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A compact watershed model system
Journal of Hydrology, 24 (1975) 155--169 © North-Holland Publishing Company, Amsterdam -- Printed in The Netherlands
A COMPACT WATERSHED MODEL SYSTEM
T.R.E. CHIDLEY and I.M. GOODWILL University of Aston, Birmingham (Great Britain) John Dossor and Partners, York (Great Britain) (Accepted for publication March 26, 1974}
ABSTRACT Chidley, T.R.E. and Goodwill, I.M., 1975. A compact watershed model system. J. Hydrol., 24: 155--179. This paper describes a framework for implementing watershed simulation and drainage network studies on a small computer. The method is illustrated by reference to a catchment model built for a large catchment in Botswana.
INTRODUCTION F o r m a n y y e a r s t h e t w o a u t h o r s have b e e n writing c o m p u t e r p r o g r a m s f o r d i f f e r e n t o r g a n i z a t i o n s ( G o o d w i l l et al., 197 0; Chidley, 197 0, 19 71; G o o d w i l l , 1 9 7 2 ) t o s i m u l a t e c a t c h m e n t b e h a v i o u r . F r e q u e n t l y o n l y small c o m p u t e r s h a v e b e e n available w h i c h m e a n t t h a t p r o g r a m s h a d to be e f f i c i e n t and c o m p a c t While w o r k i n g in d i f f e r e n t s i t u a t i o n s and w i t h d i f f e r e n t h y d r o l o g i s t s it was a p p a r e n t t h a t n o single m o d e l w o u l d m e e t all r e q u i r e m e n t s . T h e writers believe t h a t k n o w l e d g e o f t h e s u b j e c t is t o o sparse f o r t h e fixing o f d e v e l o p m e n t b y c o n c e i v i n g a general m o d e l w h i c h can, b y a d j u s t m e n t o f p a r a m e t e r s , be m a d e to w o r k in a n u m b e r o f basically d i f f e r e n t situations. T h e b e s t s t r a t e g y is to devise or recall a m o d e l w i t h t h e f e w e s t n u m b e r o f p a r a m e t e r s a n d simplest s t r u c t u r e t h a t will m e e t w i t h t h e r e q u i r e m e n t s . O t h e r p r o b l e m s w h i c h h a v e b e e n e n c o u n t e r e d are with d a t a i n p u t a n d outp u t . T h e d a t a available are usually d i f f e r e n t in f o r m a t a n d s c o p e f r o m j o b to job. O u t p u t r e q u i r e m e n t s vary, s o m e a n a l y s t s r e q u i r i n g d e t a i l e d statistical results, o t h e r s r e q u i r i n g to use t h e d a t a f o r f u r t h e r calculations, reservoir simulation f o r instance. I f o n e uses w a t e r s h e d m o d e l s w h i c h have b e e n p u b l i s h e d as c o m p u t e r c o d e o n e finds t h a t i n p u t a n d o u t p u t is an integral p a r t o f t h e p r o g r a m s t r u c t u r e a n d it is d i f f i c u l t to m o d i f y it w i t h o u t c o m p l e t e l y r e w r i t i n g the program. T h e s y s t e m d e s c r i b e d herein uses e i t h e r disc- or m a g n e t i c - t a p e based c o m puters. D a t a b a n k s can be b u i l t u p q u i t e i n d e p e n d e n t l y o f the w a t e r s h e d m o d elling s y s t e m . T h e d a t a is c o p i e d o n t o t h e d a t a b a n k s and results are d i s p l a y e d
156 using separate routines from those used in the modelling phase. The standard display routines for the watershed model results can be used for reporting on any o th er data in the bank. An account is given of how the program was used to study the r u n o f f from a c a t c h m e n t in Botswana. The main model was based u p o n the work of Boughton (1968) with modifications introduced by the authors to m eet the special requirements of the situation. THESYSTEM The actual model is based upon a generalized system concept. It is assumed that any hydrologic process can be broken down into one or more sub-processes or systems which transform a set of inputs to a set of outputs. A set of parameters and the specification for the transformation comprises each sub-system. The o u t p u t and input are linked by the modeller. There is a whole hierarchy of sub-systems which could be specified. This system permits only two types of sub-system, one to transform precipitation to r u n o f f and seepage to groundwater and a n o t h e r to route this r u n o f f through a channel network. The sub-systems for processing soil moisture, etc., which are lower down the hierarchy, are incorporated by the modeller into the first t y p e o f sub-system. The linking o f sub-systems is based upon the assumption that any river or drainage c a t c h m e n t upstream o f a given poi nt has a dendritic pattern. This means that there are no closed loops or bi-furcations in the stream pattern. To