LA-UR-22-32970
Accepted Manuscript
Single‐Well Injection‐Drift Test to Estimate Groundwater
Velocity
Paradis, Charles
Van Ee, Nathan
Hoss, Kendyl
Meurer, Cullen
Tigar, Aaron
Reimus, Paul William
Johnson, Raymond
Provided by the author(s) and the Los Alamos National Laboratory (2023-02-16).
To be published in: Groundwater
DOI to publisher's version: 10.1111/gwat.13184
Permalink to record:
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Methods Note\
Single-well injection-drift test to estimate groundwater velocity
Charles Paradis1,*, Nathan Van Ee2, Kendyl Hoss1, Cullen Meurer1, Aaron Tigar3, Paul Reimus4,
Raymond Johnson3
1
Department of Geosciences, University of Wisconsin at Milwaukee, Milwaukee, Wisconsin
2
School of Freshwater Sciences, University of Wisconsin at Milwaukee, Milwaukee, Wisconsin
3
RSI EnTech, LLC, United States Department of Energy Office of Legacy Management Support
Contractor, Grand Junction, Colorado
4
Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, New Mexico
*Corresponding author
E-mail: paradisc@uwm.edu
Conflict-of-interest: none
Financial disclosure: none
Journal: Groundwater
Article Type: Methods Note
This is the author manuscript accepted for publication and has undergone full peer review but has
not been through the copyediting, typesetting, pagination and proofreading process, which may
lead to differences between this version and the Version of Record. Please cite this article as doi:
10.1111/gwat.13184
This article is protected by copyright. All rights reserved.
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Paradis Charles J (Orcid ID: 0000-0002-1072-3988)
A simple algebraic equation is presented here to estimate the magnitude of groundwater
velocity based on data from a single-well injection-drift test thereby eliminating the timeconsuming and costly extraction phase. A volume of tracer-amended water was injected by
forced-gradient into a single well followed by monitoring of the conservative solute tracers under
natural-gradient conditions as their up-gradient portions drifted back through the well. The
breakthrough curve data from the single well during the drift phase was analyzed to determine
the mean travel times of the tracers. The estimated mean up-gradient travel distance back
through the single well and the mean travel times of the tracers were used in a simple algebraic
equation to estimate groundwater velocity. The groundwater velocity based on the single-well
injection-drift test was estimated to be approximately 0.64 feet per day. Two transects of
observation wells were used to monitor the natural-gradient tracer transport down-gradient of the
injection well. The one-dimensional, or dual-well, transport of the tracer from the injection well
to the nearest down-gradient observation well indicated that the groundwater velocity was 0.55
feet per day. The two-dimensional, or multi-well, transport of the center of mass of the tracers
indicated that the groundwater velocity was 0.60 feet per day; the dual- and multi-well results
were in excellent agreement with those from the single-well and validated the simple algebraic
equation. The new single-well method presented here is relatively simple, rapid, and does not
require an extraction phase.
2
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Abstract
A typical single-well injection-drift-extraction test involves the following steps: 1)
forced-gradient injection of a conservative tracer into a single well, 2) natural-gradient drift of
the tracer down-gradient from the single well, and 3) forced-gradient extraction of the tracer
from the single well (Fig. 1). During the injection phase, the tracer is transported radially about
the well due to the forced-gradient injection (Fig. 1). During the drift phase, the tracer is
transported horizontally away from the well due to the natural-gradient drift (Fig. 1). During the
extraction phase, the tracer is transported back to the well due to the forced-gradient extraction
(Fig. 1). Conceptually, the rate at which the up-gradient portion of the injection fluid (Fig. 1)
drifts back through the single well under natural-gradient conditions is proportional to the
groundwater velocity (Fig. 2). Therefore, it is conceivable that the drift-phase breakthrough
curve could yield valuable insight into the magnitude of groundwater velocity without the need
to conduct the extraction phase. Eliminating the extraction phase would be beneficial by
decreasing the time needed to conduct the test and by avoiding potential costs of personnel,
equipment, supplies, and disposal of the extraction fluid if the test was conducted at a
contaminated site. The study described herein hypothesized that a simple algebraic equation can
adequately estimate the magnitude of groundwater velocity based on the breakthrough curve data
from a single-well injection-drift test.
