Ecological Modelling, 26 (1984) 285-311 285 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
A C O N C E P T U A L M O D E L O F U N I T - M A S S R E S P O N S E F U N C T I O N
FOR NONPOINT SOURCE POLLUTANT RUNOFF
FRANCO ZINGALES
Cattedra di Chimica, Facolta' di lngegneria, Universita' di Padova, via Marzolo 9, 3.5131 Padova (Italy)
ALESSANDRO MARANI
lstituto di Chimica Fisica, Facoha' di Chimica lndustriale, Unwersita' di Venezia, S. Marta 2137, 30124 Venezia (Italy)
ANDREA RINALDO
lstituto di ldraulica, Universita' di Padova, via Loredan 20, 35131 Padooa (Italy)
and GIUSEPPE BENDORICCHIO
lstituto di Chimica Industriale, Facoha' di lngegneria, Universita' di Padova, via Marzolo 9, 35131 Padova (Italy)
(Accepted for publication 25 July 1984)
ABSTRACT
Zingales, F., Marani, A., Rinaldo, A. and Bendoricchio, G., 1984. A conceptual model of unit-mass response function for nonpoint source pollutant runoff. Ecol. Modelling, 26: 285-311.
A conceptual mathematical model of unit-mass response function (UMRF) of nonpoint source pollutants (NPSP) is presented. The physical aspects of the mathematics involved yield explanation of some basic mechanisms observed in nature for various environmental re- sponses, like inherent hysteretical effects (rising-falling flow dependence) and linear pollu- tant load/discharged water volume correlations. The rationale of certain empirical relation- ships is discussed, and reasonable explanations are given for frequently occurring cases of lack of correlation for water quality field data.
The mathematical model substantiates screening or planning modelling techniques pro- posed in the literature, provided that reference weighted pollutant concentrations in the runoff are used. A conceptual form of such an average concentration arises also from the mass transfer and balance relationships investigated.
The model is designed to portray consistent NPSP balances (with interphase mass transfer) via an extension of Nash' conceptual scheme of rainfall-runoff transformations. The limited number of parameters and their clear physical meaning make this approach suitable to NPSP simulation and forecasting. The derived form of UMRF is simple, thus calling for microcomputer software applications.
Although the proposed model structure consists of a lumped parameter formulation, the approach may serve as an alternative to empirical routing procedures often implemented in distributed parameter codes. The techniques adopted make use of standard mathematics of hydrological and chemical models, and therefore the related high degree of sophistication developed for simulation and forecasting can be exploited by the present approach for ecological modelling.
Results of applications to the case study of the Venice Lagoon (Italy) are presented and discussed.
INTRODUCTION
A necessary component of water quality management studies is the analysis of nonpoint source pollution (NPSP). Determination of source magnitudes (which, unlike pollution from point sources, is associated with the early phases of the hydrological cycle), evaluation of control measures and selection of possible control programmes are primary goals for research progress, whereas the basic approach for the studies may be experimental or based on mathematical modelling. Nevertheless, NPSP is difficult and ex- pensive to measure both in its runoff or water quality components. Further- more, short-term water quality monitoring is seldom sufficient to char- acterize pollution variability, in particular as far as forecasting is concerned. Thus mathematical modelling is often the only approach for the study of NPSP which is consistent with budget and time constraints.
The historical development of modelling as a tool for quantitative descrip- tions of water quality processes has progressed along two main routes: the first, referring to the research of hydrologists and agricultural engineers, deals with runoff and leaching models; the second deals with instream or receiving water models, mainly investigated by environmental engineers.
In the paper attention is concentrated on the former research field for which general reviews of significant contributions have been recently pub- lished (Overcash and Davidson, 1980; Haith, 1981; Novotny and Chesters, 1981; Jol~nkai, 1983; J~rgensen, 1983).
In earlier hydrological contributions, the authors have been searching for experimental relationships between rainfall and runoff via the so-called runoff coefficient methods. Among the foremost contributions in this field, the U.S. SCS (Soil Conservation Service) equation still stands the test of time for its applicability and worldwide diffusion (e.g. Chow, 1964).
The development of unit hydrograph concepts, which had opened new frontiers towards a better understanding of flood generation mechanisms
287
(e.g. Chow, 1964; Dooge, 1977), has also brought new impulse to NPSP research. Among the leading contributions, it seems worthwhile mentioning the conceptual form of linear instantaneous unit hydrograph (IUH) pro- posed by Nash (1957). The basic idea consists of a schematization of the watershed into a cascade of linear reservoirs, by means of which the two basic mechanisms of flood generation and routing - storage and propagation - can be accurately modelled.
The need for deeper insight into the nature of flood generation mecha- nisms arose in the late sixties. The foremost achievements are related to the application of continuity principles and kinematic considerations for over- land and subsurface flow (e.g. the Stanford Watershed Model, Crawford and Linsley, 1966).
The built-in nonlinear nature of the IUH (e.g. Blank et al., 1971; Rao and Delleur, 1971; Amorocho, 1973) has been investigated, while the alternative charge of nonlinear effects on net precipitations has drawn further attention somehow relevant to the rationale of the present work (e.g. Dooge, 1977; D'Alpaos and Rinaldo, 1980).
The achievements of wide fields of modern hydrology (mainly referring either to ARMA and ARIMA or Box-Jenkins models, or to stochastic approaches) have not yet been exploited for NPSP simulations, even though some aspects seem promising and are likely to draw interest from NPSP research in the near future.
