1 Introduction

La vitesse de chute de sédiments cohésifs en suspension dans les cours d'eau est très supérieure à celle des grains d'argile individuels. Ceci s'explique par le fait que la plus grande partie de la masse sédimentaire en suspension se rassemble sous la forme d'agrégats [9]. La vitesse de chute synthétise certaines caractéristiques de ces agrégats, telles que la taille, la forme et la masse volumique.

La vitesse de chute intervient dans deux phénomènes sédimentaires de première importance, qui sont la répartition verticale des matières en suspension (MES) et leur flux de dépôt. La vitesse de chute varie principalement en fonction de la concentration locale C [4,5,11]. Pour les suspensions diluées, la vitesse de chute augmente avec C. Pour les suspensions très concentrées $(C>\sim 10\text{kg}{\text{m}}^{\mathrm{-3}})$, la vitesse de chute diminue quand C augmente (vitesse entravée). Cette étude porte essentiellement sur la vitesse de chute de suspensions diluées.

La plupart des auteurs caractérisent la vitesse de chute par sa valeur médiane locale $W_{50\%}$, qui est toujours inférieure à la vitesse de chute moyenne locale $\overline{W}$, laquelle donne une meilleure description de la dynamique verticale des sédiments. Dans les modèles hydrosédimentaires, la vitesse de chute est usuellement représentée par un scalaire dont la valeur dépend de C [3–5,7,10,11] ; cependant, compte tenu du grand nombre d'agrégats divers se trouvant simultanément en suspension, la vitesse de chute est en réalité une variable stochastique. Une variable aléatoire W est définie comme la vitesse de chute d'un élément constitutif des MES de masse unitaire infinitésimale.

2 Théorie

Le Tableau 1 donne, pour plusieurs valeurs de r, les valeurs de $\sigma [W]/\overline{W}$, de $\overline{W}/W_{50\%}$, et de $W_{90\%}/W_{10\%}$ (où $W_{N\%}$ représente la valeur de W ayant N% de probabilité de ne pas être dépassée par la masse sédimentaire). La Fig. 1 montre la fonction de répartition $F(w^{\ast })$ associée à (8) en termes de $w^{\ast }$, qui est représenté à l'échelle logarithmique. La fonction de répartition $F(w^{\ast })$ représente la proportion en poids des matières en suspension avec des valeurs de $W^{\ast }$ inférieures à $w^{\ast }$.

Table 1

Values of $\sigma [W]/\overline{W}$, $\overline{W}/W_{50\%}$, and $W_{90\%}/W_{10\%}$ linked to several values of r

Valeurs de $\sigma [W]/\overline{W}$, de $\overline{W}/W_{50\%}$, et de $W_{90\%}/W_{10\%}$ en fonction du paramètre r

r	$\sigma [W]/\overline{W}$	$\overline{W}/W_{50\%}$	$W_{90\%}/W_{10\%}$
0.25	2.0000	5.7243	11116
0.50	1.4142	2.1981	171.3
1.00	1.0000	1.4427	21.85
2.00	0.7071	1.1966	7.31
4.00	0.5000	1.0893	3.83
∞	0.0000	1.0000	1.00

À titre d'exemple, la Fig. 2 montre, pour $r=2$ et un nombre de Péclet $\mathit{Pe}=\overline{W_a}d/K_z=2$, avec $K_z$ constant sur la verticale, quelques aspects de la distribution verticale des MES. La Fig. 2a représente la répartition verticale de $C/C_a$. La Fig. 2b montre, pour cinq valeurs de $z°=z/d$, la distribution locale des MES $\mathit{SC}(w°)$ rapportée à la concentration au fond $C_a$ en termes des valeurs possibles w° de la variable aléatoire $W°=W/\overline{W_a}$ décrivant une vitesse de chute non locale adimensionnelle. Le produit $\mathit{SC}(w°)dw°$ représente la quantité de masse par unité de volume ayant des valeurs de W° comprises entre $w°-dw°/2$ et $w°+dw°/2$. Enfin, la Fig. 2c montre la variation verticale de $\mathit{SC}(w°)$ rapporté à la distribution au fond des MES ${\mathit{SC}}_a(w°)$, pour quatre valeurs de w°. Pour chaque valeur w° de W°, la distribution locale des MES est régie par la loi exponentielle en z, $\mathit{SC}(w°)={\mathit{SC}}_a(w°)\exp (-\mathit{Pe}w°z°)$. Une autre propriété de la fdp gamma définie par (6) est que, quand l'Éq. (5) est complétée par (9), le rapport $\mathit{SC}(w°)/{\mathit{SC}}_a(w°)$ est décrit par la forme la plus connue du modèle de Rouse–Vanoni, et ceci d'une façon exacte pour chaque valeur w° de W°.

