Locating and quantifying spatially distributed groundwater/surface water interactions using temperature signals with paired fiber-optic cables

Authors


Abstract

[1] Diurnal temperature fluctuations of surface water, as result of solar heating, function as a tracer that continuously exchanges energy between streams, streambed sediments, and discharging groundwater. Analytical solutions exist to estimate discharge by extracting the amplitude ratio between pairs of subsurface temperature time series measurements. The research presented here adds to the expanding body of heat tracing literature by applying the amplitude-shift time series discharge estimation method to pairs of distributed temperature sensor (DTS) fiber-optic cables. A pair of DTS fiber-optic cables is placed in an experimental streambed, one over the other, with a small vertical separation to measure continues heat-based vertical streambed fluxes along the entire length of cable, thus eliminating a long series of point measurements. This study utilized time series data from synthetic data sets, modeled numerically using COMSOL Multiphysics, and physical data sets, modeled in a 10 m long sandbox model to assess the viability of this new distributed flux quantification method. Discharge estimated with spatially averaged temperature data are accurately approximated where groundwater flow is uniform and the temperature signal is constant at the streambed surface. Error is introduced where focused groundwater discharge exists, resulting in temperature profiles that vary laterally throughout the streambed. Spatial averaging inherent to DTS data results in dampening of flux measurements over focused discharge zones, as temperature is averaged as a result of the measurement technique. This leads to underestimating peak flux at localized discharge zones and overestimating discharge measurements away from these locations. The spatial integration of the DTS as well as the sampling interval and cable position can lead to error in calculated groundwater fluxes. Results demonstrate the potential advantages and disadvantages of using paired fiber-optic cables to quantify high-resolution groundwater discharge to streams at the reach scale.

1. Introduction

[2] Quantifying streambed flux from surficial aquifers is crucial for efficient development and management of groundwater resources. Conceptual models of streambed seepage generally assume uniform exchange along the length of a stream [e.g., Winter et al., 2006], however, heterogeneity in sediments or fractured bedrock below the streambed can cause focused groundwater discharge [Alley et al., 2002; Conant, 2004; Schmidt et al., 2007]. Identifying patterns of focused discharge in the streambed are important for quantifying groundwater discharge and nutrient transport to streams as well as supporting effective stream restoration. Localized discharge can help support hot spots of biogeochemical activity [McClain et al., 2003], as well as locally regulate stream temperature [Alexander and Caissie, 2003; Rosenberry et al., 2000]. Unfortunately, it is difficult and labor intensive to identify these areas throughout a large stream reach with existing methods. Developing tools to better locate and quantify fine-scale zones of groundwater discharge throughout large-field sites will allow for more cost effectively restoration of stream ecosystems.

[3] Currently, there are a wide range of methods utilized to estimate groundwater surface water (GW/SW) interactions, such as seepage meters [Lee, 1977; Murdoch and Kelly, 2003], shallow piezometers [Lee and Cherry, 1978; Baxter et al., 2003; Rosenberry et al., 2008], differential gauging [Ruehl et al., 2006], tracer injection [Harvey et al., 1996; Gooseff and McGlynn, 2005], and heat tracing [Anderson, 2005; Constantz, 2008]. Each technique has its own spatial and temporal scales, assumptions, uncertainties, costs, and limitations. Seepage meters, shallow piezometer nests, and temperature sticks are capable of accurately estimating single point in space flux. To gain a more thorough understanding of discharge throughout the entire stream hundreds of these instruments would need to be deployed. This study investigates a new combination of measurement techniques capable of quantifying discharge over a large area on a fine spatial and temporal scale, expanding the use of heat as a tracer.

[4] The use of heat as a naturally occurring tracer is expanding rapidly with new analytic solutions to quantify discharge [Stallman, 1965; Hatch et al., 2006; Keery et al., 2007]. Developments in waterproof measurement devices have made accurate, reliable data collection both easier and more affordable [Anderson, 2005]. These equations and instruments leverage the continuous exchange of energy that is transferred throughout a streambed from solar heating. By extracting the amplitude ratio or diurnal phase-shift from pairs of time series thermal data collected in a streambed, discharge can be estimated [Goto et al., 2005; Hatch et al., 2006; Keery et al., 2007]. Conditions upon which this method is founded assume vertical and steady flow of water, and quasi-steady (cyclic) flow of heat, through homogeneous, isotropic, fully saturated sediments, and temperature measurements collected in a vertical profile. Recent studies have focused on assessing the method's validity under nonideal conditions. Numerical modeling studies suggest nonvertical flow conditions are the greatest source of error in estimating flux [Lautz, 2010; Roshan et al., 2012]. The influence of bedform induced hyporheic flow on temperature gradients was demonstrated numerically to impact vertical temperature profiles [Cardenas and Wilson, 2007]. Studies have also considered how transient groundwater flow effects flux estimation [Lautz, 2012; Jenson and Engesgaard, 2011]. Sensitivity analyses have demonstrated the uncertainty of flux calculations due to inaccurate thermal properties and sensor spacing [Munz et al., 2011; Schornberg et al., 2010; Shanafield et al., 2011].

