mudline1.pdf

XVIII. INTERACTION BETWEEN SURFACE WAVES AND

MUDDY BOTTOM SEDIMENTS

Joseph N. Suhayda Ports and Waterways Institute Louisiana State University

Baton Rouge, Louisiana 70803

ABSTRACT

Surface wave-induced bottom pressure fluctuations produce shear stresses in

soft muddy bottom sediments that cause the sediments to undergo oscillatory

motion. This motion can be described as a "mud-wave" and causes surface wave

properties to vary from those that occur over a rigid bottom. Theoretical studies

have attempted to describe this interaction using a variety of soil models, i.e.,

viscous fluid, elastic solid, viscoelastic material and nonlinear viscoelastic.

Although the experimental basis for evaluating the validity of these assumptions is

incomplete, it appears that a nonlinear viscoelastic soil model is required to

describe the observed behavior. An example of the interaction of hurricane waves

and soils found offshore of the Mississippi Delta is considered in detail. The soil

is described using a model which is nonlinear in relating shear strain to shear

stress and damping ratio. The surface wave-mud wave interaction for hurricane waves

is significant and causes wave heights of 70 ft (21.3 m) and 80 ft (24.4 m) in deep

water to decrease to values of from 10 ft (3.0 m) to 25 ft (7.6 m) at a water depth

of 50 ft (15 m). Soil response during this wave-mud interaction is greatest at

water depths of between 150 ft (45.7 m) and 250 ft (76.2 m). Maximum soil movements

of 1.5 ft (.46 m) are predicted to occur under hurricane waves. As a means for

making rough calculations of the wave-mud interaction a simplified technique for

making engineering predictions is presented. The technique is based upon a non-

linear stress-strain and damping-strain soil model and predicts surface wave

attenuation, soil shear stress and shear strain profiles.

INTRODUCTION

This paper presents a review of available information concerning the inter-

action of surface waves and muddy bottom sediments. Muddy bottom sediments occur in

a variety of coastal zones in the United States and in the world such as the

Guianas, the northern coast of China and in southwest India. The presence of muddy

bottom sediments has a profound effect on hydrodynamic processes, particularly on

surface waves. The purpose of this paper is to document the current state of

Lecture Notes on Coastal and Estuarine Studies Estuaririe Cohesive Sediment Dynamics Vol. 14

Copyright American Geophysical Union. Transferred from Springer-Verlag in June 1992.

402

knowledge of wave-mud Interactions, demonstrate the significance of this interaction

in offshore design wave forecasts and to present a simplified technique for pre-

dicting the attenuation of surface waves propagating over muddy bottom sediments.

MUDDY COASTS

Muddy coasts represent a type of coastline where fine grained sediment is

consistently present in the nearshore waters and on the shoreline. This type of

coast occurs at all latitudes and is often associated with large deltas, lagoons and

estuaries. In general, the offshore area has a smooth, low sloping profile, with

very turbid water occurring from the shoreline to several kilometers offshore.

Usually a mud flat, which is exposed at low tide, occurs in front of a vegetated low

backshore. In the United States, muddy coasts occur extensively in Louisiana and

Florida.

Morphology

Coasts having extremely high concentrations (i.e., 10,000 mg/1) of suspended

material in the nearshore water can have a wide range of geometries and forms. Some

of the most common shoreline features are the presence of nearshore mudbanks, beach

ridges and vegetation, i.e., tidal marsh, meadow, mangroves or nipa palms. The

vegetation is usually fronted by an unvegetated flat. There is rarely any coarse

material on the flats or forming beaches along the shoreline. A small scarp may

occur at the shoreline if the coast is undergoing erosion and retreat. Coarse sedi-

ment may occur as a deposit at the base of the scarp. When storm, hurricane or

typhoon waves attack the coast, this coarse lag is cast over the vegetated marsh and

forms thin linear detrital deposits. In addition to the coarse lag, erosion of the

vegetated shoreline yields large quantities of organic debris that adds to the

material in suspension in offshore waters.

A second kind of muddy coast consists of bare mud flats with only a few

scattered salt tolerant plants on its surface. This shoreline is most commonly

found in high tide regions or in areas where severe climates prevail (arid, semi-

arid or arctic). The shoreline position constantly changes as a result of water

level variations induced by tides, wind setup or atmospheric pressure changes.

Slopes on these flats are extremely low, ranging from 0.01 to 0.0005. A variety of

drainage patterns can be found on the flats, whose configuration changes due to the

influence of tides and coastal currents. On some bare mud flats thin narrow beach

ridges composed of sand or shell debris are found behind the normal high tide

shoreline.



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Offshore of muddy coasts the bottom may be covered by patches of "fluid mud".

This "fluid mud" is a thixotropic gel formed when suspended sediment concentrations

become very high, between about 10,000 and 250,000 mg/1 (40). This mud has the

consistency of yogurt and remains as an intact mass for long periods of time (i.e.,

months to years). Where fluid mud is present the bottom may be difficult to define

because mud concentration and soil density gradually increases from the water column

to the underlying consolidated sediment.