Transfer to NPSP simulation has come also from the vast literature on heterogeneous chemical reactors, even though some built-in drawbacks have limited the exchange between ecology and chemical engineering. Reactions in soil and waterways are, in fact, very complex and not yet fully under- stood, so that simple overall kinetics would perhaps suffice for accuracy of NPSP interest. Furthermore, the models investigated by chemical engineers use forced inflow rates which are hardly tailored to natural systems (Bird et al., 1960; Himmelblau and Bischoff, 1968; Hill, 1977).
The development of NPSP models, related to the mentioned progress of hydrological and chemical research, was recently reviewed in a thorough state-of-the-art contribution (Jol~nkai, 1983; J~rgensen, 1983), which classi- fies differing modelling approaches according to the degree of sophistication. The first order approximation consists of the identification of relationships between runoff yield of pollutants/runoff or runoff/concentration curves. Interestingly enough, experimental verification of field data brought to light significant differences in the relationship between rising or falling discharge and concentrations, so that hysteretical effects have been empirically taken into account (Smith and Stewart, 1977). As discussed in the theoretical spinoffs of the model presented in this paper, only pollutant load/dis- charged water volume relationships seem to fit linear correlation schemes
288
(Betson and McMasker, 1975; Smith and Stewart, 1977). More complex NPSP and land runoff models have been classified into
three categories (Jol~nkai, 1983). The first category refers to hydrological rational approaches (ranging from runoff coefficient to SCS equation meth- ods). Screening or planning models (Haith and Dougherty, 1976; Haith, 1981, 1982) are very important and flexible tools for planners and re- searchers, mainly because they do not require specialized data sets, water quality data or even computer programs. Such models are designed to identify potential nonpoint source problems and likely control measures and programmes.
The second category of modelling approaches is related to water quality components of IUH models. It is possible, in fact, to link directly unit hydrograph concepts to pollutant runoff-water runoff via definition of a unit-mass response function (UMRF). Such a function is defined as a mass flux against time in response to a rainfall event of unit intensity and duration (mathematically defined as a Dirac-delta function) uniformly dis- tributed over the watershed. Some experimental determinations of UMRF by standard techniques have already been carried out (as reported, e.g., in Jolfinkai, 1983).
The third group consists of a number of basically deterministic hydrology, sedimentology and water quality models which can be used to simulate NPSP loadings to surface waters (Crawford and Donigian, 1974; Donigian et al., 1977; Frere et al., 1980; Beasley, 1976; Novotny et al., 1978; Kniesel, 1978; Beasley et al., 1977). These models are designed to show the watershed impact or sensitivity to spatial placement of soils, crops and practices, thus allowing the user to describe thoroughly the physical setting of the watershed which is stlbdivided into regions of homogeneous land use and terrain.
Stochastic approaches for modelling of loadings and receiving water body have also been developed (e.g. Bog~rdi and Duckstein, 1978) and success- fully used on case studies. The techniques are powerful and lead to im- portant applications, although long-term reliable monitoring is needed for testing and calibration (Hartigan et al., 1983).
Quite recently, critical disposal or restricted-use areas contributing to NPSP have also been pointed out (Gburek, 1983).
From a conceptual standpoint it seems that neither approach can be uniquely defined as optimal. Lumped parameter approaches have the ad- vantage that inputs are simple to obtain and that their formulation and implementation is straightforward. Inherent nonlinearities can be accounted for either by nonlinear transfer function or by identification of net rainfall excesses. Distributed parameter approaches easily account for spatial hetero- geneity of the watershed allowing thorough description of the basic mecha- nisms of flood generation at the cost of defining several parameters. When
289
accurate long-term monitored data are available all techniques may prove valid within the accuracy of technical interest.
The research presented in this paper deals with a conceptual form of the unit-mass response function of NPS pollutant runoff to rainfall impulses uniformly distributed over a watershed. Recent literature (Jol~nkai, 1983) has recognized the significance and relevance of UMRF approaches both from the theoretical and the practical standpoint.
The mathematical model has been used for simulation of soluble phase field data gathered' within a reclaimed agricultural watershed discharging into the Lagoon of Venice (Italy) (Zingales et al., 1980, 1981b, 1982), as a first step towards a complete simulation of NPSP from agricultural runoff for the 2000 km 2 mainland discharging into the Lagoon. Further support for the need of novel approaches issues from the particular hydraulic and topographic conditions of these watersheds, whose complexity (with further complications induced by the subjacency of most reclaimed basins to the average sea level) proved unsuitable to simulation by existing distributed parameter approaches (Zingales et al., 1981b).
The structure of the model, the basic assumptions and conceptual schema- tizations, together with the resulting response functions are illustrated, while theoretical spinoffs and comparisons with the wide literature are discussed.
MATHEMATICAL MODEL
The rationale for the model lies in the attachment of a water quality component, based on simple mass balances, to the Nash conceptual hydro- logic model of rainfall-runoff transformations (Nash, 1957). Like the Nash rainfall-runoff transfer function, the structure of the model consists of a lumped parameter approach, even though a network of homogeneous areas (simulated by IUH-UMRF techniques) may serve as a submodel of distrib- uted parameter codes.
The interest for the procedure seems to be twofold. In fact: - Limits and validity of the actual conceptual schemes, proposed for water quality features of watershed simulations, depend upon the bounds imposed by the Nash hydrologic model, which has extensively proven both its theoretical and practical value. - The analytical form of the derived unit-mass response function allows for important theoretical considerations. For instance, hysteretical effects found in experimental curves of pollutant concentration/discharge relationships are explained, and a theoretical basis is given for certain absences of correlation.
