Optimization of depths of channels

ARCHIEF

Hogescljo0i

De lit

--waterloopkundig laboratorium

deift hydraulics laboratory

optimization of depths of channels

J. Strating, T. Schilperoort and H.G. Blaauw

publication no. 278

September 1982

optimization of depths of channels

J. Strating, i Schilperoort and H.G. Blaauw

Deift Hydraulics Laboratory

ABS TRACT

The determination of the required depth of approach channels to harbours is a problem of great concern. Nonetheless, there are no sophisticated design methods. In this paper, a design method is described which overcomes many objections to previous methods.

Using a probabilistic approach, spatial and temporal changes of the depth-determining factors can easily be incorporated into the final design. Further-more, since a probabilistic approach enables the quantification of

uncertain-ties, both the safety level for the use of the channel and the accessibility of the port can directly be related to the channel depth. This feature allows the channel depth to be optimized from an economical point of view.

Several factors contribute to the growing problems of accessibility of existing and new ports, e.g.:

- scale enlargement in shipping in the past two decades; - port developments on hostile coasts;

- creation of deep-water ports in areas without natural provisions for deep-draught vessels; and

- the growing need for shipping safety, especially for shipping of dangerous bulk cargoes such as liquefied gasses.

The accessibility of many ports has to be ensured by large-scale dredging operations. Capital and maintenance dredging may require large sums of money and, consequently, the impact on the economy of these ports can be dramatic. Therefore, it is justifiable that the design of entrance channels be given considerable attention; both width and depth aspects should be studied. This

2. _{Discussion of design methods}

The main factors which influence the depth of an entrance channel are:

- draught and other characteristics of the vessels to be received; - vertical ship motions in waves;

- squat and trim;

- hydrographic factors (vertical tide, bottom position);

- morphological factors (sedimentation or siltation, both in relation to the dredging cycle); and

- safety level of channel transits,and the downtime of the channel. Futhermore, many secondary factors may also influence the channel depth.

To find an optimal channel depth, these factors have to be studied in an integrated way. [n principle, two methods of integration can be distinguished: - deterministic methods;

- probabilistic methods.

In practice, various combinations of these basic methods are used.

Any method of channel depth design should be related to the design of the entire port. Therefore, the design of approach channels should be part of an

integrated port design, such as described in [i] and [2].

It should yield a relation between channel depth on the one hand, and safety of channel transits and port accessibility on the other, thus allowing the integrated design procedure to lead to the economically optimal depth. In order to find such a relationship, a probabilistic integration of depth deter-mining factors is essential. Yet, even recent manuals of port design [3,4,5] still rely on deterministic approaches like teat proposed in one of tne earliest channel depth design descriptions, [6].

The core of the deterministic approach is shown in Figure 1: the required channel depth is obtained by adding the separately quantified contributions of all the relevant factors. The quantifications are often only based on rules of thumb. The use of these rules of thumb, which in the most favourable situation are based on generally applicable research and experience, makes it difficult to take local circumstances (such as wave climate, channel alignment, sea bed characteristics, etc.) into account in a particular design. Improvements could be obtained if those contributions were computed for representative local extreme conditions.

However, a principle drawback of a deterministic suuuuation is that a

trade-off between channel depth on the one hand, and safety of channel transits and accessibility to the port, on the other hand, cannot be made, since this requires the quantification of uncertainties in a statistical sense.

An advantage of the deterministic approach is that the feasibility of the application of a more sophisticated channel depth determination for a spe-cific port design can be quickly and cheaply determined. This is of special

interest in the initial design phase [1].

A probabilistic approach integrates design factors in such a way, that all possi-ble circumstances during the lifetime of the object to be designed,contribute

to the design in a weighted form. The weights are proportional to the frequen-cy of occurrence of the circumstances. Using this approach in a channel

depth design, uncertainties with respect to safety of channel transits and port accessibility can be quantified.

Obviously, to find the optimal channel depth this method is to be preferred. Its main drawback, however, is the amount of statistical data required on all relevant aspects.

In practice, the required amount of data often hampers a full application of the probabilistic approach. Hence, several mixtures of determInistic and

probabilistic approaches have been presented in literature,which are all based on rather pragmatic arguments (see e.g. [7], [8]).

Consequently, depth-safety and depth-accessibility relationships are still dif-ficult to obtain with these techniques. A description of a fully probabilistic method is given in [9]. However, some objections, mostly of a methodological nature, can be made to this method. This becomes obvious if a detailed compa-rison is made between [9] and the method presented here.

This paper provides a detailed description of a probabilistic method which in-corporates all relevant prototype information in a systematic way. The method was developed by the Deift Hydraulics Laboratory and applied to the design of various entrance channels.

