### : RIJKSWATERSTAAT

### . COMMUNICATIONS

### No. 18

### EXPERIENCES

### WI11I

### MATHEMATlCAL MODELS

### USEDFOR

### WATER

### QUALITY AND QUANTITY

### PROBLEMS

### BY

IR. J. VOOGT

A D

DR. IR. C. B. VREUGDE UIL

1974

B 2032(1 18

RIJKSWATERSTAAT COMMUNICATIONS

### EXPERIENCES WITH MATHEMATICAL

### MODELS USED POR WATER QUALITY

### AND QUANTITY PROBLEMS

by

IR. J. VaaGT

Chief Engineer, Rijkswaterstaat, Data Processing Division. The Hague

DR. IR.C. B. VREUGDENHIL

Head Mathematics Branch, Laboratory 'De Voorst', Delft Hydraulics Laboratory

RIJKSWATERSTAAT

DIRECTIE WATERHUISHOUDING EN WATERBEWEGING

THE HAGUE - NETHERLANDS

This article is the text of a discussion paper presented at the Symposium on the Use of Computer Techniques and Automation for Water Resources Systems, Washington, 1974, organized by the Uni-ted Nations Economie Commission for Europe.

The views in this article are the authors' own,

### Contents

page

5 Summary

7 1. Introduction

9 2. Computational methods

13 3. Quantity problems in one dimension

13 3.1 Water management in drainage networks

17 3.2 Water management in the Delta region

21 3.3 High discharges along the Rhine branches

23 4. Quality problems in one dimension

23 4.1 Cooling water circulation

24 4.2 Water quality in storage basins

28 4.3 Waste water in estuaries

31 5. Two-dimensional problems

31 5.1 Waste concentrations on the coast

31 5.2 Waste water in estuaries

36 5.3 Steady flow in lakes

37 6. Morphological aspects

38 7. Conclusions

**Summary**

The use of models for solving water quality and quantity problems is not restricted to mathematical modeIs, as other types also have their possibilities and drawbacks. Mathematical methods, however, have certain properties which make them partic-ularly suitable for water management studies.

The examples of mathematical models dealt with in this paper involve one or two spatial dimensions (in the horizontal plane) and time. Research on two-dimensional modeIs in the vertical plane and three-dimensional modeIs is not far advanced enough for them to be available for routine use. Quality parameters that do not affect the flow of water are the only ones considered in this paper. Methods of evaluating the effects of buoyancy and stratification are peing developed but are not yet operationaI.

Situations which can be investigated by means of present-day mathematical models are:

a. Flow of water and dispersion of heat or dissolved substances in canals or systems of canals (networks), either for steady or for unsteady flows,

b. The same phenomena in shallow lakes, estuaries or seas characterized by two-dimensional flow in the horizontal plane.

One-dimensional flow models have been used to study water management in several
networks of canals. The studies concern both design problems, such as the location
and capacity of new pumping stations, and decision problems connected with the
operation of a system of pumping stations and sluices. The one-dimensional technique
has also been successfully applied to the Oosterschelde estuary, much of which
con-sists of natural channels. Similar methods are being used to determine the distribution
of flow in the major beds of river systems when designing dikes. **In the latter type of**

computation the flow is assumed to be quasi-steady.

Once the flow in channels or rivers is known, either from one-dimensional models as described above, or from other sources, the water quality can be studied by means of convection-dispersion equations for the transport of dissolved substances or heat. Examples are the cooling-water circuit of a power plant (steady flow), the quality of water in artificial or natural storage basins (unsteady flow, including tidal effects) and the distribution of waste water in estuaries (considering net-flow only; dispersion is taken to include tidal mixing). The coefficient of dispersion involved in these models depends very greatly on the schematization and is empiricaI.

Two dim@Rsionalmethods tQ(}, havebeendevelopetl fot-the spreadiflgofl'OllUtllfltS.

Empirical dispersion coefficients are also required for tidal mean situations but they can largely be dispensed with by studying the two-dimensional tidal flow together with the waste distribution. Some results obtained from a comprehensive model of the Westerschelde estuary are given.

Lastly, attention is drawn to morphological changes which may be caused by human interference with the regime of rivers, estuaries, or coastal seas for water-management purposes.

In all the applications described above the mathematical formulation of water move-ment is quite well known (that for vertically two-dimensional or three-dimensional models is less so, because the turbulence structure is then involved). The description of quality parameters depends on the extent of our knowledge of biological and bio-chemical processes and that is still very little. Close cooperation with mathematicians can be very useful in further studies of these processes.

