Nuclear Instruments and Methods in Physics Research B 301 (2013) 23–28
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Nuclear Instruments and Methods in Physics Research B
journal homepage: www.elsevier.com/locate/nimb
Neutron tomography of axisymmetric flow fields in porous media
A.J. Gilbert, M.R. Deinert ⇑
Department of Mechanical Engineering, The University of Texas at Austin, United States
a r t i c l e
i n f o
Article history:
Received 5 January 2013
Received in revised form 19 February 2013
Available online 26 February 2013
Keywords:
Preferential flow
Wetting front
Neutron radiography
Image analysis
Fingered flow
Computed tomography
a b s t r a c t
A significant problem in the study of fluid transport in porous media is the ability to visualize the structure of the flow field when moisture contents vary rapidly in space and time. Here we present a method
for determining the radial and vertical saturation profiles within axisymmetric preferential flow fields
using neutron radiography. Flow fields such as these are surprisingly common in nature and determining
the three-dimensional structure of their wetting front region has proven difficult. In this work, the moisture profiles are determined using a simple algorithm for algebraic computed tomography, which gives
the three-dimensional structure of the moisture profile with a temporal resolution that is limited only by
the desired noise level. The algorithm presented can be translated to radiography done using X-rays or
light and is applicable to any rotationally symmetric object.
Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction
A significant obstacle in the study of fluid transport in porous
media is the ability to visualize flow field structure and how it varies with time [1]. This is a particular limitation for research on preferential flow fields. Here, wetting fronts become unstable,
resulting in axisymmetric flow paths that translate rapidly through
unsaturated media [2,3]. These types of flow fields are common in
nature and are responsible for the rapid movement of pollutants
into the subsurface, as well as problems in oil recovery from aquifers [4].
The standard techniques for measuring moisture contents within preferential flow fields have either used light box imaging [5,6]
or X-rays where a thin, fan beam is used to scan the flow field by
varying the location of the chamber or beam [7,8]. The former technique is limited to experiment chambers that are thin enough for
the flow fields to span their width and the latter is slow because
of the need to move the experiment chamber between measurements in order to capture the full profile of the flow field’s moisture content. Computed tomography (CT) provides a cross
sectional view of the object under study. Modern CT scanners
use well collimated, closely spaced, high efficiency detectors for
counting the particles that pass through the object under study.
As a result, standard medical CT scanners offer good resolution,
around 80–100 lm for X-ray CT with 100–500 lm thick slices [9]
and 2–5 mm for c-ray CT [10]. Sensitivity to changes in mass den-
⇑ Corresponding author. Tel.: +1 206 300 4147.
E-mail address: [email protected] (M.R. Deinert).
0168-583X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.nimb.2013.02.011
sity of 3% have also been reported for X-ray CT but a range for c-ray
CT’s mass resolution is not available at this writing [11].
What CT-systems typically lack is good temporal resolution.
Imaging times range from a few minutes to over an hour depending on the scanner design. While X-ray and c-ray CT have been
used for imaging flows in porous media and multiphase flows in
general, the work has been limited to studies requiring low temporal resolution [9,11,12]. X-ray interrogation is also limited because
the fluid of interest and porous medium through which the fluid is
moving can exhibit similar levels of X-ray attenuation, thereby
limiting image contrast. Magnetic resonance imaging (MRI) is another option that has been explored, but it cannot always be used
to establish absolute values for the moisture content within a system, especially at low saturation levels [13,14].
Neutron radiography is an alternate technique for imaging fluids in porous media that combines a high spatial resolution (typically better than 1 mm) with a field of view sufficient to capture
the whole flow field and the ability to image through large volumes
of dry media at video frame rates [15]. The technique works especially well when the fluid under study is hydrogenous, such as
water or oil, since neutrons are heavily attenuated by hydrogen
[1,12,16–18]. In contrast, the neutron attenuating properties of
many porous materials (SiO2 sands in particular) are small relative
to the fluid, thereby increasing the contrast of the fluid under study
above the background porous material.
