l d
and
nso
asy
nsor
es. T
e se
ta f
1. Introduction
to the s
e meth
remen
ds. The
namics
of app
hroma
motion alone. In a porous medium, the presence of fluid/matrix
interfaces impedes the Brownian motion so that apparent diffusion
rates depend on the length and time scales used in making the
measurement. But in the presence of flow, the dispersion of mole-
cules speeds up as other mechanisms for separating initially adja-
cent molecules take over and the rate of dispersion rises
significantly above the diffusion ‘‘baseline”. These include mechan-
ical dispersion due to stochastic variations in velocity induced by
long history, dating back to the original suggestion by Hahn [1],
and by Carr and Purcell [2] that, in the presence of magnetic field
gradients, the spin echo would be flow sensitive. In 1972, Hayward
et al. [3] carried out pioneering work in the use of ensemble-aver-
aged Pulsed Gradient Spin Echo NMR to study laminar flow in a
pipe. was carried out by. In 1996 Lebon et al. [4] used PGSE NMR
methods to measure the displacements of molecules in the sto-
chastic flow as fluid was forced through a random porous medium
and in 1997, Seymour and Callaghan [5] carried out an extensive
PGSE NMR investigation of timescale dependence of dispersion.
* Corresponding author. Fax: +64 6 350 5164.
Journal of Magnetic Resonance xxx (2010) xxx–xxx
Contents lists availab
ne
w.e
ARTICLE IN PRESSE-mail address: Paul.Callaghan@vuw.ac.nz (P.T. Callaghan).recovery, catalysis, environmental waste management, groundwa-
ter flows, geothermal venting and processes relevant to animal and
plant physiology. In all these applications one of the most impor-
tant underlying physical phenomena is that of fluid dispersion,
the process whereby molecules that start together in the same
vicinity become separated as a result of translational motions. In
this paper we provide details regarding a new experimental tool
for the measurement of dispersion physics from a novel stand-
point, that of the ‘‘non-local dispersion tensor”, whereby spatio-
temporal correlations in the flow field are revealed.
Of course molecular separation, and hence dispersion, occurs in
thermal equilibrium, and in the absence of fluid flow, by Brownian
ven infinite spatial and temporal resolution, MRI is capable of
revealing this field. However, time and spatial resolution are finite
and indeed traded off in MRI. By contrast, Pulsed Gradient Spin
Echo (PGSE) NMR techniques which obtain an ensemble average
signal from an entire sample, are ideally suited to measuring trans-
port properties in porous media. While molecular positions are not
measured in PGSE NMR, their displacements over well-defined
time scales are determined. The advantage of trading away spatial
localisation in such ensemble averaging is a significant gain in the
available displacement length scales and time scales over which
the motion may be probed.
The application of spin-echo methods to the study of flow has aThe application of NMR methods
media is now well-established. Thes
time measurement, diffusion measu
local inhomogeneous magnetic fiel
methods for understanding fluid dy
important because of the wide range
materials play a role, for example c1090-7807/$ - see front matter  2010 Elsevier Inc. A
doi:10.1016/j.jmr.2010.01.006
Please cite this article in press as: M.W. Hunter e
(2010), doi:10.1016/j.jmr.2010.01.006tudy of fluids in porous
ods include relaxation
t and measurement of
development of new
in porous materials is
lications in which such
tography, filtration, oil
advection along tortuous paths and flow bifurcations, diffusive
(Taylor) dispersion arising from molecular diffusion across stream-
lines and holdup dispersion which arises from the presence of dead
end pores. Like diffusion, dispersion involves stochastic processes
that necessitate the language of statistical physics.
A complete description of fluid behaviour is given by a knowl-
edge of the time-dependent Eulerian flow field, in other words,
the velocity vðr; tÞ, at all points in space and time. In principle, gi-PGSE NMR measurement of the non-loca
in porous media
M.W. Hunter
a
, A.N. Jackson
a,b
, P.T. Callaghan
a,
*
a
MacDiarmid Institute for Advanced Materials and Nanotechnology, School of Chemical
Wellington 6001, New Zealand
b
The British Library, Boston Spa, West Yorkshire, United Kingdom
article info
Article history:
Received 30 November 2009
Available online xxxx
Keywords:
Non-local dispersion tensor
Velocity auto correlation function
abstract
The non-local dispersion te
ment information in a pre-
be used to measure this te
the data analysis procedur
displacement resolution, th
ulated echo attenuation da
Journal of Mag
journal homepage: wwll rights reserved.
t al., PGSE NMR measurementispersion tensor for flow
Physical Sciences, Victoria University of Wellington, P.O. Box 600,
r provides a fundamental description of velocity correlations and displace-
mptotic dispersive system. Here we describe in detail how PGSE NMR may
, outlining the pulse sequences needed for signal superposition, as well as
he sequence is inherently two-dimensional, the first dimension giving the
cond giving correlation information. The technique is verified against sim-
rom a lattice-Boltzmann simulation.
 2010 Elsevier Inc. All rights reserved.
le at ScienceDirect
tic Resonance
lsevier.com/locate/jmrof the non-local dispersion tensor for flow in porous media, J. Magn. Reson.

