1
Diffusion and flow dynamics in porous media by nmr spin echo
JanezJanez StepiStepiššniknik,,University of Ljubljana, FMF, Physics Department and J Stefan IUniversity of Ljubljana, FMF, Physics Department and J Stefan Institute, Ljubljana, nstitute, Ljubljana,
SloveniaSlovenia
Faculty of Mathematics and Physics
2
Molecular motion in heterogeneous structures by NMR
Faculty of Mathematics and Physics
MEDIA:•materials for distillation, filtration,catalysis….
•chromatographic sieves
•colloids and bio-colloids (viruses, bacteria)
•soil, concrete, rocks or clays
•biologic tissues etcpore size:
10-2 – 10-10 mMETHOD:
Analysis and interpretation of diffusion and flow measurement by the gradient spin echo:
•averaged propagator approach
•Gaussian phase approximation
(Torrey method)
(Hahn-Carr-Purcell method)
3
Spin in magnetic field- magnetic resonance
B
Faculty of Mathematics and Physics
( ) µrBµ×=
∂∂ tt
,γ
µ=magnetic moment B=magnetic field
( ) BtieE LL γωωτ =≈
µ
4
Spin in non-uniform magnetic field
G
Faculty of Mathematics and Physics
applied or internal magnetic fields
( )rBBB inho +=
( )( )[ ]∫
≈
τω
τ 0, dttti
eEr
( ) rGrB .=inh
magnetic field gradient
5
NMR imaging
Faculty of Mathematics and Physics
Amplitude T2 Flow-area
Magnetic resonance imaging of tomato plant stem
leaves
roots
3 m
m
stem
6
Spin echoπ
rf+G(t)
τ
π/2
q(t)
Encoding of spin motion by spin echo
Faculty of Mathematics and Physics
Hahn, E. L., Spin-echoes, Phys. Rev., 80, 580-94 (1950)
G
Pha
se g
ratin
g
( )
( ) ∫
∑
=
∫≈
⎟⎠⎞⎜
⎝⎛⎟
⎠⎞⎜
⎝⎛
t
eff
i
i
dttt
dtttieE
0
')'(
.0
Gq
vq
γ
τ
τ
Applied MFG
7
( )∫
≈⎟⎠⎞⎜
⎝⎛⎟
⎠⎞⎜
⎝⎛
τ
0i dtt.tiγ
τvq
eE
Molecular motion and spin echo: approaches
G
q(t)
π
time
G(t)
τ
Spin echoπ/2
Faculty of Mathematics and Physics
( ) ( )( )mrDmrBm∇∇+×=
∂∂ tt
,γ
m=magnetization density D=self-diffusion tensor
H.C. Torrey. Bloch equations with diffusion terms, Phys. Rev., 104, 563-565, 1956
∑∫−
≈i
i dttte
τ
0)(..)( qDq free diffusion
( ) ∫=t
eff dttt0
')'(Gq γ
∑i
E.L.Hahn, Spin-echoes, Phys. Rev. 80, 580-94 (1950),
H. Y. Carr and E. M. Purcell, Effects of diffusion on free precession in nuclear magnetic resonance, Phys. Rev. 94 , 630-38 (1954)
8
∑∫−
=i
diqZeDeD
R
Rτ
τπ4
2
41
Spin echo as Fourier transform of probability function
Faculty of Mathematics and Physics
( ) ∑∑ −=∫−
=i
i
i
i Dqedttt
eqE ττ
τ
2)(..)(, 0
qDq
free isotropic diffusion
J. Karger and W. Heink. The propagator representation of molecular transport in microporous crystallites , J. Magn. Reson. 51, 1-7 (1983)
Spin echo
q(t)
π
time
G(t)
τ
π/2
Short PGSE
q-space Fourier transform
Probability function