describe the t o p o l o g y of the resultant tree-like structure a channel ordering system has been devised. The r u n o f f f r om each c a t c h m e n t can be determined i n d ep en d en tly o f any ordering scheme, but channel routing has to be carried o u t in the o u t e r m o s t branches of the tree first, accumulating the flow as one approaches the trunk. In order to explain the philosophy of the system furt her a h y p o t h e t i c a l example o f its application is shown. Each c a t c h m e n t in Fig.1 is t h o u g h t of as a black box, the input to the box is precipitation. The only o u t p u t of interest in this case is the runoff. The r u n o f f from each c a t c h m e n t flows through a channel considered to lie within the catchment. The flow f r om the channels in the o u t e r m o s t branches of the tree is considered to run through the channel lower down in the tree. Fig.2 shows the system f o r m a t of the p r o t o t y p e c a t c h m e n t in Fig.1. The channel itself can be treated as a distributed parameter model, for instance, it could be divided up into a n u m b e r of segments with parameters to indicate geom et ry o f each segment. Flows f r om upstream channels would enter the most upstream segment of a reach, while inflow from the current c a t c h m e n t would be p r o p o r t i o n e d o u t over the length of the current channel. However, in this distributed parameter case the model would only consider what went in at the beginning o f the channel and what left at the end. Alternatively more r u n o f f models and channel connections could be used.
157
Fig.1. Schematic arrangement of a typical catchment. Numerals denote sub-catchments; letters denote main channels; dotted lines are main channels.
rcl)n
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I 01 co.c,m.ot too0., channel cf
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Fig.2. Block diagram of catchment.
For a given time increment a n y b o x or circle in Fig.2 can be written functionally as:
Qi = fi (Xi, Pi, Ei) where Qi is the o u t f l o w from the ith b o x , X i are an array o f parameters and initial values associated with the ith b o x , Pi is the precipitation or inflows to the b o x as the case may be and E i is the p o t e n t i a l evaporation rate. The initial values are u p d a t e d after the f u n c t i o n fi is called. The o u t p u t at p o i n t C in the system can be expressed as:
Qi = fi (Xi,
Pi
,El )
Qc = fc { Xc,fB(XB,f2(X2,P2,E2),EB),f3(Xa,P3,Ea),Ec } The c o m p u t e r program is told in which order it will find the successive
158
catchments and channels and when a run of the model is requested it simply calls the appropriate functions which operate on the given data. Obviously the input files must have been loaded either in a previous run of part of the system or earlier in the current run. Results can be displayed as part of the current run or later. In the form used for the Botswana study the analyst modifies his program accordingly. Work is current on incorporating the system into the problem orientated language GENYSYS. THE PROGRAM
The computer program consists of a series of modules which could be subroutines or self-standing programs. As to what they are depends upon the computer system being used. One series of modules provides means of inputting the parametric and control data to the model and loads the data banks with the meteorologic data and possibly the historic runoffs. Whilst another series performs the watershed model calculations. A third series provides means of analysing results, performing parameter sensitivity analysis for instance. The fourth and final series displays data and results from the data bank. Fig.3 shows the structure of the data base used. There are two main files, one to contain r u n o f f and the other precipitation and potential evaporation data. The latter is assumed to be uniform over the basin though this need not necessarily be so. Each line {record) in the data bank contains some identifying information about the spatial and temporal location of the data in that record. This is followed by a string of data which could typically be the rainfall for each day for a particular catchment for a particular month. In order to obtain machine efficiency and minimum processor size the user normally processes a batch of records at a time for each location. For m o n t h l y data this batch would comprise a year of data. If a record contained 24 hourly Precipitation data fi1¢ r¢c no
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Renoff data file. re c no.
