View source for Siltation in harbors and fairways



==Introduction==
This article addresses the siltation in semi-enclosed harbor basins and fairways in open water with sediments from the surrounding waters. In its most basic form, siltation occurs when the sediment transport capacity is locally exceeded by the supply of sediment. 


==Siltation in semi-enclosed harbor basins==

For harbor basins, siltation with fine (cohesive) sediments is in general more problematic than siltation with coarser sediments (sand), as the siltation rates with fines are often larger, and at times, these sediments are contaminated as well. Thus we focus on siltation with fine sediments, which are generally carried by the flow in suspension (see [[Dynamics of mud transport]]).

The harbor basin and the ambient water system exchange sediment-laden water by the three mechanisms described below. Because the basin is semi-enclosed, no net inflow or outflow of water occurs (over a long-enough period). However, there is a gross exchange of water – sediment-rich water from the ambient system is exchanged with sediment-poor water from the basin itself. In zero-order approach we assume that the sediment flux may be treated as the product of water flux and SPM (suspended particulate matter) concentration, corrected for a trapping efficiency (the fraction of sediment that enters the harbor and deposits). 

A harbor basin can be situated along a river, a lake, a tidal river, a coast, an estuary etc. We always assume that the water body in front of the harbor entrance flows with a characteristic velocity <math>U</math>. In a tidal river and along a coast, tides play a major role, whereas in estuaries, and along some coasts, density currents are also important.

[[Image: HanFig1.jpg|thumb|450px|right|<small>Figure 1: Visualization of exchange processes between harbor basin and surrounding (courtesy Vanlede and Dujardin<ref> Vanlede, J. and A. Dujardin, 2014. A geometric method to study water and sediment exchange in tidal harbors. Ocean Dynamics 64:1631–1641 DOI 10.1007/s10236-014-0767-9</ref>  and de Boer and Winterwerp<ref> De Boer, W.P. and J.C. Winterwerp, 2016. The role of fresh water discharge on siltation rates in harbor basins. Proceedings, PIANC-COPEDEC IX, Rio de Janeiro, Brazil </ref>).</small>]]

The three relevant mechanisms exchanging sediment-laden water between the ambient water body and the harbor basin are (see also Fig. 1):
# Horizontal exchange: large scale circulations in the harbor’s mouth driven by flow separation, entrainment and stagnation effects at the downstream side of the basin’s entrance (see also Fig. 6). This mechanism always plays a role in flowing ambient water. Net inflow/outflow of water is always zero.
# Tidal filling: during rising tide, sediment-laden water flows into the basin, while during falling water, the same amount of water flows out of the basin, containing less sediment, though. Over a tidal period, no net amount of water is exchanged. This mechanism plays a role in tidal rivers and along open coasts.
# Density currents: gradients in salinity induce density currents with a near-bed current (which contains the majority of the sediment) against the direction of that gradient. Salinity-induced density currents play a role in estuaries and coastal systems with salinity gradients (see [[Estuarine circulation]]). Eysink (1989<ref name=E>Eysink, W.D. 1989. Sedimentation in harbour basins. Small density differences may cause serious effects. In: 9th International Harbour Congress, Antwerp, Belgium</ref>) emphasized the importance of this mechanism. Density currents can also be induced by gradients in SPM and temperature.

Note that these mechanisms interact. For instance, during tidal filling, the wake induced by flow separation in the harbor’s mouth is deflected into the basin, and horizontal exchange thus becomes less effective in exchanging water between the basin and the surrounding water body.

[[Image: HanFig2.jpg|thumb|400px|right|<small>Figure 2: Timing of water exchange processes over a tidal period, assuming that salinity and velocity are in phase. </small>]]

These three mechanisms are illustrated below for an estuary, where salinity variations are more or less in phase with ebbing and flooding of the tide. In general, water level and tidal velocity are out of phase, and in the following we assume a phase difference of <math>45^{\circ}</math>. During flood, flow separation occurs at the down-estuary side of the harbor basin, and the wake is deflected into the basin during rising tide, and out of the basin during falling tide (Fig. 6). Thus, relatively, little water is exchanged during falling water and ebb, which is therefore neglected in this zero-order estimation. 

