# A Simple 1D Model of Bedload Transport in Torrents: The Manival Stream Downstream its Debris Basin (S.E. France)

Coline Tissot, Guillaume Piton, Alain Recking, Vincent Mano, Samuel Bacon, Clément Misset, Damien Kuss

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Coline Tissot, Guillaume Piton, Alain Recking, Vincent Mano, Samuel Bacon, et al.. A Simple 1D Model of Bedload Transport in Torrents: The Manival Stream Downstream its Debris Basin (S.E. France). 12th International Conference on Fluvial Hydraulics, Sep 2024, Liverpool, United Kingdom. pp.207-2014, 10.1201/9781003475378 hal-05048729

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# A Simple 1D Model of Bedload Transport in Torrents: The Manival Stream Downstream its Debris Basin (S.E. France)

C. Tissot, G. Piton, A. Recking

Univ. Grenoble Alpes, INRAE, CNRS, IRD, Grenoble INP, IGE, 38000, Grenoble, France

V. Mano, S. Bacon, C. Misset

Service de Restauration des Terrains en Montagne de l'Isère, 9 quai Créqui, 38000 Grenoble, France

D. KUSS

SYMBHI, Syndicat Mixte des Bassins hydrauliques de l'Isère, 9 Rue Jean Bocq, 38000 Grenoble, France

**ABSTRACT:** EVOFOND is a one-dimensional hydro-sedimentary numerical model designed to study bedload transport in mountain rivers and torrents. This simple tool is here tested on three bedload transport experiments conducted on a small-scale model representing the Manival torrent (S. E. France). Despite simplified hydraulic calculations, EVOFOND reproduces reasonably well the trends in deposition and erosion when realistic boundary conditions are applied. The choice of a representative transported grain size for torrents subject to high transport intensities over coarse bed material is discussed.

## 1 INTRODUCTION

In mountainous regions, floods on alluvial fans are devastating because of their strong capacity to transport sediment (Badoux et al. 2014). This transport, in the form of bedload, can considerably modify the channel morphology, notably through bed aggradation, lateral shifts in the active channel, or bank erosion. As a result, the geometry of the river and the hydraulics change. The latter conditions sediment transport, which in turn affects the torrent morphology once again. There follows a feedback loop highlighting the difficulty of estimating the height of water and the river bed level reached during a debris flood (Church and Jakob 2020). Therefore, the study of a river in a mountainous context is complex, especially since field measurements are challenging to obtain due to sudden events and dangerous data acquisition. Historical, geomorphological and simple hydraulic cross-analysis is thus the first step of any study of steep slope streams. Numerical models supplement these expert's statements by helping to bound the magnitude and location of deposition and erosion during events associated with high sediment supply from the catchment (Jakob 2021).

