Use of heat as tracer to quantify vertical streambed flow in a two-dimensional flow field

Transcription

1 WATER RESOURCES RESEARCH, VOL. 48, W10508, doi: /2012wr011918, 2012 Use of heat as tracer to quantify vertical streambed flow in a two-dimensional flow field Hamid Roshan, 1 Gabriel C. Rau, 1 Martin S. Andersen, 1 and Ian R. Acworth 1 Received 26 January 2012; revised 14 August 2012; accepted 27 August 2012; published 5 October [1] Analytical solutions to the heat transport equation in porous media have been developed in the past to estimate surface water-groundwater interactions. These solutions, however, are based upon simplifying assumptions that are frequently violated in natural systems. A nonvertical flow field, inherent to most field settings, can violate the one-dimensional (1-D) flow assumption and lead to erroneous velocity estimates. In this study, we have developed a 2-D heat and mass transport finite element-based numerical model for a stream aquifer cross section experiencing flow-through. Synthetic multilevel streambed temperature time series were generated with the model using a sinusoidal temperature boundary. The temperature data was used to quantify the vertical flow velocity with a 1-D analytical solution based on the amplitude decay and phase shift of temperature with depth. Results demonstrate that erroneous vertical components of fluid velocity can be obtained by the 1-D analytical solution when the true vertical velocity approaches zero and the flow regime becomes almost horizontal. The results also illustrate that the amplitude ratio method performs quite poorly on the gaining side of the stream where the only reliable method is phase shift. On the losing side of the stream, both methods can be employed but a better estimation is obtained from the amplitude ratio method. In general, amplitude ratio and phase shift data should be used in conjunction to maximize the information of the system. Citation: Roshan, H., G. C. Rau, M. S. Andersen, and I. R. Acworth (2012), Use of heat as tracer to quantify vertical streambed flow in a two-dimensional flow field, Water Resour. Res., 48, W10508, doi: /2012wr Introduction [2] Surface and subsurface temperature fluctuations have been extensively used as a natural tracer to quantify exchange flows between surface water and groundwater in shallow sediment systems [Constantz et al., 2002; Hatch et al., 2006; Keery et al., 2007; Rau et al., 2010; Silliman et al., 1995; Stallman, 1965]. The conductive-convective heat transport equation is usually linearized by assuming quasi-transient heat flow and steady state fluid flow conditions to derive the one-dimensional (1-D) analytical closed form solution which relates the temperature changes to mass flux in porous media [Hatch et al., 2006;Silliman et al., 1995]. The boundary conditions relevant for this analytical solution are in the form of sinusoidal fluctuations on the top and constant temperature at infinity [Stallman, 1965;Suzuki, 1960]. In such an analysis, two main limitations seem to be the sources of concern. First, the accuracy of the tool with which the temperature can be measured is important since 1 Connected Waters Initiative, Water Research Laboratory, National Centre for Groundwater Research and Training, University of New South Wales, Sydney, New South Wales, Australia. Corresponding author: H. Roshan, Connected Waters Initiative, Water Research Laboratory, National Centre for Groundwater Research and Training, University of New South Wales, 110 King St., Manly Vale, NSW 2093, Australia. (hamidrooshan@yahoo.com) American Geophysical Union. All Rights Reserved /12/2012WR random errors are likely to have a potentially large impact in low-velocity fields. This can be the reason why the flux estimate is very sensitive to probe spacing in field measurements [Shanafield et al., 2011; Soto-López et al., 2011]. A second concern is the nonuniformity in physical properties of the system. The nonuniformity in physical properties can be in the form of heterogeneity in sediment thermal properties [Ferguson and Bense, 2011], violation of the 1-D flow field [Lautz, 2010] or a nonsinusoidal temperature signal at the stream boundary [Lautz, 2010]. [3] The effect of measurement accuracy on data acquisition has been briefly discussed by Shanafield et al. [2011]. They introduced a known degree of uncertainty into the temperature datasets and estimated the error which is likely to occur during analysis. They found that analytical solutions can reproduce velocities above 1.25 m d 1 correctly despite introduced uncertainty in sensor spacing, thermal diffusivity and the accuracy of the temperature sensors. The effect of heterogeneity in the streambed hydraulic conductivity has been investigated by Schornberg et al. [2010] and they concluded that this can significantly affect the accuracy of velocity estimates particularly for low-flow velocities when high contrast between high and low-permeable geological units exist. [4] Field investigations have also demonstrated that the temperature measurements at a particular depth can be a function of the horizontal location where the temperature measurements are made in the streambed [Jensen and Engesgaard, 2011; Shanafield et al., 2010]. For instance, W of16