Theoretical support for experimental finds and some empirical rela- tionships tailored to pollutant load/discharged water volume correlations
290
RAINFALL
0 16 3,2 ~8 6.L 80 9.6 II.2TIME OqF z'-"~
~ J I I
On ,q,.
C/CE r
08 [[~j m~ m j 0.4
0 I L I J ] 0 16 32 l,.8 6.• 8.0 9.GTIME
0 16 32 L,8 6L 8,0 9.6 If2 TIHE
Oq F . - - - - . / / /
f "* I I I I I I I 1 I I OF 16 3.2 i.8 6.1, B.0 9.6 11.2 TIME
• O,q . - -
I L ~ ' ] - - / I I I I I I I ~ l l . q 0 16 3.2 L8 6.1. 80 9.6 11.2 TIME
Fig. 1. Conceptual model of hydraulic and chemical behaviour of the watershed.
291
issues from the present results, hence pointing at the inadequacy of certain empirical treatments.
The model is derived from the following set of conceptual assumptions - The system soil-groundwater-surface water can be modelled by a two-
phase reactor. - The watershed can be modelled by a cascade of continuous stirred tank
reactors (CSTR), which embody Nash' schemes for water discharges. - The discharge outflowing the i th CSTR has a linear dependence on the
volume of mobile phase via a runoff coefficient K (Nash, 1957). - An interphase equilibrium exists between the concentrations of fixed and
mobile phases. - The driving force for interface mass transfer is proportional to the
difference between actual and equilibrium concentrations. - Equilibrium concentrations can be regarded as constant in the time scale
of rainfall-runoff transformation lags.
The resulting conceptual schemes of hydraulic and chemical components for watershed simulations are illustrated in Fig. 1, whereas Figs. 2 and 3 outline the mathematical models for mobile and fixed phases. Figure 2 shows the ith CSTR, which would differ from the sketch of the first reactor of the cascade only because of the inflowing discharge. Water flow rates into the first CSTR are, in fact, the net precipitations while in flowing chemical discharge - the rate of nutrients lost by the atmosphere - is regarded as negligible (Frere et al., 1980; Kniesel, 1978).
A noteworthy spinoff is that the response of the system to finite impulses of precipitation can be obtained via convolution of the unitary response with the impulse function. This holds true, in particular, because of the linearity assumed for the conceptual schemes.
The chemical mass transfer (as, e.g., well summarized in Novotny and Chesters, 1981; JCrgensen, 1983) is ruled by:
OC/Ot = h( C E - C) (1)
where C and C z are current and equilibrium concentrations in the mobile phase, respectively. The equilibrium concentration C E is a time dependent parameter connected with the number of chemical species supplied to the soil (partially uptaken by the crop within the plant root zone), with current reaction processes and with complicated removal mechanisms due to earlier rainfall events. The equilibrium relationship between the two phases governs the conceptual schemes of Figs. 2 and 3, thus allowing a separate modelling of the phenomena in fixed and mobile phases. Furthermore, according to Eq.
292
1 the mass transfer coefficient h accounts for effects of interphase specific surfaces.
The transfer function of the watershed to rainfall events distributed in time according to f(~') is built by convolution as"
u(t; n, K)= (AK/r(n))(Kt)"-l exp(-Kt) (2)
Q(t; n, K)=fotU(t-'r; n, K) f(r) d"r (3 )
where u(t; n, K) = unit hydrograph (after Nash, 1957), as a function of the parameters n and K; F(n) = gamma function of argument n; A = area of
h (CE-C): .qH (F-F E) Fig. 3. Mathematical model of mobile and fixed phase interactions.
293
the watershed; f(~-)=impulse hyetograph, and Q(t; n, K ) = w a t e r dis- charge as a response to net precipitations distributed according to f(~-).
Whenever equilibrium concentrations C E are regarded as constant in the time scale of unit-mass response function (UMRF), both the hydrograph and mass-graph can be extended to the case of noninteger values of CSTRs.
Furthermore, hydrologic simulation of different mechanisms of runoff formation (surface runoff, interflow, groundwater flow) has long been han- dled by a schematization of the watershed into two (or more) cascades of reservoirs in parallel, aiming at realistic portrayal of different concentration times shaping the final hydrograph (i.e. Dooge, 1977). The analogy for water quality components may be related to combined models of reactor en- gineering (Himmelblau and Bischoff, 1968).
The unit mass response function (UMRF) derived from the above as- sumptions is obtained by integration of the mass balance equations for the CSTR cascade. The results, whose integration details are reported elsewhere (Bendoricchio and Rinaldo, 1982; Zingales et al., 1982) in terms of xth pollutant flow rate qX and total quantity ZX, are:
qX(t; n, K, hX, CEX )
=CEX(Q(t; n , K ) - ( K / ( K + h X ) ) " Q ( t ; n , K + h X ) } (4)
C(t; n, K, hX, CEX )
= CEX{1 - ( K / K + hX)"(Q(t; n, K+ hX)/Q(t; n, K))} (5)
foqX(t; n K, hX, CEX ) dt ZX(t; n, K, hX, CEX ) = '
=CEX{V(t; n, K ) - ( K / ( K + h X ) ) " V ( t ; n, K+hX)) (6)
where X = Xth pollutant index; CqX(t; n, K, hX, C E X ) = Xth pollutant flow rate due to water discharge Q; C(t; n, K, hX, CEX ) = Xth pollutant concentration; ZX(t; n, K, hX, CEX ) = Xth pollutant total load at time t; V(t; n, K) = water volume at time t; hX= mass transfer coefficient for the Xth pollutant; and CEX = equilibrium concentration for the Xth pollutant.