3 Méthodes et résultats

La Fig. 3 montre les fréquences cumulées expérimentales de $W^{\ast }=W/\overline{W}$ en termes de ses valeurs possibles $w^{\ast }$ (avec $\overline{W}$ calculé par (3) et (10)). Les deux répartitions suivantes n'ont pas été représentées : celle du test n°1 à mi-profondeur (erreur expérimentale) et celle du test n° 5 au fond (concentration excessive, incompatible avec la méthode du tube de décantation). La Fig. 3 montre aussi la fonction de répartition $F(w^{\ast })$ résultant de l'intégration de le fdp définie par (8), avec r calculé par (11), c'est-à-dire sans ajustement spécifique à chaque test. La vitesse de chute moyenne des sédiments du fond $\overline{W_a}$ (m s⁻¹) et le paramètre r sont rattachés à $C_a$ (kg m⁻³) par les lois (10) et (11), propres à la vase de Loire.

4 Conclusions

The settling velocity of cohesive suspended sediments in the natural open surface flows is much higher than that of the elementary clay grains. This is due to the fact that the greatest part of the mass of sediments consists of aggregates, whose weight is largely higher than that of the individual grains [9]. The settling velocity includes a certain number of the characteristics of these aggregates, such as their size, shape and density.

The dynamics of the cohesive sediment is strongly dependent on the settling velocity of the suspended matter. This parameter intervenes in two sedimentary processes of first importance: the vertical distribution of the suspended matter and its deposition rate.

Settling velocity varies mainly with the local suspended sediment concentration C. A great number of field and laboratory measurements [4,5,11] showed that for diluted suspensions, settling velocity increases regularly with the concentration. On the other hand, for the highly concentrated suspensions starting from a concentration of about 10 kg m⁻³, this velocity decreases when the concentration increases because the sediments are blocked in their fall (hindered velocity). What follows deals with the study of the settling velocity of diluted suspensions for concentrations lower than about 10 kg m⁻³.

Most of the researchers characterize settling velocity by its median value. In all the cases, the median settling velocity $W_{50\%}$ is inferior to the mean settling velocity, which multiplied by the concentration gives the downward vertical solid flux, linked to gravity. The local mean settling velocity $\overline{W}$ gives the best description of the downward vertical dynamics of the cohesive sediments.

Settling velocity in sedimentary models is usually represented by a scalar whose value depends mainly on the local suspended sediment concentration [3–5,7,10,11]. However, taking into account the great diversity of aggregates that one can simultaneously find in suspension, the settling velocity is actually a stochastic variable. A random variable W is defined as the settling velocity of a particle of the suspended aggregates of infinitesimal elementary mass.