2 Shanafield et al. [2010] observed that 1-D analytical solutions could fit the temperature data for a probe at only one side of the channel. [5] Finally, the effect of longitudinal and lateral flow has also been investigated [Jensen and Engesgaard, 2011; Lautz, 2010; Shanafield et al., 2010; Swanson and Cardenas, 2010] and the results clearly show that the 1-D analytical solution produces errors in the vertical velocity estimates when a horizontal flow component exists. The nonvertical flow induced by current-bed forms and their effects on temperature distribution in sediments has also been previously investigated [Cardenas and Wilson, 2007a. 2007b]. They found that the generated flow patterns could violate the 1-D assumption for using heat as a tracer. In addition, Lautz [2010] made a comprehensive analysis of the deviations from 1-D flow along a hyporheic flow path. However, there appears to be no comprehensive analysis of the effect of nonvertical flow on estimates of vertical velocities obtained from 1-D analytical solutions across a stream channel when it is subject to flow-through, i.e., gaining conditions at one side and losing at the opposite side [Sophocleous, 2002; Woessner, 2000]. Such a flow system should be expected when streams are cross-cutting the groundwater hydraulic gradient, in highly meandering systems as suggested by both field and modeling studies and by anisotropy in geologic materials [Cardenas et al., 2004; Harvey and Bencala, 1993; Huggenberger et al., 1998; Peterson and Sickbert, 2006; Winter and Pfannkuch, 1984] and potentially for systems where significant groundwater abstraction is happening on one side of the stream. This type of surface water-groundwater interaction causes a highly variable flow field with both losing and gaining conditions and severe deviations from the 1-D flow field. In this study the flowthrough case is used as a particularly demanding test case in order to investigate the use of 1-D analytical solutions for using heat as a tracer for surface water-groundwater interactions. [6] First, the 2-D heat-mass transport equation is solved by means of a finite element scheme for a wide range of gradients across the stream to generate realistic flow fields and temperature distributions for a flow-through system. Second, by using the simulated streambed temperature distributions in the 1-D analytical solutions estimates of the vertical flow velocity were calculated. Finally, these estimates were compared to the numerically simulated velocities to assess the impact of horizontal flow on analytical vertical velocities estimates. 2. Mass and Heat Transfer Equation 2.1. Fluid Flow Equation [7] The fluid flow equation in porous media is defined as [Ahmed, 2000]: ¼r! ~ k rp ; (1) where ~ k is the permeability tensor. p, cm, and are the fluid pressure, fluid compressibility, fluid viscosity and porosity, respectively. Note that for a homogeneous porous medium the permeability tensor reduces to a constant value Heat Transfer Equation [8] The conduction-convection dispersion equation can be expressed as [Kaviany, 1995; ¼r ~D e rt 1 rð~vtþ; (2) where T is the temperature, ~D e is the effective thermal diffusivity matrix and ~v is the Darcy velocity vector. Also, ¼ c f c f, where and c are the average density and the heat capacity of saturated porous medium and f and c f are the density and heat capacity of the fluid. The effective thermal diffusivity is defined as the thermal diffusivity at zero flow plus a hydrodynamic dispersion function: ~D e ¼ ~ ke c ¼ k 0 c þ f ð~ ;~vþ; (3) where ~ ke is the effective thermal conductivity matrix, k 0 is the thermal conductivity at zero flow, and ~ is the thermal dispersivity matrix. It is well-known that the dispersion is a function of grain size [Green et al., 1964] and therefore should be experimentally determined. In this study a dispersion function based on extensive laboratory experimentation with natural materials is used [Rau et al., 2012]: f ð ~ ;~vþ ¼ ~ f c 2 f c ~v : (4) [9] It is also noteworthy that, in the range of velocities investigated in this study, the thermal dispersivity did not induce any change in heat transport results. This is consistent with findings of Rau et al. [2012] for homogeneous well-sorted coarse sand. 3. Model Description [10] Conceptually, a flow-through stream aquifer continuum is modeled in which groundwater is flowing perpendicular to the channel. Conditions along the third dimension (along the stream) have already been modeled by Lautz [2010] and are considered constant here, and consequently the model can be reduced to 2-D. The model geometry is shown in Figure 1 with a stream cross section in the middle of the domain surrounded by sediments. Also presented in Figure 1 are the measurement lines where temperature data are recorded. [11] A constant pressure (hydraulic head of the stream water) is applied on the top stream elements and a no flow boundary is set for the top elements from the edge of the stream toward the left and right boundaries. A differential pressure (representing a hydraulic head) is then applied on the left (incoming flow) and right (outgoing flow) sides of the domain to create a differential head, which simulates a groundwater flux across the stream. The bottom boundary of the domain is considered as a no flow boundary. Considering that the stream water head P3 is constant, when the P1 and P2 boundary pressures (Figure 1) are modified, different flow regimes are created in the streambed that in turn influence the heat flow. 2of16

