The modelling of cohesive sediment transport

The management of coastal zones and estuaries requires more and more accurate and detailed knowledge of cohesive sediment (mud) transport processes to cope with various problems (e.g. wetland protection and restoration, maintenance of navigation channels, dredging and dredged material relocation, effects of construction works on siltation and turbidity levels, dispersion of pollutants, etc.). Detailed mathematical models, including full three-dimensional codes, are necessary tools for the development and application of this knowledge. Presently, this is becoming practically feasible in view of the current developments in soft- and hardware. The physical understanding and mathematical description of the processes, however, is still lagging behind, especially with respect to the presence of concentrated near-bed suspension layers, as explained below.

PHYSICAL PROCESSES

The behaviour of mud in coastal zones and estuaries is the result of horizontal and vertical sediment fluxes, i.e. transport, water-bed exchange processes, including the formation of concentrated suspensions, and processes within the bed.

This whole (semi-)cycle is particularly complicated, because it does not necessarily have to be closed and the processes are interdependent. For instance, settling does not always lead to deposition, i.e. when entrainment dominates, and the sediment then remains in suspension; the consolidation processes within the bed are affected by the flocculation processes in the water column and in turn the erosion processes are governed by the consolidation processes and mediated by biota. The actual transport is the net effect of the interactions of different processes, which, until now, have been studied mainly in isolation.

Concentrated benthic suspensions (CBS) and fluid mud

The capacity to transport sediment in suspension by currents and waves is limited by the amount of energy available in the flow. In many cases the suspended matter is not well mixed over the water column and stratification occurs due to settling when the turbulent energy decreases, resulting in a concentrated near bed sediment suspension. They can be maintained by the turbulent energy of the flow when there is equilibrium between the depositional flux and the vertical turbulent transport flux (e.g. Wolanski et al., 1988). These layers are often thin and therefore frequently remain undetected. The concentrations in these layers can be of the order of 5-10 g/l, but also much higher. Because they include a high proportion of the mobile fine sediment, the total amount of sediment that is transported in these sheet flows can still be enormous. Research has revealed that these near-bed layers are the major mechanism of the transport of fine-grained sediments in coastal zones and estuaries (e.g. Faas, 1984; Odd et al., 1992). Therefore, they must be considered in transport models. These near-bed layers will hereafter be called concentrated benthic suspensions (CBS). This terminology is used for the regime where the sediment/water mixture behaves as a fluid. The concept has a broader meaning than that of “fluid mud”, which is associated with high concentration suspensions with a sharp mud/water interface, which can be detected by echo sounding (Parker & Hooper, 1994). The term “fluid mud”, moreover, is often also used for soft consolidating mud layers, which do not behave as a true fluid, and therefore can be misleading.

At concentrations above a critical value of a few 10 g/l, the particle interactions start to modify the properties of the suspension, i.e. the particle interactions lead to hindered settling and at higher concentrations the rheological behaviour of the suspension becomes non-Newtonian (Faas, 1984). To maintain CBS of higher concentrations (above the critical value), much more energy is required as turbulence is being damped by buoyancy effects and by the dissipation of energy due to interparticle collisions and its resulting effect on the floc structure. Therefore, when the energy level is too low to maintain the CBS, this layer will deposit and form a denser fluid mud layer.

The interface between a CBS layer and the water column above can be very distinct, but also very unstable. Helmholtz-Kelvin instabilities may occur (internal waves), leading to interfacial mixing, which contributes to the entrainment of sediment into the water column and of water into the CBS layer (i.e. diluting it) (Scarlatos & Mehta, 1992; Mehta & Srinivas, 1992; Le Hir, 1994; Winterwerp & Kranenburg, 1994). The vertical fluxes involved in this process are not yet fully quantified.

Pressure gradients (e.g. induced by density differences) may cause gravity current flow of CBS. Lower concentration gravity currents are very unstable and generate turbulence. Density currents of dense, visco-plastic fluid mud layers on the other hand are generally laminar.