3. An outline of the probabilistic approach

The core of the probabilistic approach is the determination of the probabi-lity of the ship's keel contacting the channel bed during one channel

transit (i.e. a bottom touch). This probability is a measure for the expected number of bottom touches in a certain period of time and provides an

indica-tion of the safety level in the use of the channel.

Once this probability is known, the derivation of relations between depth, safety level and harbour downtime becomes straightforward.

The probability of touching and, hence, the expected number of bottom touches is determined by four categories of processes (see Table 1). By successively introducing the effects of these processes, this probability is derived as follows:

cateor

1: wave-ìnduced (random) vertical sh motions

Considering only the effect of wave induced vertical ship motions, the probability of touching in one channel transit can be derived, (i.e. boundary conditions such as: type of ship, sailing velocity, water depth, wave climate, etc. are fixed)

cateor2:rocesses inducing randomly chanin boundary conditions within

each channel transit

Next, those boundary conditions are allowed to vary which may change randomly within each channel transit, e.g. a non-flat channel bed or random water-level variations.

cate qor3:rocesses inducin

random l chanjin boundary conditions between each channel transit

Then processes are introduced which may cause different condi-tions for different channel transits, but which may remain constant during one transit, e.g. wave condition, type of ship, sailing velo-city, water depth, etc.

At this stage the arrival process of ships at the channel en-trance(s) has to be incorporated.

cae2rL_L.2rocesses

inducin deterministical ly chanin boundar conditions

ithin each channel transit

Finally, the influence of deterministically changing boundary conditions on the final design is treated. This is especially important if a channel consists of several sections each having specific properties, like their orientation which causes a known variation in wave conditions.

4. Category I process

The probability that the ship touches the channel bed due to wave-induced motions, equals the probability that the vertical downward displacement, Zc(t)Of the most critical point on the ship's keel (i.e. the point with the highest probability of touching) exceeds the net keel clearance,, during the transit. In the literature, several methods have been presented to compute this exceedance probabílíty. These methods will be discussed below. The characteristic properties of z(t) which will be used are defined in Figure 2.

In the sequel, the process z(t) is considered to be Gaussian and ergodic. Possible non-stationarities,due to deterministic influences,can sometimes be accounted for when dealing with the fourth category of processes.

4.1 _{Theory of extrema of stochastic processes}

The theory of extrema of stochastic processes provides an expression for the probability that the absolute maximum of Zc(t) will not exceed i in

one transit, provided that the second spectral moment m2 of Zc(t) is finite, see [10], [11]:

Pr (t) i O t ( T = . exp [_V2nm T - F

(;T )

c max pj 2 _p

'(k)-1

max

Here, T denotes the average duration of one transit. Moreover,

2

(ka) -

exp

{-}

/2rrm' o o KC = J (x) dx

(1.2)

(1.3)

in which m denotes the zeroth spectral moment (i.e. the variance) of If z (t) < i then no bottom touches will occur. If A(i;T ) is defined

c max

_-

p

as the number of bottom touches in T , then: p

F

(k;T)=Pr[A(i;T)0],

max

E{A} =

m2

12

. T

.

exp---ru _p ru

o o

4.2 A Poisson process description

The probability of touching may be computed using a Poisson process descrip-tion if, besides the previous condidescrip-tions, the level of KC is so high that the number of exceedances of this level within disjunct time intervals are

inde-pendent. This condition is approximately fulfilled if:

KC//m » I

Pr

[(;T)

o] = Pr

[(l;Tp)

= k] = I - F

(k;T )

max p

k= I

It should be noted that the theory of extrema does not provide an expression for the probability of A = k for arbitrary k. It does, however, provide an expression for the average E(A} in T:

o o

For k = 0, (4.1) equals:

Pr

[A(i;T)

= O] = exp

This expression is similar to (1.1) because for KC/Vm » i,

(k)

i,.

4.3 A binomial description

Poisson distributed independent events can be approximately described by a binomial distribution, provided the number, N, of occurring events is high and the probability of occurrence p small.

(2)

(3)

that is if the number of exceedances is small.

The probability of k bottom touches in period T

A()T

}k

is then given by:

Pr

[A(i;T)

= k] - exp

-(k)T }

(4.1)

- k

According to (3) the intensity X(KC) is given by: m2

X(KC) = /2Trm2 . (KC) = exp

ru

Hence, for I//m » I and a sufficiently large Ip' (4.1) can be approximated by: Pr

[A(;T

) = k N k N-k p

j

= with N.p = For K = 0, (5.1) yields: Pr

[A(;T)

= ]

(J)N

= exp (-pN) p = exp

-2

KC m o

Moreover, it approximately holds that:

p -)- 0,

N

-The average number of bottom touches in T now becomes:

p

EA} =

Pr

[A(;T

) = k] . k N.p = A(i).T

k=0 p p

which is again similar to (3).

However, these expressions are not directly applicable as the values of N and p are still unknown. To calculate them an additional assumption

should be made.

