# Integral Flow Modelling Approach for Surface Water-Groundwater Interactions along a Rippled Streambed

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Geometry and Mesh

#### 2.2. Numerical Model

^{f}are averaged only over the pore space volume. The conservation of mass and momentum are defined after Oxtoby et al. [34] as:

^{3}); t is time (s); p is pressure (Pa); μ is the dynamic viscosity (Ns/m

^{2}), g is the gravitational acceleration (m/s

^{2}) and D an additional drag term (kg/(m

^{2}s

^{2})). The drag term was developed by Ergun [37] and accounts for momentum loss by means of fluid friction with the porous medium and flow recirculation within the sediment. To consider flow recirculation, an effective added mass coefficient is included after van Gent [38]. The porous drag term is defined as:

_{p}(m) as effective grain size diameter.

#### 2.3. Turbulence

#### 2.4. Boundary and Initial Conditions

^{3}/s for case 1–5 and to 0.25 m

^{3}/s for case 6, whereas for the air phase a total pressure condition is defined with a total pressure of 0 Pa. The total pressure condition specifies the given total pressure for outflow and the dynamic pressure subtracted from the total pressure for inflow. Next to the air phase at the inlet, the total pressure definition is applied for the upper boundary and at the outlet. The streambed is surrounded by walls. Consequently, the velocity is set to 0 m/s with a no flow condition.

^{−2}m/s and medium gravel of about 10

^{−1}m/s. These values are only estimations and are not considered in our simulations. An initial water level of 1 m is set from the inlet up to the weir structure (see Figure 1).

#### 2.5. Validation

## 3. Results and Discussion

#### 3.1. Reference Case

^{3}/s, the ripple length to 20 cm and the height to 5.6 cm. Figure 6 shows the pressure distribution and velocity vectors at the investigated ripple (see Figure 1) for case 1 with a sandy and a gravel sediment. The solver solves the pressure term p_rgh as the static pressure minus the hydrostatic pressure (ρgz with z as coordinate vector). The highest pressure is observed at the last third of the upstream face of the ripple. Low pressure is present at the ripple crest and the first two-thirds of the upstream face as well as downstream the crest. As these pressure differences lead to hyporheic exchange, flow occurs in downstream and upstream directions from high to low pressure. The described flow paths fit well to the results by Fox et al. [49], where the exchange of water between surface and subsurface was illustrated based on tracer experiments in the laboratory at a rippled sandy streambed. Also Thibodeaux and Boyle [50], Elliott and Brooks [14] and Janssen et al. [51] came to similar results from laboratory experiments with triangular bedforms. Fehlman [52] and Shen et al. [53] presented non-hydrostatic pressure distributions at triangular bed forms which were also similar to our results with pressure peaks at the middle of the stoss face, pressure minimum at the crest with low pressure remaining at the lee face until the pressure increases again at the stoss face of the following ripple. The description of the principal pressure pattern at the observed ripple in our simulations is valid for the sand as well as for the gravel, though the pressure values differ. Due to the higher resistance of the sand compared to gravel, higher pressure gradients are observed. Conversely, it behaves in terms of subsurface velocities: higher velocities are determined in the gravel sediment compared to the less permeable sand.

^{−2}m

^{3}/s/m

^{2}than for the sand with 2.7 × 10

^{−3}m

^{3}/s/m

^{2}. This fits to the flume experiment results by Tonina and Buffington [17], where larger hyporheic exchange was claimed for gravel compared to sandy sediments.

#### 3.2. Ripple Dimension

^{−3}m

^{3}/s/m

^{2}and 1.81 × 10

^{−2}m

^{3}/s/m

^{2}than for the reference case with 2.7 × 10

^{−3}m

^{3}/s/m

^{2}and 1.84 × 10

^{−2}m

^{3}/s/m

^{2}.

^{−3}m

^{3}/s/m

^{2}) compared to the reference case (case 1) with sand (2.7 × 10

^{−3}m

^{3}/s/m

^{2}). For the gravel the opposite is true (case 3). 1.7 × 10

^{−2}m

^{3}/s/m

^{2}and case 1: 1.8 × 10

^{−2}m

^{3}/s/m

^{2}). The extremely high turbulence between the ripples for the sand could be an explanation for that. The results for cases 2 and 3 with gravel and sand show, that a general statement about the influence of the ripple size is not possible, as there is a complex relation between the size and the material leading to different turbulence and pressure distributions, where also a threshold can be conceivable. Tonina and Buffington [16] presented results from a laboratory experiment with a pool-riffle channel and came to the same conclusion that hyporheic exchange does not necessarily decrease with lower bed form amplitudes. Closer investigations with more simulations including additional ripple size variations would be necessary for a more profound interpretation.

