A 56-sensor, 3-D temperature array with three heat pulse sources (also known as the Hot Rod) was developed to measure the flow direction and magnitude up to 200 mm below the water–sediment interface in the streambed (Fig. 1). The central carbon-fibre rod (260 mm long with a diameter of 12 mm) has three equally spaced, 60 W heating elements along its length at positions of 65, 140 and 215 mm below the base plate and are referred to as heat injection depth R1, R2 and R3, respectively (Fig. 1). Eight stainless steel rods (6 mm diameter and 298 mm long) housing seven equally spaced (38 mm apart) temperature thermistors (Maxim DS18B20; precision 0.06∘) are arranged cylindrically around the carbon-fibre rod and at two fixed spacings of 28 and 47 mm (Table S1 in the Supplement). The central rod and thermistor sticks are fixed to a rigid circular aluminium base plate which is attached to a collapsible handle. The material, dimensions and spacing of the rods to the base plate were designed to reduce flexibility and minimise disturbance to the sediment material on insertion, as this was a limitation in previous studies. A critical design feature was to ensure that the rods stayed parallel to one another and that the spacing did not vary during installation, as it was designed to be used in a range of sedimentary environments from fine silt to coarse gravels.

A terminal program is used to communicate with the data logger and to control the sampling routine of the Hot Rod (e.g. sampling frequency, duration and power output of the active heat pulse, selected heating element used and logging period). A sealed 12 V lead acid battery together with a power supply regulator is used to maintain a constant 12 V output to the heating elements. The power delivered to the three heating elements can be adjusted in the logger program from 0 to 100 % to provide greater flexibility to the required active heat pulse. The input current from the power supply regulator is also recorded each time the temperature is measured to ensure a tight control on the actual power being delivered through the heating elements to the surrounding material and therefore reducing uncertainty in the analysis routine.

An important aspect of the design was that it could be rapidly and easily deployed to capture large spatial data sets along a reach of stream or across a pool–riffle sequence. Installation requires gently pushing (or lightly tapping with a shockless impact hammer) the device into the streambed ensuring that there is a sufficient gap between the top of the sediment and the underside of the base plate to prevent streamflow constriction (the top thermistor is set at 44 mm below the baseplate so a gap of ∼30 mm puts the first thermistor just below the sediment–water interface). The impact of the installed device on flow velocity and direction is expected to be minimal given that the volume of the device relative to the volume of the measurement area is less than 4 %.

Once installed and equilibrated with the surrounding sediment, the logger program is executed and the ambient temperature is measured at each thermistor (T0), directly followed by activation of the selected heat element for the chosen duration. The data logger records the temperature differential (ΔT=Tt-T0) at the chosen sample frequency to clearly discern the timing and location of the breakthrough curve at each of the thermistors. Field and laboratory tests showed that short, 1 min heat pulse injections and a 20–30 min temperature response monitoring period are appropriate for estimating the dominant direction and flux magnitude in sandy streambeds. Specific details of the analysis routine that was used are described in the following section.

A laboratory sand tank was used to provide a controlled environment on the hydraulic regime to test the performance of the Hot Rod and so that the different flux calculation methods could be compared. A total of 36 combinations of flow direction, magnitude and depth of heat pulse were used. The dimensions of the sand tank for each of the scenarios varied slightly according to the fixed boundary conditions (Fig. 3). Four flow scenarios were tested: (1) horizontal flow from left to right (inflow and outflow occurred over the entire saturated cross-sectional area of the sediment volume on the left and right boundaries of the tank), (2) diagonal flow from the top left to bottom right (inflow was through a 20 mm horizontal inlet slit on the top left boundary and outflow through a 20 mm horizontal slit, 85 mm above the base of the right boundary), (3) upward flow (inflow occurred over the cross-sectional area of the base of the tank and outflow at the top of the sediment at overflow points above the sediment on the left and right boundaries), and (4) downward flow (inflow was distributed over the cross-sectional area of the sediment surface and outflow via the cross-sectional area of the tank base). A steady state flow regime for each scenario was maintained by constant heads at the inflow and outflow ports of the tank and the use of a peristaltic pump with a highly accurate ultrasonic flowmeter (Atrato ultrasonic flowmeter; 0.05 % linearity on flow less than 5 Lmin-1) to ensure a constant discharge rate. The flowmeter data were also used to determine the Darcy flux for the different flow conditions. Fine perforated mesh was used to contain the sediment and to provide the necessary flow conditions along each of the tank boundaries. The hydraulic conductivity and thermal hydraulic conductivity of the graded, saturated sand were measured using a KSAT meter (UMS GmbH, Munich, Germany) and KD2 Pro (Decagon, Washington, USA), respectively. Three different hydraulic gradients and discharge rates for the four flow scenarios were conducted to capture low- (∼10-6 ms-1), moderate- (∼10-5 ms-1) and high-flow conditions (∼10-4 ms-1). The Hot Rod was positioned in the middle of the sand tank with thermistor stick sensor number one orientated perpendicular (90∘) to the flow direction for all of the scenarios. To validate the spatial arrangement of the sensors, the horizontal flow scenario was repeated with thermistor stick sensor number one rotated to be 45∘ to the direction of flow. In addition to these flow scenarios, a no-flow experiment was conducted to evaluate the analysis routine, where the boundary conditions meant that there was stagnant water.

