# Modelling Fine Sediment Dynamics: Towards a Common Erosion Law for Fine Sand, Mud and Mixtures

^{1}

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## Abstract

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## 1. Introduction

_{e}in N·m

^{−2}) and several empirical relationships between the erosion flux E (in kg·m

^{−2}·s

^{−1}) and excess shear stress (i.e., the difference between the actual shear stress (τ in N·m

^{−2}) and the critical value τ

_{e}, either normalized by the latter or not). The parameters of such an erosion law involve bed characteristics, which may concern electrochemical forces, mineral composition, and organic matter content [2], or pore water characteristics [3]. They also depend on the consolidation state [4], and may be altered by biota effects [5]. Bulk density has often been proposed as a proxy for characterizing the bed, but the plasticity index has also been suggested [6], as well as the undrained cohesion and the sodium adsorption ratio [3].

_{e}), involving (or not) a threshold value τ

_{e}.

_{0}is an erodibility parameter in kg·m

^{−2}·s

^{−1}and n a power function of the sediment composition. In Equation (1), E

_{0}and τ

_{e}are functions of the sediment composition and its consolidation state in the case of cohesive sediments, or functions of the particle diameter and density in the case of non-cohesive sediments. Assuming a similar erosion law for the whole range of mixed sediments, the problem becomes the assessment of the critical shear stress for erosion on the one hand, and the erosion factor E

_{0}on the other hand, in the full transition range between cohesive and non-cohesive materials. Following Van Ledden et al. [12], the proxy for such a transition could be clay content. An alternative proxy could be mud content (hereafter referred to as f

_{m}), considering that the clay to silt ratio is often uniform in a given study area [13].

_{e}) increases when mud is added to sand, either because of electrochemical bonds which take effect in binding the sand grains or because a cohesive matrix takes place between and around sand grains (e.g., [6,7,14,15,16,17]). Literature on erosion rate for mixed sediments is much less abundant, but a significant decrease (several orders of magnitude) in the erosion rate with mud content is most often reported (e.g., [14,17,18]). For instance, Smith et al. [17] presented laboratory measurements showing a decrease of about two orders of magnitude when the clay fraction in the mixture increased from 0% to 5–10%, and up to one order of magnitude more when the clay fraction increased from 10% to 30%.

_{mcr}

_{1}, they considered a non-cohesive regime where the erosion flux of any class of sandy and muddy sediments remains proportional to its respective concentration in the mixture, but is computed according to a pure sand erosion law (with a potential modulation of the power applied to the excess shear stress). Starting from a value characteristic of a pure sand bed, critical shear stress in this first regime is either kept constant [1], or linearly increases with the mud fraction f

_{m}[19,20]. Above a second critical mud fraction f

_{mcr}

_{2}, these authors defined a cohesive regime: Waeles et al. [19] and Le Hir et al. [1] formulate the erosion law using the relative mud concentration (the concentration of mud in the space between sand grains), considered as more relevant than the mud density in the case of sand/mud mixtures, which was in agreement with observations from Migniot [21] or Dickhudt et al. [22]. Between f

_{mcr}

_{1}and f

_{mcr}

_{2}, they ensured the continuity by prescribing a linear variation of E

_{0}and τ

_{e}between non-cohesive and cohesive erosion settings. Carniello et al. [23] used a two-stage erosion law built by Van Ledden [24]: below a critical mud fraction f

_{mcr}, the erosion factor of the sand fraction is steady and the one for the mud fraction varies slightly according to the factor 1/(1 − f

_{m}), while above f

_{mcr}the erosion rate is the same for sand and mud fractions and logarithmically decreases according to a power law. Regarding the critical shear stress for erosion, it first slightly increases with f

_{m}and then varies linearly to reach the mud shear strength. Dealing only with the critical shear stress, Ahmad et al. [25] proposed an alternative to the Van Ledden [24] expression: without any critical mud fraction, τ

_{e}varies linearly with f

_{m}for low values of f

_{m}and more strongly for high f

_{m}values, using a parameter representing the packing of the sand sediment in the mixture. Generally speaking, the transitional erosion rate between the two regimes is poorly documented.

## 2. Strategy and Modelling Background

#### 2.1. Strategy for Assessing an Erosion Law, and Its Application to the BoBCS

#### 2.2. Measurements Used for Erosion Law Assessment and Model Validation

_{10}of 7.5 µm, d

_{50}of 163 µm, 5.1% clay (% < 4 µm) and 25% mud (% < 63 µm) contents), and exhibits some gradients around the station, with muddy facies to the north, and more sandy ones to the south (Figure 1).

^{3}·s

^{−1}, and maximum flow rate of 2240 m

^{3}·s

^{−1}), and rather large swells (H

_{s}(significant wave height) peaks > 3 m, T

_{p}(peak period) 10–18 s).

^{−1}using water samples. The backscatter index (BI) from the AWAC profiler was evaluated from the sonar equation [29], following the procedure described by Tessier et al. [30], in particular by considering the geometrical attenuation for spherical spreading, the signal attenuation induced by the water, and the geometric correction linked to the expansion of the backscattering volume with increasing distance from the source. Given that SSC derived from the turbidity sensor did not exceed 100 mg·L

^{−1}, the signal attenuation caused by the particles was disregarded when estimating backscatter [30]. Then, an empirical relationship was established between the BI of first AWAC cell and SSC measurements of turbidity sensor, following Tessier et al. [30]:

^{2}of 0.78 with c

_{1}= 0.42794 and c

_{2}= 32.8907. Changes in SSC concentrations in the water column could thus be quantified (Section 4).

