A Novel Biomorphodynamic Model to Enhance Bedload Transport Modelling in Emergent and Submerged Rigid Vegetation

Abstract

Riparian and floodplain vegetation play a key role in controlling flow resistance, sediment transport, and channel morphology, shaping the dynamics of riverine ecosystems. Accurately representing these vegetation–flow–sediment interactions in numerical models is essential for predicting system responses and supporting sustainable river management. This study introduces an enhanced biomorphodynamic model in the open-source framework openTELEMAC, which combines multiple vegetation friction approaches with a method for predicting sediment transport in vegetated flows. The modular structure of the framework enables flexible configurations for different vegetation types (rigid or flexible) and flow conditions (emergent or submerged) by selecting suitable vegetation friction approaches, improving usability and extending model applicability. Model performance is evaluated using two laboratory experiments on bedload transport through emergent and submerged rigid vegetation. Simulations reproduce the measured bed and water surface profiles with high accuracy, yielding low goodness-of-fit errors (RMSE ≈ 0.5–1.8 cm, MAE ≈ 0.5–1.6 cm). The results highlight the sensitivity of predictions to vegetational input parameters such as the drag coefficient. Overall, the enhanced biomorphodynamic model advances the representation of vegetation–sediment interactions and provides an adaptable, open-source tool for eco-hydraulic and morphodynamic research.

1. Introduction

The presence of riparian and floodplain vegetation in river reaches significantly alters the flow and sediment transport dynamics in alluvial channels, effectively serving as ecosystem engineers [1,2,3]. Riparian vegetation affects sediment transport and flow in complex ways, increasing hydraulic roughness and consequently influencing water depth, flow velocity [4,5,6] as well as the mean bed shear stress [6,7,8]. The accurate representation of hydraulic roughness is a key aspect in hydraulic and morphodynamic modelling, as it directly affects flow resistance and energy dissipation. Previous studies have demonstrated the broader implications of roughness and vegetation characteristics for floodplain hydraulics and flood wave behaviour, e.g., [9,10,11], emphasizing the importance of accounting for vegetation-induced flow resistance in model applications. Regarding morphodynamic patterns, the density of plant stems significantly impacts erosion and deposition patterns at the patch scale [12], with sparse vegetation patches causing turbulence and material resuspension [13,14], and denser patches resulting in reduced flow velocities and material deposition [13,15,16,17]. At the reach scale, vegetation plays a critical role in reducing local bed erosion [1] and shaping channel patterns [18,19,20,21]. These effects of riparian and floodplain vegetation on flow and sediment dynamics need to be assessed to inform river management practices.

Accurately representing and quantifying flow-sediment-vegetation interactions remains an ongoing research challenge, particularly in determining reliable sediment transport rates in vegetated flows [22]. A key parameter in sediment transport modelling is the shear stress due to grain roughness (so-called skin friction), which determines the initiation and rate of sediment movement [23,24,25]. However, in vegetated flows, the skin friction is more complex to obtain, since the total bed shear stress ${\tau }_t$ acting on the bed is composed of the components skin friction ${\tau }_b$, the form drag ${\tau }_{bf}$ caused by bedforms such as dunes and ripples, and the vegetation-induced drag ${\tau }_v$ [26,27,28].

Recent research has focused on partitioning this total bed shear stress into its individual components to better understand and model sediment transport in vegetated environments [26,29]. Numerous studies have shown that vegetation reduces skin friction, which is decisive for sediment transport, with denser vegetation producing a stronger effect [26,27,29]. Thereby, laboratory experiments play a critical role in advancing this understanding by revealing the underlying physical mechanisms and supporting the development of empirical and theoretical approaches to estimate bedload transport in vegetated flows. Many of these methods propose removing the contribution of vegetation-induced shear stress from the total flow resistance in order to isolate the bed shear stress component that directly governs sediment transport [8,22,26,27,29,30,31,32,33].

In addition to experimental research, numerical simulations provide a powerful and efficient tool for investigating hydrodynamic and morphodynamic processes. Two-dimensional (2D) numerical modelling enables the simulation of complex flow conditions in a cost- and time-effective manner. In most modelling approaches, sediment transport is predicted based on the total bed shear stress, which is assumed to be the primary driver of bedload transport [34,35,36]. Consequently, many studies have focused on accurately defining bed shear stress in the presence of vegetation, particularly by partitioning the total bed shear stress into components representing skin friction, form drag, and vegetation-induced drag [8,29]. This partitioning is crucial to prevent overestimating sediment transport, particularly when vegetation-induced resistance is substantial [22]. Therefore, several modelling efforts have aimed to incorporate these effects into hydromorphodynamic simulations. Guan and Liang [37] developed a 2D model that distinguishes between flexible and rigid vegetation by representing the former with increased roughness coefficients and the latter with an explicit drag force. Their approach also includes a shear stress partitioning scheme to isolate the vegetation drag from the stress acting directly on the bed. Caponi and Siviglia [20] presented a model that accounts for both above- and belowground vegetation feedbacks in gravel-bed rivers. Building on Bertoldi et al. [38], they incorporated biomass density into the formulation of the Strickler roughness coefficient and adjusted the critical Shields parameter to more accurately capture the influence of vegetation on sediment entrainment [39]. Within the openTELEMAC framework, a numerical software for solving the shallow water equation, Li et al. [40] and Li et al. [41] implemented a biomorphodynamic modelling approach, hereafter referred to as the Li-BMDM. The Li-BMDM considers the hydraulic resistance of vegetation as a drag force in the hydrodynamic module TELEMAC-2D [35] and incorporates its effect on sediment transport by modifying the apparent Shields parameter ${\theta }_v$ and the dimensionless sediment transport rate ${\Phi }_v$ in the morphodynamic module GAIA [25]. This adjustment is based on the approach proposed by Armanini and Cavedon [42] for emergent vegetation and extended by Bonilla-Porras et al. [43] to account for submerged conditions, hereafter referred to as the Bonilla-Porras method. More information about the Li-BMDM is provided in Appendix A.1 and the original publications [40,41,43,44].