[5] This research represents the initial lab testing of a new combined method to locate and quantify spatially discrete zones of groundwater discharge using the amplitude-shift time series method over extended domains. This method utilizes time series data collected using a distributed temperature sensor (DTS) from stacked lengths of fiber-optic cable, one cable on top of the other, that run parallel along an experimental streambed with a small vertical separation. To test this method, a streambed was modeled numerically and physically. COMSOL Multiphysics (COMSOL, Inc., Burlington, MA) was selected to numerically model groundwater flow and heat transport, while a 10 m sandbox model was constructed to test the method physically. The objective of this study is to assess the viability of using pairs of DTS time series profiles to quantify and locate zones of discrete groundwater discharge throughout an entire stream reach. To evaluate the viability of the DTS/amplitude method, a series of discharge scenarios were conducted in a controlled laboratory setting where all variables were known. To assess the spatial resolution of the method and its ability to distinguish multiple localized flow zones, two discrete discharge zones were systematically moved closer together. The physical domain was further instrumented with piezometers and additional thermal probes to evaluate the new method. Time series were filtered using dynamic harmonic resonance and processed using the amplitude time series method introduced by Hatch et al. [2006] using VFLUX, a Matlab script [Gordon et al., 2012].

2. Background

2.1. Fiber-Optic Distributed Temperature Sensing: Two-Dimensional Qualitative

[6] DTS is an example of a temperature logging technology that has been adapted to hydrologic study with exciting implications [Selker et al., 2006a; Tyler et al., 2009]. By measuring variations in backscattered light returning along a fiber-optic cable, the DTS is capable of collecting temperature measurements at a fine scale (1–2 m) over distances up to 30 km with a precision as low as 0.01°C. As result, the DTS is an attractive tool to obtain continuous, extended temperature measurements. While traditional temperature sensors utilized by GW/SW investigations are point measurements, capable of collecting data at only one location, DTS facilitates synoptic measurements along a stream reach.

[7] Use of DTS has focused mainly on spatially variable, qualitative measurements of discharge to streams. Researchers have installed DTS along streambeds, at the sediment water interface, to identify gaining zones of streams [Lowry et al., 2007; Selker et al., 2006b]. This technique identifies spatially discrete zones of discharge based on the assumption that groundwater is cooler than surface water, during the summer months, and dampens the amplitude of time-series measurements collected at the sediment interface (Figure 1a). Infiltrations, or losing zones, are not detected by this method however, as downward flux has no effect on the temperature at the GW/SW interface. When used in combination with a coupled groundwater/heat balance model DTS measurements can be used to quantify groundwater discharge on a reach scale [Westhoff et al., 2007; Briggs et al., 2011].

Figure 1.

(a) Conceptual model of fiber-optic DTS measurements for identify gaining reaches. Arrows represent magnitude of groundwater discharge and the shaded boxes at the streambed surface represent the spatial averaging performed by the DTS cable. (b) Conceptual setup for collecting pairs of time series measurements using fiber-optic DTS data. These data pairs are processed with amplitude ratio flux calculation to measure flux throughout an entire streambed on a fine scale.

