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PAPER
Push-pull tests for estimating effective porosity: expanded analytical
solution and in situ application
Charles J. Paradis
1
&Larry D. McKay
1
&Edmund Perfect
1
&Jonathan D. Istok
2
&
Terry C. Hazen
1,3,4,5,6,7
Received: 17 March 2017 /Accepted: 12 September 2017
#The Author(s) 2017. This article is an open access publication
Abstract The analytical solution describing the one-
dimensional displacement of the center of mass of a tracer
during an injection, drift, and extraction test (push-pull test)
was expanded to account for displacement during the injection
phase. The solution was expanded to improve the in situ esti-
mation of effective porosity. The truncated equation assumed
displacement during the injection phase was negligible, which
may theoretically lead to an underestimation of the true value
of effective porosity. To experimentally compare the expand-
ed and truncated equations, single-well push-pull tests were
conducted across six test wells located in a shallow, uncon-
fined aquifer comprised of unconsolidated and heterogeneous
silty and clayey fill materials. The push-pull tests were con-
ducted by injection of bromide tracer, followed by a non-
pumping period, and subsequent extraction of groundwater.
The values of effective porosity from the expanded equation
(0.6–5.0%) were substantially greater than from the truncated
equation (0.1–1.3%). The expanded and truncated equations
were compared to data from previous push-pull studies in the
literature and demonstrated that displacement during the in-
jection phase may or may not be negligible, depending on the
aquifer properties and the push-pull test parameters. The re-
sults presented here alsodemonstratedthe spatial variability of
effective porosity within a relatively small study site can be
substantial, and the error-propagated uncertainty of effective
porosity can be mitigated to a reasonable level (< ± 0.5%).
The tests presented here are also the first that the authors are
aware of that estimate, in situ, the effective porosity of fine-
grained fill material.
Keywords Groundwater flow .Heterogeneity .Hydraulic
properties .Hydraulic testing .Tracer tests
Introduction
The effective porosity of saturated porous media is a funda-
mental hydrogeological parameter for modeling the fate and
transport of dissolved-phase contaminants in the subsurface.
Reliable modeling is highly dependent on accurate character-
ization of effective porosity. Field-scale tracer-based methods
are particularly attractive to estimate effective porosity be-
cause they are based on direct measurements of the in situ
transport of a dissolved-phase constituent. The single-well
push-pull test method has been developed to estimate effective
porosity and has been successfully applied in situ (Istok
2013). However, the current analytical model (Hall et al.
1991) assumes the transport of the tracer during the push
phase is negligible, which may or may not be an appropriate
Electronic supplementary material The online version of this article
(https://doi.org/10.1007/s10040-017-1672-3) contains supplementary
material, which is available to authorized users.
*Ter ry C . Ha ze n
tchazen@utk.edu
1
Department of Earth and Planetary Sciences, University of
Tennessee, Knoxville, TN 37996, USA
2
School of Civil and Construction Engineering, Oregon State
University, Corvallis, OR 97331, USA
3
Biosciences Division, Oak Ridge National Laboratory, Oak
Ridge, TN 37830, USA
4
Department of Microbiology, University of Tennessee,
Knoxville, TN 37996, USA
5
Department of Civil and Environmental Sciences, University of
Tennessee, Knoxville, TN 37996, USA
6
Center for Environmental Biotechnology, University of Tennessee,
Knoxville, TN 37996, USA
7
Institute for a Secure and Sustainable Environment, University of
Tennessee, Knoxville, TN 37996, USA
Hydrogeol J
DOI 10.1007/s10040-017-1672-3
assumption in all cases. Theoretically, neglecting to account
for the transport of the tracer during the push phase may lead
to an underestimation of effective porosity. In this study, the
analytical solution to describe the displacement of a tracer
during a push-pull test was expanded to account for the push
phase and then applied in situ toestimate the effective porosity
across six test wells located in a shallow, unconfined aquifer.
Effective porosity can be qualitatively defined as the vol-
ume of the void spaces through which water or other fluids
can travel (by advection) in a rock or sediment divided by the
total volume of the rock or sediment (Fetter 2001). Domenico
and Schwartz (1998) explained that effective porosity implies
some connectivity through the porous medium and is more
closely related to permeability than is total porosity. The def-
inition and conceptualization of effective porosity has led to
the use of more descriptive terms such as mobile porosity,
kinematic porosity, and dynamic porosity. Determining the
appropriate value of effective porosity for groundwater
models can be challenging, due in part, to the spatial hetero-
geneity of porous media. Field-scale tracer-based studies have
shown that effective porosity in granular porous media can
range from 40% (alluvial sediments; fine sands, and glacial
till) to 0.4% (layered medium sand) and in fractured porous
media from 60% (fractured dolomite and limestone) to 0.5%
(fractured chalk) (Gelhar et al. 1992). There is also increasing
evidence that effective porosity is dependent on the scale at
which it is assessed, which suggests that field-scale methods
may be more appropriate to inform groundwater models (Li
1995;Stephensetal.1998).
Methods to estimate effective porosity typically rely on
calculating proxy parameters such as specific yield (Meinzer
1923a) or correlating grain-size distribution and soil-water
characteristic curves to representative values of specific yield
(Meinzer 1923b). Estimation-based methods have the disad-
vantage of being indirect but are relatively simple to conduct.
Methods to calculate effective porosity typically rely on
conducting tracer-based tests and interpretation of subsequent
breakthrough curves (Stephens et al. 1998). Tracer-based
methods have the advantage of being direct but can be rela-
tively difficult to conduct, especially at the field scale.
Moreover, the interpretation of breakthrough curves requires
careful consideration of the properties of the tracer and the
porous medium—for example, tracer mass transport mecha-
nisms such as: (1) sorption to the porous medium, (2) diffu-
sion from mobile to immobile pore water, (3) volatilization to
the unsaturated zone, and (4) degradation or transformation
are not truly representative of the void spaces through which
water can travel by advection, i.e., effective porosity (Davis
et al. 1980; Turnadge and Smerdon 2014).