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Fig.3. Layout of disc file storage.
159
records of data one may then elect to process 30 records, or 1 month, at a time. Furthermore, while one batch of data is being processed the machine is set to search for the location on the backing store device of another batch. THE CATCHMENT
In order to illustrate the use of the system its application to a catchment study carried out by the authors is described. The catchment is the Mahalapshwe in eastern Botswana (see Fig.4). The area above the gauging site at Madiba covers an area of 727 km 2. The river drains an area that has a comparatively high population of cattle and is, therefore, over-grazed. The over-grazing causes denuding and trampling of the land; these factors help to increase overland flow causing high flood peaks. The catchment is situated in an area which shows little relief except for isolated kopjes (small rocky hills), characteristic of granite topography in this part of the world. The land slopes gently eastward towards the Limpopo. The area is underlain by granite rocks of the basement complex. The catchment is bounded in the southwest by the Shoshong hills which rise to a considerable height above the adjacent plain. The catchment has broad flat valleys in the northern and eastern regions but in the southwest has steeper incised streams which rise on the northern slopes of the Shoshong hills. The mean elevation of the catchment is approximately 1,150 m. The Mahalapshwe is typical of the rivers in eastern Botswana in having a dry sandy bed. The groundwater in the catchment is largely confined to these sands which are recharged whenever floods occur. This groundwater is depleted by pumping, evaporation, sub-surface flow downstream and downwards seepage through faults in the underlying rocks. The infiltration capacity of the sand is high, it is evident t h a t it has to be fully saturated before r u n o f f can take place. Away from the streams the ability of the ground to store water is poor due to the shallow soils. The soils consist largely of sandy loams. The natural vegetation is pasture and acacia woodland. At the end of the dry season the pasture is thin due to over-grazing and trampling by the cattle. This together with a hard soil crust reduces infiltration and increases storm runoff. The climate is semi-arid and characterized by summer rainfall. Approximately 80% of the rainfall occurs within the 5-month period from November to March. Most of the remainder fails in October and April.The rainfall is derived from the easterly air streams originating in the Indian Ocean. The annual rainfall is highly variable and the distribution t h r o u g h o u t the season is erratic. During the period October 1959--October 1969 the m a x i m u m annual rainfall was 913 mm and the minimum 229 mm. The annual coefficient of variation for this record is 33~%. Mean daffy temperatures at Mahaiapye vary from about 12°C in winter to 25°C in summer. Relative h u m i d i t y is low. Potential evaporation is high at 1,800 mm/year. The prevailing wind is from the east or northeast. It is usually light and variable but sometimes strong in winter.
160
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Fig.4. Mahalapshwe river catchment. Rain gauge network 1970--71.
