Magnetic Resonance Printed in the USA.
Imaging, Vol. IO, pp. 741-746, All rights reserved.
1992 Copyright
0
0730-725X/92 $5.00 + .OO 1992 Pergamon Press Ltd.
l Session: Contribution
MAGNETIC RESONANCE IMAGING: APPLICATIONS OF NOVEL METHODS IN STUDIES OF POROUS MEDIA P. MANSFIELD, R. BOWTELL, S. BLACKBAND,” AND D.N. GUILFOYLE~ Magnetic Resonance Centre, Department of Physics, University of Nottingham, University Park, Nottingham, NG7 2RD, England NMR imaging is finding broad applications in nonbiological areas including the study of fluid flow and fluid ingress in porous media. The porous media include at the one end mineral rocks and various building materials through various solid plastic materials to foodstuffs at the other end of the spectrum. The fluids within these various media range from crude oil and water mixtures, and water itself, to a range of organic solvents in plastic materials. This paper is concerned with the flow and ingress of water through Bentheimer sandstone and Ninian reservoir specimens, and also in solid nylon blocks. Keywords:
NMR imaging; Porous media; Echo-planar imaging.
INTRODUCTION
cerned has a high proton content, as in water and many organic solvents. This paper will concentrate on modifications of ultra-high-speed imaging techniques for the study of fluid flow in rock samples and also the application of simple projection imaging for the study of translational diffusion and localized relaxation times in homogeneous plastic materials.
The unprecedented success of NMR imaging applications in medicine has led recently to a reevaluation of imaging techniques for applications of NMR imaging in materials science and in studies connected with the oil industry and oil recovery processes. There are also clear applications of NMR imaging in the study of fluid ingress in building materials and other porous media. In all applications referred to above, the materials generally have physical characteristics that make them unsuitable for study using conventional MRI methods. The physical characteristics referred to relate to the solid nature of some of the materials when interest is in the solid matrix itself. In situations in which the interest is the study of fluid ingress within porous media, the nonresonant or nonobserved solid matrix often has a large magnetic susceptibility that at high field induces undesirable static field inhomogeneities. The properties of interest to industry relate to the quantification of fluid flow through porous media, especially rock samples of different types, and the study of translational diffusion process and associated relaxation time phenomena. NMR imaging is uniqueIy suited to the study of these aspects when the fluid con-
The study of fluid flow through porous media is of considerable inierest to the oil industry. A nondestructive technique like NMR imaging is valuable as a means of studying the efficiency of the oil recovery process. Details of the behavior of liquids in solid porous rocks as found in oil reservoirs can be modeled and studied in the laboratory. We have combined the ultra-high-speed echo-planar imaging (EPI) method’ with a flow-encoding sequence to produce flow images in which the signal intensity is a function of localized fluid velocity.2*3 The basic technique is illustrated in Fig. 1, which shows the flow-encoding phase prefixed to a block diagram of the EPI experiment. The flow-encoding phase comprises a sequence of short RF pulses of the form 90,-
*Present address: Magnetic Resonance Royal Infirmary, Hull, HU8 9HE, UK.
tPresent address: BP Research Centre, Chertsey Rd., Sunbury on Thames, Middx., TW16 7LN, UK.
Centre,
FLOW
Hull 741
IN POROUS
MEDIA
742
90X
Magnetic Resonance Imaging 0 Volume 10, Number 5, 1992 180,
g”x,Y
RF
Gv
l-l
1
EPI
I
EXPERIMENT
GS
I Fig. 1. Flow-encoded timing sequence showing RF pulses, flow encoding gradient GV, and spin-phase scrambling gradient Gs applied prior to a standard EPI interrogation sequence.
~-180,-7-90~,~where the subscripts X, y refer to the RF phase of the 90” and 180” pulses. Between the pulses a flow-encoding gradient GV is applied, and its effect is to add a phase shift /3 given by 0 = 2yGyr2 V = /cV
(1)
where V is the fluid velocity, 7 the velocity-encoding period, and y the gyromagnetic ratio. Following the last 90” pulse an additional spin-phase scrambling gradient Gs is applied in order to remove any unwanted signal immediately prior to the EPI experiment. The EPI sequence acts as a spin interrogation phase recording the state of the spin magnetization. The effect of the velocity-encoding sequence on the magnetization components is summarized in Fig. 2.
2
A
M,sinp
,I
1’ //
/
Y
Fig. 2. Positions of magnetizations MS, and MY just before application of the second 90” RF pulse of Fig. 1. Phase change of this puke determines which magnetization component is destroyed by the spin-phase scrambling gradient and which component is re-stored along the z axis for subsequent interrogation by the EPI sequence.
