Exploring morphological equilibrium and flow resistance in gravel-bed rivers: insights from channel-scale experimental analysis

Explorando o equilíbrio morfológico e a resistência ao fluxo em rios de leito de cascalho: insights a partir da análise experimental em escala de canal

Abstract

Our aim is to recreate the distinct patterns observed in gravel-bed river channels for studies involving physical modeling by designing a morphological configuration and investigating bed resistance. We used a 10x0.56x0.4m experimental channel that scaled-down a natural river at a ratio of 1:20, utilizing sand as the bed material. Employing both field data and measuring instruments such as a laser profiler and pitot tube, we conducted a total of 12 tests spanning 27 hours. In the initial 15-hour phase, we closely monitored deformations and sediment transport under various flow rates. Subsequently, we adjusted the flow rates and proceeded with testing. The results indicated that a larger diameter of the movable bed led to increased flow resistance, evident in parameters like n, f, and C. Additionally, we observed the deposition of fines within intergrain voids and noted an increase in k_s with reduced discharge volume, illustrating the influence on flow dynamics and sediment transport. Furthermore, this preliminary study provides valuable insights into the physical processes governing water and sediment transport, which are crucial considerations for future physical simulation studies.

Keywords:
Physical modeling; Movable bed; Sediment transport; Bed roughness; Flow resistance

Resumo

Nosso objetivo é recriar os padrões distintos observados nos canais de rios de leitos de cascalho para estudos envolvendo modelagem física, projetando uma configuração morfológica e investigando a resistência do leito. Utilizamos um canal experimental de 10x0.56x0.4m que reduziu uma proporção natural do rio em 1:20, utilizando areia como material de leito. Utilizando dados de campo e instrumentos de medição como um perfilador a laser e tubo de Pitot, conduzimos um total de 12 testes ao longo de 27 horas. Na fase inicial de 15 horas, monitoramos de perto as deformações e o transporte de sedimentos sob várias taxas de fluxo. Posteriormente, ajustamos as taxas de fluxo e prosseguimos com os testes. Os resultados indicaram que um maior diâmetro do leito móvel levou a um aumento da resistência ao fluxo, evidente em parâmetros como n, f e C. Além disso, observamos a deposição de finos dentro dos vazios entre grãos e notamos um aumento de k_s com volume de descarga reduzido, ilustrando a influência na dinâmica do fluxo e no transporte de sedimentos. Além disso, este estudo preliminar fornece insights valiosos sobre os processos físicos que governam o transporte de água e sedimentos, os quais são considerações cruciais para futuros estudos de simulação física.

Palavras-chave:
Modelagem física; Leito móvel; Transporte de sedimentos; Rugosidade do leito; Resistência ao fluxo

INTRODUCTION

Definition of morphological equilibrium conformation

The bed surface of an alluvial river serves as the primary interface between flow rate and available sediment for transport. Therefore, it is a critical component of the river system, exerting influence on flow hydraulics (Hardy et al., 2010; Ferraro et al., 2019), flow resistance (Rickenmann & Recking, 2011), sediment transport rate, and grain size distribution (Wang et al., 2011). Additionally, due to the dynamic interaction among the bed, flow, and transported sediment, the channel bed is free to adjust to changes in discharge (Polvi, 2021; Rachelly et al., 2022) and sediment supply (Dietrich et al., 1989; López et al., 2023). In pursuit of greater stability, the riverbed surface freely adapts to accommodate available sediment, resulting in the formation of a distinct bed architecture (Ferguson, 2007; Nelson & Dubé, 2016).

An example of this can be seen in river restoration projects, which often involve constructing new channels or riverbeds that have not been significantly altered by water action. According to Cooper & Tait (2009), this approach may not faithfully replicate the hydraulic and ecological components of a gravel-bed river; instead, the bed should be constructed through sequential cycles of sediment transport and deposition.

Numerous studies have employed ideal gravel bed configurations to isolate the effects of different surface deposit settings (e.g., Schmeeckle et al., 2007; Ferro, 2018; Smith et al., 2023), but few have developed techniques to recreate the bed conditions found in natural gravel-bed rivers. Recent studies combine physical modeling with computational fluid dynamics (CFD) modeling, as exemplified by Kumar & Afzal (2023). These authors aimed to determine erosion at vertical wall encounters under conditions dominated by strong currents, both in combined wave and current scenarios and in current-only scenarios. Their experimental findings demonstrate that the initial phase of erosion development is faster under the combined flow of waves and currents, but the equilibrium erosion depth is greater in current-only conditions.

Bahmanpouri et al. (2021) also conducted a numerical and experimental investigation. In this case, the authors evaluated sediment transport and the morphological evolution of the erodible bed induced by dam-break flows. Experiments were conducted in a specially designed channel for the dam-break flow process. Based on an initial flat bed, the effect of three different bed compaction rates was evaluated. Experimental data suggested that an increase in the bed compaction rate resulted in a decrease in erosion and sedimentation depth, as well as the sediment transport rate. Additionally, the increase in the bed compaction rate led to an increase in the wave front celerity, reducing the erosion rate and erosion depth.

Conversely, using only physical experiments, Alomari et al. (2018) investigated the effect of a diversion channel on the erosion of the main downstream channel, while Friedl et al. (2018) studied sediment replenishment with artificial gravel deposits as a method to compensate for sediment deficits in rivers and improve their ecological conditions.