2.2. Amplitude Time-Series Method: One-Dimensional Quantitative

[8] While DTS is primarily used as a qualitative tool, point measurements of streambed temperatures are traditionally used as a quantitative tool to measure GW/SW discharge. Based on the one-dimensional governing equation for heat transport, GW/SW discharge can be computed using paired time-series temperature records from a streambed [Goto et al., 2005; Hatch et al., 2006; Kerry et al., 2007]. The one-dimensional governing equation derived by Stallman [1965] used to describe advection and dispersion of heat in saturated sediment is

display math(1)

where T (°C) is temperature, which varies with time, t (days), and depth, z (m). ρw (kg/m3) and cw (KJ/(kg·K)) are density and specific heat of the fluid. ρ(kg/m3), c (KJ/(kg·K)), and λ (W/(m·K)) are the density and specific heat, and thermal conductivity of the rock/fluid matrix. Specific discharge is described by q (m/d). This equation can be applied under the following assumptions: fluid flow is steady and uniform in a vertical profile, heat characteristics of the sediment and fluid are constant in space and time, and temperature of the water at every point in the sampled area is equal to the temperature of adjoining rock at all times [Stallman, 1965].

[9] Derived from the one-dimensional Stallman [1965] equation governing heat transport, two approaches have been adapted for calculating seepage flux [Hatch et al., 2006; Kerry et al., 2007]. By quantifying changes in the amplitude and phase-shift between pairs of subsurface temperature time-series data, rates of groundwater discharge can be estimated. Over a normal range of streambed thermal properties, sensor spacing, and normal diurnal cycle the amplitude-shift method allows for consistent estimation of discharge rates between −5 and 3 m/d [Hatch et al., 2006]. Flux estimations based on changes in diurnal phase-shift between sensors pairs can calculate discharge within ±10 m/d [Hatch et al., 2006]. The amplitude-shift method is most sensitive to low seepage rates, where heat conduction influences the thermal profile. Conversely, the phase-shift method is more effective when flow is greater than ±1 m/d, and subsequently convection governs the thermal profile [Hatch et al., 2006]. The following research focuses exclusively on the amplitude-shift method as discharge rates throughout the study will remain below 1 m/d.

[10] Hatch et al. [2006] analytically defines the relationships between streambed thermal properties, seepage flux, and the amplitude ratio between diurnal peaks from a deep and a shallow temperature time series (Ar = Ad/As):

display math(2)

where math formula, v (m/d) is the velocity of the thermal front, P is the period of temperature variation [P = 1/frequency], Δz (m) is the spacing between two measurement points in the streambed, and ke (m2/d) is the effective thermal diffusivity [ke = λ/ρc].

2.3. Combined One-Dimensional Quantitative and Two-Dimensional Qualitative

[11] Combining the one-dimensional quantitative and the two-dimensional qualitative DTS methods provides a new combined method for measuring flux throughout an entire stream on a fine spatial and temporal scale. As described above, the one-dimensional method focuses on measuring variations in temperature time-series amplitudes between pairs of subsurface sensors to determine specific discharge at point locations. DTS has primarily been used in hydrologic studies as a qualitative tool, capable of distinguishing gaining zones from nongaining throughout an entire stream. By installing evenly spaced pairs of fiber-optic cable that run parallel to each other at two depths below the streambed with one on top of the other, distributed temperature time-series data can be collected in a vertical profile along the length of a stream reach. Applying the amplitude-shift method to each pair of overlapping time-series measurements allows for quantitative flux estimates longitudinally down a stream. This will enable researchers to pin-point and measure groundwater discharge on a fine scale (Figure 1b). A similar method utilizing layered fiber-optic cables to collect distributed temperature time series was recently applied in a recharge basin to quantify aquifer recharge [Becker et al., 2013].

3. Numerical Modeling Methods

[12] In order to test the feasibility of using paired fiber-optic cables to quantify groundwater discharge along a stream reach a forward analytical model was used to derive expected temperature signals in the streambed with the flexible finite-element analysis solver COMSOL Multiphysics. The numerical model fully couples two time-dependent partial differential equations (PDE) to solve the governing equations for both heat transport equation (1) and groundwater flow equation (3):

display math(3)

[13] Each PDE is controlled by selecting and defining the model's boundaries, with Dirichlet (variable) or Neumann (specified flux) conditions.

[14] The numerical model domain is a 10 m × 1 m rectangle, which corresponds to the dimensions of the physical model described in the next section. COMSOL discretizes the domain in a mesh of Lagrange-quadratic triangular elements with node spacing less than 0.02 m. The upper boundary is described as a Dirichlet boundary condition for both PDEs. The governing equation for groundwater flow treats the upper boundary as a constant head of 0 m. In the case of the heat transport PDE, the boundary is described by a diurnal cycle of temperature, which represents solar heating of surface water:

display math(4)

where T(t) represents changing temperature with time, Tave is the average about which the temperature fluctuates, Tamp is the amplitude fluctuations, and τ is the period of fluctuations. Period τ = 8 h (0.333 days), Tave = 30°C, and Tamp = 4°C in all simulations, which match the physical model.