The simple algebraic equation to estimate groundwater velocity from a typical singlewell injection-drift-extraction test was first described by Borowczyk et al. (1966) as:
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Introduction
v = groundwater velocity [L/T]
Qe = extraction pumping rate [L3/T]
te = time elapsed to extract one half of tracer mass [T]
b = saturated aquifer thickness [L]
n = aquifer effective porosity [L3/L3]
td = time elapsed during drift phase [T]
Equation (1) has since been expanded twice, first by Leap and Kaplan (1988) to include
the effect of the natural-gradient groundwater velocity during the extraction phase, and then by
Paradis et al. (2019a) to include the same effect during the injection phase. The current simple
algebraic equation of groundwater velocity is:
where:
ti = time elapsed during injection phase [T]
Hall et al. (1991) noted that if the magnitude of the local hydraulic conductivity and
hydraulic gradient are known, the equations by Leap and Kaplan (1988), and subsequently
Paradis et al. (2019a), could be rewritten without the effective porosity term as:
where:
4
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where:
i = local hydraulic gradient [L/L]
More recently, Wang et al. (2021) compared the results of a finite-element numerical
model to those from the simple algebraic equations by Hall et al. (1991) and Paradis et al.
(2019a) and found that errors in the simple algebraic equations increased with decreased
groundwater velocity and increased dispersivity. These recent findings further indicated the
importance of the following assumptions that underlie the simple algebraic equations: 1)
advection is the dominant transport mechanism, 2) groundwater velocity is steady state, and 3)
the aquifer is homogeneous, isotropic, and confined with a constant thickness.
The typical single-well injection-drift-extraction test may be modified by eliminating the
extraction phase. This modified test is a single-well injection-drift test. This test is typically
used to estimate reaction rates within the vicinity of the well by plotting the dilution-adjusted
breakthrough curve of a reactive tracer as the up-gradient portion of the injection fluid drifts
back through the single well under natural-gradient conditions; this type of test is described in
detail by Istok (2013). The methodology to generate dilution-adjusted breakthrough curves was
subsequently improved by Paradis et al. (2019b). However, the data from a single-well
injection-drift test has not been coupled to a simple algebraic equation to estimate groundwater
velocity.
The objective of this study was to determine if the magnitude of groundwater velocity
could be estimated based on the breakthrough curve data from a single-well injection-drift test
using a simple algebraic equation, thus eliminating the time-consuming and costly extraction
phase. To this end, a field tracer study was conducted and the groundwater velocity estimated
5
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K = local hydraulic conductivity [L/T]
multi-well tests.
Simple Algebraic Equation
The mean up-gradient travel distance of a conservative tracer during the injection phase
can be estimated as:
where:
r̄ = mean up-gradient travel distance [L]
Vi = injection volume [L3]
b = aquifer saturated thickness [L]
n = aquifer effective porosity [L3/L3]
Equation (4) assumes that the transport of a conservative tracer is due to divergent and
cylindrical advection during the forced-gradient injection phase in a confined, homogeneous, and
isotropic aquifer and that both the natural-gradient and dispersion are negligible.
The mean travel time of the conservative tracer during the drift phase can approximated
using the trapezoidal rule from the breakthrough curve data as given by:
where:
Ci = tracer concentration at the ith time step [M/L3]
ti = time at the ith time step [T]
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from the single-well injection-drift test was compared to results from natural-gradient dual- and
Equation (5) assumes that it describes the transport of a conservative tracer as its upgradient portion drifts back though the injection well during the natural-gradient drift phase.
