Thermal skin effect of pipes in streambeds and its implications on groundwater flux estimation using diurnal temperature signals

Authors


Abstract

[1] Heat tracing is widely used for quantifying groundwater flux across the river-streambed interface. This study investigates the potential for a thermal skin effect (TSE) whereby temperature measured inside a pipe buried in sediment is lagged and damped compared to temperature outside of the pipe, violating the assumption that monitored temperatures are representative of the saturated sediment. Numerical simulations show that 1–3 inch diameter pipes lead to an additional lag of tens of minutes to almost 1.5 h when diurnal temperature signals originating from the river and propagating into the sediment are monitored. The lag increases with pipe diameter and diminishes when steel is used instead of PVC. The temperatures taken from inside the pipe lead to substantial errors when they are used as targets for fitting with a forward heat transport model. The errors are large when the fit is made to the timing of the temperature signals but is small when the fit is made to the amplitude of the signal while ignoring the timing. This is because the TSE leads to little damping of the amplitude of the temperature fluctuations and mostly exhibits as extra lag. Therefore, methods that take advantage of the amplitude of the signal, such as time series analysis using amplitude ratios, are not sensitive to the TSE. Time series data can be corrected by subtracting the response time of the water-filled pipe or by adding the response time as a phase shift to analytical models.

1. Introduction

[2] Several important and interesting hydrologic phenomena are set at the interface between different reservoirs in the hydrologic cycle [National Research Council Committee on Hydrologic Science, 2004] such as that between surface water and groundwater [Sophocleous, 2002]. Processes occurring along these interfaces lead to strong coupling between surface water and groundwater and have consequences on processes internal to both reservoirs. Localized interfacial exchange processes affect larger-scale, sometimes regional, solute [Battin et al., 2008; Cardenas, 2008; Harvey and Fuller, 1998] and energy budgets [Arrigoni et al., 2008; Poole and Berman, 2001; Webb et al., 2008]. It is therefore critical to characterize and quantify the exchange of mass and energy between surface water and groundwater.

[3] There are numerous methods for quantifying flux across the interface between saturated sediment and its overlying water column [Rosenberry and LaBaugh, 2008]. Perhaps, the most direct way to measure groundwater flux is through seepage meters, or through concurrent observations of hydraulic conductivity and head gradients for simple Darcy equation calculations. However, there are challenges to using and installing seepage meters in flowing water [Rosenberry, 2008], and the typically small head gradients in these environments require careful measurements [Kennedy et al., 2007]. Another approach is to inject dissolved tracers either in the river or in the sediment and to monitor their arrival at some observation points in the other reservoir after the tracer has crossed the interface. This approach is tedious and expensive and does require some level of luck in terms of correct placement of receptors especially when these are in the sediment. This is partly because river sediment is heterogeneous and dissolved tracers may take tortuous multidimensional paths.

[4] An attractive alternative to using introduced dissolved tracers is using heat as a natural tracer [Constantz, 2008]. Heat is free, omnipresent, and, because of the high thermal diffusivity of saturated sediment, any thermal signals from the river are transported readily into the sediment. Temperature is also easy to measure using inexpensive even custom-built devices. Monitoring temperature is sometimes an end in itself when the problems of interest relate to thermal energy budgets [Webb et al., 2008] or are thermally sensitive.

[5] Heat has been used as a tracer of subsurface fluid flow since the 1960s; the reader is referred to a recent review by Anderson [2005] for a thorough treatment of the subject. Constantz [2008] summarizes recent applications of heat tracing more specific to the scope of this study: heat tracing in sediment underneath surface water bodies. Because of the many advantages it offers, there are now numerous excellent examples of heat tracing [Constantz 2008, and references therein] and various models for heat tracing in streambeds [Conant, 2004; Hatch et al., 2006; Keery et al., 2007; Schmidt et al., 2006] which indicates its growing popularity and use.