The concentration of fine grained sediment in the water column of a muddy coast

can undergo exceptionally large variations in short periods of time. Sediment con-

centrations in nearshore waters of muddy coasts are given in Table 1 (40). In the

Gulf of Thailand, suspended sediment concentrations at middepth in 10 m of water

varied from 52,000 mg/1 during ebb tide, to 2,100 mg/1 during slack tide, to 6,000

mg/1 during flood tide. Waves have caused suspended sediment concentrations to vary

from 1,500 mg/1 during low waves (wave height 1 m) to 6,200 mg/1 during high waves

(wave height 4 m). These high concentrations of sediments can cause a significant

increase in the viscosity of the water.

Wave-Bottom Interactions

As surface waves propagate over muddy bottom sediments an interaction occurs

which does not occur when waves pass over a sand or rock bottom. This interaction

involves the physical movement of the muddy sediments in a mass. Waves cause bottom

pressures which may be larger than muds can support. Under wave action the bottom

muds are alternately exposed to high and low pressures, that cause the muds to

oscillate. This movement can be visualized as a "mud-wave", as illustrated in

Figure 1.

The height of the mud-wave depends upon the geotechnical properties of the muds

and the amplitude and wavelength of the bottom pressures. Heights of mud-waves

range from a fraction of a centimeter under low surface waves to a meter under storm

waves. The mud-wave can have a significant effect on the surface wave because it

represents a boundary which moves and can absorb energy. Very large losses of

energy from surface waves can occur when mud-waves are generated. The wave height

can decrease by 10% in a distance of as little as a few tens of meters.

REVIEW OF THE LITERATURE ON WAVE-MUD BOTTOM INTERACTIONS

The present understanding of wave-mud interaction has developed over the last

30 years. The first studies occurred during the 1950's and were based upon field



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TABLE 1. ( a , b)

a. SURFACE SUSPENDED SEDIMENT CONCENTRATION

IN NEARSHORE MUDDY COASTAL WATERS

Location Concentration (mg/1)

Maximum Minimum

Louisiana Coast 6.4 x 101

1.0 x 10

East China Sea 7.0 x 101

5.0 x 10

Venezuela Coast 1.0 x 102

1.0 x 10

Gulf of San Miguel 2.0 x 102

6.0 x 101

Dutch Wadden Sea 6.2 x 102

5.0 x 101

Gulf of Thailand 9.7 x 102

1.0 x 10

Gulf of Ho Pai 1.0 x 103

1.0 x 102

British Guiana Coast 2.6 x 103

5.0 x 10

3 1 Surinam Coast 3.7 x 10 4.5 x 10

b . SURFACE SUSPENDED SEDIMENT CONCENTRATIONS IN RIVER

AND ESTUARINE WATERS ALONG MUDDY COASTS

Location Concentration (mg/1)

Maximum Minimum

Harlingvliet Estuary 1.2 x 102

3.8 x 101

(Netherlands)

Po River Plume 1.1 x 102

7.0 x 10

Ems Estuary 1.8 x 102

1.0 x 101

Thames Estuary 2.0 x 102

1.0 x 10

Mississippi River 3.1 x 102

4.0 x 10

(South Pass)

Chao Phya River 6.9 x 102

1.4 x 101

Surinam River 9.2 x 102

6.0 x 10

Bristol Channel 1.3 x 103

3.0 x 101



405

Figure 1. Schematic diagram showing the surface wave and the "mud-wave" along the water sediment interface.

observations by earth scientists. The interest in wave-mud interaction signifi-

cantly increased as a result of the development of offshore sites for oil

production. Mud-wave interactions were and are an important consideration in

developing design criteria for offshore oil production platforms. Theoretical and

laboratory studies were conducted and some predictive models were developed during

the 1970's. Present research is focused upon conducting field and laboratory

experiments to test existing predictive models and upon developing new predictive

models.

The first reference to wave interaction with soft bottom sediments is Ewing and

Press (14). It was reported that a kind of wave motion was likely to occur in soft

sediments and that it would affect the characteristics of the overlying surface

waves. They suggested that a "liquid" bottom could change the magnitude of wave-

induced bottom pressures from the values which would occur over a rigid bottom.

