I

/,.C.M. 1982/E:40

Marine Environmental Quality Commitlee

This paper not to be cited without prior reference to the author

MATHEMATICAL MODELLING FOR EVALUATING

WATER QUALITY OF AQUATIC ENVIRONMENTS

Martine SOMVILLE

Introduction

International treaties regulating water quality at Belgianborders are on the point to be layed down. Concerning theScheldt at the Belgian - Dutch border, the norms asked at lowtide will at first be 4 mg/l dissolved oxygen and 4 mg N/lammoniacal nitrogen. In 1987,+they will respectively be raisedto 5 mg/l 02 and 2.3 mg N - NH 4 /1-

The mean actual oxygen concentration of the Scheldt at theBelgian - Dutch border can be described by figure 1.

This figure shows on an average the distance from the mouthwhere sampies characterized by 4 mg/l 02 are observed. Thearrows represent the interval estlmates at 90 per centconfidence. Figure 1 shows that 4 mg/l 02' i.e. the firstnorm to be respected has hardly never been attained between 1973and 1978. The place of occurence of 4 mg/l 02 is locatedabout 10 km downstreams thc Belgian - Dutch border.

This situation is the result of the organotrophic activitywh ich consumes dissolved oxygen to oxidize the organic loadcarried by the stream and the regeneration of oxygen by naturalrearation and mixing with seawater.

On the other hand, the distribution of ammonium in theScheldt estuary in the surroundings of the Belgian Dutchborder is gover~ed partly by the dilution of upstreams water,loaded in ammonlum, into paar seawater but also by themicrobiological process of nitrification which oxidizes ammoniuminto nitrate (Billen 1975, Somville 1978).

The importance of variousmicrobiological on the water quality

processes: hydrodynamical,of the Scheldt is then

Vniversite Libre de Bmxelles, Laboratoire d'Oceanologie, avenue F.-D. Roosevelt, 50, B-1050 BRVXELLES (Belgium).

(

2

60

E.x

s:: 50:;0E.sw 40uc

~ •'6

30

1913 1974 1975 1976 1977 1978

Fig. I: Evolution with time of the mean oxygen concentration at theBelgian-Dutch border.The arrows represent the interval estimates at 90 per centconfidence.

evident. However, the primordial factor affecting water qualityis the organic matter concentration, concentration which can bereduced greatly by epuration of domestic and industrialsewages.

A question is then immediatly asked: to satisfy the normsfixed by the international treaties, which epuration must berealised?

Is it sufficient to realise a elasssieal (primary +secundary) treatment or is it necessary to eliminate thenutrients from the effluent, operation which rises considerablythe cost of the treatment plan.

The appropriate answer to such problem can be readilyobtained by the use of deterministic mathematical models. Frominitial conditions like organic load, oxygen, nitrogeneoncentrations, these models can predict water quality of astream and the influenee of environmental parameters on it.They constitute the only approach which summarize the naturalprocesses and therefore are a valuable management tool.

The objectives of this paper is first to describe aphysiologieal model of heterotrophie activity in an ideal riversubmitted to a punctual organic load.

This model ignoring dispersion processes will howeverdescribe a situation comparable with the Scheldt where a highorganie load is observed and whose degradation, once theavailable oxygen exhausted, requires the reduetion of other

,

3

r iver:theoxidants present inFe(OH)3 and S04-'

As shown by the study of Somvi11e and Oe Pauw (in press) onthe evolution of water qua1ity of the Sche1dt water on a 10years per iod, the hydrodynamies of the estuary inf1ueneesgreat1y the water qua1ity. In particu1ar the river discharge isone of parameters determinating the oxygen profile in theSehe1dt.

A seeond mathematiea1 model will thence be presented for thenitrification process.