Equations 2, 3 and 6 constitute a set of easy-to-use tools for simulation and forecasting of flood events of water and pollutants. It seems of some interest, in particular for forecasting purposes, that the number of parame- ters involved is limited and that their physical meaning is clear.
Owing to its simple analytical formulation, optimal calibration of the parameters via iterative procedures is relatively cheap and, the overall approach is deemed suitable to microcomputer software applications to- wards which a large portion of modern ecological modelling is oriented.
294
PHYSICAL ASPECTS OF THE MATHEMATICAL DEVELOPMENTS
Some mathematical aspects of the relationships found seem noteworthy. Hysteretical effects are experimental features of chemical solutions, which hold a sort of memory of the state system of which they are part. In particular, flow rates of pollutant concentrations depend upon state variables of soil layers and water bodies together with typical residence times of the process. Therefore concentrations of pollutants are nonuniquely related to water discharge according to the rising or falling stage of water discharge, with effects somehow similar to inertial effects in flood wave propagations in open channels.
From the physical standpoint hysteretical effects may be due to interac- tions with channel bottom material or to runoff state variables.
The former consists of a phenomenon described by Smith and Stewart (1977) by assuming constant supply of chemicals from a point source. An heuristic explanation has been proposed: water surface profiles are im- portant as the waterways, which fall more slowly in the lower reaches, allow deposition of materials (pollutant or other, which are then flushed and transported by higher flows, so that loads increase with the flow. It appears that some particulate materials are released both by increased erosion and increased flushing at higher flow rates (Fig. 4).
The latter can be related to chromatographic effects developed by the exchange with solid phase.
The q/Q and C/Q curves (calculated for sample values of parameters) induced by isolated rainfall events (assumed respectively as a Dirac-delta function 6(t), 5 h 'slug' and Heaviside step function of intensity p) clearly show the phenomenon (Figs. 5 and 6) while the response to two impulses of precipitation at 12- and 48-h intervals substantiates the fact that the latter event modifies the cycle of the former response (Figs. 7 and 8).
J ~ CONSTANT SUPPLV
FLOW Fig. 4. Load/flow relationship of a river during rising and falling flow (after Smith and Stewart, 1977).
295
10 I
05-
CONCENTRATION VS FLOW RATE (Single ra,nfail) , I , I , I , I ~ I ,
Oirac- # impulse J "stug"(5 h x) . . . . /
/ PARAMETERS
A:155 Krn; ~ f p:1Omm /
/ K:0~90' I
01 02 03 06. Q (mJ/$) -0.6
i B
0~-
03-
02-
01-
0 ,
LOAO VS FLOW RATE (single rainfall)
/ . . . . . . . . . . . O,rac-6 ~mpulse
-~'~ I ~ I t I ' I ' 01 02 03 04 OlmVs) =06
Fig. 5. Hysteretical effects shown by the present model: response to Dirac-delta and 5-h 'slug' impulses (values of the parameters are relative to a case study, Zingales et al., 1982).
1.0
A o~
E
(:r
¢._)
CONCENTRATION AND LOAD VS FLOW RATE (Single rainfall)
, I , I , I , I ,
C/CE ,//:,
qlC~Ap
PARAMETERS
K: 0,69 d -I I
Heaviside step function
0 i ~ ' ~ ' J ' I t I l i b I I i 0 0.2 O.& 0.6 Q(mVs) - lO
Fig. 6. Hysteretical effects shown by the present model: response to Heaviside step function.
296
A direct consequence of Eqs. 4 and 5 (Figs. 5 to 8) on field data acquisition systems is due to the need of gathering discharge-weighted measurements of concentration. Concentration differences between rising and falling stages and the actual lags in the assessment of equilibrium concentrations with respect to discharge peaks are, in fact, two aspects of the same phenomenon. Mass transfer kinetics required by water and pollutant mass/momentum balances fully justify the indicated hysteretical effects.
A uniquely determined linear relationship can be pointed out, according to the model proposed, between total quantities of Xth pollutant ZX and the total runoff volume V. In fact, according to Eq. 5:
For a sequence of j rainfall events, j = 1, 2 . . . . . m, the total quantity transported Z,o t is given by:
tgZ
Z t o t = E (ZX)/=[1-(K/(K+hX))"] ~ CEX(V)/ (8) ,1=1 j = l
hence for C E X independent of j, a tight linear relationship between Z,o t and E~'_-I (V) / i s found. Possible deviations from the relationship may be due to truncation of the integral, Eq. 6, which is actually introduced whenever time-averaged (or smoothed) pollutant and water discharges are considered.
The concept of reference weighted concentration of pollutants in the runoff (Haith and Dougherty, 1976) is therefore amenable to interesting
05-
0 L
CONCENTRATION VS FLOW RATE (two rainfaUs,12h, interva[}
02 0~, 05 OB Q(mVs) - - - - 1 2 02-
LOAD VS FLOW RATE (tworainfaU.s,12 h mtervat
/ /
/
J p~=lO m m ~- pz= 10 mm 08- K:0.69 d
n=156 h=30 d -~
05-
Ot,-
J t ~ I J t L I L 0 2 0 & 0 6 018 0 (m~/s) - - - ~ ) 2
Fig. 7. Hysteretical effects shown by the present model: response to two impulses at 12-h intervals.
297
confirmation by mass exchange and balance relationships, provided that long-term, or event-based, time scales are assumed. According to Eq. 5, such a reference concentration is:
c= fn~CQ dt/L~Q dt = CEX[1-( K/( K + hX)) ' ] (9)
which is a conceptual form of Haith' average concentration (Haith and Dougherty, 1976; Haith, 1981, 1982).
