# The Effect of Stream Discharge on Hyporheic Exchange

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Site

^{2}and an elevation range of 114–405 m a.s.l. Hydrological parameters were measured in Krycklan Catchment at 15 stations located in different parts of the catchment [43]. Krycklan Catchment is characterized by a cold and humid climate, featuring deep snow cover during the entire winter [44]. The 30-year (1981–2007) mean annual precipitation is 614 mm/year, 30–50% of which falls as snow [45]. The Quaternary deposits mostly consist of glacial till with a thickness of up to tens of meters [42].

#### 2.2. Field Measurement

#### 2.3. Field Data Analysis

#### 2.4. Modeling Framework

^{®}software to simulate a two-dimensional longitudinal transect along the center of the stream network. The Brinkman–Darcy equation (Equation (1)) was used with the continuity equation to describe the subsurface flow in the hyporheic zone near the streambed surface:

^{3}] is the water density, ε [–] is the Brinkman porosity,

**u**[m/s] is the subsurface velocity vector, t [s] is the time, p [pa] is the water pressure, μ [pa.s] is the water’s dynamic viscosity, μ

_{e}[pa.s] is the water’s effective viscosity, k [m

^{2}] is the intrinsic permeability of the sediment, and ($\frac{\mathsf{\mu}}{\mathrm{k}}u)$ is the Darcy term.

^{−9}[m

^{2}] representing the streambed sediment [18]) for the entire subsurface region. The second permeability scenario was performed by applying a decaying intrinsic permeability in the top meter of the subsurface region (starting with k = 10

^{−9}[m

^{2}] for the top surface decaying to k = 10

^{−12}[m

^{2}] at a depth of 1 m) and a constant intrinsic permeability (k = 10

^{−12}[m

^{2}] refers to the glacial sediment soil type) for the rest of the subsurface region (deeper than 1 m from the top surface). The depth-decaying function for the soil permeability was defined based on the findings of Morén et al. [18]. They performed an experiment in a small stream in Sweden (Tullstorps Brook) and measured the hydraulic conductivity at two depths (i.e., 3 and 7 cm) every 100 m for a reach of 1500 m. The permeability k(z) of the top meter of the stream was described according to [17]:

_{0}[m

^{2}] is the intrinsic permeability at the streambed interface, z [m] is the depth from the streambed, and c [m

^{−1}] is an empirical decay coefficient. An exponential function was fitted to the measured data of Morén et al. [18] with a lower hydraulic conductivity limit of K = 10

^{−6}[m/s] (corresponding to an intrinsic permeability of k = 10

^{−12}[m

^{2}]) for a depth of 1 m from the streambed surface (Figure S2, Supplementary Materials).

^{2}] is the gravitational acceleration, and ${d}_{i}\left(x\right)$ [m] is the water depth for each flow discharge. The water depth was quantified using the corresponding mean discharge value for each flow discharge (see Section 2.3). Furthermore, an open boundary was assumed for the upstream and downstream sides, whereas a no-flow boundary was considered for the bottom surface of the model. Each model was run to a steady state flow condition based on hydrostatic head boundary condition on the top surface.

## 3. Results

#### 3.1. Water Temperature

#### 3.2. Modeling Results

_{HF,max}). Figure 5 shows the calculated maximum depths of the hyporheic fluxes for the case of constant permeability (Figure 5a) and decaying permeability in the top meter of the flow domain (Figure 5b). The results indicate that D

_{HF,max}varied during the three flow discharges. In particular, the minimum D

_{HF,max}is highly affected by stream water flow intensity (the range of D

_{HF,max}is affected by the order of 10 and 100 in constant and decaying permeability scenarios, respectively). High-flow discharges have lower values of D

_{HF,max}than the base-flow discharge, whereas the minimum D

_{HF,max}for the base flow is higher than that for low-flow discharge, regardless of the permeability scenario applied. The median value of D

_{HF,max}varies slightly between different flow discharges, with the low-flow discharge having the highest median value of D

_{HF,max}among all the flow discharges, i.e., 4 m and 0.03 m in constant and decaying permeability scenarios, respectively (Figure S3 in the Supplementary Materials). However, in the case of decaying permeability, the median value of D

_{HF,max}(Figure 5b) is in the range of centimeters which is substantially lower than in the constant permeability case (Figure 5a) that is in a range of meters. Additionally, the logarithmic ranges of interquartile of the maximum depth of the hyporheic fluxes in the constant permeability case (Figure 5a) are smaller than those in the decaying permeability scenario (Figure 5b) for all flow discharges.