Therefore, the magnitude of the groundwater velocity can be estimated as:
Equation (6) is a simple algebraic equation to estimate the magnitude of groundwater velocity
based on the data from a single-well injection-drift test. Equation (6) was informed and inspired
by the utility and simplicity of Equation (1) that was developed by Borowczyk et al. (1966) to
estimate the magnitude of groundwater velocity based on data from a typical single-well
injection-drift-extraction test. Conceptually, Equation (6) should adequately estimate the
magnitude of groundwater velocity based on the data from a single-well injection-drift test.
Field Tracer Study
The field tracer study was conducted at the United States Department of Energy Office of
Legacy Management Processing Site in Riverton, Wyoming. A detailed description of the site
was provided by Dam et al. (2015). In brief, the site is underlain by a shallow and unconfined
aquifer that is comprised of highly permeable sands and gravels and is contaminated with
uranium. The depth to the water table is typically less than 10 feet below ground surface and the
direction of the hydraulic gradient is generally to the southeast. A series of temporary wells (1.5inch diameter PVC) were installed by direct push and screened (5-foot interval) at the water table
(Fig. 3). The wells were installed as three transects oriented perpendicular to the general
direction of the hydraulic gradient (Fig. 3). A 100-gallon solution of tracer-amended (500 mg/L
iodide and 500 mg/L pentafluorobenzoate (PFB) as NaI and KPFB, respectively) water was
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t̄ = mean travel time [T]
hours. Iodide and PFB were used as relatively high- and low-diffusive conservative solute
tracers, respectively, to investigate the potential for dual-porosity solute transport, i.e., matrix
diffusion (Paradis et al. 2020). For example, if matrix diffusion were to occur it would likely
result in a relatively lower peak concentration and longer mean travel time of the high-diffusivity
tracer, iodide, as compared to the low-diffusivity tracer, PFB. The injection well fluid was
continuously recirculated in a closed loop to ensure a uniform vertical distribution of the tracers.
The change in hydraulic head of the injection well was measured periodically using an electronic
water level meter and indicated a slight, but stable, increase of a few centimeters during injection
followed by a rapid return to static conditions immediately after injection. The injection phase
was followed by periodic groundwater sampling of the tracers in the injection well and in the
down-gradient transects of observation wells (Fig. 3) under natural-gradient conditions for 18
days. Groundwater samples were collected under low-flow and low-volume conditions as to not
disturb the natural gradient. Groundwater samples were filtered (0.45 μm) and preserved (4°C)
in the field and shipped to RSI EnTech, LLC (Grand Junction, Colorado) and Los Alamos
National Laboratory (Los Alamos, New Mexico) for quantification of iodide by ion
chromatography and PFB by high-performance liquid chromatography, respectively; iodide was
also measured in the field using an ion-specific electrode for rapid results to better track the
transport of the injection fluid back through the injection well and down-gradient in the transects
of observation wells. The two-dimensional center of mass of the tracers down-gradient of the
injection well was quantified as the centroid of their plan-view concentrations at snap shots in
time as:
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injected into well 1001 (Fig. 3) using a peristaltic pump at a constant rate for approximately 7.5
x̅ = center of tracer mass in the x dimension [L]
y̅ = center of tracer mass in the y dimension [L]
Ci = tracer concentration at the ith coordinate step [M/L3]
xi = tracer location in the x dimension at the ith coordinate step [L]
yi = tracer location in the y dimension at the ith coordinate step [L]
Single-well Results
The breakthrough curve data from the injection well (well 1001 in Fig. 3, data in Table
S1 in Supporting Information) during the natural-gradient drift phase resembled that of one half
of a bell-shaped curve (Fig. 4) as expected based on the conceptual model (Fig. 2). Both iodide
and PFB tracers showed relatively good overlap with nearly identical mean travel times of 1.18
and 1.17 days, respectively (Fig. 4), and suggested that matrix diffusion was negligible. The
injection volume (100 gallons or 13.37 cubic feet), aquifer saturated thickness (5 feet), and
approximated effective porosity (0.38 based on permeable sands and gravels as described by
Dam et al. (2015)) was entered into Equation (4) to estimate the mean up-gradient travel distance
of the tracers during the injection phase at approximately 0.75 feet. The mean up-gradient travel
distance and mean tracer travel times were entered into Equation (6) to estimate the magnitude of
groundwater velocity at approximately 0.64 feet per day.
