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Experimental evaluation of the applicability of phase, amplitude,
and combined methods to determine water flux and thermal diffu-
sivity from temperature time series using VFLUX 2
Dylan J. Irvine, Laura K. Lautz, Martin A. Briggs, Ryan P. Gordon, Jeffrey M. McKenzie
Accepted at Journal of Hydrology. Accepted on 21st October 2015
Article will be available at: http://www.journals.elsevier.com/journal-of-hydrology
The VFLUX 2 program is available at: http://hydrology.syr.edu/vflux.html
Abstract
Vertical fluid exchange between surface water and groundwater can be estimated using di-
urnal signals from temperature time series methods based on amplitude ratios (Ar), phase
shifts (∆φ), or combined use of both (Ar∆φ). The Ar, ∆φ, and Ar∆φmethods are typi-
cally applied in conditions where one or more of their underlying assumptions are violated,
and the reliability of the various methods in response to non-ideal conditions is unclear.
Additionally, Ar∆φmethods offer the ability to estimate thermal diffusivity (κe) without
assuming any thermal parameters, although the value of such output has not been broadly
tested. The Ar, ∆φ, and Ar∆φmethods are tested under non-steady, 1D flows in sand
column experiments, and multi-dimensional flows in heterogeneous media in numerical mod-
eling experiments. Results show that, in non-steady flow conditions, estimated κevalues
outside of a plausible range for streambed materials (0.028 to 0.180 m2d−1) coincide with
time periods with erroneous flux estimates. In heterogeneous media, sudden changes of e
with depth also coincide with erroneous flux estimates. When (known) fluxes are variable
in time, poor identification of ∆φleads to poor flux estimates from ∆φand Ar∆φmethods.
However, when fluxes are steady, or near zero, Ar∆φmethods provide the most accurate
flux estimates. This comparison of Ar, ∆φand Ar∆φmethods under non-ideal conditions
provides guidance on their use. In this study, Ar∆φmethods have been coded into a new
version of VFLUX, allowing users easy access to recent advances in heat tracing.
1. Introduction
Anderson (2005), in her review of heat as a groundwater tracer, called for groundwater tem-
perature to be measured and interpreted more widely. Since the Anderson (2005), review,
the study of surface water-groundwater interactions has been an area that has seen a large
increase in the use of water temperature data. Following on from seminal work by Stallman
(1965) and important steps from theory to application by Lapham (1989) and Goto et al.
(2005), diurnal temperature signals in particular have become widely used. Temperature
time series approaches, based on variations in diurnal signals, have been used to investigate
fluid exchange between streams and aquifers in a range of settings (Lapham 1989; Stone-
strom and Constantz 2003; Hatch et al., 2006; Rau et al., 2010; Briggs et al., 2013; Briggs et
al., 2014; McCallum et al., 2014; Birkel et al., 2015), to determine changes in streambed hy-
draulic conductivity with time (Hatch et al., 2010), and to produce spatial maps of exchange
fluxes (Lautz and Ribaudo, 2012; Gordon et al., 2013; Irvine and Lautz, 2015). The use
of heat as a tracer to determine the fluid exchange between surface water and groundwater
is not restricted to stream settings. For example, the use of heat as a tracer has also been
applied in wetlands (e.g. Hunt et al. 1996; Bravo et al. 2002), and in lakes (e.g. Krabbenhoft
1
and Babiarz 1998). Similarly, the use of temperature signals is not restricted to analysis of
the diurnal signal, as annual signals contain similar fluid flux information but are expected
to propagate to deeper depths (e.g. Lapham, 1989), and thermal profiles without periodic
signals can be numerically simulated (e.g. Koch et al. 2015).
Temperature time series can be collected at discrete points along a vertical profile (e.g.
Hatch et al., 2006), or near-continuous temperature with depth profiles using fiber-optic
high-resolution distributed temperature sensing (e.g. Vogt et al., 2010; Briggs et al., 2012).
While temperature time series are straightforward to collect in streambed sediments, the cal-
culation of fluxes can involve complex steps, including the isolation of the diurnal component
of the temperature signal from other signals and stochastic variation. There are also multiple
analytical solutions based on diurnal signals presented in the literature available to solve for
fluid flux (e.g. Hatch et al., 2006; Keery et al., 2007; McCallum et al., 2012; Luce et al., 2013).
The VFLUX MATLAB routines (Gordon et al., 2012, http://hydrology.syr.edu/vflux.html)
address many of the complexities in processing temperature time series data by integrat-
ing data management, signal processing, fluid flux modeling, and post-analysis based on
straight-forward user input. VFLUX (version 1.2.5 and earlier versions, from herein referred
to as VFLUX 1) is an all-in-one modelling system that synchronizes multiple temperature
time series, isolates and extracts the amplitude and phase of the signals for the period of
interest using advanced signal processing techniques, calculates fluxes at high spatial and
temporal resolution, and is able to quantify the influence of uncertainty in thermal parame-
ters. VFLUX 1 includes analytical solutions by Hatch et al. (2006) and Keery et al. (2007),
both of which provide approaches to calculate time series of flux using either the amplitude
ratio (Ar) or phase shift (∆φ) of diurnal signals between two temperature time series.
The assumptions of a number of 1D analytical methods (e.g. Lapham, 1989; Hatch et al.
2006; Keery et al. 2007) include that fluid flow is vertical and one-dimensional, that the
temperature signal is sinusoidal, that there is no mean thermal gradient in the streambed
with depth, and that streambed properties are homogeneous. The influence of non-ideal field
conditions (where one or more of the abovementioned assumptions are not met) on the use of
the Hatch et al. (2006) methods have been widely investigated. Lautz (2010), Roshan et al.