Gade (17), theoretically investigated wave-induced bottom sediment movement

with the sediment modeled as a viscous fluid. Small amplitude waves were assumed to

exist at both the sea surface and at the water/sediment interface or mudline. The

model predicted that the surface wave would decay exponentially with distance

traveled. The rate of decay had a maximum value when the dimensionless ratio,

1/2

h/(a/2v) , had a value of 1.2, where a is the angular frequency of the wave, v the

sediment viscosity and h the thickness of the mud. Examples were presented com-

paring wave decay over a sand bottom with wave decay over a mud bottom. For a wave

with a height of .61 m and a period of 8 sec in a water depth of 1.22 m , the wave

height would be reduced to 79% of its original value after traveling 300 m over a



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sandy bottom. Over a mud bottom only .46 m thick and having a viscosity of .46

m /sec, the same wave would be reduced to less than 37% of its original height after

traveling the 300 m . For a thickness of mud of 1.83 m the wave height would be

reduced to 2% of its original height after traveling 300 m . It was concluded that

even for a relatively thin layer of fluid-like mud the dissipation of wave energy to

bottom motion viscosity effects exceeds by far the energy loss due to bottom

friction. Effect due to muds having some rigidity were addressed in later studies.

Subsequently the problem was investigated with sediment modeled as a linear

viscoelastic material or a Voigt solid (Gade (18)). The material properties which

specify a Voigt solid are an elastic modulus and a viscosity. It was found that the

amplitude of the mudline wave was determined by the elastic properties of the sedi-

ment and was not noticeably influenced by sediment viscosity. However, it was also

found that sediment viscosity determined the rate of dissipation of wave energy.

The values used for sediment shear modulus and viscosity, the Voigt constants, were

hypothetical since no actual data were available.

Henkel (20), examined the conditions under which surface waves could cause soft

sediments to fail at large depths (tens of meters) below the mudline. Using linear

wave theory over a rigid bottom to estimate wave-induced bottom pressures he found

that hurricane storm waves could cause Mississippi Delta sediments to fail at depths

as great as 30 m . Sediment shear strengths in the delta area were described as

increasing at an approximately linear rate of 2-4 psf per foot (1.5 to 3 kPa per m)

of depth below the mudline.

Mitchell (28) conducted laboratory studies of wave induced sediment insta-

bility. The wave induced shear stresses when combined with gravity stresses

produced mass movement of sediment over about two thirds of the soil profile.

Because of the failure of an oil platform off of the Mississippi Delta during

hurricane Camille, 1969, due to a submarine mudslide, the interest in wave-mud

interactions involving extreme wave heights (2,3) significantly increased during the

1970's. The immediate need was to predict the amount of mud motion occurring under

hurricane waves. Several reports are available which documented the affect of wave-

induced mudslides on offshore structures (1,4,6,7,23,32,34,43).

Wright and Dunham (42) presented a simplified finite element model to study the

wave-induced soil motion problem. The model Is static, two-dimensional and includes

effects of large vertical deformations of the seafloor boundary as well as the non-

linear stress-strain characteristics of the sediments. Input to the model are the

bulk modulus and its variation with increasing numbers of stress cycles and stress



407

magnitude. Sediment vertical motion of .76 m was predicted to occur during hurri-

cane wave conditions at the seafloor in a water depth of 100 m. Even at a depth of

30 m below the seafloor the predicted vertical sediment motion was about .3 m.

Laboratory tests were reported by Doyle (13) on wave interaction with soft

sediments. It was reported that bottom sediment movement showed a considerable

variation with depth below the mudline. Wave-induced bottom pressures were consid-

erably altered from those expected over a rigid bottom. Measured bottom pressures

were only about 27% of predicted values. It was reported that considerable remold-

ing of the sediments took place in the zone of soil movement. The distribution of

wave-induced shear stress with depth was given indicating that the maximum stress

was about .37 times the bottom pressure and occurred at a depth of .16 times the

wavelength.

Carpenter, Thompson and Bryant (8) described the viscoelastic properties of

marine sediments. In this study the shear resistance of the soil was related to the

rate of shear strain. The soils were shown to behave as non-Newtonian viscous

material. The viscous property was indicated to be related to the liquid limit of

the soil at a depth below the mudline.

Esrig, Ladd and Bea (15) presented the results of a laboratory study of

Mississippi Delta sediments under simulated wave loading. Data relate the shear

strength of the sediments and their elastic properties and the liquidity index.

These data indicate that the shear strength decreases as a function of decreasing

liquidity index. The ratio of the elastic modulus and the shear strength is shown

to be a nonlinear function of axial strain and the number of cycles of wave loading.

The ratio decreases with increasing number of cycles and increased strain.

The development of a sediment failure profile was investigated by Pamukcu and

Suhayda (31) using a numerical wave/sediment model. The soils where modeled using

critical state parameters so that stresses, strains and soil property changes could

be described for the first few cycles of loading. It was found that soil properties

under waves could be forced to the critical state after from one to a few cycles of

wave action.

The first reported field measurements of wave-induced seafloor motions were

presented by Suhayda, et al. (36). Simultaneous measurements of bottom oscillations

and wave characteristics were made in East Bay, Louisiana, in a water depth of

19.5 m. Bottom motions were small (< 1 cm) under waves having a height of 1 m and a

period of 5 sec. The bottom sediments moved in an elastic response with the

seafloor being depressed under the wave crest, that is, the mud acceleration was



408

upward under the wave crest. Additional analysis of this data was presented in

(41).

