Coastal Engineering 64 (2012) 102–112

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Coastal Engineering

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Mud-induced wave damping and wave-induced liquefaction

Johan C. Winterwerp a,b,⁎, Gerben J. de Boer a,b, Gert Greeuw c, Dirk S. van Maren a,b

a Deltares (Delft Hydraulics), Marine and Coastal Systems, PO Box 177, 2600 MH Delft, The Netherlandsb Delft University of Technology, Faculty of Civil Engineering and Geosciences, Section of Environmental Fluid Mechanics, PO Box 5048, 2600 GA Delft, The Netherlandsc Deltares, Geo-engineering, PO Box 177, 2600 MH Delft, The Netherlands

⁎ Corresponding author at: Deltares (Delft HydraulicsPO Box 177, 2600 MH Delft, The Netherlands.

E-mail address: han.winterwerp@deltares.nl (J.C. W

0378-3839/$ – see front matter © 2012 Elsevier B.V. Alldoi:10.1016/j.coastaleng.2012.01.005

a b s t r a c t

a r t i c l e i n f o

Article history:Received 29 October 2010Received in revised form 20 January 2012Accepted 23 January 2012Available online 21 February 2012

Keywords:Fluid mudLiquefaction of mudCyclical triaxial testsWave dampingWadden Sea

This paper describes the results of a study on potential wave damping in the Dutch Wadden Sea by viscouswave energy dissipation within soft and/or liquefied mud layers. The results are relevant for wave heightboundary conditions for levee design along the mainland coast.From a site inspection, we observed that the mud fields along the Wadden Sea coast are quite dynamic, andseem to migrate in alongshore direction during storm conditions. A site along the Wadden Sea coast withextensive soft mud deposits, known as Zwarte Haan, was selected for more detailed laboratory and modelanalyses. Both field and laboratory measurements on sediment cores indicated strengths of a few kPa, i.e.slightly above the Liquid Limit. To assess the sensitivity of the mud deposits to liquefaction induced byincoming waves, we carried out cyclical triaxial tests in the laboratory on a series of samples taken fromthe field. The stresses prescribed were established from the wave heights and periods expected duringstorm conditions. However, even at stresses well beyond these values, the triaxial tests did not indicate fullliquefaction of the mud samples; only some frictional losses were monitored.These results are in contradiction with laboratory studies, where liquefaction has been measured frequently.However, our observations corroborate with other recent observations in literature, indicating that wavedamping does not occur during a storm, but afterwards. It is hypothesized that fluid mud, dissipating waveenergy, is formed by so-called auto-saturation and/or from deposition of mud eroded during that storm,and not from liquefaction during the storm. Wave damping then occurs through viscous dissipation in thefluid mud layers until these mud layers regain their initial strength through consolidation.Next, we assessed the degree of wave damping to be expected at the beginning of a storm, when the bed isstill unaffected and during/after storm conditions, when fluid mud is formed. We used the measured mudproperties and those typical for fluid mud, as found in the literature. This assessment was done throughsimulations with a new version of SWAN, referred to as SWAN-mud. These simulations suggest significantwave damping under fluid mud conditions, but only 10% damping at the beginning of the storm; moreover,the degree of wave damping over a muddy bed may decrease during the storm, as the soft mud deposits maybe eroded by the waves.Our analyses suggest that wave cannot liquefy soft mud beds under natural conditions, and that fluid mudlayers, damping incoming waves by viscous dissipation, are formed from mud eroded by the shear flowinduced by the waves' orbital motion. Note that in-situ pore water pressure measurements are required tocorroborate this hypothesis.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

The literature describes many studies on pronounced damping ofwaves along muddy coasts. For instance, Kurup (1972) reports almostcomplete wave dissipation over the mudbanks off Kerala coast, India.A few years later, in 1976, Tubman and Suhayda (1976) carried outmeasurements in the East Bay near the Mississippi mouth at two sta-tions 3.5 km apart at 20 and 3.5 m water depth, respectively. The bed

), Marine and Coastal Systems,

interwerp).

rights reserved.

consisted of thick layers (several 10 m) of soft mud with a strengthclose to the Liquid Limit (1.75–2.36 kPa). From an estimation offriction-induced wave damping over a sandy seabed, Tubman andSuhayda concluded that the wave energy must have been dissipatedby about 50% within the soft mud over this 3.5 km trajectory.

Wells and Kemp (1986) measured wave heights in the coastal areaof Suriname at three stations approximately 21.1, 11.7 and 4.3 km offthe coast at 7.1, 4.7 and 3.1 m water depth, respectively. This coastalarea is characterized by layers of soft Amazon mud on top of a thickpackage of well-consolidated clay (thickness exceeding a few 100 m).They observed 88% and 96% dissipation of the wave energy travelingfrom the most seaward monitoring station towards the shore. It is

103J.C. Winterwerp et al. / Coastal Engineering 64 (2012) 102–112

noted that dissipation was observed over the entire energy spectrum,though the longer waves dissipated more.

More recently, Sheremet and Stone (2003) analyzed wave measure-ments over a five day period in early 2001 at two stations at the 5 m iso-bath offshore Atchafalaya Bay. One station was located over a sandyseabed, while the other was placed over a muddy seabed. Thoughthese stations were 150 km apart, the incoming wave conditions andlocal wind climate were so similar that the wave conditions at these sta-tions should be comparable. They observed significantly lower waveheights (more than 70% smaller) at the muddy site in comparison withthe sandy site. This damping was attributed to dissipation of waveenergy in the mud layers. Moreover, they also observed wave dampingthroughout the entire energy spectrum, which was unexpected as theauthors presumed that the smaller waves would not be affected by themuddy seabed. Further measurements in March 2006 at the Atchafalayasite (Jaramillo et al., 2009; Sheremet et al., 2011) revealed a 50% wavedamping during the waning of a storm in the measuring period.

In the south of Brazil, Pereira et al. (2011) applied video techniquesat Cassino Beach, observing ~1 m fluidmud deposits on the foreshore ofthe beach, over a width of 100–400 m. When these mud deposits werepresent, no breaking waves were observed, and reduction in waveheight was estimated at 40%.