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and
The breakthrough curve data from the nearest down-gradient observation well (well 1005
in Fig. 3, data in Table S2 in Supporting Information) resembled that of a bell-shaped curve with
a notable skew to the right-hand side, i.e., tailing towards later travel times (Fig. 5). Both iodide
and PFB tracers showed relatively good overlap with nearly identical mean travel times of 5.61
and 5.60 days, respectively (Fig. 5), and further suggested that matrix diffusion was negligible.
The distance between injection well 1001 and observation well 1005 was 3.09 feet (Fig. 3) and
resulted in a dual-well estimate of the magnitude of groundwater velocity at approximately 0.55
feet per day.
Multi-well Results
The two-dimensional tracking of the center of mass of the tracers down-gradient of the
injection well (wells in Fig. 3, data in Tables S3 and S4 in Supporting Information) showed a
nearly linear path to the southeast over the course of 10 days (Fig. 6). Both iodide and PFB
tracers showed relatively good overlap (Fig. 6) and even further suggested that matrix diffusion
was negligible. The distance of the center of mass of the tracers with respect to the injection
well versus time elapsed also showed a nearly linear relationship and indicated that the multiwell magnitude of groundwater velocity was approximately 0.60 to 0.61 feet per day (Fig. 7).
The estimated groundwater velocity from the single-well data (≈0.6 ft/day) as compared to the
dual-, and multi-well data (≈0.5 to 0.6 ft/day) showed excellent agreement.
It is important to note that only the multi-well data can yield insights into the direction of
groundwater velocity as evident by the two-dimensional tracking of the center of mass of the
tracers that verified transport was indeed along the direction of the natural hydraulic gradient and
nearly on-center with the down-gradient observation well 1005 (Fig. 6). That is, neither the
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Dual-well Results
of the natural hydraulic gradient must be assumed to represent the direction of groundwater
velocity.
Conclusions
The results of this study supported the hypothesis that a simple algebraic equation can
adequately estimate the magnitude of groundwater velocity based on the breakthrough curve data
from a single-well injection-drift test. The simple algebraic equation, when coupled to data from
single-well injection-drift test, resulted in an estimated groundwater velocity that was in
excellent agreement with the true groundwater velocity based on both dual- and multi-well
natural-gradient tracer tests. This straight-forward method is more advantageous than a typical
single-well injection-drift-extraction test because it eliminates the time-consuming and costly
extraction phase. Conversely, eliminating the extraction phase prevents the ability to perform a
proper mass balance of the injected versus the extracted tracer masses. Therefore, it must be
assumed that the full portion of the up-gradient injection fluid drifts back through the single well
under natural-gradient conditions.
Acknowledgments
The authors would like to thank the editors and peer reviewers of the original and revised
versions of this manuscript for their constructive comments that greatly improved the clarity and
contribution of this research. The authors would also like to thank the United States Department
of Energy Office of Legacy Management for their broad support of applied groundwater
research.
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single- or dual-well data can verify the direction of groundwater velocity and thus the direction
Additional Supporting Information for this article may be found in an online document
that contains tables S1 through S4 as referenced in the text above. Supporting Information is
generally not peer reviewed.
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Supporting Information
Borowczyk, M., J. Mairhofer, and A. Zuber. 1966. Single-well pulse technique. Proceedings,
Symposium on Isotopes in Hydrology, Vienna.
Dam, W. L., S. Campbell, R. H. Johnson, B. B. Looney, M. E. Denham, C. A. Eddy-Dilek, and
S. J. Babits. 2015. Refining the site conceptual model at a former uranium mill site in
Riverton, Wyoming, USA. Environmental Earth Sciences 74, no. 10: 7255-7265. Doi:
10.1007/s12665-015-4706-y.
Hall, S. H., S. P. Luttrell, and W. E. Cronin. 1991. A method for estimating effective porosity
and groundwater velocity. Ground Water 29, no. 2: 171-174. Doi: 10.1111/j.17456584.1991.tb00506.x.