Following the initial 90” pulse applied along the x axis in rotating reference frame, static components MS,,, and dynamic components of the image MV moving with velocity V will be aligned initially along the y axis. However, at the end of the flow-encoding sequence, the moving component will have acquired a phase /3 so that immediately prior to the second 90” pulse the state of the magnetization is as indicated in Fig. 2. That is, there will be two dynamic components of MV, namely MV sin 0 along the x axis and MV cos 6 along they axis. If a pixel in an image has both a static and a dynamic component, the effective magnetization of that particular pixel is Me,,. However, for situations as considered here, parts of the image are either static or moving but never both. In this circumstance the effective pixel magnetization for flowing spins is given by Mv. The second 90” pulse carrier phase is either x or y. If it is applied along the x axis, the y component MV cos /3 is rotated back along the z axis. The remaining x component is then destroyed by Gs, and the EPI experiment then interrogates the z component of magnetization producing an image that contains both static parts proportional to MS,,, and flowing parts proportional to Mv cos 0. If the last 90” pulse is applied along the y axis, the x component MV sin /3 is stored along the z axis and the gradient Gs destroys any static or dynamic spin components along the y axis prior to application of the EPI readout experiment. The interrogation phase then produces a wholly dynamic image proportional to Mv sin /3. By performing two experiments, two images may be obtained and combined to give a velocity map throughout a crosssection of the object. This flow-encoding strategy works well for flow studies in biological systems where susceptibility problems at 0.5 T are minimal. However, in mineral rock specimens the induced susceptibility problems are considerable so that the signal from water and oil contained within the specimen can be severely distorted due to locally induced inhomogeneous magnetic fields. This problem can affect both the flow-encoding phase and also the spin interrogation or EPI phase. In order to overcome this problem, it has been necessary to introduce modifications to both the simple flow-encoding sequence as outlined in Fig. 1 and the standard EPI sequence. These modifications and adaptations comprise the introduction of additional 180” RF pulses in the flow- encoding phase, and the introduction of 180” RF pulses in the EPI experiment to replace the bipolar gradient modulation customarily used in medical imaging experiments. Both modifications are described in greater detail elsewhere.4 Using these modified sequences we have been able
Novel MRI methods in studies of porous media 0 P.
MANSFIELD
ET AL.
143
INGRESS OF WATER INTO SOLID NYLON
0
V (mm
set)
2.25
Fig. 3. Measured velocity distribution for water flowing through a Bentheimer sandstone sample.
to produce cross-sectional flow-encoded images through a Bentheimer sandstone outcrop specimen and a Ninian reservoir sample.4 The in-plane spatial
resolution currently obtainable is 1.O-3.0 mm with a slice thickness of between 10 and 25 mm. The numerical velocity data obtained from a flowencoded image can be used to form a velocity distribution graph. An example of a velocity distribution curve is shown in Fig. 3 for the case of water flowing through a Bentheimer sandstone sample. The distribution is centered around an average velocity through the center of the core of 0.3 mm/set and agrees with measurements of flow rate taken by collecting the exit water from the core. The velocity distribution is in effect a measure of the pore size distribution within the core. In the example given our velocity distribution suggests a very uniform pore size within the specimen.
The ingress of fluids into porous solid systems is important in a wide range of applications covering uptake of water in building materials; the diffusion of water and other fluids in a range of plastic materials and in the oil industry; and studies of oil and water in mineral rocks.5 NMR imaging offers a valuable nondestructive method of studying fluid ingress so that diffusional processes can be modeled. We have used NMR imaging to study water uptake in blocks of nylon 6.6 over a range of temperatures and at selected pressures. A full account of this work is presented elsewhere6 so that only a brief overview is given here. Two NMR imaging techniques were used for this work. Our early data were obtained by a steady-state free precession (SSFP) method modified to give a one-dimensional projection profile of the water concentration in a rectangular block of nylon.7~8 More recent data were obtained using a selective pulse and read gradient (SPRG) procedure. For the SSFP data the spatial resolution was 240 pm and for the SPRG data the resolution was 160 pm. In both cases the slice thickness was around 5-10 mm. The block dimensions were approximately 2.5 x 2.5 x 6.0 cm3. Since the blocks were totally immersed in water, ingress could occur in all six faces. However, with a block length of 6.0 cm, diffusion effects from the end faces into the central imaging slice were negligible. The blocks were removed from the water bath and surface water wiped off before the imaging examination. During the imaging process, which lasted about
* c
Fig. 4. NMR images of water uptake in rectangular nylon 6.6 blocks. (A) Water ingress after a 4-min immersion in water at 100°C. (B) Water ingress after a 50-min immersion in water at 100°C.