The current scientific understanding is that waves are damped byviscous dissipation in fluidmud layers, their viscosity being a few ordersofmagnitude larger than that ofwater (e.g. Gade, 1958;Maa andMehta,1990). The soft mud layers would liquefy within a short time period (ofthe order of minutes, or less) owing to large normal and shear stressesinduced by incoming waves, as was concluded from various laboratorystudies, amongst which those by Sakakiyama and Bijker (1989); DeWitand Kranenburg (1997); and Soltanpour et al. (2003). Thus a pictureemerges in which the first waves generated during a storm periodwould liquefy the soft mud bed, where upon subsequent waves wouldlose their energy rapidly by viscous dissipation. If so, coastal mud layerswould considerably contribute to coastal protection.

In anticipation of accelerated sea-level rise and in response to newinsights in the wave climate on the North Sea, Rijkswaterstaat (DutchMinistry of Transport and Public Works) initiated a study in TheNetherlands to improve wave height predictions along the DutchWadden Sea coast under storm conditions, generating boundary condi-tions for designing levee heights. The Wadden Sea in The Netherlandscan be characterized as a shallow tidal lagoon, protected by barrier

North SeaAmeland Tida

Wad

den

Sea

Afsluitd

ijk

Afsluitd

ijk

Fig. 1. Fines content in theWadden Sea Db16 μm, which is a proxy for the seabed mud contethe Dutch Wadden Sea coast (mainland).

islands; this lagoon has a width of about 20 km in the west, narrowingto about 8 km in the east. North Sea generated waves can penetratethrough the tidal inlets in between the islands, but lose their energy rap-idly in the shallow basin. Moreover, waves are also locally generated;these waves are important in particular during storms from northerndirection.

Overall, the Wadden Sea seabed is predominantly sandy. However,large and thick of layers soft mud are encountered along the mainlandcoast. Dike inspectors have reported visual observations on wavedamping over some of thesemud patches during storm conditions. Un-fortunately, we do not have any details on these observations. However,if wave damping would indeed occur, the design heights of primary seadefenses landward of these mud fields may be lowered. Therefore, afeasibility study was executed identifying possible wave dampingalong theDutchWadden Sea coast. This study addresses two questions:1) Can soft mud layers along the coast liquefy, given the local sedimentproperties andwave conditions? and 2) Howmuchwave damping is tobe expected over the liquefied and non-liquefied soft mud deposits?During liquefaction effective stresses reduce, while pore water stressesincrease. At full liquefaction, no effective stresses remain, and thepore water stress equals the total hydrostatic stress, i.e. of the totalsediment–water mixture. Liquefaction can be induced by waves,when the wave-induced stresses destroy the granular or cohesive skel-eton of the sediment bed. Liquefaction is an undrained process, andwater content does not change.

Section 2 of this paper describes a field inspection along the DutchWadden Sea coast, and summarizes the sediment properties that wereestablished. In Section 3, we discuss how the sensitivity to liquefactioncan be assessed from cyclical triaxial tests, and we present the resultsof such tests on samples from the Wadden Sea. Computations on theexpected wave damping, given the sediment properties, summarizedin Sections 2 and 3, are carried out with the new SWAN-mud model(Kranenburg et al., 2011), the results of which are presented inSection 4. In Section 5 we discuss the results and anticipate on the roleof softmud layers onflooding risks along theWadden Sea coasts, and ex-trapolate the results to other muddy coasts elsewhere in the world.

2. Field survey and sediment properties

On the basis of existing information, and inspection of satellite im-ages and aerial photographs, we have selected five locations along the

l Inlet

nt (mud content is equals to 2–3 times the fines content), and inspected locations along

Table 1Summary of the field campaign, showing sample locations and sample names, the av-erage yield strength in the top 40 cm, obtained with a hand vane (LL indicates close tothe Liquid Limit, i.e. around 1 kPa), and a description of the sediments in the top meter;location Zwarte Haan.

Location Samplenumber

Yield strength intop 40 cm (kPa)

Description

Z1 10b LL 1 m of mud (clay 17%) with a thin sandylayer 10 cm below the surface

Z2 6a LL 1 m of mud (clay 17%), weakly consolidatedZ3 LL 20 cm of mud (9% clay) on top of a 20 cm

thick sandy layer, on 60 cm of mud(12% clay)

Z4 7 5.7 40 cm of muddy fine sand with mud layerson sandy clay.

Z5 LL 40 cm of mud (10% clay) with sandy layerson 20 cm muddy sand, on 20 cm mud(15% clay), on muddy sand.

Z6 LL 20 cm mud with sand laminations, on30 cm muddy sand, on 20 cm mud(15% clay), on muddy sand

104 J.C. Winterwerp et al. / Coastal Engineering 64 (2012) 102–112

Friesland and Groningen coast where wave-induced liquefaction andmud-inducedwave dampingmight occur (Fig. 1). These sites were visit-ed in August 2008, measuring extension and thickness of the mudflats,and taking samples for further analyses. The area near Zwarte Haanseems to be persistently accreting, which is confirmed by local reportson 1.5 m of mud deposition over the last 50 years. This deposit tapersin seaward direction, extending some 500 to 1000 m offshore. NearHolwerd, only shoreline samples were taken, revealing a weakly consol-idated, relatively fresh mud layer with thickness of several tens of cm.Mud deposition rates in the fairway near Holwerd towards the islandof Ameland have increased over the past years. However, the seawardextent is limited owing to a nearby tidal channel, and profound anthro-pogenic influences. A weakly consolidated mud layer is found nearWierum. This layer is only several 100 mwide, and seems to be deposit-ed very recently, probably during thewinter storms of 2007/2008. Beforethatwinter, the areawas predominantly sandy. The area nearModdergat(translation: muddy hole) used to be muddy, as its name suggest, but at

Fig. 2. Sample locations

present the bed surface is sandy. The area in-between Marnewaard andWestgat has been quitemuddy aswell, but at present themud layers arerelatively thin and sandy.