Despite the advantages of adjusting sediment transport modelling in vegetated flows, the Li-BMDM offers a limited approach to representing the hydraulic resistance of vegetation in hydrodynamic simulations, focusing solely on drag force modelling for rigid cylinders and utilizing only one method to adjust bedload transport in vegetated flows, specifically, the Bonilla-Porras method. Moreover, the Li-BMDM lacks flexibility and user-friendliness, particularly when representing the hydraulic resistance of flexible or foliated vegetation, which can significantly influence sediment transport [45]. In contrast, Folke et al. [46], Folke [47], Dallmeier et al. [48] and Farina et al. [49] have integrated a total of ten vegetation friction approaches into the openTELEMAC system. These are experimentally derived methods to quantify the influence of plant characteristics and hydraulic conditions on the hydraulic resistance of vegetation, thereby enabling the determination of total hydraulic roughness [50,51,52,53,54]. Within this framework, the total hydraulic roughness is calculated using the linear superposition principle proposed by Einstein and Banks [55], where the vegetation-induced friction coefficient ${\lambda }^{″}$ is added to the bottom friction coefficient $\lambda$’ to yield the total Darcy–Weisbach friction coefficient $\lambda$ [47]. Applying these vegetation friction approaches, the total hydraulic roughness is dynamically computed at each time step by incorporating various plant-specific parameters and current hydraulic conditions. They differentiate between rigid vegetation, e.g., [28,51,53,56], and flexible, foliated vegetation [50,57,58,59] under emergent and submerged flow conditions. Several studies [46,47,48,49,60] have demonstrated that vegetation friction approaches are effective for calculating the hydraulic resistance of a broad range of plant morphologies and hydraulic conditions. Besides that, the method of vegetation friction approaches offers the benefits of user-friendliness, as input parameters can be easily defined in an ASCII file, and developer-friendliness, as the setup of each vegetation friction approach can be easily adapted or extended.

Therefore, this study aims to advance methodologies for modelling bedload transport in vegetated flows by developing a comprehensive and user-friendly framework within the openTELEMAC modelling system. The specific objectives are to (1) introduce an enhanced biomorphodynamic model in openTELEMAC by integrating vegetation friction approaches in hydromorphodynamic modelling, (2) provide a modular structure for the biomorphodynamic model that improves user accessibility, broadens its range of applications, and serves as a foundation for future developments, and (3) test the model’s applicability by evaluating the performance of different vegetation friction approaches within the enhanced framework and comparing them to the drag force approach under emergent and submerged flow conditions. Overall, the enhanced framework lays the groundwork for future development of biomorphodynamic modelling in openTELEMAC, enabling more accurate predictions of sediment transport processes when vegetation is present and supporting sediment management strategies in real-world vegetated flow scenarios.

2. Materials and Methods

2.1. Development of the Novel Biomorphodynamic Model

The enhanced biomorphodynamic modelling framework was developed to improve the representation of vegetation–flow–sediment interactions in openTELEMAC. Similar to the Li-BMDM, it accounts for vegetation effects through two coupled processes:

• vegetation-induced hydraulic resistance, represented using vegetation friction approaches, and
• adjustment of sediment transport using suitable methods.

This new framework, hereafter referred to as flexBMDM, features a modular structure that enables flexible combinations of hydraulic resistance and sediment transport formulations under vegetated flow conditions.

Within the modular architecture of flexBMDM, users can select among various vegetation friction approaches, allowing a more flexible and physically consistent representation of vegetation effects across a wide range of hydraulic and ecological settings. This adaptability enables explicit consideration of plant traits—such as stem flexibility, foliage, and submergence level—thus extending the model’s applicability to both natural and restored river systems. An overview of the framework is presented in Figure 1 and described in detail below.

The key advantage of flexBMDM lies in this modular and extensible design. Multiple vegetation friction approaches can be variably combined with different morphodynamic formulations for sediment transport in vegetated flows. Although the current implementation employs a single sediment-adjustment method (the Bonilla-Porras method), the framework provides a robust foundation for future developments. To the authors’ knowledge, no previous study has systematically combined multiple vegetation friction approaches with sediment-adjustment methods. The flexBMDM therefore establishes an adaptable platform for eco-morphodynamic modelling and paves the way toward a best-practice framework for simulating vegetation–flow–sediment interactions.

The flexBMDM framework operates by first computing the hydrodynamics using TELEMAC-2D. As a prerequisite, plant-specific parameters must be defined according to the selected vegetation friction approach. For instance, the approach by Baptist et al. [51] is suitable for rigid, tree-like vegetation, while the hybrid approach proposed by Folke et al. [59] and Box et al. [58] performs well for flexible, leafy vegetation, capturing plant-induced hydraulic resistance across a wide range of flow conditions [47,48]. Based on the defined vegetation parameters and hydraulic conditions, the vegetation friction approach computes the vegetation-induced Darcy–Weisbach friction coefficient $\lambda ″$. TELEMAC-2D then determines the total Darcy–Weisbach friction coefficient $\lambda$ as the superposition of the bottom friction $\lambda \prime$, representing grain and form roughness, and the vegetation-induced friction $\lambda ″$ [61]. This total friction coefficient $\lambda$ is subsequently used to determine hydraulic parameters such as flow velocity components $u$ and $v$ and water depth $h$. This procedure is performed at each time step, thereby updating the total friction coefficient throughout the simulation.

The hydrodynamic parameters computed in TELEMAC-2D are passed to the morphodynamic module GAIA to compute bed shear stress, which governs sediment transport. Within the flexBMDM, the vegetation friction coefficient $\lambda ″$ and the vegetation-induced drag force ${\tau }_v$ are dynamically updated at each time step. To prevent double-counting of vegetation effects, only the skin-friction component ${\tau }_b$, representing grain roughness, is used for morphodynamic calculations. The total bed shear stress ${\tau }_t$, a sum of skin friction ${\tau }_b$ and vegetation drag ${\tau }_v$, is then applied in the Bonilla-Porras method, in which the Shields parameter ${\theta }_v$ is modified to account for vegetation-induced changes in sediment mobility. For a detailed description of the Bonilla-Porras and its implementation in openTELEMAC, we refer to Bonilla-Porras et al. [43], Li et al. [40] and Li et al. [41].