Tidal filling is in phase with rising and falling water. The salinity in the estuary in front of the harbor basin is more or less in phase with flood and ebb – for a harbor basin situated down-estuary (close to the sea) salinity in the estuary is larger than in the basin during flood, while the opposite is true during ebb. Hence, a near-bed sediment-rich density current flows into the basin during flood, while, owing to siltation in the basin, a near-bed sediment-poor density current flows out of the basin during ebb. This phasing is sketched in Fig. 2, showing that sediment import into a harbor basin is unevenly distributed over a tidal period.

[[Image: HanFig3.jpg|thumb|450px|left|<small>Figure 3: Zero-order mass balance for harbor siltation, where <math>F_s</math> = siltation rate, and <math>\alpha</math>  an efficiency parameter (here the trapping efficiency) – in the following this efficiency is accounted for through the settling velocity of the sediment in the basin (which thus becomes the effective settling velocity).  <math>S, \, h</math> are the harbor surface and depth, respectively.</small>]]

Eysink<ref name=E/> was the first to quantify the siltation rate in harbor basins, using a zero-order assessment (see also <ref>Van Rijn, L.C., 2005. Principles of sedimentation and erosion engineering in rivers, estuaries and coastal seas. Aqua Publications, The Netherlands </ref><ref>Winterwerp, J.C. and W.G.M. van Kesteren, 2004. Introduction to the physics of cohesive sediments in the marine environment, Elsevier, Developments in Sedimentology, 56</ref>). It is based on the zero-order sediment balance, sketched in Fig. 3, where it is assumed that the SPM-values in the harbor basin <math>c_e</math> (and thus the siltation rate) are proportional to the ambient SPM-value <math>c_a</math>. The equilibrium solution to this simple differential equation is given in equation (1):

<math>c_e=\large\large\frac{<Q>}{<Q>+\alpha S W_s}\normalsize  c_a  ,       \qquad (1) </math> 

in which <math>S</math> = projected basin’s surface. The exchange flow <math>Q</math> is determined by the three processes described above:

<math>\large\large\frac{1}{T}\normalsize \int_0^T Q dt \equiv <Q>=<Q_t>+<Q_e>+<Q_d>  , \qquad(2) </math>

where <math><Q_t></math> is the gross water exchange by tidal filling (angular brackets implies averaging over tidal period), <math><Q_e></math> is the gross water exchange by horizontal circulation (entrainment), and <math><Q_d></math> is the gross water exchange by density currents, <math>\alpha=1-u_h^2/u_{cr}^2 \;</math>, <math>u_h</math> is a characteristic velocity in the harbor basin, and <math>u_{cr}</math> is a threshold velocity below which sediment can permanently settle of the basin’s bed.
 
[[Image: HanFig5.jpg|thumb|250px|right|<small>Figure 5: Cartoon of shipping-induced siltation, especially important in lakes and canals. </small>]]

Eysink<ref name=E/> proposed a number of coefficients to quantify these gross water exchange rates:


<math><Q_t>=\large\frac{V_t}{T}\normalsize, \quad <Q_e>=f_e A U - f_{e,t}<Q_t> , \quad <Q_d>=f_d A  \sqrt{ \large\frac{\Delta \rho_s g h_0}{\rho}\normalsize } - f_{d,t} <Q_t> , \qquad(3)</math>

where <math>V_t</math>= tidal volume of harbor basin, <math>T</math>= tidal period, <math>A</math>= cross section harbor entrance, <math>U</math>= characteristic velocity along the harbor entrance, <math>\Delta \rho_s</math> = characteristic salinity-induced density difference across the harbor entrance, <math>h_0</math> is local water depth, the coefficients <math>f_{e,t},  f_{d,t}</math> reflect a reduction in exchange efficiency during rising tide.

The following empirical coefficients (Fig. 4) were proposed by Eysink<ref name=E/>, showing also the dependence of <math>f_e</math> on the harbor entrance orientation with respect to the flood direction.

[[Image: HanFig4.jpg|thumb|350px|left|<small>Figure 4: Empirical coefficients for harbor siltation. Below: <math>f_e</math>-values for different harbor entrance orientations with respect to the flood direction (to the right).</small>]]


Next to these three exchange mechanisms, also shipping itself can induce sediment import into harbor basins, in particular in lakes and canals, where the above-mentioned mechanisms are small. Fig. 5 sketches this process, for which however no general quantification exists.