**1.**

**Introduction**

The use of models of whatever kind is prompted by the fact that they enable us to study and understand processes under existing or future conditions without interfering with the actual prototype situation. Some indications of the possibilities offered by various kinds of investigations connected with the design and operation of facilities, not only for water management, but also more generally, are given in this treatise. The fol1owing diagram indicates the possibilities.

### I

type of i nvestigation### H

measu rem ent i n prototype (ex isti ng situations only)### H

compilation and extrapolation of existing information### ~

model studies_{I}

### H

hydraulic model: model of the physical process studied, built to suitable scales so th at essential processes are represented satisfactorily~ analogue model: model of the physical process by analogy withsome other physical process which is easier to turn into a model

### y

mathematical model: solution of mathematica I equations representing the physical process### ~

digital solution### H

solution by analog computer, i.e. by means of electronic circuits designed to solve differential equations### ~

hybrid solution: combination of digital and analog techniquesAttention~drawDto

### too raGt

that measurement in protmype isalro requiredwhenmodel studies are performed.

Model studies enable us to investigate design situations, provided we can have
reason-abie confidence in the results obtained from the model. To ensure this, we need some
kind of mathematical formulation for all types of model. The main thing about
*mathematical models is that they require quantitative relationships between the*
physical quantities involved. A quantitative relationship can be very useful to estimate
scale effects in hydraulic models too [3]. The various types of model are operated in
much the same way. All of them require some kind of schematization, they should be
calibrated and verified carefull1y, they need the same data and design criteria, and the
results are interpreted in the same way with respect to costs or objectives and
con-straints. Mathematical modeis, on the other hand, have some properties which render
them particularly suitable for water management problems. They are:

(i) Mathematical models are not subject to scale effects which is an advantage

especially in cases where the scale conditions are unknown or difficult to satisfy; on the other hand it is not always easy to devise a satisfactory numerical representation of the equations.

(ii) In view of (i) no scale problems can arise when various interacting phenomena are being studied within the same model. Mathematical models are therefore very suitable for studying such interactions.

(iii) Mathematical models have great flexibility and can be readily consulted; each model is represented by a record of data on punched cards or magnetic tape; it is easily stored and recalled.

(iv) The part played by the computer can be extended to include (a) systematic optimization of the situation with respect to specified objectives and constraints, and (b) decisions for and possibly the automatization of the control of the pro-cesses involved.

The possibilities of optimization and automatic control have not yet been realized to a great extent. For well-defined situations, however, systematic optimization (such as the design of sewage networks) is quite feasible. The optimization of process control is also being realized, in industry and elsewhere, and so is the automatic optimal control of processes. Itis essential for such applications that the objectives (minimum cost, maximum safety, etc.) and constraints (limits to process control variables, criteria set by environmental considerations, etc.) be formulated very clearly. At present our knowledge of physics and computational facilities are restricting the range of application of mathematical models. Nevertheless, mathematical models can also be devised for situations that are not fully understood. A more or less phenomeno-logical description must then be resorted to, the empirical parts of which should be verified very carefully against prototype data. A semi-empirical model of this kind can be very useful if restricted to situations where it is applicable, being used, for example, on an operational basis for forecasting or management purposes.

**2.**

**Computational methods**

Most operational mathematical models run on digital computers.

Exceptions are these for one-dimensional flow problems for which an analog computer can be used (paragraph 3.2).

These models fall into two categories : one-dimensional and two-dimensional. The two-dimensional ones refer to the horizontal plane. Two-dimensional models in the vertical plane are not yet available for practical use. The same is true of three-dimen-sional models, the deve10pment of which has only recently been undertaken.

Consequently, stratified flows, e.g. cooling water problems near an outlet or
salinity-induced density currents in an estuary, cannot be dealt with adequately at the moment.
The one-dimensional flow models are based on two equations for the unknown
*water-level ( and the unknown discharge Q:*

the continuity equation

*aA* *aQ*

- + - = 0

*at* *ax*

and the momentum equation

in which

*A* = cross sectional area

### =

time*x* = location

C = Chézy coefficient for bottom roughness

g = acceleration due to gravity

*R* = hydraulic radius

*W _{x}* = wind force component along the river.

The solution of those equations is usually obtained by finite-difference methods [7,20] For one-dimensional water quality models the convection-diffusion equation

*a* *a* *a (* *ac)* *Ac*

*-* *(Ac)*

### + -

*(Qc) - -*

*AD -*

### + -

### =

0*at* *ax* *ax* *ax* }'

is used, in whieh

*c*

### =

eoneentration*D*

### =

dispersion eoefficient*y*

### =

relaxation time.If several eonstituents (possibly interaeting) are eonsidered, this equation holds for eaeh constituent, provided the last term is replaeed by the term

*[K] Aë*

*[K]* being the reaetion matrix. This eonvection-diffusion equation is also solved by
finite-differenee methods [2].

