that developed in a 3D cylindrical space was observed by 2D imaging. Investigations of such 3D phenomena using 3D imaging are unfortunately difficult due to the short time of the infiltration process relative to image acquisition time and poor S/ N. Consequently, future studies in this area should be based on fast data acquisition MRI protocols. In addition to these experiments, one-dimensional ( 1D) and 2D imaging techniques were employed to detect unsaturated flow patterns, to track the movement of wetting fronts, to obtain water content profiles, and to estimate sorptivities and steady-state infiltration rates in an undisturbed core (46 millimeters X 80 mm) of the sandy-loam soil during infiltration7 The results showed preferential flow phenomena, as well as air entrapment in the soil. It is important to contrast the strengths and weaknesses of 1D and 2D MRI methods. Clearly, the 2D method has the advantage of enhanced spatial coverage, but
t
Studies of soil-water transport by MRI
limits the time resolution per image slice. In this research, the temporal resolution was enhanced to 2 s by reducing the spatial resolution to 1D. Clearly, reduced image dimensionality could lead to ambiguities if applied to heterogenous samples. Nevertheless, 2D images using a short repetition time and prefocused rf pulses in a SE sequence are feasible. In this work, the alternating-pulsed-field-gradient MR technique has been used to obtain displacement and velocity spectra of steady-state saturated flow through a column packed with glass beads.” The displacement spectra obtained by PFG MR correspond to travel-distance probability-density functions (pdf) for initial conditions of a concentration impulse in a column with zero concentration. These spectra show strong dispersion-time dependence and differ from Gaussian-shaped pdfs for short dispersion times. These data provide estimates of the dispersion-time dependence of transverse and longitudinal dispersion coefficients. The longitudinal dispersion coefficient reaches its long-term behaviour slower than the transverse one. Long-term values obtained from MR data fall between the values calculated using the existing empirical relationships of Gunn16 and of Wakao and Funazkri.17 A model based on three components of apparent velocity and dispersion coefficient has been developed for the description of dispersion-time dependent displacement spectra for water flow through the bead column.‘5 The short-distance component corresponds to convectiondispersion-diffusion within necks between particles, whereas the long-distance component represents the macroscopic convection-dispersion process. This research shows that PFG MR flow spectroscopy is a simple but potentially useful method for study of flow and hydrodynamic dispersion in porous media, especially for investigations of time-dependent phenomena. A similar technique was also used to obtain maps of macropore flow velocities and dispersion coefficients in cross-sections of steady-state saturated flow through a packed bead column and a clay soil column. The following conclusions can be drawn from this work. First, MRI is a potentially useful tool for a wide range of studies of soil-water transport, especially with respect to preferential flow in soils. Secondly, soils with an iron content of ~3% and a water content of >20% can be studied using MRI. The results reported here suggest that MRI can achieve measurements and allow inferences that would be difficult to obtain by any other technique. Because the spatial and temporal resolutions of MRI are restricted by the dynamic features of the transport process under investigation and by the MR properties of the soil water, it is clear that it is now important to investigate roles for faster MRI
l
M.H.G. AMIN ETAL.
881
protocols. Such methods will facilitate higher spatial and temporal resolution. It is obvious that advances in MRI of other porous media, especially of rocks, provide a further source of important methodology that should now be applied to soil. Acknowledgments-This
work was made possible by a munificent
benefaction to LDH andTAC fromDr. HerchelSmith,andpartially supported by NERC Research Grant GR3/9369 and a PGRA Grant to MHGA. Thanksarealsodueto Drs.M. Cislerova andT. Vogel
for helpfuldiscussions andprovidingthe sandy-loam soil sample, andto Dr. N. Herrod,Mr. J.A. Derbyshire, andMr. D. Xing for providing software for MR data processing and analysis.
