Water Resources Management 5: 199-208,
1991.
0 1991 Kluwer Academic Publishers. Printed in the Netherlands.
REGIONALISATION
OF PHYSICALLY-BASED
APPLICATION
TO
Laboratory
199
WATER
BALANCE
UNGAUGED
CATCBIIENTS
MODELS
IN
BELGIUM.
G.L. Vandewiele.
Chong-Yu Xu
of Hydrology,
Vrije Universiteit
Brussel
Pleinlaan
2, 1050 Brussels,
Belgium
W. Huybrechts
Institute
of Nature Conservation
Kiewitdreef
3, 3500 Hasselt,
Belgium
ABSTRACT. More than 60 catchments
from Northern Belgium ranging in size
from 16 to 3160 km2 have been studied by means of a physically-based
stochastic
water balance model. The parameter values derived from calibration
of the model were regionally
mapped for the study region.
Associations
between model parameters
and basin lithological
characteristics
were established
and tested.
The results
show that the simple
conceptual
monthly water balance model with three parameters for actual
evapotranspiration,
slow and fast runoff
is capable either
to generate
monthly streamflow
at ungauged sites or to extend river flow at gauged
sites.
This allows a fairly
accurate estimation
of monthly discharges
at
any location
within the region.
Key
words.
Water
balance
model,
ungauged
catchments,
Belgium.
1. INTRODUCTION
One of the main objectives
in the development of a conceptual
model
of the rainfall-runoff
process is to provide a model that can be used in
ungauged catchments to generate a record of runoff
for planning and
design purposes. Various kinds of rainfall-runoff
models have been
developed since the early 1930’s. However, the need of prior calibration
by using an available
streamflow
record of sufficient
length makes it
difficult
to apply existing
models to ungauged catchments.
It is therefore appropriate
to emphasize the study of correlating
model parameters
to measurable physical catchment characteristics.
The research reported
in this paper aims:
(11 to develop a model that simulates monthly hydrographs of river
flow on the basis of minimum available
meteorological
data and parameters that can readily
be derived from catchment maps.
(2) to establish
quantitative
relationships
between model parameters and the most meaningful
physical
basin characteristics.
(3) to present the regional
distribution
of model parameters and
water balance terms simulated
by the model for the Flemish region of
Belgium.
2. STATE OF THE ART ON RELATING LUMPED SYSTEMMODEL PARAMETERSAND BASIN
CHARACTERISTICS
Regionalisation
used to facilitate
has been for many years a standard hydrologic
tool,
extrapolation
from sites at which records have been
200
G. L. VANDEWIELE
ET AL.
collected
to others at which data are required
but unavailable
(Riggs,
1973). Considerable
effort
has been invested in representative
and
experimental
basins to predict
hydrologic
variables
and parameters of
complex distributed
models in terms of catchment characteristics
(e.g.
Aston and Dunin, 1979). However this is not the case for lumped parameter models. Owing to the simplifications,
the physical significance
of
model parameters and relations
is reduced by the effects
of spatial
and
time variations.
Moreover, variations
of rainfall
in time and space,
which cannot usually be modelled,
also affect
the values of derived
parameters.
From these points of view, it is not surprising
that little
success has been reported
in relating
lumped model parameters and catchment characteristics.
Few exceptions
are reported
by Jarboe and Haan
(1974) who used multiple
linear regression
methods to relate each of the
four parameters of Haan’s model and measurable watershed characteristics;
Magette et al. (19761 used Jones’ (19761 procedure to fit a
subset of six parameters of the Kentucky watershed model, and were ab Nle
to obtain acceptable
multiple
regression
equations
using indices of
fifteen
watershed characteristics:
Weeks and Ashkanasy (19831 related
parameter values of the Sacramento model to six’catchment
characteristics,
the resultant
regional
parameters were found to be
satisfactory.
Although only few hydrologists
would confidently
compute the discharge hydrograph from rainfall
data and the physical description
of the
catchment,
this is a practical
problem which must often be faced by
practicing
engineers.
(Nash and Sutcliffe,
19701.
3. A SIMPLE LUMPED SYSTEMWATERBALANCEMODEL ON A MONTHLYSCALE
The natural
hydrologic
phenomena are very complex, but it often is
possible to simplify
the rainfall
runoff
process description
and still
obtain satisfactory
model performance.
“It appears that three to five
parameters should be sufficient
to reproduce most of the information
in
a hydrological
record.”
(Beven, 19891.