AVAILABLE DATA No great a m o u n t o f data was available for the study and this illustrates the typical developing c o u n t r y application of watershed models. One reason for the model study was to e x t e n d the r u n o f f records, particularly to acquire i n fo r matio n on variability. Long term daily rainfall records exist for two
161 stations in the area, Mahalapye and Kalamare; b o t h lie outside the c a t c h m e n t b o u n d ar y . The collection o f rainfall data for the study c o m m e n c e d in D ecem ber 1968 with the installation of 15 gauges, 3 of which were o f the recording type. A fu rth er 15 recording gauges were added during the course o f the study up to late 1970. These gauges were, on the whole, reliable yielding accurate and c o n tin u o u s data. U n f o r t u n a t e l y n o t all of the recording gauge data could be used because all long t er m records are on a dally basis. Any missing data was estimated by a simple three-station arithmetic average because there was insufficient data available to use the normal ratio m et hod. R u n o f f f r o m the c a t c h m e n t is measured only at Madiba. Measurement of r u n o f f at this point started in the 1968--69 wet season with a staff gauge. The record can be e x t e n d e d back one year because it is know n t hat r u n o f f did not occur in the 1 9 67- - 68 wet season. In S e p t e m b e r 1969 a Leupold and Stevens water-level recorder with a pen trace was installed. In January 1970 construction was started on a concr e t e weir at Madiba, the work was com pl et ed the following June. The structure was designed as a sand storage device rather than a gauging weir. However, it did give a greater degree of cont rol for calibration of the water-level record. The control was calibrated during the 1970--71 wet season using an O t t cableway located a b o u t 1 km upstream of the weir. Ten daily mean values of potential evaporation for the model were c o m p u t e d by means of the Penman m e t h o d using the climatological record at Mahalapye. THE MODEL USED Several basic watershed models were tried in the study and of these the one based u p o n the work of B ought on seemed to be the most suitable, An extra sub-program was added to allow for the infiltration and r u n o f f from the sand bed o f the river. The model, in c o m m o n with most other c a t c h m e n t models, is a b o o k keeping process using, in this case, a time increment of 1 day. The precipitation is distributed over various storages, it is then routed out of the model as surface or u n d e r g r o u n d flow. The various processes are as follows.
Input Depth of daily precipitation weighed by the Thiessen m e t h o d and potential evaporation were the only time series data used. Snow does n o t occur on the c a t c h m e n t so is not included in this model.
Storages Interception store Part of the precipitation reaching the ground is intercepted by the vegetation and retained there. For modelling purposes this storage has to be full before
162 precipitation can reach the land surface. Evaporation from this store is at the potential rate. The true limit of interception storage for any given type of vegetal cover is n o t constant but largely depends on the time of year and the intensity and duration of rainfall. In the model it is assumed that interception has a constant value.
Upper-soil store Any excess precipitation that has passed through the interception store enters the upper-soil store. Boughton assumes that this store can only be depleted by evapotranspiration. Because evaporation and transpiration account for a high proportion of the water lost from a catchment, their accurate evaluation is important. Most of the loss occurs as transpiration from vegetation. Although it is now relatively easy to compute potential evaporation and transpiration rates, it is n o t easy to compute rates under conditions other than potential. For this model Boughton uses a simple m e t h o d based on the work of Denmead and Shaw (1962) and Slatyer and Denmead {1963). They have shown that the ratio of actual to potential transpiration depends n o t only on the soil-moisture level but also on the prevailing potential rate.
Depression store This store is called the drainage store by Boughton. When the precipitation reaches the land surface it collects in surface depressions as soon as the uppersoil store has been filled. When the depression store is full, the remaining water will either run off as overland flow or, alternatively, infiltrate into the subsoil store. Boughton modified the exponential equation of Horton (1939) and proposed that infiltration could be obtained from:
f= fi* tanh (P/fi*) where P is the excess moisture in the drainage store, f is the infiltration rate and fi* is the infiltration capacity according to Horton, which is a function of the soil-moisture c o n t e n t of the soil at the given time. Using this equation it is seen that the r u n o f f Q is the remainder of the excess moisture after infiltration has been satisfied.
Subsoil store Moisture enters the subsoil store by infiltration from the depression store. The rate of infiltration is controlled entirely by the moisture level in the subsoil store and the available moisture in the depression store. The subsoil can lose moisture by drainage to the groundwater and by evaporation. The drainage c o m p o n e n t is estimated by assuming that the subsoil store drains as a linear reservoir. The evaporation loss is computed in exactly the same manner as from the upper-soil store.