REFERENCES 1. Gummerson,R.J.; Hall, C.; Hoff, W.D.; Hawkes,R.; Holland, G.N.; Moore, W.S. Unsaturatedwater flow within porousmaterialsobservedby NMR imaging.Nature 281:56-57; 1979. 2. Bottomley, P.A.; Rogers,H.H.; Foster, T.H. NMR imaging showswater distribution and transport in plant root systemsin situ. Proc. Natl. Acad. Sci. USA. 83:8789; 1986. 3. Liu, I.-W.Y.; Wong, S.T.S.; Waldron, L.J. The application of nuclear magneticresonanceimaging to study preferential water flow through root channels.In: S.H. Anderson;J.W. Hopmans(Eds.) Tomographyof SoilWater Root Processes.Madison, WI: ASA, Inc. and SSSA, Inc.; 1994:pp. 135-148. 4. Amin, M.H.G.; Hall, L.D.; Chorley, R.J.; Carpenter, T.A.; Richards,KS.; Bathe, B.W. Magnetic resonance imaging of soil-water phenomena.Magn. Reson.Imaging 12:319-321; 1994. 5. Roberts,T.P.; Carpenter,T.A.; Hall, L.D. Designand applicationof prefocusedpulsesby simulatedannealing. J. Magn. Reson.89:595-604; 1990. 6. Amin, M.H.G.; Hall, L.D.; Chorley, R.J.; Carpenter, T.A.; Richards,K.S.; Bathe, B.W. Visualisationof static and dynamic water phenomenain soil using magnetic resonanceimaging. In: V.P. Singh; B. Kumar (Eds). Subsurface-WaterHydrology. Dordrecht: Kluwer Academic Publishers,1996:pp. 3- 16. 7. Amin, M.H.G.; Chorley, R.J.; Richards, K.S.; Hall, L.D.; Carpenter,T.A.; Cislerova, M.; Vogel, T. Study of infiltration into a heterogeneoussoil using nuclear magneticresonanceimaging.Hydro. Proc. (in press). 8. Saffman,P.G.; Taylor, G. The penetrationof a fluid into a porousmediumof Hele-Shawcell containing a more viscousliquid. Proc. R. Sot. Lond. [A] 245:312-331; 1958. 9. Hill, D.; Parlange,J.Y. Wetting front instability in layeredsoils.Soil Sci. Sot. Am. Proc. 36:697-702; 1972. 10. Philip, J.R. Stability analysisof infiltration. Soil Sci. Sot. Am. Proc. 39:1042-1049; 1975. 11. Chuoke,R.L.; van Meurs, P.; vander Poel,C. The instability of slow, immiscible, viscous liquid-liquid dis-
882
Magnetic ResonanceImaging l Volume 14, Numbers 7/8, 1996
placements in permeable media. Trans. Am. Inst. Min. Metall. Pet. Eng. 216:188-194; 1959. 12. Chang, W.-L.; Biggar, J.W.; Nielsen, D.R. Fractal description of wetting front instability in layered soils. Water Resources Res. 30: 125- 132; 1994. 13. Feder, J. Fractals. New York: Plenum Press; 1988. 14. Davies, ES.; Roberts, T.P.L.; Carpenter, T.A.; Hall, L.D.; Hall, C. Visualization of viscous fingering by nuclear magnetic resonance imaging. J. Magn. Reson. 96:210-214; 1992.
15. Amin, M.H.G.; Gibbs, S.J.; Hall, L.D.; Chorley, R.J.; Richards, K.S.; Carpenter, T.A. Study of flow and hydrodynamic dispersion in a porous medium using PFG MR. Proc. R. Sot. Lond. [A] (in press). 16. Gunn, D.J. Axial and radial dispersion in fixed beds. Chem. Eng. Sci. 42:363-373; 1987. 17. Wakao, N.; Funazkri, T. Effect of fluid dispersion coefficients on particle-to-fluid mass transfer coefficients in packed beds. Chem. Eng. Sci. 33:13751384; 1978.