# 1-D Vertical Flux Dynamics in a Low-Gradient Stream: An Assessment of Stage as a Control of Vertical Hyporheic Exchange

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Site: Little Kickapoo Creek

^{−6}m/s [42], but within the streambed, the mean hydraulic conductivity is 2 × 10

^{−3}m/s [32,36]. Regionally, LKC is considered a gaining stream that has a baseflow of 0.1 m

^{3}/s supported by groundwater [37]; locally, Bastola and Peterson [32] calculated upward vertical fluxes of 10

^{−7}m/s.

^{−8}m/s [42]. The sharp contrast between the hydraulic conductivity values of the Tiskilwa and the Henry Formations, forces flow through the overlying Henry Formation [38,42]. Therefore, the Tiskilwa Formation is considered a confining layer [38,42].

#### 2.2. Data Collection

#### 2.3. Flux Calculations

_{e}is the thermal diffusivity, n is the porosity, ν

_{f}is the fluid velocity, and γ is the ratio of the volumetric heat capacity of the streambed (C

_{s}) to the volumetric fluid heat capacity (C

_{w}) [7,15]. The second term in the equation is the conduction-dispersion term. Conduction is dependent on K

_{e}and the change in thermal gradient [31]. K

_{e}is dependent on thermal dispersivity (β), baseline thermal conductivity in the absence of fluid flow (λ

_{o}), ν

_{f}, and the C

_{s}[7,19,20] (Table 1).

_{f}representing the fluid flux (q), which Hatch et al. [7] and Keery et al. [5] solved for flux using the attenuation of the amplitude (amplitude ratio) of sinusoidal temperature signals propagating vertically into the streambed.

#### 2.4. Statistics: Stage and Vertical 1-D Flux

## 3. Results

#### 3.1. Trends in Raw Temperature Data and Vertical 1-D Flux

#### 3.2. Correlation of Stage and 1-D Flux

## 4. Discussion

#### 4.1. Low-Gradient Stream Conceptual Model—Temperature and Flux dynamics

^{3}/s that can rise to 4 m

^{3}/s with increases in runoff and tile drainage input [34,37]. The 1-D flux data illustrate the strong upwelling component but also reveal occurrence of downwelling water.

^{−1}to 10

^{−2}m/d across the 4 m vertical streambed domain, with lower rates during the wet winter and higher rates in the dry summer due to evapotranspiration [32]. The magnitude of flux rates produced in the VS2DH model aligns with the rates calculated in this study, and the direction of advective heat transfer is consistent with the groundwater upwelling from 15 to 120 cm seen in the streambed. Downward flux rates in the top 15 cm of the streambed are not observed in the VS2DH model, but the VS2DH model has a coarser spatial resolution of 1 m compared to the 30 and 60 cm resolution used in VFLUX. The coarse resolution smooths out discrete changes in flux seen at 15 cm due to spatial calculation frequency being too small the VS2DH model. Bastola and Peterson [32] reported the influence of the stream on the temperature to a depth of 30 cm suggesting the near-surface hyporheic zone is more dynamic than the deeper substrate. This study exhibits a stronger capability to see small spatial changes in fluxes in the LKC hyporheic zone. Thermal envelopes illustrated a stream influence in the top 50 cm depth of the streambed [32] similar to the groundwater transition zone seen in this study. Overall, LKC temperature models display upward groundwater advection dominating the LKC streambed, and LKC solute models point to hyporheic mixing conditions in the near-subsurface with deeper groundwater upwelling.