In this equation the discharge is known either from observations or from a one-dimensional flow model.

The two-dimensional flow model used in the Netherlands is based on what is ealled the Leendertse-method [10].

Here the unknowns are the waterlevel , and the two eomponents *u* and *v* of the

vertieally averaged veloeity. For the sake of eompleteness the eontinuity equation and the two momentum equations are given here

*0'* *0* *0*
*- + - ( H u )*

### +

*-(Hv)*= 0

*ot*

*ox*

*oy*

*ou*

*ou*

*ou*

*0'*

*gV*2

*W*

_{x}### -

### +

*u -*

### +

*v -*

*- fv*

### +

*g -*

### + - -

*u - Ah*V

*U - - -*

### =

0*ot*

*ox*

*oy*

*ox*

C2

_{H}

_{H}*ov*

*ov*

*ov*

*0'*

*gV*2

*W*

_{y}*;;-t*

### +

*u*::lx

### +

*v -*

### +

*fu*

### +

g -### + - -

*v -*

*Ah*V

*V -*- - = 0

*u*

*u*

*oy*

*oy*

*C2*

_{H}*in whieh*

_{H}*H*

### =

distanee from surfaee to bottom*f*

= Coriolis parameter
*V*

### =

magnitude of veloeity vector### =

*(u2*

### +

*V2)-!-Ah*

### =

horizontal eddy viseosity eoeffieient.For two-dimensional quality problems a set of eonveetion-diffusion equations

*o*

*0*

*0*

*0 (*

*Oë)*

*0 (*

*Oë)*

*-*

_{ot}

_{ot}

*(Hë)*

### +

*-(Huë)*

_{ox}

_{ox}

### +

*-(Hvë) - -*

_{oy}

_{oy}

_{ox}

_{ox}

*HDx*-

_{ox}

- -_{ox}

_{oy}

_{oy}

*HD*-

### +

*[K] Hë*

### =

0 y*oy*

The solution technique used in these two-dimensional models is a finite-difference method related to the alternating-direction-implicit method.

Like the one-dimensional equation the two-dimensional convection-diffusion equa-tion can be solved if the flow field does not proceed from a flow model but has been ascertained differently; assuming a uniform flow field the equation could be solved analytically for certain dispersion coefficient formulae.

**•** **MAIN** **PUMPINGSTATION**

**3.**

**Quantity problems in one dimension**

3.1. Water management**in**drainage networks

There are drainage networks in many parts of the Netherlands that serve more than one purpose. Their primary purpose is, of course, to get rid of surplus of water but when there is a shortage of water the system can be used to supply fresh water for irrigation. In addition the canals ean be used to get rid of pollutants or to prevent salt ground-water intrusion. Careful operation of the systems is required to maintain acceptable quality conditions, using the limited amount of fresh water available to keep them flushed.

The quantitative aspects of water management in a large number of these drainage networks have been studied by means of a computational'method for one-dimensional steady or unsteady flows in networks of canals [19, 20]. Some of the applications are diseussed below.

The Rijnland network (figure 1) had to increase its pumping capacity [16]. The siting and capacity ofthe new pumping station had to be determined; it had to satisfy design criteria for two extremes:

1. To ensure the delivery of water from the network at all nodes;

2. To enable the system to be flushed by using the fresh-water intake near Gouda. The water levels and the distribution of flow throughout the network had to be con-sidered. The mathematical model shown in figure 1 was calibrated and verified eare-fully, using readings taken in the prototype. A number of possible sites were con-sidered; the site near Halfweg turned out to be the best. The same model was used for studying several other matters, such as:

- The effects of filling in same of the canals in the cities of Haarlem and Leiden for road-improvement purposes ;

- The effects of the sudden collapse of a dike (causing water to run into the adjoining lower polders) on water levels and the stability of dikes in other parts of the system. The model can also serve as a basis for water quality computations. For example, when the model is operational, it can be used in case of emergeney, say, in the event of the accidental release of a dangerous pollutant, as an aid to deciding what measures to take. Computations could also be carried out for maintaining an acceptable water quality throughout the network during periods of water shortage.

A similar model for quantitative water management has been developed for the main

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canal system in the polders of North Holland (figure 2). There are several sluices and pumping stations in this system with which the flow in the network can be controlled. The model was calibrated and verified fairly thoroughly using the figures for a rainy period lasting seven days in 1960. Using the same data as a design condition, future situations were studied together with the effects of the proposed measures. For exam-ple, figure 3 shows the water levels at a central check point (Spijkerboor). When the level exceeds N.A.P. -0.10 m., the polders are forbidden to discharge water into the main system. Discharge may be resumed when the water level has fallen below N.A.P. -0.20 m. again. lt is evident from figure 3 that this situation is almost obviated by using the new pumping stations. Other methods of water management can be studied in advance in a similar manner.