These studies share the commonality of not utilizing a preliminary phase of shaping the movable bed to represent the natural watercourse. In this context, Buffin‐Bélanger et al. (2003) and Spiller et al. (2015) devised a molding procedure that accurately replicates the complexity of gravel bed surfaces. Molds were created from a field section of a gravel bed, and replicas were produced from these molds and placed in the channel. While this method generates a surface with the geometric structure of a natural gravel bed, it's important to note that it is impermeable, the bed is static, and it doesn't reproduce subsurface vertical sorting.

In this context, for a better understanding of the water-sediment interaction, physical modeling of a movable bed necessitates a preliminary phase of experiments enabling the creation of a riverbed flume. This flume should exhibit morphological features, sediment distribution, and flow paths that closely resemble its natural conditions (Cooper & Tait, 2009).

Flow resistance in gravel-bed rivers

In mountainous rivers, when sediments are at rest or transported with low intensity, the presence of sediment particles on the riverbed surface and their impacts on discharge are considered as the bed resistance of the channel. The resistance characteristics of mountainous rivers, composed of coarse bed material, are quite distinct from those of alluvial rivers with fine bed particles. One of the significant influencing factors is roughness, which is responsible for providing flow resistance in mountainous rivers (Bathurst, 1985; Maxwell & Papanicolaou, 2001).

In Brazil, there are few studies focused on the understanding of gravel-bed rivers. However, in recent years, the characteristics of some of these rivers have been examined, particularly in the southern region of the country (e.g., Menezes & Kobiyama, 2023; Paixão et al., 2023; Paixão & Kobiyama, 2022; Bartels et al., 2021). Both studies emphasize the occurrence of different bed morphologies and the influence of macro-roughness in these Brazilian rivers.

These studies have a wide range of applications, such as in flood risk assessment. As highlighted by Vázquez-Tarrío et al. (2024), who conducted a thorough review of the scientific literature addressing the implications of sediment transport in flood risk assessment, most studies on the influence of sediment transport on flood risk have focused on mountain streams due to their high sensitivity to changes in sediment supply. Additionally, the research has concentrated on explaining historical changes in channel conveyance capacity and flood frequency based on long-term trends in sediment supply, and recent developments in hydrodynamic and morphodynamic numerical models provide opportunities to explicitly incorporate sediment transport into flood risk analysis. However, the review does not address how small-scale physical models can also be useful for this issue, when combined with numerical models and field observations.

In this context, the work by Sturm et al. (2018a, 2018b) highlights, through a reduced-scale physical model (1:30 scale), the influence of sediment transport and deposition on floods in the Austrian Alps and their impact on buildings exposed to this phenomenon. Compared to clear water, sediment transport and deposition significantly influence impact forces due to sediment accumulation near buildings in the alluvial fan (Sturm et al., 2018b).

During floods, increased flow velocity mobilizes sediments from upstream sources, transporting them downstream and leading to sediment deposition in lower-energy environments. On the other hand, floods can also erode and reconfigure existing sediment deposits, altering riverbed morphology and channel geometry. According to the work by Hamidifar et al. (2024), which studied the role of sediment transport in flood mapping, sediments transported during floods influence channel erosion and deposition patterns, affecting the magnitude and extent of flood events and subsequent sediment transport dynamics. This reciprocal relationship underscores the complex interaction between floods and sediment dynamics, highlighting the need for a comprehensive understanding (Hamidifar et al., 2024).

From this perspective, the interaction between water and sediments in channel experiments can provide detailed insights into the hydro-sedimentological processes responsible for creating a channel geometry that closely resembles a natural river. This occurs because the water in the natural watershed environment, with its distinct velocity and behavior, interacts with the channel bed, inducing flow resistance. This resistance, as measured in physical models, is associated with the size, shape, and gradation of sediment particles, as well as the geometric form and flow magnitude of rivers. To calibrate this relationship, most studies use measured data from a specific river or river section (Recking et al., 2008).

There exists a multitude of formulas due to varying choices of roughness size and data differences. For steep channels with high relative flow submergence, certain researchers (e.g., Aberle & Smart, 2003; Ferguson, 2007) have found that flow resistance correlates with relative submergence and have noted that this correlation follows a logarithmic pattern. However, for channels with lower gradient and extensive bed granulometry, as is the case in this study, this behavior may not be the same. The corresponding hydraulic roughness, as a singular characteristic roughness scale, is frequently determined based on the equivalent grain size of the riverbed.

Although, it is important to highlight that in granular beds composed of a single layer on a flat surface, each bed elevation is directly related to the grain size; therefore, bed roughness can be adequately expressed through particle size. In contrast, in beds with multiple layers, surface particles nestle into cavities formed by the underlying particles. In natural beds, with a mixture of particle sizes, these formed cavities have varying dimensions, and a specific characteristic grain diameter will be less representative of the bed surface protrusions compared to the case of a single-layer bed (Smart et al., 2002).

This becomes more evident when considering that the majority of flow resistance equations were developed for fixed-bed channels. In this regard, predicting flow resistance and velocity distribution in movable-bed channels is considerably complex. Two reasons can be cited: a) By increasing the bed shear stress beyond a critical value, bed sediments begin to move. Consequently, the shape of the alluvial bed can change. Depending on flow conditions and other influential factors due to sediment particles, this type of bed deformation increases flow resistance compared to rigid-bed channels. This makes it challenging or even impossible to predict flow resistance with a constant ratio; b) Sediments transported both as bedload and suspended load can alter flow characteristics, such as velocity distribution, particularly near the bed where sediment concentration is higher. Therefore, it is necessary to consider sediment transport, variation in bed roughness, and their interaction (Abrahams & Li, 1998).