Because of its sensitivity to small variations in water density,
neutron imaging has been used to study fluid transport in the porous membranes of fuel cells, fluid transport in geological media,
and for measurement of fluid transport coefficients, e.g. [1,18–
22]. Neutron radiography has also found recent application in the
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A.J. Gilbert, M.R. Deinert / Nuclear Instruments and Methods in Physics Research B 301 (2013) 23–28
Water feed line
Image plane
Neutron beam
Uncollided
neutrons
k
Experiment
chamber
z
Underflow drains
i
Image of flow with
intensity I(i,k)
x
(a)
j'th ring
j+1'th ring
rj+1
si
A
rj
B
C
D
i'th ray
(b)
Fig. 1. Experimental schematic of the axis symmetric flow inspection. (a) Schematic of the experimental setup. (b) Illustration of the axis symmetric configuration used for
determining the radial moisture distribution within a preferential flow field. Path length through the j’th ring is equal to length BC. Path length through the j + 1’th ring is
equal to length AB + CD. In general, the path length of the i’th ray through the j’th ring is given by Sij ¼ 2½ðr 2j s2i Þ1=2 ðr 2j1 s2i Þ1=2 .
Here we present a method for determining the radial and vertical moisture profiles within axisymmetric preferential flow fields
using neutron radiography. Unlike conventional neutron tomography, which requires attenuation data taken from multiple angles,
e.g. [21], the present technique is applicable to single view radiographs, provided that the object to be reconstructed is symmetric
about an axis of rotation. The algorithm can be easily translated
to radiography done using X-rays or light and gives temporally
averaged snapshots of the three-dimensional structure of the flow
field.
1
I/Io
R2 = 0.99
0.8
2. Methods
0.6
0
0.2
0.4
0.6
0.8
1
1.2
exp(-Σt d)
Fig. 2. System calibration using a step standard. Calibration curve using exexperimental I/Io data-vs-exp(Rt d), where d is the thickness of a polystyrene
standard whose linear neutron attenuation coefficient is 2.45 {1/cm}. The slope of
this line corresponds to a, the system calibration constant.
study of phase change in capillary systems [23,24], as well as the
composition of building materials [25]. Neutron radiography has
been shown to have potential for applications in treaty verification,
where the contents of containers need to be verified without their
being physically opened [26].
The data obtained from neutron radiography can be used to
determine the cross sectional moisture distribution within a flow
field. This is particularly easy when the flows are axis symmetric.
This is typically the case with preferential flow fields in porous
media, and a simple algorithm for algebraic tomography [27] can
be adapted to determine the cross sectional structure. It was
shown previously that the response of the imaging system used
in this work had the form [28]:
dði; kÞ ¼ ln
ðIði; kÞ=Io ði; kÞ 1Þ
a
þ1
Rt
ð1Þ
where d(i,k) {cm} is the amount of water through which the neutrons moved before being registered at location (i,k) on the image
A.J. Gilbert, M.R. Deinert / Nuclear Instruments and Methods in Physics Research B 301 (2013) 23–28
25
Reactor Pool
Reactor core
Graphite reflector
Neutron
beam
Graphite scatterer
(a)
Flat reflector face
Top view
(b)
High density concrete
Borated polyethylene
Experiment
chamber
Neutron beam
Collimator and
sapphire filter
Imaging system
Sliding lead shutter system
Fig. 3. Sketch of neutron source and imaging hutch. (a) Schematic of the TRIGA Mark II research reactor that was used as a neutron source. The reactor core was surrounded
by a graphite reflector with cylindrical beam ports. The radiography facility made use of a tangential beam port which did not look directly into the core, which reduces the
presence of gamma ray photons in the beam. A graphite plug was used to scatter neutrons down the beam port. A sapphire filter was used in the collimator to remove high
energy neutrons and gammas. (b) Schematic of the beam hutch containing the experiment chamber and imaging system. High density concrete and borated polyethylene
were used as shielding.
plane, Fig. 1a. The fractional change in image intensity is I(i,k)/Io(i,k),
Rt is the total neutron attenuation coefficient of water, and a is a
system calibration constant that was determined using a calibration
standard, Fig. 2.
The images used in this study are digital so they can be viewed
as two-dimensional data arrays, the image brightness at point (i,k),
I(i,k), corresponds to the pixel intensity at that location (here integer values ranging from 1 to 256). For a given vertical position in
the flow field:
dðiÞ ¼
X
hj Sij
ð2Þ
j
where hj fcm3water =cm3soil g is the average moisture fraction in the j’th
ring and Si,j {cm} is the path length of the i’th neutron ray traveling
through the j’th ring, Fig. 1b. Here the flow field is considered to be
made of concentric rings, each of which has a uniform moisture
content, see Fig. 1b. Eq. (2) can be expressed in matrix form as:
di ¼ Sij hj
ð3Þ
Here the matrix elements Sij are given by:
h
i
Sij ¼ 2 ðr 2j s2i Þ1=2 ðr 2j1 s2i Þ1=2
ð4Þ
For a given set of di measurements, the radial moisture profile, hj,
can then be reconstructed by inverting the Si,j matrix in Eq. (3). Because of the symmetry of this system of equations, it is always over
determined and the best solution for the moisture content vector is
found in a least squares sense.