'Slug' effects of intensity p and duration T are also well described by the model, whereas the resulting hydrographs become incomplete gamma func- tions (y(n, Kt)= fo kt x "-I e x p ( - x ) dx):
Q(t; n, K)=(Ap/KT)y(n, Kt) fort~<T (8)
Q(t; n,K)=(Ap/KT)[v(n, K t ) - y ( n , K ( t - T ) ) ] f o r t > T
CONCENTRATION AND LOAD VS TIME (two rainfaLLs L,B h intervaL)
E " I . / "
\ p1:lO mm
qo. Cz pz:lO mm - - - K:OGg d
7 h:30d I
0," ' ' ' I ~ ~ I''' I L ~ ~ I ' ' ~
0 20 (,0 GO 80 (h) ~ 1 0 0
LOAO VS FLOW RATE (two rainfails.t,8h ,nterval
l o - t - - , t , t , I - , I , , I I . "
/ /
E p,=10 mm / I - ~=I0 mm / I
K:o69d' / / n:ISG
)t,
}.2
O, ~ ~ t ' I t t I I i 02 O~ OG OlmV$) ~ 1.0
CONCENTRATION VS FLOW RATE (tworainfatLs./& h~ tnter'val.
# , , , i , , , t , , , t , , , i , , ta
~Z I pl:TO ram / I p~:10 mm I K=069 d" /
05- I n:lSG
C ~ I ~ I I I I l I I I I I I I l I
0 0.2 OiZ, O!G O l m V s ) - - - ~ 1 0
Fig. 8. Hysteretical effects shown by the present model: response to two impulses at 48-h intervals.
298
Among the spinoffs of the model, it seems worthwhile mentioning the dependence of pollutant runoff q upon water discharge Q, which represents a crucial point for the appropriateness of the scheme and, of course, needs further experimental confirmation. Such a relationship, which is built in the model assumptions, may yield success of quantitative descriptions of NPSP processes provided that sufficient information and noncontroversial experi- mental data are available. Typical examples of experimental trends have been shown in the literature - see e.g. Jolhnkai, 1983, figs. 8.9 and 8.26 for the Lake Balaton experience. Actual experimental evidence gathered from the experience of the Lagoon of Venice (Zingales et al., 1980) seem to support the conceptual modelling features. In fact, a screening phase of pollutant versus water volumes correlation has been performed, with particu- lar emphasis on smoothing of experimental data on time scales of flood events. The results, yielding support for the modelling procedure proposed, are presented in Figs. 9 and 10 for gauged and integrated data.
Nevertheless, in most experimental and modelling tests of concentration/ flow rate relationships it is not explicitly stated what might influence hysteretical effects on frequently encountered cases of absence of correlation - e.g. in Jol~mkai (1983, fig. 8.9) the relationship turns to a quasi-linearity (fig. 8.26) if reference is made to load/volume curves. From Eq. 6, in fact, one might infer that only measurements at large times from the impulse would respect the linearity of the relationship, for which:
Q(t; n, K)=. Q(t; n, K+ hX) (9)
Hence it seems that conceptually no basic (and simple) flow versus con- centration (Jolhnkai, 1983) or pollutant load versus discharge relationships would suffice in portraying the features of the physical phenomena.
1 8 0 [ _ , I I I I I I ' I ' i ,
-~" 1600~_ •
C L~'I , I ~ I , i i I ~ I I
1 0 0 1 5 0 3 0 0 1 5 0 o 5o o'o 2"0 ~- W a t e r f l o w rate ( m 3 k m - 2 d -1 ) 1 0 2
-- ~ ' t ' ' t ' 1 0 0
I - W a t e r f l o w r a r e ( m 3 k m - 2 d - 1 ) l O 2
Fig. 9. Measured (left) and integrated (right) total N data for the case study of Zuccarello, Italy (after Zingales et al., 1981a).
299
These facts may explain quite deeply the severe scatter of experimental data (e.g. Jolhnkai, 1983) shown on pollutant concentration versus water flow rate curves. In fact, according to the mass balance presented in the paper, the phenomenon of scatter would rather spring from intrinsic proper- ties of the process than from built-in experimental measurement errors.
~ , 1 0 u t I I ~ I '
f - L -
4o' I I ~ I [ I I i [ n 150 200 250 3 0 0 350 4 0 0
Water f l o w r a t e ( m 3 km-2 d-~)102
~12~- ~,. •
• _ o ~
i OA • ~
0 • ~ - 0 ~ p ~ o u i , I I I I l l l ~ [ i l i l I
0 10 20 30 4 0 50 60 70 ~yQ 9 0 100 Wate r f l ow ra te ( m 3 krn -2 d -~ )102
Fig. 10. Measured (left) and integrated (fight) PO4-P data for the case study of Zuccarello, Italy (after Zingales et al., 1981b).
120 o E IOO
?