(2012), and Cuthbert and Mackay (2013) investigated the influence of multi-dimensional flow
fields on flux estimates in homogeneous porous media. Lautz (2010) identified that multi-
dimensional flow was a greater source of error compared to the presence of thermal gradients
in the streambed or non-sinusoidal temperature signals. Roshan et al. (2012) showed that
errors in flux estimates can be reduced by locating sensors as close to the streambed surface
as possible, and suggested that multiple arrays of sensors should be deployed horizontally to
evaluate the presence of horizontal flow. Cuthbert and Mackay (2013) suggested that dif-
ferences in fluxes estimated on the basis of Aror ∆φmay indicate the degree of complexity
in the flow field. Lautz (2010) showed that the presence of a mean thermal gradient in the
streambed was not a significant source of error in downwelling zones; this was recently con-
firmed for upwelling zones by Briggs et al. (2014). Shanafield et al. (2011) and Soto-Lopez
et al. (2011) investigated the influence of uncertainties in thermal parameters and sensor
precision on flux estimates. Shanafield et al. (2011) identified that uncertainties in thermal
parameters and noise in temperature measurements (due to sensor accuracy) were particu-
2
larly an issue for estimating upwelling fluxes. Soto-Lopez et al. (2011) identified that sensor
resolution has a greater influence on the accuracy of flux estimates based on Arcompared to
∆φ. Irvine et al. (2015) investigated the effect of streambed heterogeneity on flux estimates
under downwelling conditions, and showed that on average, the Armethods produced lower
errors in flux estimates relative to ∆φin heterogeneous media.
Clearly there is somewhat conflicting evidence regarding which diurnal signal-based method
may be most reliable for field conditions, which is complicated by the fact that fluxes esti-
mated using Arand ∆φalone rarely agree (e.g. Hatch et al., 2006; Lautz, 2012; Briggs et al.,
2012; Roshan et al., 2012; Cuthbert and Mackay, 2013; Irvine et al., 2015). More recently
presented analytical solutions to the 1D heat transport equations by McCallum et al. (2012),
and Luce et al. (2013) simultaneously use both Arand ∆φto calculate flux, thereby elimi-
nating the possibility of obtaining different flux estimates when using Arand ∆φin isolation.
Benefits of the McCallum et al. (2012) and Luce et al. (2013) combined methods (Ar∆φ)
include the fact that no estimate of thermal conductivity is required for the estimation of
flux (thereby reducing the uncertainty due to uncertainty in thermal parameters), and both
methods can also produce an estimated time series of thermal diffusivity (κe, m2s−1). The
Luce et al. (2013) methods can also be used to calcualte vertical sediment thickness. Beyond
their intrinsic utility to heat transport modeling, κeestimates shows promise as an indica-
tor of poor flux estimates because it is a parameter expected to be relatively homogeneous
in time and space; large spatial or temporal changes in κefrom the Ar∆φmethods could
suggest errors in flux estimates. Using field data, McCallum et al. (2012, in their figure 4)
suggest that variations in e in time can be used to identify time periods where flux estimates
are unreliable.
The influence of non-ideal field conditions (e.g. 3D heterogeneity in sediments and flux) on
Ar∆φmodel-based flux estimates has not been investigated in great detail. Irvine et al.
(2015) compared the use of the Hatch et al. (2006), McCallum et al. (2012), and Luce et
al. (2013) methods in 2D numerical simulations with heterogeneous streambed properties.
The Irvine et al. (2015) study was restricted to estimates of downwelling, and they did not
explore the utility of κeoutput from the McCallum et al. (2012) and Luce et al. (2013)
methods. They showed that (for downwelling conditions), on average, the Ar∆φand Ar
methods produce fluxes with smaller errors than the ∆φmethod. The influence of non-
sinusoidal temperature signals, uncertainty in thermal parameters, and non-steady flows on
Ar∆φflux estimates was investigated by Rau et al. (2015). Using synthetic 1D data sets,
Rau et al. (2015) showed that errors in flux and κeestimates during non-steady flow periods
can be attributed to limitations in filtering methods.
It is important to note that all field sites are ’non-ideal’, in that one or more of the above-
mentioned assumptions of 1D analytical methods will likely not be met. It is therefore
important to identify the impact of these non-ideal conditions on flux estimates from using
known (numerically simulated and controlled laboratory) data. For this study, we investigate
how flux estimates are influenced by non-sinusoidal temperature signals, non-steady flows,
multi-dimensional flows, streambed heterogeneity and uncertainty in thermal parameters.
The Hatch et al. (2006), Keery et al. (2007), McCallum et al. (2012) and Luce et al. (2013)
methods were used to determine fluxes from temperature time series. All data are assessed
3
with an updated version of VFLUX (version 2.0.0, from herein referred to as VFLUX 2)
which includes the McCallum et al. (2012) and Luce et al. (2013) Ar∆φmethods to deter-
mine time series of flux, κeand sediment thickness.
2. Methods
The investigation is divided into two sets of experiments. First, temperature time series
from the Lautz (2012) sand column experiments are re-examined. The analysis of the Lautz
(2012) experiments allows for the determination of the influence of a non-sinusoidal tem-
perature boundary condition, and non-steady flow rates on known flux estimates from real
temperature data. Second, the influence of multi-dimensional flows and the influence of
streambed heterogeneity on flux estimates are demonstrated using a 3D numerical model
simulation.
In the sections below, first we present the equations of heat and fluid flow (Section 2.1),
followed by the analytical solutions included in VFLUX 1 (Section 2.2) and VFLUX 2 (Sec-
tion 2.3). Finally, we outline the experimental set up of the sand column (Section 2.4) and
numerical model simulation (Section 2.5).
2.1. 1D Equation of heat and fluid flow
The Hatch et al. (2006), Keery et al. (2007), McCallum et al. (2012) and Luce et al.
(2013) methods are analytical solutions to the one-dimensional heat transport equation (e.g.
Stallman, 1965):
∂T
∂t =κe
∂2T
∂z2−qCw
C
∂T
∂z ,(1)
where Tis temperature (◦C), tis time (s), qis the fluid flux (m s−1) and Cwis volumetric
heat capcity of water (J m−3◦C−1), Cis volumetric heat capacity of saturated sediment (J
m−3◦C−1), and zis depth (m). κeis calculated by:
κe=λ0
C+β Cw|q|
C!=λn
wλ1−n
s
ncwρw+ (1 −n)csρs
+β cwρw|q|
ncwρw+ (1 −n)csρs!(2)
where λ0,λwand λsare the thermal conductivities of the bulk saturated medium, water
and the solid phase (W m−1◦C−1), βis thermal dispersivity (m), nis porosity (-), cwand
csare specific heat capacities of water and the solid phase (J kg−1◦C−1), and ρwand ρsare
densities of water and the solid phase (kg m−3). Rau et al. (2012) show that the influence
of βis negligible over the range of q(e.g. ±3.0 m d−1), and sensor spacing (∆z, e.g. up
to 1.0 m) considered in a groundwater-surface water interaction context, and so the second
term on the right hand side of Eqn. 2 is often neglected.
