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## Hydrodynamic routing modeling techniques

The Navier-Stokes equations (Navier, 1822) of fluid are a formulation of Newton’s law of motion for a continuous distribution of matter in the fluid state, they are central to applied research (Doering and Gibbon, 1995). The Navier-Stokes equations are a set of nonlinear partial differential equations that describe the flow of fluids. These equations are used tomodel the movement of air in the atmosphere, river hydraulics, ocean currents, water flow in pipes, as well as many other fluid flow phenomena.

### Saint-Venant equations

In 1843 Barré de Saint-Venant published a derivation of the incompressible Navier-Stokes equations that applies to both laminar and turbulent flows. The 1D Saint-Venant equations based on the continuity (Eq. 2.5) and momentum (Eq. 2.6) equations formalize the main concepts of river hydrodynamic modeling.

The basic derivation assumptions of the Saint Venant Equations (Abbott, 1979; Chow, 1959) are the following:

• The flow is one-dimensional, i.e. the velocity is uniform over the cross-section and the water level across the section is represented by a horizontal line.

• The streamline curvature is small and the vertical accelerations are negligible, so pressure can be considered as hydrostatic.

• The effects of boundary friction and turbulence can be accounted for through resistance laws analogous to those used for steady state flow.

• The average channel bed slope is small so that the cosine of the angle it makes with the horizontal may be replaced by unity.

#### Hydrological routing modeling techniques

Hydrological routing techniques have been and continue to be an important tool to better understand the dynamical interactions within the water cycle. It is an essential component of integrated water resources management involving environment and ecology. Regional scale hydrological modeling routing techniques have been used in a number of applications, for instance:

1- Integrated basin hydrology: Runoff Routing provides a basis for comparing and validating estimates of streamflow with observed hydrograph data. As most variables describing the state of the surface are not directly observable, river discharge is an appropriate measurement to assess model qualities (Nijssen et al., 2001).

2- Export from continental surface to the sea: The simulated fresh water flux into the oceans alters their salinity and may affect the thermohaline circulation (Wang et al., 1999; Wijffels et al., 1992).

3- Climate changes scenarios: Estimates of river discharge from climate change simulations can also be used to assess the impact of climate change on water resources and the hydrology of the major river basins (Arora et al., 2001; Milly et al., 2005; Oki and Kanae, 2006).

4- Influence of river discharge on freshwater ecosystems (Azevedo et al., 2010; Carpenter et al., 1992).

In general, hydrological routing approaches are based on the conservation of mass (Eq. 2.11) coupled with empirical relationships replacing the momentum equation.

#### Stream-aquifer connectivity and exchange directions

The streams and aquifer units are hydraulically connected (Winter, 1998). Their interactions are complex, difficult to measure, and affected by natural processes and human activities (Sophocleous, 2002; Winter, 1995). Stream-aquifer interactions occur in different landscapes and environments. Streamflow gain or loss can be persistent along a hydraulic year what means that the stream is either always gaining water from aquifer units (exfiltration process) or losing water towards aquifer units (infiltration process). In other environments, flow direction can vary along a stream, some reaches infiltrate towards aquifer units when the water table altitude in the vicinity of the reach is lower than the altitude of the in-stream water level while other reaches gain water from aquifer units when the altitude of the water table in the vicinity of the reach is higher than the altitude of the stream-water surface (Fig. 2.1).

Furthermore, flow direction can change within a timeframe as a result of individual storms, rapid snowmelt, release of water and temporary flood peaks moving downwards the stream. losing during high-stage period, (b) a gaining stream where in-stream water levels are lower than the surrounding watertable, (c) a contiguous losing stream, (d) a perched losing stream (graphics from Winter et al, 1998).

To summarize, stream-aquifer interactions connectivity can be classified on the basis of several key aspects outlined in Fig. 2.2.

**Convolutional Neural Network**

principle. Fully-connected DNN are not appropriate to process input like im-ages. Indeed, to process a three-dimensional image (width, height, color channel), the whole set of pixels must be considered, which drastically increases the num-ber of parameters to process. As we have seen, too much complexity quickly leads to overfitting. It is with the objective of directly processing images as input to a network that the CNN were developed (LeCun, Bengio, et al., 1995). They thus allow, based on the same learning process, to effectively reduce the number of parameters to be updated during training.

During image processing by a CNN, the input image can be seen as a three-dimensional function img(w, h, d) with w, h and d spatial coordinates. A pixel cor-responds to a coordinate and its intensity corresponds to the amplitude img(w, h, d). The information of an image is then processed according to a given pixel and those surrounding it.

convolution. In the field of image processing, the analysis process is based on the use of filters (also called kernels). They allow to detect the characteristic patterns of the image in a spatialized way. To do this, the filter is applied to the whole image by convolution i.e. it is slid over the whole image. As described in Figure 2.8, the convolution operation is equivalent to calculating an element-wise multiplication between each component of the filter and the input on which it overlaps. This results in a matrix called feature map that captures the applications of the filters over the entire input image i.e. it stores the filter activations. The filter is also three-dimensional. The height and width of a filter is smaller than the input image in order to be able to capture « local » information. However, the depth of the filter corresponds to the depth of the input. The stride parameter corresponds to the filter advancement step after each operation in terms of number of pixels for each dimension.