~ 60
~ 40
2 O z
%
}ao
~ 1 6 ' E
~ lO
_o 6 h
O. 4
~ 2 0 O- 0
P2 Watershed s i m i I I r u I l I ' I I , I ' ~ 200
*- ~ leo E 160
140
12c
loo
• • ~ 60
• ~ 40
z 20
100 2 0 0 300 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 ~_ O, Water f low ra te (m 3 km -2 m o n t h -1 )102
P2 Watershed sirn i I I , I , I , I i I i _
" 1
i I
• o •
L• I " T ! 01 i I i I • 111 I i I i I t 100 200 300 400 5 0 0 6 0 0 700 8 0 0
Wate r f l ow r a t e ( m 3 krn -2 month -1 )102
P2 Watershed re¢ ' I r I ' I i I , I i i n I ' I 1 ~ I ~
5 0 100 150 2(30 250 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0 Water f low ro te ( m 3 km -2 month -1 )102
w? r h d 7 , , 91- - -
~- O [ ~ 1 " ~ - I • I i I I I , I t I t [ i I I i ,
0 - 0 5 0 100 150 200 250 300 350 400 450 500
Water f low r e t e ( m 3 km -2 month "-1 )102
Fig. 11. Simulated (left) and recorded (right) data of total N and POa-P for the P2 watershed (after Donigian et al., 1977).
300
When integral load/water volume relations are plotted, a trend toward linearity is often shown. The residual deviations can be further ascribed to possible time dependence of C E X not exploited in the model, to nonuniform spatial distribution of rainfall and pollutant loads over the watershed, to truncation of the integral Eq. 6 and finally to errors of measurement.
Figures 11 and 12 show an interesting elaboration of experimental and calculated data obtained in the literature (Donigian et al., 1977, tables 11.1 and 11.2, pp. 110-113) relative to P2 and P6 watersheds. Correlations between experimental pollutant volumes with water volumes integrated over a month show a notable linearity which is less evident in the calculated results.
Interestingly enough, particular forms of load/f low or concentration/flow relationships have been proposed in order to make up for the dependence on rising or falling flow conditions. For instance a load-flow relationship investigated by Hock (Jolhnkai, 1983) is:
L = aQ + b + f Q ( Q - Q_~) (10) where L = load; Q = water discharge; Q_ ~ -- the water flow rate correspond-
~ 150 P 6 W a t e r s h e d s i m ~ " P 6 W a t e r s h e d r e c
E
125 ' E
10C
v 7 5
5 0
2 5 z
' l ' l , l , l ' l ' l ' l ' l ' l ' l , - e •
~ 2 5 0 . , I ' I ' I ' I ' I ' I ' I ' I '
~ 2251 - /
~15o L- • • / " -
,25 b J ~ I00~- J
75L- 7" "
osot- -_- z 2 5 ~ . .
~ 0 ~ , L , [ , J , I , I , 1 , a , , , S o l O O 200 3 0 0 4 o 0 5 o 0 6oo 7oo 80o g o o ~- W a t e r f l o w r a t e ( m S k m - 2 m o n t h - 1 ) l O 2
"~ J , I , I , l i t , I i I J I , I i I , 00" 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 W a t e r f l o w r a t e ( m 3 k m - 2 m o n t h -1 ) 102
P 6 W a t e r s h e d s i r n P 6 W a t e r s h e d r e c 6 0 ~ _ 1 1 . . . . . I ' I ' I ' i ' I ' , ' i ~ 6 0 L ' I ' , ' I ' , ' i ' , ' i ' ' ' 0
4 + . ¢4op -
. • ; : : k
L 'I0 ~-1 -- o~ • O. • • • --
~" n I , I i I , I i I i I , I s I i I J I ~ I i I i I I i I i 0 ~ 0 I 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 2 ~ 0 100 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 e. W a t e r f l o w r a t e ( m 3 k m - 2 m o n t h -1 )102 W a t e r f l o w r a t e ( m 3 k m - 2 m o n t h - 1 ) l O 2
Fig. 12. Simulated (left) and recorded (right) data of total N and PO4-P for the P6 watershed (after Donigian et al., 1977).
301
ing to the preceding time unit; and a, b, f are dimensional constants, which represent a sort of Markov process. In the same pattern, Porter (Jolhnkai, 1983) proposed a concentration against flow relationship in which the rate of discharge variation is accounted by the relationship:
C = a o + a l Q + a 2 d Q / d t (11)
a 0, a 1, a 2 being suitable constants, which is somehow analogous to the hydraulic procedures for the account of inertial effects in flood routing.
Furthermore, screening or planning models, referring to the classic ap- proach proposed by Haith and Dougherty (1976) (Jolhnkai, 1977; Rinaldo, 1982; Jolfinkai, 1983) seem to suggest that the flow-averaged concentrations for large time scales are realistically linear with total water quantities.
Most calibration data and some computed results of distributed parameter models (Beasley, 1976; Donigian et al., 1977, Kniesel, 1978) yield linear correlations between total pollutant loads and total water volumes, creating further interest in the use of the conceptual UMRF proposed as a routing element of complex codes.
Finally Whipple and Hunter (1977) found that there are considerable seasonal differences in the relationship between BOD pollution loading and discharge for small suburban watersheds in New Jersey. Nonetheless, for a given season, BOD load has been found to be closely proportional to total streamflow volumes, thus yielding further evidence of the validity of the physical aspects underlying the mathematical manifolds of the conceptual schemes.
A further mathematical property of the model lies in the lack of singulari- ties in the parameter domain which can be pointed out by error propagation analyses. As a consequence, the approach is deemed particularly suited to parameter identification via automatic techniques.
FURTHER RATIONALE FOR THE MODEL
The physical meaning of some assumptions underlying the present scheme needs to be discussed. In fact, the concept of lumped parameter UMRF of elementary areas embodies mass balances, hence implicitly accounting for sorption of pollutants by the soil, plant uptake and, possibly, decay and transformation.
