Review and assessment of models for predicting the migration of radionuclides from catchments

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ad.2003.11.004

Journal of Environmental Radioactivity 75 (2004) 83–103

www.elsevier.com/locate/jenvrad

Review and assessment of modelsfor predicting the migration of radionuclides

from catchments

Luigi Monte a,�, John E. Brittain b, Lars Hakanson c,Jim T. Smith d, Marcel van der Perk e

a ENEA CR Casaccia, via P. Anguillarese, 301, 00100 Rome, Italyb LFI, Natural History Museums & Botanical Garden, University of Oslo, Norway

c Institute of Earth Sciences, Uppsala University, Swedend CEH Dorset, UK

e Utrecht University, The Netherlands

Accepted 19 November 2003

Abstract

The present paper summarises the results of the review and assessment of modelsdeveloped for predicting the migration of radionuclides from catchments to water bodies.The models were classified and evaluated according to their main methodological approa-ches. A retrospective analysis of the principles underpinning the model development inrelation to experimental finding and results was carried out. It was demonstrated that mostof the various conceptual approaches of different modellers can be integrated in a general,harmonised perspective supported by a variety of experimental evidences. Shortcomings andadvantages of the models were discussed.# 2003 Elsevier Ltd. All rights reserved.

1. Introduction

Models for predicting the migration of radionuclides from catchment basins to

fresh water bodies are essential for the assessment of the contamination levels of

aquatic ecosystems. The radionuclide transport through a catchment is governed

by two main classes of processes (Kivva and Zheleznyak, 2000):

L. Monte et al. / J. Environ. Radioactivity 75 (2004) 83–10384

. Hydrological processes:

1. Water flow2. Sediment erosion, transport and deposition

. Physico-chemical interaction processes between radionuclides in dissolved formwith the rocks in the catchment, and in particular, with soil particles.

The flowing and balance of water in a catchment is controlled by many mechan-isms: a) interception of precipitation by surfaces, such as the above ground veg-etation; b) storage of water in depressions; c) snow and ice storage; d) infiltrationthrough the soil; d) overland flow; e) interflow through the unsaturated zone and f)groundwater flow through the saturated zone.

Similarly, the interaction of radionuclide with soil and bedrock involves a greatdeal of phenomena of physical and chemical nature.

The complexity of the processes occurring in catchments is reflected in the diffi-culties encountered in constructing models for predicting the migration of toxicsubstances. It is a considerable challenge to develop general models that can bereliably applied to catchments in different geographical areas with differentenvironmental conditions and for different possible contamination scenarios.

A fundamental concern in radioecology is to model and predict the so-called‘‘peak and the tail’’ concerning radionuclide concentrations in ecosystems relatedto a given accidental fallout (see IAEA, 2000). Peak concentrations are generallyrelated to short-term conditions in the hours, days or weeks after the fallout,whereas the ‘‘tail’’ concentrations are generally governed by the long-term con-ditions in the catchment areas regulating fixation, percolation, surface and groundwater transport, resuspension and biotic fluxes of radionuclides (Strand et al.,1996).

The main aim of this paper is to review and assess the most common methodol-ogies and approaches that are used to quantify fluxes of substances from land tosurface water. The reported results have been discussed in the frame of the networkEVANET-HYDRA (Evaluation and Network of EC-Decision Support Systems inthe field of Hydrological Dispersion Models and of Aquatic RadioecologicalResearches) financed by the European Commission (Contract N

vFIGE-CT-2001-

20125).

2. Results and discussion

2.1. Retrospective analysis of empirically based approaches for modellingradionuclide migration from catchments: Pre-Chernobyl studies

A retrospective analysis can be helpful for the present assessment. Since thebeginning of the sixties, researchers have tried to analyse the quantitative behav-iour of radionuclide migration from catchments in a pragmatic way, taking advan-tage of the available experimental results.

85L. Monte et al. / J. Environ. Radioactivity 75 (2004) 83–103

Studies on the transport of radionuclides through runoff were initiated as soonas the fallout from nuclear weapon tests in atmosphere became a significant sourceof environmental contamination (Menzel, 1960; Yamagata et al., 1963). Heltonet al. (1985) reviewed the parameter values used in models for predicting themigration of radiocaesium, radiostrontium and plutonium from catchments. Theanalysis was done on the basis of extensive investigations in a number of catch-ments in Europe, North America and Japan. These investigations accounted forthe research carried out before the Chernobyl accident. The assessed parametervalues refer to the model structure described, for instance, by Carlsson (1978)whereby it was assumed that an initial fraction k1D(t) of radionuclide depositedper second and per square metre (D(t), Bq m�2 s�1) was instantaneously trans-ferred to the water body. Thereafter, the remaining part of deposited radionuclideaccumulated in storage compartment S(t), representing the amount of radionuclideper square metre of catchment (the radionuclide inventory at time t, Bq m�2) avail-able for delayed release. Radionuclides are, therefore, washed off from S(t) with arate constant k2. Fig. 1 shows the model structure.