4.4 A binomial description for narrow band spectra

For k//m » 1, exceedances of KC can only occur at the highest local maxima, i.e. at the peaks of z(t). Hence, the distribution of peaks is needed to

calculate N and p. Theoretically, however, this distribution is only known if Zc(t) is a narrow-band process.

Provided that the spectral width parameter of z(t) approaches zero, the

peaks of zc(t) follow a Rayleigh-dIstribution, from which:

(5.1)

(5.2)

T N = =

T

z m p

in which T denotes the average zero-crossing period.

A combination of expressions (6) and (7) yields:

N.p

12

11k2]

. exp

T m

<2

_j.- À(KC)

p 0

This expression is identical to expression (4.2). This theory is applied, for instance, in [9].

4.5 Choice of description

The previous sections have provided a number of expressions for the probabi-lity of k bottom touches in a given period of time. The assumptions made in deriving these expressions are summarized in Figure 3.

The assumptions are not all logical: adoption of a binomial description implies that the maxima in the record are independent. This is contradictory to the assumption that [ - O, which implies a mutual dependance of the maxima.

It makes little difference which theory is used for practical design

purposes. The results will be almost identical since, in practice, the number of N will be sufficiently large and the value of p sufficiently low in normal channel transits. However, in experiments, conditions may be such that a high probability of touching results in which case the theory of extrema of stochas-tic processes could be advantageous. A drawback of this theory is that there is no expression for the probability of an arbitrary number of bottom touches in period T. Therefore, the channel depth determination has to make use of a Poisson description, which does provide such expression.

5. Category 2 processes

5.1 Non-flat channel bed

A non-flat channel bed influences the required channel depth in two ways: - by influencing the hydrodynamics of ship motions; and

- by influencing the probability of touching directly via fluctuations in KC.

The first influence is usually neglected. If desired, deterministic assumptions can prevent a non-conservative effect (e.g. a conservative value of the water depth in the determination of ship motions should be adopted).

Several bed forms should be distinguished when tryíng to account for the second influence:

dredging inaccuracies;

mega ripples with characteristic lengths of several metres to some de-cades of metres;

sand dunes with characteristic lengths of several decades of metres to some hundreds of metres.

re i and ii: Both dredging inaccuracies and mega ripples are stochastic

vari-ations in the bed level, with short characteristics lengths. In general,

there will be a difference in order between their characteristic length and a ship's length. These fluctuations in cannot influen-ce the probability of touching significantly because of this

difference. Hence, a flat channel bed should be defined on the top of these fluctuations (the so-called nautical bottom level).

re iii : If the charateristic lengths of sand dunes exceed several ship lengths, a definition of KC with respect to a plain channel bed on the top of these dunes will cause a higher safety level than re-quired. Hence, other methods are required to incorporate the

effect of large dunes in the depth optimization.

A rough method to account for this effect is proposed in [9]. This method makes use of a definition of KG with respect to the mean bed level and an addition of the variance of the bed level to the variance of the vertical ship motions m. This yields both an increase in motion variance according to

m' m

o o b

and an approximate Increase In net keel-clearance according to

= _{+ 2} (9.2)

Whether or not this approach yields a reduction of the probability of touch-ing depends on the ratio of the relative increase of i and the relative in-crease of m. Usually, however, an inin-crease of this probability will be found, whereas a decrease is expected. This is a serious shortcoming of this method, which can be explained by the fact that only the variance of the bed-level

fluctuation is taken into account. Instead, the total frequency structure of both the bed level and the vertical ship motions should be considered simul-taneously. According to [10], this can be accomplished by extending the ana-lysis described in Section 4.1 to time varying exceedance levels.

5.2 Tidal variations of the water-level

In areas with tidal variations in the water-level, the water depth may vary during a channel transit.

If the duration of a transit is small compared to the time the water depth exceeds its minimum required value for navigation (i.e. a large tidal window)*,

then the variation of the water level within a transit can be considered a random variable with variance G . Therefore, this effect can be

incor-w

porated in the optimization by increasing the variance of Zc(t) according to:

rn" = m + G2 (n)

o o w

The index n indicates the dependancy of cy on the actual average water level during the transit (see Figure 4).

The variation of this average water-level between different channel transits belongs to the category 3 processes and will be dealt with in Section 6.

If there is a small tidal window, the position of a ship in the channel can be related almost deterministically to the water-level. This allows the channel

This minimum required depth will result from down-time considerations as outlined in Section 9.

(9.1)

to be divided into several reaches, each having a different average

water-level. The variation in these average water-levels belong to the category 4 processes (see Section 7). The variation of the water-level within each reach can be treated according to (10).

Finally, since the variation in the average water-level between different channel transits will now be small, it can be incorporated in m as an addi-tional variance like cr2.