#### 2.3. Hydrodynamics Models (Waves, Currents)

#### 2.3.1. Brief Description

_{w}in N·m

^{−2}). The wave-induced shear stress was computed according to the formulation of Jonsson [38], with a wave-induced friction factor determined following Soulsby et al. [39]. Then, the total bottom shear stress τ was computed from the estimated τ

_{w}and from the current-induced shear stress (τ

_{c}in N·m

^{−2}) provided by the hydrodynamic model, according to the formulation of Soulsby [40], i.e., accounting for a non-linear interaction between waves and currents. Both wave and current shear stresses were computed by considering a skin roughness length z

_{0}linked to a 200 µm sand, representative of the sandy facies widely encountered on the BoBCS (z

_{0}= k

_{s}/30 = 2 × 10

^{−5}m with k

_{s}the Nikuradse roughness coefficient).

#### 2.3.2. Hydrodynamic Validation of the Model

_{S}), surface temperature and salinity (T

_{S}and S

_{S}, respectively), and current intensity and direction (Vel

_{INT}and Vel

_{DIR}, respectively).

_{s}were around 1.5 m and occasionally exceeded 4 m in stormy conditions. Peak periods (not illustrated here) ranged from 8 to 18 s (around 10 s on average). Even during “calm” periods, H

_{S}values are generally no lower than 0.8 m. Figure 2a illustrates the ability of the model of Boudière et al. [35] to describe H

_{S}over the period, with a root mean square error (RMSE) of 0.24 cm and a R

^{2}of 0.95. Measured and modelled T

_{S}and S

_{S}are illustrated in Figure 2b,c respectively, and demonstrate the correct response of the model with respect to observations with RMSE/R

^{2}values of 0.5 °C/0.86 for T

_{S}and 2 PSU/0.7 for S

_{S}. The weaker correlation obtained for S

_{S}is mainly due to model underestimation around 2 January 2008 (i.e., about 5 PSU). It should be underlined that the model accurately reproduces the abrupt change (decrease) in surface temperature and salinity on 11 December 2007, linked to the veering of the Loire river plume caused by easterly winds. This plume advection led to stratification which in turn influenced the vertical profiles of currents in terms of direction and intensity (Figure 2e,g, respectively), which are well represented by the model. More generally, the model provides an appropriate response regarding the direction and intensity of the current (Figure 2d,f) over the entire water column throughout the period. For instance, measured bottom current velocities (0.11 m·s

^{−1}on average, max of 0.44 m·s

^{−1}) are correctly reproduced by the model with a RMSE of 0.05 m·s

^{−1}.

#### 2.4. Sediment Transport Model

_{rel mud}). Given that the surficial sediment in our study area is weakly consolidated (erosion actually occurs in surficial layers only, typically a few mm or cm thick) and mainly composed of fine sand, consolidation was disregarded and C

_{rel mud}was set at a constant value of 550 kg·m

^{−3}(representative of pre-consolidated sediment according to Grasso et al. [41]). In addition, bed load was not taken into account in the present application, assuming that surficial sediment in our study area is mainly composed of mixtures of mud and fine sand. In our case, three sediment classes (for which the mass concentrations are the model state variables) are considered: a fine sand (S1), a non-cohesive material with a representative size of 200 µm, and two muddy classes (M1 and M2), which can be distinguished by their settling velocity, in order to be able to schematically represent the vertical dispersion of cohesive material over the shelf. The sediment dynamics was computed with an advection/dispersion equation for each sediment class, representing transport in the water column, as well as exchanges at the water/sediment interface linked to erosion and deposition processes. Consequently, the concentrations of suspended sediment, the related horizontal and vertical fluxes, and the corresponding changes in the seabed (composition and thickness) are simulated. This section details the way of managing sediment deposition, seabed initialization, and technical aspects linked to vertical discretization within the seabed compartment. The erosion law establishment and the numerical modelling experiment aiming to fit an optimal setting will be addressed independently in Section 3.

#### 2.4.1. Managing Sediment Deposition

_{i}for each sediment class i is computed according to the Krone law:

_{s,i}is the settling velocity, SSC

_{i}is the suspended sediment concentration, and τ

_{d,i}is the critical shear stress for deposition. In the present study, the latter was set to a very high value (1000 N·m

^{−2}) so it is ineffective: considering that consolidation processes are negligible near the interface and that, as a result, the critical shear stress for erosion remains low (Section 3), deposited sediments can be quickly resuspended in the water column if the hydrodynamic conditions are sufficiently intense, which replaces the role played by the term between parentheses in Equation (3) [1].

_{s,S}

_{1}) is computed according to the formulation of Soulsby [40]. The one related to the mud M1 (W

_{s,M}

_{1}) is assumed to vary as a function of its concentration in the water column (SSC

_{M}

_{1}) and the ambient turbulence according to the formulation of Van Leussen [42]:

_{s,M}

_{1}is limited by minimum (W

_{s,min}) and maximum (W

_{s,max}) values, respectively set to 0.1 mm·s

^{−1}and 4 mm·s

^{−1}, the latter being reached for SSC ≥ 700 mg·L

^{−1}(thus ignoring the hindered settling process which actually does not occur in the range of SSC over the shelf). Lastly, the mud M2 linked to ambient turbidity in the water column is assumed to have a constant very low settling velocity set to 2.5 × 10

^{−6}m·s

^{−1}.