When the sediment concentration in a dense suspension exceeds a second critical value, the gel point or structural density, the flocs form a continuous network structure and develop effective stresses: a weak saturated soil is formed. At rest this structure will slowly collapse under its own weight and the interparticle bonds will increase in strength: it consolidates. The compression of the bed and the increase of its strength will be determined both by the development of effective stress and by separate time-dependent processes such as creep and thixotropy (Sills, 1994 & 1995). However, under the influence of internal or external forces (e.g. wave action, particularly in storm conditions), the structure of a weakly consolidated bed may fracture and eventually break-up into mobile aggregates and the sediment behaves again as a dense non-Newtonian suspension: the bed is liquefied under shear forces or fluidized under excess pore pressures. The resulting fluid mud layers can easily be eroded, entrained into the water column and transported as gravity currents: this is a second mechanism for the formation of CBS layers (Mehta et al., 1994).

The transition between CBS and the bed is poorly defined at the moment. The erosion happens by different processes depending on the developing rheology of the CBS and on the degree of consolidation. The entrainment of the CBS contrasts with the erosion of the bed by failure of the interparticle bonds of the flocs on the bed.
Flocculation of fine sediment

The formation of concentrated benthic suspensions, their structure and subsequent evolution is governed by the settling flux of suspended sediment towards the bed. This is the product of the mass concentration and the settling velocity. Both the settling velocity and the size of the flocs are determined by a balance between the forces of particle coagulation and disruption. Coagulation is a function of sediment concentration, salinity, organic material content, particle mineralogy and physical processes like Brownian motion, differential settling and turbulent shearing. At low concentrations a small amount of shear will help to bring small flocs together to form larger ones. Higher shear will tend to pull the flocs apart, and which is enhanced by collisions at high concentrations. The flocs most prone to disruption are the largest ones (macro flocs) that have the lowest density, but which have the greatest settling velocity and contain the greatest mass. Their break-up creates a number of smaller micro flocs which, though they together contain the same total mass, individually have higher densities, but lower settling velocities. The aggregation and disruption functions have not been independently determined for natural suspensions. This is a major shortcoming in predictive models which currently only use relationships between settling velocity and concentration. Though these relationships are determined by laboratory measurements in still water on natural samples, and empirically account for mineralogy, salinity and organic content, they are only a first approximation. Because of the interaction between the processes due to concentration and turbulent shear, their combined effect can only be derived from in-situ measurements.

THE MODELLING OF COHESIVE SEDIMENT TRANSPORT

We identify the research needs for cohesive sediment transport processes from the requirements of the transport models which are used by the managing authorities and engineering consultants.

The modelling of cohesive sediment transport requires the numerical solution of the basic conservation equations of mass, momentum and turbulent energy. Several model parameters are supplied by semi-empirical closure equations for input quantities which depend on position and time (Teisson, 1994), amongst which: the settling velocity of flocs, the eddy diffusivity and the sediment flux at the bed (deposition, erosion). For practical applications fine-sediment suspensions can be modelled as a continuous (single) phase fluid (Le Hir, 1994).

In the state-of-the-art engineering models for simulating the transport and fate of CBS layers, a three-layer approach is often applied in which the lower layer is formed by the consolidated bed, the upper layer by the water column and the layer in between by the fluid mud (e.g. Odd & Cooper, 1989; Kusuda & Futawatari, 1992). The exchange of mass between the two upper layers is described through deposition and entrainment formulae, whereas the exchange of momentum is limited to an adaptation of the internal friction coefficient. The exchange between the two lower layers is described in the form of consolidation and erosion formulae. All formulae are highly empirical often with dimensional empirical coefficients, indicating little physical background. This is cumbersome, as the transport and fate of fluid mud layers, predicted with these models, is highly dependent on the actual value of these coefficients. For instance, the settling velocity is taken constant in general, whereas in reality it is strongly affected by the level of turbulence. The entrainment process was studied by Mehta & Srinivas (1993) and Winterwerp & Kranenburg (1994). The latter derived an integral entrainment model by a rigorous integration of the equation for kinetic turbulent energy over the water depth. However, even in that case many empirical coefficients are still required, especially for the fluid mud properties.