[6] There are a variety of methods for measuring streambed temperatures. Regardless of the measuring device (typically a thermistor or a thermocouple), the instruments may be directly buried in the sediment or deployed in screened and unscreened pipes for protection and for easy extraction of the instruments and data downloading [Constantz, 2008]. Examples for both types of deployment are found in the work of Stonestrom and Constantz [2003]. Buried probes are more typical in ephemeral rivers when digging is possible when the channel is dry. For perennial rivers, it is more typical to deploy probes inside pipes or piezometers.

[7] A recent model for analysis of streambed temperature time series takes advantage of periodic variations in temperature such as that due to diurnal warming and cooling cycles [Hatch et al., 2006]. This requires that the lag or phase shift and damping of the diurnal signal originating from the river are accurately measured in the sediment. Because of the relatively small time scales involved, compared to say multiday or even seasonal signals, the method of probe deployment may impact the accuracy of the measurements; that is, incorrect damping and phase shift are monitored which leads to erroneous estimates of groundwater and heat fluxes. Probes in the middle of a pipe may detect a signal that is further lagged and damped. In essence, the pipe and the water that fills it leads to a thermal skin effect (TSE) analogous to that in well hydraulics when a low-permeability zone is created around a well due to drilling fluids imbibed by the sediment, but of course instead of damped and delayed pressure response, the TSE is for temperature.

[8] Alexander and MacQuarrie [2005] demonstrated using modeling and field experiments that well or pipe construction does not result in any differences in temperature between that of saturated sediment and that measured by a probe inside a pipe. However, they looked at time scales that are much longer than would be required for analysis using diurnal signals. For example, they looked at equilibration of sediment temperature to a 5°C increase maintained at the top of the sediment over five days for their “shorter duration” studies. Their probes were buried under tens of centimeters of saturated sediment which has an unsaturated section near the top. The temperature perturbation was introduced by warming the air above the unsaturated zone. While their statistical analysis yielded no significant differences between measured temperatures (inside and outside the pipe) and modeled temperatures (inside the pipes), their results suggest some systematic errors between the model results and observations (see their Figure 7). This may suggest numerical dispersion issues; the modeled temperatures in the probes suggest that they are equilibrating faster than the sediment. Nonetheless, the time scales in their study of the perturbation and response are different to that of diurnal signals propagating from a stream into the streambed.

[9] It has been shown previously that small thermistors in small diameter pipes do not exhibit any substantial TSE (see Figure 7 in chapter 4 of Rosenberry and LaBaugh [2008]) when the sediment is saturated. However, the temperature measured by thermistors inside a pipe can be very different when the riverbed becomes dry. This is because air is a stronger insulator than water, i.e., it results to more TSE. This suggests that the TSE needs to be more systematically studied.

[10] In this study, the TSE due to pipe construction and geometry are analyzed within the context of heat tracing in streambeds using diurnal signals. The analysis is conducted using numerical modeling. The goal of the study is to characterize and quantify any thermal skin effects and to determine its possible consequences on estimated groundwater and heat fluxes.

2. Methods

[11] The TSE is analyzed by running numerical simulations and then using the results to calculate vertical groundwater fluxes using the model proposed by Hatch et al. [2006] and by fitting the analytical solution to the numerical model results. The simulation results are compared to theoretically calculated temperatures without TSE.

2.1. Heat Transport Simulations

[12] Heat transport in a water-filled pipe driven into saturated river sediment (Figure 1) is simulated using the finite element method implemented in COMSOL Multiphysics. Since the water in the pipe is considered stagnant (i.e., there is no advective transport), heat transport is governed by the heat equation

equation image

where T is temperature, t is time, λ is thermal conductivity, ρ is density, and c is specific heat capacity. A two-dimensional slice is considered owing to axial/radial symmetry (Figure 1) and the governing equation is appropriately modified for this geometry. The pipe which is the exterior of the domain is considered to be constructed of PVC or Steel whose thermal properties are listed in Table 1; the interior is water filled. The pipe dimensions correspond to 1, 2, and 3 inch diameter pipes for schedule 40 and schedule 80 wall thicknesses (these are the same for PVC and steel pipes, see Table 2). The river follows diurnal warming and cooling via

equation image

where z is the distance from the top boundary (or sediment-water interface (SWI)), Tave is the average about which the temperature fluctuates, A is the amplitude of the fluctuations, and τ is the period of the fluctuations. Period τ = 24 h, Tave = 20°C and A = 5°C in all simulations. The penetration of diurnal temperature signals originating from the river and conducted downward through the sediment is modeled by imposing the following temperature distribution along the exterior of the pipe which is in thermal equilibrium with the sediment:

equation image

where v is the rate of penetration of the thermal fronts, κe is the effective thermal diffusivity of the saturated sediment, and

equation image
Figure 1.