Hall et al. (1991) developed a relatively simple tracer-based
method to calculate effective porosity based on conducting
and interpreting the data from a single-well push-pull test. A
single-well push-pull test is conducted by injecting (push
phase) a volume of water containing a tracer into a single well,
followed by a non-pumping period (drift phase), and subse-
quent extracting (pull phase) of groundwater from the same
well in order to generate a breakthrough curve (Istok 2013). A
single-well push-pull test has the threefold advantage of being
direct, simple, and field scale. The Hall et al. (1991)method
was theoretically developed for a confined, homogeneous,
and isotropic aquifer but was experimentally validated at the
field scale in an unconfined, heterogeneous, and sandy
aquifer. Hall et al. (1991) compared the effective porosity
calculated from a single-well push-pull test to a dual-well
natural-gradient test and found that both tests yielded similar
values. However, the Hall et al. (1991) method assumed that:
(1) the transport of the tracer during the push phase was neg-
ligible, and (2) the uncertainty in the calculation of effective
porosity was negligible. Moreover, the Hall et al. (1991)ap-
plication was limited to a single well. Although the
assumptions and spatially limited application by Hall et al.
(1991) may have been valid and appropriate, respectively,
for their case study, they may not be appropriate at other sites
with variable aquifer properties, other push-pull test parame-
ters, and different study objectives.
The purpose of this study was to utilize the single-well
push-pull test method to characterize the magnitude and spa-
tial variability of effective porosity within a shallow, uncon-
fined aquifer. The novelty of this study was threefold: (1) the
expansion of the Hall et al. (1991) analytical solution to in-
clude the transport of the tracer during the push phase, (2) the
performance of an uncertainty analysis for the calculation of
effective porosity, and (3) the assessment of the spatial vari-
ability of effective porosity within the study site.
Materials and methods
Theory
The volume of water injected into, or extracted from, an aqui-
fer at a constant pumping rate, is given by:
V¼Q
jj
tð1Þ
where:
Vvolume of water [L
3
]
Qconstant pumping rate [L
3
/T]
telapsed time during pumping [T]
By convention, the pumping rate (Q) is positive dur-
ing injection and negative during extraction. If the aqui-
fer is confined, homogeneous, and isotropic, and if the
ambient groundwater flow is negligible, the cylindrical
volume of water injected into, or extracted from, a fully
penetrating well, is given by:
Hydrogeol J
V¼πr2bneð2Þ
where:
rradius of water [L]
bsaturated aquifer thickness [L]
n
e
effective porosity [dimensionless]
If the saturated aquifer thickness is constant, equating
Eqs (1)and(2), and rearranging gives:
r¼Q
jj
t
πbne
1=2ð3Þ
Equation (3) describes the leading- or trailing-edge position
of a particle of water within an expanding or contracting cy-
lindrical volume of water as it is injected into, or extracted
from, an aquifer.
Darcy’s law can be written to include effective porosity as:
v¼
−Kdh
dr
ne
ð4Þ
where:
vaverage linear groundwater velocity [L/T]
Khydraulic conductivity [L/T]
dh/drhydraulic gradient [L/L]
Equation (4) describes the average linearvelocityofaparti-
cle of water within an aquifer due to ambient groundwater flow.
Velocity, in general terms, is given by:
v¼Δr
Δtð5Þ
where:
Δrtraveled distance [L]
Δtelapsed time [T]
Equation (5) can be rearranged to give:
Δr¼vΔtð6Þ
Equation (6) describes the average position of a particle of
water within an aquifer due to ambient groundwater flow. The
one-dimensional (1D) displacement of the center of mass of a
tracer, after completion of the injection, drift, and extraction
phases of a push-pull test, is zero (Fig. 1). The displacement of
the center of mass of the tracer is given by:
r1þr2þr3¼0ð7Þ
where:
r
1
displacement during injection [L]
r
2
displacement during drift [L]
r
3
displacement during extraction [L]
The displacement of the tracer during: (1) the injection
phase, is due to injection pumping (r
i
) and ambient ground-
water flow (r
a1
), (2) the drift phase, is due to ambient ground-
water flow (r
a2
), and (3) the extraction phase, is due to extrac-
tion pumping (−r
e
) and ambient groundwater flow (r
a3
)
(Fig. 1). The components of the displacement of the center
of mass of the tracer during the push-pull test can be substitut-
ed in Eq. (7)togive:
riþra1
ðÞþra2
ðÞþ−reþra3
ðÞ¼0ð8Þ
where:
r
i
displacement due to injection pumping [L]
r
a1
displacement due to ambient groundwater flow [L]
r
a2
displacement due to ambient groundwater flow [L]
r
e
displacement due to extraction pumping [L]
r
a3
displacement due to ambient groundwater flow [L]
The components in Eq. (8) can be substituted by their cor-
responding equations given in Eqs. (3)and(6)togive:
Qi
jj
ti
πbne
1=2þvΔta1
"#
þvΔta2
ðÞþ−Qe
jj
te
πbne
1=2þvΔta3
"#
¼0
ð9Þ
The components in Eq. (9), due to injection (first term) and
extraction (fourth term), represent the leading- or trailing-edge
position of the tracer within an expanding or contracting cy-
lindrical volume of water, whereas the components due to
ambient groundwater flow (vΔt
a1
,vΔt
a2
,andvΔt
a3
), repre-
sent the average displacement of the tracer. The average dis-
placement of the tracer, due to injection, occurs when one-half
of the mass of the tracer has been injected and is given by:
Qi
jj
τi¼Qi
jj
ti
2ð10Þ
where:
Q
i
injection rate [L
3
/T]
τ
i
time elapsed from the start of water injection until the
center of mass of the tracer is released [T]
In volumetric terms, Eq. (10) can be rewritten to give:
Vi¼Qi
jj
τið11Þ
where:
Vi= volume of water injected until the center of mass of the
tracer is released [L
3
].