## 4. Discussion

#### 4.1. Temperature Variation in the Streambed Sediment

#### 4.2. The Role of the Heterogeneity of Sediment Permeability in Hyporheic Exchange

## 5. Conclusions

## Supplementary Materials

**a**) constant intrinsic permeability (k = 10

^{−9}[m

^{2}]) for the entire subsurface region and (

**b**) decaying intrinsic permeability (starting from k(z = 0) = 10

^{−9}[m

^{2}] at the surface–subsurface water interface and decaying exponentially to k(z = –1) = 10

^{−12}[m

^{2}] down to a depth of 1 m). In the case of vertically varying permeability, a constant permeability (k = 10

^{−12}) was used for depths larger than 1 m.”, Table S1: “Correlation between discharge and mean water depth using five flow data for every 50 m distance from upstream along the main river”, Table S2: “Estimated mean water depth for different flow condition every 50 m from upstream”.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Map showing the experimental subcatchment and its topography, the main river (dark blue color), tributaries (cyan color), Lake Stortjärn (solid light blue region), and hydrological stations C5 and C6. Also shown are the locations of the V-notch weir (red flag) and temperature lances (yellow stars).

**Figure 2.**Relationships between discharge and water depth at a spatial resolution of 50 m (one of the discharge values at a distance of 550 m from the upstream station (shown with a magenta star) was not considered in the analysis due to its unrealistic value, which might be due to measurement error).

**Figure 3.**Water temperature time series measured at different locations along the stream network for different flow discharges. Each color represents the temperature recorded by sensors at increasing depth. Colors range from brown to blue as depth increases. Temperatures were not properly recorded during the first 20 h by the temperature stick located 350 m from the upstream station (panel g), and these were therefore neglected. High-flow discharge corresponds to high flow 1 period of Table 2.

**Figure 4.**Vertical envelopes of temperature dynamics during base-, low-, and high-flow discharges at different monitoring locations. The envelopes indicate the interquartile range (shaded area) and the median (red line).

**Figure 5.**Boxplots showing the maximum depth of hyporheic fluxes under various flow discharges assuming (

**a**) constant intrinsic permeability (k = 10

^{−9}[m

^{2}]) for the entire subsurface region and (

**b**) a decaying intrinsic permeability (starting from k(z = 0) = 10

^{−9}[m

^{2}] at the surface–subsurface water interface and decaying exponentially to k(z = −1) = 10

^{−12}[m

^{2}] at one meter depth). In the case of a vertically varying permeability, a constant permeability (k = 10

^{−12}) was used for depths larger than one meter. The second row of the horizontal axis (i.e., numbers), are the ranges of stream flow discharge along the stream for each flow regime. D

_{HF,max}: deepest point of the streamlines.

**Figure 6.**Box and whisker plots of hyporheic fluxes residence time under various flow discharges assuming (

**a**) constant intrinsic permeability (k = 10

^{−9}[m

^{2}]) for the entire subsurface region and (

**b**) decaying intrinsic permeability (starting from k(z = 0) = 10

^{−9}[m

^{2}] at the surface–subsurface water interface and decaying exponentially to k(z = −1) = 10

^{−12}[m

^{2}] down to a depth of 1 m). In the case of vertically varying permeability, a constant permeability (k = 10

^{−12}) was used for depths larger than 1 m. The second row of the horizontal axis (i.e., numbers), are the ranges of stream flow discharge along the stream for each flow regime. τ: residence time of the particles released at the streambed interface.

**Figure 7.**Cumulative distribution function of the length distribution of spatially coherent upwelling/downwelling stretches at the streambed interface under various flow discharges, assuming (

**a**) constant intrinsic permeability (k = 10

^{−9}[m

^{2}]) for the entire subsurface region, and (

**b**) a decaying intrinsic permeability.

**Table 1.**Estimated stream discharge values based on measured streamflow data and drainage area (Q), and measured mean water depth (d) for different flow discharges at 50 m spatial resolution.

Distance from Upstream Station [m] | Drainage Area [m^{2}] | Snow Melt Flow (22 May 2017) | Summer Base Flow (27 June 2017) | Low Flow (18 August 2017) | High Flow (19 August 2017) | Autumn Base Flow (30 August 2017) | |||||

Q [L/s] | d [cm] | Q [L/s] | d [cm] | Q [L/s] | d [cm] | Q [L/s] | d [cm] | Q [L/s] | d [cm] | ||