(left) Conceptual model, (middle) boundary and domain conditions for numerical models, and (right) close-up of finite element mesh with triangular elements.

Table 1. Thermal Properties of Materials Used in Simulations
PropertyWaterPVCSteelSedimenta
  • a

    Bulk properties for saturated sediment used for determination of boundary conditions for the pipe exterior (equation (3)).

  • b

    This value is an average since κ scales with thermomechanical dispersion.

ρ (kg/m3)1000140078702155
c (J/(kg K))420016704601761
λ (W/(m K))0.60.19801.81
κ (m2/s)1.43 × 10−78.13 × 10−82.21 × 10−52.72 × 10−6b
Table 2. Dimensions of Different Diameter Pipes
Radius (cm)1 Inch1.25 Inch1.5 Inch2 Inch3 Inch
Schedule 40Schedule 80Schedule 40Schedule 80Schedule 40Schedule 80Schedule 40Schedule 80Schedule 40Schedule 80
ri1.331.211.751.622.041.912.632.463.903.68
ro1.671.672.112.112.412.413.023.024.444.44

[13] The thermal front velocity is v = vf/γ, where vf is the fluid pore velocity which is equal to the Darcy (or groundwater) flux q divided by porosity, and γ is the ratio of volumetric heat capacity of the bulk saturated sediment to that of water. Equation (3) is the analytical solution for the steady flow one-dimensional advection-conduction-dispersion equation for a semi-infinite domain with equation (2) as the top boundary condition [Goto et al., 2005; Hatch et al., 2006; Stallman, 1965]. Simulations with various values for v are run for three days with the entire domain beginning at T = Tave. The last day is analyzed, giving a spin-up time of two days. Temperature at z = 10, 30 and 50 cm from the SWI at the axis of symmetry coinciding with the middle of the pipe and the external boundary in equilibrium with the sediment, are compared. The external temperature follows the analytical solution.

2.2. Calculation of Seepage Using Amplitude Ratios

[14] Hatch et al. [2006] proposed using the ratios of the amplitude (Ar) of the temperature fluctuations or the ratios of phase shifts from temperature observations at two different depths for calculating seepage velocities; both can be simultaneously used. They propose an iterative calculation for the seepage flux since it appears in implicit form with respect to amplitude (see equation (5)) and phase shift ratios. Based on the analysis of Hatch et al. [2006] and our own experience, Ar is a more robust predictor of flux so the phase shift ratio is not used here. The numerical simulation results are recorded every 10 min and this is sufficient for accurately capturing the transient signals. The amplitude for one period at one location is determined simply by picking the minimum and maximum T. The Ar is therefore readily calculated between multiple points. The symbolic math software Maple is then used to solve for the thermal front velocity based on the implicit equation

equation image

Note that v is also embedded in α via (4).

2.3. Fitting the Analytical Solution to Estimate Groundwater Fluxes

[15] Equation (3) is fitted to the results of numerical modeling while using all known hydraulic and thermal sediment properties; that is, there is no calibration involved. The only parameter that is varied to achieve a reasonable fit is the Darcy flux (or fluid velocity). The fitting is done manually and with visual inspection since the actual flux is known and this provides a very close initial guess.

3. Results and Discussion

[16] Temperatures are presented and analyzed in normalized form which is calculated as follows:

equation image

The range for T* from is therefore from −1.0 to 1.0.