2.2. Validation of the flexBMDM

2.2.1. Overview

In this study, the flexBMDM framework is evaluated using two numerical test cases based on laboratory experiments, one focusing on bedload transport through emergent rigid vegetation and the other on submerged rigid vegetation.

Table 1. Specifications of the two numerical test cases with the laboratory experiments they are based on, the flow conditions that are captured, the applied frameworks and a short description of the aim of each test case.

Laboratory Experiment	Flow Conditions	Applied Frameworks	Aim
Armanini and Cavedon [42]	emergent	Li-BMDM flexBMDM-WULI flexBMDM-BAPT	Validation of the flexBMDM against the Li-BMDM
Bonilla-Porras et al. [43]	submerged	flexBMDM-WULI flexBMDM-BAPT	Demonstrating the influence of different vegetation friction approaches

provides an overview of these test cases, including the corresponding experimental datasets, the flow conditions, the modelling approaches applied, and the specific objectives of each case. The numerical test cases and their aim are described in detail in the following sections.

One key advantage of the flexBMDM framework is its ability to incorporate multiple vegetation friction approaches, allowing sediment transport in vegetated flows to be calculated in accordance with the morphological characteristics of the vegetation. The framework’s flexibility enables a more accurate representation of vegetation-induced hydraulic resistance. In this study, two vegetation friction approaches are applied within flexBMDM, both representing vegetation as rigid cylinders. The first approach, proposed by Baptist et al. [51] (hereafter referred to as BAPT), has demonstrated broad applicability and reliable performance across various flow and vegetation conditions [47,48]. The second approach (hereafter referred to as WULI) is newly implemented in openTELEMAC and the flexBMDM and is based on the drag force concept employed in the Li-BMDM. Originally developed by Wu et al. [62], this method combines a simplified drag force approach with the approach of Stone and Shen [28] to estimate flow velocity under emergent and submerged conditions [40,62]. A detailed derivation of the WULI vegetation friction approach is provided in Appendix A.2.

Since the drag force approach used in Li-BMDM and the WULI vegetation friction approach implemented in the flexBMDM framework are based on the same physical concept but differ in their numerical implementation, both frameworks are applied to the emergent test case to enable a consistent comparison. This setup serves to validate the flexBMDM framework and to confirm a correct implementation. Hence, the results of Li-BMDM and flexBMDM-WULI are expected to be identical, thereby providing numerical validation of the flexBMDM framework.

Table 2. Total Darcy-Weisbach friction coefficients of the used vegetation friction approaches and their labels used according to TELEMAC-2D for emergent and submerged flow conditions. $\lambda \prime$ denotes the bottom friction coefficient, while the rest of the equations describe the hydraulic friction due to vegetation $\lambda ″$. The parameters are defined as follows: $C_D$ is the drag coefficient [-], $m$ is the number of stems per unit area [1/m²], $D$ is the stem diameter [m], $h_v$ is the plant height [m], $h$ is the water depth [m], $\kappa$ is the von-Kármán-constant [-], $u_v$ is the flow velocity acting on the vegetation [m/s], $u_m$ the depth averaged flow velocity [m/s].

Reference	Label	Emergent	Submerged
Baptist et al. [51]	BAPT	${\lambda }^{\prime }+4C_{D}mDh$	${\displaystyle \frac{4}{{\left[\frac{1}{\sqrt{\frac{{\lambda }^{\prime }}{4}+C_{D}mD{h}_v}}+\frac{1}{\sqrt{2}\kappa }ln\frac{h}{h_v}\right]}^2}}$
Wu et al. [62], Li et al. [40]	WULI	${\lambda }^{\prime }+4C_{D}mDh$	${\lambda }^{\prime }+4C_{D}mD{h}_v{\displaystyle \frac{\left|u_v\right|u_v}{u_m^2}}$ With $u_v={\displaystyle \frac{1-D\sqrt{m}}{1-\frac{h_v}{h}D\sqrt{m}}}{u}_m{\left[{\displaystyle \frac{h_v}{h}}\right]}^{{\displaystyle \frac{1}{2}}}$

shows the Darcy–Weisbach friction coefficient of the two vegetation friction approaches BAPT and WULI, applied in this study, for emergent and submerged conditions. Both approaches yield the same formulation of the Darcy–Weisbach friction coefficient under emergent flow conditions, since they share a common physical basis that represents vegetation as rigid cylinders. Consequently, flexBMDM-WULI and flexBMDM-BAPT are expected to produce identical results under emergent conditions. This consistency aligns with other established approaches, such as those of Luhar and Nepf [52], van Velzen et al. [63] and Huthoff et al. [56].

Under submerged flow conditions, however, the physical formulations of the BAPT and WULI approaches differ, leading to variations in simulated water depth, flow velocity, and bed morphology. Therefore, the submerged test case is used to demonstrate the influence of the BAPT and WULI vegetation friction approaches on hydromorphodynamic behaviour.

2.2.2. Emergent Rigid Vegetation

To validate the functionality and performance of the flexBMDM against the Li-BMDM and, hence, validate the use of vegetation friction approaches in biomorphodynamic modelling, the test case adopted by Li et al. [40] for validating the Li-BMDM is applied in this study. This test case represents a numerical model of the laboratory experiments conducted by Armanini and Cavedon [42], which investigated bedload transport through emergent rigid vegetation. The experimental setup consisted of a straight, rectangular flume measuring 15 m in length and 0.5 m in width. Aluminum cylinders with a diameter of 1 cm were used to represent the vegetation. For the numerical test cases, the laboratory runs were used where the flume was divided into four zones with varying plant densities (see Figure 2 and

Table 3. Scenarios for validating the flexBMDM. The numerical simulations are based on the laboratory tests of Armanini and Cavedon [42]. The values can be found in Armanini and Cavedon [42]. The parameters are defined as follows: $m$ is the number of stems per unit area [1/m²], ${\Omega }_v$ is the areal density of vegetation [-], $Q_w$ is the water discharge [m³/s], $Q_s$ is the bedload transport rate [kg/s], $h$ the water depth [m], and $i$ the average bed surface slope at steady-state conditions [-].