163 The sand river
The surface r u n o f f from Boughton's model is passed directly into the sand o f the river bed. After various attempts at modelling the i n f l o w / o u t f l o w process of such a river, the following m e t h o d was devised. The r u n o f f in mm over the whole c a t c h m e n t is converted to an equivalent r u n o f f in millimetres over an assumed area of sand-river bed. This r u n o f f is then added to the soilmoisture level in the sand, if the uppe r limit of this is exceeded then the excess water is added to the surface r u n o f f reservoir. The actual daily r u n o f f is computed by assuming t hat the surface r u n o f f reservoir depletes linearly in proportion to its contents. A small c o m p u t a t i o n a l p oi nt here is t hat when as in this case one has long sequences with no input to the r u n o f f reservoir the c o m p u t e r will go on reducing the contents of the reservoir until it finally underflows the machine's capacity to register a number. Evaporation from the surface r u n o f f reservoir is considered to take place at the potential rate. Evaporation from water stored in the sand is large when the water level is near the surface. Experiments by Wipplinger (1958), in southwest Africa suggest th a t the evaporation is very high when the water level is less than 600 mm from the surface and below 750 m m it almost ceases. If there is no recharge or pumping then, according to Wipplinger, the water level will drop to this level within a m ont h. Experiments in Rhodesia confirm t ha t evaporation losses cease when the water level approaches 1 m below the surface. In Rhodesia it was found that the time taken to reach this level was m uc h longer than that found by Wipplinger. Personal observations by H y d e (1969), confirm t hat the Rhodesian times are more likely to be applicable in Botswana. In view of these experiments evaporation from the sand was assumed to decrease linearly as the water level d r o p p e d below the sand surface. At a certain pre-specified level below the surface evaporation is cut off, this level is one of the parameters that requires evaluation in the calibration process. In the model evaporation does n o t take place from the sand until the surface r u n o f f reservoir is em pt y. THE CALIBRATION PROCESS The p o wer of the system structure was utilized by dividing the c a t c h m e n t into five sub-catchments. The B ought on model was used for the r u n o f f model and the special sand river routing was used to link each catchment. Rainfall for each sub-catchment was determined by the Thiessen m e t h o d . This highlights one o f the major problems associated with this work. The combined r u n o f f and stream routing model has 16 parameters that need calibration, with 5 sub-catchments this needs 80 parameters in all. Because only one hydrograph existed for the c a t c h m e n t and this was of p o o r quality it was decided t h a t the equivalent parameter values in each sub-catchment would be kept equal. If any hydrographs had existed within the c a t c h m e n t then these
164 TABLE I Finally calibrated parameter values Parameter symbol
CEPMX USMAX DRMAX SSMAX SDRMX FO FC AK AAC H PCUS DR SS CEP US AC RST SK PCT R SR MAXEE RFFP
Sub-catchment No. 1 2
3
4
5
2.5 7.5 2.0 133.0 75.0 20.0 10.0 0.03 0.95 8.9 0.75 0.0 0.0 0.0 0.0 206.80 400.0 0.45 0.00004 100.0 0.0 150.0 0.99
2.5 10.0 2.0 133.0 75.0 20.0 10.0 0.03 0.95 8.9 0.75 0.0 0.0 0.0 0.0 164.00 750.0 0.60 0.0001 100.0 0.0 500.0 0.99
2.5 12.0 2.0 133.0 75.0 20.0 10.0 0.03 0.95 8.9 0.75 0.0 0.0 0.0 0.0 65.30 750.0 0.75 0.0006 175.0 0.0 500.0 0.99
2.5 12.0 2.0 133.0 75.0 20.0 10.0 0.03 0.95 8.9 0.75 0.0 0.0 0.0 0.0 122.60 1200.0 0.75 0.0004 175.0 0.0 950.0 0.99
2.5 7.5 2.0 133.0 75.0 20.0 10.0 0.03 0.95 8.9 0.75 0.0 0.0 0.0 0.0 168.50 400.0 0.45 0.00004 100.0 0.0 150.0 0.99