==Minimization of harbor siltation costs ==
Harbor siltation necessitates regular dredging maintenance which constitutes a major cost factor for harbor operation. Several measures have therefore been devised and experimented to tackle the siltation problem. Different types of measures can be distinguished: 
* Entrance flow optimization;
* Sediment trap;
* Water Injection Dredging;
* Recirculation avoidance;
* Beneficial reuse of dredged sediment.

====Entrance flow optimization to keep sediments out====
Entrance flow optimization is a commonly applied measure to reduce harbor sedimentation. It is an effective measure in cases where flow separation and concomitant horizontal eddy formation is the major siltation cause (as shown in Fig. 6 for a harbor located on a tidal river). Measures consist of adapting the flow pattern at the harbor entrance such that flow separation is suppressed. This can be achieved by constructions that reduce the channel flow velocity at the upstream harbor entrance during flood. The effect of a current deflection wall and sill is shown in Fig. 7 left panel; this solution has been applied at the Köhlfleet harbor basin in Hamburg and at the Deurganckdok at Antwerp.  Another solution is a permeable pile-groin construction as shown in Fig. 7 right panel; it proved very effective in a laboratory experiment<ref>Van Schijndel, S.A.H. and Kranenburg, C. 1998. Reducing the siltation of a river harbour. Journal of Hydraulic Research, 36(5): 803-814</ref>. The design of a flow adaptation construction is highly dependent on the site characteristics; the design should be tested in physical or numerical simulation models to optimise effectiveness. When planning harbor construction or harbor extension, it is highly recommended to investigate the optimal layout (length, width, orientation, inflow configuration) to minimize siltation.  

{| border="0" align="center"
|-
| valign="top"|
[[File:HarborEddy.jpg|thumb|left|450px|<small>Figure 6: Schematic picture of the horizontal flow pattern at the entrance of a tidal harbor when flow adaptation constructions are absent. Left panel: Ebb. Right panel: Flood. The exchange flow at the harbor entrance during flood enhances sediment inflow. Flow separation generates primary and secondary eddies where sediments are deposited. </small> ]]
| valign="top"|
[[File:HarborEddyWeakening.jpg|thumb|left|450px|<small>Figure 7: Schematic picture of flow adaptation constructions that counteract the formation of horizontal exchange flows at the harbor entrance during flood. Left panel: Sill and current deflection wall. Right panel: Upstream permeable groin. </small>]]
|}

====Sediment trap====
The purpose of sediment traps is to collect the sediments entering the harbor in a restricted space to reduce the area where dredging is needed. A sediment trap is an artificial local bottom depression where sediments preferentially settle. The trap location must be chosen carefully. Settling of sediment from the water column is promoted by the reduced flow velocity at the sediment trap. However, the trapping efficiency is greatest for sediments moving along the bottom, such as fluid mud. Several harbor basins along the Meuse in Rotterdam have been equipped with sediment traps. A model simulation of mud sedimentation in a harbor basin with sediment trap is discussed in [[Sediment deposition and erosion processes#Basin sedimentation]]. 


[[File:WaterInjectionDredging.jpg|thumb|right|300px|<small>Figure 8: Cartoon of water injection dredging. </small>]]

====Water injection dredging (WID) to keep sediment moving====  
The purpose of water injection dredging is to create density currents near the bed by fluidizing fine-grained sediments. The fluidized sediment (fluid mud) can leave the harbor by gravity-driven flow or by tide-induced flow and entrainment<ref> Kirichek, A., Cronin, K., de Wit, L. and van Kessel, T. 2021. Advances in Maintenance of Ports and Waterways: Water Injection Dredging. In: Sediment Transport - Recent Advances. A.J. Manning (ed.) DOI: http://dx.doi.org/10.5772/intechopen.98750</ref> (Fig. 8). Fluidized sediment also flows more easily toward sediment traps; the effectiveness of sediment traps will thus be enhanced by applying WID. Another potential benefit of WID is the reduction of the yield stress of bottom sediment. This increases the effective nautical depth by enabling vessels to navigate through the WID-induced fluid mud layer<ref> Kirichek, A., Chassagne, C., Winterwerp, H. and Vellinga, T. 2018. How navigable are fluid mud layers? Terra et Aqua, 15</ref>. See also the article [[Nautical depth of harbors and fairways]]. <br clear=all>

====Minimization of recirculation from dredge spoil disposal sites====
Dredged materials, if not contaminated, are often disposed in the water body some distance from the dredged site. If the site is too close or not well chosen a considerable part of the dredged material may circulate back to the harbor. For well-chosen disposal sites recirculation will be prevented or strongly reduced. However, transport of dredged material to remote disposal sites increases the costs of maintenance dredging. An alternative is to cap disposed fine materials with clean course sediment. It should be noted that, according to the OSPAR convention, sea disposal of dredged spoil is subject to licensing to ensure that environmental change beyond the boundaries of the disposal site are as far below the limits of allowable environmental change as practicable<ref>OSPAR 1998. Guidelines for the Management of Dredged Material. Ministerial Meeting of the OSPAR Commission Summary Record OSPAR 98/14/1-E, Annex 43</ref>.