Scenario	Zone	$\mathit{m}$ [1/m2]	${\mathit{\Omega }}_{\mathit{v}}$ [-]	${\mathit{Q}}_{\mathit{w}}$ [m3/s]	${\mathit{Q}}_{\mathit{s}}$ [kg/s]	$\mathit{h}$ [m]	$\mathit{i}$ [-]
1.R20	A	200	0.0157	0.020	3.757 × 10−3	0.155	0.0080
	B	100	0.0073			0.136	0.0044
	C	50	0.0039			0.127	0.0037
	D	0	0			0.102	0.0014
2.R9	A	200	0.0157	0.009	1.213 × 10−4	0.115	0.0049
	B	100	0.0073			0.097	0.0026
	C	50	0.0039			0.093	0.0018
	D	0	0			0.080	0.0008
3.R10	A	200	0.0157	0.010	5.078 × 10−4	0.111	0.0056
	B	100	0.0073			0.099	0.0028
	C	50	0.0039			0.098	0.0023
	D	0	0			0.076	0.0010
NP1-NV	A	200	0.0157	0.010	9.457 × 10−5	0.112	0.0050
	B	100	0.0073			0.096	0.0026
	C	50	0.0039			0.088	0.0020
	D	0	0			0.078	0.0011
NP2-NV	A	200	0.0157	0.020	3.368 × 10−3	0.132	0.0101
	B	100	0.0073			0.110	0.0056
	C	50	0.0039			0.100	0.0037
	D	0	0			0.092	0.0019

). Fine, uniform sand with a density of ${\rho }_s=2591 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ and a median diameter $d_{50}=0.5 \mathrm{m}\mathrm{m}$ was used in the experiments, which were simulated numerically. All experiments were conducted until equilibrium under steady-state conditions were achieved. A detailed description of the laboratory experiments is provided by Armanini and Cavedon [42], while the numerical model setup and calibration procedure are described by Li et al. [40] and Li [64].

Since the laboratory experiments focused on bedload transport in emergent rigid vegetation, the flexBMDM framework was applied in this test case using only vegetation friction approaches suitable for rigid vegetation, as outlined in Table 1. The WULI approach was selected to demonstrate the performance of the flexBMDM in comparison to the Li-BMDM. As all vegetation friction approaches for rigid vegetation implemented in TELEMAC-2D perform similarly under emergent flow conditions [65], the results obtained with the BAPT approach are expected to be equivalent to those using the WULI implementation within the flexBMDM framework. For each of the five laboratory configurations (see Table 3), three distinct simulations were carried out, resulting in a total of 15 simulations for this test case.

In all simulations, the Van Rijn equation [66] is used to compute sediment transport. Additionally, the skin friction correction method is activated to ensure that the influence of vegetation-induced hydraulic resistance is excluded from the calculation of bed shear stress and instead considered only in the adjustment of the Shields parameter. For the inflow boundary condition, a constant discharge and sediment transport rate are applied based on the values provided in Table 3 for each experimental setup. At the outlet, a constant water depth is specified for the hydrodynamic boundary, while free sediment outflow is prescribed for the morphodynamic boundary. The mesh is defined with an element edge length of approximately 5 cm, and a time step of 0.05 s is used to maintain numerical stability. A uniform mesh resolution of 5 cm was selected as a compromise between numerical accuracy and computational efficiency. Previous experience in modelling straight flume experiments has shown that this resolution is sufficient to capture the relevant morphodynamic trends, as uniform flow conditions are established without vortex formation or structural changes in the bed geometry. Simulations are run until steady-state conditions are reached, which may require up to 90 days of simulated time. Steady-state conditions are defined here, when the bedload transport rate that exits the domain is equal to the amount entering the domain and therefore no further bed level changes occur.

To assess the performance of the numerical simulations, we analyzed the following results. Bottom and free surface elevations were obtained under steady-state conditions along a longitudinal section at the centre of the channel. This evaluation enabled validation of each simulation’s results against those of the Li-BMDM. Furthermore, we analyzed the water depth and free surface slope in each section of the flume (see Figure 2) and compared the results of all simulations to measured values. To assess the model performance, the goodness-of-fit metrics Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Bias of the simulated and measured parameters were calculated. This approach allows us to validate the performance of the flexBMDM against the Li-BMDM while also assessing its accuracy in replicating measured values.

2.2.3. Submerged Rigid Vegetation

When simulating bedload transport through rigid emergent vegetation, vegetation friction approaches yield similar results, as they are based on the same underlying principles [46,47]. Therefore, a second numerical test case was set up to investigate the influence of the two different vegetation friction approaches, WULI and BAPT, in the flexBMDM on bedload transport under submerged vegetation conditions, where their behaviour begins to diverge. This analysis demonstrates the broad applicability of the flexBMDM framework and highlights that certain vegetation friction approaches may provide improved results compared to the one employed in the Li-BMDM, which is equivalent to the flexBMDM-WULI setup.

For the validation of the flexBMDM in submerged flow conditions, data from laboratory experiments of Bonilla-Porras et al. [43] were utilized. They conducted sixteen experimental runs in a 15 m long and 0.5 m wide channel, including three runs without vegetation and thirteen runs with vegetation at two different densities (200 and 150 stems/m²). In these experiments, vegetation was modelled using rigid aluminum cylinders with a diameter of 1 cm. A uniform and poorly graded sediment mixture with $d_{10}$, $d_{50}$ and $d_{90}$ being 0.3, 0.41 and 0.48 mm is used in all runs. A comprehensive description of the experiments is provided by Bonilla-Porras et al. [43] and Bonilla-Porras [44].

A numerical mesh representing the laboratory experiment, which is shown in Figure 3, was generated using the Blue Kenue™ Software (Version 3.3.4) [67]. The mesh features an average element edge length of 5 cm, resulting in a total of 3666 nodes and 6710 triangular elements. The mesh element size is uniform in the whole domain, meaning there is no distinction in mesh resolution between vegetated and unvegetated areas. As described in the emergent case, a uniform mesh resolution of 5 cm was selected here as a compromise between numerical accuracy and computational efficiency, allowing for the capture of the relevant morphodynamic trends. The vegetated area, which is highlighted in yellow, begins 0.75 m downstream of the inlet boundary. Since Bonilla-Porras et al. [43] and Bonilla-Porras [44] did not specify the dimensions of the inflow section, its length was defined to be sufficient for the establishment of fully developed flow conditions. Moreover, the precise extent of the inflow region is not critical, as the focus of the study lies within the vegetated area.

