# Modelling of Groundwater–Surface Water Interaction Applying the Hyporheic Flux Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Study Area

^{2}(7067 km

^{2}) (53°13′02″ N 22°25′52″ E) and a length of 164 km [27]. The study area is located in the upper part of the Biebrza river Catchment.

^{3}s

^{–1}and 4.61 m

^{3}s

^{–1}) [30]. It is quite probable that the peatland flow does not have direct contact with mineral/alluvial aquifer. In hydrogeological terms, the riverbed is muddy and has an uncompacted bottom. In addition, there is the presence of Reed peat on the banks along the whole 6 km stretch of the meandering river [31]. It is interesting to notice that during the last glaciation, “the Valley served as an ice-marginal valley to glacial waters, whereas after climate warming, the peat formation processes took place there” [28]. The study area is dominated principally by arable land and meadow and in minor quantity covered by forest. Peat is the dominant type of bog. The Upper Biebrza is constructed with peat deposits with a thickness of 3–6 m. The basin is distinguished by extensive sand dunes surrounded by peat bogs, which formed as a result of Aeolian processes. Large swampy areas are cover by birches, spruce, and a large proportion of boreal species. The study area has small urban areas represented basically in small villages and small cottages and communities dedicated to agriculture.

^{−1,}with an average of 0.97 mm day

^{−1}. Between 6 September 2018 and 25 June 2019, there are 293 days, of which 106 days had precipitation [32].

#### 2.2. AHF Model

^{−1}). ${D}_{{a}_{i,j}}$is the thickness of the aquifer beneath river sediments cell location (i,j) (L), ${W}_{r}$ is the half-width of the river (L), and ${W}_{rs}$ is the half-width of river sediments (L). Applying the assumptions of Nawalany et al. [25] in this domain, it is possible to establish the relationships described in Equations (1) and (2), where ${\mu}_{i}=\frac{\left(i+1\right)\pi}{{D}_{a}}\mathrm{with}i=1,2,\dots ,N$.

^{−1}). This relationship allows estimating an expression for ${\lambda}_{k}$.

#### 2.3. Implementation of Biebrza River Groundwater Model Using River Package (RIV)

#### 2.4. Implementation of AHF Model Using Seepage Package in Biebrza River Model

^{3}T

^{−1}) where ${I}_{i,j}$ is the rate flux applicable to the map area $DEL{R}_{j}\ast DEL{C}_{i}$ of the cell (LT

^{−1}). Then, the AHF model fluxes will be implemented through the ${I}_{i,j}$ as a seepage. It is shown in Equation (19).

## 3. Results

^{3}day

^{−1}.

^{3}day

^{−1}, while for the minimum river flux where the river is recharging, the aquifer the magnitude is even higher, reaching 10.5 m

^{3}day

^{−1}.

^{3}day

^{−1}per cell. Therefore, even when on average, the MODFLOW and the MODFLOW+AHF model seems to be very similar, this analysis demonstrates some critical differences for some stress periods simulation.

^{3}day

^{−1}of variance (Figure 12).

## 4. Discussion and Recommendation

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Flow domain in the aquifer in riparian valley (

**a**) approach suggested by Nawalany et al. [25] for the river interaction, (

**b**) Modular Three-Dimensional Finite-Difference Groundwater Flow (MODFLOW) scheme of a cell.

**Figure 4.**The sensitivity analysis of the model. Comparison between bottom flow using the MODFLOW approach and the AHF model applying different values of d_s for one modelling cell.

**Figure 6.**Diagram of the piezometers that showed improvement and worsening applying the AHF model (EPSG = 2180).

**Figure 7.**Maximum differences between head at the river cell calculated by MODFLOW versus MODFLOW + AHF, (

**a**) Along the river. (

**b**) Scatter graph of river cell head.

**Figure 9.**Variation of the water balance of river leakage for the bottom and banks of the AHF model.

**Figure 10.**Comparison analysis of maximum, minimum, and average river/aquifer flux in the interaction between river and aquifer for the MODFLOW model and the MODFLOW + AHF model.

**Figure 11.**Range of variability of the river/aquifer flux for each river cell between the MODFLOW model and the MODFLOW+AHF model during the whole simulation.

**Figure 12.**Periods with a maximum difference of river/aquifer flux between the MODFLOW and the MODFLOW+AHF model.

**Table 1.**Variables used for the analytical hyporheic flux equation (AHF) model to study the sensitivity of the model.

${k}_{a}$ | m/day | 10.0 |

${D}_{a}$ | m | 20 |

${k}_{s}$ | m/day | 0.86 |

${w}_{r}$ | m | 4 |

$b$ | m | 12 |

${\mathsf{\Phi}}^{*}-{H}_{r}$ | m | 1.5 |

$lengthriver$ | m | 15 |

**Table 2.**Average balance for MODFLOW and MODFLOW + AHF model between 06/09/2018 and 25/06/2019 in m

^{3}/day.

Modflow | Modflow + AHF | ||||
---|---|---|---|---|---|

IN: | OUT: | IN: | OUT: | ||

STORAGE | 1821.5 | 2811.4 | STORAGE | 1818.4 | 2814.5 |

Leakage bottom | 19 | 1564.3 | |||

Leakage banks | 3.2 | 264.6 | |||

RIVER LEAKAGE | 21.5 | 1834.4 | RIVER LEAKAGE | 22.1 | 1828.9 |

RECHARGE | 3216.3 | 0 | RECHARGE | 3216.3 | 0 |

ET | 0 | 413.6 | ET | 0 | 413.6 |

TOTAL | 5059.3 | 5059.4 | TOTAL | 4946.8 | 4946.8 |

PERCENT DISCREPANCY | 0.00% | PERCENT DISCREPANCY | 0.00% |

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**MDPI and ACS Style**

Diaz, M.; Sinicyn, G.; Grodzka-Łukaszewska, M.
Modelling of Groundwater–Surface Water Interaction Applying the Hyporheic Flux Model. *Water* **2020**, *12*, 3303.
https://doi.org/10.3390/w12123303

**AMA Style**

Diaz M, Sinicyn G, Grodzka-Łukaszewska M.
Modelling of Groundwater–Surface Water Interaction Applying the Hyporheic Flux Model. *Water*. 2020; 12(12):3303.
https://doi.org/10.3390/w12123303

**Chicago/Turabian Style**

Diaz, Manuel, Grzegorz Sinicyn, and Maria Grodzka-Łukaszewska.
2020. "Modelling of Groundwater–Surface Water Interaction Applying the Hyporheic Flux Model" *Water* 12, no. 12: 3303.
https://doi.org/10.3390/w12123303