Environ. Sci. Technol. 2007, 41, 8453–8458
Probe for Measuring Groundwater
Velocity at the Centimeter Scale
W . L A B A K Y , † J . F . D E V L I N , * ,‡ A N D
R.W. GILLHAM§
Schlumberger Water Services, Waterloo, Ontario, Canada,
Department of Geology, Lindley Hall, University of Kansas,
1475 Jayhawk Boulevard, Lawrence, Kansas 66049, and
Deptartment of Earth Sciences, University of Waterloo,
Waterloo, Ontario, Canada
Received June 29, 2007. Revised manuscript received September
20, 2007. Accepted September 21, 2007.
A novel method of measuring small-scale groundwater
velocities in unconsolidated noncohesive media uses the
travel time of a tracer pulse between an injection port and
two detectors located on the surface of a cylindrical probe, called
a point-velocity probe (PVP), as the basis for velocity estimation.
The direction and magnitude of the water velocity vector
were determined to within (9% of magnitude and (8° in
direction, on average, in ten laboratory tank tests conducted
with the PVP, when the velocities were between 5 and 98 cm/
day. Numerical simulations supported the accuracy of the
underlying theory for interpretation of the PVP data and indicated
that the technology is capable of measuring velocity at a
very fine scale (0.5 cm around the circumference). The benchtop
and modeling investigations indicated that the probe is
moderately sensitive to the condition of the porous medium
immediately next to the cylinder surface, suggesting that
challenges exist for the deployment of the instrument in the
field.
Introduction
The measurement of groundwater velocity is fundamentally
important for the assessment of risk associated with groundwater pollution. Typically, this quantity is estimated based
on a 1-D Darcy’s Law calculation to obtain the specific
discharge, q (L/T), corrected for the porosity, n (dimensionless)
v)
q K ∆H
)
n n ∆x
(1)
where v is the average linear groundwater velocity (L/T), K
is the hydraulic conductivity (L/T), H is hydraulic head (L),
and x is distance in the direction of flow (L) (all units are
given in generalized form where L is length, T is time). The
direction is usually obtained assuming flow is perpendicular
to contoured groundwater levels.
Since the introduction of permeable reactive barriers
(PRBs), there has been growing interest and recognition that
both chemical and biological processes can exert important
effects on groundwater flow (1–8). Conversely, flow velocity
is known to affect chemical processes in a PRB because it
* Corresponding author phone: 785-864-4994; fax: 785-864-5276;
e-mail: [email protected].
†
Schlumberger Water Services.
‡
University of Kansas.
§
University of Waterloo.
10.1021/es0716047 CCC: $37.00
Published on Web 11/15/2007
 2007 American Chemical Society
directly relates to the residence time of fluids in the PRB (9).
Furthermore, the flow in aquifers and PRBs is threedimensional and not constrained to single paths, so simple
one-dimensional column tests will not reproduce the outcomes of the chemical, biological, and hydrogeological
interrelationships accurately. Thus, the study of chemical or
microbiological processes without the context of flow is
limiting, and field methods to investigate these dependencies,
in three-dimensions, are needed.
With the considerations above in mind, it is noteworthy
that the conventional Darcy’s Law approach for determination of groundwater velocity (eq 1) is not well suited to an
accurate evaluation of flow conditions on the scale of a few
meters or less, such as those in a PRB. There are two major
sources of uncertainty, stemming from scale and heterogeneity, that come from the Darcy-based method of velocity
estimation: (a) accuracy of estimated K, which has been
researched extensively for engineered and natural porous
media (10–13), and (b) accuracy of hydraulic gradient, ∆H/
∆x. With regard to the latter, Darcy-based calculations have
been shown to be subject to large errors when the maximum
well spacings are small or when the hydraulic gradient is
small, such as at small sites or sites with highly permeable
aquifers (hence low hydraulic gradients) (14). Detailed
measurements of K, employing methods such as highresolution slug testing (15, 16) or borehole flowmeters (17),
do not solve the problem completely because uncertainties
in the gradient remain large.
An alternative to the use of eq 1 for estimation of
groundwater velocities is to measure the velocities directly.
This is perhaps most simply done by injecting a tracer into
the subsurface and tracking its progress through the aquifer
(18). These tests are time and labor intensive, and provide
velocity estimates that are averages, usually from injection
point to monitoring point, rather than the actual velocities
at the monitoring points.
Other methods for direct velocity measurement are based
on more highly localized tracer movement. Tracers include
heat, colloids, radioactive tracers, and saline or dye tracers.