3 Figure 1. (a) Cross section of stream experiencing flow-through; (b) Model domain including boundary conditions. P1 to P3 are pressure boundary conditions and q ¼ 0 is a no flow boundary. [12] A finite element code was developed in FORTRAN to solve equations (3) and (4). The depth of the domain is set to be 3 m whereas the widths of stream and ground surface are considered to be 2 and 4 m. The three meter depth was chosen as no further changes in heat transport are observed when the depth surpasses three meters in a flow through system of the dimensions used here. Two-dimensional 4-noded isoparametric elements are used for the simulation of both pressure and temperature. The Characteristic Galerkin discretization method is employed in solving the heat transport equation to avoid numerical oscillations resulting from sharp fronts created by the convection equation [Lewis et al., 2004]. Due to the fact that heat fluxes through the boundaries are unknown for transient simulations, they are usually treated as a zero Neumann condition at some distance far from the domain of interest; however, this condition can cause heat accumulation and consequently an increase in temperature especially close to the boundary. For such conditions the so-called open flow boundary condition is used which means that when heat reaches the boundary of the domain it can pass through the boundary without any effect. In this way the open boundary acts like an infinite numerical mesh in which heat can propagate until dissipated completely at a distance far from the heat source. This can be achieved by implementing an open boundary assumption in the weak form of the finite element discretization [Padilla et al., 1990, 1997]. The open boundary condition is applied to all boundaries except the surface elements. The sediment is considered as a homogenous porous medium. The solution of Eqs. (3) and (4) requires an initial pressure and temperature distribution within the flow domain: Tðx; y; tþ ¼T ini ðx; yþ and pðx; y; tþ ¼p ini ðx; yþ for t ¼ 0; (5) where the initial domain pressure and temperature are set to 0 Pa and 15 C, respectively. A Dirichlet type boundary condition is applied to the top stream boundary of the domain for temperature as 2t Tðx; 0; tþ ¼T 0 þ T cos ; (6) Pr where T is a known function in time and space at all points on the specific boundary, and Pr is the period of temperature variation (Pr ¼ 1 day). T 0 and T are the average ambient temperature and the amplitude on the surface (T 0 ¼ 15 C and T ¼ 5 C). A constant head of 2 m is applied on the top stream elements and different heads are applied on the left and right side of the domain to generate different gradients (0.0125, 0.025, 0.05, and 0.125). The water head of 1.5 m and 2.5 m are also applied on right, left and the bottom of the domain s boundaries for the cases of losing and gaining, respectively. The physical properties of water and sediment used in this study are summarized in Table 1. The numerical scheme has been validated with the appropriate analytical solutions. Detail of numerical validation can be found in Roshan and Aghighi [2011] and Koh et al. [2011]. [13] To mimic typical field installations [Jensen and Engesgaard, 2011; Rau et al., 2010], virtual temperature Table 1. Data Used for Thermal and Physical Properties of Water and Sediment Parameter Unit Symbol Value Solid Thermal Conductivity W(mC) 1 k s 7.0 Water Thermal Conductivity W(mC) 1 k f 0.6 Water Specific Heat Capacity J(kgC) 1 c f 4183 Solid Specific Heat Capacity J(kgC) 1 c s 940 Water Density kg m 3 f Solid Density kg m 3 s 2650 Porosity 0.2 Longitudinal Thermal Dispersivity s l Transverse Thermal Dispersivity s t 0.4 Permeability m 2 ~ k of16

4 probes were positioned on horizontal lines at 0.2 m, 0.4 m, and 0.8 m below the entire streambed cross section to continually record the temperatures (M. O. Cuthbert and R. Mackay, Impacts of non-uniform flow conditions on stream aquifer exchange flux estimates made using streambed temperature time series and vertical head gradients, submitted to Water Resources Research, 2012). Vertical flow velocities are quantified across the stream from the modeled thermal time series as measured by the vertical pairs of surface and subsurface temperature probes in the model domain. The 1-D analytical method proposed by Hatch et al. [2006] was used to obtain the fluxes. The quantification is based on the phase shift and amplitude decay of the thermal signal between the streambed surface and the positions of the temperature probes. The vertical thermal front velocity (v t ¼ v=) derived from temperature amplitude ratio A r between streambed and a measurement point at depth is defined as v t ¼ 2D rffiffiffiffiffiffiffiffiffiffiffiffiffi e z ln A þ v r þ 2 t ; (7) 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ¼ v 4 t þ 8D e 2 P, D e is the effective thermal diffusivity, and z is the vertical distance between two measurement points. Similarly, the vertical thermal front velocity can also be calculated from the phase shift in temperature signal observed between two measurement points: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v t ¼ 2 4D 2 e : (8) Pz [14] In this study a vertical pair of observation points, one located at the surface and the other in the streambed, was used for velocity calculation. Using this scheme, apparent vertical velocities (v z,a ) are calculated analytically from the different temperature sensor locations using amplitude ratios (A r ) and phase shifts () with temperature time series obtained from the different numerical simulations. An upward velocity (gaining condition) is considered to be positive whereas a downward velocity (losing condition) is negative. [15] A problem arises when comparing the analytical vertical velocity estimates with the numerically generated flow field since the analytical estimates are for a discrete interval between two measurement points whereas the vertical velocity component from the 2-D flow field changes continually with depth. Therefore it was decided to compare with an average vertical velocity from the numerical flow field (henceforth the average true vertical velocity). This value is calculated as an average from the measurement point (probe location) up to the surface: v ;avg ¼ 1 N X N i¼1 v i ;t ; (9) where can be z or x, representing the vertical and the horizontal axes, respectively. For instance, v z,t and v z,avg are the true vertical velocities of each numerical node and the average true vertical velocity. N represents the number of vertical nodes from the measurement point to the surface. In Figure 2 the averaging procedure to obtain the true vertical velocity is illustrated. Figure 2a shows the real velocity field between 0.2 m depth up to the surface at x ¼ 1.2 m for a horizontal gradient of Additionally, the true vertical velocities at the surface and different depths along the stream cross section are presented in Figures 2b and 2c. Obtaining an average vertical velocity from each measurement point up to the surface gives the average true vertical velocity presented in Figure 2d. [16] In addition, the deviations between the analytically estimated vertical velocities and the true vertical velocities are calculated by the full-scale deflection error which is defined as: 2 3 apparent vertical velocity ðv z;a Þ 6 7 FSD ¼4 average true vertical velocity ðv z;avg Þ 5100: Max javerage true vertical velocityðv z;avg Þj (10) [17] When FSD is used for the method the absolute value of average true vertical velocity (v z,avg ) is employed due to the fact that the method does not indicate the flow direction. 4. Results 4.1. Losing Versus Gaining Conditions [18] The 2-D velocity fields in the streambed under losing and gaining conditions obtained by the numerical simulations are presented in Figures 3a and 3b, respectively. From Figures 3a and 3b it can be seen that the velocity is not uniform over the cross section. Even in the center of the stream the velocity varies vertically from ms 1 (1.728 md 1 ) at 0.8 m depth to ms 1 (3.456 m D 1 )at the surface. It should be noted that the velocity distribution is a function of stream and domain dimensions in both losing and gaining conditions if hydraulic boundary conditions are assumed unchanged. For example, a deeper domain or a wider stream causes more uniform vertical flow especially toward the center of the domain for both losing and gaining conditions. For streams that are not extremely wide purely vertical flow is unlikely to occur in natural systems for losing or gaining conditions where the groundwater table is continuously connected to the steam water level at the banks. Under these conditions a degree of horizontal flow will always occur, especially near the banks. Only for a situation where the groundwater level is well below the streambed (a disconnected losing system) will there be a truly 1-D vertical downward flow across the entire width of the streambed. The presented conceptual model and stream geometry therefore represents the most realistic flow field for losing and gaining conditions. [19] Figure 4 illustrates the average true vertical velocity as well as the full-scale deflection (FSD) error in % for the vertical velocity estimates at three different depths, 0.2 m, 0.4 m and 0.8 m, for losing and 0.2 m depth for gaining conditions obtained from A r and. [20] Only the shallow depth is shown for the gaining condition because the heat variations are not transported very far due to the upward flow. From Figure 4 it can be seen that the FSD error for the different depths under losing 4of16