The first reported field measurements of wave decay over muddy sediments were

presented by Tubman and Suhayda (38). The wave induced bottom pressure, amplitude

and phase of the mudline wave and the decay of the surface wave were measured in

East Bay. The energy loss rate was about 100 times that predicted for a sandy

bottom. The actual work done on the mudline wave was measured and found to agree

with the magnitude required to explain the anomalously high wave decay. The bottom

pressure wave was found to be out of phase with the bottom mud wave by about 202

degrees, so that the trough of the mud-wave was about 22 degrees out of phase with

the pressure wave crest.

Wells (40) presented the results of field studies on the extensive mud flats of

Surinam. It was found that wave heights in very shallow water maintained a height

to water depth ratio, H/h, of .23, due to the effect of wave attenuation by the soft 2

mud. A mud viscosity of up to 210 cm cm/sec (20,000 times the viscosity of water)

was measured for the fluid muds. The water content of the mud was about 380% (ratio

of the weight of water to weight of solids) and had a specific gravity of 1.17. The

distance over which the wave height decayed by 10% was between 150 and about 600 m.

Several theoretical studies have addressed wave propagation over a deformable

bottom. These studies will form the basis for the methodology to be developed in

this study. Mallard and Dalrymple (26) developed an analytical solution in which

the soil beneath the water was regarded as an elastic solid. They showed that the

phase speed of the surface waves would be modified from that given by classical

rigid bottom theory.

Suhayda (37) derived the linear wave theory over a bottom boundary that is not

rigid. The bottom movement was assumed to be described by a wave of the same

frequency as the surface wave but having an arbitrary phase shift. The results show

that mudline waves can cause changes to wave properties such as the velocity profile

and dispersion over a moving bottom boundary.

In an additional paper, Dalrymple and Liu (10) treated the bottom muds as a

viscous fluid, as had been assumed by Gade (17). The problem was solved for inter-

mediate as well as shallow water. Comparisons with the laboratory experiments of

Gade (17) showed that a mud viscosity could be found which would produce agreement

between the observations and the predictions of the theory. The kinematic viscosity 2

of the mud required to do this was .0026 m m/s or about 1,000 times the viscosity of

water.



409

The effect of soil inertia was investigated by Dawson (11), where it was shown

that for an incompressible sediment, the wave speed, particle velocity and pressure

were affected by the soil inertia.

Hsaio and Shemdin (22) considered the muds as a viscous fluid and compared wave

attenuation caused by muds with the attenuation occurring on sandy coasts, i.e.

bottom friction and percolation. The wave attenuation rates caused by muds were up

to 100 times larger than either bottom friction or percolation. Solutions were not

developed into useful forms, but were presented graphically.

Schapery and Dunlap (32,33) presented a method for predicting wave induced

seafloor motions using a viscoelastic model for bottom sediments. This method has

been utilized extensively by the oil industry to make predictions of seafloor

motions during hurricanes. The important soil parameters required are the shear

strength of the soil, unit weight, liquidity index, shear modulus at small strains,

nonlinear stress-strain behavior and hysteretic damping of the soil. The method has

been incorporated into a computer model and predictions of wave attenuation as a

function of distance can be made on a site specific basis. This computer model is

proprietary and was not intended to be used to make shallow water wave height

forecasts during hurricanes. It does not include the effects of wind Input of

energy, for example.

The elastic properties of submarine sediments have been the focus of recent

work by Coleman, Dawson and Suhayda (9) and Dawson, Suhayda and Coleman (12). They

used previously measured sediment movement data to determine the insitu shear

modulus of marine sediments in the Mississippi Delta. Using wave theory for waves

on an elastic bottom, the calculated shear modulus was 3,000 to 6,000 kPa. The

shear modulus decreased with increasing wave frequency. The ratio of the shear

modulus to the shear strength of the sediments under non-storm conditions was about

100.

Forristall, et al., (16) reported the first measurements of storm wave decay

for the Mississippi Delta. Measurements were made of wave heights during hurricane

Frederic at a station in 312 m of water and at a station in 20 m of water in East

Bay, Louisiana. The deep water significant wave heights reached a peak of about

8.4 m while at the same time the waves in East Bay reached only 1.7 m. Refraction

analysis was reported to change the directional spectrum by about 20 percent. Wave

shoaling, for shallow waves, increased wave height. These results clearly

demonstrated that wave decay previously observed during moderate conditions also

occurs during hurricane storms.



410

A recent theoretical paper by MacPherson (25) has considered wave attenuation

over a seafloor consisting of a linear viscoelastic material. Water depth and

sediment thickness can take on any value, rather than be limited to shallow water.

Several different types of wave-bottom interaction were identified, depending upon

the relative importance of the sediment elastic and viscous properties. The paper

presents the process of wave attenuation as an exponential decay of wave height

given as a function of input wave parameters, water depth and soil properties.