**Table of contents :**

**CHAPTER 1. INTRODUCTION **

**CHAPTER 2. INTEGRATED MODELING OF HYDROSYSTEMS: A FOCUS ON STREAM-AQUIFER INTERACTIONS**

Résumé en Français

Abstract.

2.1 Introduction

2.2 Surface routing modeling

2.2.1 Hydrodynamic routing modeling techniques

2.2.2 Hydrological routing modeling techniques

2.3 Stream-aquifer modeling

2.3.1 Stream-aquifer connectivity and exchange directions

2.3.2 Importance of coupled stream-aquifer models for interdisciplinary investigations in hydrologic sciences

2.3.3 Stream-aquifer interactions modeling: tackling the challenges

2.3.4 Scaling issues in stream-aquifer interactions modeling

2.3.5 In-stream water levels fluctuations importance to stream-aquifer interactions

2.4 Conclusions

**CHAPTER 3. PRINCIPLES AND FUNCTIONING OF THE HYDROLOGICAL PLATFORM EAUDYSSÉE AND THE HYDRAULIC MODEL HEC-RAS **

Résumé en Français

Abstract

3.1 Introduction

3.2 Principles and functioning of EauDyssée platform for hydrosystem modeling

3.2.1 Surface component mass balance

3.2.1.1 Production function

3.2.1.2 Surface runoff routing: ISO module

3.2.2 Unsaturated zone component – NONSAT

3.2.3 Saturated zone component: SAM (Simulation des Aquiferes Multicouches).

3.2.4 The regional river routing component RAPID

3.2.5 Stream-aquifer interactions

3.3 The hydraulic model HEC-RAS

3.4 Statistical criteria used to assess model performances

3.4 Conclusions

**CHAPTER 4. IMPACT OF IN-STREAM MORPHOLOGY ON SIMULATED DISCHARGE AND WATER LEVELS: SEREIN RIVER CASE STUDY **

Résumé en Français

Abstract

4.1 Introduction

4.2 Domain of application: Serein River

4.3 The construction of the Serein River hydraulic model

4.3.1 Selection of the model’s temporal and spatial computational factors

4.3.2 Manning’s roughness coefficient (n) calibration

4.4 Impact of river morphology on river stage and discharge

4.4.1 Scenario GS-I: Removing cross sections that contain two conveying arms (islands)

4.4.2 Scenario GS-II: Only three surveyed cross sections used to represent the geometry of the Serein River

4.4.3 Scenario GS-III: Uniformly generalizing one surveyed cross section along the river reach

4.4.4 Scenario GS-IV: Excluding the floodplains from the geometry representation of the Serein River

4.4.5 Scenario GS-V: Replacing each irregular surveyed section by an equivalent regular trapezoidal section

4.4.6 Scenario GS-VI: Representing the river geometry by a trapezoidal section obtained from average surveyed information of 20 surveyed cross sections

4.4.7 Scenario GS-VII: Representing the river geometry by a triangular section obtained from average surveyed information

4.4.8 Scenario GS-VIII: Representing the river geometry by a rectangular section obtained from average surveyed information

4.5 Synthesis of geometry scenarios results

4.6 Conclusions

**CHAPTER 5. AN UPSCALING METHODOLOGY FOR SIMULATING RIVER STAGES AND STREAM-AQUIFER INTERACTIONS: OISE RIVER BASIN CASE STUDY **

Résumé en Français

Abstract

5.1 Framework strategy to account for river stages fluctuations

5.1.1 Upscaling from local hydraulic modeling to regional hydrological modeling .

5.1.2 The QtoZ water level fluctuation module

5.2 Implementation of the EauDyssée stream-aquifer coupling framework strategy.

5.3 Domain of application: The Oise basin

5.4 The Oise basin initial hydro(geo)logical model: low frequency behavior Running and calibrating the initial EauDyssée version at regional scale to produce runoff and groundwater contributions to the hydraulic model

5.4.1 The Oise basin hydro(geo)logical model description

5.4.2 Surface water budget characterization

5.4.3 Hydro(geo)logical model initialization strategy

5.4.4 Recalibration of the Oise initial hydro(geo)logical model: low frequency behavior

5.5 The construction of the Oise River hydraulic model

5.5.1 Oise hydraulic model calibration of Manning’s roughness coefficient

5.5.2 Local to regional scale upscaling example

5.6 EauDyssée simulations after applying the upscaling methodology

5.6.1 Simulated discharge and river stage by the regional hydro(geo)logical model EauDyssée

5.6.2 EauDyssée hydrogeological model simulations: high frequency behavior

5.7 Impact of in-stream water level fluctuations on stream-aquifer interactions at local and regional scale

5.7.1 Stream-aquifer exchanges

5.7.1.1 Local scale analysis

5.7.1.2 Regional scale analysis

5.8 Quantification of stream-aquifer exchange

5.9 Conclusions

**CHAPTER 6. CONCLUSIONS AND FUTURE WORK ANNEXE A: SEREIN RIVER MORPHOLOGICAL DATA VS. SIMULATED RATING CURVES **

**ANNEXE B: RESUME LONG DE LA THESE EN FRANÇAIS **

**REFERENCES **