In summary, in the state-of-the-art models the interaction between the water movement and the fluid mud layer is neglected, the properties of the sediment in suspension (settling velocity) and of the fluid mud (viscosity and yield strength) are given by highly empirical relationships and the occurrence and effects of CBS is even entirely ignored.

Turbulence modelling of sediment-laden flow

Turbulence models of various levels of sophistication have been developed, from simple algebraic models to Reynolds stress models (Rodi, 1980; Schumann & Gerz, 1995). Turbulence models for estuarine flows should take into account the damping effect of vertical density stratification. Existing models satisfy this requirement in one way or another. Stratification may not only result from the ubiquitous salinity gradients, but also from gradients in the sediment concentration. In the latter case an interaction exists between turbulence and sediment transport. This interaction in near-bed layers has been analysed numerically by Le Hir (1994) (algebraic mixing-length model), Uittenbogaard et al. (1996) (k-epsilon two-equation model) and Teisson et al. (1992) (Reynolds stress model), amongst others. Uittenbogaard (1995) argues that the production of turbulent kinetic energy by internal waves should be included when modelling CBS. Turbulence modelling in the case of fluid mud is complicated by non-Newtonian rheological behaviour and low Reynolds number effects. As sediment transport processes occur on widely different length and time scales, the numerical aspects of turbulence modelling are complex and require special attention.

In summary, turbulence modelling is needed to determine the quantities required by fine-grained sediment transport models, i.e. to estimate the floc settling velocity and deposition rate, entrainment rates of suspended sediment, surface erosion rates, interfacial stability and mixing at lutoclines and the damping of turbulent energy in concentrated suspensions.

Modelling flocculation

A conceptual model of the effect of shear and concentration on median floc settling velocity has been proposed by Dyer (1989). A heuristic formulation for the process, based on laboratory studies, has been advocated by van Leussen (1994), which relates flocculation and break-up to the dissipation rate of turbulent kinetic energy. This formulation has been incorporated into a numerical estuarine model by Malcherek et al. (1994) and appears to be a major improvement. Flocculation models are most easily formulated in a Lagrangian framework. However, for implementation in general sediment transport models an averaged Eulerian formulation is needed, which requires additional turbulence modelling (e.g. Casamitjana & Schladow, 1993).

The various coefficients need to be obtained from field measurements. This approach requires simultaneous measurement of the settling velocities of the various floc size fractions, the suspended sediment concentration, and the characteristics of the turbulence such as the turbulent shear stresses, the turbulence energy and the eddy dissipation rate. An additional factor that has to be quantified is the influence of organic constituents that can act as ‘glue’ in the flocculation process.

A number of new techniques have recently been developed for in-situ measurements of floc size and settling velocity with minimal disruption (Eisma et al., 1990; van Leussen & Cornelisse, 1993; Fennessy et al., 1994; Dyer et al., 1996). These apparatus have been compared with Owen Tubes in an intercomparison experiment during which it became apparent that the Tube performance was dependent on the operator and the sampling protocol, and that the flocs were disrupted by the sampling. The new techniques are able to distinguish individual flocs, and can provide information on the spectra of floc size, settling velocity and effective density under different conditions.

Modelling the interaction between concentrated benthic suspensions and water

A further understanding and considerable improvement of the modelling of CBS layers are only possible by taking the three-dimensional (3D) effects fully into account. Considerable progress has been made recently by Le Hir (1994), Le Normant (1995), Malcherek (1996) and Galland (1996), amongst others, who studied the three-dimensional behaviour of suspended sediment transport and concluded that simulation of observations is indeed only possible by taking the 3D effects into account. Entrainment of CBS by a turbulent flow can be modelled in a similar way as the entrainment of a dense fluid (Winterwerp & Kranenburg, 1996).

