aquifer exchange flux - Wiley Online Library

PUBLICATIONS Water Resources Research RESEARCH ARTICLE 10.1002/2014WR016741 Key Points: River temperature was used to estimate reach-scale groundwater inflow River temperature is most sensitive to weather parameters Only a wide range of groundwater inflow can be estimated with river temperature Supporting Information: Supporting Information S1 Correspondence to: Y. Xie, yueqing.xie@flinders.edu.au Citation: Xie, Y., P. G. Cook, C. T. Simmons, and C. Zheng (2015), On the limits of heat as a tracer to estimate reach-scale river-aquifer exchange flux, Water Resour. Res., 51, 7401–7416, doi:10.1002/2014WR016741.

On the limits of heat as a tracer to estimate reach-scale river-aquifer exchange flux Yueqing Xie1, Peter G. Cook1, Craig T. Simmons1, and Chunmiao Zheng2,3 1 National Centre for Groundwater Research and Training, School of the Environment, Flinders University, Adelaide, South Australia, Australia, 2School of Environmental Science and Engineering, South University of Science and Technology of China, Shenzhen, China, 3Department of Geological Sciences, University of Alabama, Tuscaloosa, Alabama, USA

Abstract For the past few decades, heat has been used to estimate river-aquifer exchange flux at discrete locations by comparison of river and groundwater temperature. In recent years, heat has also been employed to estimate reach-scale river-aquifer exchange flux based only on river temperature. However, there are many more parameters that govern heat exchange and transport in surface water than in groundwater. In this study, we analyzed the sensitivities of surface water temperature to various parameters and assessed the accuracy of temperature-based estimates of exchange flux in two synthetic rivers and in a field setting. For the large synthetic river with a flow rate of 63 m3 s21 (i.e., 5.44 3 106 m3 d21), the upper and lower bounds of the groundwater inflow rate can be determined when the actual groundwater inflow is around 100 m2 d21. For higher and lower fluxes, only minimum and maximum bounds, respectively, can be determined. For the small synthetic river with the flow rate of 0.63 m3 s21 (i.e., 5.44 3 104 m3 d21), the bounds of the groundwater inflow rate can only be estimated when the actual groundwater inflow rate is near 10 m2 d21. In the field setting, results show that the inflow rate must be less than 100 m2 d21, but a lower bound for groundwater inflow cannot be determined. The large ranges of estimated groundwater inflow rates in both theoretical and field settings indicate the need to reduce parameter errors and combine heat measurements with other isotopic and/or chemical methods.

Received 27 NOV 2014 Accepted 19 AUG 2015 Accepted article online 21 AUG 2015 Published online 12 SEP 2015

1. Introduction Reliably quantifying the exchange flux between a river and the adjacent groundwater is essential for conjunctive management of surface water and groundwater. A number of methods have been used for quantifying reach-scale groundwater discharge into rivers, including differential flow gauging, tracer dilution gauging, environmental tracers, geophysics, temperature, and numerical modeling [Briggs et al., 2012; Cook et al., 2006; McCallum et al., 2012; Harrington et al., 2014; Lowry et al., 2007; Slater et al., 2010]. Kalbus et al. [2006] conducted a comprehensive review of methods for quantifying groundwater-surface water interaction, while Constantz [2008] and Cook [2013] reviewed the use of heat and chemical tracers, respectively. Among these methods, heat has gained much popularity in the past few decades for tracing water movement from point to basin scales [Anderson, 2005].

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The majority of studies that use heat to estimate groundwater-surface water exchange flux are based on point measurements [e.g., Anibas et al., 2009; Becker et al., 2004; Duque et al., 2010; Hyun et al., 2011; Luce et al., 2013]. These studies typically compare the diurnal temperature signal in the surface water and the subsurface and estimate the vertical water velocity based on either the amplitude attenuation and/or phase lag of the temperature signal. A few studies have also attempted to estimate groundwater-surface water exchange flux based on downstream changes in river temperature [Loheide and Gorelick, 2006; Westhoff et al., 2007; Loinaz et al., 2013; Brookfield et al., 2009]. This approach is potentially much more suitable for obtaining catchment or river-basin-scale estimates of exchange flux, since surface water temperature can be measured at large scales using fiber optic cables [Westhoff et al., 2007] or thermal scanners operated from aircraft [Loheide and Gorelick, 2006]. However, river temperature is affected by a large number of processes, including heat exchange not only between surface water and groundwater, but also between surface water and the atmosphere (i.e., solar shortwave radiation, atmospheric longwave radiation, stream back radiation, convection heat flux, and latent heat flux). Thus, models that seek to estimate groundwater inflow from river temperature data must consider these processes. However, how sensitive surface water