The spatial pattern of mud accretion (Zwarte Haan, Holwerd, andWierum) contrasts with areas of mud erosion or zero accretion(Moddergat, Marnewaard). The sedimentary structure of the topmeter of the sediment deposits in a large part of the surveyed studyarea is stratifiedwith 20–30 cm thickmud layers, with some sand in be-tween; sometimes these sand layers measured a few dm as well. Boththe vertical structure and spatial distribution suggest that periods ofmud accretion and erosion succeed each other: large patches of mudare remobilized during storms and migrate eastward at timescales ofseveral years to decades. From these observations we conclude thatthemud deposits along theDutchWadden Sea coast are not very stable,and are characterized by considerable temporal variations. Moreover,this would imply that wave dissipation by mud may occur only locally.An exception to this cyclical behavior occurs at the area near ZwarteHaan, where a thick mud deposit accretes persistently. This behaviormay be related to the closure of the Zuiderzee (Fig. 1): after the con-struction of the Afsluitdijk, the tidal basins still develop towards anew morphologic equilibrium. Therefore, at Zwarte Haan mud-induced wave damping may occur currently, and in coming decades.An additional advantage of this site is the ongoing deployment of fourdirectional and non-directional wave-riders anchored in the TidalInlet of Ameland; Zwarte Haan is located at the head of this inlet. Inthe remainder of this paper we focus on the Zwarte Haan site, thoughwe discuss the liquefaction behavior of two samples from Wierumsite, as thesewere presumed soft enough to liquefy underwave stresseslarge enough.

Table 1 summarizes the observations on soil properties at ZwarteHaan (Fig. 2) during the field inspection, whereas Table 2 providessome soil mechanical characterizations. The samples tested are around,or above the Liquid Limit. Fig. 3 shows photographs of the samples, aftertransportation to the laboratory. Particle size distributions, determinedwith a Micometrics Sedigraph upon chemical dispersion, showmediandiameters around 5 μm for two samples, and around ten times larger forone other Zwarte Haan sample (e.g. Fig. 4). The undrained shearstrength cu was measured locally with a torvane, and in the laboratory

at Zwarte Haan site.

Table 2Sediment properties of tested samples collected near Zwarte Haan.

Location ρ D50 OC CaCO3 PI τpeak τres

kg/m3 μm [%] [%] [%] [kPa] [kPa]

Z1 1320 3 12.8 17.4 112 17 7Z2 1340 3 13.4 13.0 108 14 3Z4 1790 57 3.6 11.4 n.a. 27 4

ρ = wet density; D50 = median grain size; OC = organic content; CaCO3 = calciumcarbonate content; PI = plasticity index; τpeak and τres are peak and residual vanestrengths — τres is equivalent to the undrained shear strength cu.

105J.C. Winterwerp et al. / Coastal Engineering 64 (2012) 102–112

using a rotoviscometer on sediment cores taken during the August visit.The in-situ torvane measurements were not very successful since thestrength of the sediment was too close to the Liquid Limit (Table 1),while laboratory measurements reveal residual strengths of a few kPa(slightly exceeding the Liquid Limit, see Table 2), whereas the peakstrengths were much larger.

3. Cyclical triaxial tests

In this section, we discuss the response of the seabed to cyclicalstresses, as induced by surface waves. This response is measuredthrough cyclical triaxial tests in the laboratory on mud samples takenduring our site inspection. The frequency of the stress cycles followsfrom an analysis of the local wave climate. The waves in the WaddenSea, adjacent to the Frisian and Groningen coast are locally generated,and depth-limited, hence Hs≈0.4×h, where h is water depth, varyingfrom about 1 to 5 m. Typical wave periods amount to 3–4 s, so wetake T=4 s (Van der Westhuysen and Van Vledder, 2007). In thatcase, k≈0.25/tanh{0.25h}, where k is the wave number. Note thatunder extreme conditions, somewhat longer and higher waves mayoccur, but for the present analysismoremoderate conditions are appro-priate, with a peak period of around 4 s. All triaxial tests have thereforebeen performed at a frequency of 0.25 Hz.

We carry out the cyclical triaxial tests at the principal stresses givenin Eq. (1), assuming a coefficient of lateral stress K0=0.5, a typical valuefor normally consolidated soils. The amplitude of these stresses, to beset in the triaxial cell, is derived in Appendix A. Ideally, the thicknessof the samples in the triaxial cell should be varied, as the thickness ofthe mud deposits in the Wadden Sea, δm, varies. However, because theexperimental program was limited, the thickness of the sample in thetriaxial cell was kept constant at 7 cm. Note that it is expected that atthese in-situ δm/ω-values, wave damping can be close to maximum(e.g. Kranenburg et al., 2011), depending of the fluidmud's viscosity νm.

The samples were pre-consolidated for 16 h in the triaxial cell by im-posing an isotropic stress p0 under drained conditions, dissipating all

Fig. 3. Photographs of Wadden Sea samples, column 10 is from location Z1, column 6from location Z2, and the columns 4, 5, 7, 8 and 9 are from location Z4.

water overpressures. This consolidation stress is kept low, i.e. between1 and 2 kPa, since the in-situ stresses are low as well. Note that triaxialtests are normally performed at stress levels well above 10 kPa, hencewe operate the triaxial cell close to its lower operational conditions. Fur-thermore, we set the phase angle ψ=0, ignoring viscous effects to firstorder (see also Appendix A). Table 3 and Fig. 5 present the principalstresses σ1 and σ2 (back pressure) to be imposed in the triaxial cell,using the following formulation:

σ1 ¼ p0 þ S1 cos ωtð Þ þ 2ffiffiffi3

p

3S2 þ S3 cos ωtð Þ þ 1

3S4

� �1=2þ ∫

h

−z

ρwgdz′

σ2 ¼ p0 þ S1 cos ωtð Þ−ffiffiffi3

p

3S2 þ S3 cos ωtð Þ þ 1

3S4

� �1=2þ ∫

h

−z

ρwgdz′ with

S1 ¼ pwe−kz

S2 ¼ pwkze−kz

S3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipwkze

−kz 1−K0ð Þ∫z

ρb−ρwð Þgdz′s

S4 ¼ 1−K0ð Þ∫z

ρb−ρwð Þgdz′

ð1Þ

Fig. 4. Grain size and cumulative grain size distribution measured with Sedigraph:A): sample 10b (location Z1)B): sample 6a (location Z2)C): sample 7 (location Z4).