Table 4. Data of the laboratory experiments conducted by Bonilla-Porras et al. [43], which are used as input parameters for numerical simulations. The parameters are defined as follows: $Q_W$ is the water discharge [m³/s], $Q_S$ is the bedload transport rate [kg/s], ${\Omega }_v$ is the areal density of plants [-], $i_{b0}$ is the initial bed slope [-], d is the sediment layer thickness [m], h is the tailwater depth [m], $C_D$ is the drag coefficient derived by Bonilla-Porras [44] [-] and $h_v$ is the plant height used for the simulations [m].

Test	${\mathit{Q}}_{\mathit{W}}$ [m3/s]	${\mathit{Q}}_{\mathit{S}}$ [kg/s]	${\mathit{\Omega }}_{\mathit{v}}$ [-]	${\mathit{i}}_{\mathit{b}0}$ [-]	$\mathit{d}$ [m]	$\mathit{h}$ [m]	${\mathit{C}}_{\mathit{D}}$ [-]	${\mathit{h}}_{\mathit{v}}$ [m]
BP01	0.0230	7.13 × 10−5	0.0157	0.0000	0.09	0.17	0.5610	0.07
BP02	0.0200	1.07 × 10−3	0.0157	0.0000	0.13	0.165	0.4964	0.03
BP03	0.0180	3.17 × 10−3	0.0157	0.0010	0.14	0.16	0.3764	0.02
BP04	0.0200	6.02 × 10−3	0.0157	0.0020	0.14	0.15	0.4057	0.02
BP05	0.0200	9.67 × 10−3	0.0157	0.0025	0.14	0.135	0.3145	0.02
BP06	0.0220	1.34 × 10−2	0.0157	0.0030	0.14	0.135	0.2324	0.02
BP07	0.0220	1.33 × 10−2	0.0157	0.0030	0.15	0.155	0.1404	0.01
BP08	0.0220	1.33 × 10−2	0.0157	0.0030	0.16	0.175	0.1584	0.001
BP09	0.0220	2.95 × 10−2	0.0157	0.0030	0.14	0.115	-0.0540	0.02
BP10	0.0210	1.45 × 10−2	0.0157	0.0045	0.16	0.16	0.2199	0.001
BP11	0.0210	1.58 × 10−2	0.0118	0.0045	0.15	0.16	0.2715	0.01
BP12	0.0210	6.55 × 10−3	0.0118	0.0045	0.15	0.17	0.4245	0.01
BP13	0.0200	1.87 × 10−2	0.0118	0.0060	0.14	0.15	0.3348	0.02
BP14	0.0180	4.81 × 10−3	0.0000	0.0020	0.23	0.25	0.0000	0.00
BP15	0.0200	9.57 × 10−3	0.0000	0.0020	0.23	0.25	0.0000	0.00
BP16	0.0220	3.07 × 10−2	0.0000	0.0020	0.23	0.23	0.0000	0.00

presents the input parameters used for the numerical simulations, which were derived from the laboratory experiments conducted by Bonilla-Porras et al. [43]. At the upstream inlet boundary, a prescribed discharge and sediment transport rate were imposed, while the downstream boundary condition was defined by a fixed water depth corresponding to the tailgate height. The initial water depth over the whole domain was also set equal to this tailgate height. The numerical meshes for each simulation were adjusted to reflect the initial bed slopes reported by Bonilla-Porras et al. [43], and the erodible layer was defined according to the sediment layer thickness of each experimental configuration. Since the sediment used in the experimental runs was assumed to be uniform and poorly graded, a single grain size class with a median diameter $d_{50}$ of 0.41 mm was applied.

Bonilla-Porras et al. [43] reported a cylinder height of 18 cm; however, visual inspection of the result plots in Bonilla-Porras [44] indicates a vegetation height of 16 cm. To ensure consistency with the experimental results, a total vegetation height of 16 cm was used. Since the flexBMDM framework requires only the vegetation height above the bed surface as input, the vegetation height was calculated as the difference between the total vegetation height and the sediment layer thickness. For example, in simulation BP01, the resulting input vegetation height $h_v$ was 7 cm.

The drag coefficient is one of many input parameters for various vegetation friction approaches, also including the approaches BAPT and WULI used here. Bonilla-Porras [44] derived the drag coefficients using a momentum balance based on experimental measurement data. However, the drag coefficients obtained for cases BP07, BP08, and BP09 were unphysically low, which led to numerical instabilities. Consequently, these test cases were excluded from the analysis.

Table 5. Input parameters for the vegetation friction approaches WULI and BAPT within the flexBMDM. The parameters are defined as follows: $C_D$ is the drag coefficient derived by Bonilla-Porras [44] [-], $D$ is the diameter [m], $m$ is the number of stems per square meter [1/m²], $mD$ is the hydrodynamic density [1/m], and $h_v$ the vegetation height [m].

WULI					BAPT
Test	${\mathit{C}}_{\mathit{D}}$ [-]	$\mathit{D}$ [m]	$\mathit{m}$ [1/m2]	${\mathit{h}}_{\mathit{v}}$ [m]	${\mathit{C}}_{\mathit{D}}$ [-]	$\mathit{m}\mathit{D}$ [1/m]	${\mathit{h}}_{\mathit{v}}$ [m]
BP01	0.5610	0.01	200	0.07	0.5610	2.0	0.07
BP02	0.4964	0.01	200	0.03	0.4964	2.0	0.03
BP03	0.3764	0.01	200	0.02	0.3764	2.0	0.02
BP04	0.4057	0.01	200	0.02	0.4057	2.0	0.02
BP05	0.3145	0.01	200	0.02	0.3145	2.0	0.02
BP06	0.2324	0.01	200	0.02	0.2324	2.0	0.02
BP10	0.2199	0.01	200	0.001	0.2199	2.0	0.001
BP11	0.2715	0.01	150	0.01	0.2715	1.5	0.01
BP12	0.4245	0.01	150	0.01	0.4245	1.5	0.01
BP13	0.3348	0.01	150	0.02	0.3348	1.5	0.02
BP01	0.5610	0.01	200	0.07	0.5610	2.0	0.07
BP02	0.4964	0.01	200	0.03	0.4964	2.0	0.03
BP03	0.3764	0.01	200	0.02	0.3764	2.0	0.02

summarizes the vegetation-specific parameters corresponding to the rigid cylinders used in the BAPT and WULI vegetation friction approaches for the successfully completed simulations. For the BAPT vegetation friction approach, the hydrodynamic density $mD$ is another input parameter. It is defined as the product of the vegetation diameter $D$ and the stem density $m$ (stems per square meter). The cases BP14, BP15, and BP16 are not included in this table, as no vegetation was present in these experiments.