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temperature is to these processes in numerical models is still unclear. Westhoff et al. [2007], Loinaz et al. [2013], and Brookfield et al. [2009] treated all weather parameters as calibration parameters, whereas Loheide and Gorelick [2006] assumed that weather parameters were perfectly known from various sources. However, most weather parameters cannot be estimated accurately. Hence, whether surface water temperature can be used as a reliable tool to estimate reach-scale groundwater-surface water exchange flux will depend upon (i) the sensitivity of river temperature to groundwater exchange, relative to its sensitivity to other parameters, and (ii) the accuracy with which these parameters can be estimated. Sinokrot and Stefan [1994] analyzed the sensitivity of stream temperature to different parameters and concluded that river temperature is more sensitive to solar radiation and air temperature than to other parameters. However, their model assumed that the stream did not gain or lose any water during the simulation period, and hence water temperature did not vary with distance. They were therefore not able to examine the sensitivity to groundwater exchange. Westhoff et al. [2007] conducted a sensitivity analysis in a calibrated model with different governing equations, particularly for longwave radiation and latent heat flux. They found that cloud and canopy cover was the most influential parameter on river water temperature in their specific case. However, the only parameters that were allowed to vary during calibration were fraction of solar radiation reaching the streambed, fraction of diffuse solar radiation, cloud and canopy cover, and conductive layer thickness. In this study, we examined the conditions under which heat can be used as a reliable tracer to estimate river-aquifer exchange flux at the reach scale. To achieve this, a model was first developed to simulate water flow and heat transport in rivers. A sensitivity analysis was then conducted on two 32 km long synthetic rivers, representing typical upland and lowland rivers. Next, hypothesis testing was performed to determine whether a temperature time series simulated using a particular set of parameters could be equally explained using different values of the exchange flux, without other parameters changing by more than their allowable uncertainty ranges. Finally, the same hypothesis testing method was applied to field data from the Heihe River in northwest China, to determine the accuracy with which river-aquifer exchange flux can be determined.

2. Modeling 2.1. Numerical Approach Heat transport in a river is controlled by several concurrent processes: heat exchange with the atmosphere, heat exchange with the underlying subsurface, flow dynamics, and dispersive mixing. Heat exchange with the atmosphere consists of solar shortwave radiation, atmospheric longwave radiation, river surface back radiation, convection heat flux due to turbulent and molecular heat exchange, and latent heat flux caused by evaporation and condensation. Heat exchange with the underlying subsurface consists of riverbed conduction, heat flux associated with river-aquifer exchange, and heat flux due to hyporheic exchange. Equations governing these processes are documented in Appendix A. In this study, a new numerical code RiverHeat incorporating these processes was written using C11 in order to simulate both river flow and heat transport simultaneously. A fully implicit finite difference scheme was adopted and the Newton-Raphson technique was used to linearize nonlinear equations. The assembled matrix was then solved iteratively using the Gauss-Seidel matrix solver method with a constant time step. The performance of this code was evaluated by comparing results of several 1-D flow and transport examples to those produced by the physically based numerical code HydroGeoSphere [Therrien et al., 2010]. Results produced from both RiverHeat and HydroGeoSphere are identical if convection and latent heat fluxes are not included. In our code, different empirical equations were used to calculate convection and latent heat fluxes because these equations are more commonly used than those in HydroGeoSphere. Examples of this comparison can be found in the supporting information. 2.2. Model Setup Two synthetic 1-D longitudinal models representing two contrasting types of perennial rivers (upland and lowland) were established. Here it is assumed that heat was well mixed within the river both transversely and vertically, as has been assumed in many surface water-groundwater interaction studies using temperature or chemical tracers [Cook et al., 2006; Loheide and Gorelick, 2006; Westhoff et al., 2007]. It is also assumed

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Figure 1. Temperature variation in space and time of (a) lowland large river and (b) upland small river.