Yamamoto (44) has described the interaction of a spectrum of waves with bottom

sediments having poro-viscoelastic properties.

Pamukcu (29,30) presented profiles of sediment dynamic properties of material

from a boring taken in the Mississippi Delta. The shear modulus was found to depend

upon the magnitude of shear strain, so that the sediment becomes less stiff as it is

moved. The shear modulus was found to be about 250 times the shear strength of the

sediment.

Kraft, et al. (24) presented the results of predictions of soil response to

ocean waves. The predictions were made using the model developed by Schapery and

Dunlap (32) and soil properties typical of the Mississippi Delta. Results of the

study indicate that the wave induced stresses on the muds can alter the soil prop-

erties during the loading event. Thus soil properties at the initial stage of a

storm may not be the same as during the later stages of the storm. Quite large soil

displacements are predicted during extreme wave conditions.

Summarizing the state of knowledge of wave-mud interactions, it is evident that

while there is a great deal of isolated theoretical and experimental work completed,

there is not available at this time a comprehensive theoretical description of the

interaction which has to be verified with conclusive experiments in the laboratory

and in the field. The material properties of the muds that are critical for accu-

rately describing the interaction are both viscous and elastic and depend in a

nonlinear way upon axial and shear strains, axial and shear rates of strain and

number of cycles of loading. The theoretical models have not included nonlinear

material properties in the analysis. The theoretical models indicate rates of wave

decay can be very large for certain assumed values of input soil properties.

However, these properties have not been well documented for many field sites. What

little documentation of these properties that exists has been primarily in the area

of the Mississippi Delta. For this area there seems to be a correlation between the

viscoelastic properties of the sediments and standard soil parameters such as shear

strength and liquid limit. This information is used in the model presented in this

paper.



411

DATA ON WAVE DECAY OVER MUDS

There have been relatively few measurements made of wave decay over muds,

although the range in wave heights involved is considerable.

There have been a number of reports of hurricane induced wave heights in the

shallow water areas of Louisiana. Unfortunately, most of the data are the result of

visual estimates or inferred wave heights. The wave data which follow represents a

survey of available wave data roughly in the order of decreasing importance.

Bea (3) and Bea and Aurora (5) reported wave heights occurring within East Bay

Louisiana during hurricanes Betsy, 1965 and Camille, 1969. The wave heights were

estimated by experienced engineers from the damage sustained on offshore platforms.

These hurricanes are considered great hurricanes having respective central pressures

of 27.8 and 26.7 inches of Hg. The path of both of these storm was northward

directly toward East Bay. Betsy had a landfall to the west of East Bay, while

Camille veered eastward having a landfall along the Mississippi coast near Pass

Christian. The storms each passed within 20 miles of East Bay.

Bea (3) reported maximum wave heights in East Bay resulting from hurricane

Camille at several locations. At a water depth of 100 m, the maximum wave height

was 20 m. Near South Pass of the Mississippi river, where the water depth was 12 m,

the observed maximum wave heights were less than 4.5 m. For a wave height equal to

4.5 m, the wave height to water depth ratio would be 4.5/12 or .375. Within East

Bay, where the water depth was also about 20 m, the maximum wave height was less

than 3 m. The wave height to water depth ratio was at most .167.

Within East Bay during hurricane Betsy, the maximum wave height was estimated

to be about 7 m at a water depth of 20 m, giving a wave height to water depth ratio

of about .35. The offshore maximum wave height during hurricane Betsy was measured

to be about 17 m, at a water depth of 76 m. The deep water maximum wave height for

Betsy was estimated to be about 23 m.

These results are very important for several reasons. These storms were of

extreme intensity and passed very close to an area of offshore Louisiana where a

considerable amount of information is available concerning bottom sediment geologic

and geotechnical properties. Also, offshore wave height measurements were made at

several locations near East Bay and estimates of maximum wave heights were reported

in shallow water. These data sets can be used to test the validity of any theoreti-

cal models developed to predict the decay of surface waves over muds.



412

Additional measurements of hurricane wave heights in East Bay have been

reported as part of the SWAMP study (16) (Forristall, et al., 1980). During this

project wave heights were measured over several years at the Cognac platform and at

various locations in East Bay. The Cognac platform is located about 27 km southeast

of East Bay at a water depth of 312 m. The most Important data collected during the

study was from hurricane Frederic, 1979. Hurricane Frederic had a central pressure

of 27.8 inches of Hg and passed within 150 km to the east of the platform. The

maximum significant wave height measured at the platform was 8.4 m while the maximum

significant wave height measured at a water depth of 33. 5 m, was 1.7 m. Wave

spectra were presented from the deep water and the shallow water measurement sites.

These data are for a hurricane that is as intense as the 100 year storm, however it

passed to the east of East Bay. These data do provide direct measurements document-

ing wave height decay in the area of East Bay.

