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A process-based sand-mud model
Fine SedimentDynamicsin the Marine Environment J.C. Winterwerpand C. Kranenburg(Editors) 9 2002 Elsevier Science B.V. All rights reserved.
577
A process-based sand-mud model M. van Ledden a aFaculty of Civil Engineering and Geosciences, Delft University of Technology, Stevinweg 1, P.O. BOX 5048, NL-2600 GA DELFT, The Netherlands.
Predicting the bed composition in estuaries and tidal lagoons is of great practical importance. At present however, horizontal and vertical bed composition variations are often neglected in sediment transport and morphological models. In this paper a process-based sand-mud model is proposed, the model behaviour is analysed and model results are compared to field measurements. In general, it can be concluded that with such a processbased model, governing time scales and dimensionless parameters can be derived which can significantly increase the physical understanding. Furthermore, an expression is derived for the equilibrium mud content at the bed surface when both deposition and erosion occur during the tidal period. In this expression, the settling velocity for mud, the mud concentration and the erosion rate form an important dimensionless parameter. For low parameter values (< 10), sharp transitions are to be expected between areas with a very low mud content and areas in which the mud content can vary between 0 and 100%. The existence of a sharp transition is confirmed by field data from the Westerschelde estuary (The Netherlands). Finally, model results suggest that a local hydrodynamic parameter is not very useful for predicting the mud content in areas exposed to relatively low bed shear stress. Apart from the local hydrodynamics, the local mud concentration, settling velocity, mixing properties of the bed and the sample depth detemaine the local mud content. KEY WORDS sediment distribution; sand; mud; bed composition; numerical analysis; modelling,
1. INTRODUCTION Predicting the distribution of non-cohesive and cohesive sediments in estuaries and tidal lagoons, which leads to zonation of sand, mud and mixed deposits, is of great practical importance. It determines which areas collect nutrients and pollutants, as these tend to adhere to, or be part of, cohesive sediments and which areas form the right habitat for flora and fauna. Based on the rule-of-thumb "the calmer the water, the finer the bed sediment", a local hydrodynamic parameter is often used for predicting the mud content at a certain location (WL ] Delft Hydraulics, 1998). However, a useful relationship between the mud content and a local hydrodynamic parameter is not known at present. Probably, a more process-based approach is a successful way for increasing the understanding and predictability of the sandmud distribution.
578 In the present process-based sand models (Van Rijn, 1993) and mud models (Teisson, 1997) one representative grain size is used for calculating sediment transport and morphological changes, and the bed composition is often not taken into account. The goal of this paper is an attempt to bridge the gap between sand and mud models by proposing a process-based sand-mud model. First, knowledge of the incorporated processes is discussed (section 2). Thereafter, the process equations are presented (section 3) and the model behaviour is analysed (section 4). Finally, model results are compared to field measurements and the usefulness of a local hydrodynamic parameter for predicting mud content is discussed (section 5).
2. PROCESS KNOWLEDGE
2.1. Incorporated processes In estuaries and tidal lagoons, bed sediments are continuously picked up, transported and deposited. Due to different sediment properties, the flux and direction of transport of noncohesive and cohesive sediments are often not equal (Aubrey, 1986; Dronkers, 1984). The result is a strongly varying mud content in the bed in both horizontal and vertical directions. A large number of sediment processes are involved in the distribution of sediments. Erosion and deposition are the most obvious processes, but consolidation, flocculation and physical and biological mixing within the bed may also affect the bed composition. In an ideal processbased model, all these processes should be taken into account for predicting the bed composition in time and space. However, the present knowledge of these processes for mixtures of non-cohesive and cohesive sediments is limited and constructing a process-based sand-mud model with all these processes is far beyond the present state-of-the-art. Simplifications are therefore inevitable and consequently, the validity of the presented sandmud model herein is limited. Moreover, the limited availability of data is another problem for verification of the model results. In our modelling approach, the tide is assumed to be the dominant forcing for water motion. Short waves and density currents are neglected. Therefore, a depth-averaged modelling approach is assumed to be sufficient. Furthermore, the mud concentration is assumed to be low (< a few hundred mg/1). Flocculation and consolidation are not taken into account. For the deposition, erosion and mixing processes, process equations are needed to describe the behaviour of non-cohesive and cohesive sediments. Recently, laboratory experiments with mixtures of sand and mud have been reported in the literature. The observed behaviour of the different processes for sand-mud mixtures is discussed separately in the following subparagraphs in order to support the process equations used in the model. These equations are presented in section 3.
2.2. Deposition Deposition experiments showed that in the case of sediment concentrations below the gel point, non-cohesive sand particles and cohesive mud flocs behave more or less independently during deposition (Toffs et al., 1996). Around the gel point, however, mud flocs form a network in the water column and sand particles are trapped. Typical gel point concentrations are in the range of 30 - 180 g/1 (Winterwerp, 1999). It seems reasonable that for lower concentrations the separate deposition formulae for mud and sand are valid and can be applied within our model.
579 Table 1 Critical mud content and clay content for the experiments Experiment Clay type Critical mud content Critical clay content Toffs (1995) Kaolinite 4% 3% Torfs (1995) Montmorillonite 13% 4% Panagiotopoul0s eta 1. (1997) . . . . . . Combwich .........................30% ................................ 10.8% 2.3. Erosion Recently, erosion experiments were carried out with mixtures of non-cohesive and cohesive sediments (Torfs, 1995; Panagiotopoulos et al., 1997). By adding mud to a sand bed, Torfs (1995) observed that the non-cohesive behaviour of the bed was increasingly suppressed. Above a critical mud content (% < 0.063 mm), the bed behaved cohesively. However, the critical mud content was not the same for different sand-mud mixtures. Panagiotopoulos et al. (1997) observed a critical clay content (% < 0.002 mm) of 10.8%. The experimental results are summarised in Table 1. The critical clay content for the experiments of Torfs (1995) was calculated by a re-analysis of these experiments. Despite the great difference in the critical mud content, the critical clay content is about the same for the experiments of Torfs (1995). Previously, Dyer (1986) and Raudkivi (1990) stressed the role of clay particles in the erosional behaviour of mixed beds. Dyer (1986) gave a transition range for non-cohesive to cohesive behaviour from 5 - 10% clay content by dry weight. This range agrees with the experimental results. Torfs (1995) also measured the critical shear stress for erosion of sand-mud mixtures with a varying mud content. The critical shear stress is plotted in Figure 1, both as a function of the mud content and the clay content. The clay content was calculated by a re-analysis of the experimental data. The critical shear stress differs significantly for the two sand-mud mixtures at the same mud content, but is about the same at the same clay content. The observations of the transition between non-cohesive and cohesive behaviour as well as the critical erosion shear stress suggest that the erosion process is dominated by the clay content and not the mud content itself.
3.5
9
3 ~2.5
0 O
2
+
.
.
.
+ o , []
.
.
1
. . . . . . . . . .
l
.........
t_ . . . . . . . .
KaoliniteMud Kaolinite Clay MontmorilloniteMud MontmorilloniteClay
+
0[]+
1.5
.,,,~ I,,-i
w0.5 0 0
;
1'0 1'5 2'0 2; 3'0 Mud content [%], Clay content 1%1
Figure 1. Measured critical bed shear stress vs. mud and clay content.
580 From a modelling point of view, the traditional erosion formulae for sand and mud alone must be adapted because of this behaviour. Depending on the clay content in the bed, two regimes must be distinguished: a non-cohesive and a cohesive regime. The transition between both regimes is given by a clay content of 5 - 10%. A critical mud content can also be used, because bed composition measurements suggest a strong correlation between the clay content and silt content (% 0.002 - 0.063 mm) within the bed (Van Ledden, 2000). However, it must be kept in mind that a critical mud content seems to be a site specific value, depending on the clay/silt ratio.
2.4 Mixing Mixing within the bed can have a physical as well as a biological origin. Physical mixing in a sand bed occurs by small-scale bed features and it is generally assumed that the intensity of mixing increases with increasing velocity and decreases further into the bed (Armanini, 1995). Mixing experiments for sediment beds of non-cohesive and cohesive sediments have not been undertaken to the author's knowledge. However, it can be argued that physical mixing must also be dependent on the mud content within the bed. Erosion experiments showed that small-scale bed features are suppressed when the mud content increases (Torfs, 1995). Therefore, physical mixing probably decreases with increasing mud content. Two types of biological mixing are often distinguished (Boudreau, 1997): local mixing and non-local mixing. Local mixing is caused by organisms which move through the bed and therefore mix the sediment particles. Non-local mixing is caused by organisms that transport a specific sediment fraction from one level to another in the bed. An example is the Heteromastus filiformis, which transports only fines from ten centimetres depth to the bedwater interface (Herman, 2000). Physical mixing and local, biological mixing can be modelled as a diffusion process. For both mixing mechanisms, appropriate mixing coefficients must be defined. Non-local mixing by organisms causes sources and sinks of specific sediment fractions at certain depth within the bed. In our modelling approach, non-local mixing is neglected.
a. Local behaviour
b. Numerical model
Wiae Erosion
Deposition
U = 0 sin(cot) h Cmud, Csand
Cout
l i Zb "
r
Mixing
Pm,O
E,~ ,
~D, ~
J=O
z,I, iZI;IIII IIIiZII II;I ZZZIZ i
•
II PmJ 9 .If al 9 T ,,......................."........................................................ .V.........I J = J
Figure 2. Model set-up.