The simulations were conducted using the following numerical parameter settings: a time step of 0.05 s was applied to ensure numerical stability with the given element mesh size and the resulting flow velocities (approx. 0.2–0.6 m/s); the k–ε turbulence model was selected to represent turbulence effects; sediment transport was calculated based on the van Rijn equation; the skin friction option was enabled, ensuring that bed shear stress is computed solely based on grain-scale friction; the sediment slide option was activated, with the sediment friction angle set to 32.1°; and the bedload transport rate was corrected using the formulation proposed by Koch F.G. and Flokstra C [68]. All simulations were run until steady-state conditions were reached, which is defined here, when the bedload transport rate that exits the domain is equal to the amount entering the domain and therefore no bed level changes occur anymore.

The three test cases without vegetation (BP14, BP15, and BP16) were used for hydraulic and morphodynamic calibration. The bottom friction was defined using Nikuradse’s law [69]. As a starting point for the hydrodynamic calibration, an initial estimate of the equivalent sand roughness was set to $k_s=3\times d_{50}=3\times 0.00041=0.00123 \mathrm{m}$. However, the calibrated value of $k_s$ was 0.019 m. This comparably larger roughness value is considered reasonable, as it accounts not only for skin friction but also for bedform-induced roughness. As observed in the experimental results of Bonilla-Porras, the bed surface at equilibrium exhibits pronounced ripple formations with dimensions significantly larger than the grain size. These bedforms contribute substantially to the overall hydraulic resistance through form drag, which is not captured by grain-scale roughness representations such as Nikuradse’s law alone [70].

For the morphodynamic calibration, an initial critical Shields parameter of ${\theta }_c=0.0317$ was calculated based on the median grain diameter ($d_{50}=0.41 \mathrm{m}\mathrm{m}$) and the equations according to Brownlie [71]. Following calibration, a final critical Shields parameter of ${\theta }_c=0.038$ was adopted for all three simulations. The resulting bed elevations and free surface profiles for the cases BP14, BP15, and BP16 are provided in the Appendix B.2.

The simulation results were evaluated using the same methodology as for the emergent vegetation test case. Bottom and free surface elevations were plotted along a longitudinal section located at the centreline of the flume at steady-state conditions. As the experimental data provided by Bonilla-Porras are only available for a 10 m section, the simulation results were likewise evaluated and displayed over this same extent to ensure consistency. Within this 10 m section, bed slopes and water depths obtained from the simulations were compared against the corresponding measured values. For both parameters, the goodness-of-fit metrics Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Bias were calculated to quantitatively assess model performance for both scenarios.

3. Results

3.1. Emergent Rigid Vegetation

Figure 4 presents the longitudinal profiles of free surface and bottom elevations at steady-state condition from the 1.R20 experimental run. Both flexBMDM simulations produce results identical to those of Li-BMDM, accurately capturing the bed elevation patterns across varying vegetation densities. This agreement confirms that flexBMDM reliably simulates bedload transport processes and the resulting bed morphology in vegetated flows. In this figure, the impact of varying vegetation density is also evident in the changing slopes and water depths along the longitudinal profile of the channel. Additional results from the other experimental runs are available in Appendix B.1.

The free surface slopes and water depths at steady-state conditions for each vegetated section across all simulations are compared with the results of the experimental measurements of Armanini and Cavedon [42] in Figure 5. The final free surface slope corresponds to both the bed and energy slopes; therefore, only the free surface slopes are evaluated. The simulation results show good agreement with the measurements, with values clustering closely around the line of perfect agreement. Moreover, the three configurations Li-BMDM, flexBMDM-WULI, and flexBMDM-BAPT produce identical results. This consistency is also reflected in the goodness-of-fit metrics, which are therefore reported globally for all three simulations. The low RMSE, MAE, and Bias values indicate a high level of agreement between simulated and measured data, with the slightly negative Bias suggesting a minor tendency toward underestimation. A similar pattern is observed in the comparison of final water depths for each vegetated section. The identical outcomes of Li-BMDM, flexBMDM-WULI, and flexBMDM-BAPT are evident from the overlapping results and identical goodness-of-fit criteria. However, the agreement with measured values is slightly weaker for water depths, with RMSE and MAE values indicating an average deviation of approximately 1.6 to 1.8 cm. While the scatter plots show a relatively balanced distribution of data points around the line of agreement, the negative Bias again indicates a general underestimation by the framework. Nonetheless, considering that the Li-BMDM has been previously validated against these measurements by Li et al. [40], the results can still be regarded as satisfactory.

3.2. Submerged Rigid Vegetation

Figure 6 shows the longitudinal profiles of the bed and free surface elevations for experimental run BP04. The measured values, represented as an average profile based on the data of Bonilla-Porras for the initial bed elevation as a black dotted line and for the final bed elevation as a black solid line, are shown alongside the simulation results from flexBMDM-WULI (green lines) and flexBMDM-BAPT (blue lines). Since the WULI and BAPT vegetation friction approaches compute the hydraulic resistance of vegetation differently in submerged flow conditions, the resulting free surface and bottom elevation, as well as bed slopes and water depths (see Figure 7), do not overlap. The flexBMDM-BAPT provides a closer match to the measured data than flexBMDM-WULI. Nonetheless, the results demonstrate that both vegetation friction approaches implemented in flexBMDM produce satisfactory agreement with the experimental observations. Additional results from the other experimental runs are available in Appendix B.2.