gnet
ARTICLE IN PRESSMuch prior PGSE NMR work has focused on attempts to mea-
sure ‘‘asymptotic dispersion”, the behaviour which applies when
molecules in the flow field have moved sufficiently to sample a
representative elementary volume (REV) of the locally heteroge-
neous porous medium, i.e., the smallest volume containing all mor-
phological features which exist in the porous medium with their
global statistical weighting [6]. Seymour and Callaghan made com-
parisons, with literature data from other methods, of flow-rate
dependence of non-dimensionalized, asymptotic dispersion coeffi-
cients, measured transverse and longitudinal to the flow direction,
while subsequent studies by other groups [7–9] extended this
work.
In recent years attention has focused on flow field fluctuations
that precede asymptotic dispersion. A fundamental correlation
time defining the temporal structure of the velocity field is, s
v
,
the duration of flow around a characteristic length scale. In general,
for a medium with pore size or pore spacing given by size d, this
correlation time may be written
s
v
¼
d
hvi
ð1Þ
where, for a pore space fraction (porosity) /; hvi¼v
tube
=/;v
tube
being the mean velocity deduced from the volume flow rate assum-
ing that the flow area is the total cross section of the pipe. Another
characteristic time, s
D
, is the time for molecules to diffuse across a
pore. Using a variant of the PGSE method which allowed for inde-
pendent dispersion encoding at two separated time intervals, Khra-
pitchev and Callaghan [10,11] made measurements of the velocity
autocorrelation function, over a range of Peclet numbers, thus pro-
viding an experimental determination of s
v
. In 2007, Hunter and
Callaghan [12] extended this Double PGSE method so as to measure
the non-local dispersion tensor [13], thus obtaining the spatio-tem-
poral structure of dispersive flow while at the same time preserving
information regarding different directional components. The pres-
ent paper presents practical details of the PGSE NMR technique
used for the measurement of the non-local dispersion tensor, a
quantity recognised as being of fundamental importance in the
NMR characterisation of fluid dispersion [5,9,10,14–21].
In what follows we describe the non-local dispersion tensor and
provide a detailed description the NMR method used for its mea-
surement including the signal superposition method, as well as
the data analysis procedures. We demonstrate the method experi-
mentally for dispersive, low Reynolds number ðReÞ, flow in a ran-
dom bead pack of mono-sized spheres. In order to test our NMR
method independently, we have carried out a lattice-Boltzmann
simulation of the flow field through an independently generated
beadpack. This flow field is used to simulate dispersion by allowing
virtual tracer particles to flow and diffuse through the pore space.
These tracers can then be used to estimate the non-local dispersion
tensor in the simulated flow. The same code can also be used to
simulate the NMR experiment, thus providing a means of compar-
ing the tensor that results from the NMR data analysis protocol
with that obtained from the tracer particle analysis.
2. The non-local dispersion tensor
To explain the non-local dispersion tensor, it is helpful to begin
by defining a steady state Eulerian flow field v
E
ðr; tÞ¼v
E
ðrÞ and a
stationary Lagrangian flow ensemble v
L
ðtÞwith mean flow hvi. The
fluctuating (zero mean) parts of the velocities, u
E
ðrÞ and u
L
ðtÞ are
thus defined by
v
E
ðrÞ¼u
E
ðrÞþhv 2Þ
2 M.W. Hunter et al. / Journal of Maand
v
L
ðtÞ¼u
L
ðtÞþhv 3Þ
Please cite this article in press as: M.W. Hunter et al., PGSE NMR measurement
(2010), doi:10.1016/j.jmr.2010.01.006The asymptotic dispersion tensor, D