2.1. Neutron source
The neutron source for this work was a TRIGA Mark II research
reactor rated for a maximum steady state operating power of
500 kW. A collimator with a sapphire crystal was used to create
a uniform beam in the tangential beam line used for this work,
Fig. 3a. The sapphire helped to filter out both gammas and higher
energy neutrons. The L/D for this work was 110 ± 1.0 and the cadmium ratio was 21.0. The radiography system was contained inside of a beam hutch made from high-density concrete radiation
shielding and lined with borated polyethylene to cut down on scattered neutrons, Fig. 3b. The unattenuated neutron flux at the image
was determined using gold foil activation and found to be
5 106 {n/cm2/s}. Additional information on the neutron source
and imaging system used for this work can be found in [1,28].
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A.J. Gilbert, M.R. Deinert / Nuclear Instruments and Methods in Physics Research B 301 (2013) 23–28
A 16-gauge aluminum chamber with interior dimensions of
35 cm height by 20 cm width by 1.35 cm depth was loaded with
20–30 industrial quartz (Unimin) with a mean grain size of
0.6 mm and packed with vibration. The media was prepared with
a thorough rinse with distilled water followed by drying. The total
porosity was 0.33, determined by weighing the chamber and using
a bulk density of 2.65 gm cm3 for SiO2. The interior surfaces of the
chamber were sprayed with Dow-Corning Silicone-20 release compound to render them hydrophobic. Degassed, demineralized
water was used for the experiments and the peristaltic pump
was calibrated by plotting its flow rate-vs-time. The chamber
was leveled so that it stood dead vertical. The saturated moisture
content hsat {cm3/cm3} was determined gravimetrically and found
to be equal to 0.31 (air entrapment causes the difference between
porosity and saturated moisture content).
In order to illustrate differences in flow field structure, preferential flow fields were generated using point infiltration into dry
sand at 10 ml/min and 1.3 ml/min.
4.0
3.5
v = 1.05 cm/sec. R2 = 0.99
Q= 10 ml/min
Distance {cm}
3.0
2.5
2.0
1.5
v = 0.19 cm/sec. R2 = 0.99
Q = 1.3 ml/min
1.0
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
Time {sec}
Fig. 4. Wetting front position-vs-time for low and high flow rate experiments. It is
shown here that the high and low volume flow fields move at constant velocities.
1
r = 0.045cm
r = 0.135cm
r = 0.225cm
r = 0.270cm
r = 0.315cm
r = 0.360cm
r = 0.405cm
r = 0.450cm
0.9
0.8
0.7
θ/θsat
0.6
0.5
2.3. Images used for measurements
A 1000 frame sequence of the dry chamber was taken, averaged
and used as the Io(i,k) image for each infiltration experiment.
Images of the infiltration experiments were captured at 10 frames
per second. The progression of the wetting front was measured
by recording the location (i,k) on each image array for which
I(i,k)/Io(i,k) 6 0.98. The horizontal position of the preferential
flow field varied little during infiltration, which is typical with
this type of flow field although the two flow fields produced
significantly different wetting front velocities, Fig. 4.
The neutron intensity ratio I(i,k)/Io(i,k) was measured at each
pixel from 0.25 cm in front of the wetting front to 4.5 cm behind
it for each of 100 successive images. This 10 s interval of data
was then averaged at each pixel to obtain an average d(i,k), which
was used in Eq. (3) to obtain hj. The width of the flow field was
measured in all experiments and in all cases found to be less than
the chamber thickness.
0.4
2.4. Image noise and mass resolution
0.3
The frame-to-frame noise in a given pixel depends on the neutron flux at a specific location on the image plane. Images are acquired in 33 ms intervals and for the reactor power used in this
study, I/I0 had a standard deviation of 0.009 at 85% beam attenuation. This degree of attenuation is larger than that encountered
when imaging the flow fields in this study. The noise from neutron
counting followed a Poisson distribution. The system registers variations in water density within the experiment chamber of less than
0.007 cm3/cm3 and has previously been shown to account for mass
to 1% [1,28].
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
Distance from leading edge of wetting front {cm}
Fig. 5. Radial distribution of moisture within the 1.3 ml/min flow field. At each
radial position the moisture content can be seen to rise from zero to a peak value
before falling again. This is an indication that the fluid within the drainage portion
of the flow field is moving at roughly the same velocity.