Silt screens are sometimes used as a physical barrier to sediment entering a harbor. However, they hinder navigation and are  only partially effective. Silt screens can be useful to mitigate the environmental impact of engineering works by reducing the turbidity outside working areas in estuaries. Air bubble curtains have also been tested for their capacity to reduce harbor siltation. Air bubble curtains do not interfere with navigation but their performance is limited. 

====Beneficial Uses====
There is a wide variety of beneficial uses of dredged material depending on the physical and chemical characteristics. Some beneficial uses may generate revenues; examples are:
* Engineered uses - land creation and improvement, beach nourishment, offshore berms, capping material and backfill; 
* Uses in agriculture, horticulture, aquaculture and in construction material; 
* Environmental enhancement - restoration and establishment of wetlands, terrestrial habitats, nesting islands, and fisheries.
However, several uses involve costs for dewatering and decontamination of dredged material<ref> Maher, A., Scott Douglas, W., Jafari, F. and Pecchioli, J. 2013. Processing and Beneficial Use of Fine-Grained Dredge Material: A Manual for Engineers. New Jersey Department of Environmental Protection, Rutgers Center for Advanced Infrastructure and Transportation, report 193-RU2763</ref>.


==Siltation in fairways==

Here we consider an artificial dredged navigation channel from deep water to a coastal harbor, thus crossing a (relatively) shallow nearshore zone. Siltation in such fairways occur with fine (silt) and coarse (sand) sediment, which is transported as suspended load and/or as bed load. Full morphodynamic analyses are required for fairways in estuaries to assess the siltation rates, as the navigation channel and ambient water system strongly interact. This is for instance the case in the Western Scheldt where fairway deepening influences the morphodynamic development of this estuary<ref> Jeuken, M.C.J.L. and Z.B. Wang. Impact of dredging and dumping on the stability of ebb–flood channel systems. Coastal Engineering 57, 553–566</ref> .

[[Image: HanFig6.jpg|thumb|250px|left|<small>Figure 9: Refraction of current oblique to navigation channel. <math>U</math>=current velocity and <math>h</math>=depth.</small>]]




Fig. 9 depicts the refraction of a current crossing a navigation channel at an angle. To understand this picture, the scales of the system have to be considered. A cross-current experiences a sudden increase in water depth (the width of the channel is small compared to the system’s dimensions), thus the flow decelerates locally, as the water flux does not change. However, along the channel, the channel dimensions are much larger than the channel width. Given a constant pressure gradient along the channel, the larger depth within the channel reduces the effective hydraulic drag accelerating the current. Thus, depending on the angle of incidence <math>\alpha_0</math>, the flow accelerates of decelerates. A semi-quantitative analysis is presented below.
<br clear=all>

The flow velocity in the channel as a function of relative channel depth and angle of incidence (with <math>C</math>=[[Bed roughness and friction factors in estuaries|Chézy coefficient]], <math>h</math>=depth) can be derived from simple geometric arguments:

<math>\large\frac{U_{y,1}}{U_{y,0}}\normalsize =\large\frac{h_0}{h_1}\normalsize ,\quad \large\frac{U_{x,1}}{U_{x,0}}\normalsize =\large\frac{C_1}{ C_0} \left(\frac{h_1}{h_0} \right)^{1/2}\normalsize , \qquad(4)</math>

where <math>C \approx C_1 \approx C_2</math> and 

<math>U_1=U_0 \large\frac{h_0}{h_1}\normalsize [ \sin^2 \alpha_0 + \large \left (\frac{h_1}{h_0} \right )^3\normalsize \cos^2 \alpha_0 ]^{1/2} , \quad \tan \alpha_1=\large \left (\frac{h_0}{h_1} \right )^{3/2}\normalsize \tan \alpha_0. </math> 

The subscripts <math>x, y</math> indicate the along-channel and cross-channel projections of the velocity, respectively.  