Until now, significant efforts have been directed towards the first subject, primarily in estimating flow resistance using velocity equations based on power and logarithmic laws. Some researchers also emphasize that the bed shear coefficient, or the Chézy coefficient, should be divided into two parts: particle roughness and bed form resistance.

Furthermore, an important question arises regarding the effect of sediment movement on flow resistance, specifically whether sediment movement as bedload can increase flow resistance compared to clear water or not. This question has given rise to two general viewpoints among researchers. Gao & Abrahams (2004) observed that bedload movement can lead to reductions in near-bed velocity, mean velocity, and an increase in the roughness coefficient, depending on the thickness of the moving sediment layer.

The analysis by Di Stefano et al. (2020) revealed that, for mixtures of large-sized particles, the friction factor can be accurately estimated by disregarding the effect of average concentration of suspended sediments. However, for mixtures of small-sized particles, the friction factor decreases as the average sediment concentration increases. Gladkov et al. (2021) concluded that in simulating sediment transport and calculating channel deformations in rivers, it is convenient to employ Chézy coefficient calculation dependencies to assess the bed sediment mixture's roughness. However, studies involving other flow resistance factors such as the Manning coefficient (n) and the Darcy-Weisbach friction factor (f) have not been evaluated. An analysis encompassing a comprehensive set of variables could potentially offer a deeper comprehension of processes governing discharge and sediment transport in rivers with extensive granulometry.

In this context, the primary objective of this exploratory study is to develop a morphological configuration that replicates the typical patterns observed in gravel-bed river channels. This effort seeks to provide crucial fundamental insights for analogous investigations in Brazilian river systems. Starting with the disposition of sedimentary material on an inclined plane, we seek to establish a channel that reproduces characteristics of gravel-bed rivers by injecting flow for a specific period until we reach a practically invariable bed geometry, discerned by measuring the sediment transported. Subsequently, taking advantage of this initial condition, our study moves on to explore the bed's resistance characteristics. It is our aspiration that the initial morphological configuration of the riverbed will provide valuable insights into the processes of interaction between the flow and the bed, thus promoting the progression of physical modeling studies for gravel-bed rivers in Brazil.

MATERIAL AND METHODS

Experimental setup

The experiments were conducted in an experimental channel located in the Fluvial Pavilion of the Density Current Research Center (NECOD), at the Hydraulic Research Institute (IPH) of the Federal University of Rio Grande do Sul (UFRGS). The rectangular cross-section channel has dimensions of 10m in length, 0.56m in width, and 0.4m in height (Figure 1). The channel's lateral walls are constructed from glass, enabling the observation and recording of flow characteristics as well as hydrodynamic data. The liquid supply circuit comprises a triangular weir, a downstream reservoir, and a pump for water recirculation. The channel's inclination is adjustable via a support point where a hydraulic jack is employed.

Figure 1
Schematic representation of the experimental channel: (a) cross; (b) longitudinal view.

Adjacent to the water inlet, there is a region of transition and flow calming on the water surface. Over the subsequent 4.5 meters, a fixed bed (FB) was constructed using sand (D₅₀ = 1.0mm) to allow for flow development before entering the movable bed (MB) section. The MB region (effective testing area) spans a length of 2 meters for data collection. Subsequently, there is another FB region (1 meter), and finally, the sediment collection box for material transported from the MB (Figure 1).

Similarity analysis

Inspiration for selecting the sediment grain composition for the experiments was drawn from the Forqueta River (located in Maquiné/RS). The work by Menezes (2021) provides detailed insights into a stretch of this river, which served as a basis for the development of the physical model. Concerning the characteristics of this river, the mean D₅₀ of the riverbed is 63.8 mm, classified as very coarse gravel. It has an average width of 10 meters and exhibits three distinct morphologies (step-pool, cascade, and plane-bed).

Classic Froude similarity rules for open-channel flow of liquids were applied.

$Fr=\frac{v}{\sqrt{gh}}$ (1)

The geometric scale factor (r) was established based on the dimensions of the experimental setup to be used (width: 0.56 m)

$r=\frac{L_{model}}{L_{natural}}$ (2)

$r^{\frac{3}{2}}=\frac{Q_{model}}{Q_{natural}}$ (3)

$r^{\frac{1}{2}}=\frac{U_{model}}{U_{natural}}$ (4)

By non-dimensionalizing all involved quantities, it is possible to demonstrate that the scale factor for particle diameters can be taken as equal to the r, provided the model is undistorted and the densities of natural and model sediment are the same. Table 1 presents the scaling relationships considering the geometric scale of 1:20. As highlighted, this scale was determined due to the limitation of the experimental channel, which is only 0.56 m wide. Additionally, this scale allows both vertical and horizontal scales to be the same.

Table 1
Scale relationships between the model and the equivalent prototype.

	S	L	D 50	Qmed	U
Model 1:20	0.00937m/m	0.56m	1mm	67L/min	0.22m/s
Prototype	0.00937m/m	11.2m	20mm	2m3/s	1.0m/s

Inlet flow

The injected flow was supplied from a downstream reservoir with a capacity of 2,000 liters. The system operates through flow recirculation. To control and measure the flow rate (Q), a piezometer coupled with a triangular weir (60°) installed at the channel inlet is used to determine the water level on the weir (H). The obtained value of Q is cross-validated with the measured Q at the channel outlet, thus calibrating the weir discharge coefficient.