( ) ( )∫= RRqRq diePNE .,, ττ - propagator
Gq δγ=
9
Probability distribution of flow in porous media
Faculty of Mathematics and Physics
Spin echo
q(t)
π
time
G(t)
τ
π/2
Short PGSE
( ) ( )∑∫⎟⎠⎞⎜
⎝⎛ −
=i
ii d
iePE rrrqrrq 0,|,, ττ
q-space Fourier transform of spin echo can provide probability distribution of flow dispersion
K.J. Packer, J.J. Tessier, The characterization of fluid transport in a porous solids by pulsed gradient stimulated echo NMR, Mol. Physics 87, 267-72 (1996)
10
ox
Restricted diffusion and propagator method
Faculty of Mathematics and Physics
Propagator of restricted diffusion
( ) ( ) ( ) ττ DeuuP
PDtP
2'0,|,'
2
krrrr *k
kk
−=
∇=∂∂
∑
( ) ( ) ( )∫∑∫ =+= RRqRRRqrRrq diePNdiePEi
ii τττ ,0,|,,
( ) ( )∑ +=i
iiPN
P 0,|,1, rRrr ττ
11
ro( ) ( )
22
0longτ, ∫=≈∞
→Vol
dieSE rq.rqq
like Fraunhofer diffraction in optics
P. T. Callaghan, A. Coy, D. MacGowan, K. J. Packer and F. O. Zelaya, Diffraction-like effects in NMR diffusion studies of fluids in porous solids, Nature , 351, 467-9 (1991)
In the long time limit:
( ) ( ) ( ) ττ DeuuP2
'0,|,' krrrr *k
kk
−= ∑
( ) ( ) ( ) Dτ2-, * kqqqk
kk eSSE ∑≈τ
Spin echo of restricted diffusion
Spin echo diffraction-like effect in porous mediaand propagator method
Faculty of Mathematics and Physics
naq π2=
12
B. MANZ, J.D. SEYMOUR, AND P. T. CALLAGHAN, PGSE NMR Measurements of Convection in a Capillary, JOURNAL OF MAGNETIC RESONANCE 125, 153–158 (1997)
Flow and spin echo diffraction-like effect
Faculty of Mathematics and Physics
13
ox
Spin echo of restricted diffusion with propagator method
Faculty of Mathematics and Physics
Propagator of restricted diffusion
( ) ( ) ( ) ττ DeuuP
PDtP
2'0,|,'
2
krrrr *k
kk
−=
∇=∂∂
∑
( ) ( )∫= RRqRq diePNE ττ ,,
( ) ( )∑ +=i
iiPN
P 0,|,1, rRrr ττ
ensemble average averaged propagator
14
( ) ( ) ( )∑ ∫−=
i
ii dzziqePE rrrq .|,, ττ
Spin echo by the cumulant expansion
Faculty of Mathematics and Physics
( )( )
....43126
3
31222
14
144
33311233
222122
11
=−−+−=
∆=−=+−=
∆=−=−=
=−==
iiiiiiii
iiiiiiii
iiiiii
iiii
MMMMMMMK
ZZZMMMMK
ZZZMMK
ZzzMK
( )( ) ( )
∑∆−
=i
iZqZiqeqE
....21.
,22
i τττ
Gaussian phase approximation (GPA)
zq.=q.r
Cumulant expansion( ) ( )
∑∑
==i
nni
n
Kniq
e 1 !τ
( ) ( )( ) rrr dzzPM niini −= ∫ |,ττ
Taylor series( ) ( )∑∑
=
=i n
ni
n
Mniq
0 !τ
15
Loca
l norm
aliz
ed
pha
se s
hift
=d2
=d2
=d2
(x,t)qa
Diffusion between plan-parallel planes
Local n
orm
aliz
ed
atte
nuatio
n
β(x,t)q a2 2
=d2
=d2
=d2
Distribution of phase and attenuation in the pore: short PGSE with the Gaussian phase approximation
Spin echo
q(t)
π
time
G(t)
τ
π/2
Short PGSE
( )( ) ( )
∑∆−
=i
iZqZiqeqE
....21.