_{vol sort}= 0.58) was attributed to sediment when only one class of non-cohesive sediment is present. However, the bed concentration can reach a higher value if several classes are mixed (C

_{vol mix}= 0.67). These typical volume concentrations [45] led to mass concentrations assuming a fixed sediment density ρs (2600 kg·m

^{−3}) for all classes. In the case of simultaneous deposition of sand and mud, the sand is first deposited at a concentration which depends on surficial sediment composition: in the case of mixed sediment, the sand is first mixed with the initial mixture until C

_{mix}(= C

_{vol mix}.ρ

_{s}= 1742 kg·m

^{−3}) is reached. The remaining sand is deposited with the concentration C

_{sort}(= C

_{vol sort}.ρ

_{s}= 1508 kg·m

^{−3}), from which an increase in the thickness of the layer can be deduced. The same kind of deposition occurs when the surficial sediment is only comprised of sand. In addition, the thickness of any layer is limited to dz

_{sed, max}, a numerical parameter of the model. Any deposition of excess sand leads to the creation of a new layer (see Section 2.4.2). Next, mud is deposited: it progressively fills up pores between the sand grains until either C

_{mix}or C

_{rel mud}is reached. Considering these criteria, mud is mixed within the initial and new deposits starting from the water/sediment interface. If any excess mud remains after the mixing step, it is added to the upper sediment layer, contributing to its thickening.

#### 2.4.2. Sediment Discretization within the Seabed

_{sed, ini}) is introduced, and the seabed is vertically discretized in a given number of layers of equivalent thicknesses (dz

_{sed, ini}). An optimal vertical discretization of the sediment was assessed by Mengual [46]. By means of sensitivity analyses, it was shown that beyond a 1/3 mm resolution within the seabed compartment (i.e., thickness of each layer dz

_{sed, ini}), the SSC response of the model in the water column did not change anymore. According to the conclusions drawn from this sensitivity analysis, the initial sediment thickness h

_{sed, ini}was set at 0.03 m, corresponding to 90 sediment layers of thickness dz

_{sed, ini}(1/3 mm).

_{sed, max}, corresponding to a parameter of the model. Nevertheless, a maximum number of layers in the seabed compartment (N

_{max}) needs to be defined in order to make computational costs acceptable. While N

_{max}is reached, a fusion of the two sediment layers located at the base of the sedimentary column occurs. By this way, the creation of new deposited layers becomes once again possible. The parameters dz

_{sed, max}and N

_{max}control changes of the “sediment vertical discretization” during the simulation, and are likely to influence the sediment dynamics. To prevent any variations in the SSC response of the model linked to changes in the seabed vertical resolution, the maximum thickness of sediment layers, dz

_{sed, max}, was set at the same value than the initial one dz

_{sed, ini}.

#### 2.4.3. Sediment Facies Initialization for the Application to the BoBCS

^{−3}) on the seabed over the entire shelf. This particle class enables the representation of a non-negligible part of the ambient turbidity of a few mg·L

^{−1}near the coast.

_{s}of the mixed sediment, a relationship could be derived between C

_{rel mud}(relative mud concentration set at a constant value of 550 kg·m

^{−3}) and the bulk sediment density (C

_{bulk}). The latter can be written as:

^{−3}becomes unlikely, so that ${C}_{bulk}$ is maximised by ${C}_{max}$, which corresponds to a dense sediment (C

_{sort}or C

_{mix}depending on the properties of the sediment mixture, i.e., well-sorted or mixed sediments). This formulation could be validated by comparing with recent data in two sectors of the BoBCS [48]: bulk densities of 1540 kg·m

^{−3}and 1380 kg·m

^{−3}were obtained for 85% and 75% of sand, respectively, while the application of Equation (5) with sand fractions f

_{s}of 0.85 and 0.75 led to C

_{bulk}values of 1668 kg·m

^{−3}and 1346 kg·m

^{−3}in fair agreement. Sediment concentrations linked to each particle class were then deduced according to their respective fraction.

## 3. Erosion Law Setting: Building and Numerical Experiment

#### 3.1. General Formulation

_{0}(erodibility parameter, kg·m

^{−2}·s

^{−1}), τ

_{e}(critical shear stress, N·m

^{−2}), and n (hereafter referred to as “erosion-related parameters”), which are set at different values depending on the mud content f

_{m}(<63 µm) of the surficial sediment. The concept of critical mud fraction is retained, with the definition of a first critical fraction f

_{mcr}

_{1}below which a non-cohesive behaviour is prescribed, and of a second one, f

_{mcr}

_{2}, above which the sediment is assumed to behave as pure mud. Two sets of erosion-related parameters will be defined below and above these mud fractions (see Section 3.2 and Section 3.3 for sand and mud, respectively). As already mentioned in Section 1, a transition in erosion-related parameters has to be prescribed between f

_{mcr}

_{1}and f

_{mcr}

_{2}(between pure sand and pure mud parameters) to manage the erosion of “transitional” sand/mud mixtures. Such a transition is investigated in Section 3.4.

#### 3.2. Pure Sand Erosion

_{mcr}

_{1}, the non-cohesive erosion regime is prescribed by defining a first set of erosion-related parameters linked to pure sand erosion in Equation (1): E

_{0,sand}, τ

_{e,sand}, and n

_{sand}. Le Hir et al. [1] suggested to compute f

_{mcr}

_{1}as a function of the sand mean diameter D (f

_{mcr}

_{1}= α

_{0}× D with α

_{0}= 10

^{3}m

^{−1}), leading to a value of 20% considering the fine sand S1 of 200 µm (considered as a reference value hereafter). However, the model erosion dynamics is likely to significantly vary while the surficial mud fraction f

_{m}is close to f

_{mcr}

_{1}. Such a sensitivity is addressed in Section 4.2.

_{0,sand}, τ

_{e,sand}, and n

_{sand}were deduced from numerical simulations and empirical formulations. According to Van Rijn [49] and many other numerical models (e.g., [23,24]), the best fit for n

_{sand}is 1.5. The critical shear stress τ

_{e,sand}was determined from the Shields critical mobility parameter computed according to the formulation of Soulsby [40], leading to a value of 0.15 N·m

^{−2}for a sand of 200 µm.