The average displacement of the tracer, due to extraction,
occurs when one-half of the mass of the tracer has been re-
covered and isgiven by integration of the concentration versus
volume data, i.e., the breakthrough curve, as:
Hydrogeol J
Me¼1
2∫V1
V0CVðÞdV ð12Þ
where:
M
e
one-half of the mass of the recovered tracer [M]
V
0
volume of water recovered at the start of extraction
pumping [L
3
]
V
1
volume of water recovered at the end of extraction
pumping [L
3
]
C(V) concentration of the tracer (C)[M/L
3
] as a function of
the volume (V)[L
3
] of water extracted
The corresponding volume at which one-half of the mass of
the tracer has been recovered is given by evaluating the solu-
tion to Eq. (12)atM
e
by:
Me¼MVe
ðÞ ð13Þ
where:
MVe
ðÞ mass of the tracer (M) [M] as a function of volume
(Ve)[L
3
] at which one-half of the mass of the tracer
has been recovered
It is important to note that the solution to Eq. (12)canbe
estimated numerically, as opposed to solved analytically, and
doing so would allow for estimating Ve. The corresponding
times at which Viand Veoccur are given as:
τi¼Vi
Qi
jj ð14Þ
τe¼Ve
Qe
jj ð15Þ
Substituting Viin Eq. (11), Vein Eq. (13), τ
i
in Eq. (14),
and τ
e
in Eq. (15)forQ
i
τ
i
,Q
e
τ
e
,Δt
a1
,andΔt
a3
in Eq. (9),
respectively, gives:
Vi−Ve
πbne
1=2þvτiþtdþτe
ðÞ¼0ð16Þ
where:
t
d
=Δt
a2
(time elapsed from the end of water injection until
the start of water extraction) [T]
Equation (16) describes the average position of the cen-
ter of mass of the tracer during the injection, drift, and
extraction phases. Rearranging Eq. (16)tosolveforaver-
age linear groundwater velocity gives:
v¼
Ve−Vi
πbne
1=2
τiþtdþτe
ðÞ
ð17Þ
Equating Eqs. (17)and(4), and solving for effective poros-
ity (n
e
)gives:
ne1 ¼πbK2dh
dr
2τiþtdþτe
ðÞ
2
Ve−Vi
ð18Þ
Equation (18) describes effective porosity (n
e1
)asa
function of the aquifer properties, e.g., saturated thickness
(b), hydraulic conductivity (K), and hydraulic gradient (dh/
dr), and the transport of the center of mass of the tracer
during the injection (Vi,τ
i
), drift (t
d
), and extraction (Ve,
τ
e
) phases. Equations (17) and (18) are very similar to the
Leap and Kaplan (1988) and Hall et al. (1991) equations,
respectively.
From Leap and Kaplan (1988):
v¼
Ve
πbne
1=2
tdþτe
ðÞ
ð19Þ
From Hall et al. (1991):
ne2 ¼πbK2dh
dr
2tdþτe
ðÞ
2
Ve
ð20Þ
However, Eqs. (17)and(18) account for the transport of
tracer during the injection phase (Vi,τ
i
), whereas Eqs. (19)
and (20) do not. If the transport of the tracer during the injec-
tion phase is truly negligible, then Viand τ
i
are equal to zero,
and Eqs. (17)and(18) are equivalent to Eqs. (19)and(20),
Fig. 1 Plan-view depiction of the
center of mass of a tracer at the
end of the injection (1), drift (2),
and extraction (3) phases,
r
i
= displacement due to injection,
r
a
= displacement due to ambient
groundwater flow,
r
e
= displacement due to
extraction
Hydrogeol J
respectively. If the transport of the tracer during the injection
phase is not truly negligible, then Viand τ
i
are greater than
zero, and Eq. (17) will yield lower values of average linear
groundwater velocity than Eqs. (19), and (18) will yield higher
values of effective porosity than Eq. (20).
Study site
The study site is in Area 2 of the Oak Ridge Integrated Field
Research Challenge (OR-IFRC) site at the Department of
Energy’s Oak Ridge Reservation (ORR) in Oak Ridge,
Tennessee, USA (Fig. 2). A typical geologic profile of Area
2 consists of approximately 6 m of unconsolidated and het-
erogeneous materials comprised of silty and clayey fill (most-
ly fine grained soil and clay-rich residuum), related to
historical construction activities, underlain by undisturbed
and clay-rich weathered bedrock (Moon et al. 2006; Watson
et al. 2004; Fig. 3). Slug tests indicated that the hydraulic
conductivity of the fill materials was approximately two or-
ders of magnitude greater than the weathered bedrock, e.g.,
10
−6
versus 10
−8
m/s, respectively (Fig. 3). The study site
contains 13 monitoring wells (FW218–FW230), six of which
were used as test wells (FW220–FW225), and one of which
was used as a source well (FW229) for groundwater injectate
for the single-well push-pull tests, as discussed in section
‘Effective porosity’(Fig. 2). The test wells were installed by
direct push coupled with continuous electrical resistivity pro-
filing. The test wells are constructed of 1.9-cm inside diameter
schedule-80 polyvinyl chloride (PVC) pipe and are screened
from 3.7 to 6.1 m below ground surface (mbgs; Fig. 3). The
Fig. 2 Plan-view maps of the study site: acountry map showing study
site location in the southeastern United States, barea map showing study
site location in Area 2 of the OR-IFRC, and cstudy site map showing well
locations, groundwater-level elevations, and groundwater-level elevation
iso-contours (m amsl = meters above mean sea level)
Hydrogeol J
test wells are screened within the fill materials and were ver-
tically terminated at contact with the undisturbed weathered
bedrock; the contact with undisturbed weathered bedrock was
determined by a substantial increase in difficulty of advancing
the direct-push drill string and a concomitant and notable in-
crease in electrical resistivity (Fig. 3). The test wells are fully
screened across the unconfined aquifer (Fig. 3). The source
well is constructed of 5.1-cm inside diameter schedule-40
PVC pipe and is screened from 3 to 7.5 mbgs. The shallow
groundwater aquifer is unconfined and the depth to ground-
water is approximately 3.5 mbgs (Fig. 3). The site-wide aver-
age magnitude and direction of the hydraulic gradient, as de-
termined graphically, is approximately −0.045 m/m and to the
south/southwest, respectively (Fig. 2).