[15] The boundary conditions on either end of the model are considered Neumann boundaries in both sets of governing equations and the specified flux is set equal to zero. With regard to the heat transport PDE, the entire bottom boundary of the model is classified as a Dirichlet boundary and represents average groundwater temperature at depth, which is considered 10°C in this model. The bottom boundary of the model simulating groundwater flow is classified as either Neumann or Dirichlet, depending on the discharge scenario. For example, when water is discharging from a designated segment, it is classified as a Dirichlet boundary and is assigned a constant head value of 0.1 m. This facilitates an upward gradient of 0.1, resulting in vertical upward flow from the bottom boundary to the top. Alternatively, when a segment is designated as a no-flow zone the condition is reclassified as a Neumann boundary, where flux is set to zero.

[16] The model was benchmarked using the Stallman analytical solution [Stallman, 1965] and the Matlab toolbox VFLUX (Vertical Fluid Heat Transport Solver). VFLUX [Gordon et al., 2012] calculates one-dimensional vertical fluid flow (seepage flux) through saturated porous media, using heat transport equations developed by Hatch et al. [2006] and Keery et al. [2007]. To calculate flux, VFLUX uses temperature time-series data measured by paired sensors in a vertical profile. Using dynamic harmonic regression, the program filters each time series and isolates a fundamental temperature signal identified by its period of oscillation. Finally, the program calculates vertical flux between pairs of temperature sensors by measuring the amplitude ratio and phase-shift between the filtered temperature signals, according to Hatch et al. [2006] and Keery et al. [2007].

[17] To compare the numerically modeled thermal profiles with data collected with the FO-DTS the COMSOL data were spatially averaged in the same fashion. Horizontal thermal values were exported from the surface (0 m) and at depth (0.1 m) to mimic the horizontally stacked DTS cables. The numerical thermal data were then averaged over 1.5 m, and reported every meter to reflect the spatial integration and sampling interval specific to the DTS unit used in the physical model.

4. Physical Modeling Methods

4.1. Dimensions, Materials, and Construction

[18] To assess the viability of the combined DTS/amplitude-shift method a physical model was built to simulate a streambed and underlying aquifer. The physical model has a volume of 3.75 m3; it is 5.0 m long, 0.6 m wide, 1.25 m tall, and open on the top (Figure 2). There is a 1.05 m tall centerboard inside of the physical model, which is attached at one end. The centerboard extends 4.7 m along the physical model, dividing the sandbox model in two halves, effectively doubling the length of the experimental design while maintaining a relatively small footprint (Figure 2). The bottom of the physical model is divided into ten rectangular discharge bays. Each bay is 1.0 m long × 0.3 m wide and 0.2 m tall. Each bay has a perforated PVC pipe that is used to deliver water to each 1 m discharge location (Figure 2). Discharge is controlled by a system of valves that allow the user to dictate which bay receives pressure exerted from the constant head bath. The constant head bath is constructed from a 55 gallon drum with an overflow spout allowing 0.1 m of head change across the physical model (Figure 3). There are five piezometer nests centered in the middle of bays 1–5. Each nest has a piezometer screened in a vertical profile descending from the surface every 20 cm (i.e., 20 cm, 40 cm, 60 cm, and 80 cm, Figure 3). An overflow spout is located 1.15 m from the bottom and discharges water out of the system (Figure 3).

Figure 2.

Three-dimensional schematic of the sandbox dimensions. Demonstrates bay numbering, and how the domain was effectively doubled.

Figure 3.

Side view of physical model showing sediment distribution and location of fiber-optic cable, temperature loggers, and piezometers.

4.2. Sediment

[19] The physical model is filled with approximately 5 tons of sediment. Within each discharge bay on the bottom of the model there is 0.15 m of gravel where the piping discharges to each bay. The gravel acts as a highly conductive unit where the water is expelled from the pipe and spread throughout the bay before it discharges uniformly into the next layer. Above the gravel is 0.85 m of sand intended to mimic a sandy streambed. This layer consists of homogeneous fine sand and has a mean grain size of 0.155 mm. As the sand was added to the physical model, it was tamped down every 0.1 m to ensure even packing. The hydraulic conductivity of the sand was estimated using the Kozeny-Carman equation. The hydraulic conductivity was further constrained to approximately 3.0 m/d measuring discharge from the overflow spout, treating the entire physical model as a single permeameter.