#### 4.2. Stage as a Control of 1-D Vertical Flux

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Stallman, R.W. Steady One-Dimensional Fluid Flow in a Semi-Infinite Porous Medium with Sinusoidal Surface Temperature. J. Geophys. Res.
**1965**, 70, 2821–2827. [Google Scholar] [CrossRef] - Suzuki, S. Percolation measurements based on heat flow through soil with special reference to paddy fields. J. Geophys. Res.
**1960**, 65, 2883–2885. [Google Scholar] [CrossRef] - Conant, B., Jr. Delineating and quantifying ground water discharge zones using streambed temperatures. Ground Water
**2004**, 42, 243–257. [Google Scholar] [CrossRef] [PubMed] - Silliman, S.E.; Booth, D.F. Analysis of time-series measurements of sediment temperature for identification of gaining vs. losing portions of Juday Creek, Indiana. J. Hydrol.
**1993**, 146, 131–148. [Google Scholar] [CrossRef] - Keery, J.; Binleya, A.; Crook, N.; Smith, J.W.N. Temporal and spatial variability of groundwater–surface water fluxes: Development and application of an analytical method using temperature time series. J. Hydrol.
**2007**, 336, 1–16. [Google Scholar] [CrossRef] - Schmidt, C.; Conant, B., Jr.; Bayer-Raich, M.; Schirmer, M. Evaluation and field-scale application of a simple analytical method to quantify groundwater discharge using mapped streambed temperatures. J. Hydrol.
**2007**, 347, 292–307. [Google Scholar] [CrossRef] - Hatch, C.E.; Fisher, A.T.; Revenaugh, J.S.; Constantz, J.; Ruehl, C. Quantifying surface water-groundwater interactions using time series analysis of streambed thermal records: Method development. Water Resour. Res.
**2006**, 42. [Google Scholar] [CrossRef] [Green Version] - Becker, M.W.; Georgian, T.; Ambrose, H.; Siniscalchi, J.; Fredrick, K. Estimating flow and flux of ground water discharge using water temperature and velocity. J. Hydrol.
**2004**, 296, 221–233. [Google Scholar] [CrossRef] - Lapham, W.W. Use of Temperature Profiles Beneath Streams to Determine Rates of Vertical Ground-Water Flow and Vertical Hydraulic Conductivity; 0886-9308; United States Geological Survey: Reston, VA, USA, 1989; p. 35. [Google Scholar]
- Briggs, M.A.; Lautz, L.K.; Buckley, S.F.; Lane, J.W. Practical limitations on the use of diurnal temperature signals to quantify groundwater upwelling. J. Hydrol.
**2014**, 519, 1739–1751. [Google Scholar] [CrossRef] - Jensen, J.K.; Engesgaard, P. Nonuniform Groundwater Discharge across a Streambed: Heat as a Tracer. Vadose Zone J.
**2011**, 10, 98–109. [Google Scholar] [CrossRef] - Allander, K. Trout Creek—Evaluating ground-water and surface water exchange along an alpine stream, Lake Tahoe, California. In Heat as a Tool for Studying the Movement of Ground Water Near Streams; Stonestrom, D.A., Constantz, J., Eds.; United States Geological Survey: Reston, VA, USA, 2003; Volume Circular 1260, pp. 35–45. [Google Scholar]
- Luce, C.H.; Tonina, D.; Gariglio, F.; Applebee, R. Solutions for the diurnally forced advection-diffusion equation to estimate bulk fluid velocity and diffusivity in streambeds from temperature time series. Water Resour. Res.
**2013**, 49, 488–506. [Google Scholar] [CrossRef] [Green Version] - McCallum, A.M.; Andersen, M.S.; Rau, G.C.; Acworth, R.I. A 1-D analytical method for estimating surface water–groundwater interactions and effective thermal diffusivity using temperature time series. Water Resour. Res.
**2012**, 48. [Google Scholar] [CrossRef] - Anderson, M.P. Heat as a ground water tracer. Ground Water
**2005**, 43, 951–968. [Google Scholar] [CrossRef] [PubMed] - Goto, S.; Yamano, M.; Kinoshita, M. Thermal response of sediment with vertical fluid flow to periodic temperature variation at the surface. J. Geophys. Res. Sol. Earth
**2005**, 110. [Google Scholar] [CrossRef] - Young, P.C.; Pedregal, D.J.; Tych, W. Dynamic harmonic regression. J. Forecast.