Magnetic Resonance Imaging 0 Volume 10, Number 5, 1992
144
1 min, the blocks were kept close to the water bath temperature by blowing warm air through the sample probe. Figure 4 shows NMR images of the nylon blocks following immersion in water at 100°C for 4 (Fig. 4A) and 50 (Fig. 4B) min. The SPRG procedure was modified to produce a set of T,- and T,-weighted water signal projection profiles. For T, measurement the SPRG procedure was preceded by a 180”-T-Read inversion-recovery sequence. For T2 measurement, the SPRG procedure was preceded by a 90”-7-180”-r-Read spin-echo sequence. The time delays, T and 7, were varied to weight the profiles accordingly. For these experiments the imaging procedure lasted about 5 min. Figure 5 shows relaxation time weighted water profiles in rectangular nylon blocks: Fig. SA shows T,-weighted profiles with delay times T = 190, 72, and 43.3 msec, and Fig. 5B shows T,-weighted profiles with spinecho delays r = 3.17, 5.33, and 7.49 msec. If the blocks are homogeneous it transpires that the water ingress can be described by the one-dimensional Fickian diffusion equation’
ac ap(c)ac/ax) ax at=
D(C, T) = Doll + K(T)(C/Co)]exp(
-E/kT)
(3)
where D,, exp (-E/kT) is the diffusion coefficient at C = 0 and K(T) = K is a constant that is essentially independent of T and is found to have a value K = 2. Figure 6 shows the measured diffusion coefficient D versus C/C, for a number of temperatures. Also plotted is the empirical expression, Eq. (3). Equation (3) can be explained in terms of a twophase model comprising “bound” water molecules with a concentration C, having a diffusion coefficient Db, and “free” water molecules with a concentration C’ having a diffusion coefficient Of. In this model the total water concentration C is given by
(2)
where C is the fluid concentration, which is a function of position x and time t, and D(C) is the diffusion coefficient. For a Fickian process the diffusion profiles obtained at different times can be reduced to a master curve using the variable n = x/t “* instead of x. When D(C) is a function of concentration only, Eq. (2) may be integrated by the method of Matano” to yield the diffusion coefficient. The relaxation time data can also be reduced to graphs of Tl or T2 versus
(4
n which when combined with the unweighted graph of C versus r] eventually yields the required graphs of T, or T2 versus C/c,. From the In D(C) versus l/T plots we are able to extract the activation energy E, which is found to be essentially independent of temperature T. We also find that over the temperature range studied, that is, 20-lOO”C, the diffusion coefficient fits the expression
c=
c,+
c,
.
(4)
The effective diffusion coefficient is then given by D=D/[(F-
l)(z)
+ I]
in which
&Lb Df .
(6)
@I
Fig. 5. Relaxation time weighted water concentration profiles obtained from rectangular nylon blocks following immersion in water at 358 K. (A) Tt-weighted profiles for delays T of 190, 72.2, and 43.3 msec. (B) Tz-weighted profiles for echo delays r of 3.17, 5.33, and 7.49 msec. Water ingress from both faces of the block is observed.
Novel MRI methods in studies of porous media 0 P.
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MANSFIELD ET AL.
145
Eq. (9), Do/Df = f , a result consistent with the experimental ratios of the free and bound relaxation rates. Nylon forms crystalline and amorphous regions. In interpreting our results we assume that water can penetrate only the loosely structured amorphous regions. In these regions water can attach itself strongly to adjacent carbonyl groups contained within the amorphous amide (AA) sorption centers, that is, the “bound” water, or much more weakly through coupling between HN and CO groups, that is, the “free” water. There will also be some interstitial unbound water included in the free component. Counting bound and free water molecules, there could be as many as three water molecules per AA sorption center. Of course, there could be none. We therefore think of clusters of water molecules. The average number of water molecules per cluster is given by
0.8
c/c,
Fig. 6. Theoretical curves of D versus C/C,,, Eq. (3), fitted to the experimental data over the range of temperatures indicated.
Combining Eqs. (3) and (5) allows us to express the bound concentration C, as a function of C as
where we assume there is a maximum of one bound water molecule per cluster. Let CbSbe the bound saturated concentration when all the AA sites have one bound water molecule. In this case the average number of water molecules per available AA is given by
PA‘&= -
That is to say, the bound concentration is not fixed but must vary with total concentration in order to fit with the experimental data. The constants A and B are given by
c
.
(12)
cbs
From Eqs. (11) and (12) we form the fraction R of available AA sites occupied by water molecules given by
A=
[l- ($)]/[l-($)]
(8)
R+ bs
and
($1
B=AK(~)/[l-
(9)
where
a=-
Db Do
.