3.2. Water management**in the Delta region**

Large barrier dams are being built in the south-west of the Netherlands. They are part of the Delta project. As the works have a marked effect on water movements, models have been built to study the effects during construction and after completion of the dams. Because of the importance of the dams different models have been and are being used; hydraulic and mathematical models, the latter being both digital and analog. Two examples of models applied to water management are dealt with below, one pertaining to the closure of the Oosterschelde and the other to the operation of the Haringvliet sluices.

The last estuary to be closed under the Delta project is the Oosterschelde (figure 4). A hydraulic detail model was built to study possible answers to such questions as:

- Where is the barrier dam to be located?

- By what method (e.g. gradual closure by means of a cableway, caissons) should closure be effectuated?

The boundary values for the model had to be obtained from an overall model, so a hydraulic model of the whoIe Oosterschelde basin was built [12]. A one-dimensional mathematical network model for the same region was only developed recently [17]. lts accuracy appeared to be comparable with that of the hydraulic model (figure 5). The one-dimensional approach was succesful because the Oosterschelde is mainly composed of gullies separated by shallow areas.

lt should be borne in mind however that only highly experienced people can schematize one-dimensional estuary models.

The northern Delta basin (figure 6) received its final configuration in 1971 when the Haringvliet was closed. Since then the northern basin's only open link with the North Sea has been the Rotterdam Waterway. As most of the fresh water discharges of the rivers Rhine and Maas reach the sea through this basin, it is evidentthatthe sluices in

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the Haringvliet dam are major instruments for controlling the water in the basin [9]. Water management is the most important consideration when a discharge programme for the sluices is being prepared. But shipping and the safety of the low-lying areas must also be taken into account; shipping would be hindered by excessive current velocities in the narrow channels, and the safety of the low-lying areas during a storm surge is directly dependent on the water level in the entire basin at the onset of the surge. Water management in the basin is a multi-purpose undertaking and the need to counter the intrusion of salt water through the Rotterdam Waterway is almost always a major consideration. Intrusion is lessened when the fresh water discharge through the Waterway is increased; the fresh water discharge can be controlled by operating the Haringvliet sluices.

A vast number of tidal computations involving many different upland discharges had to be performed when the discharge programme for the northern Delta basin was being prepared. An analog computer known as the 'Deltar' designed for one-dimen-sional tidal computations in networks was used for the purpose.

An example of fresh water discharge through the Rotterdam Waterway as a function of the Upper Rhine discharge and the actual discharge through the sluices is given in figure 7. As a fresh water discharge of 1.500 m3/sec is desirabie to curb salt-water intrusion, the sluices are not opened until this figure is reached. Itis evident that the situation has greatly improved since the Haringvliet project was completed.

**3.3.** **High discharges along the Rhine branches**

The main purpose ofthe Delta project is to prevent flooding by the North Sea but the dikes along the river Rhine and its branches have also had to be raised to prevent inundation during high Upper-Rhine discharges.

Theoretically, the ultimate heights ofthose dikes should be such that they will prevent inundation during a certain very high discharge called the design discharge.

The usual stage-discharge relation curve could not be used to determine the heights, because figures for very high discharges were not available, so a method of computing water levels during the design discharge [15] was devised; it is called 'stream-lane' computation. The term 'stream-lane' is a reference to the configuration of the Rhine and its branches. They are composed of minor beds (the 'summer-bed') and forelands. The forelands are between the main dikes but are separated from the minor bed by 'summer dikes' (figure 8). When the river discharge is very high, the forelands are flooded and are part ofthe major bed ofthe river.

The minor bed and the forelands are taken as separate stream-lanes when computing the discharge accompanying a certain fall in a river reach. The discharge in each stream-lane is computed with the Chézy formula and also if necessary with formulas for broad-crested weirs; the latter are used for forelands where cross dikes are found. The preset fall is adjusted iteratively until the computed discharge through the river

Figurc M. Configuration of a rivcr reach

reach cquals Ihe design discharge. Thc waler levels can be delermined from the falls lhus computcd.

Originally a longhand mClhod, stream-Iane compulalion has been compulcrized bccause it is so frequently uscd; il has lO be done for cvcry project to he carried oul between the winter dikes ifit is likely to raise the water level. Compensating \Vork is then needed lO prevem the water levels from raising. The effecls ofthe compensating \Vork are al50 compUlcd by lh is method.

**4.**

**Quality problems in one dimension**

**4.1.** **Cooling water circulation**

A mathematical model of the cooling water circulation has been made for a projected thermal power plant at Lake Ketel (near where the river IJssel debouches into Lake IJssel) (figure 9).

