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

^{1}

^{2}

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## 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

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**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