[20] Thermal properties of the sediment were measured using a KD2 Pro (Table 1), with the SH-1 sensor (Decagon Devices, Pullman, WA, USA). The sensor's thermal conductivity accuracy is ±10% at conductivities between 0.02 and 2 W/m·K, and the volumetric heat capacity accuracy is ±10% at conductivities above 0.1 W/m·K.

Table 1. Thermal and Hydraulic Properties of Sand Box Sediment and Water
ParametersSymbolValue/Range (10–34°C)Units
Effective thermal conductivityλ1.644W/(m·K)
Volumetric heat capacity of sedimentρgcg2.78MJ/(m3·K)
Specific heat of watercw4.180–4.195KJ/(kg·K)
Density of waterρw999.7–994.08kg/m3
Viscosity of waterμw112,924–62,121m2/d
Porosityn0.34
Hydraulic conductivityK3.0m/d

4.3. DTS Installation

[21] In order to measure spatially distributed changes in temperature within the streambed the sandbox is instrumented with fiber-optic cable. Lengths of parallel cable span the 10 m domain of the physical model and are centered at the surface (0 m) and at mean depths of 0.128 m and 0.8 m below the surface (standard deviation of 0.01 m) (Figure 3). Within the cable sheathing are four individual optical fibers that are paired in a “double-ended” configuration. This setup allows the DTS to send a laser in both directions along the cable to minimize drift in interpretations of temperature along the cable length [van de Giesen et al., 2012]. A four-channel Sensornet Oryx SR DTS (Sensornet Ltd., Elstree, Hertfordshire, UK) unit was used to detect variations in the returning light frequency and to extrapolate a temperature signal every meter along the cable. The Sensornet Oryx SR FO-DTS collects measurements with 1 m sampling intervals, which is assumed to provide a ∼1.5 m spatial integration (P. Bennet, personal communications, 2013). For example, if the cable reports a temperature at 5.0 m the DTS has averaged temperature measurements collected from 3.5 to 5 m along the cable. A 0.75 m correction shift is applied to all measurement positions to better represent where the temperature value is collected. This recording technique may be unique to Sensornet units and should be investigated before applying the DTS to future studies. To assure accuracy and correct for drift of the double-ended configuration a dynamic calibration scheme was set up with two 20 m coiled reference sections of cable. One coil submerged in a hot water bath, and a second set of coils in a cold water bath. The hot water bath was maintained with a basic heater, while the cold water bath was stocked with ice. The DTS integrated over 20 min sampling intervals and report temperature every 40 min for each channel pair.

4.4. Additional Temperature Data Loggers

[22] In addition to the DTS, point measurements are collected in each bay using nested pairs of thermocouples. The sensors are centered directly over each bay at the surface and at 0.1 m depth, recording temperature every minute. The discharge calculated from the thermal record from each pair of sensors was compared with the discharge calculated by the DTS values at the same location. These additional measurements allow us to assess the use of DTS measurements for thermal discharge studies.

[23] Two types of thermocouples were used: duel node HOBO Pro v2 2X External Temperature Loggers U23-003 (Onset Computer Cooperation, Pocasset, MA, USA), and Omega Engineering RDXL4SD 4-Channel Datalogger Thermometers (OMEGA Engineering, Inc., CT, USA). The HOBO measurement accuracy is 0.2°C over a range of 0–50° and has a sensor resolution of 0.02°C. These loggers were buried with the DTS cable. The Omega Thermometers were connected to Type K Thermocouple probes on rods with 0.1 m spacing. The Type K thermocouples have a range of −50 to 1000°C and have a measurement accuracy of 0.5°C with a resolution of 0.1°C.

4.5. Temperature Control

[24] In order to simulate the daily heating and cooling of the stream, the sandbox model uses a custom heating system designed by Omega Engineering, which includes two industrial heating elements, a temperature controller, and software to program and monitor the temperature signal. Two 4kW immersion heaters (MT-230A/208V) screw into the side of the sandbox model, extending 0.3 m into the surface water. Circulation pumps flush water past each heater to keep the surface water temperature well mixed. The heaters were programmed to produce a repeating sinusoidal Temperature Signal, based on equation (4), in the surface water, oscillating from 26° to 34°C to mimic a daily temperature cycle. The period was compressed from a normal 24 h period to 8 h as to expedite data collection. As result, the averaged stream temperature is also warmer than typically found in the field as to facilitate a more rapid cooling of the surface water required by the shorter 8 h period.