581 3. MODEL EQUATIONS 3.1. Local behaviour In this paper, only the so-called 'local behaviour' is discussed. This situation is sketched in Figure 2a. It is assumed that the behaviour of the sand and mud concentration in the water column, the bed level and the bed composition are dominated by deposition, erosion and mixing within the bed only. Non-local sediment transport processes (advection, diffusion) are assumed to be of minor importance and are neglected in the local model analysis. Because of this local approach, bed load transport is not taken into account. For sand in suspension, the local approach seems to be valid, because its settling time is generally much smaller than the tidal period. For mud however, the settling time is often of the order of the tidal period and a local approach for mud is not realistic. Therefore, horizontal advection and diffusion are also taken into account for mud in a parameterised way (see section 3.3). A process-based model basically consists of three parts: a water motion module, a sediment transport module and bed module. For each module, the process equations are discussed separately in the following subsections. Because of the local approach, the derivatives in the horizontal directions are neglected within the governing equations. 3.2. Water motion The water depth h is taken constant and the varying depth-averaged flow velocity U during the tide is given as a function of time (Figure 2b): U - (J sin(cot)
(1)
where U is the amplitude of the velocity, co the angular frequency of the tidal period and t time. The bed shear stress rb is given by: g 2 ~ =p-Uu
(2)
where p is the water density, g the gravitational acceleration and C the Ch6zy-coefficient. The Ch6zy-coefficient C strongly depends on the bed features and composition. For sandy environments, typical values range from 40 to 60 ml/2/s. For muddy environments, the Ch6zycoefficient is often much larger, ranging from 60 to 100 mVZ/s (Van Rijn, 1993). Although the roughness strongly depends on the bed composition, the Ch6zy-coefficient is assumed to be constant at present (C = 60 m~/Z/s). 3.3. Sediment transport For sediment transport, two sediment fractions are taken into account: a sand fraction (subscript s) and a mud fraction (subscript m). For the description of the sand and mud transport in suspension, the depth-averaged advection-diffusion equation is used. For a local approach, the equation reduces to:
582 ~c, = E , - D, c~
(3)
where ci is the depth-averaged volumetric concentration of fraction i, Di the deposition rate of fraction i and E, the erosion rate of fraction i. The terms E~ and D~ on the right hand side give the net exchange of sediment fraction i between the bed and the water column: the downward deposition flux D~ and the upward erosion flux Et (Figure 2b). As was discussed in section 3.1, the above equation can be applied for the sand fraction, but is not realistic for the mud fraction. In a next section, it will be shown that the local model behaviour shows trivial or non-realistic solutions by applying (3) for the mud fraction. Therefore, an extra transport term is introduced to account for horizontal advection and diffusion. The mud concentration equation becomes: c~c,,
~=Em-D,,,+k,,(c,,u,-Cm) dt
(4)
where Cout is the mud concentration outside and km is a transport coefficient. The last term on the right hand side in (4) is a transport term by which the mud concentration Cm is affected by the concentration outside the model Cout by a transport coefficient km (Figure 2b). When the inner concentration is lower than the outside concentration, it is assumed that by advective and diffusive transport mud is imported from outside and vice versa. Exchange of sediment between the bed and the water column occurs by deposition to and erosion from the so-called 'exchange layer'. The exchange layer is the layer at the bed surface with index j = 0 (Figure 2b). As was concluded from the laboratory experiments, the erosion and deposition flux for sand and mud must be prescribed for two regimes: a non-cohesive and a cohesive regime (see section 2.3). A critical mud content within the bed can be used for the transition between both regimes. In this model, the critical mud content for the transition between both regimes is denoted as Pm,r and the mud content in the exchange layer, denoted as Pm, O, determines the erosion regime (Figure 2b). The erosion and deposition terms for both regimes are discussed successively below. Non-cohesive regime (Pm,O
E,.-D,. = ~ . , ( c ..... - c , )
(5)
where ~ is a form coefficient, ws the settling velocity of the sand fraction and Ce,s the depthaveraged equilibrium concentration of the sand fraction. The form coefficient y takes into account the vertical distribution of the velocity and concentration profile (Galappatti, 1983). The depth-averaged equilibrium concentration (Ce,s) is given by the Engelund-Hansen transport formula:
583 0.05
c.... =
U4
2Acd, 3 h
(6)
where A is the specific gravity and ds the sand diameter. In contrast to other sediment transport formulae, no threshold for the initiation of motion is included in the equilibrium concentration of Engelund-Hansen. A sediment transport formula with a threshold for sand can also be applied for the depth-averaged equilibrium concentration, e.g. Van Rijn's transport formula. However, it is not expected that including a threshold will significantly change the observed behaviour, discussed in the next paragraphs. Therefore, the Engelund-Hansen transport formula is considered to be sufficient as a first step. Further research is recommended in order to study the erosion process of sand-mud mixtures in detail and propose more realistic erosion formulations. For the erosion process of mud, the erosion formula of Partheniades is assumed to be valid. Because the mud content can vary, the mud content in the exchange layer is added to the traditional formula. During erosion experiments in the non-cohesive regime, clay and silt particles are easily washed out from the top layer (Murray, 1977; Torfs, 1995). Thus, the critical erosion shear stress in this regime (Te,nc) can be very low. For mud deposition the wellknown deposition formula of Krone is applied. The exchange of mud is given by:
m: m0 F ll'r
[1-
"]
(7)
where Pm, o is the mud content in the exchange layer, M the erosion rate, re,nc the critical erosion shear stress for non-cohesive regime, H the Heavyside function, Wm the settling velocity of mud and rd the critical deposition shear stress.
Cohesive regime (Pm,O>Pm, crit) In the cohesive regime, the erosional behaviour of sand and mud is not independent. It is assumed that sand and mud particles are eroded in the same way, but once in the water column behave independently again. The Partheniades formula seems the most obvious formula for describing the erosion of mud and sand particles. This formula is adapted by introducing a parameter for the sand and mud content in the exchange layer. The deposition formulations for sand and mud remain the same as for the non-cohesive regime. The exchange of sand and mud in the cohesive regime are respectively given by: 9
(8)
(9)
584 where p~,o is the sand content in the exchange layer and Ze.cis the critical erosion shear stress for the cohesive regime. 3.4. Bed level and composition The bed level change is determined by the flux of sediment to and from the bed from both sand and mud. The bed density is assumed to be constant in time and space. The bed level change is given by:
3zh = P" [D,, - Em + D,. - E,. ]
Ph
'
(10)
"
where Z b is the bed level, p~ the sediment density and Pb the bed density. Armanini (1995) proposed a continuous formulation for modelling the bed composition. The local change in sand content in a Langrangian co-ordinate system can be given by an advection-diffusion equation:
Op, + - - P, -s. c? Oz -ff "
=0
(11)
where p~ is the sand content, z the distance from the bed surface (positive downwards) and ez a mixing coefficient within the bed. It is important to note that in (11) the origin of the vertical coordinate z is at the bed surface and z is positive in downward direction (Figure 2b). Conversely, the bed level rate OZb/Ot is positive if the bed level rises. The first term in (11) represents the local change in sand content. The net flux of sand by bed level variations is given by the second term. The third term is the net flux by small-scale variations in the bed level, e.g. tipples. Armanini (1995) suggested that the mixing coefficient ez in (11) decreases with increasing distance below the bed surface, because the influence of the small-scale variations decreases. The mixing coefficient was assumed to be an exponentially decreasing function: ot I
s. = ~,0 e
u,
(12)
where ez,0 is the mixing coefficient at the bed surface, ds the sand grain size and al a coefficient. The mixing coefficient ez, O was assumed to be proportional to the friction velocity u. and the sand grain size ds:
4.0 = a0u, d,
(13)
where ao is a coefficient. The advection-diffusion equation (11) needs two boundary conditions. At the bed surface (z = 0) the sediment flux is prescribed. This boundary condition reads:
585
1
(14,
z=0
Deep below the bed surface the diffusive flux is assumed to be zero. The boundary condition is given by:
+
c2 c~ z=-
=0
(15)
For the mud content within the bed, the same equation can be solved. In our model however, only two sediment fractions are applied. Therefore, the mud content can be easily expressed in terms of the sand content: (16)
p,, = 1-p.,.
4. MODEL ANALYSIS 4.1. Scale parameters The model described in section 3 consists of four dependent variables: two sediment concentrations (cs) and (Cm) in the water column, the bed level (Zb) and the mud content within the bed (Pm)- For analysing the behaviour of this model, it is important to derive the governing external and internal scale parameters, which are elaborated below. The external forcing is characterised by two parameters: the amplitude of the bed shear stress gives the level of the forcing ( ?h ) and the time variation of the forcing is characterised by the tidal period (Ttide). Within the model, three critical shear stress parameters must be defined: rd for mud deposition, re, nr for non-cohesive mud erosion and re,r for erosion of sand and mud in the cohesive regime. These parameters can be made dimensionless by using the maximum bed shear stress. In general, the critical shear stresses are divided by the level of forcing. For this purpose, the reciprocal term is used, because the level of forcing can be zero: r~ = ~-~ h
~,,c= ~b
~,c = ~-~b
By using these scale parameters, three situations can be distinguished. These situations are shown in Figure 3. The bed shear stress is plotted during half a tidal period for the three situations. Also the three critical shear stresses are given with horizontal lines. The symbols denote whether or not mud erosion or deposition takes place. Sand erosion and deposition can always occur, because no threshold shear stress is included. The model results of the three situations are discussed in the next subsections. Besides the extemal scale parameters, intemal time scales also govem the model behaviour. These time scales can be derived from the underlying equations. The time scales are used for explaining the model behaviour and given in the next subsections.
586 o * []
Mud deposition Non-cohesive erosion Cohesive erosion Situation 3
!,oc m ... .'C.e, . nf'............~....... ~......y..........................
0
0.1
0.2
0.3
0.4
0.5
Time t/T [-] Figure 3. Overview of situations.