In most cases, such as in the point dilution method (19), the
colloidal borescope (20), the Geoflowmeter (21), and the laser
Doppler velocimeter (22), the instrument is installed in a
well. The advantages of this are that many locations can be
tested repeatedly with a small number of instruments and
without additional drilling, assuming that there are available
wells. The disadvantages include the need for empirical
calibration steps to account for flow distortion near the well
screens. These can contribute uncertainty to the measurements. Over- or underdevelopment of well screens may also
introduce bias to the measurements.
Direct velocity measurements can also be made without
a well. The SPFS technique uses a dedicated probe, installed
in direct contact with the porous medium, that measures
the velocity using the temperature distribution around a
heated cylinder (23). The technique can yield a velocity vector
in three-dimensions without the use of any chemical tracers.
However, the probes are about 0.75 m long and not suitable
for point-scale measurements. The cost of the probes is also
prohibitive for multiple installations in many cases.
The purpose of this work was to develop and test a simple,
inexpensive probe suitable for centimeter-scale in situ
measurements of groundwater velocity in noncohesive
porous media, such as sand, without the need for a calibration
step. The introduction of such an instrument makes it
possible to investigate dependencies of flow on chemical
and microbiological processes, and vice versa, at a scale
VOL. 41, NO. 24, 2007 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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Replacement of ω by
2v∞ sin θ
r
and ωapp by
vapp
r
and evaluation of the integrals in eq 4 leads to (27)
v∞ )
νapp × γ
2(cos R – cos (R + γ))
(5)
Similarly, eq 3 can be evaluated to give (26)
0.5
× ln
v∞ ) vapp ×
γ
[ ]
( R +2 γ )
R
tan ( )
2
tan
(6)
Equations 5 and 6 can also be used to estimate the flow
direction. Since the velocity next to the cylindrical surface
is different at neighboring points along an arc, a probe
constructed with two detectors will yield different apparent
velocities at each detector. Moreover, the difference between
the apparent velocities is a function of the R angle (Figure
1b). This fact can be exploited to estimate R. Since v∞ is the
same for both detectors, eq 5 (or 6) expressed with variables
from detector 1, vapp1, γ, can be equated to eq 5 (or 6) with
variable values from detector 2, vapp2,, γ2. In the case of
equation 5, this leads to (27)
FIGURE 1. Plan view schematics of the point-velocity probe. (a)
The velocity at the probe surface is a function of the angle, θ,
to the flow direction. (b) A PVP device consists of an injection
port, i, and two detectors, d1 and d2, for the measurement of the
velocity magnitude and direction.
comparable to that of multilevel sampling, that is, the
centimeter scale. This scale has been shown to be relevant
for the investigation of chemical and biological processes in
unconsolidated sand (24).
Theory
Flow around a smooth, solid cylindrical surface in the absence
of a porous medium occurs as depicted in Figure 1a with the
velocity, vθ, everywhere on the cylinder surface obtainable
from (25)
vθ2 ) 4v∞2
2
sin (θ)
(2)
where v∞ is the average linear groundwater velocity unaffected
by the presence of the cylinder and θ is the angle defined in
Figure 1a. If a tracer is released at location i and is detected
at location d1 (see Figure 1b), then the apparent velocity of
the tracer over the path traveled can be calculated from (26)
r
vapp )
r
∫
∫
R+γ
R+γ
(3)
dξ
√
R
4v∞2
2
sin ξ
where ξ is an integration variable, r is the radius of the cylinder
(L), and vapp refers to an apparent velocity (L/T). If the velocity
is expressed as an angular velocity, ω (1/T), defined as ω )
v/r, then eq 3 can be rewritten as an average, apparent angular
velocity
∫
)
∫
R+γ
ωapp
ω dξ
R
R
8454
9
(4)
R+γ
(7)
which can be rearranged for the quantity R
[
R ) tan–1
νapp1γ(cos γ2 - 1) + νapp2γ2(1 - cos γ)
νapp1γ sin γ2 - νapp2γ2 sin γ
]
(8)
Equation 6 cannot be rearranged to solve directly for R, but
it can be written in a form suitable for use with a nonlinear
optimizer (28)
[ ] [
R+γ
tan
vapp1
2
ln
γ
R
tan
2
(
()
)
(
R + γ2
tan
vapp2
2
)
ln
γ2
R
tan
2
()
)
]
(9)
Equations 6, 9 and 7, 8 produce similar estimates of v∞ and
R except in the vicinity of stagnation points that exist at θ
) 0° and θ ) 180° (see Figure 1a). Care should be taken not
to position the probe in the flow system at these angles.
Instruments equipped with multiple injection ports and
detectors located around the cylinder surface could overcome
this limitation.