Table 3Stress conditions for triaxial tests.

g 9.81 [m/s2] Acceleration of gravityρw 1020 [kg/m3] Water densityh 4 [m] Water depthHs 1.6 =0.4×h [m] Significant wave heightT 4 [s] Peak wave periodω 1.57 [s−1]k 0.33 [m−1] Wave numberpw 4.0 [kPa] Pressure on bedz 0.3 [m] Depth below bed surfaceK0 0.5 [−] Coefficient of lateral stress

p0 [kPa] S1 [kPa] S2 [kPa] S3 [kPa] S4 [kPa]0.35 3.64 0.36 0.31 0.26

Table 4Triaxial test results of sediment samples collected near Zwarte Haan (6A, 7, and 10B)and near Wierum (12A and 13B).

Sample ρb σcon D50 umax u/σcon

kg/m3 kPa μm kPa [−]

6A 1340 1.7 3 1.6 0.947 1790 2.1 57 1.5 0.7110B 1320 1.6 3 1.3 0.8112A 1650 1.9 52 1.7 0.8913B 1430 1.2 5 0.9 0.75

ρ = wet bulk density; σcon = consolidation stress; D50 = median grain size; umax =maximum pore pressure due to cyclic loading.

106 J.C. Winterwerp et al. / Coastal Engineering 64 (2012) 102–112

Five samples have been selected for cyclic triaxial testing: three ofthese were collected near Zwarte Haan and two near Wierum. Table 4gives an overview of the sample properties, showing bulk densitiesranging from 1320 to 1790 kg/m3. This range can be attributed to vari-ations in sand and clay contents, as can be seen from the D50 values.Samples 6A, 10B and 13B have around 10–50% clay content, and lessthan 10% (fine) sand. The purpose of the triaxial tests was to assesswhether the samples would liquefy under the cyclic stresses to beexpected under storm conditions. The seven cmhigh samples, mountedin the triaxial cell, were prepared directly from thefield cores. The cyclicvertical loadwas increased during a 24 hour period from1 kPa to a levelof about 6 to 9 kPa, see Table 4. During the tests, the porewater pressurewithin the samples was measured continuously. An example of such asequence of loadings is shown in Fig. 6.

None of the samples liquefied, even at cyclic stresses well beyondthose to be expected under storm conditions. As an example, Fig. 7shows the response of the pore water pressure during the experimentson sample 6A. The pore pressure level approaches the consolidationstress of 2 kPa (full liquefaction occurs when the pore water stressequals the total stress, e.g. ρbgz), but a small difference remains. Thissmall difference seems enough to prevent liquefaction. Table 4 showsthat du/dσcyc, the relative variation in pore pressure during a loadingcycle, does not exceed 15%. Also at stresses three times larger, liquefac-tion was not observed.

It must be concluded, that the samples taken from the locationsZwarte Haan and Wierum do not liquefy under the stress levels ap-plied, which are even higher than to be expected under storm condi-tions. The two strain levels measured at 4 kPa, together withmaximum cyclical stresses, are presented in Table 5, showing thatthe soft clay samples 6A and 10B are very compressible and largestrains develop at cyclic loads beyond 4 kPa. In fact, these high strains

0

2

4

6

8

10

0 1 2 3 4

time [s]

pri

nci

pal

str

ess

[kP

a]

sigma 1sigma 2

Fig. 5. Example of total stress variation for conditions in Table 3, with total hydrostaticpressure (p0=h+z) added; σ2 = back pressure, σ1 = external loading (principalstresses).

(23 and 37%) show that the samples did fail to a large extent. Howev-er, no (full) liquefaction was observed, as noted above.

Therefore, the analysis has been re-focused on the deformation,pore pressure development and damping at various cyclic stresslevels. A well-known measure for damping in materials is the so-called damping ratio D, which is defined in f.i. Dutch standards ASTM-3999 for soils, using cyclic triaxial tests. D is the ratio of the energy dis-sipated by internal friction in one cycle divided by π times the totalenergy stored in the cycle, e.g. Fig. 8. For soils, this parameter is straindependent, and values ranging from 5 to 25% have been found for clays.

Fig. 9 shows an example of a cyclic stress–strain curve with hys-teresis. From Table 5 it can be inferred that the damping ratio D isnot very sensitive to the sample density, but increases with cycle am-plitude. On average, a damping ratio of about 10% was found.

4. Mud-induced wave damping

In this section, we assess the impact of soft mud deposits on thewave evolution in the Wadden Sea, using the parameters predictedby the laboratory experiments, assuming full fluid mud conditions,and conditions in between. The first set of parameters would be rele-vant for conditions at the beginning of the storm, whereas the secondfor conditions during and/or after the storm, which, however may notoccur in the Wadden Sea, e.g. Section 5. We use the recent SWAN-mud model (Kranenburg et al., 2011) to establish mud-inducedwave damping, and apply values for the fluid mud parameters fromliterature. In this paper, we evaluate the wave conditions for thestorm of 8–9 February, 2004.

The original SWAN-model, as used for assessing boundary condi-tions for levee design, was developed by Van der Westhuysen andVan Vledder (2007). Fig. 10 presents the computational domain and

Fig. 6. Cyclic stress development for test 6A.

Fig. 7. Pore pressure stress development for test 6A (see also Fig. 6).

Fig. 8. Definition of damping rate D from triaxial tests (D=ΔW1/4πW).

107J.C. Winterwerp et al. / Coastal Engineering 64 (2012) 102–112

underlying bathymetry. This model contains 286×412 computationalpoints, of which about 75% are active.

The wave boundary conditions for this model are obtained fromtwo wave buoys anchored about 5 km off the tidal inlet. However,for the February-storm, data from one of the buoys was unreliable,so that the other buoy was used for the entire northern boundary.The remaining boundary conditions are obtained through nestingfrom a larger model, covering almost 200×60 km2 of the North Seaand Wadden Sea. Next to incoming waves, the tidal basin under con-sideration is in particular subject to locally generated waves.