4.2. Situation 1
In situation 1 erosion of mud from a non-cohesive bed as well as erosion of sand and mud from a cohesive bed are not possible, because the maximum bed shear stress is always lower than the critical bed shear stress for erosion during the tidal period. However, deposition of mud is possible, because the bed shear stress is lower around slack water than the critical bed shear stress for deposition. Sand deposition is always possible, because the deposition term does not depend on the local hydrodynamics. Erosion of sand from the non-cohesive bed can occur, because the equilibrium concentration formulation does not have a threshold shear stress (Figure 3). 9
Mud concentration
For the mud concentration, only two time scales are important, the deposition time scale (h/wm) and the transport time scale (h/kin), because erosion does not play a role in this situation. For a constant water depth, the behaviour of the mud concentration is governed by the dimensionless ratio k,,,/Wm. In Figure 4, results of the mud concentration are given for different values of k,,,/Wm. The mud concentration is made dimensionless by using the outside concentration Cout. It can be observed that the (tidally averaged) mud concentration in the water column tends to a finite value for all ratios of k,/wm. When the ratio k,,,/w,,, is higher, the equilibrium value is also higher. The equilibrium mud concentration is always lower than the concentration outside Cont. The (tidally averaged) equilibrium mud concentration can be explained from a balance between the inward horizontal flux due to transport and an outward downward flux due to deposition over one tidal period. The equilibrium concentration can be solved analytically when the adaptation time for settling (h/w,,) is at least in the order of the tidal period.
587 =
~
~1.5~
/ ~
k/Wm=5-O
/
k/Wm= 1.0
t'
'
--0.8
k/Wm-O.:
...........................
1 0.4 0.2
~r/" rT~
O
-'
Wzb/Wm =zb-
1.0
_, ......... Tzb/Tm= 5.0
i
5'0 100 150 200 2 3 5 Time t/T [-] Time t/T [-] Figure 4. Mud concentration for different Figure 5. Mud content in exchange layer for values of k,,/wm, different values of Tzb/Tm. In this case, a more or less constant concentration can be assumed during the tidal period and the equilibrium concentration Cm,eq is given by:
c ....q = - - C1 o u
Win_
km a o + 1
,
aD =
i( 1 -- ~1u ) ( 2
arcsin
(~//
+ / X/~d- 1 1
(17)
ka/Xd ; )
where aD is a deposition coefficient and Cm,eq the equilibrium mud concentration. The coefficient aD is solved analytically and is a function of the dimensionless shear stress rd" By using this expression, it can be seen that for the equilibrium mud concentration in the water column three situations exist: 1. km << Wm In this case, the supply of mud from outside is very low with respect to the deposition of mud. Therefore, the equilibrium mud concentration is very small, for reaching a balance between the deposition and transport flux during the tide. 2. km~wm
Because the settling velocity Wm and the transport coefficient km have the same order of magnitude, the flux by deposition and transport have the same order of magnitude for an equilibrium mud concentration between 0 and Cout. 3. km
>>
Wm
In this case, the supply of mud from outside is very fast with respect to the deposition of mud. Therefore, the equilibrium concentration is about equal to the concentration outside Cout. From the equilibrium mud concentration equation (17), it can easily be seen that when no transport (km = O) is included, the model gives a trivial solution. The only flux is caused by
588 deposition and the final mud concentration is zero. The total deposition flux depends on the initial mud concentration. 9
Sand concentration
The governing time scale for the sand concentration is h/ws. This time scale is in general much smaller than the tidal period. For the non-cohesive regime, the sand concentration shows a small phase shift with respect to the equilibrium concentration Ce,s because of its adaptation time (h/ws). For the cohesive regime, no erosion of sand can occur. If the initial sand concentration is non-zero, the sand concentration will drop very fast to zero and does not change anymore. 9 Bed level Due to the continuous deposition of mud particles and no erosion, the bed level is continuously increasing due to accretion. The net sand exchange during one tidal period is equal to zero. In case of a non-cohesive bed, the speed of the bed level rise is thus determined by the equilibrium mud concentration and the settling velocity. The bed level rise time scale is given by: T . Pe . zh
8 .
Ps (Zl)Wmem,eq
Ph . 811 Ps Cout CLDWm
+ ~ 11
km
(18)
where Tzb is the time scale of bed level rise and ~ the total thickness of the bed layer. In (18) the parameter 3 is the total thickness of the bed layer, in which the bed composition changes. It is obvious that for an increasing transport coefficient (k,,), settling velocity (win) and outside concentration (Co,a), the time scale of bed level rise (Tzb) decreases. 9
Bed composition
Because of the continuous deposition of mud, the equilibrium composition consists of a mud bed without sand. However, for the development of the bed composition two time scales are important: the time scale of bed level rise (18) and the mixing time scale (Tin). The mixing time scale is given by: 82
Tm= ~
(19)
~z,0
where Tm is mixing time scale and ez, Othe mixing coefficient at the bed surface. In Figure 5 the mud content in the exchange layer is given for different ratios of Tzb/Tm. In Figure 6 and Figure 7 the bed composition profile is given in a fixed co-ordinate system at different times for Tzb/Tm = 0.2 and 5.0 respectively. The depth within the bed is made dimensionless with the constant thickness of the bed layer 3. Due to the fixed co-ordinate system and the continuous bed level rise in this situation, the bed layer in which the bed composition is computed, shifts upward in time (Figure 6 and Figure 7).
589 2.5
,',1.5
25II~,,
+ t/T=0 - * - t/T = 40 t / T = 80 - + - t / T = 120 t / T = 160 t/T = 200
t~----0- ..... }'
"'--'*-- f i T = 4 0 2 +t/T=80
'
'
I ]
--~ t/T= 120 I "7"
,,4
---a- t / T = 160 I
~.2
~_~.5 ..................... __~ tcr = 2ool,~ ~"
~
~~.-~
~
0.5 :e
0.5
0.2
0.4
0.6
133
0.8
0'.2
" 014
016
018
Mud content [-]
Mud content [-] Figure 6. Bed composition for Tzb/Tm = 5.0.
Figure 7. Bed composition for Tzb/Tm = 0.2.
It can be observed that when the ratio of bed level rise time scale to mixing time scale is relatively large (Zzb/Tm = 5 . 0 ) , the time needed to reach a 100% muddy exchange layer is relatively long (Figure 5). Moreover, the bed composition profile with depth is more or less constant (Figure 6). For the opposite situation (Tzb/Tm = 0.2) a 100% muddy top layer is reached very fast (Figure 5) and the bed composition profile shows a stratified profile (Figure 7). The explanation for this behaviour is that when the net downward flux due to deposition is much larger than the downward transport by mixing, all mud deposited on the exchange layer is conserved in the upper part of the bed. For this situation, the bed composition profile is strongly stratified and a 100% muddy exchange layer is reached very fast. 4.3. S i t u a t i o n 2
In situation 2 erosion of sand and mud from a cohesive bed is not possible. Thus, the situation for the cohesive regime is equal to situation 1. However, mud and sand erosion from a non-cohesive bed is possible (Figure 3). The behaviour of mud concentration, bed level and bed composition in this regime is discussed below. 9
Mud concentration
In the non-cohesive regime the mud concentration in the water column always tends to the mud concentration outside Com. An equilibrium is reached when the outward deposition flux is equal to the inward erosion and transport flux during a tidal period. When it is assumed that the concentration in the water column and the mud content in the top layer are more or less constant during the tidal period in the equilibrium situation, the equilibrium mud concentration is given by:
m
WmO~D "-}-km
'
7r,
7r,
,no
where aE, NC is the erosion coefficient for non-cohesive regime and Pm, eq, O the equilibrium mud content in exchange layer. The coefficient aE, NC is solved analytically and is a function of the
590 dimensionless critical shear stress for erosion in the non-cohesive regime (rc,,c)- Thus, compared to the equilibrium mud concentration in situation 1 (17), an extra term is introduced due to erosion in (20). However, the equilibrium mud c o n t e n t (Pm, eq,O) is not known yet. In the discussion of the bed level and composition below, it will be shown that the equilibrium mud content is equal to:
Pm,eq,O =
(21)
WmCm'eq al) M O{E,N(.
By using expression (21) in equation (20), the final equilibrium mud concentration in the water column is: WmCm,eq
aD
M
O[,E,NC
M a ~.,Nc + k c o. , c .... q =
(22)
= c,,,,,
Wma D + k
Thus, the equilibrium concentration in this situation is always equal to the concentration outside. However, this is only valid for the non-cohesive regime. When the cohesive regime is reached, only transport and deposition (see situation 1) determine the equilibrium. In this case, the equilibrium concentration of situation 1 is reached. 9
Bed level and composition
For the non-cohesive regime, equilibrium is reached when the total flux of sand and mud to and from the bed are equal during one tidal period. When this situation is reached, the bed level remains constant. At this equilibrium, the mud content in the exchange layer for the noncohesive regime has also an equilibrium value. An equilibrium exists because of a balance between the erosion and deposition flux for mud during one tidal period. Assuming a more or less constant mud concentration and mud content during one tidal period, the equilibrium mud content can be expressed by: 1 7'
WmCm,eq ~
![1- Ya]/-/[1- Ya]dt WmC m : ~eq
Pm,eq,O :
_ ~_D
(23)
M lr 0
As already mentioned, the equilibrium mud concentration is about equal to the concentration outside. Thus, the equilibrium mud content in the exchange layer is: _ WmCout 0{'1)
P ....q,0 - - -
M
aE,NC
(24)
591 In the final expression for the equilibrium mud content (24), the deposition flux during one tide is recognized in the numerator, the erosion flux for a 100% mud bed (Pmud, O = 1.0) during one tide in the denominator. It is important to note that when the equilibrium mud content is larger than the critical mud content for the transition between non-cohesive and cohesive behaviour (Pm, eq,O >Pmud, crit), the non-cohesive equilibrium mud content will never be reached. When starting with a noncohesive bed, the mud content in the top layer tends to the equilibrium in the cohesive regime. When reaching the critical mud content the behaviour changes into a cohesive behaviour. When no erosion of the cohesive bed is possible, the final situation is a total mud bed. In fact, the same situation as in situation 1 is reached with only deposition and no erosion. 4.4. Situation 3
In situation 3, the model behaviour is strongly comparable to situation 2. The only extra possibility is erosion in the cohesive regime (Figure 3). Therefore, two equilibria exist for the mud content in the exchange layer (24), indicated with indices n c and c: a non-cohesive equilibrium Pm, O,eq, nc and a cohesive equilibrium pm, o,eq, c. In analogy to the non-cohesive equilibrium, the cohesive equilibrium is also defined by (24), because the only difference in the erosion and deposition formulae is the critical erosion shear stress. For the cohesive equilibrium (Pm, O,eq,c) the dimensionless critical shear stress for erosion in the cohesive regime (rr must be used. In situation 3, the final equilibrium depends on the values of the noncohesive and cohesive equilibrium compared to the value of the critical mud content Three different situations can be distinghuished and are summarised in Table 2.