The differential equation controlling S(t) is:

dSðtÞdt

¼ �ðk þ k2ÞSðtÞ þ ð1� k1ÞDðtÞ ð1Þ

k is the radioactive decay constant (s�1) and k2(s�1) is the rate of removal by run-

off of the radionuclide in S(t).The radionuclide flux per square metre U(t) is, therefore:

UðtÞ ¼ k1DðtÞ þ k2SðtÞ ð2ÞValues of the parameters of the above model were obtained by many researchers

for radionuclides introduced in the environment following the atmospheric nuclearweapon tests. The parameters were obtained by experimental assessments carriedout at both regional and experimental plot scales. The results of the review by Hel-ton et al. (1985) can be summarised as follows: k1 values range from 0.5 � 10�2 to12.2 � 10�2 and from 0.1 � 10�2 to 1.9 � 10�2 for 90Sr and 137Cs respectively. k2values range from 2.2 � 10�11 s�1 to 1.0 � 10�9 (s�1) for 90Sr and from 2.1 �10�12 s�1 to 1.8 � 10�10 s�1 for 137Cs. For plutonium isotopes fewer data wereavailable, but indicating that the order of magnitude of k2 is 10�11 s�1 (Table 1).

The total deposit S(t) of a radionuclide following a single pulse deposition is anexponential function of time as seen from Eq. (1). The effective decay of the depo-sition is therefore k + k2. Following the deposition pulse, radionuclide flux shows asimilar exponential decay (Eq. (2)) (notice that, when t > 0, k1D(t) = 0 whereD(t) = Dd(t); d(t) is the so called Dirac function). The concentration of radio-nuclide in runoff water can be calculated by dividing the radionuclide flux persquare metre by the runoff water flux per square metre. As consequences, radio-nuclide concentration in runoff water is also an exponential function of time witheffective decay constant k + k2.

Data in Table 1 suggest that the range of the values of removal rate (k2) forstrontium is significantly higher that the corresponding range for caesium. There-


fore, the decline of the concentration of radiostrontium in runoff water is expected

to be faster than the concentration of radiocaesium.As Eq. (1) shows, the assessment of the radionuclide migrating from a catchment

was made hypothesising that the radionuclide removal rate (k2) is constant over

time.

Table 1

Ranges of parameter values in the review of Helton et al. (1985)

Radionuclide k
1 k2 (s�1)
137Cs 0
.1 � 10�2–1.9 � 10�2 2.1 � 10�12–1.8 � 10�10
90Sr 0
.5 � 10�2–12.2 � 10�2 2.2 � 10�11–1.0 � 10�9
239,240Pu –
10�11 (few data available)
Fig. 1. Structure of models for predicting the migration of radionuclides from catchments to water

bodies following the assessment of data from fallout arising from nuclear weapon tests.


Model (1) can be derived accounting for the role that the partition coefficient kdhas in the long term migration of radionuclides through a catchment (Joshi andShukla, 1991). The fluvial removal Fir (Bq m�2 s�1) is assumed to be proportionalto the radionuclide inventory in the watershed S(t) at instant t (Bq m�2).

Fir ¼ k2SðtÞ ð3Þ

where, as usual, k2 is the removal rate (s�1).Multiplying both sides by the area of the watershed A, we get

FrðtÞ ¼ k2SðtÞA ð4Þ

where Fr(t) is the flux of radionuclide (Bq s�1).The radionuclide inventory can be calculated according to the following equa-

tions:

dSðtÞdt

¼ DðtÞ � k þ k2ð ÞSðtÞ k2 ¼VwiðtÞVsiðtÞkdi

ð5Þ

where D(t) is the time dependent deposition rate of radionuclide in the watershed(Bq m�2 s�1), Vwi(t) is the rate of rainfall (m s�1), Vsi(t) is the water penetrationdepth (m), k is the radioactive decay constant, and kdi is the dimensionless par-tition coefficient (kdi = qkd, q = soil density in kg m�3, kd = soil-water soil par-tition coefficient in m3 kg�1).S(t) depends on the time behaviour of Vsi(t) and Vwi(t). Hypothesising that the

ratio Vsi(t)/Vwi(t) = n is constant with time the solution of Eq. (5) for a pulsedeposition is

SðtÞ ¼ Sð0Þe�ðkþ 1nkdi

Þt ð6Þ

The radionuclide flux is

FrðtÞ ¼A

nkdiSð0Þe�ðkþ 1

nkdiÞt ð7Þ

The model correlates the effective decay constant k þ 1nkdi

� �with the radionuclide

partition coefficient, kd. It accounts for this correlation by means of an inversefunction: as kd increases the effective decay constant decreases. This result wouldseem to support the previously described experimental observations. Indeed theexperimental long-term effective decay constant for 90Sr, a radionuclide char-acterised by a low value of kd, is higher than the corresponding parameters for137Cs (high kd radionuclide) as shown in Table 1.

As previously stated, a major problem with the above approach was due to thenon-pulse character of the radionuclide fallout from nuclear weapon tests inatmosphere. This prevented an accurate evaluation of the time behaviour (andconsequently of the effective decay constants) of radionuclides in water. Moreover,these quantitative assessments were based on the assumption that radionuclidesaccumulated in the catchment storage compartments S(t) were fully available formigration. This hypothesis implies a significant underestimate of k2 when such a


parameter is assessed by accounting, solely, for the radionuclide balance in thecatchment in terms of total deposit and radionuclide removal by runoff waters.