6. Category 3 processes

6.1 Variation in wave conditions

A channel should function under varying wave conditions fixed by the local wave climate. Each wave condition j can be characterized by an energy density spec-trum and a mean direction of wave propagation, and has a specific frequency of occurrence f..

J

The number of bottom touches A. at wave condition j (in the period of time T) depends on the number of channel transits B. at this condition. The

distribu-tion of B. will be linked closely to the arrival and departure patterns of ships. A Poisson description of these patterns often holds and, therefore, it is likely that B. is sufficiently described by a Poisson process. The in-tensity of this process is given by

N. (T)

= hm

j T

T-in which N.(T) denotes the number of ships transitT-ing the channel under wave con-dition j, in a period of time T. The expected number of channel transits in

this neriod and at this wave condition then equals:

EfB.} =

. f. T (12)

J

J J

The total number of bottom touches A in a period of time T is given by:

B.

A = A. = A (k)

- j3

jk=1j

In Chapter 4 it was shown that the events A.(k), being the number of bottom touches in an arbitrary channel transit at wave condition j, can be described by mutually independent identical Poisson processes with intensity X. (1(C). Hence, the conditional probability Pr [A. kB. = 9] can be calculated

direct-ly from a Poisson distribution with parameter \. (KC)

T.

Since the number of channel transits B. in a period of time T is assumed to follow a Poisson distribution with parameter 5. f. T, the probability Pr [A. = k]

can be calculated by substituting these distributions in:

Pr [A. = kl = Pr [A. = k B. = . Pr [B. =

LJ

i L-J J J L-J

Hence, A. follows a Poisson distribution in a first order approximation with parameter:

= X.(KC)

t.

f. T T (15)

J J J J p

which results also in a Poisson distribution for A with parameter:

k.(KC).f.TT =TT

X.(KC).f.

J J J

J .1 J p

pi

i

The distribution of A is given by:

Pr [A = k] =

This result is vital for the channel depth detetinination.

6.2 Other processes

There is no principle difference between the treatment of the other factors in this category and that of the variable wave conditions, if it is assumed that all factors are independent and can be described by one-dimensional pro-bability distributions. The extension to correlated processes (such as

wave condition and water-level) is straightforward when their joint distribution is known.

As an example, (16) will be extended by the incorporation of the variation in the average water-level from one channel transit to another.

Let T denote the total amount of time in period T that the th class of water-levels occurs.

Since in (16) A. (KC), KC and .

depend on n, averaging (16) with respect to n yields: E = T f. A. (Kc ) 5. T (18) j n in n Jfl n since T = T n n (14) e k:

7. Category 4 processes

Often, processes change in a deterministic way during a channel transit. For instance, in a channel consisting of legs with a different alignment,

the different angles of wave incidence in these legs induce deterministic-ally known non-stationarities in the ship motions, even when the wave field

is stationary and homogeneous.

A non-stationary process In time during a channel transit gives rise to theo-retical complications. These complications, which are often overlooked

(see [7] for example), can be overcome by splitting the non-stationary process into stationary pieces and treating these pieces separately. This

can be realized by defining different channel sections in which the process is stationary.

There are two ways of dealing with the different channel sections. Firstly, the sections can be treated fully separate from each other. A probabilistic analysis then produces a depth for each section. Secondly, once can carry out an integrated probabilistic analysis of all the sections together. To that aim, a summation should be defined like those proposed for category 3.

For example, a channel which consists of several legs, each having a specific sailing speed (squat) for the vessels and wave climate description, can be dealt with according to the following refinement of expression 18:

f. T A. (KC

)5.

T

jm pm jmn mn jn n

jm

n

in which m denotes the number of channel legs.

8. The strategy for channel depth determination

The ultimate channel depth should satisfy a predeteìmined criterion with res-pect to the exceedance probability of Ad bottom touches in a period of time T:

Pr E _{A > Ad] =} (20)

The value of must be that low that the Poisson description of the bottom touching phenomena is indeed valid.

With a given value for Ad and ct, the expressions (17) and (20) produce a cri-ticat parameter of the Poisson distribution

k=A,

Pr [A > Ad] = o. = I

-k'

k=o

(21)

can now be solved by combining the expressions (21), (16) or subsequent modifications, and (4.2). A first solution for the channel depth then becomes

a suiiuuation of

ki

and the ship's draught and squat.

It should be noted that the obtained channel depth may difter from the water depth at which ship motions were measured or computed. Significant differences should result in an iterative procedure: a second determination of ship mo-tions in the new water depth and a second determination of required water depth, etc. Such a procedure is necessary because of the strong hydrodynamic inter-actions between ship motions and the water depth.