Thus, some considerations must be drawn about the use of Eq. 1, whereas the actual relationship between the concentration of dissolved pollutant and that of pollutant adsorbed on soil particles needs some experimental con- firmation, since adsorption of, e.g., pesticides may be essentially complete within hours, while phosphate and ammonium adsorption takes longer.
302
Studies reported in the literature (e.g. Novotny and Chesters, 1981) or based on available experimental evidence for a case study at hand (Zingales et al., 1980) seem to support the actual modelling approach, in particular because first-order adsorption kinetic equations have been widely used as a model system for the description of solid and free phase concentration changes of pollutants in the soil. The experimental evidence brought by the case study at hand (Zingales et al., 1980, briefly outlined in what follows) is based on two classes of soil analyses: the first dealt with random measure- ments of grain-size distributions (and hence relative percentages of clay, silt or loam), organic carbon and pH within the upper soil layer about 0.2 m deep (horizon 0 plus horizon A); the second yielded significant information about horizons A and B total N and PO4-P. Soil-adsorbed components data collection has been performed at 0-0.2 m, 0.2-0.4 m and 0.4-0.6 m on a
2000
1500-
Iooo
5oo
0 ~ ' ' I " ~ I ,~ t J J J
0 2 ~ c~ { rag /L / - - - , , , - -
Fig. 13. Nitrogen Langmuir isotherm plotted with experimental data (after Zingales et al., 1981a).
303
monthly basis at five locations of the gauged watershed. Some results of the analyses turned out to be controversial, even though total N and PO4-P concentrations measured in the upper layer (0-0.2 m), plotted with the equilibrium concentration of the mobile phase in Figs. 13 and 14, fit acceptably a Langmuir isotherm of the type:
FEX = QobCEX/(1 + bCEX ) (12)
where F E X = the Xth pollutant concentration adsorbed by the solid phase (usually mg/1); CEX= the equilibrium solution concentration (usually mg/1); Qo = the adsorption maximum at the fixed temperature (usually rag/g); and b = an energy term related to the enthalpy of adsorption (1/mg or 1 /g or ml/mg).
25o
:::1.
t l t t t l l t l t ~ l ;
0~" %, /
150
I00-
0 • I I ! I I 1 l 1 l 11 , ~, A , t t ~ A I ~ A
0 0.5 I0 C~ (rag/t) = 2.0
Fig. 14. Phosphorus Langmuir isotherm plotted with experimental data (after Zingales et al., 1981a).
304
The total N Langmuir curve has been calculated using Q0 = 7.3 m g / g and b = 65 1/g (Fig. 13).
The theoretical curve for the Phosphorus isotherm has been calculated by an assessment of the most satisfactory combinations for values of Q0 and b found in the literature (Donigian and Crawford, 1977, Novotny et al., 1978). The thick line of Fig. 14 refers to the actual average values of Q0 and b, while the lower line has been calculated via values one standard deviation off the mean measured values. It seems therefore that a reliable value of FEX for total N and PO4-P can be readily determined by the Langmuir isotherms (with the above mentioned parameters) once related values of CEX have been identified by preliminary model calibration phases.
CASE STUDY OF THE VENICE LAGOON: PARAMETER CALIBRATION AND MODEL TESTING
Field data aimed at model testing and calibration, and verification of site conditions have been gathered during a campaign of field studies concerning the environmental conditions of the Lagoon of Venice (Italy). In particular, attention has been focused on nonpoint source pollution of the Lagoon from the agricultural watersheds of the mainland (about 2000 km 2) by identifica- tion of a pilot area for extended field experiments. The selection of the pilot watershed of Zuccarello (15.5 km2), indicated in Fig. 15, is justified by favourable conditions: - The area is characteristic of the mainland landscape as far as hydrologi-
cal-geological conditions and crop management techniques are concerned. - The watershed main elevation lies below the average sea level, therefore
full-time mechanical drainage is needed for maintenance of a safe hydra- ulic regime. The subjacency of the area to average sea levels (hence the site of a drainage station) implies that one control section (at the hydraulic plant where the excess runoff is pumped out of the system) is available and sufficient for monitoring and control of hydrology and water quality of the entire area.
- The boundary conditions are well-known, since the watershed is hydrauli- cally isolated from the external hydraulic regimes by the Zero (North) and Dese (South) banks and by a line of primary collection channels at the western boundary. Deep groundwater movement is limited by a compact clay formation and hence violations of water balances performed at the control section are negligible.
- The area is slightly disturbed by point pollution sources.
All measurements, aiming at quantitative evaluations of NPSP processes have been gathered within two annual campaigns (1976 and 1981). The
305
features investigated concern: - A preliminary study, which illustrates crop management techniques in the
area (on average corn covers 40% of the cultivated surface; grass 19%; wheat 15%; vine 10%) and number of inhabitants (about 2000 people).
- Annual loads for unit area, which have been evaluated at about 110 kg/ha (11 t / k m 2) of total N and 20 kg/ha (2 t / km 2) of total P.
- Hydrologic features. Meteorological data, currently gauged in the area by the National Meteorological Service, have been integrated by continuous rainfall measurements. Runoff at the control section is calculated via archival runs of the pumping system (time of starting and stopping of the pump; water elevation inside and outside the drainage final receiving basin; drained water volumes), whereas a continuous monitoring of the
f
/
• ¢" U,~..
l.) ~,.
, - . .,.1
~ VE PADO VA(,.., ~
/
i I ~J"L..~._.3 k
~.-.,_..~,....V "~- • ..~. ~ • ~ "
ROVIGO
0 10 20
TREVISO,.,.-,¢" / • .~ >
k.,,,~ o'' , . . . 5 ~ . j ~ "
'~. ~," '~" \ . . ~ . ~
) m
3O I
km
""7
. . . . . ~ ~ " " C ; ¢ ~ L i : . " : . : . - " "
i .:.:L:. i. '. . ' . ' " . . ~ . '
• • . . . . . . .