could have been utilized to individually calibrate the parameters of the subcatchment. In any hydrologic study using models, groundwater or surface water, the investigator should at the outset of the study formulate a possible model. Data collection procedures could then be designed to accumulate the right data. T h e M a h a l a p s h w e c a t c h m e n t is r e l a t i v e l y h o m o g e n e o u s , t h e v e g e t a t i o n a n d t o p o g r a p h y a r e u n i f o r m t h r o u g h o u t . T h e g e o l o g y is u n i f o r m , t h e w h o l e a r e a b e i n g u n d e r l a i n w i t h b a s e m e n t c o m p l e x g r a n i t i c r o c k s . T h e soils a r e n o w h e r e t h i c k , in t h e u p p e r p a r t s o f t h e c a t c h m e n t r o c k o u t c r o p s a r e c o m m o n . F o r t h i s r e a s o n t h e u p p e r - s o i l s t o r a g e w a s d e c r e a s e d in s u b - c a t c h m e n t s 1 a n d 2 t o 7 . 5 m m , a n d in s u b - c a t c h m e n t 3 t o 1 0 . 0 0 m m . In s u b - c a t c h m e n t s 4 a n d 5 i t w a s 1 2 m m . T h i s r e d u c t i o n in u p p e r - s o i l s t o r a g e w a s n e c e s s a r y in t h e u p p e r sub-catchments to increase the surface runoff. The stream channels are not homogeneous, the channel widths vary from a f e w m in t h e u p p e r r e a c h e s t o a l m o s t 50 m a t t h e d o w n s t r e a m e n d o f t h e c a t c h m e n t . T h e d e p t h o f s a n d in t h e s t r e a m b e d s v a r i e s f r o m 1 . 0 - - 1 . 5 m in t h e u p s t r e a m r e a c h e s t o a p p r o x i m a t e l y 5.0 m a t M a h a l a p y e .
165
The parameter values are shown in Table I. From this table it can be seen that the calibrated parameter values for each sub-catchment overland flow phase are the same except for the upper-soil storage which has been discussed earlier. The parameters for the channel phase are different for each channel because of the different physical dimensions of the sand beds. The meanings of the various symbols in the table are given in the appendix. RESULTS OF THE CATCHMENT STUDY
The c o m p u t e d and measured hydrographs for 1 9 6 9 - - 7 0 and 1 9 7 0 - - 7 1 are given in Figs.5 and 6. It will be seen from these that calibration could n o t be described as good. Various anomalies occur, sometimes the c o m p u t e d runoff is t o o high and sometimes t o o low. The most likely cause of error is the runoff record at Madiba. Errors in rainfall data are normally unlikely to cause very large errors in the runoff data because the weighting process and the model will usually damp them out. Although in cases where one polygon covers a large part of the
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Fig.5. A n n u a l d i s c h a r g e h y d r o g r a p h at Madiba. River M a h a l a p s h w e
JUN
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1969--70.
AUG
166
OCT
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Fig.6. Annual discharge hydrograph at Madiba. River Mahalapshwe 1970--71.
catchment then this damping effect will not be so pronounced. When a high error was observed the rainfall data were re-checked and found to be accurate. Rainfall over the catchment often occurs in intense local storms, these are sometimes large enough to produce r u n o f f but, because of their local nature, may be inadequately measured by the rain gauge network used. For a developing c o u n t r y the network in this catchment is good and is typical of what faces the analyst in these circumstances. A further possibility for error is the time increment used for computing runoff, in this case a day. Some of the floods in the Mahalapshwe catchment start and finish in less than 24 h. For the period of calibration rainfall and r u n o f f data are available as continuous records, therefore it would be quite feasible to model the catchment using a much smaller time increment still using this system. However, long term records were on a daily basis and one of the reasons for the study was to extend r u n o f f records using the long term rainfall data, therefore calibration had to be fixed on a daffy basis. Using a
SEP
167
T A B L E II List o f i n p u t variables u s e d b y p r o g r a m t o g e t h e r w i t h u n i t s Variable