Materials and Methods
dξ
R
νapp2 × γ2
νapp1 × γ
)
(cos R - cos(R + γ)) (cos R - cos(R + γ2))
dξ
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 41, NO. 24, 2007
To assess the viability of a groundwater velocity probe based
on the theory described above, numerical modeling was
conducted to generate velocities on a cylindrical surface that
could be analyzed with eqs 5-9. In addition, laboratory testing
was undertaken to verify the probe performance under
physical conditions resembling an aquifer. A prototype PVP
was constructed from two 7.5 cm long, 3 cm outside diameter
(o.d.) half-cylinders of stainless steel, held together with
machine screws (Figure 2). Half-cylinders were used for ease
of construction. One half-cylinder was hollow and contained
the injection line and detection wires. The two parts were
threaded on both ends to accept drill rods at the top end and
a drive point at the lower end.
FIGURE 3. Schematic of the tank experiments for assessing
PVP performance.
FIGURE 2. Schematic of the PVP instrument.
A saline tracer was released through a stainless steel screen
(0.055 cm mesh) welded onto a 0.6 cm outer diameter
stainless steel nut that tightened from the outside of the
cylinder against a single small orifice connected to the
injection line on the inside of the cylinder. The effective
diameter of the screen was 0.3 cm. The injection line consisted
of an L-shaped stainless steel tube, 0.3 cm o.d. connected to
a section of polyethylene tubing of the same diameter with
a Swagelok connector. The top end of the injection line was
connected to a 60 mL plastic syringe filled with salt tracer
solution. A graduated roller clamp, or a 1 mL plastic syringe,
was attached to the injection line to deliver a small volume
of tracer (0.01–0.25 mL) at the onset of each experiment. The
tracer injection volumes were small enough that they exerted
a negligible effect on the groundwater flow (see modeling
results in Supporting Information).
The detector system was designed to sense conductance
changes in the water. It consisted of 0.075 cm gauge insulated
copper wires positioned in individual grooves that were 0.3
cm apart on the surface of the probe. The copper wires in
the detector were stripped of insulation where the wires came
into contact with the porous medium. The detector wires
were connected to a conductivity meter that output the data
as µΩ, which were recorded with a data logger. The γ angles
between the injection port and each of two the detectors
(Figure 1b) were fixed at 40 and 70°, respectively.
Experimental Procedure. Laboratory tests of the probe
were performed in a tank 22 cm wide, 25 cm long, and 30
cm deep with open-water compartments at each end (Figure
3). The dimensions of this tank were considered large enough
that boundary effects on the PVP measurements would be
negligible. The central compartment was filled with homogenized, medium-sized sand from the Canadian Forces Base
(CFB), Borden, Ontario, packed to a porosity of 0.36 ( 0.01,
based on the mass of sediment used, an assumed particle
density, Fs, of 2.65 g/cm3, and the volume of the tank that
was packed. Flow was induced by pumping water from one
open water compartment to the other.
The probe was placed in the center of the tank. Two
methods of placement were used. In the first method, the
probe was lowered into the tank after it was filled with water,
and then the sand was filled in around it. This method ensured
near-uniform packing of the porous medium throughout the
tank and next to the probe surface. In the second method,
the probe was driven into the prepacked, saturated sand
using a hammer. In this case the porous medium was
expected to be altered in its packing next to the probe.
A typical tracer injection consisted of a 0.01 mL pulse of
a 600 mg/L solution of NaCl. Conductance of the water was
recorded with a data logger at intervals between 0.5 and 5
min, depending on the velocity of flow in the tank. The
average flow velocity in the tank was determined by dividing
the pump discharge (Q, L3/T) by the saturated cross-sectional
area (A, L2) and porosity (n) according to the equation
v)
Q
An
(10)
The apparent velocity values, vapp, were determined by fitting
tracer breakthrough curves from each detector to a solution
of the advection–dispersion equation (see refs 26, 27, and
Supporting Information).
Experiments were conducted using velocities ranging
between approximately 5 and 100 cm/day. In another set of
experiments, the flow velocity was maintained at about 35
cm/day, and R was varied from 5 to 110 °. Replicate
measurements (four on average) of each test were made to
determine the standard deviation of the results.