4.6. Physical Model Operation

[25] A series of five experiments were conducted, simulating a range of discharge scenarios, which are commonly observed in naturally occurring streams. The physical model was run under conditions where discharge was uniform throughout the entire domain, a no-flow scenario, and three localized flow regimes where only two bays were discharging separated by 5 m, 3 m, and 1 m. Localized discharge scenarios were simulated by manipulating the head in specific bays along the bottom of the model. Scenarios with discharge present were conducted under 0.1 m of head change inducing upward flow. Each discharge scenario was allowed to run for at least 48 h or six diurnal cycles (8 h each) allowing the temperature profile to reach cyclic steady state. A Matlab code was used to extract the time stamp from each DTS file, as well as isolate the 10 m sections of cable at the surface and at 0.128 m depth. VFLUX [Gordon et al., 2012] was used to estimate discharge between the pair of cables along the experimental domain.

[26] Calibration of particular variables was required to address inaccuracy of initial flux calculations. Using the temperature time series collected during the no-flow scenario, variables with associated uncertainty were calibrated to return zero flux values. Because the sediment is homogenous, adjusting the thermal properties of the sand were not a viable option as the flux calculations were not consistently under or over estimating flux throughout the model domain. Instead, inaccuracy in the spacing value between the paired cables was determined to be the erroneous variable. These new calibrated spacings were maintained throughout the rest of the experiment runs. Following the final experiment, the cable was carefully excavated every meter along the model to determine true cable spacing. The standard deviation of the difference between the calibrated and the remeasured spacing is 0.006 m. The discrepancy between the actual measured depth and the calibrated depth is likely the result of the 1.5 m spatial integration of our DTS instrument. (i.e., The temperature reported is averaged over a 1.5 m length of cable whose spacing fluctuates longitudinally).

5. Results and Discussion

5.1. Numerically Modeled Limitations of the DTS Method

[27] Before the physical model results are analyzed it is important to understand the limitations of spatially averaged temperature data when applied to the amplitude-shift flux estimation. Flux rates estimated using spatially averaged temperature data are not affected when groundwater discharge is uniform as the temperature profile does not deviate horizontally. During conditions with focused groundwater flow the temperature profile varies as a result of a horizontal competent to groundwater flow and error is introduced. A 1 m wide discharge conduit, 1 m from the streambed surface, was numerically modeled with 0.1 m of head gradient to evaluate the limitations of data collected by this method (Figure 4). Temperature values, calculated using the numerical model, were averaged and filtered to reflect the data collected by the DTS; the temperature record was collected along a horizontal line with a 1 m sampling interval and a 1.5 m spatial integration. Figure 4 highlights several sources of error associated with calculating flux measurements from DTS data: spatial integration, measurement offset, and nonideal sample positioning.

Figure 4.

The color bands in (a and b) represent the temperature profile as result of a 1 m focused discharge zone (red is warm and blue is cold). The lines with shaded boxes near there surface represent a FO-DTS cable and potential sample locations which lead to possible limitation associated with (a) offset (I, II, III) and (b) positioning (I, II, III). Graphs (c) (I, II, III) demonstrate numerically modeled flux calculations (in blue) as result of nonideal offset and positioning error. The conceptual sampling location illustrated in Figures 4a and 4b represent the results seen in graph Figure 4c. (i.e., When the cable is offset by 50 cm in Figure 4aIII, then the estimated flux will be represented as shown in graph Figure 4c). The gray line represents vertical Darcy flux modeled numerically. The stars in graphs Figure 4c represent the shaded reference section in Figures 4a and 4b.

[28] An adverse effect of DTS spatial integration (1.5 m) is a loss of spatial resolution as compared to the precision associated with an array of temperature rods. Even under best case scenarios, where there is no offset of the sensor pairs and the measurement is centered over the discharge conduit (Figures 4aI and 4bI), the thermal signal is dampened by the inherent nature of temperature measurements averaged over 1.5 m sections of the cable. In this configuration, some portions of the fiber-optic cable are not located directly above the discharge zone (Figure 4cI).