First, simulations are carried out with a common sandy bed, usingstandard SWAN settings, as outlined in the SWAN-manual (see alsoVan der Westhuysen and Van Vledder, 2007). SWAN is run in station-ary mode for the storm conditions of February 8, 2004, 22:30 h withwater levels and flow velocities obtained from hydrodynamic model-ing, and boundary conditions as discussed above. The computed sig-nificant wave height for reference conditions, i.e. without any effectof mud, is presented in Fig. 11, showing a strong reduction in waveheight over the tidal inlet; however, waves between 0.5 and 1 m arepredicted in the major part of the basin. The offshore waves have aperiod between 5 and about 16 s, whereas within the basin, thepeak period decreases to a few seconds (results not presented). Fora further analysis of the model results and a detailed comparisonwith observations, the reader is referred to Groeneweg et al. (2008).The simulations with mud are presented relative to this referencecondition.

The effect of soft (fluid) mud on the wave energy (e.g. wavedamping) is simulated with a new version of SWAN, a version includ-ing a two-layer schematization of mud-induced wave damping, re-cently developed. For details, the reader is referred to Kranenburg etal. (2011).

Table 5Pore pressure and damping ratio for two cyclic loads.

Sample du/dσcyc D1 σcyc1 Dmax σcycl-max ε4 εmax

[−] % kPa % kPa % %

6A 0.15 10.5 6 13.1 7.1 1.5 237 0.14 9.6 7.2 10.3 8.9 1.1 510B 0.10 9.7 6.2 14.7 7.9 2.5 3712A 0.20 11.6 5.2 12.4 6.6 1.7 1013B 0.09 9.8 4.8 11.6 6.3 2.2 16

u= pore water pressure, D= damping ratio (Fig. 8); ε4 = average strain level at 4 kPacyclic load; εmax = maximum strain level.

For the fluid mud configuration, we use the DELFT-settings, i.e. thedispersion relation described by Kranenburg et al. (2011), which isapplicable for shallow water and shallow mud layers, as encounteredin the Wadden Sea.

As discussed in Section 3, the samples taken at Zwart Haan, i.e. atthe head of the Tidal Inlet of Ameland could not be liquefied with thetriaxial tests. However, some damping (10%) was found. Therefore,three simulations were carried out:

1. Simulation 1 with soft mud parameters from the laboratory tests,i.e. ρmud=1200 kg/m3 and νmud=10 m2/s (expected to yield ca.10% damping),

2. Simulation 2 with fluid mud parameters as may be expected if themud would be fully fluid, i.e. ρmud=1200 kg/m3 andνmud=0.1 m2/s (typical values found in literature, e.g. Gade,1958; and Winterwerp et al., 2007), and

3. Simulation 3 with fluid mud parameters in between, i.e.ρmud=1200 kg/m3 and νmud=1 m2/s, closing the gap between0.1 and 10 m2/s.

Simulation 2 would represent conditions at which the mud bed isentirely liquefied, and Simulations 3 transient conditions, when themud regains its original properties, reflected by Simulation 1 condi-tions. The thickness and extension of the fluid/soft mud depositswere estimated from the field inspection — the actual applied fluidmud thickness is presented by contour lines in Fig. 11, its width mea-suring about 2000 to 3000 m.

The results of these simulations are also presented in Fig. 11 and de-tails of the computational results along a trajectory in the tidal inlet(white line in Fig. 11), showing that at the shore (2800 m from theedge of mud patch, Fig. 12c) all simulations predict similar wave

Fig. 9. Cyclic stress–strain curve test 7.

Fig. 10. Bathymetry with respect to Dutch ordnance level NAP of the SWAN computa-tional domain (the model contains 286×412 computational points, of which 75% areactive).

108 J.C. Winterwerp et al. / Coastal Engineering 64 (2012) 102–112

dissipation, even when the bed is sandy. However, the distribution ofdamping is very different. More than 90% of thewave energy is dampedover the first km of the mud patch for a viscosity of 0.1 m2/s (Fig. 12a;Simulation 2). The other two simulations predict only 10–20% dampingat that location. After 2200 m, the waves seem to have fully dissipatedover the softer mud patches (νmud=0.1 and 1 m2/s), while over thestiffer mud, and sandy bottom about 20% of the initial wave energy ismaintained.

It is further noted that all simulations predict the larger dampingat a frequency of around 0.3–0.4 Hz. Gade (1958) already showedthat the wave damping is a function of the relative mud layer thick-ness, i.e. δ� ¼ δmud=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2νmud=ω

p, where δmud = mud layer thickness.

With decreasing mud viscosity, wave damping first increases, thenpeaks and subsequently decreases. The Simulation 1 and 2 conditionsare found at the left side of the peak in damping. The reader is re-ferred to Kranenburg et al. (2011) for a more detailed discussion onthe frequency-dependency of wave dissipation over fluid mud layers.

Fig. 11. Plan view of computed significantwave height on February 8, 2004, 22:30 h for thecase with nomud (reference conditions). The dashed white line indicates the area of inter-est that is extracted for the simulations with non-liquefied mud: νmud=0.1m2/s,νmud=1m2/s and νmud=10m2/s. The thin black line depicts the perimeter of the fluidmud patch, the thick black line indicates where the mud thickness is 0.5 m. The solidwhite line indicates the ray for which more detail is shown in Fig. 12 for all 4 case.

5. Conclusions and discussion

This paper describes the results of a feasibility study to assesswhether or not soft mud deposits along the Wadden Sea coast in TheNetherlands can liquefy under wave-induced cyclical loading, and, ifso, to what extent the resulting fluid mud layers would damp thelocal wave energy. This study was initiated to establishing hydraulicboundary conditions for the design heights of levees along theWaddenSea coast. Further to a site inspection,we focus on the Zwarte Haan areaat the head of the tidal inlet of Ameland (Borndiep – AmelanderZeegat). Samples from this area (andWierum)were brought to the lab-oratory and were subjected to cyclical triaxial tests. At stresses well be-yond those expected to be induced by the local waves over the mudfields, the samples could not be liquefied according to the measuredpore water pressure response. However, induced by internal friction,some 10% damping of the total stored energy was measured.