To date, there are numerous one-dimensional models enabling the calculation of bedload transport. However, many are applicable to low-gradient rivers in plains. In torrential environments, the roughness associated with the shape of the bed, the singular topography (such as step-pool and riffle-pool patterns), and the irregular geometry is not negligible (Millar 1999, Church and Zimmermann 2007). The use of laws established in laboratory conditions becomes problematic in such a context because, by considering only grain resistance, they overestimate the transport rate by several orders of magnitude compared to measurements taken under low and moderate transport intensities (Rickenmann and Recking 2011; Nitsche et al. 2011, 2012). Despite the abundance of models for lowlands rivers, there are few available 1D models for mountain streams. The manual by Kuss (2022) lists several 1D modeling tools used in the management of torrents and mountain rivers. SEDFLOW and TOMSED can be used for bedload transport calculations over large watersheds (Heimann et al. 2015). 3ST-1D is also specified in bedload transport modelling for mountain streams, and can be used for unsteady flows (Papanicolaou et al. 2004). BASEMENT (Kammerlander et al. 2016) and RUBARBE (Paquier and Kadi 2008) were initially designed for fluvial applications and dam break studies but are applicable in a torrential context. Finally, the French torrent control service (ONF-RTM) has developed an easy-to-use tool, EVOFOND (Kuss et al. 2022), specifically for mountain streams, which is the focus point of this paper. Other tools are mentioned in the literature, including the model by Cui and Parker (2005) and the DREAM model inspired by the former, which focuses more on dam removal and is therefore limited in its application to torrential contexts.
These software programs differ on several aspects: channel geometry, hydraulic model, available bedload transport laws, model precision, and source code computational time. BASEMENT and 3ST-1D use complete shallow water equations, while SEDFLOW and TOMSED employ the kinematic wave approximation, and EVOFOND assumes uniform flow. The commonly proposed friction law is Manning-Strickler, but other options, more suited to steep slopes are available: SEDFLOW includes the Ferguson (2007) law usable with values proposed by Rickenmann and Recking (2011), which are suitable for steep slope conditions and shallow flows. TOMSED allows the use of the Smart and Jaeggi (1983) law, adapted to torrential contexts and accounting for increased flow resistance at low relative flow depths. Regarding bedload flux calculations, models assuming kinematic wave introduce a corrective term on the slope to account for energy dissipation related to form roughness. This corrective parameter is treated as an input parameter in the model. In the article by Chiari et al. (2010), a sensitivity analysis is conducted on this parameter within the context of the extreme flood in August 2005 in the Chiene watershed, in Switzerland. Several friction laws are available on SEDFLOW including a similar method to account for form roughness, the reduced slope is based on the flow-resistance partitioning approach of Rickenmann and Recking (2011) and Nitsche et al. (2011). For other models solving the complete shallow water equations, the slope of the energy line is used to compute solid transport. Additionally, several bedload transport laws used in these models are adapted for mountain rivers, including the formulas of Rickenmann (1990, 1991), as well as the formulas of Recking et al. (2016) or Piton and Recking (2017). While we still have scarce evidences of the accuracy of these models, which, despite the efforts of their developers, still have a number of empirical parameters that one must calibrate, they can support the expertise of practitioners who massively use them. They provide access to the distribution of the deposition and erosion after a flood with reasonable calculation times. This can help in later stage to map overflowing points, help the quantification of damages and to design protection structures.

The objective of this paper is to use the simplest model among those previously presented, EVOFOND, and to illustrate its application through a recent lab case study. The Manival stream, located in the Chartreuse massif near Grenoble, France, was studied on a small-scale model where multiple tests were conducted to understand its hydro-sedimentary behavior (Alliau et al. 2023). In this work, we try to reproduce numerically the results of the small-scale model with the EVOFOND software.

## 2 MATERIAL AND METHOD

### 2.1 EVOFOND Simulation Software

EVOFOND is a simplistic model that relies on two strong assumptions — uniform flow and critical Froude regime — with the aim of optimizing computation times and model stability while capturing the main trends regarding bed level changes. The tool is built mainly for steep slopes where these assumptions are reasonable first estimates (Piton and Recking 2019). It offers a wide range of bedload transport laws, all applicable to steep slope conditions (Meunier 1991; Smart and Jaeggi 1983; Rickenmann 1990; Couvert et al. 1991; Recking et al. 2016; Lefort 2018). The transversal sections can be rectangular or trapezoidal, and a minimal longitudinal profile is necessary to define the non-erodible bed level.

In the remainder of the article, all bedload transport calculations are performed using the Recking et al. (2016) threshold-less formula:

$$ \frac{Q_s}{W \sqrt{g(s-1)d^3}} = \phi = 14 \tau^{*2.5} / \left[ 1 + \left( \frac{\tau_m^*}{\tau^*} \right)^4 \right] \quad \text{with} \quad \tau_m^* = 1.5 S^{0.75} \quad (1) $$

$$ \text{And } \tau^* = S / \left[ 1.65d \left( \frac{2}{W} + 74p^{2.6} (gS)^p \left( \frac{Q}{W} \right)^{-2p} D^{3p-1} \right) \right] \quad \text{with} \quad \begin{cases} p = 0.23 \text{ if } \frac{Q}{W \sqrt{gSd^3}} < 100 \\ \text{and } p = 0.3 \text{ otherwise} \end{cases} $$

With $Q_s$ the volumetric bedload transport [m³/s], $W$ the channel width [m], $g$ the gravitational acceleration [m/s²], $d$ the diameter of the grain [m], $s$ the sediment density [-], $\phi$ the dimensionless
bedload transport, $\tau_m^*$, the mobility threshold Shields number [-] that marks the transition from partial mobility to full mobility, $\tau^*$ the Shields number [-] and S the slope [m/m].