Subsequently, a rating curve for flow was established. The Excel Solver was employed to adjust the calculated values with the observed measurements, aiming to determine the parameters a and n of Equation 3.

$Q=a{H}^n$ (5)

where a and n are dimensionless adjustment parameters of the equation. With the calculated parameters a and n, Equation 4 is derived. Thus, based on the water level reading on the weir (H in cm), the flow rate (Q in L/min) entering the channel through the weir can be determined.

$Q=0.60067H^{\mathrm{2,5}}$ (6)

Characterization of the Movable Bed (MB)

Sand (D₅₀ = 1.0 mm) was employed as the material for the MB, as shown in Table 1. The grain size distribution curve of the utilized material can be observed in Figure 2.

Figure 2
Grain size distribution curve of the material for the channel's movable bed.

Measurement of channel bed deformations

A laser profilometer was used to determine deformations in the MB. Thus, at the end of each test, the bed's topography was surveyed, enabling a comparison of bed modifications over time. The laser distance measuring equipment (model MD250 ADV) offers a precision of 0.1 mm and a reading speed of 100 measurements per second. Due to its high accuracy, it is suitable for evaluating distances, dimensions, relative positioning, and comparing elevations. It performs dynamic, point-to-point, and continuous measurements without physical contact between the device and the target object. Further details are presented in Manica (2009).

This equipment is attached to a coordinate table, which allows highly controlled movement of the profilometer along the channel, ensuring precise scanning of the bed topography. The table's workspace is 0.5 m², hence two position sheets are used: one for the first meter of the MB and another for the subsequent meter.

In order to analyze the texture of the profile generated by the data, a factor reflecting the surface roughness will be calculated, incorporating grain size and bed shape. This factor will be termed the transverse section bed elevation Oscillation Factor (FO), inspired by the factor described in Bogatin et al. (2013). The calculation is performed by dividing the length of the laser-generated surface (L_l) in a predetermined section by the width of the channel (L).

$FO=\frac{L_l}{L}$ (7)

Collector box for transported material

At the end of the experimental channel, a collector box is situated to collect the sediments transported in each test. The material is gathered, dried in a 105 °C oven, and subjected to granulometric analysis to determine characteristic diameters (D₁₀, D₁₆, D₅₀, D₈₄, and D₉₀).

Measurement of velocities and water level

For velocity measurements, a Pitot tube coupled with an inclined piezometer (30°) was employed, as illustrated in Figure 3. Velocity was calculated using Equation 6.

$U=\sqrt{2g\Delta h\sin \alpha }$ (8)

where U is the flow velocity (m/s), g is the gravitational acceleration (9.81 m/s²), Δh is the difference between the dynamic and static heights of the piezometer, and α is the angle of the piezometer, in this case, 30°.

Figure 3
(a) Pitot tube coupled with piezometer; (b) Measurement setup; (c) Pitot tube.

Instantaneous velocities were measured at two cross-sections (sections 40 and 60 cm), and velocity profiles were captured at two other sections (sections 120 and 160 cm). Both types of measurements were taken along three verticals of each section (located at 15, 30, and 45 cm from the right channel wall) (Figure 4). Water depths across the cross-sections were also recorded during velocity measurements.

Figure 4
Movable bed channel: (a) schematic showing evaluated cross-sections and respective velocity measurement points; (b) pre-test photos (inclined and ruled plane) and post-test photos (change in texture and creation of preferential pathways).

Treatment of velocity profiles

After measuring the velocity profiles, bed roughness coefficients and hydraulic resistance parameters were estimated. The velocity profile was adjusted considering a turbulent velocity distribution profile, as proposed by Prandtl (1926), prioritizing the velocities measured near the bottom.

$\frac{u}{u*}=\frac{1}{\kappa }\ln y+c$ (9)

where u is the velocity at a certain distance from the bottom (y), u∗ is the shear velocity, κ is the Von Karman constant, and c is a constant determined when y is equal to y₀, corresponding to a specific distance from the bottom where the action of the viscous sublayer occurs. The constant y₀ can be defined, in rough beds, as k_s, which corresponds to a height where interaction between water and the channel bed takes place due to bed roughness.

$\frac{u}{u*}=\frac{1}{\kappa }\ln \frac{y}{y_0}$ (10)

Or, when rewritten following algebraic handling,

$\frac{u}{u*}=\mathrm{5,75}\log \frac{y}{k_s}+B$ (11)

where B equals 8.5 for rough beds, as per Einstein (1950).

Einstein (1950) proposed a more comprehensive expression for the velocity profile in turbulent flow, incorporating a correction factor that varies depending on the flow regime being smooth, transitional, or rough (Figure 5).

$\frac{U}{u*}=\mathrm{5,75}\log \left(\mathrm{12,27}\frac{y}{k_s}x\right)$ (12)

where x is a correction factor given by Figure 5 as a function of k_s/δ, where δ is the thickness of the laminar sublayer.

Figure 5
Correction term (x) in the formula proposed by Einstein (1950).

The measured velocity profiles were fitted to the theoretical velocity profile, enabling the determination of the shear velocity at the location (u*), the roughness coefficient k_s, through which the shear stress was estimated. Table 2 presents the equations utilized for the calculation of each parameter.

Table 2
Equations used for the calculated variables.