9. The optimization of channel depth

It has been assumed that the channel is open to navigation under ail wave and tidal conditions. This assumption is not realistic in practice as it would result in deep channels with high costs for capital and maintenance dredging. It is acceptable that channels are closed for navigation for some periods. These periods altogether produce the downtime percentage of a port. Benefits of downtime reduction should be weighed against costs of dredging. This reduces

the channel depth problem to an economic optimization problem.

A second factor in this optimization is the choice of a safety criterion determined by Ad, T and a). Safety can sometimes be expressed in terms of costs and benefits, in which case the costs of downtime together with the costs of a possible contact between the ship's keel and the channel bottom (the costs

of a possible disaster) should be weighed against the costs of dredging.

This pure optimization is difficult to put into practice because of factors which cannot be expressed in terms of costs and benefits (human lives and a non-polluted environment for instance). Therefore, it is generally accepted that the first type of optimization is used only, and that a political choice is made for a generally accepted, sufficiently low value of _{Ad in period T.}

The aim of the approach outlined in this paper is to present the results in such away that the economic optimization can follow. An ultimate channel depth will not be obtained at once; on the contrary, the channel depth study pro-duces combinations of water depth and downtime. A non-zero downtime will imply that an entrance regime has to be maintained during channel operation by the port authorities.

In first instance, it is usually assumed that no downtime will occur, (DT=O), and that the intensity of the traffic in the channel remains constant, irrespective

of wave climate and tidal action (. = c).

Such a first analysis reveals the conditions of j and n which contribute most

to

d' and combination of j and n at which à. should be made zero will result

easily. A à. = O for certain conditions will result in an increasing value of . at other conditions j and n. This effect should be described.

Assuming the wave climate and the water-level to be independent, the total downtime percentage is given by:

M1

DT = f.

waves

. J

J=1

Similarly, one finds for DT

tide M7

DT.

=L

p tide n n= I o

j=1...M

i-1 k k =

k1

j =

_{M +} I

j=M+2,...N

j

a

in which S.' denotes the traffic intenisity in the channel if an entrance regime is used.

DT=DT

+DT.

-DT

*DT.

waves tide waves tide

in which DT and DT . denote the downtime percentages due to extreme

waves tide

wave conditions and tidal conditions respectively. If the first M1 out of N

different classes of wave conditions are characterized by such significant

wave heights and periods that the channel will be closed, DT equals:

waves

with p denoting the frequency of occurrence of the th class of water-levels, and M2 the number of tidal conditions for which the channel will be closed. The average number of ships that arrive at the channel entrance in a period of time T, during downtime conditions equals aTi)T with '5a being the average arrival intensity for these conditions. This number of ships will transit the channel iiirniediately after the opening of the channel. Hence, a traffic intensity peak will occur for those conditions which follow the last

downtime condition. This effect can be taken into account if the most pro-bable development of wave and tidal conditions is known. Usually, this is the case for the water-level variations in a tidal window. For wave conditions this will be more difficult, although theoretical wave development models may produce sufficiently accurate information on this point.

As an example, suppose in first instance that only the wave conditions deter-mine the downtime. If the (MI)til condition follows the last downtime

condition, the following equations hold: M

DT=kl

k

(22)

(23.1)

This is sketched in Figure 5A.

If besides wave conditions also tidal conditions determine the downtime, (24) can be extended as sketched in Figure 5B.

If the development of wave and tidal conditions is fully unknown, it is better to spread the number of _aT DT ships equally over all conditions for which the channel is open. This will lead to a change of

n'

which if each equals a'

is given by:

O , for downtime conditions

(25)

1-DT

, for open conditions

For the open conditions, this corresponds to a redefinition of T according to Tt = T(1-DT)

Since 5. is explicitly used in (17), it is possible to calculate the corre-sponding value of for several entrance regimes. A combination with (21) then immediately yields a relation between entrance regime and channel depth for a specified safety level.

IO. An application of the probabilistic method

The previously set-out lines of probabilistic determination of channel depth can best be illustrated with a recently completed case study. The case study is based on the design studies for Dos Bocas harbour, a future oil exporting

harbour in Mexico. The alignment of the channel to Dos Bocas Harbour is shown in Figure 6. A channel depth optimization was carried out for this alignment.

The probabilistic approach included the processes 1, 3.1 and 4.1 of Table 1. Other factors were either not considered or treated by deterministic

assump-tions (a fully probabilistic design was not feasible due to the lack of data). These will be left out of consideration.

A wave climate description was adopted according to Table 2. This description comprised nine classes of wave-height/wave-period combinations for five sec-tors of wave propagation directions and two channel legs, see Figure 6.

Representative wave spectra for each wave-height/wave-period class were de-fined as in Table 3. A Pierson-Moskowitch decription was adopted

for each representative spectrum. For all conditions j thus defined, motions in waves of a 250,000 dwt tanker (the representative ship size) were either computed or measured at h/T = 1.20, for a representative angle of wave incidence per given sector. A linear relation between ship motions and wave height was inherently applied in the computations. Ship motions in waves were measured for those situations in which the provisional method of incorporating viscous roll damping made the computations unreliable (beam waves). Measured Response Am-plitude Operators for beam wave conditions were used to compute responses at other wave conditions, see Table 4.