**1999**, 18, 369–394. [Google Scholar] [CrossRef] [Green Version] - Swanson, T.E.; Cardenas, M.B. Ex-Stream: A MATLAB program for calculating fluid flux through sediment–water interfaces based on steady and transient temperature profiles. Comput. Geosci.
**2011**, 37, 1664–1669. [Google Scholar] [CrossRef] - Gordon, R.P.; Lautz, L.K.; Briggs, M.A.; McKenzie, J.M. Automated calculation of vertical pore-water flux from field temperature time series using the VFLUX method and computer program. J. Hydrol.
**2012**, 420, 142–158. [Google Scholar] [CrossRef] - Irvine, D.J.; Lautz, L.K.; Briggs, M.A.; Gordon, R.P.; McKenzie, J.M. Experimental evaluation of the applicability of phase, amplitude, and combined methods to determine water flux and thermal diffusivity from temperature time series using VFLUX 2. J. Hydrol.
**2015**, 531, 728–737. [Google Scholar] [CrossRef] [Green Version] - Ward, A.S.; Fitzgerald, M.; Gooseff, M.N.; Voltz, T.J.; Binley, A.M.; Singha, K. Hydrologic and geomorphic controls on hyporheic exchange during base flow recession in a headwater mountain stream. Water Resour. Res.
**2012**, 48. [Google Scholar] [CrossRef] [Green Version] - Ward, A.S.; Schmadel, N.M.; Wondzell, S.M.; Gooseff, M.N.; Singha, K. Dynamic hyporheic and riparian flow path geometry through base flow recession in two headwater mountain stream corridors. Water Resour. Res.
**2017**, 53, 3988–4003. [Google Scholar] [CrossRef] - Ward, A.S.; Schmadel, N.M.; Wondzell, S.M.; Harman, C.; Gooseff, M.N.; Singha, K. Hydrogeomorphic controls on hyporheic and riparian transport in two headwater mountain streams during base flow recession. Water Resour. Res.
**2016**, 52, 1479–1497. [Google Scholar] [CrossRef] - Schmadel, N.M.; Ward, A.S.; Wondzell, S.M. Hydrologic controls on hyporheic exchange in a headwater mountain stream. Water Resour. Res.
**2017**, 53, 6260–6278. [Google Scholar] [CrossRef] - Sickbert, T.B.; Peterson, E.W. The effect of surface water velocity on hyporheic interchange. J. Water Resour. Prot.
**2014**, 6, 327–336. [Google Scholar] [CrossRef] [Green Version] - Harvey, J.W.; Bencala, K.E. The effect of streambed topography on surface-subsurface water exchange in mountain catchments. Water Resour. Res.
**1993**, 29, 89–98. [Google Scholar] [CrossRef] - Kasahara, T.; Wondzell, S.M. Geomorphic controls on hyporheic exchange flow in mountain streams. Water Resour. Res.
**2003**, 39. [Google Scholar] [CrossRef] [Green Version] - Storey, R.G.; Howard, K.W.F.; Williams, D.D. Factors controlling riffle-scale hyporheic exchange flows and their seasonal changes in a gaining stream; a three-dimensional groundwater flow model. Water Resour. Res.
**2003**, 39. [Google Scholar] [CrossRef] - Wondzell, S.M.; Gooseff, M.N. Geomorphic controls on hyporheic exchange across scales: Watersheds to particles. In Treatise on Geomorphology; Shroder, J., Wohl, E., Eds.; Academic Press: San Diego, CA, USA, 2013; Volume 9, pp. 203–218. [Google Scholar]
- Cardenas, M.B.; Wilson, J.L.; Zlotnik, V.A. Impact of heterogeneity, bed forms, and stream curvature on subchannel hyporheic exchange. Water Resour. Res.
**2004**, 40. [Google Scholar] [CrossRef] [Green Version] - Deming, D. Introduction to Hydrogeology; McGraw-Hill College: New York, NY, USA, 2002; p. 468. [Google Scholar]
- Bastola, H.; Peterson, E.W. Heat tracing to examine seasonal groundwater flow beneath a low-gradient stream. Hydrogeol. J.