In principle, more compartments can be used for the assessment of migration ofradionuclides from catchments. Unfortunately, for such more complicated modelsless experimental parameter values are available from the literature (Helton et al.,1985).

A typical multi-compartment model was proposed by Korhonen (1990). Soil inthe drainage area was subdivided into layers of various thicknesses. The migrationof radionuclides from one layer to another was evaluated accounting for the waterfluxes. The model hypothesizes constant rates of water infiltration. The last layer isa sink compartment and only the first layer contributes to the runoff. Following asingle pulse of deposition of radioactivity, the concentration of radionuclide in run-off water is the sum of several exponential functions of time with effective decayconstants ki + k. As in the case of other models that account for the role that kdplays in radionuclide migration, the model implies that lower values of kd are asso-ciated with higher values of decay constants ki + k. Therefore, the predicted effec-tive decay rates of 90Sr concentration in water are higher than the correspondingvalues for 137Cs.

The above approaches consider important aspects of the migration processes andrepresent the foundation for most models subsequently developed. They have sti-mulated many significant speculations about the methodologies for modelling themigration of radionuclides from catchments.

2.2. Results following the Chernobyl accident experience

The Chernobyl accident represents a line of demarcation between research car-ried out following the environmental contamination due to the nuclear weapontests in the atmosphere and new results from such an accidental pulse contami-nation event.

Following the Chernobyl accident a variety of studies focused on the evaluationof the quantitative behaviour of radionuclide transport through catchments (Hiltonet al., 1993; Santschi et al., 1990).

Monte (1995) analysed data collected by various European laboratories (Kanivietsand Voitcekhovich, 1992; Mundschenk, 1992; Maringer, 1994) and some of theresults are analysed below.

The experimental dissolved radionuclide flux in some rivers, which drain largecatchments, were fitted to the following function (radionuclide Transfer Function,TF).

urðtÞ ¼ eDXx

i

uaiðtÞbiAie�ðkrþkiÞt ð8Þ

Xx

i

Ai ¼ 1 ð9Þ


The symbols used are as follows: Ur(t) is the radionuclide flux from catchment(Bq s�1), e is the radionuclide transfer coefficient from the catchment (m�1), ai is anempirical exponent (dimensionless), D is the pulse deposition (Bq m�2), U(t) is thewater flux from the catchment (m3s�1), Ai is the weight of the ith component(dimensionless), ki are empirical parameters controlling the decay of radionuclideconcentration in water due to environmental effects (s�1), k is the radioactive decayconstant (s�1). The terms biU

aiðtÞ in Eq. (8) account for possible non-linearityeffects of radionuclide flux (concentration in run-off water) as function of the waterflux. bi are ‘‘normalisation’’ coefficients. A possible choice of bi, is as follows:

bi ¼ Uð0Þ1�ai ; Ur(t) is the radionuclide flux (Bq s�1) following a single pulse ofradionuclide deposition.

When aI = 1, the Transfer Function fits the radionuclide average flux disregard-ing the effects due to the seasonal variations of the water flow. In such a case,dividing both members of Eq. (8) by the water flux and accounting for the simpleformula:

CwðtÞ ¼UrðtÞUðtÞ ð10Þ

we get

CwðtÞ ¼ eDX

i

Aie�ðkþkiÞt ð11Þ

As seen from Eq. (11) the yearly average concentration in water comprises twoterms:

. A multiplicative ‘‘scaling factor’’ e; and

. The expression:Xi

Aie�ðkþkiÞt ð12Þ

It is quite obvious that, multiplying e by any factor K and dividing Ai by thesame factor, the transfer function does not change. Therefore it is always possibleto determine Ai according to Eq. (9) (i.e. the sum of Ai is normalised to 1).

Therefore the sumP

iAie�ðkþkiÞt characterize the ‘‘shape’’ of the curve (the time

behaviour) and the scaling factor e represents the ‘‘position’’ of the curve in thegraph ‘‘radionuclide concentration in water versus time’’. e is the ratio between theinitial concentration of radionuclide in water and the total deposition.

From the available experimental data Monte (1995) identified two main expo-nential components:

a) a
short-term component over a period of few months after the accident; b) a long-term component over a period of some years after the accident.