(10)
If we make the reasonable assumption that CY= 1, it follows that A = 1. Using the same two-phase model for the relaxation time data, we are able to show that B = 1 as well as A = 1. This means that for K = 2 in
(cO~cbs)(c/c,) N
’
(13)
Figure 7 shows the curve of R versus PAA for different values of N. Also plotted is the scaled graph e[Cb/CO] versus e[C/Co] for LY= 1, Eq. (7), where the scaling factor E = Co/& = 0.7 and has been evaluated from the measured surface concentration signal and the known physical parameters of nylon. This graph shows that water is initially taken up with one molecule per cluster, N = 1, presumably occupying the bound sites. As the relative concentration increases, the curve breaks away from the N = 1 line, eventually intersecting the N = 2 line at C/C, = 1, showing that two water molecules are taken up on some sites. At this point R is only 0.35 and PM is 0.7, which means that there are many unoccupied sites. The result suggests that it is energetically more favor-
Magnetic Resonance Imaging 0 Volume 10, Number 5, 1992
146
’ 0.6 t .sJ
/
2
a=1
3
% B 0.4 s ‘Z f 0.2 B 4 8 0 L&f 0 0.2 0 Av. No. Hz0
,/c/c,=I 0.4 MOlf2Cu~~S
0.8
0.6 Wr
AA
(PAA
1.0 = c/cb,)
Fig. 7. Graph of R versus PAa, Eq. (13), for different N values. Also plotted is the scaled graph e [ C,/C,] versus E[C/C,] for (Y= 1, Eq. (7), where the scaling factor e = c,/c,, = 0.7.
able for water to build into clusters of two water molecules at atmospheric pressure. Other high-pressure results6 not discussed here suggest that as pressure is raised the cluster size increases to possible N = 3 when C/C, is forced to be greater than 1. CONCLUSIONS We have shown that the novel technique of EPI developed primarily for medical imaging applications can be suitably adapted to the study of flow within porous rocks where large susceptibility differences induce strong local field gradients. The snapshot nature of EPI is such that it is in principle possible to measure transient flow phenomena. In the examples discussed average flow velocities as low as 0.33 mm/set have been measured by this method. The velocity distribution throughout a core can be measured by this technique and is a measure of the pore distribution size. We have also demonstrated that simple one-dimensional projection imaging can produce valuable spatially resolved diffusion and relaxation time data for
water uptake in plastic materials. We have combined’ our T,, T,, and diffusion results obtained over a range of temperatures to produce a consistent twophase model of the behavior of the diffusion coefficient D and the relaxation times as a function of concentration and temperature. This has led to new insights into the detailed process of water sorption at the available amorphous amide sites within the nylon matrix. Although at an early phase in its application to nonbiological systems, NMR imaging shows considerable promise as a nondestructive technique for the study of fluids within porous media. REFERENCES 1. Mansfield, P. Multiplanar image formation using NMR spin echoes. J. Phys. C: Solid State Phys. lO:L55; 1977. 2. Guilfoyle, D.N.; Gibbs, P.; Ordidge, R.J.; Mansfield, P. Real-time flow measurements using echo-planar imaging. Magn. Reson. Med. 18:1-8; 1991. 3. Guilfoyle, D.N.; Mansfield, P. Flow measurement of porous media by echo-planar imaging. Magn. Reson. Imaging. 9:775-777; 1991. 4. Guilfoyle, D.N.; Mansfield, P.; Packer, K.J. Fluid flow measurement in porous media by echo-planar imaging. J. Magn. Reson. 97:342-358; 1992. 5. Blackband, S.J.; Mansfield, P.; Barnes, J.R.; Clague, A.D.H.; Rise, S.H. Discrimination of crude oil and water in sand and in bore cores with NMR imaging. SPE Formation Evaluation. February: 31-34; 1986. 6. Mansfield, P.; Bowtell, R.; Blackband, S. Ingress of water into solid nylon 6.6. J. Magn. Reson. (in press). 7. Blackband, S.J.; Mansfield, P. Diffusion in liquid-solid systems by NMR imaging. J. Phys. C: Solid State Phys. 19:L49; 1986. 8. Mansfield, P.; Bowtell, R.W.; Blackband, S.J.; Cawley, M. Ingress of water into solid nylon: Diffusion studies by NMR imaging. Magn. Reson. Imaging. 9:763-765; 1991. 9. Crank, J. Mathematics of Diffusion. Oxford: Clarendon Press; 1975. 10. Matano, C. Jpn. J. Phys. 8:109; 1932.