3.1. Effects of Capping a Pipe Versus Extending It Into the River

[17] Some researchers suggest deploying pipes with caps wherefrom strings of temperature probes are hung [see Hatch et al., 2006, Figure 1]. Simulations with the cap at the SWI and with a water-filled pipe rising further into the river (this is similar to having a pipe capped above the SWI) show no differences in modeled temperatures when compared to each other (Figure 2). However, in both cases, the modeled temperatures are somewhat lagged and damped compared to what the theoretical temperature should be; this is already indicative of some TSE. The results for a 10 cm pipe extending above the SWI compared to those for a 50 cm extension show no differences. Therefore, the location of the cap above the SWI has negligible effects on temperature measurements for the SWI from inside a pipe.

Figure 2.

Comparison of theoretical versus numerically simulated temperature time series at the sediment-water interface (z = 0 cm) for cases with (a) a pipe capped at the sediment-water interface and (b) a pipe filled with water continuing upward 10 cm and 50 cm into the river. The results are from a simulation with a 1 inch diameter schedule 80 PVC pipe and the sediment is subjected to a downward groundwater flux q of 50 cm/d.

3.2. General Heat Transport Simulations

[18] Any TSE depends on the time scale and magnitude, effectively the rate of change, of the perturbation relative to the effective response time of the system which in this case is for a probe insulated by a water-filled pipe; the TSE obviously does not apply to systems at equilibrium. The response of a probe depends on the thermal mass of the “skin” surrounding it.

[19] The magnitude of the TSE is qualitatively indicated by the vertical inclination of an isotherm. In the absence of TSE, isotherms for the modeled domain (Figure 1) would be horizontal. That is, a probe located at the center of the pipe will detect the same temperature as a point outside of the pipe at the same elevation. Conversely, a strong TSE would result in closer to vertical isotherms with a perfect insulator showing vertical isotherms. A small diameter 1 inch PVC pipe can already have a TSE (Figure 3a) with inclined isotherms. The TSE is driven by not only the pipe material but also by the water filling it as isotherms are not horizontal in both the pipe and water for a PVC pipe. Highly thermally conductive steel expectedly results in little to no TSE as any deviations from horizontal by the isotherms in steel are imperceptible (compare PVC to steel in Figure 3). However, since water in larger diameter pipes leads to substantial driver of a TSE (Figures 3b and 3c), larger diameter steel pipes are also subjected to a TSE but mainly due to the water in the pipe.

Figure 3.

Snapshots of normalized temperature (equation (6)) for schedule 80 PVC (P) and steel (S) pipes for three different diameters.

[20] Increasing the diameter of the pipe or the thickness of its wall increases the response time of the system. However, the perturbation rate is also important. The TSE is further investigated by changing the time scale of “contact time” with the perturbation by increasing the downward groundwater flux. As the advection of heat increases with groundwater flux, the rate of change of the temperature perturbation increases. Modeled temperature time series at three depths, 10 cm, 30 cm and 50 cm below the SWI, are analyzed by comparing them with theoretical temperatures. 1 inch diameter PVC pipes, regardless of wall thickness, i.e., schedule 40 (Figure 4) vis-à-vis schedule 80 (Figure 5), result in little TSE. As the PVC pipe diameter becomes larger from 1 inch to 2 inch to 3 inch, the TSE increases, with the temperature signal becoming progressively lagged (Figures 4 and 5). For 2 inch and 3 inch pipes, the additional lag can be on the order of a few hours. This additional lag approaches 2 h for a 3 inch diameter pipe. The difference in PVC wall thickness between schedule 40 and schedule 80 does not induce any additional obvious TSE. The TSE is systematic, with the delay in the temperature signal arrival of similar magnitude across depth. For the studied cases, the TSE does not show strong effects on temperature damping across different pipe diameters and groundwater fluxes.

Figure 4.

Comparison of numerically modeled temperatures (Mod) at the center of the pipe with theoretical temperatures (Thl) at the perimeter of the pipe for schedule 40 PVC pipes for all three diameters and for increasing groundwater flux.

Figure 5.

Comparison of numerically modeled temperatures (Mod) at the center of the pipe with theoretical temperatures (Thl) at the perimeter of the pipe for schedule 80 PVC pipes for all three diameters and for increasing groundwater flux.