,22
i τττ
Faculty of Mathematics and Physics
16
Spin echo diffusive diffraction by the cumulantexpansion in GPA
Faculty of Mathematics and Physics
J. Stepišnik, A new view of the spin echo difusive diffraction in porous structures, Europhysics Letters, 60, 353-9 (2002)
( )( ) ( ) ( ) ( )
∫∑∆−
⇒∆−
= rrr
dZqZiq
eZqZiq
eqEi
i τττττ
,.21,..
21.
,2222
i
δγGq =naq π2=
00.1
0.2
0.3
0.4
12
3
-5
-4
-3
-2
-1
0
-5
-4
-3
-2
-1
0
π2/qa
)/( 0EELog
2/ aDτ
Diffusion between plan-parallel planes Diffraction-like effect results from
phase interference of spins scattering at opposite boundaries.
( ) ( )
( )( )τ
ττ
Zq
Ziqe
Ziqe
.cos
.. −
17
Diffusive diffraction
Faculty of Mathematics and Physics
GPA method
00.1
0.2
0.3
0.4
12
3
-5
-4
-3
-2
-1
0
-5
-4
-3
-2
-1
0
π2/qa
)/( 0EELog
2/ aDτ
( )( ) ( )
∑∆−
=i
iZqZiqeqE
....21.
,22
i τττ
Propagator method
00.1
0.20.3
0.4
12
3
-5
-4
-3
-2
-1
0
-5
-4
-3
-2
-1
0
2/ aDτ
)/( 0EELog
π2/qa( ) ( ) ( ) ( )
∫∫−= Rrrqrrrq diePE o
oo 0,|,, τρτ
18
Diffraction-like effect in spin echo time dependence
Faculty of Mathematics and Physics
Spin echo diffusive diffraction of water between packed polystyrene spheres by MGSE-Z
G.δ=4.5 10 −4Ts/m
0,0001
0,0010
0,0100
0,1000
1,0000
10,0000
0,0 0,1 0,2 0,3 0,4 0,5 0,6
time [s]
atte
nuat
ion
2 µ
15.8 µ
π2=∆zq
Spin e cho diffusive diffraction o f water betw een pa cked polysty rene sphere s by MGSE-Z G.δ =4.5 10−4Ts/ m
0,00010,0010
0,01000,1000
1,000010,0000
0,0 0,10,2 0,3 0,40, 50,6time [s]
attenuation 2 µ15 .8 µ
pa cked polysty rene sphere s by MGSE-Z G.δ =4.5 10−4Ts/ m
0,00010,0010
0,01000,1000
1,000010,0000
0,0 0,10,2 0,3 0,40, 50,6time [s]
attenuation 2 µ15 .8 µ
Spin echo atte nuation ob tained by a new appTime (D /a)τ 2
G radient ( qa/2)π
Lo g(E/E)o
P. T. Callaghan,NMR imaging, NMR diffraction.. MRI, 14, 701 (1996)
19
Averaged propagator by GPA
Faculty of Mathematics and Physics
( )( ) ( )
( )∫∑ =∆−
= RRqRq diePZqZiq
eEi
ii .,.
21.
,22
τττ
τ
0.2
0.4
0.6
0.8
diffusion between parallel planes in long time approximation
GPA method
propagator method (dotted line)
q-space Fourier transform
( )( )
( )( )( )∑
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∆
−−
∆=
i i
i
iZZ
ZP
τ
τ
τπτ
2
2
2 2exp
2
1,R
R( )
( )( )( )∫ ⎥
⎥
⎦
⎤
⎢⎢
⎣
⎡
∆
−−
∆⇒ r
r
rR
rd
Z
Z
Z τ
τ
τπ ,2
,exp
,2
12
2
2
20
GPA propagator vs. averaged propagator
Faculty of Mathematics and Physics
GPA method (line)
q-space Fourier transformation
( ) ( )( ) ( )
∑∫∆−
==i
ii ZqZiqediePE
ττττ
22.21..,, RRqRq
Corrections to the 7th order (dots)
( ) ( ) ..120
.241.