Upper and lower bounds of κethat could realistically be expected from streambed materials
can be determined from the range of parameter values that make up κe. Typical ranges of
parameter values are shown in Table 1.
Table 1: Parameters in calculation of limits on κevalues
4
Parameter, symbol (unit) Parameter
range
Source, comment
Porosity, n, (-) 0.2 - 0.5 Fitts (2013)
Specific heat capacity of solid,
cs, (J kg−1◦C−1)
731 - 1078 Waples and Waples (2004), range for
quartzite
Specific heat capacity of water,
cw, (J kg−1◦C−1)
4178 - 4216 Calculated for 1 - 40 ◦C at 1 atm using
Excel plugin (Wagner et al. 2000)
Thermal conductivity of solid,
λs(W m−1◦C−1)
2.18 - 3.39 Hopmans et al. (2002), Rau et al. (2012),
Stonestrom and Constantz (2003)
Thermal conductivity of water,
λw(W m−1◦C−1)
0.56 - 0.63 Calculated for 1 - 40 ◦C at 1 atm using
Excel plugin (Wagner et al. 2000)
Density of solid, ρs(kg m−3) 2625 - 2680 Stonestrom and Constantz (2003),
Waples and Waples (2004)
Density of water, ρw(kg m−3) 992 - 1000 Calculated for 1 - 40 ◦C at 1 atm using
Excel plugin (Wagner et al. 2000)
Thermal parameters (including n) of porous media can vary outside of the ranges presented
in Table 1. For example, Conant (2004) measured nup to 0.72, or cscan exceed 1100 J kg−1
◦C−1for some minerals (Waples and Waples, 2004), and organic-rich sediments may contrast
strongly with mineral grains (Stonestrom and Constantz 2003). However, the parameters in
Table 1 provide a reasonable parameter set to identify the limits of e that could be expected
at a field site. Other ranges of thermal properties can be found in Lapham (1989) or Stone-
strom and Constantz (2003), although it should be noted that Lapham (1989), presents bulk
thermal properties (i.e. λ0not λs), and thermal properties of the solid and liquid phases are
required for heat capacities (i.e. Cs=ρscs,Cw=ρwcw) are required for the VFLUX inputs,
with Ccalculated internally (i.e. C= (1 −n)Cs+nCw). The range of possible κevalues
calculated from Table 1 is 0.028 - 0.180 m2d−1(calculated at 20 ◦C). This range could be
reduced by measuring any of the parameters included in κe.
The methods described in Sections 2.2 and 2.3 calculate the thermal front velocity (or the
’advective thermal velocity’, Luce et al. 2013), vt(m s−1), using two temperature time series
from a vertical profile. Estimates of qare obtained by multiplying vtby C/Cw. All qcal-
culations are generally considered representative of the mid-point between two temperature
sensors. Estimates of qare based on either Ar(i.e. Ar=Ad/As, where subscript dand
srepresent the deep and shallow sensor, respectively), ∆φ(i.e. ∆φ=φs−φd), or both,
as shown below. VFLUX (1 and 2) uses Dynamic Harmonic Regression (DHR) from the
Captain toolbox (Young et al., 1999; Young et al., 2010) to determine Arand ∆φ, although
other methods are available (e.g. see Hatch et al., 2006; Rau et al., 2010; Swanson and
Cardenas, 2011; McCallum et al., 2012). Benefits of the DHR method include the fact that
variable fluxes can be calculated on a sub-daily time step (e.g. Lautz, 2012).
2.2. Existing methods in VFLUX 1
VFLUX 1 includes the Hatch et al. (2006) and Keery et al. (2007) models. The Hatch et
al. (2006) model for Aris:
q=C
Cw
2κe
∆zlnAr+sα+v2
t
2
,(3)
5
and the ∆φmodel is:
|q|=C
Cwv
u
u
tα−2 4π∆φκe
P∆z!2
,(4)
where Pis the period of the temperature signal (s), ∆zis the sensor spacing (m), and αis
calculated by α=qv4
t+ (8πκe/P )2. Eqn. 4 allows the magnitude of flux to be determined,
but not the direction of flow because phase shift is reduced with water movement in either
direction (Hatch et al., 2006; Briggs et al., 2014).
Keery et al. (2007) also provides individual solutions based on Ar(Eqn. 5) and ∆φ(Eqn.
6). The Keery et al. (2007) model for Aris:
H3lnAr
4∆z!q3− 5H2ln2Ar
4∆z2!q2+ 2Hln3Ar
z3!q+πC
λ0P2
−ln4Ar
∆z4= 0,(5)
where H=Cw/λ0. As Eqn. 5 is presented, both here, and in Keery et al., (2007, their
equation 9), λ0is included (in the Hterm), but the influence of βis not.
The ∆φmodel from Keery et al. (2007) is:
|q|=v
u
u
t C∆z
∆φCw!2
− 4π∆φλ0
P∆zCw!2
.(6)
As with Eqn. 4, Eqn. 6 allows the magnitude of flux to be calculated but not the flow
direction.
When β= 0 m, calculated fluxes using Ar(Eqns. 3 and 5) or ∆φ(Eqns. 4 and 6) are
identical (Irvine et al., 2015). Here, we consider β= 0 m for all analyses. Throughout
this paper, qcalculated from either Armethod (Eqn. 3 or 5) are presented as qAr, and q
calculated from either ∆φmethod (Eqn. 4 or 6) are presented as q∆φ.
The additions to VFLUX 2 are outlined below. No additional user input is required to run
the main routine of VFLUX 2 relative to previous releases.