The physical properties of the fill materials, in which the
test wells are screened, are poorly characterized compared to
those at other study sites located within Area 2. This is due, in
part, to environmental safety policies at the site, which is lo-
cated adjacent to highly contaminated areas. As a result of
these policies, it was not feasible to collect core samples for
evaluation of material properties during this study. However,
previous investigations at nearby sites in Area 2 indicate that
fill commonly used in this part of the ORR consisted mainly
of locally obtained silty and clayey material derived from
decomposition of shale and limestone. Fill material at the
study site is expected to be similar in composition. It should
be noted that the chemical and biological properties of the
groundwater system at the study site are better characterized.
A previous study by Paradis et al. (2016)reportedthatdespite
the high level of aquifer heterogeneity within Area 2, the
biogeochemical processes associated with the reduction and
oxidation of uranium within the study site wells (FW218–
FW227) were spatially consistent; nevertheless, the spatial
variability of the physical properties of the fill materials,
e.g., effective porosity, were unknown at the time of this study.
Hydraulic gradient
The hydraulic gradient, within the vicinity of each test well,
was estimated using ArcMap (version 10.5) software. The
depth to groundwater, relative to the top of the casing (sur-
veyed to 0.3-cm above mean sea level) of each site well, was
measured using an electronic water level indicator (Solinst)
immediately prior to conducting single-well pumping and
push-pull tests. The depth to groundwater measurements were
converted to meters above mean sea level (m amsl) and
uploaded to ArcMap, along with the coordinates (latitude
and longitude) of each site well, to create a point shape file.
The groundwater elevation data were interpolated, using the
spline tool, to create a digital elevation model (raster file) of
the water table (cell size = 0.15 m, weight = 0, all other pa-
rameters set at default). The slope of the water table was cal-
culated using the slope tool (z-factor = 1.171 × 10
−5
,basedon
the latitude of the study site). The average slope, within a 1-m
radius about each test well, was calculated using the zonal
statistics tool. The rationale for a 1-m radius, as representative
of the hydraulic conditions within the vicinity of each test
well, was based on Eq. (3) which describes the leading-edge
position of a particle of water within an expanding cylindrical
volume of water as it is injected into an aquifer, i.e., the max-
imum frontal position of bromide tracer during the injection
phase of a push-pull test. An effective porosity of 5% was
assumed, a priori. For a 20-L injection volume and a saturated
aquifer thickness of 2.5 m, the radius in Eq. (3) would be
approximately 0.25 m. It is important to note that Eq. (3)
ignores heterogeneity and the drift phase of the push-pull tests
which would lead to an underestimation of radius; therefore, a
1-m radius was assumed. The slope at each test well was
converted from degrees to hydraulic gradient values and in-
putted into Eqs. (18)and(20) to estimate effective porosity.
Hydraulic conductivity
The hydraulic conductivity, within the vicinity of each test
well, was estimated by conducting single-well pumping tests.
Single-well pumping tests were conducted according to the
methodology of Robbins et al. (2009) and Aragon-Jose and
Robbins (2011). In brief, groundwater was pumped from each
test well at a constant discharge rate using a peristaltic pump
(Geotech Geopump) and stored in a 208-L plastic drum. The
discharge rate was measured using a graduated cylinder and a
stop watch. The depth to groundwater was measured using an
electronic water level indicator (Solinst). The discharge rate
Fig. 3 Vertical-view conceptual modelof the shallow unconfined aquifer
and construction details of a test well; horizontal exaggeration is 50-fold.
Fill consists of Bsilty and clayey^material
Hydrogeol J
and depth to groundwater were measured sequentially until
steady-state conditions were achieved; steady-state conditions
were defined as a change in drawdown less than 1.2 cm over
the course of 15 min during a constant discharge rate.
Single-well pumping test data were analyzed according to
the general methodology of Robbins et al. (2009) and Aragon-
Jose and Robbins (2011). In brief, the steady-state discharge
and drawdown values, along with the construction details of
the test wells, e.g., saturated screen length and radius of well,
were used to calculate the hydraulic conductivity using the
half-ellipsoid flow equation, described analytically by
Dachler (1936). Graphs showing single-well pumping test da-
ta can be found in the electronic supplementary material
(ESM).
Effective porosity
The effective porosity, within the vicinity of each test well,
was estimated by conducting single-well push-pull tests.
Single-well push-pull tests were conducted according to the
general methodology of Istok (2013). In brief, 23 L of ground-
water (injectate) were collected from the up-gradient well
FW229 (Fig. 2) using a peristaltic pump and stored in a plastic
carboy. Three grams of potassium bromide (KBr; Sigma-
Aldrich) were then added to 20 L of the injectate and mixed
by recirculation using a peristaltic pump for a target concen-
tration of 100 mg/L bromide. During mixing of the injectate,
three samples were collected in 20-ml scintillation vials and
were analyzed for bromide.The concentration of bromide was
determined in the field using a bromide ion selective half-cell
electrode (Thermo Scientific Cat. No. 9435BN) coupled with
a double junction reference electrode (Thermo Scientific Cat.
No. 900200). The minimum detection limit for bromide was
1 mg/L and the reproducibility of bromide measurements was
±2%. Immediately prior to injection, 1 L of groundwater was
purged from the test well (approximately two test well vol-
umes) and three samples were collected and analyzed for the
background concentration of bromide. The push phase of the
test consisted of low-flow (approximately 250–400 ml/min)
injection of the 20-L bromide-amended injectate followed im-
mediately by the injection of 3 L of nonamended injectate
(herein referred to as the Bchase^) using a peristaltic pump.
The injection of the chase was conducted to clear the test well
volume (approximately 0.5 L) of the bromide-amended
injectate. The total push time (tracer plus chase) ranged from
approximately 1–1.5 h. The injectate was then left to drift in
the groundwater system under nonpumping conditions for up
to 2 h. The pull phase of the test consisted of low-flow extrac-
tion (approximately 100 to 300 ml/min) of up to 65 L of
groundwater and sequential collection of 20-ml samples
which were analyzed for bromide.
Single-well push-pull test data were analyzed according to
the general methodology of Istok (2013). In brief, the time (τ
i
)
and volume (Vi) at which the center of mass of bromide was
released were calculated by evaluating Eqs. (10) and (11),
respectively. The concentration of bromide versus the volume
and time elapsed during the pull phase of the tests were gen-
erated to calculate the volume (Ve) and time (τ
e
) at which the
center of mass of bromide was recovered. Veand τ
e
were
calculated by numerical integration of the bromide versus time
data (Thomas et al. 2008). Veand τ
e
were concomitant with
one half of the region between the bromide and volume/time
data. Graphs showing single-well push-pull test data can be
found in the electronic supplementary material (ESM).