This simulation will deseribe the bio10gical process in these1f-purification reach of the Sehe1dt estuary as well as thecomp1ex hydrodynamies of the stream.

l.-oeseription of the water qua1ity in the Sche1dt estuary

•

A typieal situation in the Scheldt estuary is deseribed infigure 2. This figure shows, for May 1976, the longitudinalprofiles at low tide of mineral species succeptib1e to bereduced or oxidized by microbiologiea1 activity. Fo110wing thef10w of the r iver, .rone ean suce~~ive1y observe the oxygenconsumption, the production of Mn (rf~ueed form ofmanganese) which never exeeed 0,304 mg Mn /1 whieh is thesolubility of MnC03 in fresh water ealeu1ated with the freeenergies tabu1atea by Garre1s and Christ (1965) and Berner(1971) for pH = 7.4 and 250 C. Afterwards, nitratedisappears, redueed iron is produeed and a sma11 eonsumption ofsulfate is observed. All these observations eorrespond to theeonsu~ption of oxidants or the produetion of reducers.Oownstreams, the opposite processes occur in the reverse order:rising of sulfate, disappearanee of redueed iron, produetion of"nitrate, eonsumption of redueed manganese and final1yreapparition of oxygen due to reaeration of the river.

These profiles show that:

1. the oxygen is rapidly exhausted by aerobic aetivity andis absent on 30 km.

2. when oxygenother oxidantsmangani-reduetion,by ferri-reduetion

is absent, heterotrophie aetivity usesin a definite order: Mn02 by

N03- by_denitrifieation, Fe(OH)3ana 504- by sulfato-reduetion.

the

3. downstreams, se1f-purifieation processes restore oxidizedspeeies in the river: in the ordre sulfo-oxidation,ferro-oxidation, nitrifieation and mangano-oxidation.

4

~ ! 20_- 400

c c.2 ·'". ~·" cc ".

)00 ·" > 1\c

]c"

I cM

'" "~,~N 200 10

100

i '"c ~.2 .· ~"~ ~~ ce I. \

i·· ·" ·:..-g •.··c

'"

o. \

\0 100

r>ist,lncc to thc sca (KM)

Fig.2: Longitudinal profiles of O2 , lln++, NO;, NO~, Fe++ and SO:concentrations at low tide in the Scheldt as a function ofthe distance to the sea for l1ay 1976.

2. Simulation of organotrophic activity

The simulation of organotrophic activity in a heavilypolluted stream presented below modelates the variousmicrobiological processes pointed out in the Scheldt estuary.The basic principle of the model is that each matabolism occursin definite ccnditions of oxygen concentration. ~

The rnicrobiological activity has been related to the organicload C carried by the river by the following relation:

\~here oABa

fl'f 2 ,f3 ,f4 ,

is the organotrophic activitythe corresponding biow~55

the optimal specific acitivityare respectively functions oforganic load (C), temperature,chlorinity and oxygen concentra­tion.

5

The f l to f 4 functions have been choosen accordingto experimental results (Somville, 1980). The principles of themodel are graphically represented on figure 3.

8tmosphere

•M

R

®

dl~,8CJ BI \~.DA DMD) CO2

~g~ .'------_-.:...- @

ll"NO::=A:Ä==>8 - ---J

•Fig. 3: Principles of the model of organotrophic activity.

Organic earbon C, present in the strean at kn 0, consumed bythe organotrophic activity oA, is partially c~iGized in CO 2by the respiratory rnetabolism i and partially netabolized by theheterotrophie biomass Bi. The organic natter oxif.aticn inCO 2 (OMO) has been expressed by IQi, Qi correspo~ding to theoXIdation of C by the metabolism i aceordinq to its particularkineties.

Organotrophic aetivity OA has been evaluated from the fluxof organie matter oxidized in CO 2 in eonsidering for eachmetabolism i the existenee of a constant ratio bet~een thebiomass synthesis Ci and the oxidation of the load in CO 2 •Organie earbon is regenerated by rnortality M.

Organie nitrogen fluxes have been eonsidered proportional tothe earbon fluxes: the ammonifieation A proportional to theorgantrophie activity OA, assimilation NO proportional to earbon

6

assimilation Ci and regeneration R nrC'!'"~r~:'· ;ial to mortality Mby means of the CIN ratio.