(Pmud,~r,).
The final equilibrium for these situations can be explained as follows: 9 Situation 3.1: the non-cohesive equilibrium is never reached, because the non-cohesive equilibrium mud content is higher than the critical mud content. When the mud content is lower than the critical mud content, the mud content increases to reach the non-cohesive equilibrium value. When the critical mud content is reached, the equilibrium changes to the cohesive equilibrium and this is also the final equilibrium. 9 Situation 3.2: this situation is comparable to situation 3.1, but in this case the cohesive equilibrium is never reached because its value is in the non-cohesive range. The final equilibrium is the non-cohesive equilibrium. 9 Situation 3.3: both equilibria can exist and the final situation depends on the initial mud content in the exchange layer. If the initial mud content is higher than the critical mud content, the cohesive equilibrium is reached. If the initial mud content is lower than the critical mud content, the non-cohesive equilibrium is reached. Table 2 Overview of equilibria in situation 3 Situation Cohesive equilibrium Non-cohesive equilibrium .
.
.
.
.
............
3.1 3.2 3.3
(Pm,0,eq, c)
(Pm,0,eq,nc)
> Pmud,crit < Pmud,crit .> pmudrcrit
> Pmud,crit < Pmud,crit
_
< pmudrcrit
Final equilibrium Cohesive Non-cohesive Depends on initial conditions
592
~ 0.8
~
..... -=_-. . . . . ~. . . . . , . _ _ . _ . . . . _ _ . _ . . _ . _ - - , .... \ t A ~\ A , 0 i A ~-~ i
WmCout/lVI
0 ", 0 i O' 0\
0.6
100
",,. .\,,.. "
o',,,,
0.4
:
"".,,
"-.
0"
2
(
'X WmCoutflVI= 10
~0.2
O0
'-
Bed shear stress [-] Figure 8. Equilibrium mud content.
5. COMPARISON TO FIELD M E A S U R E M E N T S Correlations between the maximum bed shear stress and the mud content at a certain location of the Westerschelde estuary (The Netherlands) showed the following pattern (WL [ Delft Hydraulics, 1998): for high bed shear stress, the mud content was always low (< 10%). For low bed shear stress, the mud content varied between 0 and 100% and no useful relationship could be determined. Often, a sharp transition between these regimes was observed and a critical transition value could be defined. However, physical explanations for this typical pattern are not available at present. In the previous chapter an equilibrium mud content in the exchange layer was defined (24). Similarly to the above described correlation, this equilibrium mud content can be given as a function of the maximum bed shear stress during the tide (Figure 8). The equilibrium mud content is given for different values of WmCouc/M.The maximum bed shear stress is made dimensionless by the critical erosion shear stress. It is assumed that the critical erosion shear stress for the non-cohesive regime is equal to the critical erosion shear stress for the cohesive regime. The critical shear stress for deposition must also be known to calculate the coefficient aD. The equilibrium mud content is given for two ratios of Ze/Zd = 5 (lines) and re~re= 10 (open symbols). For small values (order 1) of w,,,Cou/M a sharp transition exists between a full mud bed and a full sand bed ( Figure 8). Beyond a dimensionless bed shear stress of about 3, the mud content is less than 10%, while below a dimensionless bed shear stress of about 2, the equilibrium value is about 100%. With increasing values of w,,,Cou/M the transition becomes less sharp. For the Westerchelde area, typical values for the parameters are w,, = 5'10 "4 m/s, M = 1* 10 8 m/s and Cout= 4" 10 -5 ( ~ 100 mg/1). The parameter-value WmCout/Mhas a value of about 2. Thus, a sharp transition has to be expected in this area. This is confirmed by the observed correlation between the bed shear stress and the mud content (WL [Delft Hydraulics, 1998). It is important to note that when the critical shear stress for erosion in the non-cohesive and cohesive regime are not equal, the non-cohesive and cohesive equilibrium mud content are
593 not equal (see section 4.3). In this case, the equilibrium mud content as a function of the maximum bed shear stress is discontinuous. The discontinuity arises at the critical mud content between the non-cohesive and cohesive regime. The presented sand-mud model also implies some explanation for the observed large scatter in areas with a low bed shear stress. First, the relatively long time scale for reaching the equilibrium mud content suggests that the measured mud content at the sample points is probably not at its equilibrium value, but evolves towards an equilibrium value. The time scale for reaching the equilibrium mud content can be estimated by using (19). The parameters for the time scale Tzb are assumed to be as follows" ~ = 0.10 m, Pb = 1200 kg/m 3, ps = 2650 kg/m 3, Wm= 5 " 1 0 -4 m / s , Cout-- 4"10 -5 (~ 100 mg/1). Thus, the time scale for reaching a total mud bed is about 50 days, about two months. When mixing within the bed is included, the time scale becomes even much larger. Thus, the measured mud content at a single site not only depends on the actual hydrodynamic conditions, but also on the hydrodynamic conditions in last months. The equilibrium mud content can be seen as the upper limit for the mud content in a certain area. Second, scatter can be caused by differences between the real velocity profile and water depth during the tide and the assumed sinusoidal velocity profile and constant water depth (1). Especially for the intertidal areas, the velocity profile and water depth can be quite different from the assumed hydrodynamical situation due to flooding and drying.
6. CONCLUSIONS In this paper, a process-based sand mud model is proposed and analysed. In general, it can be concluded that with such a process-based model, goveming time scales and dimensionless parameters can be derived which can increase the physical understanding of the bed composition significantly. Furthermore, an equilibrium mud content within the exchange layer was found when both deposition and erosion occur during the tidal period (24). In this equation, the settling velocity for mud (Win), the mud concentration (Cout) and the erosion rate (M) form an important dimensionless parameter (WmCou/M). This parameter expresses the ratio between the deposition and erosion flux capacity. In earlier studies, correlations between the maximum or mean shear stress and the mud content often showed the following characteristic picture. A critical shear stress seems to exist, below which the mud content can vary between 0 and 100%. Above this value, the mud content is always low. A sharp transition is sometimes observed between both regimes. This pattern can be explained with the presented sand-mud model. Model results suggest that the sharp transition between areas with a very low mud content and other areas depends on the aforementioned dimensionless parameter. For low values (< 10) the transition is sharp, while for higher values the transition becomes more and more gradual. The observed sharp transition in field data in earlier studies follows from the low value of the dimensionless parameter for these areas. Two explanations are given for the variation in mud content between 0 and 100% at sample sites with low bed shear stress. First, the actual mud content at the sample site is probably not in equilibrium due to relatively large adapting time scales. Second, the scatter is probably caused by the difference between the actual hydrodynamic situation and the assumed sinusoidal velocity profile and the constant water depth. Finally, the model results also suggest that a local hydrodynamic parameter (e.g. maximum bed shear stress) for predicting the mud content at a certain location is not very useful for
594 areas exposed to a relatively low bed shear stress. Apart from the local hydrodynamics, the local mud concentration, the settling velocity, the mixing properties within the bed and the sample depth are parameters which determine the local mud content.
ACKNOWLEDGEMENTS
The author is grateful to dr Z.B. Wang for the extensive discussions and valuable comments. Also the comments from dr J.C. Winterwerp and prof dr H.J. de Vriend are highly appreciated. This research was supported by the Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs.
REFERENCES
Armanini, A., 1995, Non-uniform sediment transport: dynamics of the active layer, Journal of Hydraulic Research, 33(5), 611-622. Aubrey, D.G., 1984, Hydrodynamic controls on sediment transport in well-mixed bays and estuaries, In: Physics of Shallow Estuaries and Bays, ed. J. v.d. Kreeke, 245-258. Boudreau, B.P., 1997, Diagenetic Models and Their Implementation - Modelling Transport and Reactions in Aquatic Sediments, Springer Verlag. Dronkers, J., 1984, Tide-induced Transport of Fine Sediment, In: Physics of Shallow Estuaries and Bays, ed. by J. van de Kreeke, Springer Verlag, 228-244. Dyer, K.R, 1994, Estuarine sediment transport and deposition, In: Sediment Transport and Depositional Processes, ed. by K. Pye, Blackwell Scientific Publications, 193-218. Herman, P., 2000, personal communication. Murray, W.A., 1977, Erosion of coarse sand-clayey silt mixtures, Journal of Hydraulic Division, 1222-1227. Panagiotopoulos, I., Voulgaris, G., Collins, M.B., 1997, The influence of clay on the threshold of movement on fine sandy beds, Coastal Engineering, 32, 19-43. Rijn, L.C. van, 1993, Principles of sediment transport in rivers, estuaries and coastal seas, Aqua Publications, Amsterdam. Teisson, C., 1997, A review of cohesive sediment transport models, In: Proceedings of 4th Nearshore and Estuarine Cohesive Sediment Transport Conference, Wallingford, England. Torfs, H., 1995, Erosion of mud/sand mixtures. Ph.D. thesis, Katholieke Universiteit Leuven, faculteit der Toegepaste Wetenschappen, Departement Burgelijke Bouwkunde, Laboratorium voor Hydraulica. Torfs, H., Mitchener, H., Huysentruyt, H., Toorman, E., 1996, Settling and consolidation of mud/sand mixtures, Coastal Engineering, 29, 27-45. Van Ledden, M., 2000, Sediment segregation in estuaries and tidal lagoons, a literature survey, Delft University of Technology, The Netherlands. Winterwerp, J.C., 1999, On the dynamics of high-concentrated mud suspensions, Ph.D. thesis, Delft University of Technology, Delft. WL [ Delft Hydraulics, 1998, A tool for mud flat classification (Z2037.50), Prepared for: European Commision, MAST3 Programme.