2.2. Imaging system and setup
A Precise Optics neutron image intensifier (model PS93NX) connected to a Precise Optics Vidicon camera (model PVC525V) was
used for this work. The chamber to image plane separation used
for this work was 5.0 cm which limited the spatial resolution to
<1.0 mm2, defined as the ability to completely distinguish adjacent
points with a field of view of 410 cm2. The radioscopic images were
recorded using NIH ImageJ 1.61 running on a G3 Macintosh with a
Scion LG-3 interface.
3. Results and discussion
Figs. 5 and 6 show the radial and axial saturation profile for the
1.3 and 10 ml/min flows, respectively, for preferential flows that
move at constant velocity through the dry SiO2 sand. In Fig. 5, it
is shown that the low flow rate field exhibits pronounced drainage
behind the wetting front, which has been noted in many previous
studies, e.g. [3,29,30]. Fig. 7 shows the corresponding radially averaged saturation profiles for the two flow fields.
In Fig. 5, the 1.3 ml/min flow tomographic reconstructions show
much steeper saturation gradients in the centerline region of the
wetting front than do the radially averaged profiles. This feature
is not seen in the radially averaged data because the gradient in
saturation is smaller away from the centerline, which brings the
A.J. Gilbert, M.R. Deinert / Nuclear Instruments and Methods in Physics Research B 301 (2013) 23–28
27
1.0
r = 0.045cm
r = 0.225cm
r = 0.270cm
r = 0.315cm
r = 0.360cm
r = 0.405cm
r = 0.450cm
θ/θsat
0.8
0.6
0.4
0.2
0
1
0
2
3
4
5
6
7
8
Distance from leading edge of wetting front {cm}
Fig. 6. Radial moisture distribution in the 10 ml/min flow field. Note that drainage is not seen behind the wetting front. The results show a monotonically increasing
saturation at the edges of the flow field (large r), while the centerline region peaks then appears to begin to drop far from the wetting front (at the right hand side of the plot).
0.50
θ / θsat
0.40
Q = 10 ml/min
Q = 1.3 ml/min
0.30
0.20
0
0
1
2
3
4
5
6
7
8
Distance from wetting front {cm}
Fig. 7. Radially averaged saturation. The radially averaged saturation is shown for the high and low velocity preferential flow fields. The general behavior is the same as in the
tomographically reconstructed flow fields in Figs. 5 and 6. However, the averaged results indicate smaller saturation gradients in the wetting front than do the tomographs.
Also, the radially averaged results for the high velocity flow field fail to show how different parts of the moisture profile rise while others fall.
average down. For the 10 ml/min flow data, Fig. 6, an important
feature observed in the tomograph is the monotonically increasing
saturation that occurs away from the centerline. This is in contrast
to what is happening at the centerline, where the saturation peaks
and then begins to drop. By contrast, the radially averaged saturation seen in Fig. 7 is monotonically increasing.
Considerable noise is evident in the reconstructions shown in
Figs. 5 and 6. However, this comes not from the imaging system,
but from the variation in moisture content within the flow fields
as they are translated through the sand. The reconstructions in Figs. 5
and 6 represent a 10 s time average of the moisture profile within the
flow field. Even so, there is still considerable variation from one location to the next. This can also be seen, to a lesser extent, in the radial
averages in Fig. 7. To reduce this noise in the data, longer imaging
times would be needed to get a true average for the axial and radial
saturation profile, though a balance must be made between data
noise and temporal resolution. Even in the noisy data shown, the
tomographs show important differences in the flow field structure
when compared to the radially averaged profiles.
4. Conclusions
We have developed a simple method for determining the threedimensional structure of axisymmetric flow fields using neutron
radiography. The moisture content of the flow fields is determined
using a simple algorithm for algebraic computed tomography and
gives the three-dimensional structure of the flow field with a temporal resolution that is limited only by the desired noise level. The
tomographs reveal important differences in flow field structure
when compared to radially averaged profiles. The algorithm presented is easily translatable to radiography done with X-rays or
28
A.J. Gilbert, M.R. Deinert / Nuclear Instruments and Methods in Physics Research B 301 (2013) 23–28
light imaging. The type of tomography described here would work
with any rotationally symmetric object.
Acknowledgements
Special thanks to Paul Craven, reactor supervisor extraordinaire,
and to Kenan Unlu for making available the beam time used for this
study. This work was supported with funds from NRC-38-08-946.
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