[21] Steel pipes exhibit much less TSE. A 2 inch diameter schedule 80 steel pipe exhibits an additional lag on the order of 10 min (Figure 6) whereas this is on the order of an hour for the PVC pipe of similar construction (Figure 5). A 3 inch diameter schedule 80 steel pipe leads to additional lag on the order of 40 min (Figure 6) compared to 1.5–2 h for a similar PVC pipe (Figure 5). A 3 inch steel pipe has slightly less TSE than a 2 inch PVC pipe.

Figure 6.

Comparison of numerically modeled temperatures (Mod) at the center of the pipe with theoretical temperatures (Thl) at the perimeter of the pipe for schedule 80 steel pipes for all three diameters and for increasing groundwater flux.

3.3. Groundwater Flux Estimates Following the Hatch et al. [2006] Amplitude Ratio Method

[22] Usually the ultimate goal of monitoring streambed temperatures is to calculate groundwater fluxes. The consequences of a TSE, if any, are analyzed by using the modeled TSE-impacted temperatures to calculate fluxes using the Hatch et al. [2006] approach. Based on our experience and based on recommendations in the original reference, the amplitude ratio (Ar) method is more robust. Therefore, we do not use the phase ratio for analyzing the modeled temperatures. Since it is more typical to use the stream temperature, taken from an in-stream probe, as the temperature for the SWI (z = 0), the temperature for z = 0 is therefore theoretically calculated using (1) implying that the in-stream probe is not subjected to any TSE and it is directly in the stream and not inside a pipe. Two pairs of temperature locations are used: 0 and 10 cm, and 10 and 20 cm.

[23] The estimated fluxes and their corresponding percent errors relative to the actual flux are compared in Table 3. The error is calculated as

equation image

where the subscripts denote estimated and actual fluxes. The actual flux is one that is used in calculating the boundary condition outside the pipe (i.e., equation (3)) while the estimated flux is derived following Hatch et al. [2006] approach using the resulting temperature time series from inside the pipe from the forward numerical model. Large errors, close to 40%, may result when fluxes are small and when using the 0:10 cm pair from PVC pipes (Table 3). This is because the temperature data from 10 cm is subjected to a TSE whereas the data from 0 cm are not. This is not a problem when a steel pipe is used. However, the errors due to the PVC pipe markedly diminish when the deeper 10:20 cm pair is used (Table 3). The errors fall below 1% in these cases, except for two instances. Since the TSE is more or less uniform with depth and since it mostly affects the timing of the temperature signal while hardly modifying its magnitude, the Ar is unaffected by any TSE. In fact, the Ar for different pairs but with the same vertical spacing (i.e., 20:30, 30:40, and 40:50 cm) are similar. If the temperature for z = 0 cm was taken from inside the pipe, where it is also impacted by TSE, instead of from a probe in the river, the error associated with the 0:10 cm pair also drops below 1%.

Table 3. Percent Error in Groundwater Flux Estimates Following Hatch et al. [2006]
Darcy Flux q (m/d)Ar From z = 0 and 10 cmAr From z = 10 and 20 cm
PVC Schedule 40PVC Schedule 80Steel Schedule 80PVC Schedule 40PVC Schedule 80Steel Schedule.80
 1 Inch Diameter
0.0527.127.0−0.20.06−0.34−0.02
0.113.914.1−0.4−0.18−0.02−0.10
0.256.26.2−0.3−0.030.010.02
0.53.63.6−0.30.00−0.07−0.09
12.42.2−0.3−0.15−0.05−0.04
 2 Inch Diameter
0.0537.137.0−2.7−0.20−0.34−0.08
0.118.718.5−2.40.080.09−0.06
0.257.57.1−1.80.02−0.020.05
0.53.62.8−0.20.020.000.06
1−0.3−1.5−2.90.10−0.240.26
 3 Inch Diameter
0.0536.733.9−8.40.981.370.14
0.116.013.5−7.00.450.44−0.01
0.253.00.5−5.6−0.08−0.12−0.01
0.5−3.8−4.5−6.2−0.26−0.44−0.02
1−5.1−19.2−10.4−0.48−2.110.00

[24] The error in estimated flux decreases with depth since deeper temperature signals in the sediment are subjected to more lag and since the TSE leads to an added lag of constant magnitude across different depths. A small additional lag hardly affects a largely lagged signal whereas at shallower depths near the SWI with very little lag, a small additional lag severely alters the signal resulting in further miscalculation of the groundwater flux.