_{s}is the settling velocity and C

_{ref}is a reference concentration which characterizes the equilibrium. The description of C

_{ref}is generally associated with the reference height, h

_{ref}, the distance from the bed where the concentration is considered. In point of fact, this location has to be the one where the equilibrium between deposition and erosion is considered, with respective values that largely depend on the reference height, because of large concentration gradients near the bed. From the point of view of sediment modelling, this means that the deposition flux at the base of the water column has to be expressed at the exact location where the erosion flux is considered, that is, at the reference height where the equilibrium concentration is given. Van Rijn [49] used the concept of equilibrium concentration as a boundary condition of the computation of suspended sediment profile and fitted the expression:

_{ref}formulation of Van Rijn [49] in Equation (6) enables us to express the E

_{0,sand}constant in the “Partheniades” form of the erosion law as:

_{ref}= 0.02 m, and W

_{s,S}

_{1}= 2.5 cm·s

^{−1}, Equation (8) leads to E

_{0,sand}= 5.94 × 10

^{−3}kg·m

^{−2}·s

^{−1}. The relevance of this E

_{0,sand}value was assessed by simulating an equilibrium state under a steady current and by comparing the depth-integrated horizontal sediment flux with some standard transport capacity formulations. Using a 1DV version of the code, several computations of fine sand resuspension were performed under different flow intensities ($Ve{l}_{INT}$), and once the equilibrium was reached (deposition = resuspension) the total transport (Q

_{sand}) was computed as:

_{S}

_{1}(k) to the suspended sediment concentration of sand (S1) in the cell k. Sand transport rates Q

_{sand}were then compared to the rates deduced from the formulations of Van Rijn [50], Engelund and Hansen [51], and Yang [52] for similar flow velocities (hereafter Q

_{sand}

_{,VR1984}, Q

_{sand}

_{,EH1967}, and Q

_{sand}

_{,Y1973}respectively). Results are illustrated in Figure 3. The results obtained by Dufois and Le Hir [53], who also used an advection/diffusion model to predict sand transport rates for a wide range of current conditions and numerous sand diameters, have been added in Figure 3 (Q

_{sand}

_{,DLH2015}). Figure 3 shows that the sand transport rates obtained from our computations are in a consistent range regarding those obtained with other formulations or studies cited in the literature, demonstrating the suitability of our E

_{0,sand}parameter.

_{0,sand}, τ

_{e,sand}, and n

_{sand}for fine sand are summarized in Table 2 and constitute reference parameters characterizing pure sand erosion.

#### 3.3. Pure Mud Erosion

_{mcr}

_{2}(reference value of 70% according to the default value used by Le Hir et al. [1]), the cohesive erosion regime is prescribed by defining a second set of erosion-related parameters linked to pure mud erosion in Equation (1): E

_{0,mud}, τ

_{e,mud}, and n

_{mud}.

_{mud}exponent was set to 1. Given the lack of established formulation of the erosion factor for pure mud, experimental approaches are often used to calibrate it for specific materials, preferably in situ when possible. For this purpose, a specific device had been designed: the “erodimeter” is described in Le Hir et al. [7]. It consists of a small recirculating flume where a unidirectional flow is increased step by step and interacts with a sediment sample carefully placed at the bottom after transfer from a cylindrical core. When measurements are made on board an oceanographic vessel, the test can be considered as quasi in situ. On the BoBCS, erosion tests had been conducted on board the Thalassa N/O: a few of them were performed on muddy sediment samples (mud content higher than 80%). Figure 4 illustrates the critical shear stress for erosion τ

_{e,mud}, estimated to be 0.1 N·m

^{−2}, suggesting a barely-consolidated easily erodible sediment.

_{0,mud}, the range of values cited in the literature extends from 10

^{−3}to 10

^{−5}kg·m

^{−2}·s

^{−1}for natural mud beds in open water (e.g., [54]). Simulating fine sediment transport along the BoBCS, Tessier et al. [43] applied the Partheniades erosion law but used an even lower erosion constant (E

_{0}= 1.3 × 10

^{−6}kg·m

^{−2}·s

^{−1}). As a first attempt in the present study, a low value E

_{0,mud}= 10

^{−5}kg·m

^{−2}·s

^{−1}was used, and its appropriateness was demonstrated by comparing the computed erosion fluxes (E

_{modelled}) from Equation (1) (with τ

_{e}= τ

_{e,mud}and n = n

_{mud}otherwise) with measurements from erodimetry experiments (E

_{measured}) conducted on three muddy sediment samples from the BoBCS (mud fraction higher than 70%) (Figure 5).

_{0,mud}, τ

_{e,mud}, and n

_{mud}values are summarized in Table 2 and constitute reference parameters characterizing pure mud erosion.

#### 3.4. Erosion of Transitional Sand/Mud Mixtures: Selection of Transition Formulations to be Tested

_{mcr}

_{1}and f

_{mcr}

_{2}), E

_{0}, τ

_{e}, and n ranged between pure sand (E

_{0,sand}, τ

_{e,sand}, and n

_{sand}) and pure mud (E

_{0,mud}, τ

_{e,mud}, and n

_{mud}) parameters, following a transition trend which had to be specified. We defined several expressions of the erosion law for the transition between non-cohesive and cohesive behaviours as a function of the surficial sediment mud fraction (f

_{m}).

_{exp}) as a function of mud content, with a coefficient C

_{exp}allowing the adjustment of the sharpness of the transition, which becomes more abrupt with an increase in C

_{exp}:

_{exp}= (f

_{mcr}

_{1}− f

_{m})/(f

_{mcr}

_{2}− f

_{mcr}

_{1}); X

_{sand}= {E

_{0,sand}; τ

_{e,sand}; n

_{sand}}; and X

_{mud}= {E

_{0,mud}; τ

_{e,mud}; n

_{mud}}.