Table

2

Rev

iew

ofm

easu

red

valu

esofTF

para

meter

s

Riv

erR

adio

-

nuclid

e

Per

iod

ofco

llec

-

tion

offitted

data

(followin

g

theacc

iden

t)

A2(d

imen

-

sionless

)

a 2(d

imen

-

sionless

)

Sta

ndar

d

dev

iation

ofa 2

k 1+

k(s�1)

Sta

ndard

dev

iation

of

k 1+

k

k 2+

k(s�1)

Sta

ndar

d

dev

iation

ofk 2

+k

Ref

eren

ce

Po

(a)

137Cs

few

month

s2.3

�10�7

5.5

�10�

8M

onte

(1995)

(c)

Rhin

e(a

)137Cs

<2

yea

rs0.0

52

0.5

30.3

6.5

�10�7

1.3

�10�

72.7

�10�

80.6

�10�8

Monte

(1995)

Pry

pia

t(a

)137Cs

�5

yea

rs0.0

35

1.0

80.0

65.2

�10�7

6.5

�10�

71.8

�10�

80.7

�10�9

Monte

(1995)

Dnieper

(a)

137Cs

�5

yea

rs0.0

28

0.8

60.0

68.8

�10�7

1.1

�10�

71.1

�10�

80.7

�10�9

Monte

(1995)

Tet

erev

(a)

137Cs

�5

yea

rs0.9

60.1

58.2

�10�

92.0

�10�9

Monte

(1995)

Uzh

(a)

137Cs

�5

yea

rs1.0

20.1

1.5

�10�

81.8

�10�9

Monte

(1995)

Inlets

of

137Cs

few

yea

rs1.0

–1.3

1.2

+10�8

Hilto

net

al.

(1993)

Dev

oke

Wate

r(b)

Inlets

ofla

kes

137Cs

�6

yea

rs0.6

�10�7–

1.5

�10�7

7�

10�

9–

2�

10�

8Sundbla

d

etal.

(1991)

Hillesjon

and

Salg

sjon

(b)

Garo

nne,

Meu

se,

(d)M

ose

lle,

Rhone,Seine

137Cs

�18

yea

rs3.6

�10�

9–

8.0�

10�

9Vra

yet

al.

(2003)(d

)

Po

(a)

131I

few

month

s1.1

�–

10�6

6.5

�10�

8M

onte

(1995)

(c)

Po

(a)

103R

ufe

wm

onth

s4.7

�10�7

4.0�

10�8

Monte

(1995)

(c)

Pry

pia

t90Sr

�5

yea

rs0.0

48

1.4

10.0

89.0

�10�7

1.1

�10�

74.9

�10�

90.9

�10�9

Monte

(1995)

Dnieper

90Sr

�5

yea

rs0.1

66

1.4

0.0

85.2

�10�7

1.5

�10�

75.5

�10�

90.9

�10�9

Monte

(1995)

Tet

erev

90Sr

�5

yea

rs1.1

20.1

43.6

�10�

92.1

�10�9

Monte

(1995)


Uzh

90Sr

�5

yea

rs1.3

10.0

95.9

�10�

91.8

�10�9

Monte

(1995)

Irpen

90Sr

�15

yea

rs1.6

�10�

9–

Sm

ith

etal.

(2002)

Ilya

90Sr

�15

yea

rs2.7

�10�

9–

Sm

ith

etal.

(2002)

Sakhan

90Sr

�15

yea

rs3.8

�10�

9–

Sm

ith

etal.

(2002)

Glinitsa

90Sr

�15

yea

rs1.9

�10�

9–

Sm

ith

etal.

(2002)

(a)disso

lved

radio

nuclid

e;(b

)to

tal

137Cs(p

articula

te+

disso

lved

);(c

)data

fitted

toa

singleex

ponen

tialfu

nct

ion;(d

)ass

esse

sfrom

conta

min

ation

dec

linein

sedim

entand

bio

ta.



Some evaluations of the effective decay constant and of the exponent ai arereported in Table 2.

Table 3 shows the values of some measured parameters of TF for particulatecaesium. The values of a2 are larger than 1 for particulate caesium. This is due tothe increase of suspended matter and, consequently, of particulate radionuclidewith water flux. Dissolved radiostrontium shows similar behaviour whereas dis-solved radiocaesium is not significantly correlated with the water flux.

The previous discussion clearly shows that many experimental evaluations of theeffective decay rates and of the weights of the exponential components of the TFwere obtained. However, comparatively few experimental assessments are availablefor the transfer coefficients from catchment to water bodies.

Parameter k1 in the Helton assessment can be related to the parameters in trans-fer function (8) according to the following formula:

eA1 �k1k1R

ð13Þ

where R is the water runoff for unit surface (m3 m�2 s�1). Using data from Heltonet al. (1985) and supposing R = 0.5 m3 m�2 y�1 and k1~5 � 10�7 s�1 we obtainthat, for caesium, eA1 ranges from 0.03 to 0.6 m�1. Smith et al. (2002) related thevalues of eA2 and eA3 (a third longer exponential component) with the fraction oforganic soil in the catchment by assessing experimental concentrations of 137Csfrom catchments of many European lakes following the Chernobyl accident

eA2 ¼ að1� forgÞ þ bforgeA3 ¼ cð1� forgÞ þ dforg

ð14Þ

The suggested values for the parameters in the previous formulae were a = 3 � 10�3,b = 5 � 10�2, c = 2 � 10�4, d = 2 � 10�3m�1. Similar assessments for 90Srsuggest that eA1 ranges from 0.2 to 4 m�1 and that estimates of eA2 and eA3

obtained by fitting Eq. (11) to measurements from several catchments in Europeare as follows:

Table 3

Measured values of some parameters of TF (particulate caesium) (Monte, 1997)

River a
2 (dimen-
sionless)