A steady-state vertical distribution of the suspended sediment in a flow is theoretically possible if at each level z the vertical downwards mass flux due to the local mean settling velocity $\overline{W}$ is equal to the ascending mass flux linked to the turbulent diffusion of sediment. The vertical movement of fine sediments can be expressed by assuming a Fickian mass diffusion as:

The vertical profile of the concentration depends on the expression used to describe $K_z$. For instance, when $K_z$ is depth-invariant, if the bottom concentration is taken at $z=a=0$, then ${\lambda }_z={\lambda }_a+z/K_z$. If the probability density function (pdf) $f_W(w)$ of W for every level z is in accordance with the following gamma distribution:

If a non-dimensional local settling velocity $W^{\ast }$ is defined in terms of the local mean settling velocity $\overline{W}$ by $W^{\ast }=W/\overline{W}$, then, the pdf of $W^{\ast }$ for every level z is in accordance with the following standardized gamma distribution:

In steady-state conditions, if, for a given sediment, the pdf of W linked to the bottom sediments is described by a gamma distribution, then, for every level z, W is also gamma distributed with the same parameter r [1]. In this case, the vertical variation of the local mean settling velocity $\overline{W}$ with the local concentration is governed by the power law defined by Eq. (3), independently of the retained expression for $K_z$. The validity of Eq. (3) can also be justified when the vertical distribution of the suspended sediment varies gradually in time, if the concentration C decreases monotonically when z increases.

These results are compatible with the usual expressions of $K_z$, which can eventually take into account the effect of a vertical stratification of the water density [5]. The most widely known form of the Rouse–Vanoni model for the vertical variation of the suspended sediment is obtained assuming the equality between diffusivity for sediment and momentum, with $K_z=\kappa U^{\ast }z(1-z/d)$, where κ is Von Karman's constant, $U^{\ast }$ is the shear velocity, and d is the total depth. In this case, the integral intervening in Eqs. (2) and (5) is given by:

Table 1 gives, for several values of r, the values of $\sigma [W]/\overline{W}$, $\overline{W}/W_{50\%}$, and $W_{90\%}/W_{10\%}$ (where $W_{N\%}$ is the value of W having $N\text{\%}$ of probability per weight of not being exceeded). For five values of r, Fig. 1 shows the cumulative distribution function $F(w^{\ast })$ linked to $f_{W^{\ast }}(w^{\ast })$ in terms of $w^{\ast }$, which is plotted in logarithmic scale. The cumulative distribution function $F(w^{\ast })$ represents the proportion per weight of suspended sediment with values of $W^{\ast }$ less than $w^{\ast }$.

As an example, Fig. 2 shows for $K_z$ depth-invariant, $r=2$, and a Péclet number $\mathit{Pe}=\overline{W_a}d/K_z=2$, some aspects of the suspended sediment vertical distribution. Fig. 2a represents the vertical variation of the concentration C related to the near-bottom concentration $C_a$. Fig. 2b shows, for five values of $z°=z/d$, the suspended sediment local distribution $\mathit{SC}(w°)$ related to the bottom concentration $C_a$ in terms of the possible values w° of the non-dimensional non-local settling velocity $W°=W/\overline{W_a}$. The product $\mathit{SC}(w°)dw°$ represents the quantity of matter per unit volume associated with values of W° ranging between $w°-dw°/2$ and $w°+dw°/2$. Finally, Fig. 2c shows the vertical variation of the suspended-sediment local distribution $\mathit{SC}(w°)$ related to the suspended sediment bottom distribution ${\mathit{SC}}_a(w°)$ for four w° families. For each value w° of W°, the vertical variation of $\mathit{SC}(w°)$ is governed by the exponential law $\mathit{SC}(w°)={\mathit{SC}}_a(w°)\exp (-\mathit{Pe}w°z°)$.

Another property of the gamma distribution defined by Eq. (6) is that when Eq. (5) is completed by Eq. (9), then, the ratio $\mathit{SC}(w°)/{\mathit{SC}}_a(w°)$ is exactly described by the most-widely known form of the Rouse–Vanoni model for each value w° of W°.

In order to study the vertical distribution of the suspended sediments and their settling velocities, a series of six tests was carried out on samples collected in the Loire estuary simultaneously at the bottom, half-depth and surface [1]. The local distribution of the suspended matter in terms of settling velocity has been examined by the bottom-withdrawal method [6]. Table 2 recapitulates the characteristic parameters of each of the six tests carried out.