The uncertainty in the experimentally imposed velocity,
or expected velocity, was determined by calculation of the
relative standard deviation of eq 10. The pump discharge, Q,
can be estimated from M/tF, where M is the mass of water
pumped over time t, and F (M/L3) is the density of water,
which was assumed to be 1.00 g/mL. Consequently, the
relative standard deviation of the expected velocity was
expressed as
sv
)
v
sh2
h2
+
sw2
w2
+
st2
t2
+
sM2
sn2
+
M2 n2
(11)
where Sv, Sh ) Sw ) (0.05 cm, St ) (0.05 s, SM ) (5 × 10-5
g, and Sn ) (0.01 are the standard deviations of the various
parameters, including thickness (h ) 28 cm) and width (w
) 22 cm) of the flow section. The uncertainty in the density
of water, F, was assumed to be negligible.
The values of v, t, and M were determined before and
after every experiment. The uncertainty in measured parameters was taken to be 0.5 times the smallest division of
the measuring instrument. The uncertainty on porosity was
calculated by propagating the uncertainty in the terms in
the following equation
Ms
(hwl)t
Fs
Vt - Vs
)
n)
Vt
(hwl)t
VOL. 41, NO. 24, 2007 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 5. Agreement between velocities input to the
SALTFLOW 3D flow and transport model and the velocities
estimated from eq 5 (see text). The slight negative bias is
thought to be related to boundary effects in the numerical
model.
FIGURE 4. Breakthrough curves at detectors 1 and 2 for
simulated tracer (symbols) at various velocities. Lines represent
the fitted solutions to the advection dispersion equation using
the FORTRAN program PULSEPE (see Supporting Information).
Note the low conductance measured in the v ) 1 cm/day
experiment is likely caused, in part, by diffusional loss of
tracer away from the probe surface during the long data
collection time.
where Vt is the total volume of the tank (L3), and Vs is the
volume of sand used (L3), assuming ( 0.05 cm in each length
dimension, h, w, and l, and ( 57 g in the total sand mass,
Ms, and negligible uncertainty in Fs (2.65 g/cm3).
On the basis of eq 11, the standard deviation values for
expected bulk velocities ranged between (0.2 and 2.7 cm/
day for velocities of magnitude of 5.3–97.3 cm/day, respectively. The average experimental relative standard deviation
on expected velocity was somewhat higher at (9%, possibly
because of installation issues and nonidealities at the
boundaries of the tank.
Numerical Simulations. Equation 2 was developed for
flow in the absence of a porous medium (25). However, the
continuity equation for flow in a porous medium is of the
same form as that used for open-channel flow (29), and on
this basis, it would seem that eq 2 should be applicable for
flow in porous media. To test this assumption, numerical
modeling was undertaken in which a simulated tracer was
released on a cylinder, and the detector responses were
converted to velocities using eqs 5 and 7. SALTFLOW 3.0, a
3D finite element code capable of simulating densitydependent flow and transport, assuming a saturated, nondeforming, isothermal medium with incompressible fluids,
was used for the simulations (30). A radial grid with 400 000
elements was used to represent the porous medium in the
vicinity of half-a PVP instrument surface. Dispersivity values
were selected to be 0.9 mm in the direction of flow and 0.1
mm in the other two principle directions, based on tracer
behavior in laboratory tests.
Results and Discussion
The evaluation of the velocity probe concept was accomplished by testing the theory with a numerical model,
SALTFLOW, and with a series of laboratory tank experiments.
The numerical simulations gave a flow regime with velocities
immediately next to the probe surface that were in close
agreement with those of eq 2 for a variety simulated porous
media permeabilities (Figure 4) (see Supporting Information).
Moreover, the application of eqs 5 and 6 to the interpretation
of tracer breakthrough curves from the simulations (Figures
4 and 5) gave estimates of v∞ that were within 3% to 7% of
the correct values. Slight negative biases were thought to be
8456
9
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 41, NO. 24, 2007
TABLE 1. Summary of Simulation Input Angles and Angles
Estimated from Equation 7 (see text)
model input (deg)
PVP analysis (deg)
20
30
45
60
90
17
28
45
61
76
related to the numerical boundary conditions. Flow directions
were similarly well estimated (Table 1).
The effect of porous media disturbance next to the
cylindrical surface because of installation procedures was
assessed by numerically simulating such zones with either
enhanced or diminished K values. The K-diminished zones,
or skins, were fixed at 50, 25, and 15% of the background K
(1 × 10-3 cm/s). Additional simulations of 50% reduced K
were conducted with skin thicknesses of 0.6, 1.5, and 3 cm
to assess the effects of skin thickness. It was found that
estimated velocities declined in direct proportion to the
decline in K of the skin. Estimated velocities were not very
sensitive to skin thicknesses. Similar findings, pertaining to
velocity overestimations, were obtained from the simulations
involving K-enhanced skins (see Supporting Information).