2.3. Additions in VFLUX 2
McCallum et al. (2012) produced direct solutions to the 1D heat transport equation that
use both Arand ∆φsimultaneously, where qcan be determined from:
q=−C
Cw ∆z(P2ln2Ar−4π2∆φ2)
∆φ√16π4∆φ4+ 8P2π2∆φ2ln2Ar+P4ln4Ar!,
(7)
Note that Eqn. 7 does not require a value of κeto determine q(hence it does not require a
priori knowledge of λ0), which offers advantages over the Hatch et al. (2006) and Keery et
6
al. (2007) models. The negative sign in Eqn. 7 is due to McCallum et al. (2012) adopting
the convention of positive flow for upwelling. The implementation of the Hatch et al. (2006),
Keery et al. (2007), McCallum et al. (2012) and Luce et al. (2013) methods in all VFLUX
releases use the convention of a negative sign for upwelling.
Another key benefit of the methods outlined by McCallum et al. (2012) is that it is possible
to calculate a time series of κefrom:
κe=∆z2P2lnAr(4π2∆φ2−P2ln2Ar)
∆φ(P2ln2Ar+ 4π2∆φ2) (P2ln2Ar−4π2∆φ2)(8)
Luce et al. (2013) also present approaches that simultaneously use Arand ∆φto determine
a time series of q(Eqn. 9), and κe(Eqn. 10):
q=C
Cw ω∆z
∆φr 1−η2
1 + η2!! (9)
and,
κe=ω∆z2
∆φ2
r(1/η +η),(10)
where ∆φris the phase shift in radians, η=−lnAr/∆φr, and ω= 2π/P .
Eqns. 7, 8, 9 and 10 have been added to VFLUX 2. Irvine et al. (2015) show that the
McCallum et al. (2012) and Luce et al. (2013) equations for qproduce equivalent results,
and hence all qcalculated by simultaneously using both Arand ∆φare presented as qAr∆φ.
Estimates of κealso are also identical for McCallum et al. (2012), and Luce et al. (2013),
and hence they are presented as κeAr∆φ.
Other changes in VFLUX 2 (e.g. the ability to determine streambed scour using a refor-
mulation of Equation 10 outlined in Luce et al. (2013), and the ability to refine the range
of plausible κefor a given field site) are described in the documentation provided with the
computer code. The VFLUX 2 program files are included in the Supplementary materials,
available with the online version of this article. The most up to date versions are available
at the following website: http://hydrology.syr.edu/vflux.html.
2.4 Sand column experiments
Lautz (2012) performed laboratory experiments to investigate the influence of non-steady q
(a step change and gradual reduction in q), and the influence of stream temperature time
series with variable amplitudes under a constant q. The Hatch et al. (2006) methods were
applied to vertical temperature time-series collected during controlled sand column experi-
ments (Fig. 1). For all experiments, actual flow rates were determined using a tipping bucket
mechanism, and temperatures were measured in the column using Omega R
t-type stainless
steel thermocouples which were located at depths of 0.025 m, 0.05 m, 0.10 m 0.15 m, 0.25
m and 0.35 m (Fig. 1). The sand column was insulated with a 2.5 cm thick Grainger R
insulating jacket. Temperature time series were generated with P= 0.25 d. In all experi-
ments reanalyzed here, flows are vertical, one-dimensional, and downwelling (positive); for
7
full details on the experiments, see Lautz (2012).
Fig 1: Configuration of the Lautz (2012) sand column experimental set up (adapted from
Lautz (2012)).
Temperature time series from the Lautz (2012) sand column study are reanalyzed to inves-
tigate the estimates of qfrom the combined Ar∆φmethods, and to explore the benefits of
the additional ability to compute κetime series. The vf luxmc.m (Monte Carlo) script from
VFLUX 2 was used to estimate qvalues as the mean of 200 realizations, using the thermal
parameter values and uncertainties (standard deviations) shown in Table 2. The vfluxmc.m
program assumes normally distributed parameter distributions defined by the user defined
mean and standard deviation. The benefit of the use of the vfluxmc.m program is that both
the mean q, as well its uncertainty based on estimated parameter values are obtained. κe
estimates do not require thermal parameters, and hence the source of uncertainty in κecomes
from the uncertainty in values of Arand ∆φ(which is not accounted for in vfluxmc.m).
Estimates of qand κewere output 12 times for each temperature cycle using sensors located
at depths of 0.05 and 0.10 m (i.e. ∆z= 0.05 m). Using data from other pairs of sensors did
not significantly alter the results.
8
Table 2: Thermal properties of sand column experiment and numerical model (From Lautz
(2012))
Parameter (symbol) Value Standard deviation
Porosity (n) 0.30 0.05
Dispersivity (β) 0.0 m 0.0 m
Baseline thermal conductivity (λ0) 3.4 W m−1◦C−10.6 W m−1◦C−1
Volumetric heat capacity of solid (Cs) 3.6 ×106J m−3◦C−10.4 ×106J m−3◦C−1
Volumetric heat capacity of water (Cw) 4.2 ×106J m−3◦C−10.2 ×106J m−3◦C−1
2.5 Numerical modeling
To investigate the influence of multi-dimensional flows and streambed heterogeneity (in hy-
draulic conductivity, Km s−1), synthetic daily streambed temperature time series were
generated using HydroGeoSphere, HGS (Therrien et al., 2006), using the model domain
from Irvine and Lautz (2015). The numerical model domain (Fig. 2) was 10 m long (x), 4
m wide (y) and 3 m deep (z). Model discretization was uniform in xand y, with ∆x= ∆y
= 0.05 m. Vertical discretization was variable, with ∆z= 0.02 m for the upper 2 m of the
domain, and 0.05 m for the lower 1 m of the domain. This discretization produced a grid
with 1.92 million elements.
Fig 2: Showing a) model dimensions, boundary conditions and locations of observation
point profiles for considered simulations (flow paths for a homogeneous simulation shown),
9
and b) considered heterogeneous ln(K) field.