3.4. Groundwater Flux Estimates From Manual Fitting of Analytical Solution

[25] Due to the TSE, simultaneous fitting of a theoretical temperature curve to the numerical simulation results to correctly capture both the amplitude and the timing of the temperature time series is practically impossible. Therefore, fluxes are calculated either by fitting a theoretical curve to match only the amplitude of the fluctuations or the timing of the time series. It also proved difficult to simultaneously fit to three curves (from 10, 30, and 50 cm) so data from each depth is used exclusively leading to six estimates per simulation (time series from three depths used for timing or amplitude fitting). Only the 2 inch and 3 inch diameter pipes are analyzed here since the 1 inch diameter pipe has minimal TSE.

[26] Since the TSE mostly affects timing by inducing extra lag, matching the timing of the theoretical temperature curve to those from the simulations exhibiting TSE results in underestimated fluxes (Figures 7 and 8). In these cases, qest is manually adjusted until the timing of the maximum and minimum temperatures for the theoretical and simulated curves are within 20 min of each other (the discrete model output is every 10 min). The error is largest for fits to data from shallower locations, with the error diminishing with depth. The error is smallest for intermediate qact. For a 3 inch pipe and when qact is 5 cm/d, even reducing the qest to 0 cm/d (upward fluxes where qest < 0 are not considered) still leads to a poor fit of the theoretical curve to the model results. The modeled temperature time series at 10 cm depth for a 3 inch PVC pipe is still lagged compared to a theoretical purely conductive case (3rd column of Figure 7). For a 2 inch pipe, the timing between the theoretical and simulated time series are similar when qest = 0 cm/d (1st column of Figure 7). The timing of temperature variations for a 2 inch pipe with TSE would suggest heat transport is conductive when there is groundwater flux of 5 cm/d outside of the pipe. A zero flux estimate obviously corresponds to an error of more than 100%. The error drops to 20% and 40% for the same schedule 40 and schedule 80 pipes, respectively, when the data from 50 cm is used.

Figure 7.

Theoretical model fits (solid curves) to numerical model results (discrete symbols) and resulting fluxes for 2 inch and 3 inch schedule 80 PVC pipes. The fitting either optimizes the fit to the temperature amplitude or timing/phase, with each resulting in a separate estimated flux. The temperature time series are from 10 cm (black diamonds), 30 cm (blue triangles), and 50 cm (red squares). The inset boxes indicate the estimate flux based on the fitting while the row labels show the actual flux.

Figure 8.

Theoretical model fits (solid curves) to numerical model results (discrete symbols) and resulting fluxes for 3 inch schedule 80 steel pipe. The fitting either optimizes the fit to the temperature amplitude or timing/phase, with each resulting in a separate estimated flux. The temperature time series are from 10 cm (black diamonds), 30 cm (blue triangles), and 50 cm (red squares). The inset boxes indicate the estimate flux based on the fitting while the row labels show the actual flux.

[27] A suitable amplitude fit is achieved when the absolute differences between the maximum and minimum temperatures of the theoretical and modeled curves are <0.01°C. The resulting qest for the 2 inch and 3 inch PVC pipes (2nd and 4th columns in Figure 7) are much closer to the actual values than when timing is the chosen target. The errors also decrease when the data from deeper locations are used. When data from below 10 cm are used, the errors are typically <10% for the different qact considered. Unlike the analysis with timing fitting, the errors with amplitude fitting increases slightly with increasing qact. The estimated fluxes from amplitude fitting are more or less consistent with each other when data from different depths are exclusively used, except perhaps using data from ≤10 cm. This suggests that simultaneous fitting to data from different depths would lead to a reasonably accurate qest, whereas estimates from timing fitting are more broadly distributed which would still result in an erroneous estimate for flux when simultaneous fitting is attempted, if a reasonable fit is even possible.