Simulations with the new SWAN-mud model have been carried outfor Zwarte Haan conditions, as here stationary mud deposits werefound, and wave data are available. We assume an overall sandy seabedwith a patch of fluid mud, with dimensions estimated from a visual in-spection of the site, with viscosities varying from νmud=0.1, 1 and10 m2/s. The latter was the measured viscosity of the Wadden Sea sam-ples, the other values have been obtained from literature, being represen-tative iffluidmudwould occur in theWadden Sea. All simulations predictalmost full damping at the shore/dike. However themajority of the com-puteddamping in case of the softerfluidmud (νmud=0.1 and1 m2/s) oc-curs more than 500 m off the dike, whereas in case of the lower viscosity90% of the damping takes place over the first km of the mud patch.

Observations by a local dike inspector, that waves are significantlydamped along the Wadden Sea coast during storm conditions, the re-sults from our field work (suggesting mud deposits with strengthsaround the Liquid Limit), and triaxial tests at loads well beyondwave-induced stresses during storm conditions, seem inconsistent.The first two observations would suggest the occurrence of fluidmud layers during storm conditions, in line with previous (laborato-ry) studies, whereas the latter revealed that fluid mud cannot beformed from liquefaction (i.e. no sufficient increase in pore waterpressure). Possible causes for these contradictions are:

• The samples taken at the end of summer, were deposited at their loca-tion at least half a year before, and therefore (too) well consolidated,

• The samples have densified considerably, gaining strength, duringsampling and the subsequent transportation to and storage in thelaboratory.

• Only a thin top layer of the bed is sensitive to liquefaction.

Though consolidation may explain the contradictions between ourresults and those published previously, the other two possible causesare unlikely.

Therefore, we propose another explanation for the apparent incon-sistency between laboratory experiments and in situ observations. If itis assumed that the results of the triaxial tests are reliable and represen-tative, one has to conclude that themud deposits along theWadden Seacoast indeed cannot liquefy, neither fresh, nor consolidated, under theinfluence of waves: the mud is just too stiff, and the wave-inducedstresses too low.

Further to this hypothesis, an alternative explanation is then re-quired for wave damping over muddy beds, as observed in the labora-tory and in the field. The current scientific understanding is that thisdamping is governed by viscous dissipation within a fluid mud layer(and possibly some plastic effects). Hence, if our hypothesis is true,such a fluid mud layer should be formed otherwise than through lique-faction of the soft mud bed. The saturation concept introduced inWinterwerp (2001) provides a possible framework for this. This con-cept implies that a turbulent shear flow can carry a limited amount ofsediment in suspension. When this sediment is cohesive and consistsof flocs with a large water content, deposition upon exceeding of the

Fig. 12. Wave damping along the white ray shown in Fig. 11. Full spectra are shown at 1700 m off the coast (upper left panel), 600 off the coast (upper right panel) and 0 m off thecoast (lower left panel) (respectively 1100, 2200 and 2800 m from edge of mud layer). The cases with fluid mud (ν=0.1, 1 and 10 m2/s) are shown in gray shades; the black linerepresents the no-mud reference case.

109J.C. Winterwerp et al. / Coastal Engineering 64 (2012) 102–112

flow carrying capacity generates a soft layer (of fluidmud) at which in-terface turbulence is damped. Then, the carrying capacity drops furtheryielding a total collapse of the turbulent energy and of the suspendedsediment concentration profile. This snowball effect is presumed togovern the formation of fluid mud in tidal flow dominated conditions.

Further to this concept, Winterwerp (2001) postulated the occur-rence of auto-saturation under wave conditions, explaining completewave damping over the mudbanks off the Kerala coast (India). At theonset of the monsoon period, waves would start to erode the thicklayers of mud on the seabed by the shear flow induced by orbital mo-tion. When the amount of fine sediment in the water column exceedssaturation conditions, fluid mud is formed upon deposition of thesediment. Auto-saturation is a much slower process than liquefaction(the latter typically a few minutes, if occurring) owing to the limitedrate of erosion, even by the largest waves.

Indeed observations by Traykovski (2008) on Louisiana mud coastand by Vinzon et al. (2010) in Cassino Beach, Brazil suggest that themud bed is not being liquefied by incoming waves. In fact, thesetwo observations suggest that during storm conditions, part of themud bed is eroded, even slightly decreasing wave damping over themuddy bed. Only after some time, when fluid mud layers are formed(from deposition), significant wave damping was measured.

Further to these observations, Rogers and Holland (2009) measuredsignificant wave damping throughout the Cassino Beach experiment,except during periods with high wave events. This suggests that thesoft mud deposits are eroded and/or become mobile, traveling awayfrom the observational site.

The detailed measurements by Jaramillo et al. (2009) at 4–5 mwater depth off Atchafalaya coast (Louisiana, USA) give further supportto our hypothesis, showing the formation of a soft mud layer duringstorm conditions, reflected by a heightening of the lutocline. By defini-tion, this observation excludes the occurrence of liquefaction, as lique-faction is an undrained process, characterized by a break-down ofcohesive bonds within the bed at constant water content. This mud

layer was mobile, as appeared from the current measurements. Wavedamping was small, probably because of the small thickness of the mudlayer (10–15 cm), and its small viscosity (concentrations of a few g/l).

These measurements were further analyzed by Sheremet et al.(2011), and they report an evolution of the surficial bed layer from con-solidated mud through liquefaction, fluidmud formation, and hinderedsettling to gelled under-consolidated mud. Maximal damping was ob-served after the storm, when no fluid or under-consolidated mud wasobserved. The observed dissipation rates could be reproduced with anon-linear wave model, assuming a fluid mud layer, but at viscositieswhich are a bit small for fluid mud, i.e. 2·10−4 m2/s. We believe thatthis inconsistency can be explained from the measuring techniqueused. The ADCP deployed operates at 1 MHz frequency. At these fre-quencies, the acoustic signal will reflect at the interface of the gelled,under-consolidated mud, or consolidating mud. It is this mud layerwhich dissipates most wave energy. Note that in the nomenclatureused in the present paper, this under-consolidated or consolidatingmud is referred to as fluid mud. Hence, it would be the fluid mudformed from deposition that dissipates wave energy, as hypothesizedin the current paper.