Variable	Equation	Equation number
u*	-
ks	-
τ	$\tau =\rho u_{*}{}^2$	(13)
δ	$\delta =11.6\frac{\nu }{u_{*}}$	(14)
ks/δ	-
h	-
U	-	(15)
Fr	$Fr=\frac{V}{\sqrt{gh}}$	(16)
Re	$Re=\frac{Uh}{\nu }$	(17)
Re*	$R{e}_{*}=\frac{u_{*}d}{\nu }$	(18)
f	$\frac{U}{u*}=\sqrt{\frac{8}{f}}$	(19)
C	$\frac{V}{u*}=\frac{C}{\sqrt{g}}$	(20)
n	$\frac{V}{u*}=\frac{R^{\frac{1}{6}}}{n\sqrt{g}}$	(21)
I	$I=\frac{\tau }{\Upsilon h}$	(22)

Experimental design

The experiments were divided into two phases:

• Phase 1: In this phase, the objective was to find a morphological bed configuration that exhibits equilibrium. This means that changes to the bed at a given flow rate do not result in variations in sediment transport. Thus, for a given liquid flow injected into the channel, a bed configuration was generated in which no variation occurred. This equilibrium condition was identified through the quantification of transported sediment. This state was considered to be similar to the behavior of a natural river, indicating that the deformable medium had acquired a configuration compatible with the flow.

During this stage, 6 tests were conducted, each divided into time periods (0, 2, 6, 8, 10, 12, and 15 hours). At the end of each test, the bed topography of the channel was surveyed, and transported sediments were collected.

• Phase 2: Evaluation of channel bed roughness. In this phase, the dynamics of water and sediment were evaluated in the presence of a specific channel bed roughness. This roughness was established after achieving phase 1 under a condition without sediment supply upstream of the section. The focus of this phase was on the flow's ability to shape (sculpt) its channel. Six tests were also conducted during this phase with decreasing flow rates, during which velocity profiles were measured as described in the velocity measurement section. At the end of each test (E1: 17h, E2: 19h, E3: 21h, E4: 23h, E5: 25h, and E6: 27 hours), the bed topography of the channel was surveyed, and transported sediments were collected. The Surfer software was used for visualization of the data obtained by the profiler.

The bed granulometry was determined after each test based on the percentage difference between the initial granulometric curve and the granulometric curve of the collected transported sediments. Thus, the relative submergence ratios were determined, and the hydraulic resistance parameters of the flow were calculated.

RESULTS AND DISCUSSION

In the first part, we will present the results obtained from the initial tests aimed at developing a channel bed that recreates the behavior and characteristics of a natural river. This bed configuration resulted from the balance between transported sediment and bed alterations.

Subsequently, we will present the analyses carried out on a previously worked bed. In this bed, the velocity profiles were examined at different Q values, all of them lower than the Q that generated the initial configuration of this channel. The focus will be on flow resistance parameters and related variables.

Evaluation of bed morphological configuration

Evaluation of flow and transported sediment

The total time for stabilization of the channel bed was 15 hours. Initially, the experiment began with a near-maximum Q of approximately 250 L/min. During this period, it was observed that the quantity of transported sediment stabilized at a constant value (~2 g/min) (Figure 6). Subsequently, the Q was increased to 318 L/min, which corresponds to a return period (TR) of 2 years in the prototype (Table 3). This Q was maintained for 5 hours until the exhaustion of sediment that was susceptible to transport in the channel. As time passed, transport reduced, indicating sediment deposition and immobilization in the channel.

Figure 6
Flow and transported sediment until stabilization of the mobile bed channel.

Table 3
Flow and transported sediment rates before bed stabilization.

Time (h)	Flow rate (L/min)	Amount of transported sediment (g)	Cumulative amount of transported sediment (g)	Bed load (g/min)
0	102.06
0.12	189.95
0.2	230.25
0.32	187.58	2256.1	2256.1	322.3
2	230.25	4245.2	6501.3	42.03
6	241.06	498.9	7000.2	2.08
8	241.06	236.8	7237	1.97
10	318.71	2214.2	9451.2	18.45
12	318.71	792.1	10243.3	6.6
15	318.71	411.7	10655	2.29

Figure 6 illustrates the inserted Q in the system and the amount of transported and collected sediments at the end of the channel. It can be observed that when the Q is increased to 318 L/min, there is an increment in the total amount of transported sediments, and thereafter, the values tend to decrease. According to Masteller & Finnegan (2017), who assessed different sediment bed accommodation timescales, at short timescales, sediment transport rates decrease as the bed stabilizes.

Regarding the characteristics of the transported material, the D₁₀, D₁₆, and D₅₀ of this material showed an increase during the initial Q (Figure 7), which coincided with the majority of the transported material (Figure 6). After 2 hours of testing, the values decreased and remained relatively constant, indicating that only a small fraction of the bed was mobilized by the flow (Kirchner et al., 1990) until the next Q increment.

Figure 7
Behavior of transported characteristic diameters throughout the morphological configuration phase.

On the other hand, with the increase in Q, the D₈₄ and D₉₀ of the transported material were larger than the other analyzed diameters, as observed in Figure 7. Subsequently, both diameters decreased until the 12-hour mark of testing. In the following 3 hours, totaling 15 hours of testing, the sediment diameter values exhibited an increase in their magnitudes, as the finer sediment available had already been transported. Paphitis & Collins (2005) and Haynes & Pender (2007) suggest that, under prior accommodation, the bed is never fully stabilized, and the grains most exposed to flow (protruding grains) are always available for transport.