. ' . . " " . . . ' , .
~'.-
Fig. 15. Layou t of the Zucca re l lo wa te r shed ( N o r t h e r n Italy). T h e d o t t e d l ine b o u n d s the m a i n l a n d d i s cha rg ing in to the Lagoon .
306
water level within the receiving basin of the drainage plant has been set up in order to account for different heads. Hence rainfall-runoff transforma- tions have been monitored on an event-by-event basis, then integrated on a daily basis. Daily measurements of nitrogen (total N, NO-N, NO2-N, NH3-N ) and phosphorus (total P, PO4-P ) concentrations in the runoff have been carried out. The sampling mechanism has been flow-weighted, so that if no excess water is drained off the watershed, no concentration of pollu- tant is shown in the measurement. The detailed description of experimental values, comparisons, statistical
analyses and model calibration is described elsewhere (Zingales et al., 1981a, 1982). As previously mentioned (Figs. 9 and 10), some of the features built in the proposed conceptual model have been verified by comparison with actual experimental evidence for soluble phases, with particular emphasis on
0
--h
4W~
s,i ,I-
o WATER
~' NITROGEN o PHOSPHORUS
3 3
/,0
-11
CI3 [3
0 50 lO0 150 200 t (DAYS)
I / O 250
Fig. 16. Simulated and recorded data for the case study of Zuccarello; discharged water volumes, soluble total N, soluble orthophosphates. Measured precipitations are also reported.
307
linear integrated pollutant load/water volume correlations. The net distribution of precipitation fit) (on which non linear effects of
rainfall-runoff transformations have been weighted) has been obtained by multiplication of rainfall intensities by consistent runoff coefficients.
Values of n, K and CEX, hX have been obtained by minimization of square residuals between measured and calculated values of water and pollutant discharges. The fitting has been performed on cumulative data in order to compensate faulty timing of rainfall and runoff as daily events, arising when rainfall durations are shorter than 24 h.
The capability of the model to fit the experimental values collected in the case study at hand is illustrated in Fig. 16 for total N and PO4-P. Similar results are obtained for the simulation of the NOR-N, NOa-N and NHa-N. The optimal values of the parameters are"
n=1 .5 6 ; K = 0 . 6 9 d - 1 ; hN=hP=O.5d-1
Average monthly runoff coefficients and equilibrium concentrations have also been determined and found in agreement with the values of the literature (e.g. Novotny et al., 1978).
From the best fit of the results, the limited sensitivity of mass transfer coefficients to the differing nature of chemical species in the mobile phase can also be inferred: the calculated values, in fact, match the experimental values gathered in analogous conditions (Novotny et al., 1978).
CONCLUSIONS
A conceptual mathematical model of unit-mass response function of nonpoint source pollutants to rainfall impulses has been illustrated in the paper. The physical aspects of the mathematics involved seem to yield accurate description of some basic mechanisms observed in nature.
It is shown that the inherent hysteretical effects (rising-falling flow dependence), gauged in nature for environmental response to various im- pulses, are fully explained by the differential mass balance and transfer relationships investigated. The theoretical inadequacy of certain runoff/pol- lutant concentration correlations is investigated and tentative theoretical explanations are given for frequently occurring cases of absence of correla- tion between instantaneous values of concentration and flow rate. Although the results are tied to the conceptual scheme of the watershed assumed for the calculation of mass and water balances, the results seem of general validity.
Theoretical support is given for the rationale of screening and planning models underlying the concept of reference weighted concentration in the runoff. A conceptual form of Haith's average concentration in the runoff
308
arises, in particular, from the mass balance/transfer relationships embodied by the actual scheme.
The model formulation is simple, the parameters have clear physical meaning and the approach, as a whole, is deemed suitable to general simulation and forecasting of non-point source pollution responses to rain- fall events of any extent.
The model seems also suited to microcomputer software applications owing to its simple formulation.
A further possibility to use the procedure seems to consist of the integra- tion of complex distributed parameter models, aimed at determining the fate of pollutants in the runoff, since this goal is mostly achieved by a set of processes simulation submodels (e.g. hydrological rainfall-runoff transforma- tions; erosion and sediment transportation submodels; pollutants generation and routing) which sometimes lead to empirical approximations. Albeit the proposed model structure consists basically of a lumped parameter formula- tion, the identification of elementary subbasins and the introduction of suitable routing lags between the "nodes" of the network connecting such areas (as commonly adopted in hydrological simulation of floods generation and routing) yields a distributed parameter structure. A further advantage of this approach is the use of Nash' conceptual form of the unit hydrograph, widely adopted in hydrological simulations. Well-known approaches availa- ble in the hydrologic literature also allow for the treatment of inherent nonlinearities.
The conceptual analytical form of the unit mass-flux response function allows to overcome the further difficulties connected with storm-by-storm variations of instantaneous hydrographs by reliable parameter identification techniques.
The application of the proposed model to NPSP of an agricultural watershed (the case study of the Venice Lagoon, Italy) supports the rationale of the research.
ACKNOWLEDGEMENTS
The present work has been supported by funds provided by the Consorzio di Bonifica Dese-Sile (Mestre-Venezia), by the Consiglio Nazionale delle Ricerche (CNR, Roma) and by the Italian Ministero della Pubblica Instruz- ione (Roma).
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