The hydraulic boundary conditions of the model were no flow boundaries on all faces, except
the top boundary, where the hydraulic head (h) was a function of x, with a gradient of
0.1 m m−1. These boundary conditions cause a flow-through hyporheic flow cell with the
(constant in time) vertical component of flux (qz) ranging between 8.83 to -6.18 m d−1(with
downwards flow denoted as positive). This range may seem high, however it is important to
note that these extreme values occur at the edges of the model (x= 0 and 10 m), and do
not feature in any of the profiles used for analysis. The thermal boundary condition at the
bottom of the domain was a constant temperature of 15 ◦C, and the top thermal boundary
condition was calculated using Eqn. 11:
T(t)=T0+ ∆T sin 2π(t−P/4)
P!,(11)
where T(t)is stream temperature (◦C) as a function of time, T0is the mean stream tempera-
ture (17.5 ◦C), and ∆Tis the amplitude of the stream temperature signal (2.5 ◦C). Eqn. 11
produced a sinusoidal temperature variation with a period of 1 day, and temperature varying
between 15 and 20 ◦C. As per Irvine and Lautz (2015), the initial temperature conditions
were established by first running the simulation for 14 days. The temperature distribution
after 14 days was then used as the initial temperature distribution for the simulations of a
further 14 days, which produced time series that were analyzed to calculate qfrom Eqns. 3,
4, 5, 6, 7 and 9, and κefrom Eqns. 8 and 10.
Observation points were horizontally located at x= 0.5, 2.0, 5.0, 8.0 and 9.5 m to record tem-
perature profiles that span the range from strong downwelling to strong upwelling (Fig.2).
In the ydirection (across the model), observation points were located at y= 1.0, 2.0 and
3.0 m. Observation points were vertically located every 0.1 m, from z= 0.0 m to 0.7 m. A
vertical observation point spacing (∆z) of 0.1 m provided up to seven flux estimates with
depth at each vertical profile.
The heterogeneous streambed Kdistribution was generated using the Sequential Gaussian
Simulator from the Geostatistical Software Library (Deutsch and Journel, 1998). The mean
Kwas 1 ×10−4m s−1, and correlation lengths were 1.0 m, 1.0m, and 0.2 m in the x,y, and
zdirections respectively, and the variance of ln(K) was 0.5. For the purposes of this study,
only one realization of the ln(K) field was considered, because the goal was to compare
the results of the Ar, ∆φ, and Ar∆φmethods in heterogeneous (rather than homogeneous)
media, rather than a detailed analysis of the influence of varied Kdistribution. It is impor-
tant to note that Gaussian simulators produce relatively smoothly varying fields, and more
sophisticated methods, such as Multiple-Point Statistics (e.g. Mariethoz and Caers, 2014)
can be used to produce fields with more connected preferential flow paths.
All thermal properties of the HGS model are as presented in Table 2. As with the Lautz
(2012) sand column data, all HGS model output were analyzed with the vf luxmc.m script,
10
using 200 realizations, with 12 qand κeoutputs per cycle (in this instance, per day). As
numerical models can produce output (e.g. temperature) with far higher precision than can
be obtained from field equipment, all temperature time series from HGS were adjusted before
being processed. All temperature time series from the HGS simulation were rounded to 2
decimal places before being read into vfluxmc.m.
The use of 3D, heterogeneous K-fields provides the opportunity to assess the performance of
temperature time series solutions with highly complex flow paths, where previous investiga-
tions have either used homogeneous Kdistributions (e.g. Lautz, 2010; Roshan et al., 2012;
Cuthbert and Mackay, 2013) or 2D model simulations (Irvine et al., 2015).
3. Results
The results below are presented as follows: first, the influence of non-steady flows (Section
3.1), and non-sinusoidal temperature signals (Section 3.2) are assessed for the sand column
experiments. Second, the influence of multi-dimensional flows in heterogeneous media (Sec-
tion 3.3) are assessed using the numerical model output.
3.1. Influence of non-steady flows
Fig. 3 shows the results from the Lautz (2012) step experiment, where three step changes
in measured q(denoted as qmeas) occur after approximately 4.5, 12.7 and 21.5 temperature
cycles. Fig. 3 shows the output from the Ar, ∆φ, and Ar∆φmethods. The gray box in Fig.
3c represents the plausible range of κecalculated from Table 1.
Fig 3: Reanalysis of the Lautz (2012) step change experiment. a) shows temperatures at
0.05 and 0.10 m, b) shows the observed flux (qmeas) through the sand column (black) with
the calculated fluxes from the use of the Ar(green), ∆φ(blue) combined Ar∆φmethods
11
(purple), c) shows the calculated κe(m2d−1) from the thermal properties from the Lautz
(2012) experiment in black, the e(purple), and the plausible range of κe(gray).
In Fig. 3b, the temperature dependency of hydraulic conductivity causes the cyclic variabil-
ity in qmeas (e.g., between cycles 4.5 and 12.7). The greatest temperature induced variation
in qmeas occurs when there is the largest amplitude in the temperature signal (e.g., compare
qmeas from cycle 21.5 to 25.9 with cycles 12.7 to 21.5). Interestingly, all three methods (qAr,
q∆φ, and qAr∆φ) produce flux estimates with these cyclic variations; however, modeled flux
changes are out of phase with measured changes, further indicating the 1D analytical models
lag real step changes in flux as observed by Lautz (2012).
To analyze the performance of the temperature time series models, the mean from the Monte
Carlo simulation (for each time step, i.e. each 1/12th cycle) is compared to the known qmeas.
Errors are calculated as ¯qAr−¯qmeas, ¯q∆φ−¯qmeas, or ¯qAr∆φ−¯qmeas (a positive error denotes
estimates of stronger downwelling than ¯qmeas ). In order to present the results in a more
meaningful way, the average error is presented between cycles 4.5 and 12.7 (labelled as time
period A), cycles 12.7 and 21.5 (time period B), and from cycle 21.5 to the end of the ex-
periment (time period C). The estimates and errors for each method (Ar, ∆φ, and Ar∆φ)
are presented in Table 3 below.