Figure 9. Schematization of Lake Ketel

Due to the geometrical configuration of the area, the direction of flow is fairly clear-cut, so a one-dimensional schematization can be used as indicated. Quasi-steady flow situations only were considered; the flow distribution was determined by means of a general-purpose computer programme for steady flow in networks.

The steady temperature distribution was determined from the balance between the convection of heat with the flowing water and the loss of heat to the atmosphere. Some of the results for certain design conditions are given in figure 10. A great number of hypothetical situations have been investigated in this way; it facilitates siting the power plant and designing the cooling circuit [13].

The manner in which changes in temperature distribution take place due to the power
plant suddenly increasing its heat production through a rise in the oudet temperature
has also been studied. The propagation of the sudden rise in temperature calculated
from a one-dimensional convection-diffusion equation is shown in figure **11.** Such
variations in temperature will usually be of short duration. For example, the effect
of two hours of excessive heat can be deduced by subtracting curves in figure **11**at a
two-hour interval, as indicated by the shading.

**OIJSSEL=300m3/s**
**e!JSSEL-3dcg**
**OZWARTE MEER= 0 m3/s**
**60 =240 m3**
**/s**
**6e _6dcg**

Figure 10. Computed isotherms; figures give temperature excess re!ative to natura! background
temperature (deg)
**[Y]HWJ** **37** **[mJ**
IN LET
Ilïl 31
- _ LOCATON
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± : : - - - - f - - - + - - - + _---t---t---r---+--+---"1

Figure 11. Temperature changes after sudden increase in oudet temperature (power plant near mouth of river Yssel)

Shaded portions : changes if increased temperature !asts only two hours.
*Diffusion coefficient D* = *20 h*

### lvi

**4.2.** **Water quality in storage basins**

Much of the drinking water for the west of the Netherlands comes from the rivers Rhine and Maas. As the water from the latter is generally better, some schemes have been devised to store water from the river Maas befare further treatment. The storage basins should be large enough to provide sufficient water during periods in which the quality of the river water is below the standard set for intake.

A system of four interconnected basins is being built in the Biesbosch area between one of the branches of the Rhine and the river Maas [21]. Basins A and Bare storage

basins, basins C and P serve as processing or mixing basins in which water is held for a time. A mathematical model has been constructed to study the capacity of the system and the strategy to be adopted when taking in river water. A diagram illustrating the system is given in figure 12. The quality of water is indicated mainly by its chlorinity,

*SUPPLY*
*p*
*RH/NE*
*c*
*AMER*
I
*B :* *A*
*Ia.b.lca+b·na.b*
*PUMPING*
*STATION*
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*CM* *CM*

Figure 12. Scheme of Biesbosch storage basins

to which a strict upper limit is set. Variations in the river discharge and in the quality of the water have been determined from historical records of observations. The intake volume and the volume of water in the storage basin are shown in figure 13 together with the chlorinity of Rhine and Maas water. The evening-out of chlorinity in the mixing basins is clearly seen.

There are several other important aspects of the quality of the water in these storage basins. A preliminary study of the growth of algae has been made [6], based on the assumption that light is the main agent limiting algae growth, and using rough empir-ical estimates of the vertempir-ical diffusion coefficient. The conditions under which the basins may be regarded as horizontally homogeneous have been inferred. The effect of artificial vertical mixing by means of air curtains has also been studied.

A second storage basin is projected using a naturallink between the rivers Waal and Maas called Andelse Maas.Itis closed at the river Waal end but is stilllinked by open water with the river Maas and it is situated in the tidal region (figure 14). A model

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Chlorinity of Rhine and Maas water

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depieting thequalityofthewater(mainly itschlorinity) shouldheprepared to enabte
a study to be made of the Andelse Maas as a storage basin. A computer programme
has been developed based on the one-dimensional convection-diffusion equation in
which the dispersion due to non-uniform velocity and concentration in a cross-section
is represented by a diffusion term [2]. There is a loek at the dosed end of the Andelse
Maas through which some Rhine water intrudes which has a different chlorinity and
can therefore be used as a tracer. Readings have been taken by the Delft Hydraulics
Laboratory and the Municipal Water Board of The Hague and the mathematical
model has been calibrated by means of them. Some results are shown in figure 15.
These are preliminary results, using a more or less schematized representation of the
river system. The effect of the longitudinal coefficient of dispersion *D* has been
in-vestigated.

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local velocity). The effect of the dispersion coefficient is not very dear. Additional dispersion however may result from temporary detention in channel irregularities, the effect ofwhich is still being studied.4.3. Waste water**in**estuaries

Estuaries are often used as dumping grounds for domestic sewage and industrial waste water, largely because oftheir self-purification properties.