that the cross sections of both rivers are rectangular in shape and subsurface thermal properties do not vary with temperature. In the base cases of both models, all input parameters except solar radiation and air temperature were fixed in space and time. These constant parameters were representative of mean values in the respective river settings. Although many of these parameters (e.g., wind speed and cloudiness) will vary both spatially and temporally, the use of constant parameters allows the sensitivity analysis to be more easily understood. The first model represents a lowland alluvial river, and is loosely based on the middle reach of Heihe River in northwest China [Chen et al., 2006; Yang et al., 2011]. These rivers tend to be very wide and shallow and have large discharges. In the base case, the model was 32 km long and 70 m wide. The river discharge rate (Qsw) at the upstream boundary was 63 m3 s21 (i.e., 5.44 3 106 m3 d21), and the average groundwater discharge rate per reach length (Qgw) was 100 m2 d21. Thus, at the downstream boundary, the total groundwater discharge accounted for 37% of the total river discharge. Weather parameters are based on climate in the Heihe River basin in late October, to facilitate comparison with the field example. Air temperature (Ta) was assumed to follow a sinusoidal wave, with daily minimum and maximum temperatures of 2 and 188C, respectively. The groundwater temperature (Tgw) was constant at 148C, higher than mean Ta (Figure 1a). The hyporheic exchange was assumed to be negligible and so the exchange flux (qhyp) was set to zero. All other relevant constant parameters are provided in the Lowland Large River column of Table 1. The second model was largely based on upland alluvial rivers [Loheide and Gorelick, 2006]. Our base case model was 32 km in length and 2 m in width, although field studies usually consider river reaches less than 2 km [Loheide and Gorelick, 2006; Westhoff et al., 2007]. This upland river model was deliberately made long to allow a relatively high proportion of groundwater flow into the river. Qsw at the upstream boundary (0.63 m3 s-1 or 5.44 3 104 m3 d21) and Qgw (1.0 m2 d21) were 2 orders of magnitude smaller than for the lowland river. Hence, the total groundwater discharge along the 32 km long reach also accounted for 37% of the total river discharge at the downstream boundary. Incoming solar radiations, Ta and Tgw, were prepared by assuming a similar location and time to the river studied in Loheide and Gorelick [2006]. Ta ranged from a daily minimum of 118C to a maximum of 218C. In this case, Tgw was set to 78C, which is less than the mean Ta (Figure 1b). qhyp was also set to zero, same as that in the first model. All other relevant constant parameters are provided in the Upland Small River column of Table 1. Both models were uniformly discretized into 640 elements with an elemental length of 50 m. A constant time step size of 60 s was used. Because the water depth in both rivers changed downstream due to groundwater inflow and evaporation, the rivers did not reach a stable temperature that did not vary with distance. The upstream boundary for each model was a constant water depth based on the desired river flow rate, and specified diurnal temperature distribution. The downstream boundary was specified with a zero-depth gradient, which forces the river stage slope to equal the channel bed slope. The temperature boundary condition was derived by running the base case with no evaporation and no groundwater discharge. This created a constant flow in the river, and at a sufficient distance produced a diurnal temperature distribution that no longer changed with distance. This distribution was then used as the upstream

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Table 1. Parameter Values and Relevant Percent Errors for Both Large and Small Rivers Lowland Large River

Relative humidity RH (2) Wind speed U (m s21) Atmospheric pressure Pa (mb) Cloud cover Cc Groundwater inflow Qgw (m2 d21) Hyporheic exchange flux qhyp (m2 d21) Groundwater temperature Tgw (8C) River width B (m) River slope So Channel roughness n (m21/3 s) Specific heat capacity of water cw (J kg21 8C21) Specific heat capacity of solid cs (J kg21 8C21) Density of solid qs (kg m23) Thermal conductivity of water kw (J s21 m21 8C21) Thermal conductivity of solid ks (J s21 m21 8C21) Effective porosity h Longitudinal dispersivity bL (m) Conductive layer thickness Lh (m) Water surface reflectivity r

Upland Small River

Value

Percent Error

Value

Percent Error

0.35 4 880 0.2 100 5 14 70 0.001 0.03 4189 800 2538 0.59 2.2 0.35 10 0.2 0.2

20% 50% 1% 50% 10% 100% 5% 10% 20% 20% 0.4% 5% 20% 2% 45% 40% 40% 20% 10%

0.7 7 1000 0.2 1 0.1 7 2 0.005 0.03 4189 800 2538 0.59 2.2 0.35 10 0.2 0.2

20% 50% 1% 50% 10% 100% 10% 10% 20% 20% 0.4% 5% 20% 2% 45% 40% 40% 20% 10%

boundary for the models that included evaporation and groundwater discharge. Since all parameters are spatially uniform, if there were no groundwater inflow or evaporation, the temperature would not change with distance. Uniform initial conditions were used and their values were assumed to equal boundary conditions at the initial time. Simulations were performed for a simulation time of 4 days. If the diurnal temperature variation at 30 km did not reach a stable condition, the models were simulated for another 4 days, and then evaluated every 4 days until this condition was met. 2.3. Sensitivity Analysis A sensitivity analysis can aid in the identification of the most influential parameters in a specific system. In accordance with sensitivity analysis results, proportional efforts can then be dedicated in order to reduce parameter uncertainties. A number of methods can be used to perform the sensitivity analysis [Hamby, 1994]. In this study, the simplest method requiring one parameter perturbation at a time was adopted for demonstrative purposes. The equation describing the sensitivity of model output to parameter P is SP 5

Tn 2To Pn 2Po

(1)

where Pn and Po are values of parameter P in the new and base cases, respectively, and Tn and To are values of the output parameter in the new and base cases, respectively. For a specific system, N 1 1 model runs are needed to calculate SP with respect to N individual parameters using the forward finite difference. To avoid the potential numerical influence from too small parameter perturbation, parameter perturbation was set at 10%. Because SP has different units with respect to different parameters, SP can be multiplied by the estimated error of parameter P (e.g., 10%) to derive sensitivity (TP). This facilitates comparison because the units of TP will not depend on parameter P. TP is given by TP 5SP 3 rP

(2)

where rP is the parameter error. Often parameter errors are not well known, and so there can be some subjectivity involved in choosing the value of rP . In this study, assumed parameter uncertainties are given in Table 1. Of course, rP is also subject to physical conditions. For example, the cloudiness may have an error of 50% on a partially cloudy day, whereas on a sunny day the relevant error can be less than 5%. In our study, because temperature varies diurnally, sensitivity statistics are calculated for four temperature statistics: minimum temperature (TPmin ), maximum temperature (TPmax ), mean temperature (TPmean ), and temperature range (TPran ). TPran equals TPmax minus TPmin . In synthetic systems, temperature time series vary in regular XIE ET AL.