Finally, we note that also the large wave damping measured in thecoastal zone of Suriname byWells and Kemp (1986)would corroborateour hypothesis. Augustinus (1978) and Allison and Lee (2004), andothers showed that the large, 30 km wide mudbanks in front of theGuiana's coastlines migrate westward through weather-side erosionby ocean waves, and subsequent deposition of the eroded sedimentson the bank's lee-sides. Transport of the fine sediments occurs in theform of a high-concentrated suspension, subject to saturation at thelee-side, forming extensive layers of fluid mud. Hence, also here, fluidmud is formed from deposition of a saturated suspension.

This picture would imply that Simulation 1 conditions, presented inSection 4, are representative for conditions at the onset of a storm, withlittle damping, whereas Simulation 2 conditions are representative forconditions in a later phase of the storm, or afterwards, when fluid

110 J.C. Winterwerp et al. / Coastal Engineering 64 (2012) 102–112

mud layers are formed by auto-saturation, or from deposition of muderoded during the storm. Simulation 3would represent transient condi-tions when the mud re-attains it original properties. The large-scalemud deposition patterns corroborate with observations that some ofthe mud fields in the Wadden Sea are strongly varying in time andspace. This is substantiated by observations by local residents, and con-served in vertical alterations of sand and much laminations typicallyone to several dm thick. Themud field sampled nearWierum, for exam-ple, was characterized by several dm thick mud layers with shearstrengths close to the Liquid Limit. These mud layers were formed dur-ing the previous storm season.

The large scale mud dynamics in theWadden Sea, the formation offairly thick layers of soft mud (few dm), and possible conditions ofsaturation and auto-saturation are subject of ongoing research. Cur-rently, we do not have proof that mud-induced wave damping mayoccur in the Wadden Sea.

Our presumption of mud erosion during a storm would also implythat for some locations, wave simulations for storm conditions shouldbe carried out assuming a sandy seabed, with energy dissipation bybed friction and wave breaking only. Only later, wave damping mayoccur over fluid mud layers.

We realize that many laboratory experiments have shown liquefac-tion of the bed, either from visual observations, or from pore watermeasurements, e.g. Sakakiyama and Bijker (1989); De Wit andKranenburg (1997); and Soltanpour et al. (2003). These laboratory re-sults are therefore in contradiction with our triaxial test analyses. Thiscontradiction may be explained by the fact that these experimentswere not carried out on unremolded soil samples. Then, these laborato-ry results may be misleading when extrapolating to real world condi-tions. We presume that liquefaction in the laboratory could occurbecause the laboratory beds were not well consolidated. In otherwords, the history of the mud deposits seems therefore of crucial im-portance. Indeed, yield strengths obtained by De Wit and Kranenburg(1997) measured a few Pa up to 20 Pa for the most consolidated sam-ples, i.e. far below the Liquid Limit, and far below the strengths mea-sured on the samples from the Wadden Sea. Unfortunately, we cannotsustain our arguments further, as it is not possible to retrieve more in-formation on these samples or from the other experiments, such asyield values, over consolidation rates, etc.

Yet, it is emphasized that our conclusions may not (yet) be general-ized, as more data are required to test our hypotheses. Moreover, sedi-ment deposits with a high content of organic material may remain softenough to induce significant wave damping. However, better than fur-ther laboratory studies, the only definite answer onwhether or not natu-ral mud beds are liable to liquefaction can be obtained solely from in-situpore water pressure measurements. Also, we recommendmeasuring theundrained shear strength of bed samples under undisturbed conditions,when wave attenuation studies are carried out.

Acknowledgments

This study was financed from the internal stimulation funds pro-vided by Deltares, further to an idea by André van Hoven. Moreover,we are grateful for assistance in the field work by André van Hoven,Marco de Kleine and Rein Lantman (all from Deltares), and by Koosde Vries (Medusa) for his guidance and input of local knowledge.Prof. A.J. Mehta commented on an earlier draft of this manuscript,and we like to thank two unknown reviewers and Todd Holland fortheir constructive comments.

Appendix A. Wave-induced stresses within the bed

In this appendix, we derive the stresses to be imposed in the triaxialtests, simulating wave-induced stresses within the bed. We start withthe wave-induced stresses on a consolidated mud surface. Then wecompute the stresses within the bed in x–z co-ordinates, where x is

the co-ordinate along the bed surface in wave direction, and z is theco-ordinate into the bed (positive downward), perpendicular to the x-co-ordinate. Finally, we derive the principal stresses and present theirvariation with time for some specific conditions.

Let us analyze the stresses in the seabed induced by regular waveswith wave height H, wave number k and wave number ω in x-direction. Then, the wave-induced water pressure Pw (we can omitthe effects of the wave-induced bed-shear stresses, which hardly gen-erate stresses within the bed) on the seabed reads (Flamant — seeYamamoto et al., 1978; Sumer and Fredsøe, 2002):

Pw ¼ ρwgH cos ωt−kx−ψð Þ2 cosh khf g ðA:1Þ

where h the local water depth, ρw is the water density, and ψ is thephase angle. The amplitude of the stresses pw on an elastic bed reads:

pw ¼ ρwgH2 cosh khf g

FF− tanh khf g with F ¼ Gk2

ρwω2 1−υð Þ ðA:2Þ

where G is the elastic shear modulus. For the time being, we assumenon-elastic bed properties (G=F=∞), so that the stress amplitudebecomes pw=ρwgH/2 cosh{kh}, e.g. Sumer and Fredsøe (2002). Thetotal stresses within the bed with respect to the equilibrium hydro-static pressure read (where superscript ▪w refers to wave-inducedstresses):