When studying the behaviour of dissolved substances it is often enough to use a one-dimensional model of the estuary. In this type of model the transport of constituents is effected by convective and diffusive processes.

As the time scale of the diffusive processes is much larger than that of the convective processes, the distribution of continuously discharged waste in the estuary often takes weeks or months to become steady.

In this process of adjustment only the net-transport over a tidal period is relevant, so a convection-diffusion equation averaged over the tidal period is used [4]. The convective transport in this equation is provided by the fresh water discharge (at all events in an unbranched estuary). The diffusive transport accounts for the tidal mixing processes, so a dispersion coefficient must be adopted which is several orders of magnitude larger than usual.

The Eems estuary is taken as an example of a river mouth charged with waste water. This estuary, situated in the north-east of the Netherlands, receives waste water from straw-board factories and potato-flour mills. It is still being discharged into the canals but a few years ago plans were worked out for a pipeline to transport the waste to the estuary. To investigate the effect of various possible outfall sites the waste

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distribution in each of them was computed [8] with the afore-mentioned convection-diffusion equation.

The empirical dispersion coefficient was determined from observation of the salinity distribution, as the salt is a conservative constituent, in fact,ithas been measured very often.

Itappeared that the dispersion coefficient needed to approximate the observed salinity distribution depended on the season (figure 16). The values are lower in dry than they are in wet weather. Tidal mixing processes seem to be stronger in wet weather possibly due to meteorological factors.

Figure 17 shows the distribution of BOD and DO for three possible locations of the outfall site.

Itshould be noted that the distribution curves have no bearing on future conditions, because since the investigations described in [8] it has been decided not to discharge the waste water until it has been partially purified.

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calculated for three different discharge points along the Eems

### 5.

### Two-dimensional problems

5.1. Waste concentrations on thc coast

When domestic sewage or industrial waste is discharged into the sea the point of discharge must be far enough from the coast to prevent harmful concentrations reaching the shore. The superposition method [1] can be used for determining the length of the outfall conduit. The continuous release of sewage or waste is taken as an infinite series of instantaneous releases, the concentration distributions of which can be computed. The concentration at a certain point is obtained by superimposing the concentrations due to the successive instantaneous releases.

The concentration distribution due to an instantaneous release must be determined from tracer tests in the coastal area concerned. By adjusting the coefficients in the mathematical model, computed and measured values can be made to tally fairly closely. Applications of the method win be found in [5]. One of them was for the design of an outfall for domestic sewage from The Hague near the bathing beach at Scheveningen. The computed values for a conservative tracer tallied fairly well with the observed values (figures 18 and 19) but problems were met when a decaying constituent (coliform bacteria) was used. Itis thought that lack of knowledge of the decay time was responsible.

5.2. Waste water in estuaries

As stated in paragraph 4.3. it is often enough to study waste water problems in estu-aries by using one-dimensional models, though they suffer from shortcomings: - no detailed information can be obtained as the concentrations are cross-sectional

averages;

- some convective transports cannot be accounted for (e.g. overflowing of shallows) while others have to be accounted for in the dispersion coefficient;

- reaction processes between constituents cannot be computed accurately (one reason being that reaeration makes the DO concentration much higher in shallow areas than in the deeper parts).

Consequently the predictive value is limited because of the semi-empirical nature of the model. If such limitations are not admissible a two-dimensional model must be used.

The Westerschelde is an example of an estuary with gullies and shallows. The dif-ference between the area fLooded at high water and that at low water is shown in

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figure 20, in which the dots represent the grid points used to compute the flow process. Domestic sewage and industrial waste water are discharged into the estuary and the water coming from the Belgian Schelde is fairly strongly poUuted.

Due to the large tidal prism, the water in this estuary is weU mixed so the two-dimen-sional water quality model described in [11] could be employed.

The flow model is set by using the water levels recorded at the western and eastern boundaries. As the location of the land/water boundaries in the shallow areas changes during the tidal cycle, the model aUows for boundary changes dependent on time. The distributions of dissolved oxygen, biochemical oxygen demand, coliform bacteria and salinity are computed simultaneously. Salinity is used to calibrate the convective and diffusive transports. Figure 21 shows a hypothetical DO-distribution along the estuary together with the velocity vectors at the grid points.

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When designing the cooling water circuit for a power plant near a lake called Bergu-mermeer the question arose as to what part of the surface would be available for atmospheric heat-dispersal. To find the answer, a two-dimensional flow computation was performed using the Leendertse-method. It appeared that the cooling water discharge generated circulation currents in the lake (figure 22) which play an impor-tant part in the cooling process.