The distribution of the oxidants results of theireonsumption by heterotrophie metabolisms Cox i and theirregeneration ROOX i by self-purifieation and reaeration.

An example of theoretieal profiles of organie earbon andoxidants eomputed by the model are represented on figure 4. Thisfigure shows that the eomputed profiles reproduee the generaltrend of experimental profiles meaf"~,:ed in the Seheldt estuary: •oxidants are eonsumed in a definite order when the load is highand regenerated in the reverse order during self-purifieation•

...- --,400

~-

•c

~c

'"e...200

300

604020

o L1__=:::;;::::..:.....z:t:======-_......._...J

o

9- 2000

.:-.

.~

i 15003

~u

~ 1000

0

~.~

~ 500

Distancc to thc 8(.'3 (km)

Fig. 4: Ca!culated profiles of 0fganic carbon. dissolved oxygen. ~ln02.

N03. Fe(OH)3. 504 and NH4 as a function of the distance from theorganie matter diseharge.

Sueh a nodel is partieularly interesting for studying troeeffect of organic carbon coneentration on water quality. ayexanple, the effect of the initial load on two parameters oftenused to assess water quality of a strean has been studied:

1. the distance necessary to reobserve 90% ofoxygen saturation.

2. the maximum defieit of oxidants in the stream.

7

Figure 5 represents Lcth of these rara~~ters a~ a functicnof the initial organic load. In thi~ fi~~rc, t~F cc~c=r~ratior.

of the different oxidants (02' r-n0 2 , NQ3'Fe(OH)3' S04) have been expressed in eq e 11susceptible to be exchanged ane sumed.

r----------------...,200

fJ eq .-/1

,-- --

100

a small 'reach of the river. Thisthe ob~ervaticns in the upstrearn(fig. 2).

•Fig. 5: Evolution of the minimal concentration of oxiJants (---) and

of the distance necessary to reobserve 90% of oxygen saturation(----) as a function of the initial organic load •

Figure 5 shows tha t tr.e biger tl:e cr<:;ar.ic l~:a<.~ the furtl:er90% oxygen saturation is cbservec. On the ether r.anc for ~~all

loads, oxygen is sufficient to oxiLize the orsanic ~atter. Forhigher loads, the other the oxiäants are sueeessively recucec.

l~oreover, these results can be e;{tra;:olat€:G tc U:e Scl:clctestuary. One can be evaluate that the biedeara{able loac i~ theupstrearn part of the estuary is actually abcut 2000 atg organicC/l. Fer this load, the model predicts (fic. 4), the using upof oxygen, Hn02 , N03-.Sulfato-reduct~on occurs onprediction is comparable withpart of the Scheldt in summer

---- --- ----- - ---- ---- --------------------------------

8

The model indicates (fig. 5) that for organic loads in therange 1000-2000watg org C/1, the most important deficit ofoxidants does not change. This would further indicate, ifextrapolation of the ideal model to the Scheldt is correct, thatan epuration of 50% of the load of the river, reducing the loadto 1000watg org C/l would not improve the oxidants content ofthe river.

Dissolved oxygen would be regenerated sooner (moreupstreams). In the Scheldt, however, regeneration of dissolvedoxygen is chiefly governed by the mixing with aerated seawater • •3. Modelisation of nitrification process in the Scheldt

a. determination opf ecophysiological parameters

Two examples ofan environmentalconcentration anddescribed.

experimental determination of the effect ofparameter: i.e. substrate (ammonium)

salinity on nitrifying activity will be

Effect of substrate concentration

The relation between the potential nitrifying activity of anenrichment culture of nitrifyers and the ammonium concentrationis shown on figure 6. This experi~ental relationship has beenrepresented by a ~lichaelis-Menten-r_~onod functior.:

[lIH4

+]

with Km equal to

[NH4+) + Km

+250wH NH 4 •

Effect of salinity

Potential nitrif4ing activities measured on short term.experiments by dark C-incorporation (Somville, 1978) atdifferent places in the estuary have shown that duringprogressive mlxlng of fresh into saline water masses, the insitu population of nitrifying bacteria tends to adapt itself tothe prevailing chloride concentration, with, however, adefinitedelay (Somville, 1980).