Table 3:qestimates and errors from step experiment in Fig. 3
Estimates (m d−1) Errors (m d−1)
Time Period
(cycles)
Period
name
¯qmeas ¯qAr¯q∆φ¯qAr∆φ¯qAr¯q∆φ¯qAr∆φ
4.5 - 12.7 A 2.93 2.93 3.52 3.35 0.00 0.56 0.43
12.7 - 21.5 B 1.56 1.61 1.93 1.67 0.06 0.38 0.12
21.5 - 26.0 C 3.67 3.11 4.44 4.06 -0.56 0.77 0.39
It is clear from Table 3 that the q∆φmethod was the poorest performing method across all
time periods considered. During time periods A and B, the qArmethod outperforms the
qAr∆φmethod, with the qAr∆φmethod producing estimates that, on average, were slightly
lower than the qArmethod for period C. However, Fig. 3 shows that the errors from the Ar
method are larger than the Ar∆φmethod immediately after the large step change in qmeas
after 21.5 cycles. If period C was calculated between cycle 22.5 and the end of the exper-
iment, the mean errors are -0.46, 0.87 and 0.59 m d−1for qAr,q∆φ, and qAr∆φrespectively
(i.e. smaller errors for qArrelative to qAr∆φ).
The time series of κeAr∆φshows cyclic variations as was the case for each of the qestima-
tion methods. However, the κeAr∆φtime series also shows rapid changes that coincide with
the step changes in qmeas. The step change in qmeas that occurs at cycle 4.5 produces a
rapid increase in κeAr∆φ, which takes the value outside of the reasonable range. The κeAr∆φ
estimates are also often outside of the theoretical range from cycle 21.7 to the end of the
experiment. In both cases, the time periods where unreasonable κeAr∆φestimates coincide
12
with large errors in qestimates. This observation supports the suggestion of McCallum et
al., (2012), that unreasonable variations in κeAr∆φcoincide with erroneous flux estimates,
with the largest occurring when the flow rate changes rapidly. The range of plausible κe
for any given site can be reduced in the VFLUX 2 post processing options to assist with
decisions concerning the reliability of flux estimates.
Fig. 4 shows the results from a gradual change in qmeas that occurs over the duration of a
temperature cycle (from cycle 8 to 9). The layout of Fig. 4 is identical to Fig. 3, with the
exception of the extent of the time axis (xaxis). The time axis spans from cycle 5 to 16, to
highlight the influence of the gradual change in qmeas.
Fig 4: Reanalysis of the Lautz (2012) gradual change experiment. a) shows temperatures
at 0.05 and 0.10 m, b) shows the observed flux (qmeas) through the sand column (black) with
the calculated fluxes from the use of the Ar(green), ∆φ(blue) combined Ar∆φmethods
(purple), c) shows the calculated κe(m2d−1) from the thermal properties from the Lautz
(2012) experiment in black, the (purple), and the plausible range of kappae(gray).
The results in Fig. 4b highlight challenges in the use of ∆φ. It is known that qmeas remains
positive (i.e. downwelling) throughout the entire column experiment, although in a field
setting it is likely that the flow direction may be unknown. If the convention suggested by
Soto-L´opez et al. (2011) is used, where Aris used to determine flow direction, and ∆φto
determine q, a sudden change in q∆φwould occur after cycle 11 as qArestimates become
negative (suggesting upwelling). Fig. 4b also shows that the qArmethod produces a more
reliable range of fluxes during the gradual change in qmeas (cycles 8 to 12).
13
As with the results shown in Fig. 3, the large errors from the qArand q∆φmethods during
the gradual change experiment coincide with κeAr∆φvalues that are outside of the practical
κerange; this indicates ∆φinformation is compromised during the hydraulic shift, introduc-
ing error into the qAr∆φmethod. As qAr∆φgradually approaches qmeas after the change in
qmeas, the qAr∆φtime series approaches the calculated κevalue calculated from the measured
thermal parameter values from Lautz (2012).
Monte Carlo error analysis for all methods is based on assumed uncertainty distributions
for select sediment and thermal parameters in the heat transport equation (n,λ0,Cs, and
Cw; Table 2). The apparent uncertainty of qAr∆φis lower than qAror q∆φfor nearly all q
estimates during the column experiments, and this is particularly apparent at lower q(e.g.
Figure 3b and 4b). The lower uncertainty of qAr∆φis expected, given that a priori knowledge
of λ0is not required for the flux calculations and therefore fewer variables are considered
uncertain in the Monte Carlo analysis for qAr∆φ. The uncertainty of qAr∆φis notably reduced
at lower q(cycles 5-7 versus cycles 12-14, Figure 4b); this is due to the smaller temperature
oscillations (∆T) that occur in the sediment-water matrix under low flow conditions. When
∆Tis small, model sensitivity to n,Cs, and Cw(the thermal parameters considered uncer-
tain in the combined methods) is minimal and uncertainty of qAr∆φvalues due to thermal
parameter uncertainty is minimized. While the Arand ∆φmethods are still sensitive to
uncertainty in λ0at low flux, λ0is not an input for the combined methods and therefore
not a source of uncertainty. In contrast, when ∆Tis large at higher flux rates, uncertainty
of the heat capacity of the sediment-water matrix (derived from n,Cs, and Cw) becomes
the primary source of error and uncertainties of the Ar, ∆φ, and combined methods are
essentially identical.
Although the Monte Carlo analysis suggests qAr∆φhas a lower degree of uncertainty at low
qmeas, we can clearly see from Fig. 4 that the qAr∆φvalues are substantially different from
qmeas, with greater model residuals than for qAr. Although uncertainty of qestimates due to
thermal parameter uncertainty has been reduced using the combined methods, uncertainty
derived from the shifting phase lag through time under transient flow conditions causes error
in qAr∆φ. For this reason, caution should be used when interpreting flux uncertainty using
only thermal parameter uncertainty analysis. The combined methods reduce the flux esti-
mate uncertainty associated with uncertainty in the thermal parameters when ∆Tis small.
But, that apparent certainty in flux estimates could be misleading, given the other potential
sources of uncertainty in the analysis.
3.2. Influence of non-sinusoidal temperature signal
Fig. 5 shows the results from an experiment with a relatively constant qmeas, but an irregular
heating of the source water.
14
Fig 5: Reanalysis of the Lautz (2012) weather experiment. a) shows temperatures at 0.05
and 0.10 m, b) shows the observed flux (qmeas) through the sand column (black) with the
calculated fluxes from the use of the Ar(green), ∆φ(blue) combined Ar∆φmethods (pur-
ple), c) shows the calculated κe(m2d−1) from the thermal properties from the Lautz (2012)
experiment in black, the e(purple), and the plausible range of kappae(gray).