95%

confidence

limit (up)

of a2

95%

confidence

limit (down)

of a2

k
2 + k(s�1) 95%
confidence

limit (up)

of k2 + k

95%

confidence

limit (down)

of k2 + k

Danube 2
.44 2.98 1.90 1 .4 � 10�8 2.2 � 10�8 6.7 � 10�9
Uzh 1
.02 1.39 0.65 1 .1 � 10�8 1.8 � 10�8 4.0 � 10�9
Teterev 1
.34 1.77 0.97 1 .2 � 10�8 1.8 � 10�8 6.0 � 10�9
Prypiat 1
.52 1.70 1.34 1 .4 � 10�8 1.6 � 10�8 1.3 � 10�8
Dnieper 1
.24 1.37 1.00 1 .2 � 10�8 1.4 � 10�8 1.1 � 10�8
Desna 1
.11 1.39 0.83 8 .9 � 10�9 1.3 � 10�8 4.7 � 10�9
Rhine 1
.12 1.97 0.27 1 .7 � 10�8 2.4 � 10�8 1.0 � 10�8
Geometric

mean

1
.34 1 .2 � 10�8


. eA2 = 2 � 10�2 m�1 (organic soils) and 5 � 10�3 m�1 (mineral soils)

. eA3 = 5 � 10�3 m�1 (organic soils) and 3 � 10�3 m�1 (mineral soils).

In a study of 90Sr concentration in 11 rivers in Italy, Monte (1997) observed an

average value of eA2 = 9 � 10�3 m�1 with a range from 6 � 10�3 m�1 to 1.6 �10�2 m�1 and suggested a value of eA3 = 4 � 10�3 m�1.

More recent studies by Smith et al. (2003) suggest the following experimental

evaluation of the parameters in Eq. (8): A1 = 0.905, A2 = 0.09, A3 = 0.005 for cae-

sium and, A1 = 0.984, A2 = 0.012, A3 = 0.004. Assessed values of e were as follows:

0.0106(%inland water) + 0.063 m�1 (caesium), 0.146(%inland water) + 0.528 m�1

(strontium).It seems clear, from the above formulae, that the migration of strontium is one

order of magnitude larger that the caesium. Such a result is supported by consider-

able experimental evidences and is in evident agreement to the results obtained fol-

lowing pre-Chernobyl assessments (see section 2.1).

2.3. From experimental assessments to model development

As discussed in a previous paper (Monte, 1997), it is possible to derive a simple

compartment model for predicting the time behaviour of the radionuclides in river

water from the transfer function:

UrðtÞ ¼ eX

i

UaiðtÞbiSi ð15Þ

where Si are solutions of the following system of differential equations:

dSi

dt¼ �ðki þ kÞSi þ AiDðtÞ ð16Þ

Such a model may be used to assess the radionuclide migration following time-

dependent deposition of radionuclides onto the catchment (Monte, 2001). Si are

the radionuclide storage compartments that may be schematically regarded as the

various environmental components involved in the migration process such as soil

layers and the vegetation cover in the catchment. A model developed by Monte

(1996), that was based on the above techniques and accounts for the water satu-

ration effects of surface layer of soil in catchments, is in agreement with the experi-

mental evidences described in the previous section (exponential components of TF,

non-linearity effects due to water flux, etc.)In addition to the described empirical approach, attempts to assess the role that

different components of a catchment can play in relation to the pollutant migration

mechanisms were also carried out.The ‘‘mechanistic’’ model ECOPRAQ (Hakanson et al., 2002), that has been

implemented in the MOIRA Decision Support System (Monte et al., 2000), com-

prises two main compartments for predicting the radionuclide inflow to a lake


from its catchment (Fig. 2):

dMIA

dt¼ �ðk þ k1 þ k12ÞMIA þDmiaðtÞ

dMOA

dt¼ �ðk þ k2ÞMOA þ k12MIAþDmoaðtÞ ð17Þ

where

k1 ¼ RIAð1�DOAWÞ k12 ¼ RIADOAW k2 ¼ ROA ð18Þ

the above terms are as follows: MIA: (� dry land) is the so called inflow areadominated by vertical transport processes, through the soil horizons, the groundwater transport and, finally, the tributary transport to the lake; MOA: (� wet-lands) is the so called outflow area dominated by a fast turnover of substances andhorizontal transport processes; Dmia and Dmoa are, respectively, the fallout inter-cepted by the inflow and outflow areas; k is the radioactive decay constant; Doaw isa distribution coefficient partitioning the flow from inflow area either to outflow

Fig. 2. ECOPRAQ-MOIRA model for catchments (Hakanson et al., 2002)


area or directly to the lake; RIA and ROA are, respectively, the migration rate fromthe inflow area and the outflow area.