0 | 234,446 | 54.48 | 31 | 2.63 | 11 | 0.19 | 3 | 25.58 | 17 | 7.36 | 9 |

50 | 236,031 | 54.54 | 31 | 2.65 | 11 | 0.19 | 3 | 25.59 | 17 | 7.42 | 9 |

100 | 243,262 | 54.81 | 31 | 2.76 | 11 | 0.21 | 2 | 25.63 | 11 | 7.48 | 7 |

150 | 250,838 | 55.09 | 32 | 2.86 | 15 | 0.23 | 18 | 25.67 | 24 | 7.55 | 26 |

200 | 258,334 | 55.37 | 28 | 2.97 | 17 | 0.26 | 21 | 25.71 | 35 | 7.61 | 25 |

250 | 262,731 | 55.53 | 32 | 3.03 | 9 | 0.27 | 8 | 25.74 | 26 | 7.67 | 14 |

300 | 265,497 | 55.63 | 33 | 3.07 | 19 | 0.27 | 7 | 25.76 | 16 | 7.73 | 13 |

350 | 267,926 | 55.72 | 27 | 3.11 | 7 | 0.28 | 6 | 25.77 | 8 | 7.79 | 10 |

400 | 291,324 | 56.59 | 27 | 3.44 | 11 | 0.35 | 9 | 25.90 | 12 | 7.86 | 6 |

450 | 293,516 | 56.67 | 29 | 3.47 | 0.35 | 7 | 25.91 | 19 | 7.92 | 14 | |

500 | 296,214 | 56.77 | 21 | 3.51 | 0.36 | 5 | 25.93 | 16 | 7.98 | 10 | |

550 | 337,040 | 58.29 | 20 | 4.09 | 0.47 | 26 | 26.16 | 34 | 8.04 | 22 | |

600 | 343,055 | 58.51 | 42 | 4.17 | 0.49 | 15 | 26.19 | 24 | 8.11 | 16 | |

650 | 357,882 | 59.07 | 32 | 4.39 | 0.53 | 8 | 26.28 | 19 | 8.17 | 13 | |

700 | 373439 | 59.64 | 29 | 4.61 | 0.57 | 19 | 26.37 | 33 | 8.23 | 27 | |

750 | 375,921 | 59.74 | 27 | 4.64 | 0.58 | 14 | 26.38 | 20 | 8.29 | 11 | |

800 | 379,197 | 59.86 | 35 | 4.69 | 0.58 | 5 | 26.40 | 15 | 8.35 | 5 | |

850 | 396,899 | 60.52 | 22 | 4.94 | 0.63 | 5 | 26.50 | 16 | 8.42 | 17 | |

900 | 405,145 | 60.82 | 15 | 5.06 | 0.66 | 9 | 26.54 | 16 | 8.48 | 12 | |

950 | 411,322 | 61.05 | 26 | 5.14 | 0.67 | 4 | 26.58 | 19 | 8.54 | 14 | |

1000 | 413,347 | 61.13 | 29 | 5.17 | 0.68 | 24 | 26.59 | 29 | 8.60 | 29 | |

1050 | 535,591 | 65.67 | 31 | 6.91 | 1.01 | 6 | 27.28 | 17 | 8.66 | 10 | |

1100 | 558,355 | 66.51 | 21 | 7.24 | 1.07 | 4 | 27.41 | 9 | 8.73 | 8 | |

1150 | 567,113 | 66.84 | 14 | 7.36 | 1.10 | 2 | 27.46 | 9 | 8.79 | 5 | |

1200 | 568,492 | 66.89 | 28 | 7.38 | 1.10 | 2 | 27.47 | 20 | 8.85 | 4 | |

1250 | 573,211 | 67.07 | 30 | 7.45 | 1.11 | 14 | 27.49 | 23 | 8.91 | 20 | |

1300 | 578,673 | 67.27 | 35 | 7.52 | 1.13 | 19 | 27.52 | 32 | 8.98 | 26 | |

1437 | 648,703 | 69.87 | 35 | 8.52 | 1.32 | 19 | 27.92 | 32 | 9.19 | 26 |

Flow Discharge | Time Period | Mean Upstream Discharge [L/s] | Mean Downstream Discharge [L/s] |
---|---|---|---|

Base flow | 03–07August | 8.71 | 10.8 |

Low flow | 07–19 August | 0.22 | 1.56 |

High flow 1 | 19 August | 25.59 | 27.96 |

High flow 2 | 21 August | 25.57 | 30.09 |

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**MDPI and ACS Style**

Mojarrad, B.B.; Betterle, A.; Singh, T.; Olid, C.; Wörman, A.
The Effect of Stream Discharge on Hyporheic Exchange. *Water* **2019**, *11*, 1436.
https://doi.org/10.3390/w11071436

**AMA Style**

Mojarrad BB, Betterle A, Singh T, Olid C, Wörman A.
The Effect of Stream Discharge on Hyporheic Exchange. *Water*. 2019; 11(7):1436.
https://doi.org/10.3390/w11071436

**Chicago/Turabian Style**

Mojarrad, Brian Babak, Andrea Betterle, Tanu Singh, Carolina Olid, and Anders Wörman.
2019. "The Effect of Stream Discharge on Hyporheic Exchange" *Water* 11, no. 7: 1436.
https://doi.org/10.3390/w11071436