654433 qiZqZqi ii +∆+∆− ττ
Fast convergence of cumulant series
21
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )( )∑∫∫
∑
∑∫
−==
=
==
=
=
i
nii
nn
nn
nn
n
n
dzzPN
dZPM
Kniq
Ne
MniqNdiePNE
rrrRR
RRqRq
|,1,
!
!.,,
1
0
τττ
τ
τττ
Cumulant expansion with ensemble average
Faculty of Mathematics and Physics
Coarse graining in
space
Taylor series
without odd terms
224
312
222
14
144
3311233
22122
11
343126
03
0
ZZMMMMMMMK
ZMMMMK
ZMMK
ZMK
−=−−+−=
==+−=
=−=
===
Cumulant series
22
( ) ( ) ( ) ( ) ( ) ( )
( )∑ ∑∑∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛−−=∆−==
+∆+∆∆+−⇒
ii
jji
ii
ii NNNN
iiNeE
4222
2432
'31''1
41''
21'1''1'
...'''.'.'.,
φφφφ
qφqqφqq
βββββ
τβτβττβττ
Cumulant expansion with respect to average over ensemble of spins
Ensemble average and low-q spin echo
Faculty of Mathematics and Physics
( )( ) ( ) ( )
∑⎥⎦⎤
⎢⎣⎡ −−
⇒i
iiiieE
ττβττ
22 '21.'.
,φqφq
q ( )2'
'
ioii
ioii
rr
rrφ
−=
−=
β
'',
'''2
βββ
∆∆<<∆
φφq
1)(. <tq ξ
GPA condition for the diffusion in closed pore
23
Gaussian phase approximation of spin echo
Faculty of Mathematics and Physics
( )( ) ( )
∑∆−
=i
ii ZqZiqeE
τττ
22.21.
,q
•Gaussian phase approximation is a sufficient condition for the spin echo of diffusion in enclosed pore as long as q|Z(τ)| <1.
24
( )( ) ( )
∑∆−
=i
ii ZqZiqeE
τττ
22.21.
Spin echo of flow dispersion in porous media with GPA
Gradient
Flow
( ) ( )
( )
Dk
ett
ddtPt
k
k
t
kjj
j
21
sin.
'0,|,''.,
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡+=
=−=
−
⎟⎠⎞
⎜⎝⎛
∑
∫∫
τ
φ
τvkbvq
rrrrrrqv
k
( )
( ) ( ) ( ) k*k
kk rvrrr
v
τt
eututP
PDPtP
−
∑ −=
∇=∇+∂∂
'0,|,'
2
Taylor`s dispersion-diffusion equation:
Faculty of Mathematics and Physics
v
jv
( ) ( ) ( )
2
222
,21
cos121,
⎟⎠⎞
⎜⎝⎛
−
−
−⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−++≈ ∑
j
k
k
t
jkjj
t
ecqDqt
v
vkq.vv
φ
ττβ τ
Summation over flow channels∑
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ −
=j
jjj
ieN
vv ,, τβτφ
25
Spin echo flow phase shift in porous media
Gradient
Flow
Electrophoresis
E=960 Vm’1
E=1770 Vm’1
E=4000 Vm’1
Experiment
S.R. Heil, M. Holz, J. Mag. Res. 135 (1998) 17-22, M. Holz, S. R. Heil. I. A. Schwab, Mag. Res, Imaging 19 (2001) 457–463
Faculty of Mathematics and Physics
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛+= +
−⎟⎠⎞⎜
⎝⎛ ..11
1sin., ττ
ττφ et vkbvqv
Theory
τ
v=1v=2v=3
ττφv
vq
⎟⎠⎞⎜
⎝⎛ ,
26
Evolution of flow dispersion propagator
Gradient
Flow
Faculty of Mathematics and Physics
Spin echo
F(t)
π
time
G(t)
τ
π/2
( ) ( ) ( )
( )∫
∑
=
−=
RRqvR
vvq
dieP
ieNE
j
jjj