[[Image: HanFig7NEW.jpg|thumb|350px|left|<small>Figure 10: Ratio flow velocity inside and outside channel according to equation 4 (black curves) and the three-dimensional approach by Jensen et al. (red curves)<ref>Jensen, J.H., Madsen, E.Ø. and Fredsøe, J., 1999, “Oblique flow over dredged channels – I: Flow description”, ASCE, Journal of Hydraulic Engineering, Vol 125, No 11, pp 1181-1189</ref><ref>Jensen, J.H., Madsen, E.Ø. and Fredsøe, J., 1999, “Oblique flow over dredged channels – II: Sediment transport and morphology”, ASCE, Journal of Hydraulic Engineering, Vol 125, No 11, pp 1190-1198</ref>.</small>]]

The ratio between the flow velocities inside and outside the navigation channel are plotted in Fig. 10 as a function of the angle of incidence of the ambient flow. Note that up to quite large angles, the flow velocity within the channel is larger than outside the channel. A larger flow velocity implies a larger transport capacity. The implications for channel self-cleansing are discussed below. 

[[Image: HanFig8.jpg|thumb|300px|right|<small>Figure 11: Cartoon of extra water to be attracted. </small>]]

A larger flow velocity through a larger cross section implies a larger specific discharge, and the extra water has to come from the channel’s surrounding waters. However, this  extra water carries also extra sediment. For simplicity, only along-channel flow is discussed, as illustrated in Fig. 11. Continuity and substitution from equation (4) yields a relation for an “effective channel width” <math>b_0</math> representing the area from which the “extra” water and sediment is attracted,

<math>\large\frac{b_0}{b_1}\normalsize =\large\frac{U_1 h_1}{U_0 h_0}\normalsize =\large\frac{C_1}{C_0} \left (\frac{h_1}{h_0} \right )^{3/2} \normalsize. \qquad(5)</math>

Ignoring settling and erosion time lag effects, the equilibrium transport <math>T_e</math> is represented by a power of the local flow velocity: <math>T_e \propto u_*^n</math>, where <math>u_*</math> = shear velocity and in which <math>n</math> = 3 represents bed load and <math>n</math> = 4 – 6 represents suspended load (see e.g. [[Sediment transport formulas for the coastal environment]]). Substitution from equations (4) and (5) yields a relation for the sediment transport capacity inside the channel in relation to the sediment transport capacity outside the channel:

<math>\large\frac{T_{e,1}}{T_{e,0}}\normalsize =\large\frac{b_1}{b_0} \left (\frac{u_{*,1}}{u_{*,0}} \right )^n\normalsize =\large\frac{b_1}{b_0} \left (\frac{U_1 C_0}{U_0 C_1} \right )^n\normalsize =\large\frac{b_1}{b_0} \left (\frac{h_1}{h_0} \right )^{n/2}\normalsize = \large\frac{C_0}{C_1} \left (\frac{h_1}{h_0} \right )^{(n-3)/2} \normalsize. \qquad(6)</math>

As <math>C_1</math> is (slightly) larger than <math>C_0</math>, the sediment transport capacity within the channel is smaller than outside in case of bed load (<math>n</math> = 3). Thus the channel will silt up. However, for suspended load, equation (6), when <math>n</math> = 4 – 6, equation (6) predicts that the fairway is self-cleansing, and possibly even eroding. Advanced three-dimensional numerical sediment transport models yield a similar prediction, at least qualitatively. This result conflicts with observed fairway sedimentation, however.

This paradox illustrates that some important phenomena are missing in the above analysis. This holds in particular for the omission of wave effects. Waves stir up and can transport large amounts of sediments in the shallows surrounding the fairway, whereas wave activity on the channel’s bed is small. [[Shallow-water wave theory|Linear wave theory]] gives a first-order estimate of the ratio of maximum wave-induced bed shear stresses:

<math>\large\frac{\hat \tau_{b,channel}}{\hat \tau_{b,ambient}}\normalsize \approx \large \left (\frac{u_{b,channel}}{u_{b, ambient}} \right )^2\normalsize \approx \large \left (\frac{\sinh kh_{ambient}}{\sinh kh_{channel}} \right )^2 \normalsize  \approx  \large \left (\frac{h_{ambient}}{h_{channel}} \right )^2 \normalsize ,        \qquad(7) </math>

where the subscript <math>b</math> indicates the near-bed value. The most right member follows from the inequality <math>k=2 \pi / \lambda << 1 /h</math>  (wavelength <math>\lambda >> h </math>).