Imaging, Vol. IO, pp. 741-746, All rights reserved.
1992 Copyright
0
0730-725X/92 $5.00 + .OO 1992 Pergamon Press Ltd.
l Session: Contribution
MAGNETIC RESONANCE IMAGING: APPLICATIONS OF NOVEL METHODS IN STUDIES OF POROUS MEDIA P. MANSFIELD, R. BOWTELL, S. BLACKBAND,” AND D.N. GUILFOYLE~ Magnetic Resonance Centre, Department of Physics, University of Nottingham, University Park, Nottingham, NG7 2RD, England NMR imaging is finding broad applications in nonbiological areas including the study of fluid flow and fluid ingress in porous media. The porous media include at the one end mineral rocks and various building materials through various solid plastic materials to foodstuffs at the other end of the spectrum. The fluids within these various media range from crude oil and water mixtures, and water itself, to a range of organic solvents in plastic materials. This paper is concerned with the flow and ingress of water through Bentheimer sandstone and Ninian reservoir specimens, and also in solid nylon blocks. Keywords:
NMR imaging; Porous media; Echo-planar imaging.
INTRODUCTION
cerned has a high proton content, as in water and many organic solvents. This paper will concentrate on modifications of ultra-high-speed imaging techniques for the study of fluid flow in rock samples and also the application of simple projection imaging for the study of translational diffusion and localized relaxation times in homogeneous plastic materials.
The unprecedented success of NMR imaging applications in medicine has led recently to a reevaluation of imaging techniques for applications of NMR imaging in materials science and in studies connected with the oil industry and oil recovery processes. There are also clear applications of NMR imaging in the study of fluid ingress in building materials and other porous media. In all applications referred to above, the materials generally have physical characteristics that make them unsuitable for study using conventional MRI methods. The physical characteristics referred to relate to the solid nature of some of the materials when interest is in the solid matrix itself. In situations in which the interest is the study of fluid ingress within porous media, the nonresonant or nonobserved solid matrix often has a large magnetic susceptibility that at high field induces undesirable static field inhomogeneities. The properties of interest to industry relate to the quantification of fluid flow through porous media, especially rock samples of different types, and the study of translational diffusion process and associated relaxation time phenomena. NMR imaging is uniqueIy suited to the study of these aspects when the fluid con-
The study of fluid flow through porous media is of considerable inierest to the oil industry. A nondestructive technique like NMR imaging is valuable as a means of studying the efficiency of the oil recovery process. Details of the behavior of liquids in solid porous rocks as found in oil reservoirs can be modeled and studied in the laboratory. We have combined the ultra-high-speed echo-planar imaging (EPI) method’ with a flow-encoding sequence to produce flow images in which the signal intensity is a function of localized fluid velocity.2*3 The basic technique is illustrated in Fig. 1, which shows the flow-encoding phase prefixed to a block diagram of the EPI experiment. The flow-encoding phase comprises a sequence of short RF pulses of the form 90,-
*Present address: Magnetic Resonance Royal Infirmary, Hull, HU8 9HE, UK.
tPresent address: BP Research Centre, Chertsey Rd., Sunbury on Thames, Middx., TW16 7LN, UK.
Centre,
FLOW
Hull 741
IN POROUS
MEDIA
742
90X
Magnetic Resonance Imaging 0 Volume 10, Number 5, 1992 180,
g”x,Y
RF
Gv
l-l
1
EPI
I
EXPERIMENT
GS
I Fig. 1. Flow-encoded timing sequence showing RF pulses, flow encoding gradient GV, and spin-phase scrambling gradient Gs applied prior to a standard EPI interrogation sequence.
~-180,-7-90~,~where the subscripts X, y refer to the RF phase of the 90” and 180” pulses. Between the pulses a flow-encoding gradient GV is applied, and its effect is to add a phase shift /3 given by 0 = 2yGyr2 V = /cV
(1)
where V is the fluid velocity, 7 the velocity-encoding period, and y the gyromagnetic ratio. Following the last 90” pulse an additional spin-phase scrambling gradient Gs is applied in order to remove any unwanted signal immediately prior to the EPI experiment. The EPI sequence acts as a spin interrogation phase recording the state of the spin magnetization. The effect of the velocity-encoding sequence on the magnetization components is summarized in Fig. 2.
2
A
M,sinp
,I
1’ //
/
Y
Fig. 2. Positions of magnetizations MS, and MY just before application of the second 90” RF pulse of Fig. 1. Phase change of this puke determines which magnetization component is destroyed by the spin-phase scrambling gradient and which component is re-stored along the z axis for subsequent interrogation by the EPI sequence.