.,,
,,,
τ
τβτφτ
Time
Co
ord
ina
teEvolution of flow dispersion propagator
through leaky plan-parallel planes
27
2-dimensional presentation of diffusion and relaxation
Faculty of Mathematics and Physics
( )( ) ( )
( )( )
∫
∑
−−⇒
−∆−=
22
2
2
2
22
,,
21
,
dTdDTDqDZiq
eTD
TZqZiq
eEi
iii
τττρ
ττττq
relaxation 1/T2
diffusio
n D
2D-Laplace transformation
Distribution of diffusion and relaxation
28
General theory: stochastic process with the cumulantexpansion method
( ) ( )∫∑ ⇒−
=⎟⎟⎠
⎞⎜⎜⎝
⎛
RRqRrr
q diePiieE
iτ
ττ ,
)0()(q,
Spin echo is Fourier transform of the averaged propagator
( ) ( ) rfiedrrfierPf ==Φ ∫Fourier transform of probability function is its characteristic function
variablestochastic=r
Faculty of Mathematics and Physics
( ) ( )
( )( ) ( )∫
∫
=Φ
=⇒
τ
τ 0
0
''',
''
dttvtfief
dttvtrt
It becomes its characteristic functional when
functionarbitrary)( =tf
Theo
ry o
f sto
chas
tic p
roce
sses
Cumulant expansion
∫∫ ∫∫∫∫ +−−
⇒
⇒
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛τ τ ττ ττ
0 0 00 00
....'''6
'21 dtdtdtct''vt''ft'.vt'ft.vtfidtdtt'.fct'vtv.tfdttv.tfi
e
29
Spin echo with cumulant expansion in the Gaussian approximation
Faculty of Mathematics and Physics
( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑
∫ ∫ ∫∫ ∫∫
∑∫
−−=
==⇒Φ
i
τ
cii
i
dtdtdtt.t'ttt.tidtdtttttdtttie
dtttieEf
...'''.'''''6
''.'.21.
.,
0 0 00 0cii
0i
0i
τ ττ ττ
τ
ττ
qv.vqvqqvvqvq
vq
( ) ( )( ) ( )ttv
ttf
ivq
≡≡
Gaussian phase approximation (GPA)
q(t)
π
time
G(t)
τ
Spin echoπ/2
spin dephasing
velocity of i-th spin
no limits with regards to the gradient pulse or gradient waveform
J. Stepišnik, Validity Limits of Gaussian Approximation in CumulantExpansion for Diffusion Attenuation of Spin Echo, PhysicaB, 270, 110-117 (1999),
30
Spin echo with the cumulant expansion in Gaussian approximation - general method
Faculty of Mathematics and Physics
q(t)
π
time
G(t)
τ
Spin echoπ/2
( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∑∫ ∫∫
∑∫ −
==ii
dtdtttttdtttie
dtttieE
...''.'.21..
0 0cii
0i
0i
τ τττ
τqvvqvqvq
Cumulant expansion of characteristic functional
π π π π π π ππ/2
q(t)
rfG(t)
Modulated gradient spin echo
31
Spectrum of motion
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) dttiettdttietq
d
iii
i
ωωωτω
ωτωωτωπ
τβ
ττ
∫∫
∫
==
=∞
0c
0
0
*
',
,..,1
vvDq
qDq
Time domain
Frequency domain
Relation of gradient spin echo to correlation function and spectrum of molecular motion
( )
( ) ( )
( ) ( ) ( ) ( )∫ ∫
∫
∑
=
=
=−
=
⎟⎠⎞⎜
⎝⎛
⎟⎠⎞⎜
⎝⎛
⎟⎠⎞⎜
⎝⎛
τ
τ
τβ
τφ
ξξτβτφ
τ
0 0c
0
''.'.