#### 3.3. Ripple Length

^{−3}m

^{3}/s/m

^{2}and 1.7 × 10

^{−2}m

^{3}/s/m

^{2}, case 1: 2.7 × 10

^{−3}m

^{3}/s/m

^{2}and 1.8 × 10

^{−2}m

^{3}/s/m

^{2}). The pressure distribution is very similar to the reference case (case 1). The decisive difference is probably again the turbulence, which is higher for large height-to-length-ratios as already described by Broecker et al. [31].

#### 3.4. Ripple Distance

^{−3}m

^{3}/s/m

^{2}and 2.7 × 10

^{−2}m

^{3}/s/m

^{2}, case 1: 2.7 × 10

^{−3}m

^{3}/s/m

^{2}and 1.8 × 10

^{−2}m

^{3}/s/m

^{2}). Broecker et al. [31] already presumed higher hyporheic exchange for this case compared to the reference case, based on the higher pressure gradients. To our knowledge, distances between the ripples were never investigated so far for hyporheic zone processes, apart from Broecker et al. [31] where only a surface water model was used.

#### 3.5. Flow Rate

^{3}/s (for case 1–5 the discharge was 0.5 m

^{3}/s). The ripple geometry is the same as for the reference case (case 1). Comparing the reference case with case 6, it is obvious that both flow discharges show qualitatively similar flow fields. The flow velocities within the ripples decrease due to lower surface water velocities. Nevertheless, there is still a layer with Reynolds numbers higher than 10, which is slightly smaller than for the reference case (see Figure 13). The hyporheic fluxes are decreased compared to the reference case (case 6: 2.9 × 10

^{−4}m

^{3}/s/m

^{2}and 1.9 × 10

^{−3}m

^{3}/s/m

^{2}, case 1: 2.7 × 10

^{−3}m

^{3}/s/m

^{2}and 1.8 × 10

^{−2}m

^{3}/s/m

^{2}). This fits with laboratory observations e.g., by Marion et al. [54] and Elliott and Brooks [14].

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Model geometry and initial condition for the water level (sediment: yellow, water: blue, air: gray); top: front view, bottom right: cross-section.

**Figure 3.**Seepage calculated with 1D and 2D analytical and numerical solutions for a rectangular dam.

**Figure 6.**Pressure distribution and velocity vectors at a sandy (

**left**) and gravel (

**right**) ripple for case 1 (Table 1). The white line indicates the sediment-water interface. The colors indicate the pressure distribution. Please note that the scaling is different in the right and the left panel. The arrows indicate flow directions of the surface and the subsurface flow. To visualize the intensity of the flow U a grey is used.

**Figure 8.**Pressure distribution and velocity vectors at a sandy (left) and gravel (right) ripple for case 2 (Table 1). The white line indicates the sediment-water interface. The colors indicate the pressure distribution. Please note that the scaling is different in the right and the left panel. The arrows indicate flow directions of the surface and the subsurface flow. To visualize the intensity of the flow U a grey is used.

**Figure 9.**Pressure distribution and velocity vectors at a sandy (top) and gravel (bottom) ripple for case 3 (Table 1). The white line indicates the sediment-water interface. The colors indicate the pressure distribution. Please note that the scaling is different in the right and the left panel. The arrows indicate flow directions of the surface and the subsurface flow. To visualize the intensity of the flow U a grey is used.

**Figure 10.**Pressure distribution and velocity vectors at a sandy (

**left**) and gravel (

**right**) ripple for case 4 (Table 1). The white line indicates the sediment-water interface. The colors indicate the pressure distribution. Please note that the scaling is different in the right and the left panel. The arrows indicate flow directions of the surface and the subsurface flow. To visualize the intensity of the flow U a grey is used.

**Figure 11.**Pressure distribution and velocity vectors at a sandy (

**left**) and gravel (

**right**) ripple for case 5 (Table 1). The white line indicates the sediment-water interface. The colors indicate the pressure distribution. Please note that the scaling is different in the right and the left panel. The arrows indicate flow directions of the surface and the subsurface flow. To visualize the intensity of the flow U a grey is used.

Case | 1 (Reference Case) | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

ripple height (cm) | 5.6 | 1.4 | 11.2 | 5.6 | 5.6 | 5.6 |

ripple length (cm) | 20 | 5 | 40 | 40 | 20 | 20 |

ripple distance (cm) | 0 | 0 | 0 | 0 | 20 | 0 |

flow rate (m^{3}/s) | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.25 |

**Table 2.**Hyporheic fluxes of a single ripple in the center of a series of ripples for case 1–6 (sand). Right and left indicate the part of the ripple right and left of the ripple crest (compare Figure 4).