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**6.**

**Morphological aspects**

If water management considerations necessitate the alteration of natura1 water cours-es, it shou1d be rea1ized that morpho10gica1 changes may ensue. The cana1ization of the Lower Rhine [18] is a case in point. Three weirs have been built in the Lower Rhine, both to distribute the water over the three branches (figure 6) and for naviga-tiona1 purposes ; the one near Driel serves to control the discharge of the Lower Rhine. This weir's backwater influence extends, however, to the two bifurcation points, so that the Waal branch will tend to draw more water. This is counteracted by artificially 10wering the bed of the Pannerden Cana1 and the upper IJssel, and by cutting off two bends near Doesburg and Rheden.

The river bed is required to be stabie under the new conditions so the sand and water regimes, especially near the bifurcations, shou1d not change. This is being investigated by means of hydraulic modeis, while the river reaches are being studied by means of a mathematical model for river bed behaviour.

The mouth of the Haringvliet on the seaward side of the dam is another examp1e. Before c10sure the tide moved free1y in and out but now the tide is prevented from entering the estuary and water is discharged from the sluices at irregu1ar intervals. Morpho10gica1 changes will occur but how they will take place is not yet certain. Two-dimensiona1 tida1 computations have been performed for the area on the seaward side of the sluices [14] to study the effects of the c10sure when the sluices are c10sed or are discharging.

Qua1itative conc1usions regarding the morphologica1 deve10pments were drawn from the computed flow patterns.

### 7.

**CODcluiiioDii**

The conclusion can be drawn from the mathematical models dealt with that in many cases it has been possible to obtain reasonabie results with fairly simple tools. A major drawback was that many of the models contained some empirical parameters which had to be determined anew for each new application. This is especially true of the disper-sion coefficient under which all kinds of processes have been classed.

To increase the reliability of predictions with mathematical modeis, the empirical part should be reduced. This can be done by building more detailed models in which the various processes are modelled more accurately. Good progress has been made in this respect as far as the two-dimensional horizontal models are concerned.

Two-dimensional vertical models are not yet available for operational use. This is mainly due to the fact that so little is known about the physics of vertical diffusion and shear stress, especially in the case of density currents.

Three-dimensional models are beset by the same problems. From the computational point of view it is reasonabie to expect that in the near future computer facilities will be available for the limited application of three-dimensional models.

It is too often evident when modelling the reactions between different constituents that an accurate quantitative description of biological and biochemical processes is lacking. These processes call for careful study.

**8.**

**References**

[1] ABRAHAM, G. and VAN DAM, G. C. - On the predictability of waste concentrations. FAO technical conference on marine pollution and its effects on living resources and fishing, Rome, 1970.

[2] BERKHOFF, J. C. W. - Transport of pollutants or heat in a system of channels, in: Hydraulic Research for Water Management. Proceedings and Information No. 18 Committee for Hydro-logical Research TNO, 1973, also Pub!. No. 100 of Delft Hydraulics Laboratory.

[3] BIJKER, E. W., STAPEL, D. R. A. and DE VRIES, M. - Some scale-effects in models with bed-load transportation. IAHR Congress Lisbon, 1957.

[4] VAN DAM, G. C. - Some formulae for a one-dimensional approach to the oxygen balance in a river or other watercourse into which oxidizable substances are discharged. Rijkswaterstaat Report MFA 6701, 1970.

[5] VAN DAM, G. C. and DAVIDS, J. A. G. - Radioactive waste disposal and investigations on turbulent diffusion in the Netherlands' coastal areas. International Atomic Energy Agency, Symposium on the disposal of radioactive wastes into seas, oceans and surface waters, Vienna, 1966.

[6] Delft Hydraulics Laboratory - Model of algae growth in the 'Brabantse Biesbosch' storage basins. Report R 705, 1972 (in Dutch).

[7] DRONKERS, J. J. - Tidal computations for rivers, coastal areas and seas. Journal ofthe Hydraulics Division, Proceedings of the American Society of Civil Engineers 95, HY 1, January 1969, 29-77.

[8] EGGINK, H. J. - Estuaries as receivers of large quantities of waste (in Dutch). Thesis Wage-ningen, 1965, also Communications of the Government Institute of Sewage Purification and Industrial Waste Treatment No. 2 (RIZA Mededelingen nr. 2).

[9] VAN EYDEN, W. A. A. and LANGEWEG, F. - The function of the Haringvliet Sluices within the water control system of the northern Delta basin. Weg en Waterbouw 31, January 1971, 30-35 (in Dutch).

[10] LEENDERTSE, J. J. - Aspects of a computational model for long-period water-wave propagation. Thesis Delft University of Technology, also Rand Memorandum RM-5294-PR, 1967. [11] LEENDERTSE, J. J. and GRITTON, E. C. - A water-quality simulation model for weil mixed

estu-aries and coastal seas: Vo!. Il, Computational procedures. The New-York City Rand Institute, R-708-NYC, 1971.