Figure 7 presents the comparison of simulated and measured bed slopes and water depths under steady-state conditions. For the bed slopes, both scenarios show good agreement with the measured data, with flexBMDM-BAPT demonstrating slightly lower deviations, as indicated by the goodness-of-fit metrics. Both approaches tend to overestimate the bed slope, resulting in a higher predicted gradient than the observed one. Similar trends are observed for the steady-state water depths. The average deviation for flexBMDM-WULI ranges from approximately 0.59 to 0.74 cm, while flexBMDM-BAPT shows slightly lower deviations of 0.50 to 0.69 cm. In this case, flexBMDM-WULI tends to underestimate the water depth, whereas flexBMDM-BAPT tends to overestimate it, suggesting more erosion and consequently higher water levels. Overall, both vegetation friction approaches implemented in flexBMDM for submerged flow conditions result in a satisfactory match between simulated and measured values. Notably, the BAPT approach provides a closer agreement with the measured data than the WULI approach.

4. Discussion

4.1. Advantages of the flexBMDM

The objectives of this study were to (i) introduce the novel biomorphodynamic model flexBMDM, which uses vegetation friction approaches for representing the hydraulic resistance of vegetation, (ii) provide a modular and user-friendly framework structure, and (iii) test the applicability of this enhanced flexBMDM under emergent and submerged conditions. The discussion below addresses these points in turn.

This study introduces the flexBMDM, an enhancement of the originally implemented Li-BMDM in openTELEMAC by Li et al. [37], to improve the simulation of sediment transport in vegetated flows in openTELEMAC. The enhancement redefines the drag force approach as a vegetation friction approach, based on the equivalence of both methods as body forces in the momentum equation. While sediment transport is still calculated using skin friction, the vegetation-induced shear stress is now computed using interchangeable vegetation friction approaches. The Li-BMDM [40,41,72] was limited to a single method for calculating hydraulic resistance (the drag force approach) and sediment transport adjustment (the Bonilla-Porras method). Given the wide range of available methods to calculate hydraulic resistance [50,51,52,53,54,56,57,58,59] and to adjust sediment transport [22,73,74,75,76] in vegetated flows, the novel flexBMDM overcomes this limitation by enabling the use of alternative vegetation friction approaches. To achieve this, the drag force approach was redefined as the WULI vegetation friction approach and integrated, along with additional vegetation friction approaches, into the flexBMDM framework. The perfectly matching results of the Li-BMDM and flexBMDM-WULI simulations confirmed the accuracy of this transformation and implementation, thereby supporting the first study objective. The novel flexBMDM not only maintains the robustness of the Li-BMDM but also enables broader application by allowing for more diverse representations of plant characteristics and hydraulic resistance mechanisms.

The integration of vegetation friction approaches into the flexBMDM was accompanied by the construction of a modular structure that improves user-friendliness. Users can define relevant plant-specific parameters in an ASCII file format, which makes the implementation more suitable for real-world study areas. From a model development perspective, the modular structure provides the benefit of easy extension and adaptation for multiple other vegetation friction approaches and sediment transport adjustment methods. These features highlight that the second study objective has been achieved.

In contrast to the validation under emergent flow conditions, the objective of applying flexBMDM to submerged conditions was to assess the influence of different vegetation friction approaches and to demonstrate the broader applicability of the flexBMDM framework. While both vegetation friction approaches, WULI and BAPT, are based on the same underlying equations under emergent conditions and therefore yield identical results (see Table 2, Figure 4 and Figure 5), their performance diverges in submerged flows (see Figure 6 and Figure 7). In such conditions, different sediment transport rates are expected, leading to distinct erosion and deposition patterns. The simulation results confirm this: the flexBMDM-WULI scenarios exhibit less erosion than the flexBMDM-BAPT scenarios, due to lower total shear stresses ${\tau }_t$. This total shear stress ${\tau }_t$, comprising skin friction ${\tau }_b$ and vegetation drag ${\tau }_v$, is used in the Bonilla-Porras method to adjust the Shields parameter ${\theta }_v$ for sediment transport in vegetated flows. The BAPT approach, applied within the flexBMDM, results in a higher adjusted Shields parameter ${\theta }_v$, indicating an increased bedload transport rate. These findings demonstrate that the choice of vegetation friction approach significantly affects the simulation outcomes, where the flexBMDM-BAPT configuration shows substantially better agreement with the measured data than the flexBMDM-WULI configuration. These results suggest that the suitability of a vegetation friction approach depends on flow conditions and vegetation characteristics. Therefore, the study successfully achieves the third objective and demonstrates the benefits of the flexBMDM.

Beyond its methodological advances, the flexBMDM framework also holds potential relevance for practical applications in river management. By allowing flexible combinations of vegetation friction approaches and sediment transport formulations, it provides a tool for assessing how different vegetation configurations influence channel stability, flow resistance, and sediment dynamics. Such analyses are crucial for understanding the impact of vegetation on riverbed evolution and for informing the design of nature-based restoration and floodplain management measures. Although the framework still requires further testing regarding the sensitivity of input parameters and validation across a broader range of hydraulic and vegetative conditions, it represents a promising step toward process-based modelling tools that can inform sustainable river management and planning.

4.2. Sensitivity of Bonilla-Porras Method

Within the Bonilla-Porras method, the vegetation drag term ${\tau }_v$ is derived from the chosen vegetation friction approach in the flexBMDM framework. In this study, two approaches suitable for rigid vegetation WULI and BAPT were tested, both requiring the specification of a drag coefficient $C_D$. Although previous studies, e.g., [5,77,78], suggest that $C_D=1.0$ is a reasonable and robust choice for rigid cylinders, this assumption led to unrealistically high vegetation drag and, consequently, excessive erosion in the simulations. This outcome contradicts the general understanding that dense vegetation reduces sediment transport, e.g., [5,26,27,29,30].