FIGURE 6. Agreement between expected velocities in the tank
experiment and those estimated using eq 5 (see text). Open
circles represent data from experiments in which the PVP was
positioned before the sand was packed. The broken line
represents the trendline through the circles. Note the trendline
has a slope very near 1, indicating good accuracy. Error bars
on the circles represent one standard deviation estimated from
replicate experiments. Open triangles represent data from
additional experiments in which the PVP was installed by
hammering. The hydraulic conductivity of the sand in the tank
was apparently reduced next to the probe as a result of
compaction during the hammering. The observed bias in the
data is clear and corresponds to measured velocities averaging
about one-third the expected value.
TABLE 2. Summary of PVP Assessment in Laboratory Tank Experimentsa
magnitude
direction
error
r (deg)
no. of
replicate pairs
expected
(cm/day)
measured
(cm/day)
expected
(cm/day)
measured
(cm/day)
magnitude
(%)
direction
(deg)
5
20
45
45
45
50
60
70
85
110
3
3
4
3
3
5
5
7
5
3
34.7 ( 1.0
32.1 ( 0.9
5.3 ( 0.2
14.2 ( 0.4
93.5 ( 2.6
35.2 ( 1.0
97.3 ( 2.7
33.6 ( 0.9
32.8 ( 0.9
31.7 ( 0.4
33 ( 2.3
38.4 ( 9.4
5.1 ( 1.3
13.6 ( 0.8
93.6 ( 3.4
35.2 ( 8.7
95.5 ( 2.6
37.9 ( 4.7
34.9 ( 2.6
43.4 ( 7.9
5(5
20 ( 5
45 ( 5
45 ( 5
45 ( 5
50 ( 5
60 ( 5
70 ( 5
85 ( 5
110 ( 5
4(3
23 ( 5
46 ( 18
57 ( 3
68 ( 7
58 ( 10
51 ( 16
68 ( 13
86 ( 2
129 ( 8
-4.7
19.6
-1.6
-4.1
0.1
-0.2
-1.8
13
6.5
37.2
-1
3
1
12
23
8
-9
-2
1
19
a
Control refers to the values expected based on the tank set up and pumping rate. Measured refers to values estimated
from eqs 5 and 8 (see text).
In summary, the modeling study indicated that equations
5–9 are suitable for application to the estimation of groundwater velocities in porous media over a wide range of
velocities typical of those found in natural geologic deposits
(1–320 cm/day), provided that a minimal disturbance of the
porous medium can be assured during installation of the
device.
In the laboratory tests where the PVP was installed before
packing, to minimize skin effects, and in which velocities
ranged between 5 and 98 cm/day, the PVP instrument was
able to provide estimates within 15% of the expected value
in 8 out of 10 tests. In all tests, the agreement between the
measured and expected values was within 50% (Figure 6 and
Table 2), and the average error was only about (9%. In
addition, the estimated flow directions were within 15° in
eight out of ten tests and did not exceed the expected angle
by more than 23 ° in any tests. The average angular error was
about (8°.
A second tank experiment, in which the probe was
hammered into position after the sand was packed, was
performed to evaluate the effect of a skin (Figure 6). In that
case the velocity estimates were noticeably different from
the expected values, with negative biases amounting to as
much as 80% ((expected – measured) × 100/expected). This
degree of error might be considered large because it means
that a measured velocity could be smaller than the actual
velocity by a factor of 5. Nevertheless, given that the
uncertainties associated the estimations of K and velocities
derived from Darcy’s Law at field sites are recognized to be
as much as an order of magnitude (13), the biases noted here
(resulting from a skin effect) are not so severe that the
measurements should be considered meritless.
In summary, the laboratory testing of the PVP device
showed that accurate measurements of velocity magnitude
and direction are possible with the instrument. To use the
technology to best advantage in field applications, care must
be taken to minimize skin effects during installation. On the
basis of the consistency of results from the modeling and
laboratory testing and the good agreement between the
algorithms developed here and the aforementioned tests,
the prognosis for the use of PVPs in aquifers is considered
very good.
Acknowledgments
The NSERC/Motorola/ETI Industrial Research Chair in
Groundwater Remediation, NSERC, CRESTech, The National
Council for Scientific Research of Lebanon (NCSR), OGSST,
and NSF under Grant No. 0134545 are acknowledged for
funding this work. Any opinions, findings, and conclusions
or recommendations expressed in this material are those of
the authors and do not necessarily reflect the views of the
National Science Foundation. John Molson provided assistance with the numerical modeling.
Supporting Information Available
Additional details of the numerical modeling, results of the
tank experiments to investigate skin effects, and details of
the curve fitting procedure. This material is available free of
charge via the Internet at http://pubs.acs.org.
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