## 2.2 Case study

### 2.2.1 Manival

The Manival stream is a trained torrent located on the eastern slope of the Chartreuse Massif. Its watershed opens up in the form of a cirque and peaks at Bec Charvet (1738m). The watershed covers an area of nearly 7.3 km². Since 1990, it has featured a large debris basin that protects downstream residences during torrential floods (Carladous et al. 2022). This basin had a high trapping efficiency leading to issues downstream, including deep bed incision. This incision threatens the banks with erosion and possible collapse. In this context, in 2017, modifications were made to the basin shape to restore sediment continuity during routine events while trapping large magnitude events. Designing such mitigation structure with a “on/off” functioning is challenging. It was thus decided to study in detail the effect of restoring the downstream material transit, to adapt the basin management if necessary. Indeed, the materials sent downstream could be stored in the streambed which could lead to malfunctions at some sensitive points (bridges, ford crossings, or bank levees) or to bed elevations increasing the probability of overflow.

### 2.2.2 Small-scale model

To address these issues, a 1:25 small-scale model was created to quantify the deposition mechanisms and hydraulic parameters at the debris basin and along the downstream section. Due to lack of room, the model does not faithfully represent the full length of the Manival. Scale limitations forced it to represent only a few key sections of the stream, from the debris basin to the bridge of the Road Number 1090 with an equivalent length of 350 m (all parameters are given at prototype scale within the paper), a slope down to the last check dam of about 12% and downstream of the later of about 5%. Bedload supply comes from the upstream of the model through a hopper equipped with an automated conveyor belt. The quantity of fed materials is determined by weighing. A real-time material weighing system measures the outgoing solid discharge at the flume outlet. LiDAR surveys are conducted before and after the tests to calculate a volumetric differential per zone (Figure 1). Additionally, multiple cameras are placed on both sides of the model.

<figure><img src="image_2.png" /><figcaption>Figure 1: Topographic screens before and after scenario 1.</figcaption></figure>
## 2.3 Scenarios and test cases

The experiments conducted on the small-scale model and simulated in the numerical model are based on three scenarios: a 30-years return period flood (Scenario 1), a 10-years return period flood (Scenario 2), and a 100-years return period flood (Scenario 3). Data for these three tests are summarized in Table 1. The incoming and outgoing solid volumes are determined by weighing, and the volume stored in the sediment deposition area is calculated using the topographic differentials before and after the test. Regarding the spatial limits of the numerical model, it does not include the debris basin, which is more akin to a 2D problem. Downstream of the open check dam thus represents the upstream condition for the EVOFOND model, and the bridge serves as its downstream limit (Figure 1). Simulations are conducted with both trapezoidal and rectangular cross-sections, for the latter using the initial width, neglecting the wider section with bank slopes.

<table><caption>Table 1: Data from scenarios 1, 2, and 3 extracted from the reduced physical model (prototype scale).</caption><thead><tr><th>Parameter (units)</th><th>Scenario 1 (Q<sub>30</sub>)</th><th>Scenario 2 (Q<sub>10</sub>)</th><th>Scenario 3 (Q<sub>100</sub>)</th></tr></thead><tbody><tr><td>Volume of water (m<sup>3</sup>)</td><td>50 783</td><td>45 333</td><td>118 299</td></tr><tr><td>Peak discharge (m<sup>3</sup>/s)</td><td>7.5</td><td>5</td><td>11</td></tr><tr><td>Sediment volumetric concentration (%)</td><td>17.5</td><td>8.7</td><td>11.4</td></tr><tr><td>Flood duration (h)</td><td>8</td><td>8</td><td>8</td></tr><tr><td>Incoming solid volume (m<sup>3</sup>)</td><td>8 875</td><td>3 927</td><td>13 513</td></tr><tr><td>Volume stored inside the debris basin (m<sup>3</sup>)</td><td>6 305</td><td>764</td><td>1 863</td></tr><tr><td>V<sub>in</sub> = Volume entering the upstream part = Volume leaving the debris basin (m<sup>3</sup>)</td><td>2 570</td><td>3 163</td><td>11 650</td></tr><tr><td>V<sub>out</sub> = Outgoing Volume (m<sup>3</sup>)</td><td>443</td><td>234</td><td>1 511</td></tr></tbody></table>

For all three tests on the small-scale model, the poorly sorted grain size distribution is kept constant. Its median diameter $d_{50}$ is 34 mm at prototype scale, while its 84% quantile, $d_{84}$ is 57 mm. For the numerical model, a sediment diameter value must be chosen. In gentle slope river and for partial transport conditions, Recking et al. (2016) demonstrated that using $d_{84}$ is the best option. However, for very high transport condition, it remains unclear which quantile to choose because the hiding and armoring of the bed that make the coarse fraction so important loose its effect as smoothing of the bed by smaller fractions appear under these conditions (Recking et al. 2022).