5 Figure 2. (a) Velocity profile at x ¼ 1.2 m from 0.2 m depth to the surface obtained from the numerical simulation for a gradient of 0.125; (b) true vertical velocity at the surface elements; (c) true vertical velocities at different depths: 0.2 m, 0.4 m, and 0.8 m; (d) the average true vertical velocity obtained by taking the average of true vertical velocities from the measurement points up to the surface. conditions are similar in the middle of the streambed for both A r and methods where the vertical velocity is more uniform (Figure 3); however, an exact agreement (FSD ¼ 0.0) is not obtained even in the center of the streambed. This is caused by the fact that the true vertical velocity from the numerical simulation is changing as a function of depth (Figure 3). This will be the case for most field settings as the true vertical velocity is not always constant with depth and is a function of parameters such as flow regime in the streambed, the width of the stream, etc Flow-Through Condition [21] In order to study the flow-through condition different horizontal gradients (0.0125, 0.025, 0.05, , and 0.125) were applied on the system by varying the hydraulic head (pressure) at the vertical inflow and outflow boundaries. The magnitudes and directions of the velocity field resulting from the upper and lower bounds of the gradients ( and 0.125) are shown in Figure 5. The figure illustrates that the change in horizontal gradient does not affect the shape of the flow field (i.e., directions), however the magnitude of the velocities is strongly influenced by changes in gradient. [22] The 2-D simulated results of heat transport in the streambed under influence of differential horizontal gradients after 2 days are also presented in Figure 6. An increase in the horizontal gradient induces a stronger heat flow toward the losing side of the stream (Figure 6). [23] As explained earlier in brief, the velocity field in real environments changes both in time and space. In this numerical study steady state flows were attained and therefore temporal changes can be neglected. [24] The apparent vertical velocities obtained from the A r and methods at different depths along the stream cross section (x-axis) with a horizontal gradient of are presented in Figure 7. Also presented in Figure 8 is the FSD error for the A r and velocity estimates calculated using the apparent vertical velocities from the 1-D analytical solution (Figure 7) and the average true vertical velocities (Figure 2d). From Figure 8 it should be noted that at the center of the stream both the A r and methods have errors in vertical velocity estimates. For instance, right in the middle of the streambed, where the average true vertical velocity approaches zero, the FSD error from the A r and methods are almost 5% and 35%, respectively, at all depths. This can be explained by the fact that in the middle of a streambed the true vertical velocity (v z,t ) approaches zero and the horizontal velocity (v x,t ) becomes strongly dominant (Figure 9). This in turn induces errors into the velocity calculation by the 1-D analytical solution, 5of16