Uncertainly analysis
The uncertainty in the measured parameters, e.g., volume
injected/extracted, pumping rate, drawdown, elapsed time,
etc. and the propagated error in the calculated parameters,
e.g., hydraulic gradient, hydraulic conductivity, and effective
porosity, were analyzed according to the BData Analysis
Toolkit # 5: Uncertainty analysis and error propagation^,by
Kirchner (2001). More specifically, the simple rules for sums
and differences, and for products and ratios, were used.
Results
Hydraulic gradient results
The static water table was relatively stable immediately prior
to, and after, conducting the single-well pumping and push-
pull tests (data not shown). The site-wide average magnitude
and direction of the static hydraulic gradient was similar to
pretest conditions, e.g., −0.045 (Fig. 2). The near-well (1-m
radius) hydraulic gradient at each test well, immediately prior
to conducting the push-pull tests, ranged from a low of −0.020
in test well FW224 to a high of −0.085 in test well FW221
(Table 1). The range of hydraulic gradient values were notably
greater than those previously reported at other test sites by
Hall et al. (1991) and Istok (2013).
Hydraulic conductivity results
During the single-well pumping tests, steady-state discharge
and drawdown conditions were achieved within a few minutes
after the tests began and were maintained for approximately
1 h (data not shown). The drawdown was typically less than
10% of the static saturated screen length (data not shown).
Static water levels were stable prior to initiating the pumping
tests and recharge to near-static water levels generally oc-
curred within 0.5 h after pumping stopped (data not shown).
The hydraulic conductivity for each test well was then calcu-
lated by inputting the steady-state discharge and drawdown
values, along with the saturated well screen length and radius,
Hydrogeol J
into the half-ellipsoid flow equation (Dachler 1936). The hy-
draulic conductivity ranged from a low of 2.1 × 10
−6
m/s in
test well FW225 to a high of 1.6 × 10
−5
m/s in test well
FW224 (Table 1). The range of hydraulic conductivity values
were within those representative of silts and fine sands
(Domenico and Schwartz 1998) and notably less than those
previously reported at other test sites by Hall et al. (1991)and
Istok (2013)(Table1).
Effective porosity results
The breakthrough curves of bromide, during the pull phase of
the tests, showed sharp and short-lived increases followed by
gradual and non-linear decreases (Fig. 4). It is important to
note that the concentrations of bromide in the test wells prior
to injection were below the minimum detection limit (~1 mg/
L) and that the concentration of bromide in the injectate was
near the target concentration (~100 mg/L; data not shown).
The time (τ
e
) from the start of the pull phase until the center
of mass of bromide was recovered ranged from a low of 0.85 h
(3,077 s) in test well FW223 to a high of 1.14 h (4,087 s) in
test well FW222 (Fig. 4; Table 2). The corresponding volume
(Ve)atwhichthecenterofmassofbromidewasrecovered
ranged from a low of 6 L (0.006 m
3
) in test well FW225 to a
high of 15 L (0.015 m
3
) in test well FW221 (Fig. 4; Table 2).
The saturated aquifer thickness (~2.4 m) was similar
among all test wells (Table 2). The drift times (t
d
)weresimilar
among five of the six wells (~1.8 h on average), whereas the
drift time in test well FW225 was notably short (~0.5 h;
Tab le 2). The percent mass recovery of bromide ranged from
a low of 41% in test well FW225 to a high of 71% in test well
FW221 (data not shown). In general, the experimental design,
aquifer properties (Table 1), and results of the push-pull tests
(Table 2) for this study were more similar to those from Istok
(2013)thanfromHalletal.(1991). However, it should be
noted that the drift times (t
d
) for this study were substantially
less than Istok (2013).
The effective porosity (n
e
) for each test well was calculated
by inputting the parameters from Tables 1and 2into the ex-
panded and truncated equations, Eqs. (18)and(20), respec-
tively. The effective porosity (n
e1
), per Eq. (18), ranged from a
low of 0.6% in test well FW220 to a high of 5.0% in test well
FW221 (Table 3). It should be noted that the negative value of
n
e1
(−0.1%) in test well FW225 indicated that one or more
input parameters for Eq. (18) were not valid; this issue is
discussed in section ‘Discussion: effective porosity’. The ef-
fective porosity (n
e2
), per Eq. (20), ranged from a low of 0.1%
in test wells FW220 and FW225 to a high of 1.3% in test well
FW221 (Table 3). The effective porosity, per Eq. (18), which
accounts for the transport of tracer during the injection phase,
was substantially larger than that of Eq. (20), which does not
account for the transport of tracer during the injection phase
(Table 3). The range of effective porosity, per Eq. (18), was
representative of the lower end of those calculated from field-
scale tracer-based studies conducted in granular porous media,
whereas the range per Eq. (20) was representative of those
conducted in fractured porous media (Gelhar et al. 1992).
The effective porosity from Hall et al. (1991), per Eqs. (18)
and (20), were almost identical (6.2 versus 6.1%, respective-
ly), whereas from Istok (2013) they were notably different,
i.e., the expanded equation (n
e1
) yielded substantially higher
effective porosity than the truncated equation (n
e2
;37versus
13%; Table 3).
Uncertainty analysis results
The percent standard errors of the hydraulic gradient (dh/dr),
hydraulic conductivity (K) and drift time (t
d
)weretypically
less than ±2% (Table 4). The percent standard errors of the
remaining parameters, e.g., saturated aquifer thickness (b)and
the times (τ
i
,τ
e
) and volumes (Vi,Ve) at which the center of
mass of bromide was released and recovered, were typically
greater than ±2% but less than ±5% (Table 4). The error-
propagated uncertainty in effective porosity (n
e1
) was less than
±0.5% (Fig. 5). It should be noted that an uncertainly analysis
of effective porosity for the studies by Hall et al. (1991)and
Istok (2013) was not possible due to the lack of available data
on the uncertainty of pumping rates, volumes injected/extract-
ed, etc.