Additional shallow water wave data is available for areas of Louisiana to the

west of East Bay. These data are not well documented, but they do give some

indication of the observed extreme wave heights. The best of these data are from

Bay Marchand, Louisiana, having a normal water depth of 2.7 m and a storm water

depth of 5.9 m. The maximum wave crest elevation observed was 4.7 m above mean Gulf

water level. This corresponds to a maximum wave height of about 2.1 m or a maximum

wave height to water depth ratio of .37. The wave crest to wave height ratio for

this wave was taken to be .7.

Other wave data have been reported for various locations in Timbalier Bay,

Louisiana, resulting from hurricane Hilda, 1964. The data represent visual observa-

tions made at different oil fields within the bay by workers during hurricane

conditions and reported to their various companies. Hurricane Hilda had a central

pressure of 28.3 inches of Hg with deep water maximum wave heights of 16-17 m. The

hurricane had a landfall near Saint Mary parish. The observations included wave

height and storm water level and are presented in Table 2. The table gives the

location of each observation site, the wave crest elevation and the mean water

elevation above the mean Gulf water level. Also given is the computed maximum value

of the ratio of the wave crest elevation above the storm water level to the storm

water height. Normal water depths at the sites are not readily available, so that

the actual water depth during the storm are not known. The data therefore provide

estimates of the maximum wave height to water depth ratios. The data indicate that

the maximum wave height to water depth ratios reached an average value of about 1.0

for locations near the normal shoreline and decreased to a value of less than .3 for

locations at the interior shorelines of coastal bays.



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TABLE 2

WAVE DATA AT VARIOUS LOCATIONS FOR HURRICANE HILDA, 1964

Location

Water Elevation

(ft)

Wave Crest Elevation

(ft)

Min. Wave Height Ratio

Bay Junop Field 6 9 .71

Bay St. Elaine Field 8 10 .36

Caillou Island 4 8 1.43

Dog Lake Field 6 9 .71

Golden Meadow 6 8 .47

Lake Barre Field 8-10 10-13 .47

Lake Pelto 8 10 .36

Lake Raccourci 8-10 10-13 .47

Ladeyrouse Field 5 7 .57

Leeville Field 5.5 7 .43

Montegut Shipyard 5 7 .57

Note: Ratio is wave crest the water elevation.

elevation minus water elevation divided by .7 times

DEVELOPMENT OF A SIMPLIFIED PREDICTION MODEL

Several models exist for predicting the response of soils to surface waves and

involve a variety of assumptions regarding wave and soil properties. The model

presented here is intended as a tool for making engineering forecasts of wave-mud

interactions. It is simplified from the model of Schapery and Dunlap (91) and is

intended to give reasonable estimates of the primary features of the wave/mud inter-

action, i.e., wave attenuation and mud stresses and strains. It is based upon a

description of the underlying physical processes that include several simplifying

assumptions (5).

Basic Hydrodynamlc Equation

The model is based upon the conservation of energy for waves given by

d(E Vg)/dx = Q (23)



414

where E is the wave energy density in joules/m, Vg is the group velocity in m/s, x

is the coordinate positive in the direction of wave propagation in m and Q repre-

sents the rate of energy loss or gain from processes in joules/m s. An expression

for the conservation of wave energy in terms of wave height can be written given

that

E = pgH2

/8 (2)

and

d(H2

Vg)/dx = 4Q/pg (3)

The process of wave decay due to wave-mud interaction represents loss of wave

energy. This loss has been modeled as an exponential decay by MacPherson (25). For

this process the rate of energy exchange is proportional to the wave energy present,

that is

4Q/pg = A H2

(4)

where A is a proportionality constant. The solutions of the energy equation for

these processes show exponential growth or decay with distance, that is

H(x) - H(0) EXP(A x/2Vg) (5)

and for

Ka = EXP(A x/2Vg) (6)

and

H(x) = H(0) Ka (7)

Mud Bottom Attenuation

The effects of muds on wave propagation are computed from a formula based upon

the results presented by MacPherson (25). What needs to be determined is the atten-

uation parameter A in equation 5. The attenuation parameter is calculated from a

formula developed for various bottom sediments of the Mississippi Delta. The

attenuation parameter for mud is (37):

A = -3.14 MS SIN(

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speed. The bottom movement parameter MS is the ratio of the amplitude of the mud-

line wave to the amplitude of the surface wave. The formula used from MacPherson is

equation 3.33 on page 729. It depends upon the phase speed of the wave and the

dynamic properties of the shelf sediments. Values of MS are computed from a formula

which was derived from equation 3.33. The derivation is as follows:

1/2 2(gh) |(PjgG + i P L P 2 g v o ) |

** r

2. 2. 2 ^ 2 . 2 2. (9)

O ( P24 O (V + G / p

2a )J

Then using y - p2v and C = ( g h )

1

/2

2 C | ( pl gG + ipjgy)|

MS ( 1 0 ) 4 a (y V + G )

MS Cpjg|(G + ipa)|

j 2 2 2a(G + y a )

(11)