Fig. 3 shows the experimental cumulative frequencies of $W^{\ast }=W/\overline{W}$ in terms of its possible values $w^{\ast }$ (with $\overline{W}$ calculated using Eqs. (3) and (10) below). Two cumulative frequencies have not been plotted: half-depth sediment for test No. 1 (experimental error) and bottom sediment for test No. 5 (too high concentration for the bottom withdrawal method). Fig. 3 also shows the cumulative distribution functions $F(w^{\ast })$ resulting from the integration of the probability density function defined by Eq. (8), with the parameters r evaluated with Eq. (11) below. That is to say that $F(w^{\ast })$ has not been fitted specifically for each test. For the Loire estuary cohesive sediments, $\overline{W_a}$ (m s⁻¹) and r are linked to the bottom concentration $C_a$ (kg m⁻³) by the following sediment-specific laws:

These laws are confirmed by eight other measurements carried out in the field using Owen tubes in the range $0.25\text{kg}{\text{m}}^{\mathrm{-3}}<C_a<3\text{kg}{\text{m}}^{\mathrm{-3}}$, and by measurements on reconstituted sediments in the range $10\text{kg}{\text{m}}^{\mathrm{-3}}<C<15\text{kg}{\text{m}}^{\mathrm{-3}}$, where the mean settling velocity remains almost constant, before the occurrence of the hindered settling velocity [1,7]. In a general way, one notes that, for $F(w^{\ast })>0.1$, the standardized probability density function defined by Eq. (8) agrees well to the experimental statistical distributions of $W^{\ast }$. The only exceptions to this rule are the results for half-depth and surface samples of test No. 4. These exceptions could be explained by a bad sampling of the suspended sediment at the bottom, resulting from the presence of a fluid mud layer at the bottom the day of the collection (the bottom reference concentration was probably overestimated). This agreement justifies here the use of Eq. (3), whose validity is in this case analytically proved in steady-state conditions [1]. The vertical distribution of suspended sediments in the Loire estuary can thus be regarded as gradually varied in time.

It must be noted that when a layer of fluid mud covers the bottom, the reference near-bottom concentration $C_a$ must be measured above this layer. In fact, the Rouse–Vanoni model in the Loire estuary can be applied only if $C_a<\sim 15\text{kg}{\text{m}}^{\mathrm{-3}}$. Above this concentration, hindered settling occurs and the aggregates are not free to make individual displacements. When a layer of fluid mud exists, the vertical sediment dynamics near the bottom could be better simulated as a two-layer system. In test No. 5, the near-bottom reference concentration has been taken equal to 15 kg m⁻³.

In steady-state conditions, if the settling velocity W of the bottom suspended-sediment mass is gamma distributed, the vertical variation of the local mean settling velocity $\overline{W}$ with the local concentration C is governed by the power law defined by Eq. (3), independently of the retained expression for $K_z$. That is a consequence of the suspended sediment sorting, which is produced by the vertical dynamics affecting the aggregates. For every value of the settling velocity of the suspended aggregates, the gamma distribution of W defined by Eq. (6) is compatible with a vertical distribution of the suspended sediments described by the generalized Rouse–Vanoni law.

When the settling velocity of the suspended sediment mass is gamma distributed, the mean settling velocity and the shape parameter r of the distribution are sufficient to describe the vertical dynamics of the whole of the suspended aggregate population. In this case, the local mean settling velocity at the bottom must be connected with the bottom concentration ($\overline{W_a}=\varphi (C_a)$, resulting from the direct effect of the concentration on the flocculation), and the vertical variation of the local mean settling velocity can be described by Eq. (3) ($\overline{W}=\overline{W_a}{(C/C_a)}^{1/r}$, resulting from the effect of the suspended sediment sorting produced by the vertical dynamics). This approach is valid for the cohesive sediments of the Loire estuary for concentrations lower than 15 kg m⁻³. Practically all the range of concentrations observed in this estuary is covered.

Finally, it is to be noted that the theoretical development presented in this article can also model the vertical dynamics of non-cohesive suspended sediments.