Unit
NY NM NT KN NYS MST MEND NCAT N NI KC NRM NIM NOPT NC NDUR MCURS CEPMX 2 USMAX 3 DRMAX 4 SSMAX 5 SDRMX 6 FO 7 FC 8 AK 9 AAC
mm mm mm mm mm mm/day mm/day
10 11 12 13 14 15 16 2c 3c 4c 5c 6c 7c
mm/day
H PCUS DR* SS* CEP* US* AC RST SK PCT R* SR* MAXEE
8c R F F P
mm mm mm mm km: mm
mm mm mm
Description N u m b e r o f years t o b e c o m p u t e d ( i n c l u d e past years) N u m b e r o f m o n t h s in a y e a r ( r e c o r d s per b a t c h ) N u m b e r o f t i m e s t e p s / m o n t h ( t i m e steps p e r r e c o r d ) Number of time steps/evaporation measurement Number of starting year Number of starting month / o n l y r e l e v a n t to N u m b e r o f finishing m o n t h monthly time steps Number of catchments Total number of catchment model parameters (including s t a r t i n g values) N u m b e r o f o p t i m i z a b l e p a r a m e t e r s ( n o t used in t h e p r e s e n t program) Number of channels Number of channel parameters Number of optimizable channel parameters If N O P T > 0 s u m o f s q u a r e s o f t h e errors will be c o m p u t e d a n d printed If NC equals 1 t h e n t h e daily results o f t h e o v e r l a n d a n d c h a n n e l flow phases are p r i n t e d T h e n u m b e r of discharge categories for a d u r a t i o n curve; if N D U R equals 0 n o d u r a t i o n curves will be p r o d u c e d If t h e value o f M C U R S is greater t h a n zero t h e n mass curves will be p r o d u c e d for t h o s e c h a n n e l s specified b y I C H A N The capacity of the interception store T h e c a p a c i t y o f t h e upper-soil s t o r e Capacity of the drainage store C a p a c i t y of t h e subsoil s t o r e Drainage c o m p o n e n t o f t h e subsoil s t o r e I n f i l t r a t i o n rate w h e n t h e subsoil m o i s t u r e level is zero M i n i m u m daily i n f i l t r a t i o n rate E x p o n e n t k in i n f i l t r a t i o n e q u a t i o n Subsoil d e p l e t i o n factor, it o p e r a t e s as a linear reservoir o n t h e subsoil M a x i m u m e v a p o t r a n s p i r a t i o n rate P e r c e n t a g e o f e v a p o r a t i o n loss f r o m t h e upper-soil s t o r e M o i s t u r e level o f t h e drainage s t o r e M o i s t u r e level o f t h e subsoil s t o r e M o i s t u r e level o f t h e i n t e r c e p t i o n s t o r e M o i s t u r e level o f t h e upper-soil s t o r e Area of the catchment (or sub-catchment) C a p a c i t y o f t h e s a n d river s t o r e S u r f a c e reservoir d e p l e t i o n f a c t o r P e r c e n t a g e o f c a t c h m e n t area t h a t is c o v e r e d b y s a n d rivers M o i s t u r e level o f t h e s a n d in t h e river M o i s t u r e level in t h e r u n o f f s t o r e T h e m i n i m u m w a t e r level in t h e s a n d at w h i c h e v a p o r a t i o n can take place A f a c t o r to s i m u l a t e p u m p i n g f r o m t h e s a n d
168 TABLE II (continued) Variable
Unit
NORDER NOUT IN XRM XCM RT EVAP ISWO
mm mm
ICHAN
ND
Description An array of values to describe the ordering of the sub-catchments Identification number of the outlet channel for each sub-catchment A two-dimensional array which indicates the identification numbers of the channels entering a sub-catchment An array into which are packed the overland flow phase parameters for all sub-catchments An array into which are packed the channel flow phase parameters for all channels An array for precipitation data An array for evaporation data If ISWO = 1 the precipitation data are on cards If ISWO = 2 then these data are on a sequential file An array of values, one for each channel; if the value for a particular channel is 1 then the outflow from that channel will have duration and mass curves produced if NDUR and MCURS have the appropriate values; if ICHAN is less than 1 then the curves will not be produced whatever the values of NDUR and MCURS The number of historic flow values input in any given year