The model presented here, which has been tested in more than 60
Belgian catchments and a number of Chinese catchments,
was found to be
able to explain with a few parameters
the natural
phenomena and to
correlate
these parameters with the physical
characteristics
of the
basin. It is a single reservoir
model (see Fig.11.
rrecipitationlp,)
'1 Phpotranspiration
Infiltration
Fig.1.
The model concept
WATER BALANCE MODELS IN BBLGIUM
201
Input series for the model are free water evaporation
(e,) and
area1 precipitation
(p+I. The precipitation
reaching the ground is
divided into three parts:
evapotranspiration,
fast runoff and infiltration. The infiltrated
water is kept in the underground
reservoir
the
output from which is the slow runoff andpart of evapotranspiration.
Table 1 shows the model equations.
Table 1. The model equations
-------^------------------------------------------rr = min((p,+m’,-,)(l-e’-‘,‘,‘),e+)
St = azm*+-l
ft = a,m’,-,(p,-e,(I-e-P,/‘,)f
d, = s++f+
m+ = mrml+p+-rr -de
_-__----------------_______________^____-----------
(1)
(2)
(3)
(4)
(5)
where
in month t, in mm
P+ = precipitation
in month t, in mm
e t = free water evaporation
in month t. in mm
rt = actual evapotranspiration
in month t, in mm
St = slow runoff
f, = fast runoff
in month t, in mm
d, = total runoff
in month t. in mm
m, = soil moisture content at the end of month t, in mm
a, to a, = model parameters all positive
(x)* = max(O,x)
A number of formulae for estimating
monthly actual evapotranspiration have been examined and compared in this research (see Vandewiele et
al.,
1991). and Romanenko’s type formula (Eq.1) was found to be one of
the best.
It allows for both precipitation
and soil moisture content as
the available
water for evapotranspiration.
Parameter a, is controlled
by the permeability
of the soil and subsurface of the basin. Slow flow
(Eq.2) is assumed to be linearly
proportional
to the catchment storage
at the end of the previous month. Parameter a a increases with catchment
permeability.
Fast runoff
(Eq.3) is thought of as the quantity
of rainfall running off within
the month considered.
Parameter a, is influenced
by the degree of urbanization,
average basin slope and drainage density
of the basin. The sum of fast and slow runoff gives the total
streamflow
(Eq.4) while Eq.5 is the water balance equation.
4. STUDY REGION AND THE USE OF RASIN CHARACTERISTICS
4.1.
Study Region
As Weeks and Ashkanasy (1983) have pointed out, in order to provide
satisfactory
results
of the regression
approach,
the regional
study
should be confined
to an area of reasonable
hydrological
consistency.
Sufficient
data should be available
to enable some confidence
to be held
in both model parameter estimation
and regression
estimation.
202
G. L. VANDEWIELE
ET AL.
The boundary of the study area is defined as the Northern part of
Belgium (about 15,000 km2). Within this area high quality
data are
available
for more than 60 gauged catchments with at least 10 years of
flow record.
It allowed for a study of regional
variation
of the water
balance terms as well as model parameters.
4.2.
The Use of Basin Characteristics
Physical properties
of a drainage basin determine
the type and rate
of hydrological
processes.
If the relationships
between these properties
and the hydrological
response could be established,
the hydrological
behaviour of basins could be predicted
without
the need for direct
process measurement. For practical
considerations,
however, it is not
possible
and in many cases not necessary to consider all physical characteristics
of a basin.
The lumped parameters of the above mentioned model may be predicted
by equations
including
as independent
variables
several catchment characteristics
(drainage density,
mean gradient
of the basin, land use and
vegetation,
lithology
and soil).
Previous studies (Xu, 1988; Huybrechts
et al. 19901 suggest that,
for the study region,
lithologic
characteristics
should be studied as a first
step in the present research.
5. RESULTS AND DISCUSSION
5.1.
Regional
Distribution
of Water Balance Terms and Model Parameters
The model, calculating
the major terms of the water balance on a
monthly time base, is calibrated
for more than 60 gauged catchments
in
Northern Belgium. It allowed for a study of the regional
variation
of
the water balance terms and model parameters.
As examples, the regional
distribution
of the slow flow index (SFI), i.e. the ratio of slow runoff
to total runoff calculated
by the model, the variation
coefficient
of
observed runoff
(Ovcf) and the parameter a, of the model are shown in
Fig.2 to 4, respectively.
’ Fig.2 shows that the importance of slow flow in the water balance
diminishes
clearly
from SE to NW. Values of SF1 range from nearly 0.9
for permeable catchments
in SE to 0.1 for impermeable catchments in NW.