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forms. Hence, TPmean can be approximated by averaging TPmax and TPmin ; however, it has been calculated explicitly. 2.4. Hypothesis Testing Parameter uncertainties will contribute to model nonuniqueness, in which different combinations of parameters can produce the same observed data. Although automatic model calibration tools such as PEST [Doherty, 2010] can be used to achieve optimized parameters with lowest root mean squared error (RMSE) [Westhoff et al., 2007], the produced Qgw may be problematic due to model nonuniqueness. Hence, model calibration does not directly provide any information on whether temperature is a reliable tool for quantifying river-aquifer exchange flux. It is known that Qgw acts to buffer temperature against diurnal fluctuations due to heat exchange with other domains [Loheide and Gorelick, 2006]. It is possible that a different combination of parameters could mimic this buffering effect, i.e., the model may be nonunique. However, if a limit is imposed on the possible ranges of model parameters (other than Qgw) then this also probably poses a limit on acceptable values of Qgw. This range of Qgw results from the inherent parameter uncertainties. As parameter uncertainties are reduced, this range of Qgw will be narrowed accordingly. In this study, a hypothesis testing approach was used to identify the possible range of Qgw in specific systems with known parameter uncertainties. The approach involved several standalone PEST calibrations, each with a different but fixed value of Qgw, and other parameters allowed to be changed within their error ranges. Thus, the hypothesis that the data could be fitted equally well using a different value of Qgw was tested. Each calibration returned an optimized model with the lowest RMSE. The variation in RMSE with Qgw allowed us to determine the range of Qgw that could explain the observed (or modeled) temperature data. The hypothesis testing approach was used for both synthetic models and for the field example introduced later in this study. The application to the synthetic models was slightly different from the field example. For a synthetic model, forward modeling with specific parameters and Qgw was first carried out to generate temperature results. These results were treated as ‘‘measured’’ temperature time series for data fitting with different values of Qgw and allowed other parameters to vary over given ranges as previously described. In these simulations, hyporheic exchange with small fluxes was included and relevant qhyp values are listed in Table 1. Because the exact value of Qgw was known, this hypothesis testing approach allowed us to evaluate the uncertainty associated with the estimation of this parameter. For the field example, the actual Qgw is unknown, and so the method only produces a range of possible values.

3. Results 3.1. Qualitative Features of Surface Water Temperature Variation Temperature variation in the two rivers differs substantially. Tsw in the small river changed over 7–88C diurnally, slightly less than the Ta variation of 108C (Figure 1b). In comparison, the large river only exhibited about 1.58C diurnal variation in Tsw, when Ta changed by 168C (Figure 1a). This difference in the Tsw variation was caused by the river discharge rates which differed by 2 orders of magnitude. The large river requires much more thermal energy than the small river to change water temperature by 18C per reach length. Despite the greater range in Ta in the large river, Tsw was less variable. The difference between Tgw and mean Ta determines the direction of overall temperature change with distance. In the large river, Tgw (148C) was higher than mean Ta (108C). Hence, mean Tsw increased downstream due to groundwater inflow. In comparison, Tgw (78C) in the small river was lower than the mean Ta (168C), thereby causing mean Tsw to decrease downstream. Tsw will vary with weather conditions. On a sunny day, a river is exposed to more solar radiation, and so Tsw is higher than the base cases for both large and small rivers (Figure 2). In comparison, an overcast sky will reduce direct solar radiation and result in lowest Tsw. Similarly, wind will reduce Tsw as stronger air convection will cause greater heat loss. As groundwater discharge increases, Tsw will become more stable and mean Tsw will approach Tgw. In both rivers, tripling Qgw resulted in approximately 28C change in Tsw (Figure 2). The constant Tgw acts to buffer

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Figure 2. Temperature variation with time at the downstream location (30 km) for (a) large river and (b) small river. In each river, the case with larger Qgw has 3 times the groundwater inflow of the base case; the overcast case assumes the cloud cover of 0.9, much higher than that in the base case (0.2); the sunny case has a cloud cover of 0.05.