[[Image: HanFig9new.jpg|thumb|350px|right|<small>Figure 12: Ratio of maximum wave-induced bed shear stresses on the channel bed and in the surrounding waters as a function of relative channel depth. </small>]]

This ratio is plotted in Fig. 12, showing that bed shear stresses on the channel bed decrease rapidly with channel depth. Moreover, wave-induced shear stresses are often larger than flow-(tide) induced stresses. Hence, ignoring effects of waves on the siltation rates in navigation channels may give an entirely wrong picture of channel’s maintenance needs. 

Other effects, amongst which estuarine circulation, may also affect channel siltation. However, these are not accounted for in the present zero-order assessment



This analysis yields two important observations:
# Channel siltation may vary strongly over the year, in particular if the local climate is characterized by seasonal variations in wave conditions,
# Channel orientation may strongly influence siltation rates, thus maintenance costs.







Literature provides some simple engineering rules for assessing channel siltation <math>F_c</math> with channel width B:

* Mayor-Mortensen-Fredsøe<ref> Mayor-Mora, R., P. Mortensen and J. Fredsoe, 1976. Sedimentation studies on the Niger River Delta. 15th ICCE, Honolulu, Hawaii </ref>:

<math>F_c = \large\frac{T_0}{\cos(\alpha_0-\alpha_1)} \normalsize \left [ 1 - \exp \large \left ( - \frac{A_m h_0 B}{h_1 \sin \alpha_1 \cos(\alpha_0-\alpha_1)} \right) \normalsize \right ] -T_1 \left [ 1- \exp \large \left (-\frac{A_m B}{\sin \alpha_1} \right ) \normalsize  \right ] \sin \alpha_1 , \qquad(8) </math>

with <math>A_m=W_s^2/ \epsilon_1 U_1 , \quad \epsilon_1=0.085 h_1 u_{*,1} </math>.


* Bijker<ref>Bijker, E., 1980. Sedimentation in channels and trenches. 17th ICCE, Sydney, Australia</ref> :

<math>F_c = \left (\large \frac{T_0}{\cos(\alpha_0-\alpha_1)} \normalsize -T_1\right ) \left[  1- \exp \large \left (-\frac{A_b B}{\sin \alpha_1} \right ) \normalsize  \right ] \sin \alpha_1 , \qquad(9) </math>

with <math>A_b=\large\frac{F_B W_s}{h_1 U_1}\normalsize, \quad F_B=\large\frac{a_1}{a_2-a_3}\normalsize </math> and

<math>a_1=\large\frac{u_{*,1}}{u_{*,0}-u_{*,1}}\normalsize, \quad  a_2=\large\frac{W_s}{0.085 u_{*,1}}\normalsize \left [1- \exp \large \left (-\frac{W_s}{0.085 u_{*,0}}  \right ) \normalsize \right ], \quad a_3=\large \left (\frac{u_{*,1}}{u_{*,0}} \right )^3 \frac{W_s}{0.085 u_{*,0}}\normalsize \left [ 1- \exp \large \left (-\frac{W_s}{0.085 u_{*,1}}  \right ) \normalsize \right ]</math>.


* Eysink & Vermaas<ref> Eysink, W. and H. Vermaas, 1983. Computational method to estimate the sedimentation in dredged channels and harbor basins in estuarine environments. Int. Conf. on Coastal and Ports Engineering in Developing Countries, Colombo, Sri Lanka; also: Delft Hydraulics, Publication No 307</ref>:

<math>F_c = \left ( \large\frac{T_0}{\cos(\alpha_0-\alpha_1)} \normalsize - T_1 \right ) \left [ 1- \exp \large \left ( - \frac{A_{ev} B}{h_1 \sin \alpha_1}  \right ) \normalsize  \right ] \sin \alpha_1 , \qquad(9) </math> 

with <math>A_{ev} = 0.03 \large\frac{W_s}{u_{*,1}}\normalsize \left ( 1+\large\frac{2W_s}{u_{*,1}}\normalsize \right ) \left [ 1+4.1 \large \left (\frac{k_s}{h_1} \right )^{0.25}\normalsize \right ]</math>.

Note that from our analysis above, the shear velocity has to be corrected for the effects of waves, otherwise misleading results will be obtained. 