Evaluation of bed modifications

In Surfer software, bed surfaces were generated and evaluated for volume changes before and after each test. The volumes were multiplied by the density of quartz (2.65) and converted into mass (grams), with this difference referred to as calculated reworked sediment. At the 8-hour and 10-hour marks, the balance of reworked sediment material was negative (Table 3). As there was no increase in Q at these times, it is assumed that there might have been a bed material reconfiguration, seeking sediment transport stabilization, since these were among the lowest values observed. However, at the 10-hour mark, there was a Q increment, suggesting sediment mobilization and deposition within 2m of the mobile bed section. This led to the establishment of preferred Q paths and accommodation of finer sediments within voids of the mobile bed, ultimately influencing increased sediment deposition in the channel.

The values of transported sediments collected at the end of each test differed significantly from the reworked sediment values calculated by Surfer, as observed in Table 4. This difference could be attributed to the fact that the material became trapped or confined within the mobile bed itself (reworked by flow within the 2m of mobile bed). This difference was more pronounced at the beginning of the test when the surface texture was finer, potentially representing the removal of fines and bed accommodation under the influence of the water column. Additionally, fines might not have been captured by the collection box and could have clouded the water in the tank. In Figure 7, the smallest values for the characteristic diameters of the transported sediment's granulometric curve were recorded after 2 hours of testing.

Table 4
Erosion vs. deposition balance through analysis of surfaces in Surfer.

Time (h)	Eroded volume (cm3)	Deposited volume (cm3)	Net volume (cm3)	Calculated reworking sediment (g)	Collected transported sediment (g)	Difference
2	14780.72	0	14780.72	39.17	6.5	32.67
6	891.01	452.98	438.04	1.16	0.5	0.66
8	128.38	992.04	-863.66*	-2.29*	0.24	-
10	142.15	2696.97	-2554.82*	-6.77*	2.21	-
12	1073.82	420.65	653.18	1.73	0.79	0.94
15	1211.27	449.27	762.01	2.02	0.41	1.61

• Note: * Negative values indicate that this quantity remained deposited on the mobile bed.

Hence, it is presumed that the bed material could be accommodating, compacting, and forming a channel more similar to a natural river, even though limited by the duration of the experiment. Previous experiments with channels have indicated that measurable compaction required several days of execution (Charru et al., 2004). For a deeper understanding of subsurface behavior, Cooper & Tait (2009) suggest quantifying the hydraulic conductivity and porosity of the studied bed.

Figure 8 illustrates the bed dynamics throughout the tests. It can be observed that a depression or pool formed at the beginning of the mobile bed (white portion of the bed) due to the interaction between the fixed bed (FB) and the start of the mobile bed (MB). However, if we observe a natural river, this behavior is typical of such rivers (also known as step-pools, as described by Montgomery & Buffington, 1997). The figure also demonstrates the occurrence of preferential flow paths, even though water occupies the entire width of the channel. Close to the 100cm mark, an iron rod is integrated into the channel structure, causing a noticeable discontinuity in the data at this specific location.

Figure 8
Changes in the channel bed (between 40 and 170 cm of the MB) during the initial bed configuration tests (greener color indicates positive elevation, while browner colour indicates negative elevation).

Subsequently, two longitudinal profiles were selected, located at 15, 30, and 45 cm from the right channel bank. Figure 9 depicts the behavior of these two longitudinal sections throughout the tests. The difference between the profiles highlights significant material removal as soon as flow begins. After this initial accommodation, there is a shift in the average elevation of the sediment bed, where subsequent bed deformations will occur. It is also noticeable that the localized erosion pool generated by the transition between roughness elements remains relatively stable throughout the tests, ceasing to be a sediment source for the channel.

Figure 9
Longitudinal profile at positions 15, 30, and 45cm (from right to left).

The evaluation of bottom diameter values showed little variation after 12 hours of testing (Figure 10). The D₁₀ and D₁₆ decreased from 0.259 to 0.255 mm and from 0.322 to 0.316 mm, respectively, suggesting the transport of this particle size class or perhaps the filling of voids in the mobile bed. The D₅₀ remained nearly constant, ranging from 1.0 to 1.18 mm. On the other hand, the D₈₄ and D₉₀ increased from 2.445 to 2.522 mm and from 2.894 to 3.0 mm, respectively. This indicates an increase in bed roughness over time.

Figure 10
Variation of bed diameters throughout the experiment.

It is interesting to note that this pattern was also observed for the transported sediments (Figure 7). In other words, the same sediment size class that increases or decreases in the channel bed is the one being transported by the channel itself. This behavior occurs due to variations in the availability of sediment size classes during the tests. These results suggest a direct link between the number of protruding grains available and the number of grains transported (Masteller & Finnegan, 2017). These grains project further from the mean bed elevation, making them available and facilitating transport (Kirchner et al., 1990). Additionally, they are exposed to higher fluid drag, resulting in increased mobility (Fenton & Abbott, 1977).

Natural rivers mostly consist of a wide range of sediment sizes, creating stable bed conditions, but this condition is not maintained all the time. This is because these rivers experience low, medium, and high flow conditions. Often, conditions are subcritical, meaning that the flow is insufficient to significantly disturb the bed, or the stresses imposed by the flow create a transient situation where the bed is adjusting to a new state as sediment supply from upstream allows (Church et al., 1998). Thus, by including a pre-channel formation phase that is morphologically analogous to a natural channel, more robust insights are provided for understanding phenomena associated with water and sediment flow in gravel-bed rivers.

Evaluation of velocity profiles and flow resistance

Flow and sediment transport behavior

The sediment transport rate throughout the tests remained very low, as depicted in Table 5. This is evident as the flow rate decreased from 317.85 L/min in Test E1 to 53.88L/min in Test E6.