Tsw against diurnal change due to variations in air temperature and solar radiation. Depending on the contrast between mean Ta and Tgw, mean Tsw can be reduced or increased with more groundwater discharge. 3.2. Quantitative Analysis of Surface Water Temperature Sensitivities Surface water temperature sensitivities to individual parameters were examined at the downstream location (30 km) using equation (2) (Figure 3). Overall, the maximum change in any of the temperature statistics for the large river is 0.328C (TPmin with respect to wind speed, U), whereas the maximum change in the small river is 0.88C (TPran with respect to channel roughness, n). Despite greater sensitivities in the smaller river, the maximum uncertainty of Tsw is a larger percentage of the diurnal variation in Tsw in the large river (21.3% of 1.58C) than in the small river (11.4% of 78C). In general, the smaller synthetic river is more sensitive to perturbations of most parameters than the larger one, as evidenced by overall larger Tp values. This is due to lower thermal inertia and a greater surface water area to volume ratio in the smaller river. The lower thermal inertia renders the smaller river more responsive

Figure 3. Sensitivity of surface water temperature statistics to various parameters for both (a) large and (b) small rivers. Tp with respect to mean is an average of Tp with respect to max and min, while Tp with respect to range is the difference between Tp with respect to max and min.

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to any change in heat flux inputs. The greater surface water area to volume ratio allows the temperature of the smaller river to increase more quickly than that of the large river for a given percent change in heat fluxes. Some parameters need special attention in comparing their effects on Tsw in both large and small river settings. In comparison, relative humidity (RH) is a much more important parameter in the small river than in the large river. Apart from lower thermal inertia and greater surface water area to volume ratio, this is also attributed to the larger absolute error in RH in the small river. (The relative errors in RH are the same for both cases, but the absolute RH is larger in the small river case, and so the absolute error in RH is also larger.) For wind speed (U), all statistics except TPmax appear to be more responsive in the smaller river. This is because minimum temperature occurs at night when there is no solar radiation, whereas maximum temperature is often reached in the afternoon around the peak of solar radiation. As solar radiation is usually a dominant heat input, a slight change in U will cause more evident change in river thermal energy and temperature at night than during the day. Mean temperature and temperature range also change accordingly, as they are influenced by the minimum temperature. In addition, TPmax with respect to U is smaller in the small river than in the large river. This is attributed to the larger RH in the small river, which leads to larger air moisture content and hence larger vapor pressure near the water surface (equation (A13)). As a result, although U in the small river is larger, TPmax with respect to U was smaller than that in the large river because of the dampening of the small difference between the actual vapor pressure and the saturated vapor pressure (equation (A12)). Sensitivities to Qgw in the small river are in the opposite direction to that in the large river. This results from the difference in Tgw relative to the mean Tsw between the two rivers. Tgw in the small river is lower than the mean Tsw, whereas Tgw in the large river is higher than the mean Tsw. Sensitivities to Tgw occur in the same direction in both rivers and the values are very close, as the ratio of Qgw to the surface water flow rate is the same in both rivers. The same percent change in Tgw will result in the same percent increase in the thermal energy input from groundwater. Although Tsw is subject to the reequilibrium of different heat flux components, the absolute values of TP are very similar in both rivers. Sensitivities of Tsw to the remaining parameters have similar patterns in both rivers, but with larger values in the small river. Both cloud cover (Cc) and reflectivity (r) act to reduce solar radiation in a linear fashion (equation (A6)). Hence, sensitivities to Cc and r show similar trends. Although Cc also affects atmospheric longwave radiation which is a power function of Ta (equation (A8)), the absolute values of sensitivities to Cc were several times larger than with respect to r. This is because of smaller percent errors assigned to r, which is usually a function of solar angle and can be determined with smaller uncertainty than Cc. Sensitivities to river width (B) and channel bottom slope (So) are opposite to sensitivities to channel roughness coefficient (n), i.e., the increase in B or So causes Tsw to change in the opposite direction to the increase in n. These three parameters influence Tsw by affecting river flow rate as indicated by equation (A3). Parameters cw, cs, kw, and Pa do not cause large temperature uncertainties, because of relatively small percent errors. In comparison, ks, Lh, and h have some effect due to higher errors of these parameters. Two other parameters qs and bL are shown to be unimportant, despite large percent errors. qs only affects volumetric thermal capacity of streambed (equation (A20)). Given that heat fluxes across the riverbed is small compared to river surface, a small variation in qs will unlikely cause a large change of Tsw. bL affects the spreading of heat during transport. As all heat fluxes occur across all interfaces and the heat balance is in steady state, bL will have a negligible effect on Tsw. 3.3. Estimating Groundwater Inflow From Surface Water Temperature Variation Hypothesis testing was first applied to the synthetic large river model. First, the Lowland Large River model (parameters shown in Table 1) was run to generate temperature time series at different locations (5, 10, 20, and 30 km). These temperature time series acted as the ‘‘measured’’ results. Note that Qgw was 100 m2 d21 in this system. Then it was assumed that Qgw was unknown, and the temperature data were fit using different values of Qgw. For this fitting exercise, model parameters except Qgw are assumed to be imperfectly known (with errors shown in the percent error column of the Lowland Large River group of Table 1). In the case shown in Figure 4, Qgw was changed to 10 m2 d21 and fixed at this value at all times during this hypothesis testing. Using PEST, all parameters except Qgw were varied within their allowable ranges in order to achieve the best fit to the ‘‘measured’’ temperature time series at the four locations. Although the

