### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Analysis of Temperature Profiles Beneath Streambeds
- 3. Analysis of Temperature Profiles Beneath Three Ephemeral Channels
- 4. Discussion
- References

[1] Continuous estimates of streamflow are challenging in ephemeral channels. The extremely transient nature of ephemeral streamflows results in shifting channel geometry and degradation in the calibration of streamflow stations. Earlier work suggests that analysis of streambed temperature profiles is a promising technique for estimating streamflow patterns in ephemeral channels. The present work provides a detailed examination of the basis for using heat as a tracer of stream/groundwater exchanges, followed by a description of an appropriate heat and water transport simulation code for ephemeral channels, as well as discussion of several types of temperature analysis techniques to determine streambed percolation rates. Temperature-based percolation rates for three ephemeral stream sites are compared with available surface water estimates of channel loss for these sites. These results are combined with published results to develop conclusions regarding the accuracy of using vertical temperature profiles in estimating channel losses. Comparisons of temperature-based streambed percolation rates with surface water-based channel losses indicate that percolation rates represented 30% to 50% of the total channel loss. The difference is reasonable since channel losses include both vertical and nonvertical component of channel loss as well as potential evapotranspiration losses. The most significant advantage of the use of sediment-temperature profiles is their robust and continuous nature, leading to a long-term record of the timing and duration of channel losses and continuous estimates of streambed percolation. The primary disadvantage is that temperature profiles represent the continuous percolation rate at a single point in an ephemeral channel rather than an average seepage loss from the entire channel.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Analysis of Temperature Profiles Beneath Streambeds
- 3. Analysis of Temperature Profiles Beneath Three Ephemeral Channels
- 4. Discussion
- References

[2] *Rorabaugh* [1954] was the first to describe correlations between stream temperature and stream loss, subsequently proposing the use of temperature measurements as an indirect method for estimating stream losses beneath the Ohio River, OH. He indicated that a groundwater model capable of quantifying heat and water fluxes appeared to be a promising tool for using heat as a tracer of shallow groundwater flow beneath stream channels. Recently, heat as a tracer of stream/groundwater exchanges has been shown to be useful for quantifying exchanges in perennial streams in humid regions [e.g., *Lapham*, 1989; *Silliman and Booth*, 1993; *Silliman et al.*, 1995]. Analysis of temperature profiles yielded point measurements of groundwater flow beneath perennial streams. In the present work, streambed temperature analysis is extended to ephemeral stream channels, where erratic streamflow patterns and a wide range of sediment moisture regimes create additional challenges in estimating shallow groundwater flow patterns. Currently continuous estimates of ephemeral streamflow are inhibited by the extremely transient nature of ephemeral flows, resulting in shifting channel geometry, and degradation of stream gage calibration (i.e., shifts in the stage-discharge rating curve). As a result, quantitative data concerning the nature of streamflow in ephemeral channels is sparse, with most data coming from intensive, short-duration field investigations. The extension of the use of heat as a shallow groundwater tracer to ephemeral channels has shown potential, due to the robust nature of temperature data acquisition.

[3] *Constantz et al.* [1994] demonstrated that large diurnal temperature variations within ephemeral channels created large diurnal variations in streambed infiltration rate, presumably due the temperature-dependence of the hydraulic conductivity in the shallow sediments. In response to short-duration streambed infiltration events, the timing and depth of streambed percolation was resolved using vertical temperature arrays installed beneath the ephemeral channels of Tijeras Arroyo, New Mexico [*Constantz and Thomas*, 1996]. A first attempt at temperature-based estimates of streambed percolation rates was performed beneath this channel, using a first approximation approach based on a heat-pulse/arrival time procedure [*Constantz and Thomas*, 1997]. These estimates of streambed percolation rates were encouraging, though not corroborated due to lack of available surface water monitoring techniques for flashy ephemeral channels. In the work of *Ronan et al.* [1998], the multidimensional nature of shallow groundwater flow beneath ephemeral channels was examined using extensive cross-channel temperature profiles within Vicee Canyon, Nevada. This labor-intensive study relied heavily on elaborate instrumentation, with multiple streamflow monitoring locations, micrometeorological data to correct for evapotranspiration (ET), periodic stream reach cross-sectional surveys, and simulation model calibration to supplement temperature profile analysis. Our recent work has focused on the accuracy of the independent use of temperature profiles without reliance on surface water data and model calibration. Investigations at numerous sites in the arid Southwest utilized longitudinal surface-temperature arrays in ephemeral channels to successfully monitor the timing and duration of streamflows at discrete locations over extensive reaches of the ephemeral channels [*Constantz et al.*, 2001].