The applied safety criterion was expressed as an exceedance probability of 10%

of I bottom touch in 30 years. The expected total number of channel transits in this period was 10,000.

With expression (21), a value of

d = 0.53 can be deduced for the above safety

criterii.

Since was supposed to be independent of tidal conditions, expression (19)

= T . f. T X. (KC ) (26)

3m

jm pm m m

An iterative solving of KC from this equation can be executed for several pos-sible entrance regimes of the channel. Such regimes can be defined by the boundary significant wave heights as a function of the significant wave period, beyond which the channel is closed for navigation. The resulting downtime can be deduced for each regime from the frequencies of occurrence of wave conditions, see Table 2. A correction for c5. due to downtime conditions was made

accor-J

ding to (24).

Results of this case study are summarized in Fígure 7. It will be clear from this figure that the required

ÏIÏ

value, and hence the required channel depth, depends on the channel operation boundaries and the inherently accepted

down-time. In this case the operation regimes (2) and (4) were most feasible: a depth increase of 1 .7 m was to be weighed against an increase in downtime of approximately 2%. It should be noted that some wave conditions do not contri-bute to the required

k.

Therefore, is equal for the regimes (2) and (3), while the inherent downtime for both regimes differs.

The limited experience with this design technique has learned that the channel depth is sensitive to variations in:

- maximum ship's draught; - tidal window;

- boundary wave conditions; and - fluctuations in the bottom level.

in this sequence of importance. The channel depth is less sensitive to all other factors mentioned and the way in which these factors are modelled. This may indicate that concessions can be made in the accuracy of modelling several

11. Conclusions

The probabilistic approach for a channel depth optimization described in this paper is based on a Poisson process description for the number of bottom touches

of a ship transiting the channel. The advantage of this method is that all main depth-determining processes can be incorporated in the design by simply averaging the intensity of the Poisson process over all possible states of these processes, with each state being weighted by its frequency of occurrence. Inherent to this probabilistic approach is the possibility of incorporating safety criteria in the design, because safety levels are often given in terms of exceedance probabilities.

Moreover, since the traffic intensity is included explicitly in the derivation

of the Poisson process intensity, it is very easy to investigate the relationship

between several regimes for use of the channel and channel depths for a speci-fied safety level. Hence, a trade-off can be made between port accessibility and dredging costs, from which an economically optimal depth can be obtained.

ACKNOWLEDGEMENT

The permission of Petroleos Mexicanos (PEMEX) and their consulant Proyectos Marinos, Mexico to include the channel depth determination for Dos Bocas in this paper is gratefully acknowledged; and the stimulating response of Ing. M. Vazquez Zamarripa has been most obliging.

NOMENCLATURE

f. frequency of occurrence of wave condition

frequency of occurrence of wave condition i in channel leg m

j index for classes of wave condition

m index for channel leg

the zero-th and second spectral moment of ship motion z(t)

m, m

corrected values for

n index for water-level

p probability of contact between the ship's keel and channel bottom

q index

t time

x coordinate along the axis of the channel

z coordinate along a vertical axis

z vertical downward displacement of a critical point on the ship's

keel

A number of bottom touches in the period of time T

Ad admissible number of bottom touches in the period of time T

number of bottom touches in wave condition j in the period of time T

A number of bottom touches in one channel transit in wave condition i

p.

A (k) same, at k-th transit

number of channel transits in wave condition in the period of

time T

DT harbour downtime due to non-navigable conditions in the channel,

in percentage of time

DT . downtime due to the vertical tide

tide

DT downtime due to waves

wave

KC net keel-clearance of the sailing vessel

N number of peaks in z(t)

T period of time

T' period of time corrected for harbour downtime

Ti total time of occurrence of wave condition i in the period of time T

T total time of occurrence of water-level n in the period of time T

T average duration of a channel transit

p

T zero-uperossing period of Zc(t)

exceedance probability traffic intensity

average traffic intensity

6. traffic intensity in wave condition j

6. traffic intensity in wave condition j and water level n

-in

6 traffic intensity corrected for downtime

E width parameter for the spectrum of zc(t)

intensity of Poisson distribution for A

J

Pj

intensity of Poisson distribution for A admissible value for

variance of bed level fluctuations

G2 variance of water-level fluctuations

BLAAUW, H.G., KOEMAN, J.W., STRATING, J.