* The fourth model is by Allersma, known as the “volume-of-cut” method, but which has never been published:

<math>h_{T*}=h_0-(h_0-h_e) \left [ 1- \exp \large \left (-\frac{v T_*}{h_0} \right ) \normalsize  \right ]</math>.

This simple model is particularly useful in case of deepening an existing channel – data on previous dredging volumes can be used for calibration of the parameter <math>v</math>. <math>T_*</math> is an arbitrary time scale (one year, five years, ..), <math>h_0</math> the initial channel depth, and <math>h_e</math> the channel’s equilibrium depth (which may be the local water depth in open water). 


==List of symbols==

{|
|- style="font-weight:bold; text-align:center; background:lightgrey"	
! width=20% style=" border:1px solid gray;"| parameter
! width=50% style=" border:1px solid gray;"| definition
! width=10% style=" border:1px solid gray;"| unit
|-	
| style="border:1px solid gray;"|	
| style="border:1px solid gray;"|
| style="border:1px solid gray;"|	
|-	
| style="border:1px solid gray;"|	<math>A</math>
| style="border:1px solid gray;"|	cross-section harbor mouth
| style="border:1px solid gray;"|	<math>m^2</math>
|-	
| style="border:1px solid gray;"|	<math>A_b, A_{ev}, A_m</math>
| style="border:1px solid gray;"|	empirical coefficients
| style="border:1px solid gray;"|	
|-	
| style="border:1px solid gray;"|	<math>b</math>
| style="border:1px solid gray;"|	channel width
| style="border:1px solid gray;"|	<math>m</math>
|-	
| style="border:1px solid gray;"|	<math>C</math>
| style="border:1px solid gray;"|	Chezy coefficient
| style="border:1px solid gray;"|	<math>m^{1/2}/s</math>
|-	
| style="border:1px solid gray;"|	<math>c_a</math>
| style="border:1px solid gray;"|	ambient suspended particulate matter (SPM) concentration
| style="border:1px solid gray;"|	<math>kg/m^3</math>
|-	
| style="border:1px solid gray;"|	<math>c_e</math>
| style="border:1px solid gray;"|	equilibrium concentration in harbor
| style="border:1px solid gray;"|	<math>kg/m^3</math>
|-	
| style="border:1px solid gray;"|	<math>c_h</math>
| style="border:1px solid gray;"|	SPM concentration in harbor
| style="border:1px solid gray;"|	<math>kg/m^3</math>
|-	
| style="border:1px solid gray;"|	<math>F_s</math>
| style="border:1px solid gray;"|	siltation rate in harbor
| style="border:1px solid gray;"|	<math>kg/s</math>
|-	
| style="border:1px solid gray;"|	<math>F_h</math>
| style="border:1px solid gray;"|	siltation rate in channel
| style="border:1px solid gray;"|	<math>kg/s</math>
|-	
| style="border:1px solid gray;"|	<math>F_B</math>
| style="border:1px solid gray;"|	empirical coefficient
| style="border:1px solid gray;"|	
|-	
| style="border:1px solid gray;"|	<math>f_d, f_e, f_{d,t}, f_{e,t}</math>
| style="border:1px solid gray;"|	empirical coefficients
| style="border:1px solid gray;"|	
|-	
| style="border:1px solid gray;"|	<math>h</math>
| style="border:1px solid gray;"|	depth
| style="border:1px solid gray;"|	<math>m</math>
|-	
| style="border:1px solid gray;"|	<math>k_s</math>
| style="border:1px solid gray;"|	Nikuradse roughness height
| style="border:1px solid gray;"|	<math>m</math>
|-	
| style="border:1px solid gray;"|	<math>n</math>
| style="border:1px solid gray;"|	power in transport formula
| style="border:1px solid gray;"|	
|-	
| style="border:1px solid gray;"|	<math>Q</math>
| style="border:1px solid gray;"|	exchange flow rate
| style="border:1px solid gray;"|	<math>m^3/s</math>
|-	
| style="border:1px solid gray;"|	<math>Q_d, Q_e, Q_t</math>
| style="border:1px solid gray;"|	same; density-, entrainment-, tide-induced
| style="border:1px solid gray;"|	<math>m^3/s</math>
|-	
| style="border:1px solid gray;"|	<math>S</math>
| style="border:1px solid gray;"|	harbor projected surface