[28] For a 3 inch steel pipe, the TSE also leads to severe underestimation of the flux by 30- > 100% when the theoretical curve is made to fit the timing of the time series (Figure 8). Errors are similar to those from the PVC pipe case when amplitude fitting is used.

3.5. Correcting for the TSE by Considering for the Pipe Thermal Response Time

[29] The delay in temperature signal arrival is clearly associated with the thermal response time of the water-filled pipes surrounding the probe. The diffusive response time for heat transport is defined as

equation image

where Γ is the response time and X is a length scale. For the water-filled pipes, the effective response time is the sum of the response time of the pipe wall and of the water. The length scales correspond to the inner radius of the pipe for the water, and to the wall thickness for the pipe. The calculated effective response time are presented in Table 4. Contrary to conventional wisdom, the values for Γ indicate that water is an important contributor to the TSE. This is consistent with the results for larger diameter pipes visualized in Figure 3 where the isotherms in water are far from horizontal. The response times also qualitatively agree with the temporal discrepancy between the theoretical curves and numerical model results in Figures 46.

Table 4. Response Times in Minutes of Different Diameter Water-Filled Pipes
 1 Inch1.25 Inch1.5 Inch2 Inch3 Inch
Schedule 40Schedule 80Schedule 40Schedule 80Schedule 40Schedule 80Schedule 40Schedule 80Schedule 40Schedule 80
PVC11.510.719.223.825.838.541.938.591.684.8
Steel10.38.517.921.224.435.340.335.388.678.9

[30] In order to take this response time into account and correct for the TSE, the numerical model results may be shifted backward in time by Γ or it may be added as a phase shift to the analytical solution to be fitted resulting in

equation image

[31] Using (9) instead of (3) practically eliminates the TSE making the theoretical and numerically modeled curves perfectly matched except for small differences in amplitude in the 3 inch diameter cases (Figure 9). Therefore, a corrected analytical solution (or time series data) is amenable for fitting to data (or vice versa). Note that no fitting was actually done for calculations in Figure 9; the theoretical curves are calculated using the actual flux. This suggests that a unique and consistent estimate for flux will result from simultaneous fitting to data from multiple depths while getting both timing and amplitude of the temperature signal correct.

Figure 9.

Comparison of numerically modeled temperatures (Mod) at the center of the pipe with theoretical temperatures (Thl) at the perimeter of the pipe modified with an additional lag equal to the corresponding response times (see Table 4).

4. Conclusions and Recommendations

[32] Temperatures monitored inside pipes buried in riverbeds exhibit a thermal skin effect which presents as additional lag to periodic transient signals particularly that from diurnal warming and cooling of rivers. The TSE increases with pipe diameter and with more insulating pipe material such as PVC vis-à-vis steel. 2 inch and 3 inch diameter PVC pipes lead to additional lag from tens of minutes to close to 1.5 h significantly modifying the true temperature signal. This leads to large errors in estimated groundwater or advective fluxes when the TSE-impacted time series is used as target for fitting with a forward transport model. It is therefore recommended that small diameter (≤2 inches) steel pipes be used for housing temperature-monitoring devices when a little corrosion is not an issue; that is, deployment is only for a few days. When housing construction that leads to TSE is necessary such as large diameter (≥2 inches) PVC pipes, it is recommended that the objective for any fitting with a forward model should be to get the amplitudes of the temperature signal correct rather than the timing since the amplitude of the temperature signals are hardly affected by the TSE. Moreover, temperature time series analysis methods which use the amplitude of the signal, such as the amplitude ratio method proposed by Hatch et al. [2006], are not sensitive to the TSE and lead to accurate flux estimates. However, when the study objective is to monitor distributed transient temperature patterns in the streambeds such as those in the work of Loheide and Gorelick [2006], Hester et al. [2009], Cardenas and Wilson [2007], and those discussed in the work of Webb et al. [2008], housing with negligible TSE should be used. Of course, the time series data or, conversely, any interpretive transport models are correctable by adding or subtracting the collective response time for materials surrounding the temperature probes.

Acknowledgments

[33] Christine Hatch and two anonymous reviewers helped improve this manuscript. This research was supported by the U.S. National Science Foundation (EAR-0836540).

Ancillary