Following the initial 90” pulse applied along the x axis in rotating reference frame, static components MS,,, and dynamic components of the image MV moving with velocity V will be aligned initially along the y axis. However, at the end of the flow-encoding sequence, the moving component will have acquired a phase /3 so that immediately prior to the second 90” pulse the state of the magnetization is as indicated in Fig. 2. That is, there will be two dynamic components of MV, namely MV sin 0 along the x axis and MV cos 6 along they axis. If a pixel in an image has both a static and a dynamic component, the effective magnetization of that particular pixel is Me,,. However, for situations as considered here, parts of the image are either static or moving but never both. In this circumstance the effective pixel magnetization for flowing spins is given by Mv. The second 90” pulse carrier phase is either x or y. If it is applied along the x axis, the y component MV cos /3 is rotated back along the z axis. The remaining x component is then destroyed by Gs, and the EPI experiment then interrogates the z component of magnetization producing an image that contains both static parts proportional to MS,,, and flowing parts proportional to Mv cos 0. If the last 90” pulse is applied along the y axis, the x component MV sin /3 is stored along the z axis and the gradient Gs destroys any static or dynamic spin components along the y axis prior to application of the EPI readout experiment. The interrogation phase then produces a wholly dynamic image proportional to Mv sin /3. By performing two experiments, two images may be obtained and combined to give a velocity map throughout a crosssection of the object. This flow-encoding strategy works well for flow studies in biological systems where susceptibility problems at 0.5 T are minimal. However, in mineral rock specimens the induced susceptibility problems are considerable so that the signal from water and oil contained within the specimen can be severely distorted due to locally induced inhomogeneous magnetic fields. This problem can affect both the flow-encoding phase and also the spin interrogation or EPI phase. In order to overcome this problem, it has been necessary to introduce modifications to both the simple flow-encoding sequence as outlined in Fig. 1 and the standard EPI sequence. These modifications and adaptations comprise the introduction of additional 180” RF pulses in the flow- encoding phase, and the introduction of 180” RF pulses in the EPI experiment to replace the bipolar gradient modulation customarily used in medical imaging experiments. Both modifications are described in greater detail elsewhere.4 Using these modified sequences we have been able
Novel MRI methods in studies of porous media 0 P.
MANSFIELD
ET AL.
143
INGRESS OF WATER INTO SOLID NYLON
0
V (mm
set)
2.25
Fig. 3. Measured velocity distribution for water flowing through a Bentheimer sandstone sample.
to produce cross-sectional flow-encoded images through a Bentheimer sandstone outcrop specimen and a Ninian reservoir sample.4 The in-plane spatial
resolution currently obtainable is 1.O-3.0 mm with a slice thickness of between 10 and 25 mm. The numerical velocity data obtained from a flowencoded image can be used to form a velocity distribution graph. An example of a velocity distribution curve is shown in Fig. 3 for the case of water flowing through a Bentheimer sandstone sample. The distribution is centered around an average velocity through the center of the core of 0.3 mm/set and agrees with measurements of flow rate taken by collecting the exit water from the core. The velocity distribution is in effect a measure of the pore size distribution within the core. In the example given our velocity distribution suggests a very uniform pore size within the specimen.
The ingress of fluids into porous solid systems is important in a wide range of applications covering uptake of water in building materials; the diffusion of water and other fluids in a range of plastic materials and in the oil industry; and studies of oil and water in mineral rocks.5 NMR imaging offers a valuable nondestructive method of studying fluid ingress so that diffusional processes can be modeled. We have used NMR imaging to study water uptake in blocks of nylon 6.6 over a range of temperatures and at selected pressures. A full account of this work is presented elsewhere6 so that only a brief overview is given here. Two NMR imaging techniques were used for this work. Our early data were obtained by a steady-state free precession (SSFP) method modified to give a one-dimensional projection profile of the water concentration in a rectangular block of nylon.7~8 More recent data were obtained using a selective pulse and read gradient (SPRG) procedure. For the SSFP data the spatial resolution was 240 pm and for the SPRG data the resolution was 160 pm. In both cases the slice thickness was around 5-10 mm. The block dimensions were approximately 2.5 x 2.5 x 6.0 cm3. Since the blocks were totally immersed in water, ingress could occur in all six faces. However, with a block length of 6.0 cm, diffusion effects from the end faces into the central imaging slice were negligible. The blocks were removed from the water bath and surface water wiped off before the imaging examination. During the imaging process, which lasted about
* c
Fig. 4. NMR images of water uptake in rectangular nylon 6.6 blocks. (A) Water ingress after a 4-min immersion in water at 100°C. (B) Water ingress after a 50-min immersion in water at 100°C.