Figure 8 illustrates this issue using results from experimental run BP04 of the Bonilla-Porras dataset. Simulations were conducted using three different drag coefficients: (i) $C_D=0.4057$, which is based on depth-averaged flow velocity [44] and used for the simulations presented above, (ii) the commonly used $C_D=1.0$, and (iii) $C_D=1.2171$, calculated by Bonilla-Porras [44] from the apparent velocity in the vegetation layer using Stone and Shen [28]. The results show that higher drag coefficients lead to a significant overestimation of erosion, while lower values produce more realistic outcomes. This suggests that, in 2D models using depth-averaged velocities, drag coefficients derived from depth-averaged flow may be more appropriate. It should be noted that the lower value ($C_D\approx 0.4$) lies outside the typical range ($C_D\approx 0.9-1.5$) reported in design guidelines [47,61]. This discrepancy indicates a mismatch between parameterizations for hydrodynamics and morphodynamics, highlighting that sediment transport simulations may require different values of $C_D$ than those commonly applied in flow resistance studies.

However, assigning different $C_D$ values to hydrodynamics and morphodynamics contradicts the user-friendly philosophy of flexBMDM, which aims to minimize parameter tuning. Moreover, since $C_D$ also influences hydrodynamic calculations, affecting flow velocity, water depth, and overall system roughness, large deviations from standard values (e.g., 1.0) can introduce significant model errors. Therefore, two possible solutions are proposed:

• Implementing a correction factor for morphodynamic drag:

As proposed by Bonilla-Porras [44], a correction factor could adjust the drag used for sediment transport while retaining $C_D=1.0$ in hydrodynamics. Bonilla-Porras [44] proposed a factor that depends on Froude and Reynolds numbers, plant density, and submergence ratio. However, this formulation results in high $C_D$ values and excessive erosion, indicating that a revised correction function is needed.

2.

Incorporate alternative sediment transport adjustment methods:

The flexBMDM architecture allows for easy extension, and future improvements could include methods based on turbulent kinetic energy, e.g., [73,75], or effective shear stress approaches, e.g., [74]. These alternatives may provide more accurate representations of sediment transport in vegetated flows, without relying on direct adjustments to the drag coefficient.

This flexibility in integrating new methods was a central motivation for the modular design of the flexBMDM framework. The flexBMDM offers an adaptable and transparent modelling environment for simulating sediment transport in vegetated flows. Its modular structure enables the representation of diverse vegetation morphologies and hydraulic conditions, and facilitates future extensions, including the implementation of more advanced sediment transport adjustment methods. Nonetheless, the reliability of morphodynamic predictions remains contingent upon the careful specification of sensitive parameters, such as the drag coefficient, and the consistent treatment of shear stress formulations.

4.3. Limitations and Prospects of the flexBMDM

In hydromorphodynamic modelling, maintaining consistency in the definition and application of shear stress is essential. Within the openTELEMAC framework, the bed shear stress governing sediment transport can be calculated either from the total friction obtained in the hydrodynamic module or solely from the grain roughness component, i.e., as skin friction ${\tau }_b$. As illustrated in Figure 1, the Bonilla-Porras method relies on the total flow intensity parameter, which in turn requires the total bed shear stress ${\tau }_t$, defined as the sum of skin friction ${\tau }_b$ and vegetation-induced drag ${\tau }_v$. To correctly apply the Bonilla-Porras method, it is therefore necessary to use the skin friction component ${\tau }_b$ as the basis for the bed shear stress. Since the hydraulic resistance of vegetation is considered in the hydrodynamics as part of the total roughness, which in turn is used to obtain the total bed shear stress ${\tau }_t$, considering the total bed shear stress ${\tau }_t$ would result in double-counting vegetation drag ${\tau }_v$, leading to an overestimation of sediment transport rates and consequently excessive erosion. To obtain reliable and physically consistent results, users must ensure that the bed shear stress calculation is based on the skin friction when employing the Bonilla-Porras method. Clarifying which shear stress component should be applied is equally important in other morphodynamic formulations (e.g., slope effect, secondary currents) and should be explicitly documented in future modelling studies.

In this study, the flexBMDM framework is applied exclusively to test cases using rigid cylinders as vegetation. Consequently, only vegetation friction approaches suitable for rigid vegetation were implemented. Applying approaches developed for flexible vegetation would not be meaningful for these cases, as the corresponding vegetation parameters are difficult to transfer and would introduce additional uncertainty, thereby limiting the interpretability of the results. The testing and validation of the flexBMDM framework for flexible and leafy vegetation are therefore identified as essential steps for future research and model development.

Furthermore, a detailed sensitivity analysis of vegetation-related parameters has not yet been conducted. While the sensitivity of the drag coefficient to the applied vegetation friction approaches is discussed in the section above, the influence of other vegetation parameters remains unknown at this stage and should be systematically examined in future studies.

5. Conclusions

This study presents a novel biomorphodynamic model, flexBMDM, developed in openTELEMAC by integrating vegetation friction approaches. While the previously existing Li-BMDM represented vegetation-induced hydraulic resistance using a drag force approach, the flexBMDM adopts vegetation friction approaches that support a broader range of plant characteristics and hydraulic conditions. The results show that these approaches preserve the accuracy of hydromorphodynamic simulations while improving the model’s applicability, user accessibility, and extensibility fundamentally.

Under emergent flow conditions, flexBMDM produces results consistent with those of the Li-BMDM. However, in submerged conditions, the choice of vegetation friction approach has a significant influence on model outcomes. The study also demonstrates that the Bonilla-Porras method for sediment transport adjustment in vegetated flows is highly sensitive to the specification of the drag coefficient, a parameter that is difficult to measure directly. This sensitivity emphasizes the need to integrate alternative sediment transport formulations that are less dependent on a single, uncertain parameter.

Overall, vegetation friction approaches enhance the physical realism of hydraulic resistance and sediment transport modelling in vegetated flows without compromising usability or flexibility. The modular structure of the flexBMDM improves accessibility, supports a broader range of applications, and facilitates straightforward framework extensions for future development. Future work should focus on validating these methods across a wider range of flow conditions, extending the framework to include flexible and foliated vegetation, and integrating additional sediment transport adjustment methods tailored to vegetated environments. These extensions lay the foundation for more accurate, process-based modelling of real-world vegetated riverine systems.