[12] VAN DER MWLEN, T. - Model investigations for the Oosterschelde. To be published in De Ingenieur (in Dutch).

[13] A power plant at Lake Ketel, De Ingenieur 85, 21,1973,436-440 (in Dutch).

[14] Rijkswaterstaat, Deltadienst - The realization and function of the northern basin of the Delta Project. Rijkswaterstaat Communications No. 14, 1973.

[15] Rijkswaterstaat, Directie Bovenrivieren - Electronic computation of water levels in rivers during high discharges. Rijkswaterstaat Communications No. 9, 1969.

[16] STAPEL, D. R. A. and DE VRIES, M. - Experience with the mathematical model of the hydraulic network of Rijnland Water Board. 13th IAHR Congress, Kyoto, 1969, also Pub!. No. 74 of Delft Hydraulics Laboratory.

[17] STROBAND, H. J. - The Oosterschelde closure project, Hydraulic investigations by tidal compu-tations. To be published in De Ingenieur (in Dutch).

[18] SYBESMA, R. P., DE VRIES, M. and ZANEN.A. -Sandtransport~relatedtotDecanaiizatiOfl of the Lower Rhine. 22nd International Navigation Congress, Paris, 1969, also Pub\' No. 78 of Delft Hydraulics Laboratory.

[19] VEENINGEN, C. - Practical applications of computations for channel networks, in: Hydraulic Research for Water Management. Proceedings and Information No. 18 Committee for Hydro-logical Research TNO, 1973, also Pub\' No. 100 of Delft Hydraulics Laboratory.

[20] VREUGDENHlL, C. B. - Computational methods for channel flow, in: Hydraulic Research for Water Management. Proceedings and Information No. 18 Committee for Hydrological Re-search TNO, 1973, also Pub\' No. 100 of Delft Hydraulics Laboratory.

[21] WIJDIEKS, J. - Transport and storage of f1uids, Symposium 'Waterloopkunde in dienst van industrie en milieu' (Hydraulics as a tooi for industry and environment), Delft, 1973, Pub\' No. 110 of Delft Hydraulics Laboratory (in Dutch).

In the series of Rijkswaterstaat Communications the following numbers have been published before:

No. 1.* *Tidal Computations in Shallow Water*

Dr.J. J.Dronkerst and Prof. dr. ir.J.C. Schönfeld

*Report on Hydrostatic Levelling across the Westerschelde*

Ir. A. Waalewijn

No. 2.

### *

*Computation of the Decca Pattern for the Netherlands Delta Works*

Ir. H. Ph. van der Schaaft and P. Vetterli, Ing. Dipl. B.T.H.

No. 3. *The Aging of Asphaltic Bitumen*

Ir. A.J.P. van der Burgh,J.P. Bouwman and G.M.A. Steffelaar

No. 4. *Mud Distribution and Land Reclamation in the Eastern Wadden Shal/ows*

Dr.L.F.Kampst

No. 5. *Modern Construction of Wing-Gates*

Ir.J.C. Ie Nobel

No. 6. *A Structure Plan for the Southern IJsselmeerpolders*

Board of the ZuyderZeeWorks

No. 7. *The Use of Explosives for Clearing lee*

Ir.J. van der Kley

No. 8. *The Design and Construction ofthe Van Brienenoord Bridge across the River Nieuwe Maas*

Ir. W.J. van der Bbt

No. 9. *Electronic Computation of Water Levels in Rivers during High Discharges*

Section River Studies. Directie Bovenrivieren of Rijkswaterstaat

No.10.* *The Canalization ofthe Lower Rhine*

Ir. A. C. de Gaay and ir. P. Blokland

No.11. *The Haringvliet Sluices*

Ir. H. A. Ferguson, ir. P. Blokland and ir. drs. H. Kuiper

No.12. *The Application ofPiecewise Polynomials to Problems of Curve and Surface Approximation*

Or. Kurt Kubik

No.13. *Systems for Automatic Computation and Plotting ofPosition Fixing Patterns*

Ir. H. Ph. van der Schaaft

No.14. *The Realization and Function of the Northern Basin of the Delta Project*

Deltadienst of Rijkswaterstaat

No.15. *Fysical-Engineering Model of Reinforeed Concrete Frames in Compression*

Ir.J.Blaauwendraad

No.16. *Navigation Locks for Push Tows*

Ir.C.Kooman

No. 17. *Pneumatic Barriers to reduce Salt Intrusion through Locks*

Dr. ir. G. Abraham, ir. P. van der Burgh and ir. P. de Vos

### I

### .~

I