_{exp}= 10 and 40 in Equation (10)), were defined to evaluate the effect of the transition trend only, using the reference critical mud fractions (f

_{mcr}

_{1}= 20% and f

_{mcr}

_{2}= 70%). The corresponding settings are named S1

_{LIN}, S1

_{EXP}

_{1}, and S1

_{EXP}

_{2}, respectively.

_{mcr}

_{1}value was successively reduced to 10%, 5%, and ~0% (with the corresponding f

_{mcr}

_{2}= 60%, 55%, and 50%), but only the exponential transition regime (with C

_{exp}= 40) was considered (simulations S2

_{EXP}

_{2}, S3

_{EXP}

_{2}, and S4

_{EXP}

_{2}respectively), as it produced better results than the other transitions (see results Section 4.1).

## 4. Results

#### 4.1. Influence of the Transition Trend between Non-Cohesive and Cohesive Erosion Modes in the Erosion Law

_{exp}= 40 in Equation (10)), between the non-cohesive and cohesive regimes (Figure 7). This first comparison was performed using the “reference” f

_{mcr}

_{1}and f

_{mcr}

_{2}values of 20% and 70%, respectively. The two erosion settings, S1

_{LIN}and S1

_{EXP}

_{2,}are illustrated in Figure 6 (Section 3.4).

_{OBC}with F

_{OBC, mud}for mud and F

_{OBC, sand}for sand, representing the total amount (integrated) of sediment that crosses the borders of the cells along the water column, as net inflow if F

_{OBC}increases or as net outflow if F

_{OBC}decreases) is illustrated in Figure 7h.

^{−1}over the study period. Four main resuspension events can be identified: the first event lasted from 1 to 5 December 2007 (Event 1), the second from 8 to 11 December 2007 (Event 2), the third from 11 to 12 January 2008 (Event 3), and the last from 15 to 17 January 2008 (Event 4). During these events, SSC values ranged between 20 and 80 mg·L

^{−1}near the seabed, and did not exceed 40 mg·L

^{−1}close to the surface (Figure 7). The rest of the time, a higher frequency turbidity signal linked to the semi-diurnal tide resuspension was recorded near the seabed with SSC in the range of 10–15 mg·L

^{−1}. Turbidity peaks were regularly detected in the surface signal whereas there was no significant increase near the seabed: these peaks are probably due to the signal diffraction caused by wave-induced air bubbles (wave mixing; see Tessier [55]). Such a phenomenon also occurs during energetic events (H

_{s}> 2 m) with a SSC signal in the surface higher than in the rest of the water column.

_{LIN}erosion setting highlighted some periods during which turbidity was overestimated, for instance in the upper half of the water column during Event 2, and several times between 30 December and 9 January (Figure 7c,e). Overestimations were particularly noticeable in the SSC series at 1.67 m above the seabed (Figure 7e): modelled SSC regularly exceeded observed SSC by 20 to 40 mg·L

^{−1}during Events 2 and 4, and even during calmer periods (e.g., between 30 December and 9 January). In contrast, modelled SSC were underestimated by a factor of 2 during Event 3. The S1

_{LIN}erosion setting led to a representation of observed SSC with a RMSE of 14 mg·L

^{−1}over the study period.

_{EXP}

_{2}erosion setting enabled a general improvement in modelled SSC with a RMSE of 10.5 mg·L

^{−1}over the period, and a correct response during the four energetic periods (Figure 7d,e). Differences in SSC between simulations during Events 1, 2, and 4 were mainly due to different erosion rates (especially due to E

_{0}in Equation (1)) prescribed for the same intermediate mud fraction in the surficial layer (in Figure 6a, this rate is clearly higher in the S1

_{LIN}setting). However, other differences, especially between 30 of December and 9 of January and during Event 3, are linked to the contrasted changes in the seabed between the two simulations.

_{EXP}

_{2}(Figure 7f,g). This difference can also be seen in Figure 7h, which highlights a significant decrease in F

_{OBC, mud}(i.e., flow of mud out of the cell) in simulation S1

_{LIN}, but not in S1

_{EXP}

_{2}. The decrease in F

_{OBC, mud}in simulation S1

_{LIN}is probably due to less mud inputs from adjacent cells, resulting in relatively more mud exported by advection and thus less mud deposition during the decrease in shear stress following Event 2. Note that this difference in seabed changes influences the sediment dynamics in both simulations throughout the period. Following Event 2, a transition in surficial sediment from muddy to sandy occurs in both simulations but at different times. In the S1

_{LIN}simulation, the transition occurs half way through the period, around 30 December, and manifests itself as a SSC peak linked to mud resuspension near the bottom (Figure 7e), and by a decrease in F

_{OBC, mud}(relative loss by advection), which does not occur in the S1

_{EXP}

_{2}simulation. Following the transition in the nature of the seabed, the S1

_{LIN}simulation regularly gives incorrect SSC responses (e.g., overestimation around 6 January, underestimation during Event 3). These results underline the potential role of advection processes in the contrasted results of the two simulations, and the need for full 3D modelling to obtain a final fit of the erosion law. In the S1

_{EXP}

_{2}simulation, the transition occurs later in the period, around 11 of January, and enables a correct SSC response regarding Event 3, associated with a decrease in F

_{OBC, mud}. It may mean that, on the one hand, setting S1

_{EXP}

_{2}allows a more accurate representation of resuspension dynamics in response to a given forcing, and on the other hand, it induces a more correct change in the nature of the seabed with respect to the variations in forcing over time. Note that despite contrasted sediment dynamics in the different simulations, the mud budget at the scale of the cell summed over the whole period led to similar trends corresponding to a relative loss by advection of around −5 × 10

^{6}kg (that is −5/2.5

^{2}kg·m

^{−2}).