6 Figure 3. Cross-sectional (2-D) velocity fields for (a) losing and (b) gaining condition after flow reached steady state condition. especially when using the method. Moving away from the center the A r method does not perform well on the gaining side of the stream as the FSD error increases toward the gaining side. Toward the losing side; the FSD error for the A r method approaches zero near the losing edge before picking up again very close to the losing edge. The error for the A r method can be explained by inspecting Figures 2, 6, and 7. At a particular location on the gaining side of the stream the apparent vertical velocity estimates crosses through zero (i.e., from gaining to losing) while the true vertical velocity has a nonzero value. [25] The FSD errors in velocity estimates as a function of horizontal distance from the gaining stream bank at 0.2 m depth for different horizontal gradients obtained from both the A r and methods are presented in Figures 10a and 10b, respectively. Also presented in Figure 10c is the average true vertical velocity at 0.2 m depth for all horizontal gradients. [26] It is evident from this figure that with a decrease in the horizontal gradients the FSD error in the velocity estimates from both the A r and methods increase significantly; however, the FSD errors from the method are generally higher than that of the A r method, especially near the center of the streambed. For example, while the highest FSD error in velocity estimate from the A r method is 60%, the FSD error in the method reaches to more than 250% at its maximum. [27] It is also noteworthy that the difference between the velocity estimates from A r and at each particular location can only give a vague indication of possible errors in the velocity estimate and it seems that there is no straightforward answer as to what extent the field velocity data can be affected by 2-D flow. Therefore, extreme caution should be taken when interpreting field data. [28] In order to explore the anisotropy in heat transport the difference in the effects of convective and conductive heat transport is investigated. First, the temperature amplitude and phase shift from conduction was calculated analytically. The resulting temperature amplitude and the phase shift were subtracted from the overall heat transport to obtain the convective portion of heat transport. In Figure 11a, the ratio of extracted temperature amplitude for convection over conduction is presented for different horizontal gradients at z ¼ 0.2 m. This data emphasizes that when the conduction dominates the heat transport the ratio approaches zero. It also shows that when the convection dominates and has the same direction as the conduction (losing condition), the given ratio diverges from zero with positive value and in the case where the convection acts against conduction; the ratio tends toward negative value. [29] Therefore it is evident from the data in Figure 11a that a decrease in horizontal gradient implies that there is a losing condition across the entire channel. While in actual fact it is known that there is a gaining condition from left bank to the stream center and a losing condition from the stream center toward the outflow (right) boundary. This can explain the errors in the vertical velocity estimates from the temperature data. Since the vertical velocity decreases with decreasing horizontal gradient it can also be pointed out from Figure 11a that the effect of anisotropic convection is more pronounced when the horizontal gradient is low (i.e., a decrease in vertical velocity). It is also seen in Figure 11a that although at the center of the streambed conduction should be the only heat transport process in the vertical direction; the horizontal flow regime induces a false pulse of heat convection giving rise to a distortion of the temperature amplitude, which will be interpreted as vertical flow by the analytical solutions. 6of16

7 Figure 4. Average true vertical velocity for both gaining and losing conditions and the full-scale deflection (FSD) error of the average true vertical velocity (v z,t,avg ) to the apparent vertical velocity estimates (v z,a ), as a function of horizontal location obtained from A r and for losing conditions at depths: 0.2 m and 0.4 m and 0.8 m and gaining condition at 0.2 m depth. [30] Surprisingly, it is observed from Figure 11b that as the horizontal gradient decreases, the rate of change in phase shift along the x-axis also reduces significantly (the ratio becomes almost constant). This explains why the phase shift method works worse for lower horizontal gradients. Having almost the same temperature amplitude across the stream channel for a low gradient (grad ¼ in Figure 11a), the time in which those amplitudes are obtained is quite similar (Figure 11b). This implies that no change in velocity estimate from method can be expected when the phase shift does not change and consequently, that the induced error will stay the same along the x-axis (Figure 10b). For higher horizontal gradients where the temperature amplitude changes along the x-axis (Figure 11a), different arrival times are expected (Figure 11b), which in turn vary the velocity estimate from the phase shift method, therefore giving a lower error at some positions along the x-axis. [31] In order to investigate the effect of stream width and the combined effect of convective-conductive heat flow on the error in the vertical velocity estimates a dimensionless anisotropic thermal Peclet number was defined as Pe z ¼ wc w Lv z;avg v z;avg ; (11) k 0 v x;avg where L is the stream width and k 0 is the bulk thermal conductivity. The FSD error in the vertical velocity estimates 7of16

8 Figure 5. Cross-sectional (2-D) velocity fields for two different horizontal gradients: and after flow reached steady state conditions. from method versus the defined Peclet number at two different horizontal locations: x ¼ 1.2 m; 2.6 m and 1.6 m; 2.8 m at the vertical position of z ¼ 0.2 m is presented in Figure 12. The velocity errors are chosen from due to its symmetry around the center of the stream. From Figure 12 it is seen that the FSD error in the vertical velocity estimates decreases with increasing Pe z. Furthermore, moving toward the center of the stream increases the FSD error in the velocity estimate. Therefore, considering all parameters in the Peclet number constant (except the stream width), then an increase in the stream width will cause the velocity error to approach zero. 5. Discussion [32] In general the flow-through system is likely to have higher errors in the velocity compare to other flow systems such as purely losing or gaining. In this study a wide range of horizontal gradients that might exist over a stream cross section have been used to generate widely variable perturbations of the propagation of surface heat signals into the streambed. This offers an exhaustive analysis of possible effects of nonvertical flow on the estimation of the vertical flow component using 1-D analytical solutions in the stream cross section. The results of the above analysis can be summarized and generalized in the following way: (1) For situations when the true vertical velocity approaches zero and the horizontal velocity becomes dominant, the FSD error in the estimated vertical velocity increases which is especially pronounced for the method (for instance, the A r and methods, for the setting of this study, can return errors up to 60% and 250%, respectively, where the true vertical velocity is in fact zero or FSD ¼ 0.0); (2) In situations dominated by horizontal flow and insignificant true vertical flow both A r and methods perform poorly especially for low horizontal gradients; (3) the A r method falls short in estimating the true vertical velocity on the gaining side of the stream but toward the losing side the error of the A r method is close to zero; (4) the error induced in vertical velocity estimates by changes in horizontal gradients is more pronounced when using the method than the A r method at different depths Effects of Horizontal Flow on Vertical Velocity Estimates [33] The analysis clearly demonstrates that deviations from a truly 1-D vertical flow field can lead to serious overestimation or underestimation ( 60% to 250%) of the vertical velocity component as it has been previously hypothesized by various studies [Rau et al., 2010; Swanson and Cardenas, 2010]. However, the effects vary with a range of conditions: the probe location in the streambed, the probe depth, and the ratio of the horizontal and vertical velocities. Each has a different impact on the A r and methods. This has led to different hypotheses about the variability in velocity measurements [Jensen and Engesgaard, 2011;Shanafield et al., 2010] and means that there is no straightforward solution as to the size of the error based exclusively on the size of the horizontal velocity component. Nonetheless, some general observations can be made. Most seriously, large apparent vertical velocities can be obtained when in reality there is insignificant or no vertical flow (up to 250% for the method). The error in vertical velocity estimate increases as the horizontal gradient decreases for both the A r and methods. [34] The results demonstrate that with increase in the horizontal gradient the ratio of true vertical to horizontal velocity stays unchanged for each numerical node, but their magnitudes increase proportionally. This increase in magnitude of horizontal and especially vertical velocities improves the velocity estimates from the 1-D analytical solutions, especially with method (Figure 10). For example, the FSD error in the velocity estimate is almost zero at the horizontal position of x ¼ 1.2 m for the results for a horizontal gradient of while it is about 10% for the 8of16