We chose the representative grain size $d$ by an inverse method in the following: The incoming volume must be equal to that supplied in the small-scale model, for example, 3163 m$^3$ for Scenario 1. Two diameter values are associated with two slopes measured inside the debris basin, which are representative of the upstream input (between 11% and 14%). It gave reasonable estimates of the measured volume at inlet (Table 2), namely $d = 17$ mm and $d = 19$ mm. These diameters used in the numerical model, correspond to a quantile closer to $d_{30}$ than to the usually used $d_{50}$ or $d_{84}$. This can be justified by the conditions of full mobility and intense bedload transport as observed by Recking et al. (2022) who shows that the transport peak is reproduced with the fine fraction.

<table><caption>Table 2: Volumes at the inlet and outlet of the studied reach (small-scale model and numerical model).</caption><thead><tr><th>Scenario</th><th>Model</th><th>Method</th><th>V<sub>in</sub> (m<sup>3</sup>)</th><th>V<sub>out</sub> (m<sup>3</sup>)</th></tr></thead><tbody><tr><td rowspan="3">Scenario 1 (Q30)</td><td>Small scale</td><td>Mesured</td><td>2570</td><td>443</td></tr><tr><td>EVOFOND</td><td>d = 0.017 m</td><td>2555</td><td>632</td></tr><tr><td>EVOFOND</td><td>d = 0.019 m</td><td>2566</td><td>549</td></tr><tr><td rowspan="3">Scenario 2 (Q10)</td><td>Small scale</td><td>Mesured</td><td>3163</td><td>234</td></tr><tr><td>EVOFOND</td><td>d = 0.017 m</td><td>2918</td><td>360</td></tr><tr><td>EVOFOND</td><td>d = 0.019 m</td><td>2941</td><td>303</td></tr><tr><td rowspan="3">Scenario 3 (Q100)</td><td>Small scale</td><td>Mesured</td><td>11650</td><td>1511</td></tr><tr><td>EVOFOND</td><td>d = 0.017 m</td><td>10965</td><td>3224</td></tr><tr><td>EVOFOND</td><td>d = 0.019 m</td><td>10700</td><td>2722</td></tr></tbody></table>
To the best of our knowledge, almost no bedload transport data are available in the literature at such high slopes (5-12%) and with flow conditions leading to high transport stages $\tau^*/\tau_c^* > 4$, i.e. above the threshold of catastrophic debris flood (sensu. Church and Jakob 2020), where $\tau_c^*$ is the critical Shields parameter for incipient motion of sediment. As a consequence, the choice of a representative grain size to compute bedload transport remains an open question so far in these conditions. Without data, one would usually apply the sediment transport equation he best trust. Here, we could deduce the representative grain size by inversing the transport equation we believe is best suited for our case.

## 3 PERFORMANCE OF THE NUMERICAL MODEL

The bedload volume measured or computed to be exported by the system is provided in Table 2. It can be seen that the numerical model overestimates slightly bedload mobility since the exported sediment volume is systematically slightly higher in the simulations as compared with the experiments. The spatial distributions of the deposited sediment are shown through longitudinal bed profiles for the three scenarios in Figure 2.

<figure><img src="image_3.png" /><figcaption>Figure 2: Final numerical results for scenarios 1, 2 and 3 with trapezoidal sections (a.) with rectangular sections (b.). Red line: final averaged channel bed, black line: initial averaged bed channel, grey line: scouring limit, blue line: final bed level from EVOFOND simulation with $d = 0.017\text{m}$, dark blue dashed line: final bed level from EVOFOND simulation with $d = 0.019\text{m}$</figcaption></figure>
The results from the numerical simulations are compared to the cross-wise average level measured on the small-scale model. Indeed, as the 1D model does not account for cross-sectional variability of the bed and assume rectangular or trapezoidal profiles, its computed bed level should be compared to the mean bed level in the section. Cross-wise (mean-min) bed fluctuations at the end of the experimental runs reached average amplitudes of 0.21 m for Scenario 1, 0.25 m for Scenario 2, and 0.5 m for Scenario 3. Such random fluctuations that could be related to grain size sorting, local turbulence or many uncontrolled and local effects are not computed in the numerical model. They contribute to uncertainties in the numerical model and give a sense of the accuracy that one can target in such complicated conditions. Overall, we believe that simple 1D numerical models have bed level uncertainties in their results that are at least of 0.5 m.