σ zz ¼ σwzz þ ∫

zρb−ρwð Þgdz′ ¼ pw 1þ kzð Þe−kz cos ωt−ψð Þ þ ∫

zρb−ρwð Þgdz′

σ xx ¼ σwxx þ K0 ∫

zρb−ρwð Þgdz′ ¼ pw 1−kzð Þe−kz cos ωt−ψð Þ þ K0 ∫

zρb−ρwð Þgdz′

σ xz ¼ σwxz ¼ pwkze

−kz sin ωt−ψð ÞðA:3Þ

We have also added the effect of isotropic stresses at depth z by thesediment's ownweight, with ρb(z) = bulk density of the sediment bed.Perpendicular to the waves in the horizontal plane (y-direction), thenormal stress is found from:

σyy ¼σw

xx þ σwzz

2þ K0 ∫

zρb−ρwð Þgdz′

¼ pwe−kz cos ωt−ψð Þ þ K0 ∫

zρb−ρwð Þgdz′ ðA:4Þ

The isotropic (p) and deviatoric (q) total stresses follow from (e.g.Winterwerp and van Kesteren, 2004):

p ¼ I1=3

q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI21−3I2

q ðA:5Þ

in which the stress invariants Ii are defined as:

I1 ¼ σxx þ σyy þ σ zzI2 ¼ σxxσyy þ σyyσ zz þ σ zzσxx−σxzσ zx

ðA:6Þ

as σxy=σyz=0. After some re-arrangement:

q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ2

xx þ σ2yy þ σ2

zz−σ xxσyy−σyyσ zz−σ zzσ xx þ 3σ2xz

qðA:7Þ

111J.C. Winterwerp et al. / Coastal Engineering 64 (2012) 102–112

Upon substitution of thewave-induced stresses into (A.3),wefind forthe total deviatoric stress q and its amplitude, q̂, including the isotropicpart:

q ¼ffiffiffi3

p p2wk

2z2e−2kz þ pwkze−kz cos ωt−ψð Þ 1−K0ð Þ ∫

z

ρb−ρwð Þgdz′ !

þ

þ13

1−K0ð Þ∫z

ρb−ρwð Þgdz′ !2!1=2

ðA:8aÞ

q̂ ¼ffiffiffi3

p p2wk

2z2e−2kz þ pwkze−kz 1−K0ð Þ ∫

zρb−ρwð Þgdz′

!þ

þ13

1−K0ð Þ∫z

ρb−ρwð Þgdz′ !2!1=2

ðA:8bÞ

Note that the deviatoric stress becomes independent of the effectivestresses in the soil if K0=1, for sandy systems K0≈0.7. The isotropictotal stress relative to the hydrostatic pressure p follows from substitu-tion of Eq. (A.6) into Eq. (A.5):

p ¼ 13

σxx þ σyy þ σ zz

� �¼ pwe

−kz cos ωt−ψð Þ þ 2K0 þ 13

∫z

ρb−ρwð Þgdz′ðA:9Þ

And the initial effective (isotropic) stress p0 prior to liquefaction:

p0 ¼ 2K0 þ 13

∫z

ρb−ρwð Þgdz′ ðA:10Þ

In the triaxial tests the vertical and horizontal principal stresses σ1

and σ2, respectively, are prescribed. These follow from ps and qaccording to:

3ps ¼ σ1 þ 2σ2 and q ¼ σ1−σ2 thus

σ1 ¼ ps þ 2q=3 and σ2 ¼ ps−q=3ðA:11Þ

Hence, σ1 and σ2 read:

σ1 ¼ pwe−kz cos ωt−ψð Þ þ 2K0 þ 1

3∫z

ρb−ρwð Þgdz′ þ

þ2ffiffiffi3

p

3

p2wk

2z2e−2kz þ pwkze−kz cos ωt−ψð Þ 1−K0ð Þ∫

z

ρb−ρwð Þgdz′ þ

þ13

1−K0ð Þ∫z

ρb−ρwð Þgdz′ !2!1=2

¼

¼ p0 þ σ ′1 ¼¼ p0 þ pwe

−kz cos ωt−ψð Þ

þ2ffiffiffi3

p

3

p2wk

2z2e−2kz þþpwkze−kz cos ωt−ψð Þ 1−K0ð Þ∫

z

ρb−ρwð Þgdz′

þ13

1−K0ð Þ∫z

ρb−ρwð Þgdz′ !2!1=2

ðA:12aÞ

σ2 ¼ pwe−kz cos ωt−ψð Þ þ 2K0 þ 1

3∫z

ρb−ρwð Þgdz′ þ

−ffiffiffi3

p

3

p2wk

2z2e−2kz þ pwkze−kz cos ωt−ψð Þ 1−K0ð Þ∫

z

ρb−ρwð Þgdz′ þ

þ 13

1−K0ð Þ∫z

ρb−ρwð Þgdz′ !2!1=2

¼

¼ p0 þ σ ′2 ¼¼ p0 þ pwe

−kz cos ωt−ψð Þ−ffiffiffi3

p

3

p2wk

2z2e−2kz þ

þpwkze−kz cos ωt−ψð Þ 1−K0ð Þ∫

z

ρb−ρwð Þgdz′þ 13

1−K0ð Þ∫z

ρb−ρwð Þgdz′ !2!1=2

ðA:12bÞ

The samples are pre-consolidated in the triaxial cell by imposingthe isotropic part p0 under drained conditions, i.e. water overpres-sures should dissipate. Furthermore, we set the phase angle ψ=0, ig-noring viscous effects. Then we derive at the following formulation:

σ1 ¼ p0 þ S1 cos ωtð Þ þ 2ffiffiffi3

p

3S2 þ S3 cos ωtð Þ þ 1

3S4

� �1=2þ ∫

h

−z

ρwgdz′

σ2 ¼ p0 þ S1 cos ωtð Þ−ffiffiffi3

p

3S2 þ S3 cos ωtð Þ þ 1

3S4

� �1=2þ ∫

h

−z

ρwgdz′ with

S1 ¼ pwe−kz S2 ¼ pwkze

−kz

S3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipwkze

−kz 1−K0ð Þ∫z

ρb−ρwð Þgdz′s

S4 ¼ 1−K0ð Þ∫z

ρb−ρwð Þgdz′

ðA:13Þ

Herewehave added the total hydrostatic pressure to guarantee pos-itive stresses. Table 3 summarizes the stresses imposed for the triaxialtests.

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