**2016**, 24, 181–194. [Google Scholar] [CrossRef] - Buyck, M.S. Tracking Nitrate Loss and Modeling Flow through the Hyporheic Zone of a Low Gradient Stream through the Use of Conservative Tracers. Master’s Thesis, Illinois State University, Normal, IL, USA, 2005. [Google Scholar]
- Peterson, E.W.; Hayden, K.M. Transport and Fate of Nitrate in the Streambed of a Low-Gradient Stream. Hydrology
**2018**, 5, 55. [Google Scholar] [CrossRef] [Green Version] - Oware, E. The Impact of Storm on Thermal Transport in the Hyporheic Zone of a Low-Gradient Third-Order Sand and Gravel Bedded Stream. Master’s Thesis, Illinois State University, Normal, IL, USA, 2010. [Google Scholar]
- Peterson, E.W.; Sickbert, T.B.; Moore, S.L. High frequency stream bed mobility of a low-gradient agricultural stream with implications on the hyporheic zone. Hydrol. Process.
**2008**, 22, 4239–4248. [Google Scholar] [CrossRef] - Peterson, E.W.; Sickbert, T.B. Stream water bypass through a meander neck, laterally extending the hyporheic zone. Hydrogeol. J.
**2006**, 14, 1443–1451. [Google Scholar] [CrossRef] - Peterson, E.W.; Benning, C. Factors influencing nitrate within a low-gradient agricultural stream. Environ. Earth Sci.
**2013**, 68, 1233–1245. [Google Scholar] [CrossRef] - Riggs, M.H.; Abert, C.C. Cumulative sand and gravel thickness in McLean County, Illinois. In Open File Series 1997-01f; Survey, I.S.G., Ed.; Illinois State Geological Survey: Champaign, IL, USA, 1998. [Google Scholar]
- Hansel, A.K.; Johnson, W.H. Wedron and Mason Groups: Lithostratigraphic Reclassification of Deposits of the Wisconsin Episode, Lake Michigan Lobe Area; Bulletin 104; Illinois State Geological Survey: Champaign, IL, USA, 1996; p. 116. [Google Scholar]
- Basu, A. Quantifying N Cycling Between Groundwater and Surface Water Using Numerical Modeling and Mass Flux Calculations. Master’s Thesis, Illinois State University, Normal, IL, USA, 2007. [Google Scholar]
- Ackerman, J.R.; Peterson, E.W.; Van der Hoven, S.; Perry, W. Quantifying nutrient removal from groundwater seepage out of constructed wetlands receiving treated wastewater effluent. Environ. Earth Sci.
**2015**, 74, 1633–1645. [Google Scholar] [CrossRef] - Van der Hoven, S.J.; Fromm, N.J.; Peterson, E.W. Quantifying nitrogen cycling beneath a meander of a low gradient, N-impacted, agricultural stream using tracers and numerical modelling. Hydrol. Process.
**2008**, 22, 1206–1215. [Google Scholar] [CrossRef] - Young, P.; Taylor, C.; Tych, W.; Pegregal, D.; McKenna, P. The Captain Toolbox. In Centre for Research on Environmental Systems and Statistics; Lancaster University: Bailrigg, UK, 2010. [Google Scholar]
- R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2013. [Google Scholar]
- Irvine, D.J.; Briggs, M.; Cartwright, I.; Lautz, L. Improved vertical streambed flux estimation using multiple diurnal temperature methods in series. Groundwater
**2016**, 55, 73–80. [Google Scholar] [CrossRef] [Green Version] - Irvine, D.J.; Briggs, M.A.; Lautz, L.K.; Gordon, R.P.; McKenzie, J.M.; Cartwright, I. Using Diurnal Temperature Signals to Infer Vertical Groundwater-Surface Water Exchange. Groundwater
**2017**, 55, 1–17. [Google Scholar] [CrossRef] - Beach, V.; Peterson, E.W. Variation of hyporheic temperature profiles in a low gradient third-order agricultural stream–A statistical approach. Open J. Mod. Hydrol.
**2013**, 3, 55–66. [Google Scholar] [CrossRef] [Green Version] - Hester, E.T.; Doyle, M.W. In-stream geomorphic structures as drivers of hyporheic exchange. Water Resour. Res.
**2008**, 44. [Google Scholar] [CrossRef] - Hill, A.R.; Labadia, C.F.; Sanmugadas, K. Hyporheic zone hydrology and nitrogen dynamics in relation to the streambed topography of a N-rich stream. Biogeochemistry
**1998**, 42, 285–310. [Google Scholar] [CrossRef]