[29] The distributed nature of DTS data makes it difficult to assure that a section of cable corresponds directly with its underlying counterpart, violating the vertical profile assumption. The potential offset experienced by a measurement is illustrated by Figure 4I, where the averaged cable sections are offset by 0.25 m (Figure 4aII) and 0.5 m (Figure 4aIII). Nonideal positioning is portrayed conceptually by Figures 4bII and 4bIII, where the measurement is displaced by 0.25 m and 0.5 m, respectively. The discharge profile is the same when the cable is either offset or not centered over the zone of groundwater discharge by the same distance. For example, if the cable is offset by 0.25 m (Figure 4aII) the discharge profile (Figure 4cII) is the same as if the measurement is positioned 0.25 m from the point of maximum flux over a discharge conduit (Figure 4bII). Once a measurement is offset by more than half of the sampling interval the adjacent temperature time series is more applicable.

[30] When studying focused discharge zones, it is best to measure the maximum flux that is taking place. In Figure 4c, the point of maximum discharge is described by the dashed line that runs through the center of the discharge conduit. When there is no offset of the cable and its measurement is positioned ideally over the point of maximum discharge (dashed line), the maximum flux is underestimated by 14% as result of temperature averaging (Figure 5). When the cable is not positioned directly over the discharge location, or if offset occurs, there is a characteristic lopsided effect to the discharge profile (Figure 4cII). As the cable reaches maximum offset (Figure 4cIII), the calculated flux appears to better fit the numerically modeled bell-shape true discharge profile. However, in this configuration the peak discharge has the largest percent error, underestimated by 27% (Figure 5) when positioning and offset are off by half of the sampling interval.

Figure 5.

Quantitative representation of the difference between true discharge and DTS measured specific discharge as result of offset, or nonideal positioning of the fiber-optic cable at a focused discharge zone.

5.2. Physical Model Data

[31] Each model experiment in the physical model took approximately three diurnal cycles for thermal conditions to approach cyclic steady state. Allowing the thermal regime in the model to come to steady state allowed stable flux calculation. All measurement values represent time averaged flux calculations from the final two diurnal cycles of each run (n = 20).

[32] Results collected from the experimental runs examining spacing between discharge locations can be seen in Figure 6. Figure 6a represents discharge estimates calculated with distributed temperature data collected from the physical model. Figure 6b represents numerical simulations of the same spacing experiments using data that has been filtered and averaged to mimic the DTS resolution and spatial integration. Flux estimations calculated from the temperature probe data centered over each bay in the physical model are shown in Figure 6c. Each of the graphs has been superimposed with the true discharge value modeled with COMSOL, which corresponds with Darcy flux measurements recorded by the piezometers in bays 1–5.

Figure 6.

Discharge scenario where discrete discharge locations are spaced (I) 5 m, (II) 3 m, and (III) 1 m (discharge locations are highlighted with light gray boxes). Amplitude ratio flux calculated from (a) physical model DTS data, (b) numerical DTS model, and (c) temperature probe data from physical model. True discharge, modeled using COMSOL, is superimposed on all graphs (gray line).

[33] The DTS data (Figure 6a: see blue shaded area), as well as the temperature probe data (Figure 6c: see blue bars), from the physical model fits closely with the results modeled numerically (Figure 6b: see blue shaded area). As was demonstrated numerically, the spatially averaged DTS data underestimates the maximum discharge flux by roughly 27%, leading us to believe that our cable was either offset or the measurement points were positioned nonideally. The spatially averaged numerical and physical DTS data differs most drastically from the actual flux (Figure 6: see gray line) at 1 m spacing between discharge zones (Figure 6III). The spatially averaged data leads to a combined peak, where two discrete discharge peaks are picked up by the temperature probes.

[34] While flux estimated from the distributed temperature measurements in the physical model fits closely with the spatially averaged data from the numerical model there are slight differences. In the physical model flux measurements do not drop off as rapidly as predicted with the numerical model moving away from the peak discharge value. This “bleeding over” of discharge results in overestimating discharge in zones were no discharge is present. The overestimation can be seen most clearly between the discharge peaks (Figure 6aI). This is potentially a result of our DTS instrument's spatial resolution being coarser than the 1.5 m spatial integration reported by the manufacturer [Rose et al., 2013].