9 Figure 6. Cross-sectional (2-D) heat distribution in the streambed for a sinusoidal temperature boundary at the top of the streambed for four different horizontal gradients: 0.025, 0.05, , and after 2 days of simulation. same gradient from A r method. However, this also depends on how close the probes are to the stream bank on the gaining or losing side as well as the installation depth of the probes. The deeper the probe is installed, the higher the potential error, especially for the method. This is easily understood by inspecting Figure 8: due to water inflow from left side of the stream (gaining) the heat propagation from the stream surface temperature signal is washed out or greatly skewed toward the losing side. As explained in Figure 11 earlier, the lower horizontal gradients are more influenced by distortion in heat flow. For instance, at low horizontal gradients like no gaining condition is observed along the x-axis according to the A r method and the temperature amplitude appears to be indicating a strictly losing condition (Figure 11). [35] This also serves to demonstrate that significant errors are more likely for deeper probes and the higher errors in the vertical velocity estimates are likely to be for flow regimes with a significant horizontal flow component (compared to vertical flow component). As a logical consequence, the smaller errors should occur at the shallower depths toward the bank on the losing side of the stream for the A r method. On the other hand, the magnitudes of errors obtained by the method are the same on both sides of the stream banks but the error increases with increasing depth. These overall observations are consistent with the findings reported by Shanafield et al. [2010] as well as Jensen and Engesgaard [2011] whose temperature data delivers different velocities calculated by the 1-D analytical solution for both sides of the streambed. The numerical results here can provide a good explanation of the field data collected by Jensen and Engesgaard [2011], although such variable flux estimates may be linked to other causes like sediment heterogeneity. Jensen and Engesgaard [2011] 9of16

10 Figure 7. Apparent vertical velocities (v z,a ), as a function of horizontal location obtained from (top) A r and (bottom) for the throughflow system with horizontal gradient of at different depths: 0.2 m, 0.4 m, and 0.8 m. observed that the vertical velocity estimates by the 1-D analytical solution give different discharges on one side of the stream and different recharges at the other, as well as high recharge values in the middle of the stream. [36] This numerical analysis may also in part explain some of the observed spatial discrepancies in velocity estimates observed at different locations or depths along a stream transect [Jensen and Engesgaard, 2011; Rau et al., 2010; Ronan et al., 1998]. It is clear from the results of the analysis that a nonvertical velocity component can introduce a high level of error into the estimation of the vertical velocity component especially for lower horizontal gradients for the side of the stream that is gaining when the A r method is used. [37] Previous studies have been advocating the use of the A r method [Lautz, 2010; Swanson and Cardenas, 2010] as most reliable based on various considerations about losing or gaining conditions; nonvertical flow field; vertical/horizontal velocity ratio; etc. Although their results and conclusions were based on variable fluid and heat transport along the direction of the stream (i.e., a hyporheic flow path) some observations are comparable. [38] The analysis in this study shows that there is no straightforward answer as to whether the A r or method is preferable and that it is actually a complex function of the heat and flow variables such as the vertical to horizontal velocity ratio, the magnitude of both vertical and horizontal velocity, the nonuniformity in flow pattern, the geometry of the channel (where banks are thermally insulated on the gaining and losing sides), etc. As mentioned previously for lower horizontal gradients, A r clearly gives smaller errors than the method especially on the losing side of the stream (Figures 8 and 10). For instance, while the error in vertical velocity estimate is almost zero at the horizontal position of x ¼ 2.8 m from the A r method for a horizontal gradient of 0.025, the method gives an error of more than 100% for the same horizontal position and gradient. For the gaining side of the stream with higher gradients the A r method overestimate the vertical flow component by up to 10% at x ¼ 1.2 m whereas the error with the 10 of 16