MS C P

xg

2a(G2

+ 2 2.1/2

y a ) (12)

32 C T M S

" , ,r2 ^ 2 2.1/2 ' (13)

2ir(G + y a )

The formula for MS, with C T = L , where L is the wavelength in m is

MS - 5.09 L/GE (!4)

where

GE - (G2

+ y V )1 / 2

( 1 5 )

The phase angle between the mudline wave and the surface wave is computed from

equation 3.34 in MacPherson,

TAN() = ya/G (1 6 )



416

The phase angle Is related to the damping ratio Dt given by

Dt = TAN(/2) (17)

From the work on the stress/strain properties and energy loss from muds and clays,

(29), the damping ratio and the shear modulus of marine clays have found to be a

function of the shear strain. The formulas relating the shear modulus and the

damping ratio to the shear strain are:

G = GS = GM/(1 + RA y) (18)

and

Dt = FI0(1 + RA Y) (19)

F I O = A r c T a n ( . 1 2 LD (l + (GM Y /SU)) ( 2 0 )

The shear modulus and viscosity are computed from nonlinear formulas as a function

of shear strain in the soil. The formulas as a function of shear strain are

GS = GM/(l + (GM Y/SU)) (21)

y = (GS T/6.28) (.12 LD 1 + (GM Y/SU)) (22)

GM = RA SU (23)

where GM is the initial tangent modulus, SU is the undrained shear strength, Y Is

the shear strain in the soil, LD is the liquidity index and RA is the ratio of

GM/SU. The liquidity index is related to the shear strength by (15)

LD = l.l/(.8 + SU/500) (24)

The shear strain in the soil is computed from the following formula

Y = Tw/GE (25)

where Tw is the wave induced shear stress in the soil. This shear stress in the

sediment depends upon the wavelength of the wave and the bottom pressure amplitude

(13).

Tw = PA (6.28 z/L) e -(6.28 z/L)

( 2 6 )



417

where z is the distance (positive) below the mudline. The maximum value of the

shear stress for a constant shear modulus profile is .37 Fa and occurs at a depth of

.16 L.

If the shear strength varies linearly with depth, given by

SU = SO + Z(S1 - S0)/D1 (27)

then the maximum shear strain occurs at a depth ZD given by

Z D - T l s m n r T ) H 1 - H r ( S i / s o - i ) } 1 / 2 - i ] ( 2 8 )

where SO and SI are the shear strengths at the mudline and at depth of D1 below the

mudline and L is the wavelength.

Measurements of the value of RA are rare and vary over a large range. Values

from laboratory measurements range between about 30 and 700. Field measurements

under storm conditions are not available, however measurements made during the

SEASWAB experiment, as presented in (22), indicate a value of GM in the range of

3,000 to 6,000 kPa for a site in East Bay. For the shear strengths found in East

Bay, this implies a value for RA of 100.

From a given value of the shear strength at the mudline and at some depth below

the mudline, the attenuation coefficient for wave decay over muds can be calculated.

Wave induced bottom pressures at a site are calculated using linear wave theory.

The bottom pressure amplitude is

Pa - H w/(2 Cosh(kh)) = H w CP/2 (29)

where H is the wave height and w is the unit weight of sea water, k is the wave

number, CP is the pressure coefficient and h is the water depth. The wave length Is

calculated from linear wave theory.

A forecast for East Bay, Louisiana is presented in Table 3. The significant

wave heights in East Bay are limited to maximum values of less than 10 ft for

nominal water depths less than about 50 ft. If the same forecast is made with the

mud attenuation removed the significant wave heights are about twice as large. It

has been found that the effects of muds become very significant for values of the

shear strengths below 200 psf, they show moderate effects at 400 psf and little

effect above 800 psf.



418

TABLE 3

COMPARISON OF PREDICTED AND OBSERVED WAVE HEIGHTS IN EAST BAY

Observed Predicted

Water Depth

(ft) Hm (ft)

Wave Hgt. Ratio

Hm Wave Hgt (ft) Ratio

60-70 20-25 .35 22.4 .34

60-70

419

linear theory for these sediment characteristics, as shown in Figure 8. The ratio

of the undrained shear strength to the average maximum shear stress along the wave-

length can be used as a safety factor against large seafloor motions. The variation

of this safety factor along the transect is shown in Figure 9. The safety factor is

close to one for water depths between 100 and 250 ft (30 and 75 m).

The wave-induced soil motions are described in Figures 10-12. The horizontal

soil movements under hurricane waves are predicted to be up to 1.5 ft (.5 m ) in

magnitude, depending upon the shear strength profile used as shown in Figure 10 and

11. Shown in Figure 12 is the profile of the horizontal soil movements. Maximum

soil movement occurs at depth below the mudline of about 20-30 ft. This is a

shallower depth than expected for the case of a constant shear strength half space.

Motion is predicted to occur to a depth of over 150 ft (30 m).