_{OBC, sand}in Figure 7h). Beyond this date, the contrasted nature of the seabed results in more regular sand resuspension in S1

_{LIN}, with an advection component leading to a relative local sand loss (in the cell). Starting from Event 3, F

_{OBC, sand}increases (i.e., relative sand inflow into the cell) in both simulations while the advection flux of mud decreases (in S1

_{EXP}

_{2}) or does not change (in S1

_{LIN}). This highlights the fact that sand and mud dynamics are likely to differ depending on the nature of the seabed in adjacent cells.

_{EXP}

_{1}, which is characterized by a less abrupt exponential transition in erosion law (C

_{exp}= 10 in Equation (10)), appears to be less accurate in terms of SSC (not illustrated here) with a RMSE of 12 mg·L

^{−1}over the study period (versus 10.5 mg·L

^{−1}in S1

_{EXP}

_{2}). The turbidity response provided by the model would be expected to be degraded while progressively reducing the decreasing trend of the transition (until a linear decrease is reached).

_{EXP}

_{2}, the latter was considered as an “optimum” setting, suggesting that the definition of an exponential transition to describe sand/mud mixture erosion between non-cohesive and cohesive erosion modes may be appropriate in hydro-sedimentary numerical models.

#### 4.2. Influence of Critical Mud Fractions

_{mcr}

_{1}and f

_{mcr}

_{2}, was assessed, starting from the “optimum” erosion setting deduced in Section 4.1 and characterized by an exponential transition between f

_{mcr}

_{1}= 20% and f

_{mcr}

_{2}= 70% with C

_{exp}= 40 in Equation (10) (i.e., S1

_{EXP}

_{2}setting). Both critical mud fractions were progressively reduced by 10% (f

_{mcr}

_{1}= 10%, f

_{mcr}

_{2}= 60%), 15% (f

_{mcr}

_{1}= 5%, f

_{mcr}

_{2}= 55%), and 20% (f

_{mcr}

_{1}≈ 0%, f

_{mcr}

_{2}= 50%). The corresponding settings, S2

_{EXP}

_{2}, S3

_{EXP}

_{2}, and S4

_{EXP}

_{2}are illustrated in Figure 6. Results linked to the application of these different settings are illustrated in Figure 8. Note that the second critical mud fraction f

_{mcr}

_{2}appears in the extension of the exponential decay (Equation (10)), but, due to the shape of the exponential trend, it does not constitute a real critical mud fraction but rather an adjustment parameter for the transition. We can thus consider that this sensitivity analysis mainly deals with the setting of the first critical mud fraction f

_{mcr}

_{1}.

_{mcr}

_{1}(Figure 8b), which results in underestimation of turbidity with respect to observed values. While no clear differences in SSC appear between S1

_{EXP}

_{2}and S2

_{EXP}

_{2}(the latter is not illustrated in Figure 8), i.e., with a reduction of f

_{mcr}

_{1}from 20% to 10%, significant SSC underestimations occur for f

_{mcr}

_{1}< 10%. The average turbidity during resuspension events is underestimated by 15–20% (respectively, 30%) in simulation S3

_{EXP}

_{2}(respectively, S4

_{EXP}

_{2}). Regarding maximum SSC, underestimations of SSC peaks are around 15–30% (respectively, 40–50%) during Events 1 and 2 in simulation S3

_{EXP}

_{2}(respectively, S4

_{EXP}

_{2}). In addition, SSC peaks during Events 3 and 4 are completely absent in these two simulations with an underestimation of about 60%. Other simulations with linear trend but low f

_{mcr}

_{1}were tested and showed no improvement compared with the settings illustrated in Figure 7 and Figure 8.

_{EXP}

_{2}and the simulation S3

_{EXP}

_{2}(f

_{mcr}

_{1}= 5%) are illustrated in Figure 8c,d. Following Event 2, contrary to results in S1

_{EXP}

_{2}, no drastic change in the nature of the seabed occurs in S3

_{EXP}

_{2}in the rest of the period, with a surficial sediment containing at least 30–40% of mud. This less dynamic change in the seabed is consistent with the lower SSC obtained in the water column. A reduction of f

_{mcr}

_{1}led to the application of a pure mud erosion law starting from a lower mud content in the surficial sediment. This mostly resulted in less erosion (E

_{0,mud}<< E

_{0,sand}) with weaker SSC and slower changes in the seabed. This is also visible in the variations in F

_{OBC, mud}(Figure 8e) which highlight the fact that the reduced sediment dynamics obtained by reducing f

_{mcr}

_{1}results in weaker gradients (SSC, seabed nature) with adjacent cells, and a less dynamic advection term over the study period.

## 5. Discussion

#### 5.1. Setting Describing Erosion of a Sand/Mud Mixture

_{mcr}

_{1}), above which a transition toward a cohesive erosion mode would start, is at least 10% mud content. Grain size analyses of numerous sediment samples (from several locations, and at different depths in the sediment) from the BoBCS revealed that the clay content (per cent < 4 µm) corresponds to 30% (±3%, R

^{2}= 0.96) of the mud content (per cent < 63 µm). Such a constant ratio between the clay and mud fractions (or between the clay and silt fractions) in a given area has been observed in many sites worldwide (e.g., [13]). Thus, the critical clay content deduced from our modelling fitting would be around 3%. Therefore, our results are in agreement with experimental results of previous studies regarding the existence and the value of a critical mud/clay fraction indicating a transition in the mode of erosion.