The relation found between the salinity of the sample andnitrifying activity at optimal salinity is presented on figure7. This experimental relation will be modelated by the straightline drawn on figure 7.

9

~>. •-:~-u1'0

Clc:

">'-...-c:

0250 1000 2000

Fig. 6: Relation between the nitrifying activity and the ammoniumconcentration for an enrichment culture of nitrifyers.

10 15

salinity (CI/I)

,....~>.-":;

:;;U1lI

Cl 50c:'>'-";:-'e

0 5

Fig. 7: Relation between the nitrifying activity at optimal salinityand the salinity of the original water mass.

10

-----------------------------

The model of nitrification described below has been realisedin collaboration with G. Billen.(U.L.B.) and J. Smitz (U.Lg-SartTilman Liege - Belgium) and has been accepted for publication byMa thema tical Modelling.

The hydrodynamies of the Scheldt is described by a simpleone dimentional model. The longitudinal distribution of anycross-section-averaged concentration c can be described by anequation of the form (Nihoul and Ronday, 1976):

where

,;Dc =~ (ac) + ~ (aue) - ~ [A ~ (ac)] = P - D- 6t 6x 6x 6x

x is the longitudinal coordinate1a is the mean cross-section (calculated as an

exponential function of x, Wollast, 1973)1u is the cross-section-averaged residual velocitY1A is the global dispersion coefficient

(including effects of tidal motions and othercomplex hydrodynamical phenomena typical of apartially stratified estuary)1

P and D are respectively the rates of production anddestruction of c as a rcsult of physical, chemicalor biological reactions1

c is the cross-section-averaged concentrationaveraged over some per iod T larger than the tidalperiod.

~ represents the hydrodynamical operator

Thc computation of the residual velocity u and of thedispersion coefficient A is obtained by the hydrodynamical modelof the estuary, the precise calibration being made on thechlorinity concentration profile (chlorinity is a conservativeparameter which concentration depends on mixing between saline.and fresh water) •

In the Scheldt estuary, the upstream water dischargepresents slow seasonal changes, and a steady-state assumption isvalid for the description of concentrations variations.

The comparison of nitrate flux from Scheldt sediments(Somville, 1980) and of nitrate production by planktonicnitrification (Somville, 1978) has shown the the latter process,accounting for more than 90 percent in the nitrate budget, wasby far the most important.

These observations have led to consider in the model thenitrification process as the result of planktonic nitrificationonly.

..

Growth rate of nitrifyers (G) andare considered as proportional tonitrifying bacteria (B):

G = K • BA=Cl. .K.B

II

nitrifying activity (A)the number of planktonic

•where K (sec- l ) is the growth constant1

Cl. is the quantity of ammonium to be oxidizedfor duplicating one bacterium, i.e. thereciprocical of the yield constant Y1

B is the concentration of nitrifyers(bacteria/l) •

The value of K is consideredparameters (namely salinity,temperature, redox potential).

to depend on environmentalammonium concentration,

The evolution of the nitrifying biomass Bhydrodynamic processes, growth and mortalityexpressed by:

resulting fromeffect, can be

B = KB - mB (1)

and the distribution of ammonium and nitrate are then expressedby:

a K B

- a K B

K can be expressed by:

(2)

(3)

K

where k is the ootioal growth constant for nitrifyingbacteria, anc f l , f 2 , f3~ f 4 are respectivelyfunctions of salin~ty, ammonium concentration, temperature, andredox potentia11 the value of these functions is 1 at optimalconditions.

Solution of the differential equations

Boundary conditions

For solving equations (1), and (2) and (3), a set limitconditions (upstream and mouth water composition) has to beknown. In the case of chemical species, ammonium and nitrate,

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these conditions are obviously the experimental concentrations.In the case of bacterial concentration, boundary conditions canbe experimental MNP counts.