Nautical contributions to an integrated port design,

SEATEC III Seminar on Asian Ports Development and Dredging, Singapore, March 1981

KOUDSTAAL, R., WEIDE, J. van der

Systems approach in integrated port planning

SEATEC III Seminar on Asian Ports Development and Dredging, Singapore, March 1981

Report of Working Group IV

International Coiuuission for the Reception of Large Ships of the Permanent International Association of Navigation Congresses,

Supplement to PIANC-Bulletin no. 35, Vol. I, 1980

United Nations Handbook for Port Development Publication TD/B/C4/175 of UNCTAD, New York, 1978

BRAY, R.N.

Dredging: A handbook for Engineers,

Edward Arnold Publication Ltd., London 1979

LEITE, J.C.

Big tankers and their reception,

lind International Oil Tanker Commission (1970-1974) of the Permanent International Association of Navigation Congresses, PIANC-Bulletin no. 16

1973

WANG, S.

Full-Scale Measurements and Statistical Analysis of Ship Motions in a Navigation Channel,

SINOEN, R.,

RIETVELD, C.F.W., VLIEGER, H. de, PUTTER,

The extension of the port of Zeebrugge (Belgium): study pattern for the design and realisation of an access-channel in open sea for the reception of large vessels in the extended port,

PIANC-Bulletin no. 37, Vol. III, 1980

BIJKER, E.W., MASSIE, W.W.,

Coastal Engineering, Volume II: Harbour and beach problems, Coastal Engineering Group, Department of Civil Engineering, Delft University of Technology, 1978

DITLEVSON, 0.

Extremes and first passages times with applications in civil engineering, Technical University of Denmark, June 1971

CRAMER, H., LEADEETTER, M.R.,

Stationary and related stochastic processes; sample function properties and their applications,

fixed per transit to be neglected

Table

I

Summary of relevant processes per category

Process CATEGORY NO. 1 2 3 4 stochastic varia-determinatic No, general description

stochastic variations within each transit tions between each transit tions within each transit

waves

wave-induced ver- _{tical ship motions}

1)

variation of wave conditions changes in wave conditions

2

channel bed

1)

changes in channel bed level variation of chan- nel bed level changes in channel bed level

3

vertical tide

1)

changes in water- level

variation of water- leveL changes iii water- level

4

sailing speed

1)

2)

variation of sail- ing speed

changes in sail- Ing speed

5

fleet of ships

1)

-type of ship and its characteristics

changes in draught 6 others accuracies! accuracies! accuracies! accuracies! minor factors minor factors minor factors minor factors

Table 2

Wave climate description

class no. significant wave period interval significant wave height interval

frequency of occurrence (% of time)

outer channel leg

inner channel leg

total N NE E W NW total N NE E W (s) (ft) -0

<T5<

6.4 66.47 -66.49 -1.1 6.5 < T < s O < H s < 3.9 13.15 3.28 4.70 2.24 0.95 1.98 13.48 4.38 5.10 1.13 0.57 1.2 6.5 < T < s 10.4 4.0 < H s < 7.9 9.01 3.56 3.36 0.21 0.13 1.75 9.61 4.98 2.51 0.52 0.10 1.3 6.5 < T < s 10.4 H s > 8.0 4.40 2.73 0.75 -0.92 3.17 2.56 0.24 -II.! 10.5 < T < s 14.4 0 < H s < 3.9 3.34 0.80 1.18 0.60 0.26 0.50 3.41 1.08 1.29 0.31 0.15 11.2 10.5 < T5< 14.4 4.0 < H < 7.9 1.41 0.55 0.53 0.03 0.02 0.28 1.53 0.78 0.40 0.09 0.02 11.3 10.5 < T5< 14.4 H > 8.0 .59 0.37 0.10 -0.12 0.32 0.25 0.04 -111.1 T> 14.5 0 < H < 3.9 .94 0.23 0.33 0.17 0.07 0.14 0.95 0.30 0.36 0.09 0.04 111.2 T> 14.5 4.0 < H < 7.9 .46 0.18 0.17 0.01 0.01 0.09 0.41 0.25 0.13 -111.3 'f3> 14.5 > 8.0 .23 0.14 0.04 -0.05 0.16 0.13 0.01

-Table 3 Characteristics of

determinant wave spectra wave spectrum no.

T(s)

H(m)

1.1 8.5 0.60 1.2 8.5 1.80 1.3 8.5 3.80 11.1 12.5 0.60 11.2 12.5 1.80 11.3 12.5 3.80 III.! 16.5 0.60 111.2 16.5 1.80 111.4 16.5 3.80

computed vOIuQs

TABLE 4 ShIp motion date

m2asurd valuQs

r---1

_{- - -}

IiriarIy xtrapoIotod from moasIJr3mqnts

N NE E N

I. .600.10-/9.0 .207.10/10.6 5C).1/1O4/l09l .2O7,0/10.6 .207.103/10.6I706.10/10.9I .32t103/11.5 .207.10/10.6 .600.10/9.0

[.2 .530.10-/9.0 186.10.2/10.6 53.10/f09I 9s3.1a/1o. .186jo2/lo.6 .186.102/1Q6 [953.102/10. .289.10.2/11.5 186.102/10.6 .530.10/9.0