Following a single pulse of deposition corresponding to an initial amountMIA(0) and MOA(0) in the two catchment compartments the solutions of Eq. (17)are as follows:

MIAðtÞ ¼ MIAð0Þe�ðk1þk12þkÞt MOAðtÞ ¼ MOAð0Þ þ k12MIAð0Þk1 þ k12 � k2

� �

�e�ðk2þkÞt � k12MIAð0Þk1 þ k12 � k2

e�ðk1þk12þkÞt

ð19Þ

consequently the flux from the catchment is

TFðtÞ ¼ MIAð0Þ k1 �k2k12

k1 þ k12 � k2

� �e�ðk1þk12þkÞt

þk2 MOAð0Þ þ k12MIAð0Þk1 þ k12 � k2

� �e�ðk2þkÞt

ð20Þ

The structure of the model is similar to the models derived from the TransferFunction approach when MOA(0) = 0. Therefore, the models can be easily harmo-nised by tuning the parameters in Eq. (17) with the parameters of the TF (11).

The ECOPRAQ model includes sub-models that relate the values of the aboveparameters to certain prevailing environmental and seasonal conditions such asprecipitation and soil characteristics in the catchment and to radionuclide proper-ties such as the fixation to soil.

As seen in the previous paragraph, the models based on the assessment of radio-nuclide mobility using the partition coefficient predict higher effective decay rates of90Sr concentration in water than for 137Cs. This is in contrast to experimental results.Such a difference may be explained (Monte, 1996) supposing that the interaction ofradiocaesium with soil particles in the catchment is, at least partially, a non-reversibleprocess. This implies that the concentration of radiocaesium in runoff water declinesmore quickly than the radiostrontium concentration. Indeed, the amount of radio-nuclide irreversibly fixed to the soil particles and that is not available to migrationincreases in the long term. It is also surprising that, despite the large differencesbetween the ranges of values of partition coefficients (and, more generally, of thechemical properties controlling the interactions of radionuclides with catchmentrocks) of radiocaesium and radiostrontium, the effective decay rates (ki) of thoseradionuclides (see Table 2) are not so dramatically different. Moreover such effectivedecay constants are of the same order of magnitude for 1/T where T is the centroid ofthe time interval during which the relevant exponential component in the TransferFunction is the most significant. Such results can be also explained by a statisticalassessment of radionuclide migration from the large number of sub-catchments con-stituting a catchment. These sub-catchments have differing properties and collectivelycontribute to the total radionuclide flux (Monte, 1998). This approach can be usefulto construct models for predicting the migration of contaminants from catchments. It

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suggests, moreover, that the transfer factor e rather than the effective decay rates ki, is

the main factor influencing the uncertainty of model predictions. It is interesting to

notice that such collective models are based on two main general hypotheses: a) a

stochastic distribution of radionuclide removal rates from sub-catchments showing

different properties; and b) an initial concentration of radionuclide in run off water

proportional to such removal rates (Monte, 2002). Therefore, the approach can be

applied to both dissolved and particulate radionuclides. Moreover, collective models

can explain the different time behaviour of radiocaesium and radiostrontium con-

centrations in run-off water by accounting, solely, for the different statistical distribu-

tions of the removal rates of these radionuclides.As seen in the previous discussion, the main idea of the described aggregated

models is to select emerging, quantifiable systemic behaviours and to relate these to

the functional components of the environment. The following step, that is yet to be

investigated, is to relate the parameters controlling these behaviours to environ-

mental conditions.It is evident that many factors and processes can influence the variability in

radionuclide concentrations in water among aquatic systems. However, they can-

not all be of equal importance for the predictions of the target y-variable and stat-

istical methods can be used to provide a ranking of different x-variables influencing

a target y-variable.For instance, using data from Hakanson (1991) (the target y-variable is the

empirical concentration of radiocaesium in 15 Swedish lakes) one can note that

fallout (Cssoil in Bq/m2) can statistically explain 59% of the variability in the target

y-variable. The next most important factor is the percentage of fine sediments in

the catchment area—the more fine sediments in the catchment, the greater the fix-

ation and the lower the transport from land to water. The third factor is the per-

centage of upstream lakes, a major component of the outflow areas—the more

upstream lakes, the quicker the transport of cesium deposited in the catchment

upstream the given lake. These results are all logical and can be interpreted in a

mechanistic manner, although they are basically statistical.Unfortunately, the assessed models do not account for the important seasonal

effects related to the ice and snow melt in the catchment. These processes can be of

significant importance in high mountain areas such as those located in the Alpine

region. It is well known that radionuclides stored in the snow pack or in glaciers

become available for migration during the melting season. As the Chernobyl acci-

dent happened at the end of April when the melting process was occurring, the

estimated transfer functions TFs and the derived models do not account for the

effects that a delayed ice and snow melt may have on radionuclide migration. How-

ever, such an effect can be, at least in principle, easily modelled by including a

radionuclide storage compartment whose content becomes available for migration

when ice and snow melt.


2.4. Reductionistic approach

The above described models are based on the so-called holistic approach(Jørgensen and Mejer, 1983).

Reductionistic models are instead aimed at predicting the behaviour of complexenvironmental systems by accounting for as many details as reasonably possible.Such details are modeled according to primary laws from fundamental disciplinessuch as physics and chemistry.

The hydrological cycle and surface water movement and balance have been thor-oughly investigated and many models have been developed (e.g., Wanielista, 1990).

The physico-chemical processes of interaction of solute radionuclides with soiland bedrock have been also the subject of many studies in past decades. An exten-sive review of the sub-models for the quantitative assessment of such processesrelevant to non-radioactive substances, but that can also be applied conceptually toradionuclides, has recently been published (Delle Site, 2000).