This picture
reflects
the geological
conditions
in the catchments.
In
the Western catchments the subsurface
is characterized
by the presence
of tertiary
clay layers. The importance of these clay layers is reducing
significantly
to the East where the subsurface consists mainly of highly
permeable sediments such as sand and chalk. Fig.3 shows a complementary
phenomenon with respect to Fig.2;
it shows that the variability
of the
observed runoff series is decreasing when the permeability
of the basin
increases.
This is normal since direct
runoff
is much more variable
than
decreasing
when the
base flow, and the former component is significantly
subsoil is more sandy.
Fig.4 shows that the evapotranspiration
parameter
al is higher for catchments with clay subsoil.
Similar regional
distribution maps for other water balance terms and parameters have been made.
which are not shown here.
WATER
Fig.2
BALANCE
MODELS
IN BELGIUM
Regional
variation
Northern
Belgium
@
203
of
slow
flow
__-_
Country
-
Catchment bouri‘daries
index
0.7 - 0.9
Regional
of
observed
(SFI)
in
_.-C Country boundaries
> 0.9
Fig.3
boundaries
-
distributioh
of
runoff
series
the
variation
(Ovcf)
Catchment
coefficient
boundaries
204
G. L. VANDEWIELE
0.006
-
@
Fig.4
0.009
0.009
-
Regional
distribution
parameter
of
the
Catchment
boundaries
evapotranspiration
al
(A)
(B)
(Cl
(D)
Fig.5
--
>
a3
ET AL.
Y
Y
Y
Y
=
=
=
=
0.0903+0.00623X
0.9996-0.734X
0.0195+0.377x
0.485-0.00318X
R
n
R
R
= .96
=-.97
= .96
=-.95
Relationships
between
SFI,
Ovcf,
estimated
variation
coefficient
for
mean.runoff
(Evcf)
and percentage
of
catchment
area
occupied
by permeable
subsoil.
(PS)
(X:
horizontalaxis,
Y: vertical
axis.
the
same
for
Fig.6
and Fig.7)
205
WATER BALANCB MODELS IN BELGIUM
5.2. Relationships
Between Lithological
Terms and Model Parameters
Characteristics,
Water balance
Three categories
of lithologic
variables
may be distinguished
for
the study region:
percentage of the catchment occupied by a permeable
subsurface
PS (e.g. sands, gravel and limestone):
percentage
of the
catchment occupied by impermeable subsurface
(e.g. clay) and percentage
of catchment area occupied by hardrock which is present in very small
areas.
Twenty-four
catchments located in the region ranging in size from
15.6 to 2163 km2 were selected for use in the study.
Some quantitative
relationships
between model parameters,
water
balance terms,
and lithological
characteristics
are shown in Fig.5 to
7. The corresponding
regression
equations
and correlation
coefficients
are also given in these figures.
From Fig.6(AI
and Fig.6(DI
one can see that parameters a, and a,
can be related
directly
to the permeability
of the subsurface(denoted
by
PS)of the catchments.
Parameter a2 is the “storage-constant”
of the
basin. It is proportional
to the speed with which the storage is
emptied.
It cannot be explained
by PS alone. Drainage density and the
depth to the impervious
layer should be investigated
for explaining
of
a,, which will be studied soon after.
From equation
(21 and (3) we see that the proportion
of fast runoff
and slow runoff
is roughly equal to (aJazI*P,
where P is the mean
precipitation.
For physical reasons we can expect that this proportion
will decrease when the percentage of the catchment area with permeable
subsoil increases.
This seems to be the case as Fig.7(A)
shows.
The relationships
shown in figures
5 to 7 seem to work in the
Belgian context.
It is clear that in subtropical
or tropical
climates
the high seasonal variability
of rainfall
will be very important,
and
these relationships
will perhaps be different
there.
5.3.
Test on Parameter
Evaluations
Using Lithological
Characteristics
Five catchments,
other than the 24 catchments used for constructing
Fig.‘5 to 7, were randomly selected for testing
the use of such relationships. The percentage
of catchment area with permeable subsoil of each
of these five catchments was determined,
Parameters a, and a, then were
determined
consulting
Fig.6(AI
and 6(D), respectively.
With the help of
Fig.7(AI
it then was possible
to compute parameter a2.
In Table 2 observed, estimated
and predicted
mean annual runoffs
are shown. The estimated
runoffs were computed
with parameter values
obtained by optimization
of the model using an observed
runoff series.