Magnetic Resonance Imaging 0 Volume 10, Number 5, 1992
144
1 min, the blocks were kept close to the water bath temperature by blowing warm air through the sample probe. Figure 4 shows NMR images of the nylon blocks following immersion in water at 100°C for 4 (Fig. 4A) and 50 (Fig. 4B) min. The SPRG procedure was modified to produce a set of T,- and T,-weighted water signal projection profiles. For T, measurement the SPRG procedure was preceded by a 180”-T-Read inversion-recovery sequence. For T2 measurement, the SPRG procedure was preceded by a 90”-7-180”-r-Read spin-echo sequence. The time delays, T and 7, were varied to weight the profiles accordingly. For these experiments the imaging procedure lasted about 5 min. Figure 5 shows relaxation time weighted water profiles in rectangular nylon blocks: Fig. SA shows T,-weighted profiles with delay times T = 190, 72, and 43.3 msec, and Fig. 5B shows T,-weighted profiles with spinecho delays r = 3.17, 5.33, and 7.49 msec. If the blocks are homogeneous it transpires that the water ingress can be described by the one-dimensional Fickian diffusion equation’
ac ap(c)ac/ax) ax at=
D(C, T) = Doll + K(T)(C/Co)]exp(
-E/kT)
(3)
where D,, exp (-E/kT) is the diffusion coefficient at C = 0 and K(T) = K is a constant that is essentially independent of T and is found to have a value K = 2. Figure 6 shows the measured diffusion coefficient D versus C/C, for a number of temperatures. Also plotted is the empirical expression, Eq. (3). Equation (3) can be explained in terms of a twophase model comprising “bound” water molecules with a concentration C, having a diffusion coefficient Db, and “free” water molecules with a concentration C’ having a diffusion coefficient Of. In this model the total water concentration C is given by
(2)
where C is the fluid concentration, which is a function of position x and time t, and D(C) is the diffusion coefficient. For a Fickian process the diffusion profiles obtained at different times can be reduced to a master curve using the variable n = x/t “* instead of x. When D(C) is a function of concentration only, Eq. (2) may be integrated by the method of Matano” to yield the diffusion coefficient. The relaxation time data can also be reduced to graphs of Tl or T2 versus
(4
n which when combined with the unweighted graph of C versus r] eventually yields the required graphs of T, or T2 versus C/c,. From the In D(C) versus l/T plots we are able to extract the activation energy E, which is found to be essentially independent of temperature T. We also find that over the temperature range studied, that is, 20-lOO”C, the diffusion coefficient fits the expression
c=
c,+
c,
.
(4)
The effective diffusion coefficient is then given by D=D/[(F-
l)(z)
+ I]
in which
&Lb Df .
(6)
@I
Fig. 5. Relaxation time weighted water concentration profiles obtained from rectangular nylon blocks following immersion in water at 358 K. (A) Tt-weighted profiles for delays T of 190, 72.2, and 43.3 msec. (B) Tz-weighted profiles for echo delays r of 3.17, 5.33, and 7.49 msec. Water ingress from both faces of the block is observed.
Novel MRI methods in studies of porous media 0 P.
0.2
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MANSFIELD ET AL.
145
Eq. (9), Do/Df = f , a result consistent with the experimental ratios of the free and bound relaxation rates. Nylon forms crystalline and amorphous regions. In interpreting our results we assume that water can penetrate only the loosely structured amorphous regions. In these regions water can attach itself strongly to adjacent carbonyl groups contained within the amorphous amide (AA) sorption centers, that is, the “bound” water, or much more weakly through coupling between HN and CO groups, that is, the “free” water. There will also be some interstitial unbound water included in the free component. Counting bound and free water molecules, there could be as many as three water molecules per AA sorption center. Of course, there could be none. We therefore think of clusters of water molecules. The average number of water molecules per cluster is given by
0.8
c/c,
Fig. 6. Theoretical curves of D versus C/C,,, Eq. (3), fitted to the experimental data over the range of temperatures indicated.
Combining Eqs. (3) and (5) allows us to express the bound concentration C, as a function of C as
where we assume there is a maximum of one bound water molecule per cluster. Let CbSbe the bound saturated concentration when all the AA sites have one bound water molecule. In this case the average number of water molecules per available AA is given by
PA‘&= -
That is to say, the bound concentration is not fixed but must vary with total concentration in order to fit with the experimental data. The constants A and B are given by
c
.
(12)
cbs
From Eqs. (11) and (12) we form the fraction R of available AA sites occupied by water molecules given by
A=
[l- ($)]/[l-($)]
(8)
R+ bs
and
($1
B=AK(~)/[l-
(9)
where
a=-
Db Do
.