The Sturt River, Adelaide, Australia, is a perennial river system receiving the majority of its input from a wastewater treatment facility. The geomorphology of the river was characterised by a narrow channel, no more than 3 m wide with 0.3–0.5 m deep sediment ranging from fine silt to coarse gravels overlying a tight, low-permeability clay. The selection of this field site was part of another ongoing investigation looking at attenuation of micro-pollutants in the hyporheic zone. The residence time in the hyporheic zone was critical in evaluating the stream attenuation modelling.

Overall, the modelled breakthrough curves closely fit the observed data from the 56-sensor array with the modelled curves capturing the rising limb, peak and tail of the measured temperature data over the sample period (Fig. 3). The variance of each parameter is included in the modelled temperature breakthrough curves; however, the uncertainty is so small that it cannot be seen without zooming in on the individual curves. Selected breakthrough curve plots are shown of the fourth vertical thermistor (158 mm depth) from each radial sensor location. Four flow scenarios are presented: (a) horizontal (Fig. 4), (b) diagonal (Fig. S1-ii in the Supplement), (c) upwards (Fig. S1-iii) and (d) downwards (Fig. S1-iv). The tests presented were conducted at a moderate flow rate. Thermistor 4 was 158 mm below the base plate and at a greater depth than the heat injection depth (R2; at 140 mm). Temperature increases associated with the heat pulse were observed at the inner sensor sticks (28 mm) more quickly than at the outer (47 mm) sensor sticks in the horizontal flow scenario (see Fig. 4d and h). The inner sensors also displayed a steeper rising and falling limb compared to the outer sensors. The temperature response reflected the sensor position relative to the dominant flow direction. For example, sensor T3–4 (Fig. 4c) was directly in line and down-gradient of the heat pulse, whilst sensor T7–4 (Fig. 4g) was in line but up-gradient of the heat pulse and therefore showed a smaller response. The breakthrough curves from the diagonal flow scenario showed a similar response in those sensors up- and down-gradient of the heat pulse (Fig. S1a). In the upward and downward flow scenarios there was very little response in the outer thermistor sensors due to the dominant vertical component of flow (Fig. S1b–c). This low response of the outer sensors was even more pronounced under higher flow conditions in the upward and downward flow scenarios. The 3-D time series videos showed clearly the migration of the heated plume vertically and highlighted the complexity of fitting multiple breakthrough curves to the most likely solution (refer to the video file in 10.4226/86/5aab1b67337bb).

Overall, the 3-D flow fields calculated from the HPS Hot Rod in the laboratory sand tank for the four flux scenarios and heat injection depths (65, 140 and 215 mm) were consistent with the flow conditions established in the tank (Fig. 5). Under left to right horizontal flow conditions (Fig. 5a and b), the modelled direction of flow from each of the heat injection depths is very similar with some slight offset to the observed flow in the y direction, which was perpendicular to thermistor stick sensor 1 (90∘ to the flow direction). There was also a slight deviation downwards in the z direction, particularly for the shallowest heat injection depth. To refute any bias in the optimisation routine and the array configuration, the Hot Rod was rotated by 45∘ for the horizontal flow scenario and it showed a very similar output to when it was orientated 90∘ to the flow direction.

Results from the other three scenarios showed that the modelled flux direction is close to parallel to the flow conditions established in the tank and the magnitude of the flux was similar at each of the heat injection depths (Fig. 5c–h). Reviewing the time series data in the 3-D plots (Supplement), the spreading of the heat pulse from the heat injection depth can be clearly detected, indicating how the heat pulse moves along the established flow line. The thermistor highlighted in blue in the 3-D plot was the sensor that showed the maximum temperature breakthrough curve and clearly shows a different orientation to the most likely flux direction (black arrow) as determined by the DREAM algorithm.

Some discrepancies in the direction of the modelled flow and differences in flux magnitude at each heat injection depth may be attributed to (1) placement orientation and the angle of the sensor positions of the Hot Rod relative to the flow conditions established in the tank, (2) boundary conditions in the tank to establish flow and (3) the fact that the optimisation routine determines the best fit of all the observed data in a 3-D volume around the heat injection depth rather than at a specific point.

The difference in flux magnitude at each heat injection depth may be attributed to the number of sensors that were used in the optimisation routine. For example, at the heat injection depth R2 (140 mm) there are three sensor arrays (an array being eight sensors positioned horizontally around the central carbon-fibre rod) vertically above R2 and four sensor arrays vertically below R2. In comparison, at heat injection depth R1 (65 mm) there is only one sensor array vertically above R1 and six sensor arrays below R2, which may limit the optimisation routines for particular flow conditions i.e. strongly upwards flow.

In the case of no-flow conditions established in the sand tank (Fig. S2), the optimisation routine fitted the measured temperature breakthrough data; however, on closer inspection of the 3-D time series plot it was evident based on the uniform heat plume around the heat injection point during the injection period that heat transport was by conduction only. Absence of clear advective movement of the heat pulse and a calculated flux less than about ∼10-6 ms-1 indicated the lower limit of the active heat pulse sensor.