Table 5
Characterization of flow rates and sediments for the conducted tests.

Tests	Time (h)	Flow rate (L/min)	Amount of transported sediment (g)	Cumulative amount of transported sediment (g)	Bed load (g/min)
E1	17	317.85	145.4	145.4	1.21
E2	19	249.94	17.5	162.9	0.15
E3	21	201.5	7.2	170.1	0.06
E4	23	146.38	34.7	204.8	0.29
E5	25	98.16	61.67	266.47	0.51
E6	27	53.88	52.79	319.25	0.44

Figure 6 illustrated the behavior over time, considering the period of stable bed formation where the Q was increased up to 317.85 L/min, and the period after bed stabilization where a successive series of flow rate reductions occurred. Sediment transport remained low despite the mobilization generated in the first 15 hours. Spatial characterization by sections will be presented in the subsequent sections.

Analysis of velocity profiles between sections

The velocity profiles obtained for the sections at 120 and 160 cm for Tests E1, E2, E3, E4, and E5 are depicted in Figure 11. For Test E6, velocity measurement was not possible due to extremely low values, rendering the measurement methodology unfeasible. It is observed that the velocity intensity decreased with decreasing flow rates, except for Test E1, which could indicate a greater influence of bed roughness during the measurement process.

Figure 11
Velocity profiles obtained for verticals (V14, V30, and V45) at sections (ST-120 and ST-160) for the conducted tests (E1, E2, E3, E4, and E5).

Figure 12 displays velocity profiles per vertical across the tests. It can be observed that during Test E1, the bed elevation values were higher, suggesting bed mobilization. However, during Test E2, these bed elevation values appear to be higher.

Figure 12
Velocity profiles for each vertical at ST-120 and ST-160 cm.

Evaluation of bed roughness

To assess the influence of bed roughness, the velocity distribution profile was adjusted, and the k_s (bed roughness coefficient) was calculated for each measured vertical. The average behavior for each section can be observed in Figure 13. Upon analyzing the figure, it is evident that as the flow rate decreases across the tests, the values of k_s increase, indicating an elevation in bed roughness. This behavior is reflected in the velocity data, where values decrease as flow rate diminishes and k_s values increase. Additionally, when considering the two sections separately, in ST-120, U values are higher compared to ST-160, while k_s values are greater in ST-160.

Figure 13
Behavior of k_s and U across the tests for the two velocity profile measurement sections.

During the tests, it was observed that this behavior may be associated with the deposition that occurred in the downstream portion of the mobile bed, closer to ST-160. In this section, finer sediments filled empty spaces, and larger sediments contributed to creating a higher level of roughness compared to the other sections. By examining the bathymetry of both sections, it can be concluded that deposition is taking place, resulting in increased roughness in this section (Figure 14).

Figure 14
Bathymetry of transverse sections (a) at 120 cm (ST-120) and (b) at 160 cm (ST-160) at the end of each test, along with the water depth measured during the test.

Figure 15 illustrates this behavior of the mobile bed during the tests. The measurements were taken at the end of each test. It is worth highlighting that from E1 to E6, a sequence of decreasing flow rates, velocities, and depths occurred, leading to the plan-view stabilization of the channel bed. This figure depicts only the section between 40 and 190 cm, where ST-120 and ST-160 are located.

Figure 15
Channel bed throughout the tests (greener color indicates positive elevation, and browner color indicates negative elevation).

In this regard, the oscillation factor (FO) of the bed elevation was also evaluated after each test, as shown in Figure 16. Directly looking at sections ST-120 and ST-160, it is notable that the oscillation value is lower for ST-160, with the exception of E4. However, as seen in Figure 15, for the last test, ks was higher in ST-120, and the oscillation factor continued to be lower for ST-160. This behavior could be related to the filling of voids by sediments of lower granulometry.

Figure 16
Oscillation factor (FO) of the bed elevation in transverse measurement sections of the channel.

Currently, there is no consensus on the ideal approach to relate bed characteristics in high-gradient or shallow flow rivers to flow resistance, primarily due to the lack of agreement on how to quantify roughness and the limited availability of combined measurements of flow velocity and bed roughness in the field. Additionally, the description of roughness often relies on a single parameter, and none of the proposed roughness measures to date can fully explain the observed variability in flow velocity between different locations.

Relationship between flow resistance for the tests

For a clearer visualization of the data, the averages for each test were extracted through the averages for each section (Figure 17). In this way, the main flow resistance parameters (n, f, and C) were observed. Except for E1, the values of n and f for the other tests increased as the Q in the channel decreased, indicating a reduction in flow energy. The average data obtained for each test can be seen in Table 6.

Figure 17
Behavior of resistance variables throughout the tests.

Table 6
Measured and calculated variables for the tests conducted in the channel.