| style="border:1px solid gray;"|	<math>m^2</math>
|-	
| style="border:1px solid gray;"|	<math>T</math>
| style="border:1px solid gray;"|	tidal period
| style="border:1px solid gray;"|	<math>s</math>
|-	
| style="border:1px solid gray;"|	<math>T_*</math>
| style="border:1px solid gray;"|	characteristic time scale
| style="border:1px solid gray;"|	<math>s</math>
|-	
| style="border:1px solid gray;"|	<math>T_e</math>
| style="border:1px solid gray;"|	equilibrium sediment transport
| style="border:1px solid gray;"|	<math>kg/m</math>
|-	
| style="border:1px solid gray;"|	<math>T_0 (T_1)</math>
| style="border:1px solid gray;"|	sediment transport outside (inside) channel
| style="border:1px solid gray;"|	<math>kg/m</math>
|-	
| style="border:1px solid gray;"|	<math>U</math>
| style="border:1px solid gray;"|	characteristic velocity
| style="border:1px solid gray;"|	<math>m/s</math>
|-	
| style="border:1px solid gray;"|	<math>u_*</math>
| style="border:1px solid gray;"|	shear velocity
| style="border:1px solid gray;"|	<math>m/s</math>
|-	
| style="border:1px solid gray;"|	<math>u_{cr}</math>
| style="border:1px solid gray;"|	critical velocity
| style="border:1px solid gray;"|	<math>m/s</math>
|-	
| style="border:1px solid gray;"|	<math>u_h</math>
| style="border:1px solid gray;"|	characteristic velocity in harbor basin
| style="border:1px solid gray;"|	<math>m/s</math>
|-	
| style="border:1px solid gray;"|	<math>V</math>
| style="border:1px solid gray;"|	harbor volume
| style="border:1px solid gray;"|	<math>m^3</math>
|-	
| style="border:1px solid gray;"|	<math>W_s</math>
| style="border:1px solid gray;"|	settling velocity
| style="border:1px solid gray;"|	<math>m/s</math>
|-	
| style="border:1px solid gray;"|	<math>x</math>
| style="border:1px solid gray;"|	coordinate along channel
| style="border:1px solid gray;"|	<math>m</math>
|-	
| style="border:1px solid gray;"|	<math>y</math>
| style="border:1px solid gray;"|	coordinate perpendicular to channel
| style="border:1px solid gray;"|	<math>m</math>
|-	
| style="border:1px solid gray;"|	<math>\alpha</math>
| style="border:1px solid gray;"|	efficiency parameter for harbor siltation
| style="border:1px solid gray;"|	
|-	
| style="border:1px solid gray;"|	<math>\alpha_0</math>
| style="border:1px solid gray;"|	flow angle ambient current
| style="border:1px solid gray;"|	
|-	
| style="border:1px solid gray;"|	<math>\alpha_1</math>
| style="border:1px solid gray;"|	flow angle within channel
| style="border:1px solid gray;"|	
|-	
| style="border:1px solid gray;"|	<math>\Delta \rho_s</math>
| style="border:1px solid gray;"|	salinity-induced density difference
| style="border:1px solid gray;"|	<math>kg/m^3</math>
|-	
| style="border:1px solid gray;"|	<math>\epsilon_1</math>
| style="border:1px solid gray;"|	empirical coefficient
| style="border:1px solid gray;"|	
|-	
| style="border:1px solid gray;"|	<math>v</math>
| style="border:1px solid gray;"|	empirical coefficient
| style="border:1px solid gray;"|	<math>m/s</math>
|}


==Related articles==

[[Dynamics of mud transport]]

[[Nautical depth of harbors and fairways]]

[[Sediment deposition and erosion processes]]

[[Sediment transport formulas for the coastal environment]]

[[Sand transport]]

[[Manual Sediment Transport Measurements in Rivers, Estuaries and Coastal Seas]]


==Further reading==

PIANC, 2008. Minimising harbour siltation, Pianc. Brussels, Belgium.

http://www.leovanrijn-sediment.com/papers/Harboursiltation2012.pdf


==References==

<references/>


{{author
|AuthorID=15072
|AuthorFullName=Johan Winterwerp
|AuthorName=Johan Winterwerp}}



{{Review
|name=Job Dronkers|AuthorID=120|
}}


[[Category:Physical coastal and marine processes]]
[[Category:Estuaries and tidal rivers]]
[[Category:Sediment]]