The flux magnitude (q) of the different flow scenarios and different flow intensities calculated based on different heat injection depths of the HPS (grey bars) compared to the fluxes determined based on Darcy's law (hydraulic gradient and hydraulic conductivity) and the measured discharge from the tank using an ultrasonic flowmeter is shown in Fig. 6 and Table S2.

The measured saturated hydraulic conductivity according to the KSAT meter for the sand tank sand was 4.95×10-4 ms-1. The measured saturated thermal conductivity of the sand was 3 Wm-1∘C-1 (using the Decagon KD2 Pro), whilst the average modelled κ0 from all of the flow scenarios in the sand tank was 3.8 Wm-1∘C-1. The thermal conductivity is strongly influenced by the porosity, and therefore some differences can be expected due to changes to the particle density with the packing of the sediment, where by thermal conductivity increases with decreasing porosity (Smits et al., 2010).

Histograms of the flux magnitude (q) and flux components in the x, y and z direction for the combination of most likely parameter values used in the DREAM algorithm were generated for each measurement. The results from heat injection depth R2 for the horizontal flow scenario is shown in Fig. S3, which shows that the distribution is tightly constrained. This is also evident in cross correlation plots of the flux magnitude, thermal conductivity and the specific heat capacity (Fig. S4).

Constraining the range of the thermal conductivity values used in the optimisation routine to the known measured thermal conductivity of the sand from the KD2 Pro instrument showed little impact on the calculated flux magnitude. However, comparing the modelled breakthrough curves from two optimisations when the range in thermal conductivity was limited to the measured known thermal conductivity (3 Wm-1∘C-1) in one model and in the other model it used a value of 3.52 Wm-1∘C-1 (for a best fit from a range of values from 2.5 to 4.5 Wm-1∘C-1) showed there were subtle differences between the modelled breakthrough curve peak and falling limbs (Fig. 7).

Our study found that the inclusion of longitudinal and transverse thermal dispersion had less than a 2 % difference on the mean calculated fluxes. The study by Rau et al. (2012) determined Darcy velocities derived from heat experimentation that included the thermal dispersivity term differed by up to 20 % when compared to solute experimentation. However, other studies in the literature have shown that there is considerable uncertainty on the magnitude of the thermal dispersivity (Anderson, 2005). Thermal dispersivity has also been found not to be scale-dependent such as solute dispersivity because heat transport happens through the pore water and through the sediment matrix (Vandenbohede et al., 2009). Therefore, given the scale that we are working at (few centimetres) and also the low velocities, the effect on the calculated flux is likely to be negligible.

The measured 3-D flow fields at the experimental site showed considerable variability in the direction of flow and flux magnitude over ∼0.20 m depth of the streambed. The flux magnitude at the six stations along the river at the three heat injection depths (0.065, 0.140 and 0.215 m) ranged from 4.2×10-6 to 4.26×10-5 ms-1 (mean: 1.6×10-5 ms-1). At each of the stations, the component of horizontal flow compared to vertical flow within the streambed was dominant. It was only at the shallowest heat injection depth, just below the stream bed surface, that there was a greater component of vertical flow. The results from two of the six stations are shown in Fig. 8, where the Hot Rod was positioned in the streambed such that the x axis of the figure was aligned to the direction of stream flow. In such environments the flow direction is driven by the surface flow and the geomorphological features of the streambed. Small bed structures such as ripples will only impact the hyporheic flow field in the uppermost centimetres of the sediment, which is a plausible explanation as to what was observed from the outputs of the Hot Rod.

Measured κ0 values for the sediment collected at the six stations from 0 to 0.2 m depth ranged from 0.9 to 2.84 Wm-1∘C-1, (mean: 2.07 Wm-1∘C-1) compared to the values that were determined by the optimisation routine, which ranged from 1.0 to 3.74 Wm-1∘C-1 (mean: 2.91 Wm-1∘C-1). Physical observation of the sediments collected at the experimental site stations showed that the streambed material was quite heterogeneous and also contained varying proportions of organic matter. Higher clay content and organic matter in the sediment would cause a higher volumetric heat capacity compared to sandy sediments. The volumetric heat capacity also increases with increasing moisture content and particle density (Abu-Hamdeh, 2003; Barry-Macaulay et al., 2013; Jury and Horton, 2004). The assumption of uniform thermal properties of the 3-D volume around the heat injection point is likely to contribute to the uncertainty in the flow direction. For example, Su et al. (2006) used numerical simulations to show that differences in the thermal properties of the sediment around a flow sensor can lead to incorrect velocity estimates and in particular differences in the horizontal and vertical flux components. Additional measurements of the streambed sediments would provide a tighter constraint on the parameter set used in the optimisation; however, the parameter estimation and uncertainty analysis routine does successfully fit the measured data to provide a reliable estimate of the flux and its direction. Refer to Banks et al. (2017) for all of the temperature data files from the sand tank and experimental site, available at 10.4226/86/5aab1b67337bb.