Up to now, the important interaction between water movement (turbulence) and sediment could only be accounted for to a minor degree, because the required physical-mathematical formulations are not yet available. In particular the interaction between (turbulent) water movement, damping of turbulence due to buoyancy effects and its influence on the vertical mixing processes, the effects of flocculation and hindered settling on the formation of fluid mud and CBS and their internal properties (non-Newtonian stress-strain relations), the influence of waves on the generation and erosion of fluid mud layers and the stability of fluid mud layers and CBS are still poorly understood.

The effects of surface waves on the fluid mud layer and vice-versa (e.g wave damping) are important phenomena. Their explicit effects, however, lie beyond the scope of this project. But the indirect effects, such as pore pressure build-up, prior to fluidisation, or interfacial mixing has to be accounted for. Also very little is known about the erosion of a consolidated mud bed by breaking waves, which may occur on tidal flats during a storm event following a period of deposition. This also falls outside the scope of this research project.

Modelling the interaction between fluid mud and the bed

For each of the processes which determine the fate of fluid mud or a mud bed, i.e. consolidation, liquefaction and fluidisation, different models have been developed over the past years: a unified theory for settling and consolidation has been developed (Toorman, 1996) and a poro-elastic model is used for fluidisation (Yamamoto et al., 1978). These models allow the computation of the density and/or stress history within the mud layer. In order to erode a mud bed, the bottom shear stress should be larger than the shear strength of the bed surface. Traditionally, erosion of a consolidated bed has been modelled using empirical formulae and the critical erosion stress has been empirically related to the yield stress, a rheological property which is a measure of the degree of bed structure (Gularte et al., 1979). The yield stress, which is difficult to measure (even in the laboratory), is determined as a function of the density. In this way an empirical link has been established between consolidation theory and rheology. A comprehensive theory, which incorporates liquefaction and fluidisation, is still lacking. An attempt in this direction has been made by van Kesteren et al. (1993), who used a geotechnical approach. As a further step in erosion modelling, this approach should be linked with a hydrodynamic approach. The common factor in the bed history is the structure, which should be parameterized. A possible approach is that of the structural kinetics theory, which has been applied successfully to the modelling of the thixotropic behaviour of mud (Toorman, 1995) and the formation of flocs (Winterwerp, 1996).

Parameterization

Since the processes are very complex, the process models and modules generally are parameterizations of the elaborated mathematical-physical descriptions. This is still a common practice to make the computer codes cost and time efficient in order to be applicable to practical managerial problems.
Measuring methodology

Validation of numerical models requires field and laboratory data. Measuring campaigns have been hampered by a lack of understanding of the physical processes and an isolated approach on the research of mud dynamics. For instance, it may appear necessary to measure turbulence properties simultaneously and some time prior to the settling velocity measurement to obtain reliable results. An improvement of measuring methodologies for cohesive sediments will be one of the spin-offs of the proposed research. Vice versa, insight in measuring methodologies and techniques will prevent the development of process formulations containing unmeasurable quantities.
Conclusions

In summary, it is hypothesized that the transport, fate and subsequent behaviour of cohesive sediment suspensions in many coastal and estuarine environments are largely governed by concentrated near bed suspensions and the structure of the aggregates involved. Classical fluid mud appearances are one of types of CBS identified in the literature. The state-of-the-art review leads to the identification of various gaps in our knowledge, of which the major ones can be summarized as follows:

  • The relationship between floc properties, such as strength, and settling velocity in turbulent flow is poorly understood. A tractable model of history effects on flocculation and floc break-up in turbulent flow is not available.
  • An entrainment model for fluid mud should account for all flow conditions depending on the degree of turbulence damping, i.e. the transition from high to low Reynolds number turbulent flow, down to laminar flow, where the rheological properties of become dominant. Such turbulence models are not available.
  • A physically based bed erosion model should take into account the interaction between turbulence and soil-mechanical properties of the bed. A general model is required which allows for the computation of the strength history of the bed, including the effects of consolidation, liquefaction and fluidisation.

Leave a Reply

Your email address will not be published. Required fields are marked *