[35] While the combined method is shown to be effective in the described laboratory experiments, there are several factors that must be acknowledged before applying it in the field. Centimeter scale variations in cable spacing lead to inaccuracy in flux calculations. As result, a procedure for accurately deploying the cable in the field must be established. The development of a combined fiber-optic cable that allows two physically connected fibers to run parallel with a rigid divider separating the two with a fixed spacing is recommended. This would allow an easy deployment of the instrumentation in a trench and avoid the error associated with inaccurate spacing. In a field setting, extensive sampling of streambed thermal properties along the sample profile will be required. Our laboratory experiment avoided any inaccuracy that could arise from heterogeneity in thermal properties by filling the physical model with homogeneous sand. In an actual streambed heterogeneity could cause significant error. The deployment of a DTS instrument with finer scale sampling interval and spatial integration would vastly improve the measurement capability of the described method. While the spatial integration of the instrument used in this study had a 1.5 m resolutions, DTS units exist that have integrations as low as 0.25 m, and will likely become more refined as the technology advances.

6. Conclusions

[36] By extracting the amplitude ratio or diurnal phase-shift between pairs of subsurface temperature time-series measurements, GW/SW interactions can be quantified. Applying the analytical one-dimensional amplitude-shift method to two-dimensional DTS data proved to accurately locate and quantify discharge on a fine scale (3+ m) in this pilot study and exhibited some integration-based issues at smaller scales (1–2 m).

[37] Discharge estimated using spatially averaged temperature data is accurate where groundwater discharge is uniform, because the temperature signal is uniform along a horizontal plane. Error is introduced when spatially variable groundwater discharge exists, resulting in temperature profiles that vary laterally at the edge of the discharge zone. Spatial averaging inherent to DTS data results in dampening of discrete discharge measurements, because temperature is averaged over sections where discharge is not uniform. This leads to underestimating peak discharge at localized discharge “hot-spots,” and overestimating discharge as measurements move away from these locations.

[38] Uncertainty associated with the precise location of DTS measurements can lead to offset temperature pairs, violating the one-dimensional vertical profile assumptions. The maximum offset or positioning failure possible with a measurement is equal to half of the sampling interval. When measurement pairs are collected with maximum offset or misalignment in positioning the localized discharge modeled in this study was underestimated by as much as 27%. When measurement pairs were synchronized, and positioned directly over the modeled discharge, flux was underestimated by 10%. Limitations of this technique could be minimized by implementing a DTS unit with finer scale resolution.

[39] It is important to understand the scale on which the new combined method is capable of reporting data. For example, a temperature pole is a highly localized measurement that can accurately estimate flux at one unique point. Tracer injection tests can quantify flux as well, however over a much larger area, sacrificing resolution for ease of data collection. The experiments conducted in the physical model as well as numerically help us to understand the resolution of this new combined method. Simulated flux profiles with different spacing between focused discharge zones were modeled to determine the scale at which the method is capable of accurately discerning and quantifying individual discharge locations. The effective scale of this combined technique is roughly equal to the spatial integration of the DTS unit. As seen by the physical and numerical modeling, when the focused discharge zones were only 1 m apart the individual peaks were washed out and the flux profile seemed to indicate only one wider discharge location.

[40] In conclusion, the combined method presented in this proof of concept study has many advantages over existing methods. Hydrogeologic tools such as seepage meters, shallow piezometer nests, and temperature sticks are capable of accurately estimating single point in space flux. To gain a more thorough understanding of discharge throughout the entire stream hundreds of these instruments would need to be deployed. Other methods such as regional groundwater modeling, heat balance modeling, and differential gauging are capable of estimating flux to a stream on a broader scale. By combining elements of the one-dimensional amplitude-shift flux estimation with qualitative two-dimensional DTS sampling, the new method can quantify streambed discharge throughout an entire stream on a fine spatial and temporal scale. Applying the time-series approach to DTS sampling will help hydrologists to better understand complex temporal and spatial dynamics of stream ecology, fishery habitats, lakes, and managed aquifer recharge (MAR).

Acknowledgments

[41] This work was partially funded by a NSF Equipment and Facilities award (#0824829). We gratefully acknowledge the assistance of Thomas Gruenauer, and Kevin Cullinan from the University at Buffalo Machine shop, in the design and construction of the physical model. We would like to thank Elizabeth Ceperley, Simon Pendleton, Katie Feiner, Laura Best, Peter Johnson, Samuel Kelley, Timothy O'Brian, Sandra Cronaur, and Carolyn Eve for assistance with model setup. Finally, we would like to thank Jim Constantz and two anonymous reviewers for their helpful comments that improved the manuscript.