In Fig. 5a, cycles 19 and 24 show the greatest change in the temperature signal. The effect
of the temperature signal at cycle 19 can be clearly observed in the q∆φ,qAr, and qAr∆φ
estimates, where all methods produce lower qestimates during this period. In the weather
experiment shown in Fig, 5b, the mean qmeas was 1.77 m d−1. The greatest errors (difference
between qmeas and mean from the Monte Carlo simulation) at cycle 19 were 0.03, 0.13 and
0.40 m d−1for qAr,qAr∆φ, and q∆φrespectively. The mean errors, averaged across all time
steps follows the same pattern ( ¯qAr<¯qAr∆φ<¯q∆φ), with mean errors of 0.07, 0.14 and 0.28
m d−1. The q∆φmethod also has the widest uncertainty bands throughout the experiment,
with a typical range of 1.03, compared to 0.48 for the qArmethod, and 0.59 m d−1for the
qAr∆φmethod. In Fig. 5c there is a change in κeAr∆φat cycle 19, although this would be
difficult to distinguish from the cyclic variations in κeAr∆φthat occur in Figs. 3c and 4c.
The findings from the reanalysis of the Lautz (2012) sand column experiments of non-steady
flows (Figs. 3 and 4, Table 3), and non-sinusoidal temperature signals (Fig. 5) highlight
that non-steady flow rates have the greatest influence of the validity of qestimates from the
use of Ar, ∆φor Ar∆φ. While qestimates are influenced by variations in the temperature
signal, it has been previously highlighted that this effect could be reduced by calculating
average qduring periods of relatively stable flows Lautz (2010).
15
3.3. Influence of multi-dimensional flow in heterogeneous media
The results presented in Fig. 6 are from the HGS model run for a longitudinal section along
the length (x) of the model at y= 1.0 m.
Fig 6: Analysis of HydroGeoSphere model output at y= 1.0 m. Left column shows the
vertical component of flux (qz) from HydroGeoSphere (black), ±95 % confidence limits for
qAr(green), ±95 % confidence limits for q∆φ(blue), and ±95 % confidence limits for qAr∆φ
(purple). Right column shows κefrom HydroGeoSphere (black), eAr∆φ(purple), and the
practical limits of κe(gray). From top to bottom, data are shown for x= 0.5 (a, b), 2.0 (c,
d), 5.0 (e, f), 8.0 (g, h) and 9.5 m (i, j).
Fig. 6 shows time averaged values from vfluxmc.m (with 200 realizations) qestimates (6a,
c, e, g, i), and κeestimates (6b, d, f, h, j). For the κepanels in Fig. 6, the κecalculated from
the HGS input parameters is shown in black (κeHGS ), the κeAr∆φfrom the McCallum et al.
(2012) or Luce et al. (2013) method is shown in purple. As Eqns. 8 and 10 are functions
of Arand ∆φ, and not C,Cwor λ0, these solutions provide a unique time series of κeAr∆φ,
rather than a distribution as is the case for qAr,q∆φ, or qAr∆φ. In all cases in Fig. 6, the
sign of q∆φis determined from the sign of qAr.
16
For the x= 0.5 m qestimates shown in Fig. 6a, all methods follow the general pattern of
qHGS . The range of qAr∆φand q∆φare essentially identical, with over-estimates of down-
welling occurring for the upper 0.2 m of the profile for qAr. At x= 2.0 m (Fig. 6b, where
vertical fluxes are more moderate compared to the x= 0.5 m profile), the qArand qAr∆φ
estimates are similar for the entire profile, with a wider distribution of flux estimates of q∆φ.
The benefits of the Ar∆φmethods are highlighted for the x= 5.0 m data (Fig. 6e), where
the uncertainty in estimates are essentially zero. For the upwelling profiles (x= 8.0 and 9.5
m, 6g, 6i) qestimates are either not possible, or highly variable below z= 0.4 m. This shows
a limitation of diurnal temperature time series methods, as flux magnitude and direction will
affect the damping of the surface signal with depth, and if the diurnal signal amplitude is less
than twice sensor precision it cannot be reliably measured. For an extensive investigation
into the limitations of diurnal signal methods for upwelling cases, the reader is referred to
Briggs et al. (2014). Here qestimates are only shown to z= 0.45 m below which the diurnal
signal is not present; errors clearly increase toward this limit of detection. In the upwelling
profiles (Figs. 6g, 6i), the combined Ar∆φmethods perform well in the upper 0.35 m. For
example, for the qAr∆φmethod, the errors from the mean from the Monte Carlo method at
z= 0.15 in the bottom row of Fig. 6 (x= 9.5, y= 1.0) was -0.11 m d−1where qHGS was
-0.74 m d−1.
The κeAr∆φestimates in Fig. 6 (b, d, f, h, j) show that spatial variations in κeestimates can
also be indicative of erroneous qestimates from the all methods. In the x= 0.5 m data (Fig.
6b), the sudden increase in κeAr∆φat z= 0.25 m occurs at the same depth as the greatest
over-estimate of qAr∆φ. For the upwelling profiles (x= 8.0 and 9.5 m, i.e. Figs. 6h and 6i)
the estimates of κeAr∆φthat either occur outside of the practical κelimits, or where κeAr∆φ
oscillates from one depth to the next also coincide with erroneous flux estimates.
The κeestimates in Fig. 6 show that spatial variations in κeAr∆φcan also be useful in deter-
mining where qestimates are reliable. However, sudden changes in κeAr∆φalso occur when
rapid changes in qoccur. For example, the change in qHGS that occurs between z= 0.27
and z= 0.35 m is 1.69 m d−1.
4. Discussion and recommendations
From the investigation into time-varying fluxes in Section 3.1, the use of the ∆φmethod
led to the greatest variability in qestimates. The use of ∆φalso offers other challenges,
including the fact that a method to determine flow direction is also required. The results
from the gradual qchange experiment (Fig. 4) highlight difficulties in selecting the sign for
flow from the Armethod, and the magnitude from the ∆φmethod. For the time-varying
qconditions tested here, the Armethod performed better that the Ar∆φmethod during
periods of (and after) non-steady flow, due to apparent adverse influence of the ∆φdata on
Ar∆φresults.