The predicted
runoffs
(only for the 5 test basins) were computed with
parameter values obtained by using Fig,6(AI,
6(D) and 7(A) as explained
above.
The agreement is good except perhaps for basin A878/2. This catchment was found to be difficult
to model.
In general Table 2 provides
The results
of Table 2
acceptable
data for many design applications.
clearly
indicate
that this method of parameter estimation
is one which
offers a possible
solution
for ungauged catchments.
206
G. L. VANDEWIELE
ET AL
A
/
/. . /t(l/
/
\
(A)
(8)
(C)
(D)
Fig.6
Y
Y
Y
y
Relationships
percentage
subsoil
I
\IY
I
I
.
=
=
=
=
between
catchment
of
=-.96
=-.97
=-.95
=-.93
SFI,
model
parameters
area
occupied
by
and
permeable
(PS)
I
I
I
I
I
I
1
1
-
\
R
R
R
R
1.179-0.00963X
1.294-1.500X
0.691-1.348X+0.625X'
0.548-0.00668X*1.68*10-=X2
I
I
I 7.5
B
7.5
,
6.0 6.0
,
i
I
I
I
I
I
I
I
I
A
I
.
\
(A)
(E)
(Cl
CD)
Fig.7
Relationships
deviation
occupied
Y
Y
Y
Y
=
=
=
=
R
R
R
R
10.521*EXP(-.0271X)
0.1966*EXP(3.222X)
0.00377+0.930x
1.140-0.00767X
between
(Mstd)
and
by permeable
(a3/a2)P,
percentage
subsoil
=-.97
= .94
= .94
=-.91
Ovcf,
model
of catchment
(PS)
s tandard
area
207
WATER BALANCB MODELS IN BELGIUM
Table
2. Comparison between observed, estimated
predicted
mean annual runoff
and
--_----__-------------------------------------------Catchment
(Code 1
Area
(km21
Observed
(cm)
----------__-----------------------------------------
AZ002
A539
A8132
D1221
A540
A535
DO099
A527
A523
A512
A528
A32
D2941
D4011
A2006
D5531
A879/4
A94
DO931
D4933
A9/2
A816
A536
D1551
Al35
A545
A1012
A529
A8?8/2
Calibration
catchments
532.0
34.51
110.0
17.36
2045.0
19.36
2163.0
21.29
99.6
17.34
645.0
22.58
314.0
21.31
115.0
25.84
93.0
24.18
93.0
29.35
51.0
22.34
19.0
24.41
15.6
26.10
243.0
24.53
1244.0
22.34
463.0
15.12
102.0
32.71
378.5
36.05
890.0
20.21
23.8
22.40
56.4
27.42
46.0
27.54
127’. 5
27.80
208.4
17.99
Test catchments
73.0
25.62
88.0
20.92
148.0
22.48
89.0
28.82
102.0
31.38
Estimated
(cm1
Predicted
(cm)
33.95
18.19
19.28
21.28
17.00
22.14
21.18
25.14
23.40
28.55
23.17
24.35
25.48
24.10
21.82
14.96
32.41
35.90
19.98
21.73
26.46
26.38
28.50
18.18
25.26
19.60
22.10
28.33
30.80
28.56
21.12
20.22
28.26
25.97
6. CONCLUSIONS
A simple physically-based
water balance model has been developed.
Regression equations
for estimating
model parameters
from lithological
characteristics
were obtained
for Northern Belgium. The model would find
more widespread applications
because the parameters could be evaluated
for situations
where there exists no observed streamflow
data for the
purposes of calibration.
The major water balance terms and parameter values derived from
model calibration
are regionally
mapped for the study region.
They
allowed for an extrapolation
of the results
to ungauged catchments and
larger geographical
units.
208
G. L. VAIWEWIELE
ET AL.
It should be emphasized that the research undertaken
up to now is
limited
by time, and only lithologic
characteristics
have been examined.
The results presented
in this paper constitute
the first
phase of an
ongoing research project
and should be regarded primarily
as a ‘report
of research in progress’.
Further research in this respect will be: (a)
studying of other meaningful
basin characteristics
as have been proposed
in section 4.2 of the paper. (b) applying more advanced statistical
techniques
for the regionalisation
study so as to identify
the similarity of the basins w.r.t.
model parameters and hydrological
behavior.
dcknowlea[gement.
This research is performed partly
in the framework of
the Interuniversity
Project “Models for integral
water management in the
Flemish region”,
coordinated
by Prof. A. Van der Beken, and sponsored by
the Flemish Government.
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