577
A process-based sand-mud model M. van Ledden a aFaculty of Civil Engineering and Geosciences, Delft University of Technology, Stevinweg 1, P.O. BOX 5048, NL-2600 GA DELFT, The Netherlands.
Predicting the bed composition in estuaries and tidal lagoons is of great practical importance. At present however, horizontal and vertical bed composition variations are often neglected in sediment transport and morphological models. In this paper a process-based sand-mud model is proposed, the model behaviour is analysed and model results are compared to field measurements. In general, it can be concluded that with such a processbased model, governing time scales and dimensionless parameters can be derived which can significantly increase the physical understanding. Furthermore, an expression is derived for the equilibrium mud content at the bed surface when both deposition and erosion occur during the tidal period. In this expression, the settling velocity for mud, the mud concentration and the erosion rate form an important dimensionless parameter. For low parameter values (< 10), sharp transitions are to be expected between areas with a very low mud content and areas in which the mud content can vary between 0 and 100%. The existence of a sharp transition is confirmed by field data from the Westerschelde estuary (The Netherlands). Finally, model results suggest that a local hydrodynamic parameter is not very useful for predicting the mud content in areas exposed to relatively low bed shear stress. Apart from the local hydrodynamics, the local mud concentration, settling velocity, mixing properties of the bed and the sample depth detemaine the local mud content. KEY WORDS sediment distribution; sand; mud; bed composition; numerical analysis; modelling,
1. INTRODUCTION Predicting the distribution of non-cohesive and cohesive sediments in estuaries and tidal lagoons, which leads to zonation of sand, mud and mixed deposits, is of great practical importance. It determines which areas collect nutrients and pollutants, as these tend to adhere to, or be part of, cohesive sediments and which areas form the right habitat for flora and fauna. Based on the rule-of-thumb "the calmer the water, the finer the bed sediment", a local hydrodynamic parameter is often used for predicting the mud content at a certain location (WL ] Delft Hydraulics, 1998). However, a useful relationship between the mud content and a local hydrodynamic parameter is not known at present. Probably, a more process-based approach is a successful way for increasing the understanding and predictability of the sandmud distribution.
578 In the present process-based sand models (Van Rijn, 1993) and mud models (Teisson, 1997) one representative grain size is used for calculating sediment transport and morphological changes, and the bed composition is often not taken into account. The goal of this paper is an attempt to bridge the gap between sand and mud models by proposing a process-based sand-mud model. First, knowledge of the incorporated processes is discussed (section 2). Thereafter, the process equations are presented (section 3) and the model behaviour is analysed (section 4). Finally, model results are compared to field measurements and the usefulness of a local hydrodynamic parameter for predicting mud content is discussed (section 5).
2. PROCESS KNOWLEDGE
2.1. Incorporated processes In estuaries and tidal lagoons, bed sediments are continuously picked up, transported and deposited. Due to different sediment properties, the flux and direction of transport of noncohesive and cohesive sediments are often not equal (Aubrey, 1986; Dronkers, 1984). The result is a strongly varying mud content in the bed in both horizontal and vertical directions. A large number of sediment processes are involved in the distribution of sediments. Erosion and deposition are the most obvious processes, but consolidation, flocculation and physical and biological mixing within the bed may also affect the bed composition. In an ideal processbased model, all these processes should be taken into account for predicting the bed composition in time and space. However, the present knowledge of these processes for mixtures of non-cohesive and cohesive sediments is limited and constructing a process-based sand-mud model with all these processes is far beyond the present state-of-the-art. Simplifications are therefore inevitable and consequently, the validity of the presented sandmud model herein is limited. Moreover, the limited availability of data is another problem for verification of the model results. In our modelling approach, the tide is assumed to be the dominant forcing for water motion. Short waves and density currents are neglected. Therefore, a depth-averaged modelling approach is assumed to be sufficient. Furthermore, the mud concentration is assumed to be low (< a few hundred mg/1). Flocculation and consolidation are not taken into account. For the deposition, erosion and mixing processes, process equations are needed to describe the behaviour of non-cohesive and cohesive sediments. Recently, laboratory experiments with mixtures of sand and mud have been reported in the literature. The observed behaviour of the different processes for sand-mud mixtures is discussed separately in the following subparagraphs in order to support the process equations used in the model. These equations are presented in section 3.
2.2. Deposition Deposition experiments showed that in the case of sediment concentrations below the gel point, non-cohesive sand particles and cohesive mud flocs behave more or less independently during deposition (Toffs et al., 1996). Around the gel point, however, mud flocs form a network in the water column and sand particles are trapped. Typical gel point concentrations are in the range of 30 - 180 g/1 (Winterwerp, 1999). It seems reasonable that for lower concentrations the separate deposition formulae for mud and sand are valid and can be applied within our model.
579 Table 1 Critical mud content and clay content for the experiments Experiment Clay type Critical mud content Critical clay content Toffs (1995) Kaolinite 4% 3% Torfs (1995) Montmorillonite 13% 4% Panagiotopoul0s eta 1. (1997) . . . . . . Combwich .........................30% ................................ 10.8% 2.3. Erosion Recently, erosion experiments were carried out with mixtures of non-cohesive and cohesive sediments (Torfs, 1995; Panagiotopoulos et al., 1997). By adding mud to a sand bed, Torfs (1995) observed that the non-cohesive behaviour of the bed was increasingly suppressed. Above a critical mud content (% < 0.063 mm), the bed behaved cohesively. However, the critical mud content was not the same for different sand-mud mixtures. Panagiotopoulos et al. (1997) observed a critical clay content (% < 0.002 mm) of 10.8%. The experimental results are summarised in Table 1. The critical clay content for the experiments of Torfs (1995) was calculated by a re-analysis of these experiments. Despite the great difference in the critical mud content, the critical clay content is about the same for the experiments of Torfs (1995). Previously, Dyer (1986) and Raudkivi (1990) stressed the role of clay particles in the erosional behaviour of mixed beds. Dyer (1986) gave a transition range for non-cohesive to cohesive behaviour from 5 - 10% clay content by dry weight. This range agrees with the experimental results. Torfs (1995) also measured the critical shear stress for erosion of sand-mud mixtures with a varying mud content. The critical shear stress is plotted in Figure 1, both as a function of the mud content and the clay content. The clay content was calculated by a re-analysis of the experimental data. The critical shear stress differs significantly for the two sand-mud mixtures at the same mud content, but is about the same at the same clay content. The observations of the transition between non-cohesive and cohesive behaviour as well as the critical erosion shear stress suggest that the erosion process is dominated by the clay content and not the mud content itself.
3.5
9
3 ~2.5
0 O
2
+
.
.
.
+ o , []
.
.
1
. . . . . . . . . .
l
.........
t_ . . . . . . . .
KaoliniteMud Kaolinite Clay MontmorilloniteMud MontmorilloniteClay
+
0[]+
1.5
.,,,~ I,,-i
w0.5 0 0
;
1'0 1'5 2'0 2; 3'0 Mud content [%], Clay content 1%1
Figure 1. Measured critical bed shear stress vs. mud and clay content.
580 From a modelling point of view, the traditional erosion formulae for sand and mud alone must be adapted because of this behaviour. Depending on the clay content in the bed, two regimes must be distinguished: a non-cohesive and a cohesive regime. The transition between both regimes is given by a clay content of 5 - 10%. A critical mud content can also be used, because bed composition measurements suggest a strong correlation between the clay content and silt content (% 0.002 - 0.063 mm) within the bed (Van Ledden, 2000). However, it must be kept in mind that a critical mud content seems to be a site specific value, depending on the clay/silt ratio.
2.4 Mixing Mixing within the bed can have a physical as well as a biological origin. Physical mixing in a sand bed occurs by small-scale bed features and it is generally assumed that the intensity of mixing increases with increasing velocity and decreases further into the bed (Armanini, 1995). Mixing experiments for sediment beds of non-cohesive and cohesive sediments have not been undertaken to the author's knowledge. However, it can be argued that physical mixing must also be dependent on the mud content within the bed. Erosion experiments showed that small-scale bed features are suppressed when the mud content increases (Torfs, 1995). Therefore, physical mixing probably decreases with increasing mud content. Two types of biological mixing are often distinguished (Boudreau, 1997): local mixing and non-local mixing. Local mixing is caused by organisms which move through the bed and therefore mix the sediment particles. Non-local mixing is caused by organisms that transport a specific sediment fraction from one level to another in the bed. An example is the Heteromastus filiformis, which transports only fines from ten centimetres depth to the bedwater interface (Herman, 2000). Physical mixing and local, biological mixing can be modelled as a diffusion process. For both mixing mechanisms, appropriate mixing coefficients must be defined. Non-local mixing by organisms causes sources and sinks of specific sediment fractions at certain depth within the bed. In our modelling approach, non-local mixing is neglected.
a. Local behaviour
b. Numerical model
Wiae Erosion
Deposition
U = 0 sin(cot) h Cmud, Csand
Cout
l i Zb "
r
Mixing
Pm,O
E,~ ,
~D, ~
J=O
z,I, iZI;IIII IIIiZII II;I ZZZIZ i
•
II PmJ 9 .If al 9 T ,,......................."........................................................ .V.........I J = J
Figure 2. Model set-up.