(.3 .23710.2/9.0 .830.10.2/10.6 4 .100Á0.511485.100/10.91 .830.10.2/10.6 .830.10.2Á0.61.485.100/1O.cl .129.10.1/11.5 .830.10.2/10.6 .237.10-2/9.0 11.1 .312.10.2/14.4 .712.10.2/14.5 I13O.100/12.1F30.100/12.9 .712.10.2/14.5 .71210.2/14.5 [i30.100/12.9 .208.10.1/15.2 .712.10-2/14.5 .31210.2/14.4 11.2 .281.10'/14.4 .641.10V14.5 I.117.101/12.911.117.101/12.91 .641.10/14.5 541.10.1/14.5 l.i17.101/12.H I .187.100/15.2 .641.10.1/14.5 .281.101/144 11.3 .125.100/14.4 .286.100/14.5

rr'

152O.lOVl2.911520.10V12.91.286.1O°/14.5 .286.100/14.51.520.101/12.QI .834.100/15.2 .286.100/14.5 .125.100/14.4 111.1 .167.101/18.6 .302.10-1/18.8 185.101/13.211.685.10.1/13.4 .302.10'/16.8 .302.10.1/18.8 $.685.10'/13.21 I---.-.---J.861101/18.0 .30210.1/18.8 .167.10Y18.6 11L2 .1s0.,00/lB.S.272.100/18.8L568.100/13.1II.568.100/13.1I.272.100/18.8 .272.10°/18.8 1 L568.10°/13.1I.775.10°/18.0 L__.._. _.j .272.100/18.8 .150.100/18.6 ]fl .3 .667.100/18.6 .121.101/18.8 253.1o1/137 I!253.lO1/13.1 t .121.101/18.8 .121.10'/18.8 L253.101/7: I

L____J.

.345.10'/18.0 .121.10'/18.8 .667.10"18.6

vertical ship move -'gross underkeel

ments due to swell, clearance

e,'., ,.+ 4,,, +,. e,,,,,,'l nominal channel1 net underkeel 1bed level j Clearance Y -sounding accuracy A

sediment deposit between two dredging campaigns

w

tolerance for dredging channel dredged lvl

FIG i The deterministic approach (acc. to [6])

absolute minimum zero-crossing period T1 peak admissible draught (ship size) w absolute maximum

FIG. 2 Definition of characteristics of Zc (t)

DESCRIPTION COMPUTATION OF

conditions

-Gaussian distribution of samples

-stationary process - finite m2 - value

theory of extrema for (Gaussian, stationary) stochastic processes condition /V »1 Poisson distribution of A ( ;Tp) k condition

T -*

E *. o

-binomial distribution of (R;Tp)=k

-Rayleigh distribution of peaks

-expression for N CATEGORY 2 > L o 'n Oc Tp1TpTp

-i-H

:

channel closed for navigation

-* t

(1-DT tidq)* -Pr

[(R

Tp) o] -E A(KC;Tp))

_Pr[(R; Tp)ek]

-E -Pr (Re; Tp)'. k]

-E {(R ;Tp))

FIG. 3 Available methods for analysis of vertical ship motion dato

CATEGORY 3

(n=n,,n2,n3...N)

water level n3 water level n2

water level n

FIG. 4 Discretization of vertical tide for T « T,,

}

-17

-16--9

-8-A. Traffic int5nsity du to downtime rsuIting from wave conditions only

-21

-20---19

---18

---B. Traffic intensity du to downtime rsuIting from wave and tidal conditions

FIG. 5 Effect of downtima on traffic intensity

--11342 m 11000 m --10000 m - 9000 m 8000 rn channI bottom - 25.5 m

7000m

_{radius of curvature} 3450 m

60

_5LT. OE

--/

O r) ¿: O

/

FIG. 6 Pr2liminary dQsign of thQ chanril dQfIcti0n ongI E

I

I

3 2 * KC 8.3 m, DT < 0.7 °f

V---Y R4.6 rn, 0.8I< DT< 1.3I

V--V

=4.6m, 1.3°f< DT<i.8°f,

£«--A KC= 2.9 m, 2.7°/< DT< 3.2/,

x---x KC= 1.7 m19.8°/< DT< 20.5

safQty critsrion

Pr A >1IT=9.47*108s} 0.10

FIG. 7 DQfInItIon of scvcrol sntrancs r2gimas with corrcsponding downtime and rQquird

R -vaIu for a spcIfiQd safety critarion 5 -A CHANNEL OPEN 8 9 10 ii 12 13 14 15 16 17 -25 -24 -23 -22 c C o

r

_L) L