Istok, J. D. 2013. Push-Pull Tests for Site Characterization. Lecture Notes in Earth System
Sciences. Springer.
Leap, D. I., and P. G. Kaplan. 1988. A single-well tracing method for estimating regional
advective velocity in a confined aquifer: theory and preliminary laboratory verification.
Water Resources Research 24, no. 7: 993-998. Doi: 10.1029/WR024i007p00993.
Paradis, C. J., L. D. McKay, E. Perfect, J. D. Istok, and T. C. Hazen. 2019a. Correction: Pushpull tests for estimating effective porosity: expanded analytical solution and in situ
application. Hydrogeology Journal 27, no. 1: 437-439. Doi: 10.1007/s10040-018-1879y.
Paradis, C. J., E. R. Dixon, L. M. Lui, A. P. Arkin, J. C. Parker, J. D. Istok, E. Perfect, L. D.
McKay, and T. C. Hazen. 2019b. Improved Method for Estimating Reaction Rates
During Push-Pull Tests. Groundwater 57, no. 2: 292-302. Doi: 10.1111/gwat.12770.
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References
Field experiments of surface water to groundwater recharge to characterize the mobility
of uranium and vanadium at a former mill tailing site. Journal of Contaminant
Hydrology 229. Doi: 10.1016/j.jconhyd.2019.103581.
Wang, Q. R., A. H. Jin, H. B. Zhan, Y. Chen, W. G. Shi, H. Liu, and Y. Wang. 2021. Revisiting
simplified model of a single-well push-pull test for estimating regional flow velocity.
Journal of Hydrology 601. Doi: 10.1016/j.jhydrol.2021.126711.
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Paradis, C. J., R. H. Johnson, A. D. Tigar, K. B. Sauer, O. C. Marina, and P. W. Reimus. 2020.
Fig. 1 Conceptual model of single-well injection-drift-extraction test showing relative
concentration of conservative tracer (C/Co) on y axis and distance relative to injection well on x
axis at end of forced-gradient injection phase, end of natural-gradient drift phase, and middle of
forced-gradient extraction phase, direction of natural-gradient groundwater velocity is from left
to right, up-gradient portion of tracer shown along negative values of distance relative to
injection well
Fig. 2 Conceptual model of drift phase of single-well injection-drift test showing relative
concentration of conservative tracer (C/Co) on y axis and time elapsed since beginning of drift
phase/end of injection phase on x axis (drift-phase breakthrough curves) as up-gradient portion
of injection fluid (Fig. 1) drifts back through the well under natural-gradient conditions for
various magnitudes (low, medium, and high) of groundwater velocity
Fig. 3 Plan-view map of well network showing injection well (1001), nearest down-gradient
observation well (1005), and surrounding transects of observation wells orientated nearly
perpendicular to direction of natural hydraulic gradient
Fig. 4 Breakthrough curve data from injection well (well 1001) during natural-gradient drift
phase for relative concentration (C/Co) of conservative tracers, iodide and pentafluorobenzoate
(PFB), versus time and mean tracer travel time used to estimate magnitude of groundwater
velocity (v)
Fig. 5 Breakthrough curve data from nearest down-gradient observation well (well 1005) postinjection under natural-gradient conditions for relative concentration (C/Co) of conservative
tracers, iodide and pentafluorobenzoate (PFB), versus time and mean tracer travel time used to
estimate magnitude of groundwater velocity (v)
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Figure Captions
and pentafluorobenzoate (PFB), for 10 days post-injection under natural-gradient conditions, the
injection well was well 1001
Fig. 7 Distance of center of mass (COM) of conservative tracers, iodide and pentafluorobenzoate
(PFB), versus time post-injection under natural-gradient conditions with respect to injection well
(1001), regression lines are linear fits (LinFit) of data and their slopes represent average linear
groundwater velocity (v)
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Fig. 6 Plan-view map of two-dimensional center of mass (COM) of conservative tracers, iodide
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