_{mcr}

_{1}in the mixture. The quality of the model response was evaluated by comparing SSC results with turbidity measurements provided by the AWAC profiler over the entire water column. Based on RMSE and average or maximum SSC values reached during resuspension events, the results provided a more accurate representation of observations while considering an abrupt exponential transition of erosion parameters (i.e., E

_{0}, τ

_{e}, and n in the Partheniades form of the erosion law, see Equation (1)). Actually, changes in SSC produced by this transition formulation mainly hold in the contrasted E

_{0}values prescribed in erosion law depending on the seabed mud fraction (E

_{0,mud}<< E

_{0,sand}; see Table 2). This result agrees with results recently obtained by Smith et al. [17] who performed erosion experiments on mixed sediment beds prepared in the laboratory (250–500 µm sands mixed with different clayey sediments corresponding to kaolinite or kaolinite/bentonite). In particular, they observed a rapid decrease in erosion rates, from 1.5 to 2.5 orders of magnitude, over a range of 2% to 10% clay content. In the present study, exponential transitions prescribed in settings S1

_{EXP}

_{1}and S1

_{EXP}

_{2}(C

_{exp}= 10 and C

_{exp}= 40 in Equation (10), Figure 6) led to a variation in the erodibility parameter E

_{0}of about 2.5 orders of magnitude over a mud content range of 10% and 40%, i.e., over a clay content ranging from 3% and 12% respectively. The best model results obtained from erosion setting S1

_{EXP}

_{2}are thus consistent with the findings of Smith et al. [17], and suggest that a rapid exponential transition may be appropriate to describe the erosion of a sand/mud mixture between non-cohesive and cohesive erosion modes in numerical hydro-sedimentary models.

#### 5.2. Limitations of the Approach and Remaining Uncertainties

#### 5.2.1. Mud Erosion Law

_{e,mud}) of 0.1 N·m

^{−2}was deduced. By combining this τ

_{e,mud}value with the minimum erodibility parameter E

_{0,mud}recommended by e.g., Winterwerp [54], i.e., 10

^{−5}kg·m

^{−2}·s

^{−1}, the application of the mud erosion law from the model (Section 3.3) led to good agreement between modelled erosion fluxes and those obtained in erodimetry experiments for comparable applied shear stresses (Figure 5). Such a lower critical stress for erosion when the mixture is muddier is opposite to trends most often published, characterized by an increase of the resistance to erosion when mud is added to sand (e.g., [6,7,14,15,16,17]). Other simulations were performed with higher τ

_{e,mud}values, 0.15, 0.2, and 0.4 N·m

^{−2}. As expected, modelled SSC was underestimated compared with observed SSC while τ

_{e,mud}increased (even for 0.15 N·m

^{−2}). Another assessment of E

_{0,mud}could have produced similar results, but we preferred to keep the shear strength provided by our experiments, the low value being justified by the fact that in our environment (erosion on a continental shelf with low bottom friction) the sediment is never remobilized at depth, and the surficial sediment remains unconsolidated.

#### 5.2.2. Initial Condition of the Sediment and Time Variation of the Seabed

_{EXP}

_{2}) was used again in a new simulation using the surficial sediment cover computed at the end of a one-year simulation used as spin up. We obtained similar SSC results with a RMSE of 11.3 mg·L

^{−1}over the study period (versus 10.5 mg·L

^{−1}in the original S1

_{EXP}

_{2}simulation). Similarities in seabed variations (thickness and composition) in the two simulations were likewise remarkable (not illustrated here). Thus, the seabed initialization prescribed at the beginning of each simulation appears to be appropriate and does not correspond to a transitional state regarding the sediment dynamics.

#### 5.2.3. Applicability of the Sand/Mud Mixture Erosion Law

_{mcr}

_{1}, above which erosion behaviour starts to change, mainly depends on clay content, the ratio between clay and mud fractions can be used. In future works, it would be interesting to explore other mud properties than grain size and sediment fractions, such as mineralogy, to represent more accurately the key role played by cohesive sediments in erosion process, especially for transitional sand/mud mixtures between the contrasted non-cohesive/cohesive regimes.

_{e}-1). Such a formulation is very sensitive to the value of the critical shear stress for erosion, which can be difficult to estimate and highly variable in the case of sand/mud mixtures. Alternatively, a formulation of the erosion flux proportional to the excess shear stress (τ-τ

_{e}) would reduce the sensitivity of erosion to τ

_{e}. It would also be in agreement with the Van Kesteren-Winterwerp-Jacobs erosion law [6,58], and deserves further investigations following the pioneering work of Jacobs et al. [6].

## 6. Conclusions

- Using an abrupt exponential transition, e.g., an erodibility parameter decrease of 2.5 orders of magnitude over a 10% (respectively, 3%) mud (respectively, clay) content range, improves SSC model results regarding measurements, compared to results obtained with linear or less abrupt exponential transitions. This conclusion agrees with recent experimental studies in the literature on the erosion of sand/mud mixtures, which mention a drastic change in erosion mode for only a small percentage of clay added in the mixture.
- A first critical mud fraction (above which the erosion mode begins to change) of 10–20% is required to ensure a relevant model response in turbidity. By reasoning in terms of the clay fraction, the corresponding critical clay fraction ranges between 3% and 6%. Once again, this conclusion agrees with experimental studies in the literature reporting that 2% to 10% of clay minerals in a sediment mixture are sufficient to control the soil properties.
- The erosion flux of mixed sediments appears to be very sensitive to the clay fraction of the surficial sediment, and then is likely to change considerably at a given location, according to erosion and deposition events.
- The need to perform 3D simulations to account for advection, which considerably influences sediment dynamics in terms of export of resuspended sediments, sediment inflows from adjacent cells, and consequent changes in the surficial seabed (nature and thickness of deposits).