**Figure 1.**Little Kickapoo Creek (LKC) watershed and the site configuration in plan view. There are six streambed multi-level samplers installed along the thalweg of the stream. The stilling well, installed along the stream bank in-line with the second multi-level sampler, monitored stage at a 15 min intervals.

**Figure 2.**(

**a**). A nested streambed sampler with four temperature sensors at 30, 60, 90, 150 cm depths. Each temperature sensor was isolated with foam sealant. (

**b**) Phase shift and amplitude attenuation of sinusoidal thermal signals with increasing depth. The amplitude of thermal signals decreases, ∆A, and the signal is delayed, shift in phase (∆ϕ), with increasing depth in the streambed (graphic modified from [7].

**Figure 3.**Stream stage and water temperature in the stream and at depths of 30, 60, 90, and 150 cm within sampler 1. Temperatures observed in sampler 1 were representative of all six samplers.

**Figure 4.**Week-long temperature time series with storm events at 30, 60, 90, 150 cm depth. There are storm events on 8/17, 8/18, 8/20, and 11/17–11/19. The summer temperature gradient is negative (

**a**), and the winter is positive (

**b**). The stream temperature signal change during the storm event propagates down to a streambed depth of 90 cm in the winter and summer.

**Figure 6.**(

**a**). Calculated Flux at 15 cm depth for all six multi-level samplers. (

**b**). Precipitation and stage data. Stage is lowest in the summer (May–July) when flux rates are higher relative to fall, winter, and spring.

**Table 1.**Parameter values used in flux calculations. The parameters representative of the Henry Formation.

Parameter | Units | Value | Source |
---|---|---|---|

Porosity (n) | 0.375 | [32] | |

Dispersivity (β) | m | 0.005 | [32] |

Thermal Conductivity (λ_{o}) | cal/sec-cm-C | 0.0033 | [32] |

Volumetric Heat Capacity of Sediment (C_{s}) | cal/cm^{3}-C | 0.239 | [32] |

Volumetric Heat Capacity of Water (C_{w}) | cal/cm^{3}-C | 1 | [9] |

Complete Time Series | Winter 2009 | Spring 2009 | Summer 2009 | Fall 2009 | Winter 2010 | |||
---|---|---|---|---|---|---|---|---|

Depth | Minimum | Mean | Maximum | Mean | Mean | Mean | Mean | Mean |

(cm) | (°C) | (°C) | (°C) | (°C) | (°C) | (°C) | (°C) | (°C) |

Stream (0) | 0.02 | 0.85 | 2.59 | 0.86 | 0.91 | 0.99 | 0.60 | 0.62 |

30 | 0.00 | 0.03 | 0.07 | 0.01 | 0.02 | 0.03 | 0.03 | 0.04 |

60 | 0.00 | 0.02 | 0.05 | 0.01 | 0.01 | 0.02 | 0.03 | 0.03 |

90 | 0.00 | 0.02 | 0.13 | 0.01 | 0.00 | 0.02 | 0.01 | 0.06 |

150 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

**Table 3.**Spearman rank correlation coefficients (ρ) for all six multi-level samplers at 15 cm midpoint depth. All ρ values were statistically significant for a 95% confidence level. The correlation coefficient in the first row is between stage and flux with no forward time step. The second row indicates the maximum ρ values between stage and flux with incremental 2 h time steps in flux up to 80 h of lag time. There are positive and negative statistically significant weak correlations between stage and flux.

Well 1 | Well 2 | Well 3 | Well 4 | Well 5 | Well 6 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Lag (h) | ρ | Lag (h) | ρ | Lag (h) | ρ | Lag (h) | ρ | Lag (h) | ρ | Lag (h) | ρ | |

No Lag | 0 | 0.14 | 0 | 0.05 | 0 | 0.09 | 0 | 0.37 | 0 | −0.11 | 0 | −0.03 |

ρ_{max} | 34 | 0.19 | 26 | 0.09 | 80 | −0.12 | 28 | 0.44 | 80 | −0.18 | 80 | 0.04 |

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**MDPI and ACS Style**

Harris, F.C.; Peterson, E.W.
1-D Vertical Flux Dynamics in a Low-Gradient Stream: An Assessment of Stage as a Control of Vertical Hyporheic Exchange. *Water* **2020**, *12*, 708.
https://doi.org/10.3390/w12030708

**AMA Style**

Harris FC, Peterson EW.
1-D Vertical Flux Dynamics in a Low-Gradient Stream: An Assessment of Stage as a Control of Vertical Hyporheic Exchange. *Water*. 2020; 12(3):708.
https://doi.org/10.3390/w12030708

**Chicago/Turabian Style**

Harris, F. Claire, and Eric W. Peterson.
2020. "1-D Vertical Flux Dynamics in a Low-Gradient Stream: An Assessment of Stage as a Control of Vertical Hyporheic Exchange" *Water* 12, no. 3: 708.
https://doi.org/10.3390/w12030708