[4] In the present work, temperature-based streambed percolation rates are compared with surface-based estimates of channel loss, where channel loss is defined as the streamflow loss in a given reach divided by channel area of the reach. A description of the analysis of temperature profiles for determination of streambed percolation is presented, followed by a description of the application of this analysis to several unpublished data sets from three ephemeral stream sites. Recently published temperature-based estimates of streambed percolation rates are combined with rates from these three sites, for comparison with concurrent surface water based estimates of channel loss for all sites.

### 2. Analysis of Temperature Profiles Beneath Streambeds

- Top of page
- Abstract
- 1. Introduction
- 2. Analysis of Temperature Profiles Beneath Streambeds
- 3. Analysis of Temperature Profiles Beneath Three Ephemeral Channels
- 4. Discussion
- References

[5] The quantitative use of heat as a tracer of groundwater fluxes requires a heat and water transport model capable of estimating the simultaneous movement of heat and pore water. A physically based analysis of heat and water transport through porous materials was introduced by *Philip and deVries* [1957]. Their analysis resulted in a comprehensive mathematical description of the coupled process of liquid and vapor water transport simultaneous with the transfer of heat in the solid, liquid and vapor phases of unsaturated porous material. Application of their analysis has demonstrated that the transport of heat and water in the vapor phase is often significant in unsaturated soils, and generally dominates in dry environments [e.g., *Scanlon and Milly*, 1994]. As the degree of water saturation increases in sediments, heat transport in the vapor phase abruptly declines as the gas phase becomes discontinuous, and then vanishes as sediments approach saturation [e.g., *Stonestrom and Rubin*, 1989]. In general, streambed sediments beneath wetted channels are sufficiently saturated to ignore macroscopic vapor transport, such that the comprehensive approach developed by Philip and deVries is unnecessary for analysis of heat and water fluxes beneath stream channels during periods of streamflow. During these periods, a simple single-phase model can represent heat and water fluxes in the streambed.

[6] *Suzuki* [1960] and *Stallman* [1963, 1965] were the first to use this single-phase approach to predict water fluxes through saturated sediments, based on measured groundwater temperatures. Their work formed the basis for examination of flow in environments ranging from deep groundwater systems [*Bredehoeft and Papadopulos*, 1965] to humid hillslopes [*Cartwright*, 1974]. Though rarely used as a tool, these pioneers convincingly demonstrated that heat is a viable quantitative tracer of groundwater flow.

[7] *Stallman* [1963] presented a general equation describing the simultaneous flow of heat and fluid in the earth. He indicated that groundwater temperatures could be used to determine the direction and rate of water movement. He also indicated that temperatures in combination with hydraulic gradients could be used to estimate sediment hydraulic conductivity. Stallman's equation for the simultaneous transfer of heat and water through saturated sediments for the one-dimensional case of vertical flow (z direction) is as follows:

where *K*_{t} is the thermal conductivity of the bulk streambed sediments in W/m °C, *T* is temperature in °C, *q* is the liquid water flux through the sediments in m/s, *C*_{w} and *C*_{s} are the volumetric heat capacity of water and the bulk sediment in J/m^{3} °C, respectively, *z* is length in m, and *t* is time in s. The equation neglects dispersion, but has been successful in predicting numerous groundwater temperature data. The value of *q* is controlled by the Darcy's equation as the product of the hydraulic conductivity and the total head gradient. When *q* is zero the equation reduces to the Fourier equation for the transfer of heat by conduction, and when *q* is large, advection dominates the transfer of heat, as well as the change of temperature throughout the porous material. Thermal parameters can be estimated given some knowledge of streambed materials. The heat capacity of the sediments can be estimated by a summation of the product of the specific heat and density of each sediment constituent, weighted proportionally to the constituent's volumetric fraction in the sediment as the following:

where *f*_{s}, *f*_{w}, and *f*_{a} are the volumetric fractions of the sediment, water, and air, respectively, c_{s}, c_{w}, and c_{a} are specific heats in J/kg °C of the sediment, water, and air, respectively, and ρ_{s}, ρ_{w}, and ρ_{a} are the densities in kg/m^{3} of the sediment, water, and air, respectively. The product of the specific heat capacity and the density is the volumetric heat capacity, which is in the range of 0.8 × 10^{6}, 4.2 × 10^{6}, and .001 × 10^{6} J/m^{3} °C for sediments, water and air, respectively [*de Vries*, 1963].