(10)
If we make the reasonable assumption that CY= 1, it follows that A = 1. Using the same two-phase model for the relaxation time data, we are able to show that B = 1 as well as A = 1. This means that for K = 2 in
(cO~cbs)(c/c,) N
’
(13)
Figure 7 shows the curve of R versus PAA for different values of N. Also plotted is the scaled graph e[Cb/CO] versus e[C/Co] for LY= 1, Eq. (7), where the scaling factor E = Co/& = 0.7 and has been evaluated from the measured surface concentration signal and the known physical parameters of nylon. This graph shows that water is initially taken up with one molecule per cluster, N = 1, presumably occupying the bound sites. As the relative concentration increases, the curve breaks away from the N = 1 line, eventually intersecting the N = 2 line at C/C, = 1, showing that two water molecules are taken up on some sites. At this point R is only 0.35 and PM is 0.7, which means that there are many unoccupied sites. The result suggests that it is energetically more favor-
Magnetic Resonance Imaging 0 Volume 10, Number 5, 1992
146
’ 0.6 t .sJ
/
2
a=1
3
% B 0.4 s ‘Z f 0.2 B 4 8 0 L&f 0 0.2 0 Av. No. Hz0
,/c/c,=I 0.4 MOlf2Cu~~S
0.8
0.6 Wr
AA
(PAA
1.0 = c/cb,)
Fig. 7. Graph of R versus PAa, Eq. (13), for different N values. Also plotted is the scaled graph e [ C,/C,] versus E[C/C,] for (Y= 1, Eq. (7), where the scaling factor e = c,/c,, = 0.7.
able for water to build into clusters of two water molecules at atmospheric pressure. Other high-pressure results6 not discussed here suggest that as pressure is raised the cluster size increases to possible N = 3 when C/C, is forced to be greater than 1. CONCLUSIONS We have shown that the novel technique of EPI developed primarily for medical imaging applications can be suitably adapted to the study of flow within porous rocks where large susceptibility differences induce strong local field gradients. The snapshot nature of EPI is such that it is in principle possible to measure transient flow phenomena. In the examples discussed average flow velocities as low as 0.33 mm/set have been measured by this method. The velocity distribution throughout a core can be measured by this technique and is a measure of the pore distribution size. We have also demonstrated that simple one-dimensional projection imaging can produce valuable spatially resolved diffusion and relaxation time data for
water uptake in plastic materials. We have combined’ our T,, T,, and diffusion results obtained over a range of temperatures to produce a consistent twophase model of the behavior of the diffusion coefficient D and the relaxation times as a function of concentration and temperature. This has led to new insights into the detailed process of water sorption at the available amorphous amide sites within the nylon matrix. Although at an early phase in its application to nonbiological systems, NMR imaging shows considerable promise as a nondestructive technique for the study of fluids within porous media. REFERENCES 1. Mansfield, P. Multiplanar image formation using NMR spin echoes. J. Phys. C: Solid State Phys. lO:L55; 1977. 2. Guilfoyle, D.N.; Gibbs, P.; Ordidge, R.J.; Mansfield, P. Real-time flow measurements using echo-planar imaging. Magn. Reson. Med. 18:1-8; 1991. 3. Guilfoyle, D.N.; Mansfield, P. Flow measurement of porous media by echo-planar imaging. Magn. Reson. Imaging. 9:775-777; 1991. 4. Guilfoyle, D.N.; Mansfield, P.; Packer, K.J. Fluid flow measurement in porous media by echo-planar imaging. J. Magn. Reson. 97:342-358; 1992. 5. Blackband, S.J.; Mansfield, P.; Barnes, J.R.; Clague, A.D.H.; Rise, S.H. Discrimination of crude oil and water in sand and in bore cores with NMR imaging. SPE Formation Evaluation. February: 31-34; 1986. 6. Mansfield, P.; Bowtell, R.; Blackband, S. Ingress of water into solid nylon 6.6. J. Magn. Reson. (in press). 7. Blackband, S.J.; Mansfield, P. Diffusion in liquid-solid systems by NMR imaging. J. Phys. C: Solid State Phys. 19:L49; 1986. 8. Mansfield, P.; Bowtell, R.W.; Blackband, S.J.; Cawley, M. Ingress of water into solid nylon: Diffusion studies by NMR imaging. Magn. Reson. Imaging. 9:763-765; 1991. 9. Crank, J. Mathematics of Diffusion. Oxford: Clarendon Press; 1975. 10. Matano, C. Jpn. J. Phys. 8:109; 1932.