The relationship between the surface wave profile and the bottom pressure and

mud wave is shown in Figure 13-15. The bottom pressure is shifted from being in

phase with the surface wave, as it would be over a rigid bottom, to a condition of

lagging the wave crest by about 90 degrees when the ratio of the shear modulus to

the shear strength decreases to a value of 50, as shown in Figure 13. The hori-

zontal and vertical mud motions also show a large phase shift and an amplitude

increase for the same G/Su value of 50, as shown in Figures 14 and 15.

UNDRAINED SHEAR STRENGTH, su ,KSF 0 0.4 0.8 1.2 1.6 Offci i i | i i i | i ii | i i i

0

u.

420

LIQUIDITY INDEX

Figure 4. Generalized profile of the liquidity index for various water depths.

Reproduced from (24) by permission of Crane, Russak & Company, Inc.

Figure 5. Data showing the nonlinear relationship between the soil shear stress and

damping ratio as a function of strain. Reproduced from (24) by permission

of Crane, Russak & Company, Inc.



421

120

100

5 60 UJ I

UJ I 4 0

20

J 1 1 1 1 1 1 1 1 1 1 1 1 1 | II 11_ STRENGTH PROFILE VARIED WITH. WATER D E P T H (CASE 1)

STRENGTH PROFILE FOR 3 0 0 F T m mm WATER DEPTH USED FOR A L L

_ WATER D E P T H S ( C A S E 2 ) -

_ s f -- (jL^- T=8sec = 1 / 1 '7 / / V

_

/ / / SL -

/ / J / 14 sec -/ 4 /

~l 11 i 1 M 1 1 1 1 1 1 1 1 1 i F 100 200 300 4 0 0 W A T E R DEPTH,FEET

Figure 6. The computed decrease in storm wave height resulting from wave/mud interaction. Reproduced from (24) by permission of Crane, Russak & Company, Inc.

>1 I i I I I I I I UJ o

t 8 0 0 1 _i CL 2 < 600 UJ a.

400h UJ cr CL

200

I I

I I I I | I I I I _

W A V E W A V E HEIGHT.ft PERIOD.sec -

70 14 60 16

I I l I ' I l ' ' ' I ' ' ' I

100 200 300 W A T E R DEPTH, FEET

400

Figure 7. The variation with water depth of the bottom pressure amplitude for storm waves. Reproduced from (24) by permission of Crane, Russak & Company, Inc.



422

Figure 8. Comparison of the wavelength as a function of wave period for various water depths according to linear wave theory (lines) and mud bottom theory (symbols). Reproduced from (24) by permission of Crane, Russak & Company, Inc.

Figure 9. Safety factor as a function of water depth for storm waves. The safety

factor is the ratio of the average bottom pressure amplitude to the shear strength of the bottom sediments. Reproduced from (24) by permission of Crane, Russak & Company, Inc.



423

| 2.0

s 1.5

: i i i i I i

i.o

<

8 Q 5

t-

r | i i i i | I I I I _

T=l6s)H

0=60FT-

T= I4s,Ho=70FH T=l2s,H

0=70FT-

T= I0S,Hq=80FT~

I I I I 100 200 3 0 0

W A T E R D E P T H , FEET 4 0 0

Figure 10. Magnitude of the predicted horizontal movement of bottom muds under storm waves. Reproduced from (24) by permission of Crane, Russak & Company, Inc.

Figure 11. Comparison of the magnitude of mudline soil movements between the depth variable shear strength profiles shown in Figure 2 (case 1) and a shear strength profile for all water depths equal to the 300 ft profile in Figure 2 (case 2). Reproduced from (24) by permission of Crane, Russak & Company, Inc.



424

MAXIMUM HORIZONTAL SOIL MOVEMENT, FEET

50

100

o: O o u. < UJ

? 150 S UJ m

200 a. UJ a

250

300

T T T T T T T n ]TTTT[ I i n r j m - y -- 7 FT - ^ J :

- / >WAVE HEIGHT I - / r / / 18.5 FT - / / -- / / -: / / / /

M '/

- / SOIL CONDITIONS I

CASE 1 I

_ CASE 2 ~

-50 FT WATER DEPTH

-14 SEC PERIOD ~

m i l Lit 111 ii I n n , h i i f

Figure 12. Shows the profile of horizontal soil movements at a site in 50 ft of water for case 1 and case 2. The stronger offshore soil strength profile (case 2) shows larger movements at the site because the wave height is larger. Reproduced from (24) by permission of Crane, Russak & Company, Inc.

77" 277

PHASE ANGLE,RADIANS

Figure 13. Shows the magnitude and phase relationship between the surface wave and the bottom pressure for various shear modulus to shear strength ratios. Reproduced from (24) by permission of Crane, Russak & Company, Inc.



425

133d'iN3W3DVldSI0 IVINOZIHOH H e c co o C N 0 -H H H u o co si r-l 01 01 u J= 01 CO T3 CO C si a

426

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427

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