[8] A more general approach to describe simultaneous heat and water transport through sediments has been to utilize an energy transport approach via the convective-dispersion equation [*Kipp*, 1987]. These coupled heat and water flow equations are included here as equations (3), (4), and (5).

where θ is the volumetric fraction of the water content; ϕ is sediment porosity, dimensionless; *D*_{h} is the thermomechanical dispersion tensor, in m^{2}/s; *q* is the water flux, in m/s, and *Q* is rate of fluid source, in s^{−1}. The left side of the equation represents the change in energy stored in a volume over time. The first term on the right side describes the energy transport by heat conduction. The second term on the right side accounts for thermomechanical dispersion. The third term on the right side represents advective heat transport, and the final term on the right side represents heat sources and sinks to mass movement into or out of the volume. The themomechanical dispersion tensor is defined as [*Healy*, 1990]:

where α_{l} , α_{t} are longitudinal and transverse dispersivities, respectively, in m; δ_{i,j} is the Kronecker delta function; ν_{i,} ν_{j} are the ith and jth component of the velocity vector, respectively, in m/s.

[9] The familiar Buckingham–Richards flow equation is as follows:

where *C*(ψ, *x*) is specific moisture capacity, which is the slope of the water retention curve in m, ψ is the water pressure in m, *h* is the total head in m, *x* is length in m, *t* is time in s, and *K* is hydraulic conductivity in m/s [*Buckingham*, 1907; *Richards*, 1931].

[10] A critical difference between *K*_{t} and *K* is the level of uncertainty in assigning values for *K*_{t} versus *K* for the same material. Both parameters vary with texture and degree of saturation; however, for the typical case below a stream of saturated sediment of a given textural class, the uncertainty in *K*_{t} is relatively small compared with *K*. For example, the saturated *K*_{t} for a sand channel is likely to vary only between 1.0 and 2.0 W/m °C, so that the value of *K*_{t} can be estimated as 1.5 W/m °C ± 0.5 W/m °C [*van Duin*, 1963]. In contrast, the saturated hydraulic conductivity of sands may vary from 10^{−2} down to 10^{−6} m/s [*Freeze and Cherry*, 1979, p. 29], and as saturation decreases values of *K* have been measured to vary from 10^{−5} m/s down to 10^{−10} m/s [e.g., *Constantz*, 1982]. Often, *K*_{t} is assigned a value based on textural information, and identification of *K* becomes the primary focus.

[11] The two dimensional forms of equations (3), (4), and (5) are solved numerically in the computer simulation code, VS2DH [*Healy and Ronan*, 1996], specifically for use in stream environments. Currently, this code is one of several codes available for simulating simultaneous heat and groundwater transport. VS2DH is restricted to environments in which heat and water transport in the vapor phase are small relative to transport in the liquid phase. There are other heat and water transport simulation codes that are more suitable than VS2DH for specific environments. For example, SUTRA [*Voss*, 1984] is excellent in wet environments, while VS2DH generally handles the nonlinear behavior of parameters during infiltration better than SUTRA. TOUGH2 [*Pruess et al.*, 1999] is excellent for environments where vapor transport of heat and water is significant. Using reasonable initial and boundary conditions, these simulation codes can be used to predict temperature patterns in stream sediments. Alternatively, an inverse approach is employed to determine *K* by matching simulated sediment temperatures to observed temperature data either by using a trial-and-error manual approach (TE), or by using a parameter estimation (PE) inverse-modeling approach. Stream channel parameters, such as stream stage and temperatures are monitored, as a basis for developing model initial and boundary conditions. Model simulation runs are performed to fit simulated sediment temperatures to observed (measured) temperatures. This fit is accomplished by using TE or PE through manual or automated adjustment of *K* for each consecutive simulations. The PE approach relies on a code designed to minimize the sum of the square deviations between model-predicted and observed temperatures PEST [*Watermark Numerical Computing*, 1998] and UCODE [*Poeter and Hill*, 1998] are two PE codes, that have proved successful in groundwater investigations. For 1-D problems, TE appears to be as efficient as PE; however, for multidimensional problems with significant uncertainty in several parameters, PE is superior.

[12] In the present work, TE was used at two sites, while PE was used at the third site by linking PEST to VS2DH in order to calibrate the model with respect to sediment temperatures. PEST uses the Gauss-Marquardt-Levenberg optimization algorithm to estimate model parameters. For the present study, the objective function may be written as the following:

where ϕ is the objective function value, _{i} is measured sediment temperatures, and Y_{i} is simulated sediment temperatures.