Investigations into the influence of a non-sinusoidal temperature boundary condition (Sec-
tion 3.2) showed that the influence of a non-sinusoidal temperature source did not lead to
significant variations (in time) in qestimates across all of the time series methods for the
temperature signals used in this study, supporting the findings of Lautz (2012). Of the three
17
methods considered here, the ∆φmethod led to the widest distribution of qestimates, with
comparable flux estimates from the Arand Ar∆φmethods. Analyses of the Lautz (2012)
sand column data demonstrates that large variations in κeAr∆φwith time can suggest that
qestimates are likely to be inaccurate. In the analyses presented here, large variations in
κeAr∆φoccurred in cases where qwas both under- and over-estimated.
In the investigations into the influence of streambed heterogeneity on qestimates (Section
3.3), again, the ∆φmethod performed poorly. Irvine et al. (2015) showed that when there
is no uncertainty in thermal parameters, that the use of the Armethod typically produced q
estimates with lower errors than the Ar∆φmethod. The results presented here, where there
was uncertainty in thermal parameters, show that the Ar∆φmethod typically produces q
estimates with smaller errors than the Aror ∆φmethods when temperature variations over
a time cycle are small (i.e. at low flux rates, like period B, Fig. 3), and this is due to the
removal of uncertainty in λ0, which is not required as an input for the Ar∆φmethods. The
results from Fig. 6 show that large spatial variability in κeAr∆φestimates also coincides
with unreliable flux estimates, or that there is insufficient strength in the diurnal signal for
reliable qestimates.
Analytical temperature time series-based methods of determining fluid exchange between
surface water and groundwater are typically applied where one or more of the underlying
assumptions of the methods are not met. Generally, the performance of the ∆φmethod was
poor (relative to the Aror Ar∆φmethods) under all conditions tested. When fluxes are
variable with time, the Armethod produces more accurate results. When flows are stable in
time, and near zero, estimates of κeAr∆φare typically very accurate in the cases investigated
here. This point could be utilized, whereby estimated κeAr∆φvalues from stable periods in
association with measured nand estimated solid-grain heat capacity (Cs) could be used to
constrain λ0, which in turn would improve flux estimates from the Armethod. This process
will require the estimation of other parameters; however, the measurement of some thermal
parameters can assist this process. Of the parameters required as input in VFLUX (1 and
2) (Cw,n,Cs,β, and λ0), nis likely the simplest to measure. For example a core sample can
be weighed before and after the sample is oven dried. The properties of water (Cw,ρw, and
λw) vary on a very small range for the temperature variations observed in streambeds. In
zones of strong upwelling (not investigated here), where phase lags are relatively short and
difficult to characterize accurately with most current sensing technology (Briggs et al. 2014),
methods that utilize both Arand ∆φinformation (e.g. Ar∆φ,κeAr∆φ) may be particularly
prone to error.
Additional uses of κeAr∆φoutput include analysis based on a plausible range of κe, strong
temporal deviations from this plausible range provides an objective metric to indicate time
periods when qestimates may be unreliable. The vflux post.m program allows for the re-
sults from a VFLUX run to be visualized. In VFLUX 2, plots are produced which show time
series of κefrom the McCallum et al. (2012), and Luce et al. (2013) methods in comparison
to a large range of plausible κevalues across all expected streambed sediment types. The
user has the option to enter a new range of n,cs,ρsand λsto reduce the plausible range of
κevalues for a particular site.
18
VFLUX 2 also includes the equation to assess streambed thickness (∆z) above a lower,
buried senor, which can be used to assess scour assuming a thermally-mixed water column
as presented by Luce et al. (2013), and Tonina et al. (2014). While these approaches
were originally designed to determine streambed scour and deposition, further opportunities
are yet to be explored. VFLUX 2 calculates ∆zusing an approach outlined in Luce et al.
(2013, their Eqn. 57) using the user input CwCsand λ0to determine κe. It is expected
that oscillations in ∆zwill occur when large changes in qoccur (as is the case with κeAr∆φ)
which may also provide a means to determine when q estimates are unreliable. However, a
systematic offset between the predicted and known ∆zmay also provide the opportunity to
refine thermal property inputs for VFLUX 2 to improve both qArand qAr∆φ, as both require
estimates of Csto convert vtto q.
Conclusions and implications for field applications
We present an analysis of the influence of non-steady flows, non-diurnal temperature signals,
and non-vertical flow in heterogeneous media on estimates of vertical water flux from temper-
ature time series methods included in the new VLFUX 2 program. Even with their inherent
complexities, the sand column and numerical model investigations presented here represent
simplified cases relative to field applications. It should be highlighted that all field sites are
non-ideal in that any of the abovementioned non-ideal conditions violate the assumptions
of the analytical solutions discussed in this paper. However, we have highlighted that the
use of the combined Ar∆φthermal diffusivity output can assist in the identification of time
periods where flux estimates may be unreliable.
There are many methods to determine groundwater-surface water exchange using tempera-
ture time series. This study clearly demonstrates limitations of methods based on ∆φalone,
and with alternative methods available, we do not generally suggest its use. During transient
flow conditions, the use of the Armethod is suggested. The Ar∆φmethods perform best
during steady flow conditions, and when qis near zero in low-flux environments. The ability
to define low fluxes (e.g. <0.1 m d−1) more precisely than with the individual methods may
benefit problems such as contaminant transport where even low flux rates are important
to water quality. The VFLUX 2 package integrates the combined Ar∆φmethods into a
familiar format without added user input, allowing for facile adoption by the surface water-
groundwater interaction community.
Acknowledgements
This material is based upon work supported by the National Science Foundation under Grant
No. EAR-0901480, the U.S. Geological Survey (USGS) National Research Program, and the
Groundwater Resources Program and Toxic Substances Hydrology Program. Any use of
trade, firm, or product names is for descriptive purposes only and does not imply endorse-
ment by the US Government. The paper benefitted from the feedback from two anonymous
reviewers, and the thoughtful review of an earlier version of the manuscript by Randy Hunt.
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