Case | Inflow Left (m ^{3}/s) | Inflow Right (m^{3}/s) | Inflow Sum (m^{3}/s) | Outflow Left (m^{3}/s) | Outflow Right (m^{3}/s) | Outflow Sum (m^{3}/s) | Total Flux ^{1}(m ^{3}/s/m^{2}) |
---|---|---|---|---|---|---|---|

1 | 2.9 × 10^{−4} | 3.8 × 10^{−5} | 3.3 × 10^{−4} | 2.1 × 10^{−4} | 1.2 × 10^{−4} | 3.3 × 10^{−4} | 2.7 × 10^{−3} |

2 | 1.4 × 10^{−4} | 6.3 × 10^{−6} | 1.4 × 10^{−4} | 3.7 × 10^{−5} | 1.3 × 10^{−4} | 1.6 × 10^{−4} | 5.1 × 10^{−3} |

3 | 6.6 × 10^{−4} | 5.3 × 10^{−5} | 7.2 × 10^{−4} | 4.9 × 10^{−4} | 2.4 × 10^{−4} | 7.3 × 10^{−4} | 3.0 × 10^{−3} |

4 | 4.0 × 10^{−4} | 6.1 × 10^{−5} | 4.6 × 10^{−4} | 3.3 × 10^{−4} | 1.6 × 10^{−4} | 4.9 × 10^{−4} | 2.2 × 10^{−3} |

5 | 4.2 × 10^{−4} | 6.0 × 10^{−5} | 4.8 × 10^{−4} | 1.7 × 10^{−4} | 2.9 × 10^{−4} | 4.6 × 10^{−4} | 3.9 × 10^{−3} |

6 | 1.2 × 10^{−4} | 2.0 × 10^{−5} | 1.4 × 10^{−4} | 9.6 × 10^{−5} | 4.8 × 10^{−5} | 1.4 × 10^{−4} | 2.9 × 10^{−4} |

^{1}Total flux = (mag (inflow left) + mag (inflow right) + mag (outflow left) + mag (outflow right))/area.

**Table 3.**Hyporheic fluxes of a single ripple in the center of a series of ripples for case 1–6 (gravel). Right and left indicate the part of the ripple right and left of the ripple crest (compare Figure 4).

Case | Inflow Left (m ^{3}/s) | Inflow Right (m^{3}/s) | Inflow Sum (m^{3}/s) | Outflow Left (m^{3}/s) | Outflow Right (m^{3}/s) | Outflow Sum (m^{3}/s) | Total Flux ^{1}(m ^{3}/s/m^{2}) |
---|---|---|---|---|---|---|---|

1 | 2.2 × 10^{−3} | 2.5 × 10^{−5} | 2.2 × 10^{−3} | 1.0 × 10^{−3} | 1.2 × 10^{−3} | 2.2 × 10^{−3} | 1.8 × 10^{−2} |

2 | 5.6 × 10^{−4} | 2.9 × 10^{−5} | 5.9 × 10^{−4} | 1.6 × 10^{−4} | 3.7 × 10^{−4} | 5.2 × 10^{−4} | 1.8 × 10^{−2} |

3 | 4.5 × 10^{−3} | 3.8 × 10^{−5} | 4.6 × 10^{−3} | 1.5 × 10^{−3} | 2.1 × 10^{−3} | 3.6 × 10^{−3} | 1.7 × 10^{−2} |

4 | 3.5 × 10^{−3} | 0 | 3.5 × 10^{−3} | 2.0 × 10^{−3} | 1.9 × 10^{−3} | 3.9 × 10^{−3} | 1.7 × 10^{−2} |

5 | 3.6 × 10^{−3} | 0 | 3.6 × 10^{−3} | 8.4 × 10^{−4} | 2.2 × 10^{−3} | 3.1 × 10^{−3} | 2.7 × 10^{−2} |

6 | 9.3 × 10^{−4} | 1.4 × 10^{−5} | 9.4 × 10^{−4} | 4.6 × 10^{−4} | 5.2 × 10^{−4} | 9.8 × 10^{−4} | 1.9 × 10^{−3} |

^{1}Total flux = (mag (inflow left) + mag (inflow right) + mag (outflow left) + mag (outflow right))/area.

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**MDPI and ACS Style**

Broecker, T.; Teuber, K.; Sobhi Gollo, V.; Nützmann, G.; Lewandowski, J.; Hinkelmann, R.
Integral Flow Modelling Approach for Surface Water-Groundwater Interactions along a Rippled Streambed. *Water* **2019**, *11*, 1517.
https://doi.org/10.3390/w11071517

**AMA Style**

Broecker T, Teuber K, Sobhi Gollo V, Nützmann G, Lewandowski J, Hinkelmann R.
Integral Flow Modelling Approach for Surface Water-Groundwater Interactions along a Rippled Streambed. *Water*. 2019; 11(7):1517.
https://doi.org/10.3390/w11071517

**Chicago/Turabian Style**

Broecker, Tabea, Katharina Teuber, Vahid Sobhi Gollo, Gunnar Nützmann, Jörg Lewandowski, and Reinhard Hinkelmann.
2019. "Integral Flow Modelling Approach for Surface Water-Groundwater Interactions along a Rippled Streambed" *Water* 11, no. 7: 1517.
https://doi.org/10.3390/w11071517