[13] This parameter estimation code matches the simulated temperature values to observed sediment-temperature values by optimizing hydraulic and thermal parameters to achieve the minimum differences between simulated and observed temperatures. As discussed above, the thermal conductivity possess significantly less variability compared with hydraulic conductivity for the same material. Therefore, thermal parameters can be specified within a narrow range, based on the literature for the particular textural class observed in the field, while the values of the hydraulic gradient and hydraulic conductivity are the primary variables available for optimizing the simulated sediment temperatures against the observed temperatures. Both hydraulic parameters are allowed to vary within physically reasonable ranges; the product of these parameters ( i.e., *q*) is determined during optimization of simulated and observed sediment temperatures. When a channel has saturated after an extended flow period, the hydraulic gradient can be determined using piezometers, such that simulation can be optimized varying only the value of *K*. Examples of both types of optimization are presented in discussions of field-site results.

### 4. Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Analysis of Temperature Profiles Beneath Streambeds
- 3. Analysis of Temperature Profiles Beneath Three Ephemeral Channels
- 4. Discussion
- References

[33] Figure 10 displays a 1:1 plot for comparison of temperature-based estimates of streambed percolation rates versus surface water-based estimates of streambed channel loss, for the three sites reported on in this study. In addition, Figure 10 includes a comparison of percolation and channel loss for earlier work of a similar nature performed in Tijeras Arroyo, New Mexico [*Constantz and Thomas*, 1997], Vicee Canyon, Nevada [*Ronan et al.*, 1998], and La Bajada, New Mexico [*Thomas et al.*, 2000]. Examination of Figure 10 demonstrates that comparative results reside in three distinct regions on the 1:1 plot. The initial results for the Santa Fe site and the results for the Tijeras site reside in a region above the 1:1 line, indicating that estimates of percolation rates were higher than channel loss for these two comparisons. This was not expected, though in hindsight might be explained by differential time-duration of measurements for percolation compared with seepage estimates. The initially rapid percolation rates were determined by analysis of the abrupt temperature response during the inception of streambed infiltration at each site. Alternatively, channel loss were determined over a longer time period, as streamflow traveled to the lower streamflow gaging locations. This resulted in a channel loss that was composed of both initially high streambed infiltration rates at the distal end of flow and exponentially lower rates in the upper end of the reach.

[34] In Figure 10, the comparison for Vicee Canyon lies close to the 1:1 line. This is reasonable since the Vicee Canyon study was an intensive study, based on 2-D vertical temperature profiles across the stream channel, and calibrated to earlier flow events. Additionally, channel loss included compensation for field-determined ET losses, so that the net seepage was reduced below the total streamflow loss per unit area. Comparisons for the La Bajada site, Bear Canyon, the Santa Clara site, and the final rates at the Santa Fe site all reside in the lower region of the plot. For these four comparisons, the surface water based estimate for channel loss were consistently 2 to 3 times greater than temperature-based estimates of percolation rates. This difference is reasonable due to the multidimensional nature of streamflow channel loss, as well as the lack of compensation for ET losses. Comparing all these sites, the results suggest that percolation rates are initially rapid, and decrease to a value that approaches 30% to 50% of channel loss for these streams. Regarding the relative contribution of ET to the gross channel loss, field results from the both Vicee Canyon study and La Bajada study included estimates of ET during summer months. These studies concluded that ET represented approximately 10% of the total channel loss at these sites during this period. This would suggest that the primary difference between the magnitude in percolation rate versus channel loss is due to the nonvertical component of shallow groundwater flow in these ephemeral channels.

[35] In conclusion, streambed temperature profiles form the basis of a tool capable of: (1) estimating the timing and duration of streamflow channel losses, (2) determining the percolation rate, and (3) estimating the magnitude of channel loss at a point (or points) in the channel. Temperature profiles provide a robust and continuous record, creating a long-term record of streamflow patterns within an ephemeral channel. The primary disadvantage is the inability of a single or limited number of temperature profiles to reflect the multidimensional, spatially variable nature of streamflow losses beneath ephemeral channels. As a future application, temperature-based estimates of streamflow duration and percolation rate may lead to estimates of stream loss for an entire reach by assuming a value for channel loss of 2 or 3 times the point measurement of percolation rate, and multiplying this rate by an estimate of the channel area of the reach. Cumulative stream loss (m^{3}) might then be estimated from the product of the stream loss and temperature-based estimate of flow duration for the reach. Though this approach possesses clear uncertainty due to spatial variability, challenges inherent in acquiring long-term streamflow patterns using current surface water-based methods suggest that this uncertainty may be acceptable for estimating stream losses beneath ephemeral channels.