581 3. MODEL EQUATIONS 3.1. Local behaviour In this paper, only the so-called 'local behaviour' is discussed. This situation is sketched in Figure 2a. It is assumed that the behaviour of the sand and mud concentration in the water column, the bed level and the bed composition are dominated by deposition, erosion and mixing within the bed only. Non-local sediment transport processes (advection, diffusion) are assumed to be of minor importance and are neglected in the local model analysis. Because of this local approach, bed load transport is not taken into account. For sand in suspension, the local approach seems to be valid, because its settling time is generally much smaller than the tidal period. For mud however, the settling time is often of the order of the tidal period and a local approach for mud is not realistic. Therefore, horizontal advection and diffusion are also taken into account for mud in a parameterised way (see section 3.3). A process-based model basically consists of three parts: a water motion module, a sediment transport module and bed module. For each module, the process equations are discussed separately in the following subsections. Because of the local approach, the derivatives in the horizontal directions are neglected within the governing equations. 3.2. Water motion The water depth h is taken constant and the varying depth-averaged flow velocity U during the tide is given as a function of time (Figure 2b): U - (J sin(cot)
(1)
where U is the amplitude of the velocity, co the angular frequency of the tidal period and t time. The bed shear stress rb is given by: g 2 ~ =p-Uu
(2)
where p is the water density, g the gravitational acceleration and C the Ch6zy-coefficient. The Ch6zy-coefficient C strongly depends on the bed features and composition. For sandy environments, typical values range from 40 to 60 ml/2/s. For muddy environments, the Ch6zycoefficient is often much larger, ranging from 60 to 100 mVZ/s (Van Rijn, 1993). Although the roughness strongly depends on the bed composition, the Ch6zy-coefficient is assumed to be constant at present (C = 60 m~/Z/s). 3.3. Sediment transport For sediment transport, two sediment fractions are taken into account: a sand fraction (subscript s) and a mud fraction (subscript m). For the description of the sand and mud transport in suspension, the depth-averaged advection-diffusion equation is used. For a local approach, the equation reduces to:
582 ~c, = E , - D, c~
(3)
where ci is the depth-averaged volumetric concentration of fraction i, Di the deposition rate of fraction i and E, the erosion rate of fraction i. The terms E~ and D~ on the right hand side give the net exchange of sediment fraction i between the bed and the water column: the downward deposition flux D~ and the upward erosion flux Et (Figure 2b). As was discussed in section 3.1, the above equation can be applied for the sand fraction, but is not realistic for the mud fraction. In a next section, it will be shown that the local model behaviour shows trivial or non-realistic solutions by applying (3) for the mud fraction. Therefore, an extra transport term is introduced to account for horizontal advection and diffusion. The mud concentration equation becomes: c~c,,
~=Em-D,,,+k,,(c,,u,-Cm) dt
(4)
where Cout is the mud concentration outside and km is a transport coefficient. The last term on the right hand side in (4) is a transport term by which the mud concentration Cm is affected by the concentration outside the model Cout by a transport coefficient km (Figure 2b). When the inner concentration is lower than the outside concentration, it is assumed that by advective and diffusive transport mud is imported from outside and vice versa. Exchange of sediment between the bed and the water column occurs by deposition to and erosion from the so-called 'exchange layer'. The exchange layer is the layer at the bed surface with index j = 0 (Figure 2b). As was concluded from the laboratory experiments, the erosion and deposition flux for sand and mud must be prescribed for two regimes: a non-cohesive and a cohesive regime (see section 2.3). A critical mud content within the bed can be used for the transition between both regimes. In this model, the critical mud content for the transition between both regimes is denoted as Pm,r and the mud content in the exchange layer, denoted as Pm, O, determines the erosion regime (Figure 2b). The erosion and deposition terms for both regimes are discussed successively below. Non-cohesive regime (Pm,O
E,.-D,. = ~ . , ( c ..... - c , )
(5)
where ~ is a form coefficient, ws the settling velocity of the sand fraction and Ce,s the depthaveraged equilibrium concentration of the sand fraction. The form coefficient y takes into account the vertical distribution of the velocity and concentration profile (Galappatti, 1983). The depth-averaged equilibrium concentration (Ce,s) is given by the Engelund-Hansen transport formula:
583 0.05
c.... =
U4
2Acd, 3 h
(6)
where A is the specific gravity and ds the sand diameter. In contrast to other sediment transport formulae, no threshold for the initiation of motion is included in the equilibrium concentration of Engelund-Hansen. A sediment transport formula with a threshold for sand can also be applied for the depth-averaged equilibrium concentration, e.g. Van Rijn's transport formula. However, it is not expected that including a threshold will significantly change the observed behaviour, discussed in the next paragraphs. Therefore, the Engelund-Hansen transport formula is considered to be sufficient as a first step. Further research is recommended in order to study the erosion process of sand-mud mixtures in detail and propose more realistic erosion formulations. For the erosion process of mud, the erosion formula of Partheniades is assumed to be valid. Because the mud content can vary, the mud content in the exchange layer is added to the traditional formula. During erosion experiments in the non-cohesive regime, clay and silt particles are easily washed out from the top layer (Murray, 1977; Torfs, 1995). Thus, the critical erosion shear stress in this regime (Te,nc) can be very low. For mud deposition the wellknown deposition formula of Krone is applied. The exchange of mud is given by:
m: m0 F ll'r
[1-
"]
(7)
where Pm, o is the mud content in the exchange layer, M the erosion rate, re,nc the critical erosion shear stress for non-cohesive regime, H the Heavyside function, Wm the settling velocity of mud and rd the critical deposition shear stress.
Cohesive regime (Pm,O>Pm, crit) In the cohesive regime, the erosional behaviour of sand and mud is not independent. It is assumed that sand and mud particles are eroded in the same way, but once in the water column behave independently again. The Partheniades formula seems the most obvious formula for describing the erosion of mud and sand particles. This formula is adapted by introducing a parameter for the sand and mud content in the exchange layer. The deposition formulations for sand and mud remain the same as for the non-cohesive regime. The exchange of sand and mud in the cohesive regime are respectively given by: 9
(8)
(9)
584 where p~,o is the sand content in the exchange layer and Ze.cis the critical erosion shear stress for the cohesive regime. 3.4. Bed level and composition The bed level change is determined by the flux of sediment to and from the bed from both sand and mud. The bed density is assumed to be constant in time and space. The bed level change is given by:
3zh = P" [D,, - Em + D,. - E,. ]
Ph
'
(10)
"
where Z b is the bed level, p~ the sediment density and Pb the bed density. Armanini (1995) proposed a continuous formulation for modelling the bed composition. The local change in sand content in a Langrangian co-ordinate system can be given by an advection-diffusion equation:
Op, + - - P, -s. c? Oz -ff "
=0
(11)
where p~ is the sand content, z the distance from the bed surface (positive downwards) and ez a mixing coefficient within the bed. It is important to note that in (11) the origin of the vertical coordinate z is at the bed surface and z is positive in downward direction (Figure 2b). Conversely, the bed level rate OZb/Ot is positive if the bed level rises. The first term in (11) represents the local change in sand content. The net flux of sand by bed level variations is given by the second term. The third term is the net flux by small-scale variations in the bed level, e.g. tipples. Armanini (1995) suggested that the mixing coefficient ez in (11) decreases with increasing distance below the bed surface, because the influence of the small-scale variations decreases. The mixing coefficient was assumed to be an exponentially decreasing function: ot I
s. = ~,0 e
u,
(12)
where ez,0 is the mixing coefficient at the bed surface, ds the sand grain size and al a coefficient. The mixing coefficient ez, O was assumed to be proportional to the friction velocity u. and the sand grain size ds:
4.0 = a0u, d,
(13)
where ao is a coefficient. The advection-diffusion equation (11) needs two boundary conditions. At the bed surface (z = 0) the sediment flux is prescribed. This boundary condition reads:
585
1
(14,
z=0
Deep below the bed surface the diffusive flux is assumed to be zero. The boundary condition is given by:
+
c2 c~ z=-
=0
(15)
For the mud content within the bed, the same equation can be solved. In our model however, only two sediment fractions are applied. Therefore, the mud content can be easily expressed in terms of the sand content: (16)
p,, = 1-p.,.
4. MODEL ANALYSIS 4.1. Scale parameters The model described in section 3 consists of four dependent variables: two sediment concentrations (cs) and (Cm) in the water column, the bed level (Zb) and the mud content within the bed (Pm)- For analysing the behaviour of this model, it is important to derive the governing external and internal scale parameters, which are elaborated below. The external forcing is characterised by two parameters: the amplitude of the bed shear stress gives the level of the forcing ( ?h ) and the time variation of the forcing is characterised by the tidal period (Ttide). Within the model, three critical shear stress parameters must be defined: rd for mud deposition, re, nr for non-cohesive mud erosion and re,r for erosion of sand and mud in the cohesive regime. These parameters can be made dimensionless by using the maximum bed shear stress. In general, the critical shear stresses are divided by the level of forcing. For this purpose, the reciprocal term is used, because the level of forcing can be zero: r~ = ~-~ h
~,,c= ~b
~,c = ~-~b
By using these scale parameters, three situations can be distinguished. These situations are shown in Figure 3. The bed shear stress is plotted during half a tidal period for the three situations. Also the three critical shear stresses are given with horizontal lines. The symbols denote whether or not mud erosion or deposition takes place. Sand erosion and deposition can always occur, because no threshold shear stress is included. The model results of the three situations are discussed in the next subsections. Besides the extemal scale parameters, intemal time scales also govem the model behaviour. These time scales can be derived from the underlying equations. The time scales are used for explaining the model behaviour and given in the next subsections.
586 o * []
Mud deposition Non-cohesive erosion Cohesive erosion Situation 3
!,oc m ... .'C.e, . nf'............~....... ~......y..........................
0
0.1
0.2
0.3
0.4
0.5
Time t/T [-] Figure 3. Overview of situations.