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Figure 4. Model calibration results of the synthetic large river. The ‘‘measured’’ temperature time series was generated from the case with Qgw of 100 m2 d21, whereas the modeled data were optimized results with a given Qgw of 10 m2 d21 using PEST. RMSE in this case is 0.148C.

optimized temperature time series was slightly lower than the ‘‘measured’’ ones overall, the RMSE is small and the calibration results can still be considered acceptable (Figure 4). The reasonable match between modeled and ‘‘measured’’ temperature time series suggests that in this specific river the buffering effect created by Qgw of 100 m2 d21 could be approximated by a different combination of parameters with Qgw an order of magnitude lower than the actual value. Then a series of similar model calibrations were conducted by fixing the tested Qgw to different values and allowing parameters to vary over the same ranges as in Figure 4 in order to achieve corresponding lowest RMSEs. From these model calibrations, the relationship between RMSE and Qgw was identified (Figure 5a). Results show that when the tested Qgw was within a factor of 2 (between 50 and 200 m2 d21) from the actual Qgw (100 m2 d21), RMSE remained very close to zero. As expected, as Qgw departed further away from the actual value (i.e., beyond the range from 50 to 200 m2 d21), the calculated RMSE increased rapidly. If the reported thermistor accuracy of 0.28C is used as a cutoff value, the plausible Qgw estimates range from 6 to 400 m2 d21. Hence, parameter uncertainties led to a large Qgw uncertainty. Qgw of 10 m2 d21 discussed above is within this acceptable range. The same hypothesis testing was carried out for two other scenarios where the actual Qgw used to produce the ‘‘measured’’ temperature time series was an order of magnitude lower (Figure 5b) and higher (Figure 5c). When the actual Qgw was small (Figure 5b), heat buffering due to groundwater inflow was relatively weak.

Figure 5. Relationships between RMSE and Qgw for the large river. The ‘‘measured’’ temperature time series for data fitting was produced from the base cases by assuming actual Qgw of (a) 100 m2 d21, (b) 10 m2 d21, and (c) 1000 m2 d21, respectively (indicated by downward triangles). Parameter ranges used for model calibration remained unchanged in all cases. The arrow in Figure 5a indicates the case presented in Figure 4.

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Figure 6. Relationships between RMSE and Qgw for the small river. The ‘‘measured’’ temperature time series for data fitting was produced from the base cases by assuming actual Qgw of (a) 0.1 m2 d21, (b) 1.0 m2 d21, and (c) 10.0 m2 d21, respectively (indicated by downward triangles). Parameter ranges used for model calibration remained unchanged in all cases.

Hence, the ‘‘measured’’ temperature time series were matched easily provided the tested Qgw was also small. In this specific case, any Qgw lower than 100 m2 d21 produced RMSE of lower than 0.28C. As the tested Qgw increased, RMSE rose quite rapidly. In contrast, when the actual Qgw was larger (Figure 5c), RMSE increased quickly as the tested Qgw was decreased below 400 m2 d21. If RMSE of 0.28C is also treated as a cutoff value for Figure 5c, the range of acceptable Qgw estimates is between 300 and more than 10,000 m2 d21. Following the same procedure, hypothesis testing was also performed in the small river (Figure 6). When the actual Qgw was 0.1 m2 d21, the estimated range with RMSE smaller than the cutoff value 0.28C was from 0.0 to 1.0 m2 d21 (Figure 6a; note that Figure 6a only shows the lowest Qgw at 0.01 m2 d21). When the actual Qgw was 1.0 m2 d21, any value lower than 2.0 m2 d21 produced RMSE smaller than the cutoff value and was therefore considered to be a reasonable estimate (Figure 6b). When the actual Qgw was large (10 m2 d21; Figure 6c), the estimated range became narrower (2–50 m2 d21). 3.4. Effect of Spatial Variation in River-Aquifer Exchange Flux on Surface Water Temperature Spatial variation in Qgw also affects Tsw. In the base case of the large river, Tsw increases steadily downstream because of continuous groundwater inflow (base case in Figure 7). However, if groundwater inflow is spatially variable, then there will be zones where temperature increases more and less rapidly. These zones will become more apparent when the spatial scale of the variation becomes large. Consider, for example, the case where groundwater inflow only occurs in the first half reach (0–16 km) but with double Qgw of the base case (case 200 m2 d21 #1). In this case, Tsw increases faster along the first-half reach but slower along the second-half reach. As groundwater inflow is more concentrated along a shorter section (cases 400 m2 d21 #1 and 800 m2 d21 #1), Tsw increases even faster along the reach with groundwater inflow. It will thus be easier to detect the presence of groundwater inflow when inflow is concentrated along short subreaches (provided that we have a sufficient density of monitoring sites), because the river temperature increases rapidly in these zones. However, it does not necessarily follow that the magnitude of Qgw can be more accurately estimated. As discussed above, zones of high flux are not necessarily more accurately quantified than zones of lower flux. Furthermore, the accuracy of estimating mean Qgw over the entire 32 km reach depends on the accuracy of estimating Qgw within both the high and low flux zones. Of course, as the spatial scale of the variation in groundwater inflow decreases, then areas of high and low exchange flux are less easy to detect, and Tsw will closely resemble that for constant inflow. For the example shown in Figure 7, this appears to be the case where zones are less than approximately 1 km in size. In these cases, Tsw provides a good indication of the mean value of Qgw. Where the magnitude and spatial scale of the variation in groundwater inflow becomes large, then the temperature at the downstream site begins to diverge from the temperature produced by constant inflow. Thus, while small-scale spatial variability in groundwater inflow will not affect estimates of mean Qgw, large-scale variability could mean that errors of estimation are even larger than estimated above. Further work is needed to more accurately quantify this effect.