Discussion
Discussion: hydraulic gradient
The range of the near-well hydraulic gradient (−0.020 to
−0.085 m/m) in the test wells was relatively small (within a
single order of magnitude) and representative of the site-wide
average (−0.045 m/m). The spatial variability of the hydraulic
gradient was expected due to the high level of aquifer
Tabl e 1 Hydraulic gradient (dh/dr) and hydraulic conductivity (K)for
tests in this study (FW220–FW225) and for tests from Hall et al. (1991)
and Istok (2013)
Test well/study dh/dr
(m/m)
K
(m/s)
FW220 −0.036 4.1 × 10
−6
FW221 −0.085 5.0 × 10
−6
FW222 −0.033 6.9 × 10
−6
FW223 −0.028 7.0 × 10
−6
FW224 −0.020 1.6 × 10
−5
FW225 −0.063 2.1 × 10
−6
Hall et al. (1991)−0.005 1.4 × 10
−4
Istok (2013)−0.015 2.8 × 10
−5
Hydrogeol J
heterogeneity within Area 2 (Moon et al. 2006; Watson et al.
2004). However, it must be noted that the near-well hydraulic
gradient was not measured directly, i.e., graphically, rather it
was estimated based on a digital elevation model as discussed
in section ‘Hydraulic gradient. Therefore, there is a level of
uncertainty in the near-well hydraulic gradient that must be
recognized; nevertheless, the model-generated values of the
near-well hydraulic gradient are likely much more representa-
tive of the near-well conditions than the graphically deter-
mined values at the site-wide scale.
Discussion: hydraulic conductivity
The steady-state discharge and drawdown conditions among
the test wells were consistent with the methodology of
Robbins et al. (2009) and Aragon-Jose and Robbins (2011).
It should be noted that the Robbins et al. (2009) study was
conducted in a confined aquifer comprised of fine sands,
whereas this study was conducted in an unconfined aquifer
comprised of silty and clayey fill. However, Aragon-Jose and
Robbins (2011) demonstrated the validity of the Robbins et al.
Fig. 4 Push-pull test data for all
six test wells (FW220–FW225)
showing concentration of
bromide (y-axis) versus time
elapsed (x-axis) during the pull
phase of the test. Error bars
represent the analytical
uncertainty (±4%)
Tabl e 2 Results from single-well push-pull tests for this study (FW220–FW225) and from Hall et al. (1991)andIstok(2013)
Test well/study b
(m)
τ
i
(s)
Vi
(m
3
)
t
d
(s)
τ
e
(s)
Ve
(m
3
)
FW220 2.34 1,800 0.010 6,600 3,948 0.014
FW221 2.60 1,740 0.010 7,320 3,984 0.015
FW222 2.31 1,890 0.010 7,200 4,087 0.012
FW223 2.33 1,950 0.010 4,980 3,077 0.011
FW224 2.24 1,410 0.010 6,600 3,349 0.014
FW225 2.42 810 0.010 1,740 3,496 0.006
Hall et al. (1991) 15.24 1,200 0.30 225,600 5,460 20.67
Istok (2013) 2.93 3,000 0.10 108,000 5,220 0.16
Hydrogeol J
(2009) method in an unconfined aquifer comprised of sandy
till and within test wells whose screens crossed the water table;
these hydrogeologic and test well conditions were very similar
to those in this study. Aragon-Jose and Robbins (2011)rec-
ommended that a valid application of the Robbins et al. (2009)
method in unconfined aquifers required minimal drawdown
with respect to the static saturated well screen length. The
drawdown in this study was typically less than 10% of the
static saturated screen length and was within the general range
of the percent drawdown reported by Aragon-Jose and
Robbins (2011;~8–12%).
There is a level of uncertainty in the measured drawdown
within the test wells that must be recognized. The total draw-
down within a well during pumping may due to a number of
components, including: (1) aquifer loss, (2) skin layer loss, (3)
gravel pack loss, (4) well screen loss, (5) up-flow loss in well
interior, (6) partial penetration of well screen, and (7) seepage
face (Houben 2015a,b). As previously discussed in section
‘Study site’, the well screens fully penetrate the unconfined aqui-
fer and were installed without a gravel pack, i.e., the well screens
are in direct contact with the fill materials. The wells were also
routinely developed by mechanical means, i.e., surge and purge,
to limit skin layer loss. The pump intake was set at mid-screen,
i.e., 50% of the screen length, to limit up-flow loss in the well
interior (Houben and Hauschild 2011); therefore, it is likely that
the drawdown during pumping, in order of importance, was
attributed to: (1) the aquifer and (2) a seepage face. A seepage
face would lead to overestimating drawdown during pumping
and underestimating hydraulic conductivity. In turn,
underestimating hydraulic conductivity would lead to
underestimating effective porosity. Nevertheless, the presence
and extent of any seepage face during pumping was not known;
however, by limiting the drawdown to approximately less than
10% of the static saturated screen length, the effects of a seepage
face were likely mitigated.
The range of hydraulic conductivity (2.1 × 10
−6
–
1.6 × 10
−5
m/s) in the test wells was relatively small (within
a single order of magnitude) and within the lower and upper
method detection limits (~10
−8
–10
−4
m/s; Robbins et al.