11 Figure 8. FSD error of the average true vertical velocity (v z,t,avg ) to the apparent vertical velocity estimates (v z,a ), as a function of horizontal location obtained from (top) A r and (bottom) for the flow-through system with a horizontal gradient of at different depths: 0.2 m, 0.4 m, and 0.8 m. method is much smaller for increasing horizontal gradients. However, the method is sensitive to changes in the horizontal gradients and to increases in depth (where the error increases dramatically on both the losing and gaining side of the stream, Figures 8 and 10). It is also evident that the vertical velocity estimate is not only a function of the ratio of the vertical to horizontal velocity but of the magnitudes of both velocities as well. [39] In general, the influence of horizontal flow on vertical velocity estimate is a puzzle which requires an analysis of all available information including knowledge of the hydraulic flow. Even so, some guidance can be extracted from the current analysis. From the center of the stream toward the gaining side the method can be used especially when higher horizontal gradients exist, however, a disadvantage is that the method will not give the direction of flow. Furthermore, if it is known from the hydraulic data that the horizontal gradient is low then the A r method should be relied upon. On the losing side, on the other hand, both methods can be used; however, if the horizontal gradients are very low then perhaps the method will not give reliable results. In addition, very close to the stream bank under high horizontal gradients, the A r method lacks in accuracy. Having no information about the hydraulics of the system, it is better to use both methods in combination. [40] The findings in this study are contrary to the conclusion drawn by Lautz [2010] who argued that the error in velocity estimate remains relatively constant as the ratio of vertical flow to horizontal flow changes using the method. For instance from the present analysis it is evident that the reduction in true vertical velocity while increase in true horizontal velocity (Figure 9) causes the errors in the velocity estimates to increase (Figure 8). Lautz [2010] also mentioned that the estimated vertical velocity from the 1-D analytical solution is always higher than the true vertical velocity under the influence of nonvertical flow fields (in losing condition) which is not supported by the present analysis that shows the vertical velocity determined with the A r method can be indeed less than the true vertical velocity in the losing condition (x ¼ 2tox¼ 2.8 in Figure 10). However, these discrepancies might come from the fact that the heat and hydraulic boundary conditions of the two models 11 of 16

12 Figure 9. The true horizontal velocity (v x,t ) and vertical velocity (v z,t ) across the stream for a horizontal gradient of at different depths: 0.2 m, 0.4 m, and 0.8 m. are different. In the work of Lautz [2010] the heat and hydraulic boundary conditions are in the form of a sinusoidal heat boundary and a linear hydraulic head distribution on the surface along the direction of surface flow to represent hyporheic flow along the stream, while the current study focuses on a flow-through system where the sinusoidal heat boundary condition is applied on the stream surface element only (not the banks) and the horizontal hydraulic head gradients are applied perpendicular to the direction of streamflow. Thus, in the current model some of the groundwater flow from the left boundary never discharges into the stream. [41] In order to make the difference clearer a simple example can be given from a heat transport point of view. The numerical model of Lautz [2010] can be approximated by a pipe having a heat signal from topside all along its length. When heat is distributed all over the boundary while the other sides and bottom boundaries are subjected to no heat flow conditions, the effect of flow on heat transport is more significant at the incoming and outgoing flow boundaries. Therefore if continuous estimation of vertical velocity (from heat) is made along a particular depth, the highest errors would be observed right at the incoming (left) and the outgoing (right) boundary. Whereas in the current study it can be assumed that only a section of the above pipe is exposed to heat signal (i.e., insulation from the banks) which in turn induces higher error in the velocity estimates especially at the left heat boundary. [42] In other words the differences in the results from the two models can be related to the way data is extracted from the numerical schemes. In the study by Lautz [2010] temperature time series were collected for a few sampling (observation) nodes on a horizontal line on losing side of the domain (but at some distance from left boundary) whereas in the current study the data were extracted from all numerical nodes on horizontal lines covering all possible probe locations across the stream. This leads to a more exhaustive analysis of potential deviations between the estimated and the actual vertical velocity across a stream and thus a better conceptual understanding of the possible impacts Implications for Field Investigations [43] In real field situations there is often limited a priori or detailed knowledge about the flow field to identify potential pitfalls and errors in deriving streambed velocities from analyzing streambed temperature data. Furthermore, the subsurface flow conditions may be dynamic and changing in response to flood events or groundwater abstraction. This means that errors in the vertical velocity estimates due to horizontal flow components may increase or diminish over time with these changes. This reiterates that the heat tracing methods should not be used in isolation, but as part of a larger toolbox that includes head gradient data, dissolved environmental tracers etc. [Kalbus et al., 2006]. By analyzing several different sources of data, potential errors can be identified or confidence can be gained in the temperature derived streambed velocities. [44] Some general recommendations for field installations and analysis of field data can be drawn from the analysis of this study. The analysis in this study indicates that the effects of horizontal flow are likely to be most pronounced for probes installed near gaining sides of the stream especially when using the A r method. As a consequence, temperature data from narrow streams are more likely to be affected than data from wider streams. For a given field location it is suggested to use three or more temperature arrays in two or more transects across a streambed. This in combination with subsequent inverse heat modeling could give a good estimation of the 3-D velocity field [Hopmans et al., 2002; Ismail-Zadeh et al., 2004; Vitrac et al., 2003]. Large variations in velocity estimates from one side of the stream to the other could be an indicator for potential errors due to significant horizontal flow components. The 12 of 16

13 Figure 10. (a) FSD error of the average true vertical velocity (v z,t,ave ) to the apparent vertical velocity estimates (v z,a ), as a function of horizontal location obtained from (a) A r and (b) for the flow-through system with different horizontal gradients: , 0.025, 0.05, , at 0.2 m depth, (c) the true vertical velocity for different horizontal gradients at z ¼ 0.2 m. analysis of this study also shows that both the A r and the methods should be used, because discrepancies between estimates of the two methods at each particular location in streambed could provide important clues as to whether the data is affected by horizontal flows. For instance, significant spatial inconsistency in the velocity estimates of A r and methods can be linked to nonvertical flow effect if the sediment is in the stream is not significantly heterogeneous. Furthermore each vertical array should have 3 or more probes over depth allowing for an analysis of the velocity variation over depth. The analysis also revealed that the errors in the velocity estimates are likely to be smaller if the 1-D analysis is carried out over shorter vertical distances (z) as indicated by Figure 8. However, the probes need to be far enough apart that A r and signals can be discerned. Thus, the problem becomes an issue of optimizing the probe distances against robust A r and data. [45] The directional flow analysis could be improved by including an artificial heat pulse method [Lewandowski et al., 2011]. This would offer more definite data which could be used for the estimation of slow directional flows within the streambed. Additionally, this could also verify the velocity estimates using 1-D analytical solutions. 6. Conclusion [46] This study is a comprehensive analysis of the performance of 1-D analytical solutions for calculating vertical 13 of 16