4.2. Situation 1
In situation 1 erosion of mud from a non-cohesive bed as well as erosion of sand and mud from a cohesive bed are not possible, because the maximum bed shear stress is always lower than the critical bed shear stress for erosion during the tidal period. However, deposition of mud is possible, because the bed shear stress is lower around slack water than the critical bed shear stress for deposition. Sand deposition is always possible, because the deposition term does not depend on the local hydrodynamics. Erosion of sand from the non-cohesive bed can occur, because the equilibrium concentration formulation does not have a threshold shear stress (Figure 3). 9
Mud concentration
For the mud concentration, only two time scales are important, the deposition time scale (h/wm) and the transport time scale (h/kin), because erosion does not play a role in this situation. For a constant water depth, the behaviour of the mud concentration is governed by the dimensionless ratio k,,,/Wm. In Figure 4, results of the mud concentration are given for different values of k,,,/Wm. The mud concentration is made dimensionless by using the outside concentration Cout. It can be observed that the (tidally averaged) mud concentration in the water column tends to a finite value for all ratios of k,/wm. When the ratio k,,,/w,,, is higher, the equilibrium value is also higher. The equilibrium mud concentration is always lower than the concentration outside Cont. The (tidally averaged) equilibrium mud concentration can be explained from a balance between the inward horizontal flux due to transport and an outward downward flux due to deposition over one tidal period. The equilibrium concentration can be solved analytically when the adaptation time for settling (h/w,,) is at least in the order of the tidal period.
587 =
~
~1.5~
/ ~
k/Wm=5-O
/
k/Wm= 1.0
t'
'
--0.8
k/Wm-O.:
...........................
1 0.4 0.2
~r/" rT~
O
-'
Wzb/Wm =zb-
1.0
_, ......... Tzb/Tm= 5.0
i
5'0 100 150 200 2 3 5 Time t/T [-] Time t/T [-] Figure 4. Mud concentration for different Figure 5. Mud content in exchange layer for values of k,,/wm, different values of Tzb/Tm. In this case, a more or less constant concentration can be assumed during the tidal period and the equilibrium concentration Cm,eq is given by:
c ....q = - - C1 o u
Win_
km a o + 1
,
aD =
i( 1 -- ~1u ) ( 2
arcsin
(~//
+ / X/~d- 1 1
(17)
ka/Xd ; )
where aD is a deposition coefficient and Cm,eq the equilibrium mud concentration. The coefficient aD is solved analytically and is a function of the dimensionless shear stress rd" By using this expression, it can be seen that for the equilibrium mud concentration in the water column three situations exist: 1. km << Wm In this case, the supply of mud from outside is very low with respect to the deposition of mud. Therefore, the equilibrium mud concentration is very small, for reaching a balance between the deposition and transport flux during the tide. 2. km~wm
Because the settling velocity Wm and the transport coefficient km have the same order of magnitude, the flux by deposition and transport have the same order of magnitude for an equilibrium mud concentration between 0 and Cout. 3. km
>>
Wm
In this case, the supply of mud from outside is very fast with respect to the deposition of mud. Therefore, the equilibrium concentration is about equal to the concentration outside Cout. From the equilibrium mud concentration equation (17), it can easily be seen that when no transport (km = O) is included, the model gives a trivial solution. The only flux is caused by
588 deposition and the final mud concentration is zero. The total deposition flux depends on the initial mud concentration. 9
Sand concentration
The governing time scale for the sand concentration is h/ws. This time scale is in general much smaller than the tidal period. For the non-cohesive regime, the sand concentration shows a small phase shift with respect to the equilibrium concentration Ce,s because of its adaptation time (h/ws). For the cohesive regime, no erosion of sand can occur. If the initial sand concentration is non-zero, the sand concentration will drop very fast to zero and does not change anymore. 9 Bed level Due to the continuous deposition of mud particles and no erosion, the bed level is continuously increasing due to accretion. The net sand exchange during one tidal period is equal to zero. In case of a non-cohesive bed, the speed of the bed level rise is thus determined by the equilibrium mud concentration and the settling velocity. The bed level rise time scale is given by: T . Pe . zh
8 .
Ps (Zl)Wmem,eq
Ph . 811 Ps Cout CLDWm
+ ~ 11
km
(18)
where Tzb is the time scale of bed level rise and ~ the total thickness of the bed layer. In (18) the parameter 3 is the total thickness of the bed layer, in which the bed composition changes. It is obvious that for an increasing transport coefficient (k,,), settling velocity (win) and outside concentration (Co,a), the time scale of bed level rise (Tzb) decreases. 9
Bed composition
Because of the continuous deposition of mud, the equilibrium composition consists of a mud bed without sand. However, for the development of the bed composition two time scales are important: the time scale of bed level rise (18) and the mixing time scale (Tin). The mixing time scale is given by: 82
Tm= ~
(19)
~z,0
where Tm is mixing time scale and ez, Othe mixing coefficient at the bed surface. In Figure 5 the mud content in the exchange layer is given for different ratios of Tzb/Tm. In Figure 6 and Figure 7 the bed composition profile is given in a fixed co-ordinate system at different times for Tzb/Tm = 0.2 and 5.0 respectively. The depth within the bed is made dimensionless with the constant thickness of the bed layer 3. Due to the fixed co-ordinate system and the continuous bed level rise in this situation, the bed layer in which the bed composition is computed, shifts upward in time (Figure 6 and Figure 7).
589 2.5
,',1.5
25II~,,
+ t/T=0 - * - t/T = 40 t / T = 80 - + - t / T = 120 t / T = 160 t/T = 200
t~----0- ..... }'
"'--'*-- f i T = 4 0 2 +t/T=80
'
'
I ]
--~ t/T= 120 I "7"
,,4
---a- t / T = 160 I
~.2
~_~.5 ..................... __~ tcr = 2ool,~ ~"
~
~~.-~
~
0.5 :e
0.5
0.2
0.4
0.6
133
0.8
0'.2
" 014
016
018
Mud content [-]
Mud content [-] Figure 6. Bed composition for Tzb/Tm = 5.0.
Figure 7. Bed composition for Tzb/Tm = 0.2.
It can be observed that when the ratio of bed level rise time scale to mixing time scale is relatively large (Zzb/Tm = 5 . 0 ) , the time needed to reach a 100% muddy exchange layer is relatively long (Figure 5). Moreover, the bed composition profile with depth is more or less constant (Figure 6). For the opposite situation (Tzb/Tm = 0.2) a 100% muddy top layer is reached very fast (Figure 5) and the bed composition profile shows a stratified profile (Figure 7). The explanation for this behaviour is that when the net downward flux due to deposition is much larger than the downward transport by mixing, all mud deposited on the exchange layer is conserved in the upper part of the bed. For this situation, the bed composition profile is strongly stratified and a 100% muddy exchange layer is reached very fast. 4.3. S i t u a t i o n 2
In situation 2 erosion of sand and mud from a cohesive bed is not possible. Thus, the situation for the cohesive regime is equal to situation 1. However, mud and sand erosion from a non-cohesive bed is possible (Figure 3). The behaviour of mud concentration, bed level and bed composition in this regime is discussed below. 9
Mud concentration
In the non-cohesive regime the mud concentration in the water column always tends to the mud concentration outside Com. An equilibrium is reached when the outward deposition flux is equal to the inward erosion and transport flux during a tidal period. When it is assumed that the concentration in the water column and the mud content in the top layer are more or less constant during the tidal period in the equilibrium situation, the equilibrium mud concentration is given by:
m
WmO~D "-}-km
'
7r,
7r,
,no
where aE, NC is the erosion coefficient for non-cohesive regime and Pm, eq, O the equilibrium mud content in exchange layer. The coefficient aE, NC is solved analytically and is a function of the
590 dimensionless critical shear stress for erosion in the non-cohesive regime (rc,,c)- Thus, compared to the equilibrium mud concentration in situation 1 (17), an extra term is introduced due to erosion in (20). However, the equilibrium mud c o n t e n t (Pm, eq,O) is not known yet. In the discussion of the bed level and composition below, it will be shown that the equilibrium mud content is equal to:
Pm,eq,O =
(21)
WmCm'eq al) M O{E,N(.
By using expression (21) in equation (20), the final equilibrium mud concentration in the water column is: WmCm,eq
aD
M
O[,E,NC
M a ~.,Nc + k c o. , c .... q =
(22)
= c,,,,,
Wma D + k
Thus, the equilibrium concentration in this situation is always equal to the concentration outside. However, this is only valid for the non-cohesive regime. When the cohesive regime is reached, only transport and deposition (see situation 1) determine the equilibrium. In this case, the equilibrium concentration of situation 1 is reached. 9
Bed level and composition
For the non-cohesive regime, equilibrium is reached when the total flux of sand and mud to and from the bed are equal during one tidal period. When this situation is reached, the bed level remains constant. At this equilibrium, the mud content in the exchange layer for the noncohesive regime has also an equilibrium value. An equilibrium exists because of a balance between the erosion and deposition flux for mud during one tidal period. Assuming a more or less constant mud concentration and mud content during one tidal period, the equilibrium mud content can be expressed by: 1 7'
WmCm,eq ~
![1- Ya]/-/[1- Ya]dt WmC m : ~eq
Pm,eq,O :
_ ~_D
(23)
M lr 0
As already mentioned, the equilibrium mud concentration is about equal to the concentration outside. Thus, the equilibrium mud content in the exchange layer is: _ WmCout 0{'1)
P ....q,0 - - -
M
aE,NC
(24)
591 In the final expression for the equilibrium mud content (24), the deposition flux during one tide is recognized in the numerator, the erosion flux for a 100% mud bed (Pmud, O = 1.0) during one tide in the denominator. It is important to note that when the equilibrium mud content is larger than the critical mud content for the transition between non-cohesive and cohesive behaviour (Pm, eq,O >Pmud, crit), the non-cohesive equilibrium mud content will never be reached. When starting with a noncohesive bed, the mud content in the top layer tends to the equilibrium in the cohesive regime. When reaching the critical mud content the behaviour changes into a cohesive behaviour. When no erosion of the cohesive bed is possible, the final situation is a total mud bed. In fact, the same situation as in situation 1 is reached with only deposition and no erosion. 4.4. Situation 3
In situation 3, the model behaviour is strongly comparable to situation 2. The only extra possibility is erosion in the cohesive regime (Figure 3). Therefore, two equilibria exist for the mud content in the exchange layer (24), indicated with indices n c and c: a non-cohesive equilibrium Pm, O,eq, nc and a cohesive equilibrium pm, o,eq, c. In analogy to the non-cohesive equilibrium, the cohesive equilibrium is also defined by (24), because the only difference in the erosion and deposition formulae is the critical erosion shear stress. For the cohesive equilibrium (Pm, O,eq,c) the dimensionless critical shear stress for erosion in the cohesive regime (rr must be used. In situation 3, the final equilibrium depends on the values of the noncohesive and cohesive equilibrium compared to the value of the critical mud content Three different situations can be distinghuished and are summarised in Table 2.