2009); the range of hydraulic conductivity was also within
that representative of silts and fine sands (Domenico and
Schwartz 1998). However, Watson et al. (2013)reportedthat
the hydraulic conductivity of the fill material, in Area 2 test
wells immediately east of the study site, was approximately
Tabl e 3 Effective porosity calculated from the truncated and expanded
solutions, (20)and(18), respectively, for tests in this study (FW220–
FW225) and for tests from Hall et al. (1991)andIstok(2013), n
e1
from
Eq. (18), n
e2
from Eq. (20)
Test well/study n
e1
(%)
n
e2
(%)
FW220 0.6 0.1
FW221 5.0 1.3
FW222 3.3 0.4
FW223 2.8 0.2
FW224 2.3 0.5
FW225 −0.1 0.1
Hall et al. (1991) 6.2 6.1
Istok (2013)3713
Tabl e 4 Percent standard errors (± %) for input parameters for Eq. (18) for tests in this study (FW220–FW224). Test well FW225 is omitted due to
invalid results
Tes t
well
dh/dr
(± %)
K
(± %)
b
(± %)
τ
i
(± %)
Vi
(± %)
t
d
(± %)
τ
e
(± %)
Ve
(± %)
FW220 1.2 1.6 5.0 2.5 2.5 1.1 3.0 3.9
FW221 1.0 0.7 5.0 2.5 2.5 0.4 1.1 2.1
FW222 1.7 1.7 5.0 2.5 2.5 0.4 1.1 5.4
FW223 1.1 0.6 5.0 2.5 2.5 0.5 1.4 3.6
FW224 0.6 0.8 5.0 2.5 2.5 0.9 2.8 2.8
Fig. 5 Effective porosity (n
e1
)perEq.(18) for tests in this study
(FW220–FW224). Test well FW225 is omitted due to invalid results
(see section ‘Discussion: effective porosity’). Error bars represent the
uncertainty
Hydrogeol J
3.8 × 10
−4
m/s. Therefore, the range of hydraulic conductivity
reported in this study was notably less (up to two orders of
magnitude) than to the value previously reported. It is impor-
tant to note that Watson et al. (2013), and Phillips et al. (2008),
also reported that the fill material was gravelly, whereas no
gravel component is known to exist within the area of this
study site. Therefore, the lack of a gravel component in the
fill material within the study site may explain the lower values
of hydraulic conductivity. In summary, the single-well
pumping test data and analysis suggested that the variability
in the hydraulic conductivity of the fill material was relatively
low and within that representative of silts and fine sands.
Discussion: effective porosity
The breakthrough curve for a non-reactive tracer released
from an instantaneous point source, as it passes a fixed point
of observation, should resemble a bell-shaped curve when its
transport is governed by advection and dispersion during
steady-state groundwater flow in a homogeneous and an iso-
tropic granular porous medium (Baetsle 1969). The break-
through curves for bromide, observed during the pull phase
of the tests, resembled bell-shaped curves that were truncated
at the leading edges (early time) and possibly skewed towards
the following edges (late time). The truncation at the leading
edge indicated that the full spatial extent of the injectate did
not move beyond the test wells during the drift phase. Ideally,
the entire injectate should drift beyond the test wells under
natural-gradient (non-pumping) conditions and then the entire
injectate should be pumped back to the test wells under
forced-gradient (extraction pumping) conditions (Leap and
Kaplan 1988). However, if the injectate drifts too far from
the test wells, it may only partially return during the pull phase
and lead to a low mass recovery of the tracer. Although it may
be tempting to suggest that the drift times in this study were
too short, it must be noted that the average percent mass re-
covery of the tracer (bromide) was far less than 100%
(60 ± 10%, data not shown). Therefore, an increase in the drift
time would have likely resulted in a lower mass recovery of
bromide, and thus a weaker signal for analysis. In addition to
advective mass transport, diffusive mass transport of bromide
from mobile to immobile pore water may partially explain the
low mass recovery of bromide; mobile to immobile diffusive
mass transport is well documented and described at the OR-
IFRC site and at the nearby west Beak Creek Valley site (Luo
et al. 2005; Mayes et al. 2003; McKay et al. 2000; Reedy et al.
1996). The extent of sorption or degradation of bromide
was likely negligible based on previous batch and column
studies which demonstrated that mass recoveries of bro-
mide from OR-IFRC soils and sediments are nearly 100%
under acidic to neutral pH (4.5–7; Hu and Moran 2005;
McCarthy et al. 2000); the pH at the study site ranges from
approximately 6.5–8(Paradisetal.2016).
With regard to the possible skewness of the breakthrough
curves towards the following edge, this suggested that mass
transport mechanisms in addition to advection and dispersion
and/or anisotropy and heterogeneity of the porous media were
present. The likelihood that the fill materials were packed in
the vertical direction suggests that permeable media at the site
were anisotropic. The variability in the magnitude of the hy-
draulic conductivity among the test wells (2.1 × 10
−6
–
1.6 × 10
−5
m/s) also indicates a certain amount of heterogene-
ity. Although a thorough investigation of advection, disper-
sion, and other mass transport mechanisms was not an objec-
tive of this study, the skewness of the breakthrough curves
towards the following edge may be attributed to numerous
small-scale heterogeneities in aquifer hydraulic properties dur-
ing radially convergent flow to a well (Pedretti et al. 2013). In
summary, the breakthrough curves suggested that the injectate
drifted some distance beyond the test wells under natural-
gradient conditions and that an adequate amount of tracer
(bromide) was recovered during the pull phase to accurately
calculate effective porosity using Eqs. (18)and(20).
The effective porosity values from the expanded equation
(0.6–5.0%) were substantially larger than those from the trun-
cated equation (0.1–1.3%) which indicated that the transport of
the tracer during the injection phase was not truly negligible.
From Hall et al. (1991), the effective porosity values were al-
most identical (6.2 versus 6.1%) which indicated that the trans-
port of the tracer during the injection phase was truly negligible.
From Istok (2013), the effective porosity values were notably
different (37 versus 13%), as in the tests presented here, which
indicated that the transport of the tracer during the injection
phase was not truly negligible. Therefore, the agreement, or
lack thereof, of effective porosity from the expanded versus
the truncated equation can clearly identify and quantify the
relative importance of accounting for the transport of tracer
during the injection phase, as shown in the tests presented here
and in those from the literature (Hall et al. 1991;Istok2013).
The negative value of effective porosity (−0.1%), using the
expanded equation for test well FW225, suggested that the
volume of water extracted until the center of mass of the tracer
was recovered (Ve) was less than the volume of water injected
until the center of mass of the tracer was released (Vi); this is
impossible due to the law of conservation of mass. An inspec-
tion of the breakthrough curve of bromide for test well FW225
shows that pumping stopped despite bromide concentrations
greater than 20 mg/L, whereas pumping stopped in the re-
maining five test wells at bromide concentrations less than
10 mg/L. Therefore, it is likely that the total pump-back time
in test well FW225 was too short to return an adequate volume
of water representative of the true center of mass of bromide.