	Variable	Tests
		E1	E2	E3	E4	E5	E6
		Average values
Observed	h (cm)	2.309	2.046	1.816	1.678	1.508	1.382
	Q (weir) (L/min)	317.85	249.94	201.5	146.38	98.16	53.88
	U obs (m/s)	0.23	0.299	0.273	0.261	0.215	0.124
	Qst (g)	145.4	17.495	7.2	34.7	61.672	52.785
	Qst (g/min)	1.212	0.146	0.06	0.289	0.514	0.44
	D10 bed	0.255	0.255	0.255	0.255	0.255	0.254
	D16 bed	0.316	0.316	0.316	0.316	0.316	0.316
	D50 bed	1.178	1.178	1.178	1.178	1.177	1.177
	D84 bed	2.523	2.523	2.523	2.523	2.523	2.524
	D90 bed	3.005	3.005	3.005	3.005	3.006	3.006
	D10 st	0.296	0.346	0.277	0.611	0.825	0.595
	D16 st	0.382	0.546	0.374	0.885	1.025	0.774
	D50 st	0.983	1.969	1.3	1.688	1.645	1.495
	D84 st	2.117	3.36	2.438	2.493	2.407	2.246
	D90 st	2.449	3.722	2.675	2.778	2.639	2.477
Derived from velocity profile adjustment	u* (m/s)	0.043	0.02	0.024	0.026	0.04	0.042
	ks	0.052	0.001	0.003	0.006	0.022	0.039
	τ (N/m2)	2.329	0.414	0.601	0.811	2.511	1.978
	δ (m)	3.70E-04	6.10E-04	5.20E-04	5.10E-04	4.40E-04	3.40E-04
	ks/δ	249.75	1.26	7.86	23.88	153.06	159.15
	U (m/s)	0.168	0.318	0.289	0.263	0.214	0.151
	Fr	0.358	0.755	0.717	0.681	0.592	0.428
	Re	3804.16	5741.25	4803.84	4007.26	2876.06	1921.05
	Re*	2897.08	14.62	91.21	276.98	1775.52	1846.15
	f	0.561	0.033	0.062	0.105	0.535	0.741
	C	15.78	52.41	40.9	35.47	25.37	14.67
	n (1/s.m1/3)	0.041	0.01	0.014	0.016	0.031	0.045
	S (m/m)	0.011	0.002	0.004	0.005	0.018	0.016

• The values correspond to the averages of the sections.

By observing some dimensionless parameters shown in Table 6, Figure 18 was constructed, where we can see that the behavior of Re and Fr shows a relationship with U/u* (R² = 0.77 and 0.89, respectively).

Figure 18
Relationship between Re and Fr versus U/u*.

Relationships between dimensionless variables q and U

Rickenmann and Recking (2011) utilized a dataset of 2890 field measurements to test the ability of various conventional flow resistance equations to predict the average flow velocity in gravel-bed rivers. The authors emphasize the use of two dimensionless parameters.

$q^{**}=\frac{q}{\sqrt{g.I.D_{84}{}^3}}$ (23)

$U^{**}=\frac{U}{\sqrt{g.I.D_{84}}}$ (24)

When plotting the obtained results on the diagram developed using these dimensionless parameters by Rickenmann & Recking (2011), the points from our study fall within the cluster of points of the results found by these authors. In Figure 19, it is highlighted that these points converge towards small (h/D <1.2) and intermediate (1.2 < h/D < 14) scale roughness. According to Ferguson (2007), Comiti et al. (2009), and Zimmermann (2010), a significant advantage of the U** versus q** representation is that equations based on q to predict flow velocity are more robust (with fewer errors in input variables) than equations based on D, especially when applied to steep and high-roughness channels.

Figure 19
Comparison between the dimensionless variables proposed by Rickenmann & Recking (2011).

Furthermore, we incorporated data from Menezes (2021), which illustrates the prototype inspiring this physical model. We noted consistent behavior in dimensionless values. Notably, certain data points along the Forqueta river deviate from the trend. This variance could stem from differing channel morphologies, such as cascades and plane-beds, as elaborated in Menezes (2021).

The understanding among these dimensionless parameters allows comparison with other studies in this field and enables the improvement of conventional equations for application in beds with extensive particle size distribution. However, there are still questions that deserve further observations and detailing for a complete understanding of the involved processes. For instance, some studies highlight that, in order to estimate the bed roughness height, D₈₄ is not always a reliable predictive parameter. Aberle & Smart (2003) achieved more favorable results by using the standard deviation of bed elevation (σ_z) as a roughness indicator in gravel-bed streams. Chen et al. (2020) and Ferguson et al. (2019), employing the same roughness indicator, also obtained promising results. However, both of them used data collected from natural rivers under a specific condition. In this regard, there is still a gap in research conducted in situations with more irregular beds, especially when using scale reductions.

CONCLUSIONS

Due to the lack of interaction with the flow, the laboratory channel bed needs a preliminary phase as it has not yet acquired geometric conformations and does not reproduce flow patterns similar to natural environments characterized by shallow flows and extensive granulometry.

Maintaining a flow rate characteristic of the modeled environment has allowed the generation of a channel bed that incorporates morphological elements, sediment distribution and flow paths that more closely resemble natural conditions. This provides a more accurate and realistic basis for physical simulation studies on rivers with variable grain size.

Looking at the velocity profiles obtained from the formed bed experiments, several important features can be highlighted: the thickening of the diameter of the moving bed which leads to increased resistance to flow (indicated by the parameters n, f and C), the deposition of fines in the empty spaces between the grains and the increase in ks as the discharge volume decreases. It was also observed that the dimension values obtained in the physical model correspond to those observed in nature.

Thus, through this preliminary study it was possible to reveal insights into the physical processes involved in water and sediment transport that need to be considered in physical simulation studies. This is important because understanding these processes is crucial for various fields, including hydrology, fluvial geomorphology and river engineering.

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Edited by

• Editor-in-Chief: Adilson Pinheiro

• Associated Editor: Fábio Veríssimo Gonçalves

Publication Dates

• Publication in this collection
  14 Feb 2025
• Date of issue
  2025

History

• Received
  14 Apr 2024
• Reviewed
  29 July 2024
• Accepted
  15 Nov 2024