, is described in terms of the
velocity autocorrelation function (VACF) of the Lagrangian veloci-
ties by [13,22]
D

¼ lim
t!1
sym
Z
t
0
dshu
L
ð0Þu
L
ðsÞi ð4Þ
where h  i represents the Lagrangian ensemble average. Note that
D

may also be defined in Einsteinian terms involving the dyadic of
mean square displacements, r
2
ðtÞ by [22]
D

¼ lim
t!1
1
2
dr
2
dt
ð5Þ
While the VACF is naturally describes in terms of the Lagrangian
ensemble of velocities, it can be easily linked with the Eulerian field
via a propagator Pðrjr
0
; sÞ which describes the conditional probabil-
ity that a fluid element initially at r will migrate to r
0
at a later time
s. Pðrjr
0
; sÞ is governed by the microscale advection–diffusion equa-
tion for the system. For any given starting probability Pðr;0Þ, the
velocity autocorrelation function can then be written
hu
L
ð0Þu
L
ðsÞi ¼
Z
dr
0
Z
dru
E
ðr; 0ÞPðr; 0ÞPðrjr
0
; sÞu
E
ðr
0
; sÞð6Þ
The integral over time in Eq. (4) means that details of the VACF tem-
poral correlations are buried in the asymptotic dispersion tensor.
Similarly, the integral
R
dru
E
ðr;0ÞPðrÞPðrjr
0
; sÞu
E
ðr
0
; sÞ contains spa-
tial correlation information buried in the VACF. This is known as
the non-local dispersion tensor [13] and is a primary quantity of
interest in any detailed description of dispersive flow. We may con-
veniently rewrite this quantity in terms of relative displacements in
time and space we may define the non-local dispersion tensor, the
VACF, and the asymptotic dispersion by
D
NL
ðR; sÞ¼
Z
dru
E
ðr; 0ÞPðrÞPðrjrþ R; sÞu
E
ðrþ R; sÞð7Þ
and
hu
L
ð0Þu
L
ðsÞi ¼
Z
dRD
NL
ðR; s 8Þ
and
D

¼ lim
t!1
sym
Z
t
0
ds
Z
dRD
NL
ðR; sÞð9Þ
We have shown [12] that the tensor D
NL
ðR; sÞ can be directly mea-
sured using PGSE NMR. The key to this measurement is to not only
encode the NMR signal with information concerning the displace-
ment propagator, but to ensure that the experiment is also sensitive
to velocities separated in space and time. The details of our ap-
proach are as follows.
3. NMR implementation
3.1. The pulse sequence
We begin with the pulse sequences shown in Fig. 1. Each is two-
dimensional in encoding gradient, and the signal superposition
resulting from this pair enable extraction of the components of
the non-local dispersion tensor. The first sequence shown in
Fig. 1(a) is termed ‘‘compensated” since mean flow effects are nul-
led in the double-PGSE dimension, while that shown in 1(b) is
uncompensated resulting in a net phase shift due to mean flow.
Note that as shown, with unique phases for the RF pulses, each
of these sequences contain a superposition of compensated and
ic Resonance xxx (2010) xxx–xxxuncompensated phase terms arising from the flow. In order to
ensure that pure compensated and uncompensated phase shifts
result, an appropriate RF phase cycle is required.
of the non-local dispersion tensor for flow in porous media, J. Magn. Reson.