Owing to the poor reliability of this largely usednumeration method (Tate 1977, Belser 1979), it appears necessaryto check the accuracy of the nitrifyer concentrations bycomparing the computed activities (a.K.[B]), based on thepossible range of a and K values and the exper imental bacter ia 1concentration, with thein situ activities measured in thestream (Somville, 1978, 1980)-.---

The comparison has shown that the MPN counts obtained in theScheldt were several orders of magnitude lower than what couldbe expected from the direct activity measurements. It wastherefore decided to evaluate nitrifyers numbers directly fromthe in situ measured activities.

Solution of equations (1), (2), (3)

Owing to the coupling of equation (1) and equation (3) bymeans of the aMmonium concentration, the first step computes thesolution of the bacteria equation (equation 1) using theexprimental ammonium profile, previously smoothed.

The complete solution of the problem is calculated by aniterative process adjusting the constant a. The convergence ofthe process is quadratic and is obtained after a few loops. Thevalues ofa and k so determined are in perfect agreement with theliterature.

The computed profiles of ammonium, nitrate bacteria andnitrifying activities are represented on figure 8 A and Brespectively for the situations of February and July 1976(values of a and k used for hese simulation are reported in table1). The corresponding experimental profiles are represented on •figure 8 C and D.

The unidimensional mathematical model of the nitrificationexposed above describes quantitatively the nitrification processin the Scheldt estuary. With the aid of a limited number ofecophysiological relations, an accurate description of the insitu situation is possible.

TAßLE I : a and k values used in the simulation

Honth

February 1976

July 1976

k

25 1O-6sec -1

7.5 10- 6

a

3 IO-6~~H:/bact

0.74 10-6

.I; 100

<-

....

..

c

~ g.C

C. .~ ~

'.

200

hdl'txlO,~,

;'

8 July 1976

t'lact

..

.~..~~..!->,

>

0.5 III

f

13

200

100

l: rl;>bruary 1976

')11

nist.lnn' tu tlw 'h';:I (kM)

Fig. 8: Calculated (A and B) and experimental profiles Cc and D)of nitrate, ammonium, nitrifying germs and nitrifyingactivities as a function of the distance to the sea forFebruary CA and C) and July 1976 (B and D).

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BIBLIOGRAPHY

Belser, L.W., (1979). Population eeology of nitrifying baeteria. Ann. Rev.Mierobiol., 33, 309-333.

Berner, R.A., (1971). Prineiples of ehemieal sedimentology. Mc Grow HilI,New York, 240 pp.

Billen, G., (1975). Nitrifieation in the Scheldt estuary (Belgium and theNetherlands). Est. Coast. Mar. Sei., 3, 79-89.

Garreis, R.M., and Christ, C.L. (1965). Solution, minerals and equilibria.Harper & Row, New York.

Nihoul, J.C.J., and Ronday, C. (1976). tlodele d'un estuaire partiellementstratifie. Application a la cireulation residuelle et a l'etude del' envasement de l' Escaut. In Modele Ilathematique de la Her du Nord.Rapport de synthese - Esutaire de l'Escaut. Vol. 10, J.C.J. Nihoul& R. lVollast (Eds), Brussels, Belgium.

Somville, M., (1978). A method for the measurement of nitrification ratesin water. Wat. Res., 12, 843-848.

Somville, M. (1980). Etude eeophysiologique des metabolismes bacteriensdans l'estuaire de l'Eseaut. These de doetorat. Faeulte des Seiences,Universite Libre de Bruxelles. Belgium.

Somville, H., and De Pauw, N. Influenee of temperature and river discharp,eon water qualit~ of the Nestern Seheldt estuary. Wat. Res. (in press).

Tate, R.L., (1977). Nitrifieation in histosols: a potential role forheterotrophie nitrifyer. Appl. Environm. llierobiol., 34, 911-914.

Wollast, R., (1973). Origine et mecanisme de l'envasement de l'estuaire del'Escaut. Rapport de synthese. Reeherehes effeetuees dans le eadre del'etude de l'envasement de l'Eseaut dirigee par le Laboratoire deReeherches Hydrauliques de Borgerhout. tlinistere des Travaux Publics.

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