(Pmud,~r,).
The final equilibrium for these situations can be explained as follows: 9 Situation 3.1: the non-cohesive equilibrium is never reached, because the non-cohesive equilibrium mud content is higher than the critical mud content. When the mud content is lower than the critical mud content, the mud content increases to reach the non-cohesive equilibrium value. When the critical mud content is reached, the equilibrium changes to the cohesive equilibrium and this is also the final equilibrium. 9 Situation 3.2: this situation is comparable to situation 3.1, but in this case the cohesive equilibrium is never reached because its value is in the non-cohesive range. The final equilibrium is the non-cohesive equilibrium. 9 Situation 3.3: both equilibria can exist and the final situation depends on the initial mud content in the exchange layer. If the initial mud content is higher than the critical mud content, the cohesive equilibrium is reached. If the initial mud content is lower than the critical mud content, the non-cohesive equilibrium is reached. Table 2 Overview of equilibria in situation 3 Situation Cohesive equilibrium Non-cohesive equilibrium .
.
.
.
.
............
3.1 3.2 3.3
(Pm,0,eq, c)
(Pm,0,eq,nc)
> Pmud,crit < Pmud,crit .> pmudrcrit
> Pmud,crit < Pmud,crit
_
< pmudrcrit
Final equilibrium Cohesive Non-cohesive Depends on initial conditions
592
~ 0.8
~
..... -=_-. . . . . ~. . . . . , . _ _ . _ . . . . _ _ . _ . . _ . _ - - , .... \ t A ~\ A , 0 i A ~-~ i
WmCout/lVI
0 ", 0 i O' 0\
0.6
100
",,. .\,,.. "
o',,,,
0.4
:
"".,,
"-.
0"
2
(
'X WmCoutflVI= 10
~0.2
O0
'-
Bed shear stress [-] Figure 8. Equilibrium mud content.
5. COMPARISON TO FIELD M E A S U R E M E N T S Correlations between the maximum bed shear stress and the mud content at a certain location of the Westerschelde estuary (The Netherlands) showed the following pattern (WL [ Delft Hydraulics, 1998): for high bed shear stress, the mud content was always low (< 10%). For low bed shear stress, the mud content varied between 0 and 100% and no useful relationship could be determined. Often, a sharp transition between these regimes was observed and a critical transition value could be defined. However, physical explanations for this typical pattern are not available at present. In the previous chapter an equilibrium mud content in the exchange layer was defined (24). Similarly to the above described correlation, this equilibrium mud content can be given as a function of the maximum bed shear stress during the tide (Figure 8). The equilibrium mud content is given for different values of WmCouc/M.The maximum bed shear stress is made dimensionless by the critical erosion shear stress. It is assumed that the critical erosion shear stress for the non-cohesive regime is equal to the critical erosion shear stress for the cohesive regime. The critical shear stress for deposition must also be known to calculate the coefficient aD. The equilibrium mud content is given for two ratios of Ze/Zd = 5 (lines) and re~re= 10 (open symbols). For small values (order 1) of w,,,Cou/M a sharp transition exists between a full mud bed and a full sand bed ( Figure 8). Beyond a dimensionless bed shear stress of about 3, the mud content is less than 10%, while below a dimensionless bed shear stress of about 2, the equilibrium value is about 100%. With increasing values of w,,,Cou/M the transition becomes less sharp. For the Westerchelde area, typical values for the parameters are w,, = 5'10 "4 m/s, M = 1* 10 8 m/s and Cout= 4" 10 -5 ( ~ 100 mg/1). The parameter-value WmCout/Mhas a value of about 2. Thus, a sharp transition has to be expected in this area. This is confirmed by the observed correlation between the bed shear stress and the mud content (WL [Delft Hydraulics, 1998). It is important to note that when the critical shear stress for erosion in the non-cohesive and cohesive regime are not equal, the non-cohesive and cohesive equilibrium mud content are
593 not equal (see section 4.3). In this case, the equilibrium mud content as a function of the maximum bed shear stress is discontinuous. The discontinuity arises at the critical mud content between the non-cohesive and cohesive regime. The presented sand-mud model also implies some explanation for the observed large scatter in areas with a low bed shear stress. First, the relatively long time scale for reaching the equilibrium mud content suggests that the measured mud content at the sample points is probably not at its equilibrium value, but evolves towards an equilibrium value. The time scale for reaching the equilibrium mud content can be estimated by using (19). The parameters for the time scale Tzb are assumed to be as follows" ~ = 0.10 m, Pb = 1200 kg/m 3, ps = 2650 kg/m 3, Wm= 5 " 1 0 -4 m / s , Cout-- 4"10 -5 (~ 100 mg/1). Thus, the time scale for reaching a total mud bed is about 50 days, about two months. When mixing within the bed is included, the time scale becomes even much larger. Thus, the measured mud content at a single site not only depends on the actual hydrodynamic conditions, but also on the hydrodynamic conditions in last months. The equilibrium mud content can be seen as the upper limit for the mud content in a certain area. Second, scatter can be caused by differences between the real velocity profile and water depth during the tide and the assumed sinusoidal velocity profile and constant water depth (1). Especially for the intertidal areas, the velocity profile and water depth can be quite different from the assumed hydrodynamical situation due to flooding and drying.
6. CONCLUSIONS In this paper, a process-based sand mud model is proposed and analysed. In general, it can be concluded that with such a process-based model, goveming time scales and dimensionless parameters can be derived which can increase the physical understanding of the bed composition significantly. Furthermore, an equilibrium mud content within the exchange layer was found when both deposition and erosion occur during the tidal period (24). In this equation, the settling velocity for mud (Win), the mud concentration (Cout) and the erosion rate (M) form an important dimensionless parameter (WmCou/M). This parameter expresses the ratio between the deposition and erosion flux capacity. In earlier studies, correlations between the maximum or mean shear stress and the mud content often showed the following characteristic picture. A critical shear stress seems to exist, below which the mud content can vary between 0 and 100%. Above this value, the mud content is always low. A sharp transition is sometimes observed between both regimes. This pattern can be explained with the presented sand-mud model. Model results suggest that the sharp transition between areas with a very low mud content and other areas depends on the aforementioned dimensionless parameter. For low values (< 10) the transition is sharp, while for higher values the transition becomes more and more gradual. The observed sharp transition in field data in earlier studies follows from the low value of the dimensionless parameter for these areas. Two explanations are given for the variation in mud content between 0 and 100% at sample sites with low bed shear stress. First, the actual mud content at the sample site is probably not in equilibrium due to relatively large adapting time scales. Second, the scatter is probably caused by the difference between the actual hydrodynamic situation and the assumed sinusoidal velocity profile and the constant water depth. Finally, the model results also suggest that a local hydrodynamic parameter (e.g. maximum bed shear stress) for predicting the mud content at a certain location is not very useful for
594 areas exposed to a relatively low bed shear stress. Apart from the local hydrodynamics, the local mud concentration, the settling velocity, the mixing properties within the bed and the sample depth are parameters which determine the local mud content.
ACKNOWLEDGEMENTS
The author is grateful to dr Z.B. Wang for the extensive discussions and valuable comments. Also the comments from dr J.C. Winterwerp and prof dr H.J. de Vriend are highly appreciated. This research was supported by the Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs.
REFERENCES
Armanini, A., 1995, Non-uniform sediment transport: dynamics of the active layer, Journal of Hydraulic Research, 33(5), 611-622. Aubrey, D.G., 1984, Hydrodynamic controls on sediment transport in well-mixed bays and estuaries, In: Physics of Shallow Estuaries and Bays, ed. J. v.d. Kreeke, 245-258. Boudreau, B.P., 1997, Diagenetic Models and Their Implementation - Modelling Transport and Reactions in Aquatic Sediments, Springer Verlag. Dronkers, J., 1984, Tide-induced Transport of Fine Sediment, In: Physics of Shallow Estuaries and Bays, ed. by J. van de Kreeke, Springer Verlag, 228-244. Dyer, K.R, 1994, Estuarine sediment transport and deposition, In: Sediment Transport and Depositional Processes, ed. by K. Pye, Blackwell Scientific Publications, 193-218. Herman, P., 2000, personal communication. Murray, W.A., 1977, Erosion of coarse sand-clayey silt mixtures, Journal of Hydraulic Division, 1222-1227. Panagiotopoulos, I., Voulgaris, G., Collins, M.B., 1997, The influence of clay on the threshold of movement on fine sandy beds, Coastal Engineering, 32, 19-43. Rijn, L.C. van, 1993, Principles of sediment transport in rivers, estuaries and coastal seas, Aqua Publications, Amsterdam. Teisson, C., 1997, A review of cohesive sediment transport models, In: Proceedings of 4th Nearshore and Estuarine Cohesive Sediment Transport Conference, Wallingford, England. Torfs, H., 1995, Erosion of mud/sand mixtures. Ph.D. thesis, Katholieke Universiteit Leuven, faculteit der Toegepaste Wetenschappen, Departement Burgelijke Bouwkunde, Laboratorium voor Hydraulica. Torfs, H., Mitchener, H., Huysentruyt, H., Toorman, E., 1996, Settling and consolidation of mud/sand mixtures, Coastal Engineering, 29, 27-45. Van Ledden, M., 2000, Sediment segregation in estuaries and tidal lagoons, a literature survey, Delft University of Technology, The Netherlands. Winterwerp, J.C., 1999, On the dynamics of high-concentrated mud suspensions, Ph.D. thesis, Delft University of Technology, Delft. WL [ Delft Hydraulics, 1998, A tool for mud flat classification (Z2037.50), Prepared for: European Commision, MAST3 Programme.