14 Figure 11. (a) The ratio of convective heat flow amplitude (AP conv ) over conductive heat flow (AP cond ) across the stream at z ¼ 0.2 m for different horizontal gradients. A positive ratio indicates losing conditions whereas a native ratio indicates gaining conditions. (b) The ratio of convective heat flow phase shift (PS conv ) over conductive heat flow (PS cond ) across the stream at z ¼ 0.2 m for different horizontal gradients. streambed fluid velocities in a stream-channel with variable flow fields using temperature data. The transport of heat into streambed sediments was investigated in a cross section of a stream subject to flow-through. A finite element-based numerical model was developed in FORTRAN to solve the conductive-convective heat transport and mass transfer equation. Temperature time series across the stream at 3 different horizontal levels in the streambed were generated with this numerical scheme and analyzed in the same manner as field data for the vertical velocity component using a 1-D analytical solution based on the amplitude ratio (A r ) and the phase shift (). [47] The results demonstrate that for low or no vertical velocity and a significant horizontal flow component (at the stream center under flow-through conditions), the analytical 1-D method produces velocity estimates that grossly overestimate the vertical velocity. The results also show that for higher true vertical velocities, better estimates of the velocity can be obtained using the 1-D analytical method. The errors in the vertical velocity estimates are more pronounced close to the gaining side of the stream and with increasing depth in the streambed especially for lower horizontal gradients. [48] The results have implications for designing field installations, deciding on probe locations and data interpretation: to minimize the velocity error, the temperature probes should be placed as close to the surface as possible and in two or more transects across the stream with 3 or more arrays in each transect and 2 or more probes in each array. This spatial set-up could give a more accurate interpretation of the temperature data for the quantification of vertical flow velocities. Furthermore, this can be used to evaluate the presence of horizontal flow. However, the flux predictions can still contain some level of uncertainty especially for locations close to stream banks with an inflowing horizontal flow component. Significant deviations in velocity estimates obtained from the A r and methods can be used to indicate whether or not there is a considerable horizontal velocity component. 14 of 16

15 Figure 12. FSD error of the average true vertical (v z,t,ave ) velocity to the apparent vertical velocity estimates (v z,a ) obtained from the method as a function of anisotropic Peclet number at two different horizontal location: x ¼ 1.2 m and 1.6 m and the vertical position of z ¼ 0.2 m. Appendix A: Finite Element Discretization [49] The standard and Characteristic Galerkin techniques are used to discretize the governing equations of mass and heat transport (equations (1) and (2)). This results in the following system of linear equations [Zienkiewicz and Taylor, 2000]:! ½ ðm1 þ th f ÞŠ½p iš¼½thf p i ðt i 1ÞŠ (A1)! ½ ðm þ tðc K 1 K 2 ÞÞŠ½T iš¼½tðc K1 K 2 ÞT i ðt i 1ÞŠ; (A2) where i is the time step; ~T is the temperature vector; ~T T ¼ðT 1 T 2...: T n Þ; T is the nodal temperature;!!! T ¼ Ti Ti 1 ; ~p is the pressure vector; ~P T ¼ðp 1!!! p 2...: p n Þ; p is the nodal pressure; p ¼ pi pi 1 ; t represents the time increment and is a scalar parameter of time discretization which can vary between 0.5 and 1.0. The matrices are defined as: Z ~C ¼ ½N T k@½n T ~K 1 ¼v d x Z Z ~M ¼ ½N T Š T ½N T ŠdV; Z fg n dv þ ½N T Š fgn dv þv d y ½N T Z k@½n T ½N T Š fgn dv; (A3) fg n dv; (A4) (A5) ~K 2 ¼ t Z 2 vd x þ t Z 2 vd x vd T fgn þv T Š fgn ½N T fgn þv ½N T Š fgn dv; Z ~M1¼ Z ~H f ¼ (A6) T N p cm N p dv (A7) T k rn p rn p dv; (A8) where N p and N T are the finite element shape functions of pressure and temperature, respectively, and V is the spatial area of an element. [50] Acknowledgments. Funding for this research was provided by the National Centre for Groundwater Research and Training, an Australian Government initiative, supported by the Australian Research Council and the National Water Commission. References Ahmed, T. (2000), Reservoir Engineering Handbook, edited, Elsevier, New York. Cardenas, M. B., and J. L. Wilson (2007a), Effects of current-bed form induced fluid flow on the thermal regime of sediments, Water Resour. Res., 43, W08431, doi: /2006wr Cardenas, M. B., and J. L. Wilson (2007b), Thermal regime of dunecovered sediments under gaining and losing water bodies, J. Geophys. Res., 112, G04013, doi: /2007jg Cardenas, M. B., J. L. Wilson, and V. A. Zlotnik (2004), Impact of heterogeneity, bed forms, and stream curvature on subchannel hyporheic exchange, Water Resour. Res., 40, W08307, doi: /2004wr of 16