Diffusion and flow dynamics in porous media by nmr spin echostepisnik/obj_clanki/Bozeman2.pdf ·...

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