.:
.)(
dtdttttt
dtttthen
pathfreeqifieE
t
iii
ii
i
ii
qvvq
vq
Faculty of Mathematics and Physics
Spin echo
ππ/2
( )tq∆
2 4 6ω
1
2
3
4
∆
012
34
q ( )2 ω
2 4 6ω
1
2
3
4
∆
32
( ) ( ) ( ) τωαωωτωπ
τβ ⎟⎠⎞⎜
⎝⎛
∞
≈= ∫ m0
2,1 DdDq
Modulated Gradient Spin Echo (MGSE)
Faculty of Mathematics and Physics
( )T
DeE πω
τωατ 2
mm =
−≈
⎟⎠⎞⎜
⎝⎛
frequencyωm=2π/T0
π π π π π π ππ/2
q(t)
rfg(t)
π/2 π π π π π π π π π π π π π π π π
q(t)
rfg(t)
T
J. Stepišnik, Analysis of NMR self-diffusion measurements by density matrix calculation, Physica B, 104, 350-64, (1981)
( )∑ −=
⎟⎠⎞⎜
⎝⎛
i
iiieEτβτφ
τ )( X
33
Velocity correlation function and self-diffusion
( ) ( ) ( ) ( ) DDdtvtvD xx ≡⇒=∞→∫ 00.
τ
τ
0
τ
Spectrum of velocity correlation
neutron scatteringrangeN
MR
ra
nge
Frequency
D(ω
)
105 109 1012
( ) ( ) ( )∫∞
=0
0. dtevtvD tixx
ωω
Velocity correlation function
neutron range
time
< v(
t) v(
0) >
sc1210 −≈τ
NM
R
rang
e
Faculty of Mathematics and Physics
( ) ( ) ( )tDvtv xx δ20. =
simple liquids – free diffusion
34
Velocity correlation function of restricted motion
( )tdtd fvv
=+α
Langevine equation
E. Oppenheim and P. Mazur, Brownian motion in system of finite size,Physica, 30, 1833--45, 1964.
J. Stepišnik, A. Mohorič and A. Duh, Diffusion and flow in a porous structure by the gradient-spin-echo spectral analysis, Physica B, 307, 158-168 (2001)
Faculty of Mathematics and Physics
Free diffusion
correlation
spectrum
200 400 600 800 1000
0.002
0.004
0.006
0.008
0.01
frequency
velocity
correlation
time
spectraNM
R
rang
e
Restricted diffusion- reflection at boundaries
velocity
correlation
time
200 400 600 800 1000
0.002
0.004
0.006
0.008
frequency
correlation
spectrum
22
2
rest 1D
ωτωτ
ωk
k
kkBD
++= ∑∞⎟
⎠⎞⎜
⎝⎛
35
Velocity correlation spectrum of water in a porous media by MGSE
Faculty of Mathematics and Physics
J. Stepišnik, P.T. Callaghan, The long time-tail of molecular velocity correlation function in a confined fluid: observation by modulated gradient spin echo, Physica B. 292, 296-301 (2000)
( ) τωατ
⎟⎠⎞⎜
⎝⎛−
≈ mDeE T
πω 2m =
Velocity correlation spectrum of water betweenclosely packed, mono-disperse, surfactant coated
polystyrene spheres of 2r=15µm.
D
200 400 600 800 1000
0.002
0.004
0.006
0.008
frequency
correlation
spectrum
NM
R
rang
e
22
22
rest 1D
ωτωτ
τω
k
k
k k
kBD+
+= ∑∞⎟⎠⎞⎜
⎝⎛
diffusion correlation time Dkk 2
1=τ
π π π π π π ππ/2 π
rfg(t)
36
Velocity correlation spectrum of water in emulsion droplets by MGSE
The frequency-dependent diffusion coefficient for free water and water confined within emulsion droplets in a highly
concentrated fresh (squares),(r~0.71µm) and aged (triangles) ,(r~1.70µm) emulsion.