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Geographic extent of the 3D model configuration with its bathymetry (in meters with respect to mean sea level) (

**a**). Initial condition for the seabed compartment (at the resolution of the model) (

**b**), over the zone surrounding Le Croisic station (indicated by a black dot). In (

**a**), the thickest white line represents the 180 m isobath, which can be considered as the external boundary of the continental shelf. In both subplots, black lines refer to the 40-m, 70-m, 100-m, and 130-m isobaths over the shelf.

**Figure 2.**Validation of the hydrodynamic model over most of the simulated period at LC station. Model results are compared with AWAC measurements in terms of (

**a**) significant wave height, (

**b**) surface temperature, (

**c**) surface salinity, current direction ((

**d**,

**e**), respectively), and current intensity ((

**f**,

**g**), respectively).

**Figure 3.**Sand (200 µm diameter) transport rates computed with a 1DV model using the pure sand erosion law, and obtained for different flow intensities (solid red curve). For identical flow intensities and sand diameter, sand transport rates deduced from empirical formulations of Van Rijn [50] (black curve), Engelund and Hansen [51] (blue curve), and Yang [52] (green curve) are illustrated. Empty red circles refer to the modelling results of Dufois and Le Hir [53] representing transport rates of a 200 µm sand under different flow intensities from an advection/diffusion model.

**Figure 4.**Erodimetry experiment conducted on a muddy sample of the BoBCS using the “erodimeter” device [7]. On the graph, the blue and pink curves represent time evolutions of bottom shear stress (N·m

^{−2}) and suspended sediment concentration (SSC in mg·L

^{−1}) during the few minutes of the experiment. The applied shear stress at which erosion begins (around 0.1 N·m

^{−2}) is illustrated by a grey band.

**Figure 5.**Comparisons of erosion fluxes deduced from erodimetry experiments conducted on muddy samples from the BoBCS and those computed from pure mud erosion law (Equation (1)) using similar shear stresses. Different symbols depict different sediment samples and labels refer to the applied shear stresses (τ in N·m

^{−2}). The solid black line represents perfect agreement between modelled and measured fluxes, and the dotted lines delimit the range linked to model overestimation or underestimation by a factor 2.

**Figure 6.**Variations in erosion-related parameters (erodibility parameter E

_{0}(

**a**); critical shear stress τ

_{e}(

**b**); and exponent n (

**c**); used in the erosion law (Equation (1)) as a function of the surficial sediment mud content in the different erosion settings tested.

**Figure 7.**Comparisons of the results of the 3D model obtained from erosion settings S1

_{LIN}and S1

_{EXP}

_{2}, and measurements made by the AWAC acoustic profiler. (

**a**) Shear stresses τ and depth-integrated currents VEL

_{INT}; (

**b**) measured SSC over the entire water column; (

**c**,

**d**) computed SSC for S1

_{LIN}and S1

_{EXP}

_{2}simulations; (

**e**) time series of measured and modelled SSC variations at the level of the AWAC first cell (1.67 m above the bottom); (

**f**,

**g**) changes in the seabed (mud fraction, thickness of the sediment layers) in the two simulations (white dotted lines represent the boundaries of the sediment layers); and (

**h**) integrated amount of SSC advected through the water column (solid lines represent mud and dotted lines represent sand).

**Figure 8.**Comparisons of the results of the 3D model obtained from erosion settings S1

_{EXP}

_{2}, S3

_{EXP}

_{2}and S4

_{EXP}

_{2}, and measurements made with the AWAC acoustic profiler. (

**a**) Shear stresses τ and depth-integrated currents VEL

_{INT}; (

**b**) time series of measured and modelled SSC at the level of the first AWAC cell (1.67 m above the bottom); (

**c**,

**d**) changes in the seabed (mud fraction, thickness of the sediment layers) in S1

_{EXP}

_{2}and S3

_{EXP}

_{2}simulations (the white dotted lines represent the boundaries of the sediment layers); and (

**e**) integrated amount of mud advected through the water column in the S1

_{EXP}

_{2}and S3

_{EXP}

_{2}simulations.

Forcing | Source |
---|---|

Initial & boundary conditions (3D velocities, temperature, salinity) | GLORYS global ocean reanalysis [34] |

Wave (Significant height, peak period, bottom excursion and orbital velocities) | WaveWatch III hindcast [35] |

Meteorological conditions (Atmospheric pressure, wind, temperature, relative humidity, cloud cover) | ARPEGE model [36] |

Tide (14 components) | FES2004 solution [37] |

River discharge (flow and SSC) | Daily runoff data (French freshwater office) |

Erosion Regime | E_{0} (kg·m^{−2}·s^{−1}) | τ_{e} (N·m^{−2}) | n |
---|---|---|---|

Non-cohesive (pure sand) | E_{0,sand} = 5.94 × 10^{−3} | τ_{e,sand} = 0.15 | n_{sand} = 1.5 |

Cohesive (pure mud) | E_{0,mud} = 10^{−5} | τ_{e,mud} = 0.1 | n_{mud} = 1 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mengual, B.; Hir, P.L.; Cayocca, F.; Garlan, T.
Modelling Fine Sediment Dynamics: Towards a Common Erosion Law for Fine Sand, Mud and Mixtures. *Water* **2017**, *9*, 564.
https://doi.org/10.3390/w9080564

**AMA Style**

Mengual B, Hir PL, Cayocca F, Garlan T.
Modelling Fine Sediment Dynamics: Towards a Common Erosion Law for Fine Sand, Mud and Mixtures. *Water*. 2017; 9(8):564.
https://doi.org/10.3390/w9080564

**Chicago/Turabian Style**

Mengual, Baptiste, Pierre Le Hir, Florence Cayocca, and Thierry Garlan.
2017. "Modelling Fine Sediment Dynamics: Towards a Common Erosion Law for Fine Sand, Mud and Mixtures" *Water* 9, no. 8: 564.
https://doi.org/10.3390/w9080564