As both classes of processes have been studied in detail, it seems reasonable, atfirst sight, to hypothesise that coupling hydrological models and physico-chemicalmodels of the interaction of dissolved radionuclides with bedrocks and soils makesit possible to develop general models for predicting the behaviour of radionuclidemigrating from catchments to water bodies. Moreover, to believe that thereliability and accuracy of these models can be increased at will by including moreand more details relevant to the descriptions of the processes taking place is a typi-cal reductionistic strategy.

Unfortunately, as experienced for many other kinds of models (Monte et al.,2003), the inclusion of more processes in a complex model does not guaranteegreater accuracy of model performance. Indeed the overall uncertainty of themodel is strongly influenced by the uncertainty of large numbers of model para-meters whose values cannot be known with a sufficient accuracy at site-specificlevel. For example, IAEA (2000) reported the description of such a kind of modelfor assessing the migration of radiocaesium from the drainage area of a lake. Themodel makes use of more than 20 primary parameters for predicting, among thevariety of processes that control radionuclide transport through a catchment,the soil erosion from each elementary ‘‘cell’’ of the drainage area. Moreover, itmakes use of other quantities for modelling other processes occurring in catch-ment. Unfortunately, for practical applications, it is often difficult to obtain site-specific values of many of these parameters, such as ‘‘the infiltration capacity of ashower of a given time duration’’ or ‘‘the bulk density of soil’’ or the ‘‘daily aver-aged infiltration capacity’’ in each catchment ‘‘cell’’.

Other examples of reductionistic models are reported in BIOMOVS II (1996).They were developed to predict the migration of radiocaesium and radiostrontiumfrom contaminated ‘‘experimental’’ plots. These reductionistic models assess thevertical migration of radionuclide through soils and the consequent horizontaltransport by accounting for fundamental processes like diffusion, influence of waterinfiltration rates on diffusion, transformation of radionuclide chemical forms, etc.,although they make use of many hypotheses (order of chemical transformation

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dynamics, number of radionuclide forms, different importance of soil layers at dif-ferent depths in view of the migration process, etc.) and of many parameters thatare assessed by specific sub-models or are estimated by ‘‘expert judgment’’ whenexperimental values are not available.

It is commonly assumed that two main factors can affect the reliability of amodel for predicting the behaviour of complex systems:

a) T
he lack of detailed knowledge at the level of the processes taking place; b) T he lack of parameter values necessary for the quantitative assessment of the
processes.

These factors are supposed to be the main reasons for the ‘‘structural’’ insuf-ficiency of a model. When an unlimited process of knowledge acquisition ispossible, one is enticed by the hypothesis that the structural insufficiency of amodel can be progressively reduced in order to increase the model performance.This is the case for hydrological models that can take advantage of the repetitiveoccurrence of seasonal events. It is presumptuous to debate if this is in general trueor false. However, what is definitely factual is the impossibility of such an ‘‘unlimi-ted’’ acquisition of knowledge for those processes that occur at environmental sys-tem levels in relation to rare events, such as accidents, for which it is notreasonable and indeed desirable that they are recurrent.

Moreover, the behaviour of complex environmental systems cannot be under-stood, simulated and investigated solely by laboratory experiments. Therefore, it isnecessary to take advantage from the limited empirical information available at thesystem level, and to consider cautiously the applications of those findings that,although substantiated at the level of laboratory experiments, are not adequatelyunderstood and supported at the level of emerging system behaviours. Laboratorystudies are based on the ‘‘analysis’’ of the processes by ‘‘dissecting’’ and ‘‘isolating’’the various components of the examined system. It is generally recognized that thecomplicated interactions among environmental components do not guarantee thatsuch ‘‘analytical’’ procedures allow one to understand those behaviours thatemerge at a systemic level.

Thus, we can conclude that the results of BIOMOVS II validation exercises donot allow the claim for a supposed higher performance of complex reductionisticmodels compared to the holistic ones.

It is essential to discuss a significant problem relevant to the development ofcatchment models: spatial variation and scale. Radionuclide migration from catch-ments can be studied and modelled at various spatial scales ranging from the largecontinental scale down to the small scale of an experimental plot. To simulateradionuclide migration at all these scales, it is essential to understand and quantifythe nature of the major hydrological pathways (e.g. length, tortuosity, and flowrates), and mixing and retention (sorption, deposition, and decay) along thesepathways. The spatial variation of the driving forces within the extent of the modelarea may be taken into account, depending on the purpose of the model. In spa-tially distributed models, the model input and output are defined at a resolution


ranging from tens of square metres to several square kilometres, whilst in lumpedmodels the model resolution equals the size of the modelled area. In most cases,the model parameter values are scale-dependent, which means that they vary withmodel resolution. The physical size to which a parameter value refers is oftentermed support (Bierkens et al., 2001). Thus, the model parameter values forlumped models are different from those for spatially distributed ones, which, inturn, are different from the values derived from field or laboratory measurementsand in most cases they do not equal the spatial mean value. Assuming equilibriumbetween the activity concentrations in the topsoil and runoff water, the activityconcentration in the runoff water can be estimated by:

Cw ¼ Cs

kdð21Þ

where Cw = radionuclide activity concentration in the dissolved phase (Bq m�3),Cs = radionuclide activity concentration in the topsoil layer (Bq kg�1), andkd = distribution coefficient (m3 kg�1). However, the spatially averaged activityconcentration in water is not equal to the mean activity concentration in the top-soil divided by the mean kd value:

Cw 6¼ Cs

kd

ð22Þ

Thus, an appropriate scaling of the kd value is indispensable. This can be achievedby geostatistical upscaling methods from the measurement support to the modelsupport or by calibration of the kd value at the model scale.