As expected, this error in the application and data analysis of
the push-pull test goes unrecognized when using the truncated
equation, as shown by a positive value of effective porosity
(0.1%) for test well FW225.
Hydrogeol J
This is the first measurement of effective porosity in a fine-
grained fill material that the authors are aware of; hence, there
is no Bexpected^range of values for effective porosity in this
type of material. The effective porosity values from the ex-
panded equation (0.6–5.0%) were more similar to those pre-
viously calculated from field-scale tracer-based studies con-
ducted in unconsolidated, heterogeneous, and fine-grained
granular porous media, whereas those from the truncated
equation (0.1–1.3%) were more similar to those from frac-
tured porous media (Gelhar et al. 1992;Halletal.1991;
Stephens et al. 1998). Based on the hydrogeology of the study
site, i.e., silty and clayeyfill, the effective porosity values from
the expanded equation are likely more accurate than those
from the truncated equation. Moreover, the push-pull tests
by Istok (2013) were conducted in a gravel and sand aquifer,
which also suggests that the effective porosity of 37% from
the expanded equation is likely more accurate than the 13%
from the truncated equation. However, it must be emphasized
that values of effective porosity are dependent on the type of
tracer and the nature of the porous media—for example, in
column experiments by van der Kamp et al. (1996), values of
effective porosity were equal to or far less than the total po-
rosity, depending on the type of solute tracer. van der Kamp
et al. (1996) attributed these findings to phenomena such as:
(1) ion exclusion, (2) enclosed pores, and (3) bound water. At
the nearby west Bear Creek Valley site, McKay et al. (2000)
conducted a multi-well natural-gradient tracer study and dem-
onstrated that the mean arrival times of colloidal tracers were
up to 500 times faster than those reported for solute tracers
from previous tests at the site conducted by Lee et al. (1992).
McKay et al. (2000) attributed these findings to transport of
the colloids through fractures, whereas the solute tracers ex-
perienced substantial diffusion into the immobile pore water
in the fine-grained matrix between fractures. This demon-
strates that different types of tracers can experience different
effective porosities in the same material and implies that even
the same solute tracer may encounter different pore regions
(mobile and immobile pore water) over the duration of a tracer
experiment. Therefore, the magnitude of the effective porosi-
ties calculated in this study may not be truly representative of
the void spaces through which water can flow.
Lastly, and perhaps most importantly, it must be recognized
that both the expanded and truncated equations were theoret-
ically developed for confined aquifers as opposed to uncon-
fined aquifers. However, the only in situ study to experimen-
tally test the validity of the truncated equation was by Hall
et al. (1991). Hall et al. (1991) arrived at similar values of
effective porosity (~6%) from both single-well push-pull and
dual-well natural-gradient tests which were conducted in an
unconfined, heterogeneous, and sandy aquifer. Therefore,
there is clearly a need to: (1) experimentally test the expanded
solution for the confined case, and (2) theoretically develop an
expanded solution for the unconfined case.
Discussion: uncertainty analysis
The error-propagated uncertainty in the calculated values of
effective porosity was relatively small (< ± 0.5%), due in part,
to the careful consideration for the precise determination of
the aquifer properties, e.g., hydraulic gradient, hydraulic con-
ductivity, and saturated aquifer thickness, and the push-pull
test parameters, e.g., the times and volumes at which the cen-
ter of mass of bromide was released and recovered. However,
the uncertainty analysis failed to capture the effects of: (1) the
presence and extent of seepage face during extraction
pumping, and (2) applying an analytical solution developed
for a confined aquifer to an unconfined aquifer. The presence
and extent of a seepage face could have been determined using
a down-well device with video capability during extraction
pumping. However, this was not possible due to the small
diameter (1.9 cm) of the wells and the presence of down-
well tubing (0.64 cm diameter) which limited the physical
space to deploy such a device. The effects of applying an
analytical solution developed for a confined aquifer to the
unconfined aquifer in this study was not known; however, as
previously discussed in section ‘Discussion: effective porosi-
ty’, Hall et al. (1991) demonstrated the validity of the truncat-
ed analytical solution, developed for a confined aquifer, as
applied to an unconfined, heterogeneous, and sandy aquifer.
Conclusions
The conclusions of this study are as follows: (1) the analytical
solution to describe the displacement of the center of mass of a
tracer during a push-pull test can be expanded to account for
displacement during the injection phase, (2) the transport of a
tracer during the injection phase of a push-pull test may not be
truly negligible, (3) the failure to account for displacement during
the injection phase may lead to a substantial underestimation of
the magnitude of effective porosity, (4) single-well push-pull tests
can be readily applied to multiple wells within a study site to
assess the spatial variability of effective porosity, and (5) the
error-propagated uncertainty in the value of effective porosity
can be mitigated to a reasonable level by careful consideration
for the precise determination of the aquifer properties and the
push-pull test parameters. Finally, it must be recognized that there
is a need to theoretically develop and experimentally test the
expanded solution presented here for the case of an unconfined
aquiferandfordifferenttypesofaquifermaterials.
Acknowledgements This material by ENIGMA –Ecosystems and
Networks Integrated with Genes and Molecular Assemblies (http://
enigma.lbl.gov), a Scientific Focus Area Program at Lawrence Berkeley
National Laboratory –is based upon work supported by the U.S.
Department of Energy, Office of Science, Office of Biological &
Environmental Research, under contract number DE-AC02-
05CH11231. The authors would like to thank Tonia Mehlhorn from the
Hydrogeol J
Oak Ridge National Laboratory and Julian Fortney from the University of
Tennessee Knoxville for their assistance and helpful suggestions during
the field-based portion of this study. The authors would also like to thank
Eriko Gordon and Emma Dixon from the University of Tennessee
Knoxville for their assistance with the final production of the manuscript.
Finally, the authors would like to thank the two anonymous reviewers for
their insightful comments and helpful suggestions which greatly im-
proved the quality of the manuscript.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://
creativecommons.org/licenses/by/4.0/), which permits unrestricted use,
distribution, and reproduction in any medium, provided you give
appropriate credit to the original author(s) and the source, provide a link
to the Creative Commons license, and indicate if changes were made.
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