Faculty of Mathematics and Physics
D.Topgaard, C. Malmborg, and O. Soderman, J. Mag. Res.156, 195–201 (2002)
244
21122
1
21
1 ...1
ωκ
ωτωτ
ωτω
DaD
BDBDDrest
+=
=+≈++
+≈
∞
∞∞⎟⎠⎞⎜
⎝⎛
Low-frequencies
κPlanar 0.30
Cylindrical 0.44
Spherical 0.36
High-frequencies
DDrest ≈⎟⎠⎞⎜
⎝⎛ω
∞D
37
Faculty of Mathematics and Physics
Velocity correlation spectrum of flows in a porous media measured by MGSE
P.T. Callaghan, J. Stepišnik, J. of Mag. Res. A 117, 118-122(1995)
Flow of water through an ion-exchange resin 50-100 mesh (100-30µm)
2
3
4
5
0 500 1000 1500Modulation frequency 1/T [Hz]
Def
f .10
-9
-Log
(E/E
0)/[
(1-
/3)]
MSSE-SS 25 ml/hMSSE-SS 50 ml/hMSSE-SS 100 ml/hMSSE-SS 0 ml/h
(beads 100-30 µm)
Gradient
Flow2 mm
π/2 π π π π π π π π π π π π π π π π
q(t)
rfg(t)
T
( )T
DeE
j
j πωτωα
τ 2m
m =−
≈ ∑⎟⎠⎞⎜
⎝⎛
P. T. Callaghan and S. L. Codd, Phys. Fluids, Vol. 13, (2001)
38
Attenuation time dependence of flows in porous media and dispersion spectrum
Faculty of Mathematics and Physics
( ) ( ) ( )∑≠ ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
+++
+−+=
02222
2
11
11
21k jkjk
kkj kvkvcD
ωτωτωτω
Flow of water through an ion-exchange resin 50-100 mesh (100-30µm)
2
3
4
5
0 500 1000 1500Modulation frequency 1/T [Hz]
Def
f .10
-9
-Log
(E/E
0)/[
(1-
/3)]
MSSE-SS 25 ml/hMSSE-SS 50 ml/hMSSE-SS 100 ml/hMSSE-SS 0 ml/h
Taylor`s dispersion-diffusion equation:
( ) ( ) ( )⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛∞
≈−= ∫ kdvkvv m
0m
ωρταωδρτατβ
2
4
6
8
frequency
Dispersion spectrum provides information about distribution of streams
39
( )∑∫⇒i
i drieP rqrr .|,τ
Summary
Faculty of Mathematics and Physics
( )( ) ( ) ( ) ( ) ( ) ( )
∑∫ ∫∫ +−
⇒i
t
dtdtttttdtttieE
...''.'..0 0
cii0
i
ττ
τqvvqvq
Gaussian phase approximation:
•links spin echo to the details of molecular motion
•can be used with any gradient sequence or gradient waveform
• can be used to describe measurement of diffusion and flow in restricted geometry when q ξ<1.
40
MRI in the geomagnetic field (50 µT)
Faculty of Mathematics and Physics
green pepper : different slices
apple grapefruit orange
41
MRI of diffusion and convection in geomagnetic field (50 µT)
Faculty of Mathematics and Physics
G=1.8 10 T/m-3
waterpropanol
ethanol
∆=0
∆= 150 ms ∆= 175 ms
∆= 200 ms ∆= 250 ms
diffusion
J. Stepišnik, M. Kos, G. Planinšič, V.Eržen, J. of Mag. Res. A 107, 167-172 (1994)
A. Mohorič and J. Stepišnik, Phys. Rev. E, 62, 6615-27(2000)
G=0G=1.8 10-3 T/m
∆=200 ms
14 c
m
convection of water in the cylinder
Expected
∆T
42
Spatially-distributed diffusion by spin echo and MRI
Faculty of Mathematics and Physics
δ δπ
∆=τ
π/2
Short PGSE
P.T. Callaghan, J. Stepišnik, Spatially-Distributed Pulsed Gradient Spin Echo NMR using Single-Wire Proximity, Phys. Rev. Letters, 75, 4532 (1995)
( ) ( ) ( ) τrFDrF ..rτ, −≈ ee
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