In some cases, the spatial variation of the processes within the model elements,which control the transport of radionuclide through catchments and river net-works, becomes so large that it may be inappropriate to represent these processesby single model parameters and variables. This may become apparent in a one-dimensional river model in which the radionuclide activity concentration in theriver water is a state variable depending on the exchange with the dissolved phaseand deposition of suspended matter and sediment-associated radionuclides. Thehydraulic part of these one-dimensional models commonly calculates the cross-sec-tional averaged flow velocity and water depth. The deposition rate of suspendedmatter depends on the shear stress, which in turn, depends on the water flow velo-city. As the flow velocities and water depths are much lower in the floodplainsaway from the main river channel than in the river channel itself, it is very likelythat processes and environmental conditions in the floodplains are, on balance,more important controls for radionuclide migration through river system thanthose in the river channel. However, the processes and environmental conditions inthe floodplains are generally poorly represented by the cross sectional averagedmodel parameters.

The model approaches used in the MARTE (Monte, 2001) and ECOPRAQmodels described above consider only long-term migration of radionuclides in thedissolved phase. The proportion of dissolved transport relative to the total radio-


nuclide transport can be estimated from:

a ¼ 1

kd SS þ 1ð23Þ

where a = proportion of dissolved radionuclide transport (dimensionless),kd = distribution coefficient [m3 kg�1], SS suspended sediment concentration inwater [kg m�3]. From this equation it appears that particulate becomes importantwith increasing suspended sediment concentration. In rivers, such high suspendedsediment concentrations frequently arise during flood events. Consequently, par-ticulate radionuclide transport may be especially important in small catchmentswhere flood events contribute substantially to total runoff. However, Van der Perkand Slavik (2003) demonstrated that the transfer of particulate matter from hill-slopes to the river network is particularly difficult to predict, because this transferoften occurs only at a limited number of sites and depends on local-scale topo-graphical features, such as for example small embankments, roads, hedges, andpuddles.

In general, models to simulate radionuclide migration through and from catch-ments are available. However, the major bottleneck is the lack of suitable modelinput data at the appropriate scale. Default parameter values can often be derivedfrom the literature, but their relationship to the model scale remains obscure.

3. Conclusions

The migration of toxic substances from terrestrial environments to surfacewaters involves many complex processes of hydrological, physical and geochemicalnature. Modelling the transfer of contaminants from catchments to water bodies istherefore a major challenge for modellers.

Models based on a detailed assessment of the above processes come across manydifficulties related to imprecise and limited knowledge of phenomena occurring atsite-specific level and to the uncertainty of the quantities that parameterise thesephenomena. It is therefore essential to develop approaches that allow one todevelop simplified and generic models characterised by reasonable accuracy.

Such approaches were implemented by many modellers taking advantage of theexperimental data following the introduction into the environment of radionuclidesfrom the nuclear weapon tests in the atmosphere during past decades and theChernobyl accident. The main ideas were based on the identification of short- andlong-term migration processes that can be modelled by first order compartmentsystems. The developed models were composed of two or three compartments.Many studies were carried out for assessing the values of the migration parametersin these models. Following the Chernobyl accident more accurate estimates ofthese parameters become available and general methodologies for their assessmentwere proposed.

The ‘‘holistic’’ approaches for modelling radionuclide migration from catchmentdescribed in the present report have given encouraging outcomes for certaingamma-emitting radionuclides and for 90Sr. The various conceptual approaches of


different modellers can be integrated in a general, harmonised perspective as

demonstrated in section 2.2. It is worthwhile to notice that these models are sup-

ported by a variety of experimental data.The usefulness of ‘‘reductionistic’’ models is quite obvious for the understanding

of the overall migration process but they seem somewhat questionable for practical

purposes requiring models that can be easily and rapidly applied to real circum-

stances.Many important lessons can be learnt from the above model assessment. First of

all, wise application and use of predictive environmental models require that users

must be fully aware of model performance mainly in relation to the output uncer-

tainty levels and how these are reflected in the decision making process.It is unwise to inspire in the potential users expectations as to the accuracy and

completeness of model output that are not supported by the experience gained dur-

ing model testing and validation. During the last decades model exercises and

applications to ecosystems contaminated by radionuclides introduced in the

environment following the major nuclear accidents have significantly contributed

to increase the expertise of modellers. Such experience is essential to the proper use

of models to solve practical problems of environmental management.

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Review and assessment of models for predicting the migration of radionuclides from catchments

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