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4. Field Example 4.1. Study Site Description To determine whether river temperature time series measured directly in the field can provide estimates of river-aquifer exchange flux, Heihe River in arid northwest China was chosen as a field example. Heihe River is the second largest inland river in China with a total length of about 800 km. It originates from the Qilian Mountains, flows across an agricultural area and then across the Gobi desert, and eventually terminates at Juyan Lake. The Heihe River plays a significant role in both providing water resources and protecting downstream arid ecosystems, preFigure 7. Temperature variation with distance under different groundwater inflow conditions. The base case is the large river with continuous groundventing the downstream region from water inflow of 100 m2 d21 over the entire reach (as in Figure 1a). All other further desertification. As an arid region, cases have the same mean value of groundwater inflow, but with inflow conpotential evapotranspiration (2000 mm) is centrated on one or several discrete sections (evenly spaced and of equal length), with zero groundwater inflow elsewhere. In the legend, the first nummuch higher than average precipitation ber is the river-aquifer exchange flux in the high flux zone, whereas the last (150 mm). Estimation of river-aquifer number indicates the number of discrete reaches that receive groundwater exchange flux is important to improve the inflow. Thus, for example, 400 m2 d21 #8 has eight zones of 400 m2 d21 inflow, each 1 km long and separated by 3 km zones with no inflow. water allocation plan. Numerous studies have been undertaken to examine the hydrological and isotopic characteristics of the Heihe River [e.g., Chen et al., 2006; Yang et al., 2011; Wu et al., 2014; Yao et al., 2015]. Our study reach is located along the middle part of the river and covers 32 km starting from the Gaoya Gauging Station. There are no tributaries and no other gauging stations along this reach. As it is a heavily cultivated region, the river has been managed for irrigation of crops through several irrigation canals. However, at the time of field work, crops had already been harvested and so no river water was diverted for irrigation. Within the study reach the river meanders and is partly braided, with a coarse-grained sand to gravel riverbed. Field work was carried out between 18 and 23 October 2013. Twenty sites were instrumented with ThermoC thermistors (Maxim Integrated, San Jose, CA) with spacing between sites increasing from chron iButtonV 50 m close to the gauging station to 8 km further downstream. As no significant temperature difference was identified between those temperature thermistors in the first 2 km, these are not used in the current analysis. As the river discharge rate was 63 m3 s21 (i.e., 5.44 3 106 m3 d21) and the river width was approximately 70 m, manual flow gauging was not possible at downstream sites. Air temperature was recorded using the same temperature sensor as those used for measuring the surface water temperature. Groundwater temperature was determined from sampling by investigating both domestic bores and from time series measurements of streambed temperature. Weather parameters are estimated from Weather Underground (wunderground.com). Riverbed thermal parameters were estimated by assuming mean river temperature and granite-related sediment [Dingman, 2002; Holman, 1997]. The mean values of these parameters and relative percent errors are the same as those given in the Lowland Large River column of Table 1. 4.2. Groundwater Inflow Estimates At the time of our field work, Ta in this part of the Heihe River basin varied over 158C, whereas Tsw in the Heihe River varied over less than 38C diurnally (Figure 8). This was attributed to the large discharge rate of the river (63 m3 s21) and relatively constant Tgw (148C). Diurnal temperature time series data were obtained for 3 days starting from 10:15 A.M. on 19 October 2013. The first-day temperature time series was used for the model to reach an equilibrium state. Hence, results presented in Figure 8 started from 10:15 A.M. on 20 October 2013.

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Figure 8. Comparison between measured and modeled temperature time series at different sites along the Heihe River. The top graph shows boundary conditions used for this model calibration. No groundwater inflow was allowed in this case.

When the value of Qgw was set to zero along the entire reach, PEST managed to fit the measured temperature time series at the six sites with an RMSE of 0.28C (Figure 8). Several larger values of Qgw were then employed and the PEST calibration was repeated. The minimum RMSE value remains close to 0.2 until Qgw reaches 100 m2 d21 (Figure 9). RMSE then rises rapidly from 100 m2 d21 onward. The relationship between RMSE and Qgw suggests that the temperature time series can be fitted well provided that Qgw lies in this range (0–100 m2 d21).