The general distribution of the computed sediment deposits is fairly in accordance with the small-scale measurements, slightly less for Scenario 1 and very convincing for Scenario 2 and 3. In all three tests, the computed profiles do not match the must upstream section of the model (> x=300-350 m). This is explained, in particular, by the fact that this zone is strongly influenced by the input boundary condition. Indeed, the upstream section of the model corresponds to the reach downstream of the open check dam. This structure, due to the presence of several openings, disrupts the hydraulics, introducing tri-dimensional flows and a complexity that is not accounted for in the one-dimensional EVOFOND code. For scenarios 2 and 3, the difference between the simulated deposit levels and the small scale model are <0.5 m for scenario 2 and < 0.4 m for scenario 3. A higher uncertainty is observed in the immediate vicinity of each dropping structure. Again, the simplistic hydraulics of EVOFOND do not allow for the modeling of hydraulic jumps that emerge at each weir chute.

For scenario 1, the deviation between the simulation and the experiments is more pronounced, with an average error of 0.8 m. Within the Q30 test, a volume of 8875 m3 was supplied at the inlet of the small-scale model, and only 2570 m3 passed the open check dam and entered the studied reach. More than 70% of the sediments was stored in the debris basin, strongly influencing the deposition dynamics in the downstream part. According to photos taken during the test (cf. Figure 3), a massive trapping of sediments occurs in the storage area. This could lead to the invalidity of the hypothesized continuous sediment supply and thus an unrealistic boundary condition. This is verified by comparing bed level manual measurements taken during the test, just downstream of the open check dam. Figure 4 shows that the recorded deposition height varies periodically during the test, indicating the existence of sediment pulses, while it evolves gradually in the EVOFOND model, suggesting that the timing of sediment input may slightly differs between both numerical and small-scale models. Finally, rectangular and trapezoidal versions were tested for each scenario. Figure 2.b) is an example for scenario 3 with rectangular sections. The rectangular version achieved a 40% higher error especially in the downstream part of the model. However, the issue of the influence of boundary conditions on the upstream part of the model is not corrected.

<figure><img src="image_4.png" /><figcaption>Figure 3: Deposition in the debris basin during the scenario 1: a) initial state, b) final state (© CNR)</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3: Variation in bed level immediately downstream of the open check dam for scenario 1.</figcaption></figure>
# 4 DISCUSSION

The modeling by EVOFOND of the tests conducted on the small-scale model raises once again the question of choosing a characteristic grain size for high bedload transport conditions. According to Recking (2010), when transitioning from flume tests to field conditions, especially with poorly sorted grain size distributions, the choice of the characteristic size of the transported grain becomes uncertain. The entire bed is set in motion, and larger grain sizes no longer dominate the overall behavior of the bed as they do in less intense stages (MacKenzie et al. 2018). At high intensity, the only known data to date are from Smart & Jaëggi (1983) and Rickenmann (1990), but they used uniform grain size distributions so this question did not emerge. Although debatable, the data presented in this article are among the very few bedload transport measurements with poorly sorted grain size distributions and high bedload transport intensities. Regarding the choice of the type of section for numerical modeling, the best option seems to be trapezoidal for the studied case. For longitudinal profiles where the features are pronounced, they will have an influence on the deposition height.

# 5 CONCLUSION

EVOFOND is a simplistic numerical tool that allows for quick calculations and intuitive use for engineers and practitioners studying one-dimensional bedload transport in mountains streams. Despite a simplified hydraulic description, the model reproduces the trends of deposition and erosion zones reasonably well, when realistic boundary conditions are applied. For scenarios 2 and 3, the order of magnitude of uncertainties is lower than 0.5 m, i.e. comparable than the measured transversal bed fluctuations. However, as soon as the assumption of a sediment concentration constant over time is no longer valid (as is the case for scenario 1); the sediment supply for the numerical model differs from that in the small-scale model, likely explaining the significant errors. One should also always keep in mind that local hydraulics and morphodynamics are not explicitly modelled with such simplified one-dimensional tools, in particular at chutes associated with high turbulence and hydraulic jumps. Additionally, the choice of representative granulometry for high-intensity and poorly sorted granulometry transport warrants further research.

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