5. Discussion Previous studies have examined reach-scale Qgw [Loheide and Gorelick, 2006; Westhoff et al., 2007] using heat as a tracer. The common practice is to treat this unknown variable as a calibration parameter which can be estimated by fitting measured temperature time series either manually [Loheide and Gorelick, 2006] or automatically [Westhoff et al., 2007]. While this approach will yield the Qgw value with lowest RMSE, it does not guarantee the estimated Qgw is the only plausible value due to model nonuniqueness. In this study, Qgw was removed from the parameter estimation process and was assumed to be a known variable in our study. By assigning Qgw different values and examining resultant model fits, the most plausible range of Qgw was able to be identified. Our study shows that when groundwater inflow was low, the lower limit of the estimated range of Qgw (Figures 5b, 6a, 6b, and 9) was not able to be determined. That is, Qgw could not be distinguished from zero. This is attributed to the low heat flux inputs to the river from groundwater, relative to those from the atmosphere. As heat flux inputs from groundwater become significant, both the upper and lower limits of Qgw can be determined (Figures 5a and 6c). However, as heat flux inputs from groundwater become greater, it is harder to determine the upper limit (Figure 5c). This is because at high groundwater inflows, groundwater inputs dominate atmospheric effects, and so the temperature of surface water becomes close to that of groundwater, with relatively weak diurnal variation. In cases where both upper and lower limits of Qgw can be defined, the uncertainty in Qgw can be estimated. When the actual Qgw was 10 m2 d21, the estimated Qgw was between 2 and 50 m2 d21 for the small river,

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and an actual Qgw of 100 m2 d21 was estimated to be between 6 and 400 m2 d21 for the large river. These ranges for Qgw are based on the relationship between Qgw and RMSE. However, although these ranges are informative, they are somewhat subjective. For the case shown in Figure 4, the data fitting is reasonable with a RMSE of 0.148C and so corresponding calibrated parameters were considered acceptable. If this RMSE value is treated as a cutoff value rather than the value of 0.28C, the most plausible range will be Figure 9. RMSE against Qgw for the simplified Heihe River model. reduced from 6–400 to 10–200 m2 d21 (Figure 5a). Cranswick et al. [2014] used identical thermistors to those used in our study, and determined a measurement accuracy of 0.098C based on comparison with thermistors certified by American National Institute of Standards and Technology. If this measurement accuracy is used as the cutoff value for the RMSE, then the estimated range will be narrowed to 20–150 m2 d21. In comparison to the estimated uncertainty of the temperature method, the uncertainty of groundwater discharge estimated using other tracer methods (isotopes and chemical tracers) has been estimated to be less than 50% [McCallum et al., 2012]. The uncertainty using temperature is greater because as much as 70% of heat input to a river is from radiative heat fluxes [Webb and Zhang, 1997], whereas input of dissolved solutes and isotopes is only from groundwater. Despite the greater uncertainty of temperature estimates, heat is still a popular technique as temperature can be measured easily and remotely. In practice, reliable estimates of groundwater exchange are unlikely to be obtained with any single tracer, and the use of multiple tracers is always to be recommended. McCallum et al. [2012] showed that adding more tracers will generally decrease predictive errors, although the extent of this decrease will vary from site to site. Sensitivity analysis results demonstrate that parameter sensitivities vary between different hydrologic systems. Despite this, it reveals that river temperature is most sensitive to weather parameters and least sensitive to thermal parameters. These weather parameters, in fact, can never be quantified with high precision, and some can vary locally (e.g., cloudiness and wind speed). The temperature method for estimating groundwater discharge may therefore be most successful when uncertainty in these parameter values is low, such as on a day with no wind, and a clear or overcast day or in a fully shaded stream. (Shading is easiest to estimate when it is zero or 100%.) The method may work even better during night times, because several parameters associated with radiative heat fluxes can be avoided. More importantly, groundwater will become the most important factor for maintaining river temperature during night times. The technique will also be most successful when the temperature contrast between air and groundwater is greatest (e.g., on a hot summer day or a cold winter day). Although sensitivity analysis provided insights into uncertainties of different parameters, it was not able to allow the examination of synergy of parameters. For example, cloud cover and reflectivity affect both solar shortwave radiation and atmospheric longwave radiation, the increase in both parameters may strongly enhance radiative flux inputs. Wind speed and relative humidity have similar impacts on latent and convection heat fluxes. Moreover, river width, Manning roughness coefficient, and channel bottom slope work synergistically by causing river flow rate to change. In addition to the effect on river flow rate, river width also scales all the heat fluxes and hence has much stronger synergistic effect than other parameters. The analysis of synergistic effects of different parameters needs to be carefully examined in a future study. This study made a number of assumptions in simplifying the modeling. In particular, the study assumes uniform groundwater inflow along the entire river reach. This assumption is valid when the spatial scale of Qgw variation is relatively small (