Note. V a r . l - - 1 6 are repeated NCAT times and are contained in XRM. Var.2c--8c are repeated KC times and are contained in XCM. * Indicates an initial value. small time increment allows the effects of rainfall intensity to be taken into a c c o u n t . T h i s is o f p a r t i c u l a r i m p o r t a n c e f o r t h e s u c c e s s f u l m o d e l l i n g o f f l a s h f l o o d s . A f u r t h e r r e l a t e d a d v a n t a g e o f u s i n g a s m a l l t i m e i n c r e m e n t is t h a t i f r u n o f f t a k e s p l a c e f o r o n l y p a r t o f a d a y o r a t t h e e n d o f a d a y a n d is t h e n carried over into the next day it can be better simulated. CONCLUSIONS The main conclusions to be derived from this piece of work are that: ( 1 ) I t is p o s s i b l e t o d e r i v e a d i s t r i b u t e d p a r a m e t e r w a t e r s h e d m o d e l t o r u n on a small computer (8K X 16 bit word plus one disc transport, minimum). (2) The system architecture can make available a wide variety of watershed model and river routing techniques. (3) Separation of input/output functions and generalization of parameters renders simple modification and extension of the system. (4) While 2-years data and a handful of flood events were probably too few for adequate calibration of the model in this case, sufficient success was obt a i n e d t o give e n c o u r a g e m e n t in d e v e l o p i n g t h e p a r t i c u l a r s t u d y as m o r e d a t a become available. (5) Problems with multi-parameter models are inevitable and must be faced i f p r o g r e s s in t h i s f i e l d o f h y d r o l o g y is t o b e m a d e . T h e m o d e l s t r u c t u r e sug-
1 69
gested here gives the analyst a powerful investigatory tool which could be incorporated into a language such as H Y D R O or GENYSYS. For those interested a listing of the program used for the Botswana study is available. AC KN OWLE DGEM ENTS
The writers wish to acknowledge the Permanent Secretary, Ministry of Mineral Resources, Republic of Botswana and the F o o d and Agriculture Organization of the U.N. for giving permission to publish this paper and also to the Officer in Charge of the F.A.O./U.N. Special Fund Project in Botswana, Mr. L.W. H y d e and the Government Senior Hydrological Engineer, Mr. B.H. Wilson, for their advice and support.
REFERENCES Boughton, W.C., 1968. A mathematical catchment model for estimating runoff. N.Z.J. Hydrol., 17(2): 75--100, Chidley, T.R.E., 1970. Report on the analysis of hydrological data in the Ganges tidal cell. Rep. Raikes Partners UNFAO Proj. 244. Chidley, T.R.E., 1971. The utilization of computers within the Botswana project. Rep. UNFAO/SF Proj. 359. Denmead, O.T. and Shaw, R.H., 1962. Availability of soil water to plants as affected by soil moisture content and meteorological conditions. J. Agron., 54: 385--390. Goodwill, I.M., 1972. The Mahalapshwe catchment model. Rep. Raikes Partners UNFAO/ SF Proj. 359, Tech. Note 29, 50 pp. Goodwill, I.M., Underhill, H. and Schenkefeld, M., 1970. Trials of a mathematical watershed model for runoff simulation. UNFAO FAO/SF: 166/GRE. Horton, R.E., 1940. Approach toward a physical interpretation of infiltration capacity. Proc. Soil Sci. Soc. Am., 5: 399--417. Hyde, L.W., 1969. Report of mission to UNDP/SF Proj. 359, Tech. Note 1, 20 pp. Slatyer, R.O. and Denmead, O.T., 1963. Water movement through the soil--plant--atmosphere system. Proc. Nat. Symp. Water Resour. Use Manage. Melbourne Univ. Press, Melbourne, 250 pp. Wipplinger, O., 1958. Storage of Water in Sand. S.W. Afr. Adm., Water Aff. Branch, Windhoek, S.W. Afr., 25 pp.
A COMPACT WATERSHED MODEL SYSTEM
T.R.E. CHIDLEY and I.M. GOODWILL University of Aston, Birmingham (Great Britain) John Dossor and Partners, York (Great Britain) (Accepted for publication March 26, 1974}
ABSTRACT Chidley, T.R.E. and Goodwill, I.M., 1975. A compact watershed model system. J. Hydrol., 24: 155--179. This paper describes a framework for implementing watershed simulation and drainage network studies on a small computer. The method is illustrated by reference to a catchment model built for a large catchment in Botswana.
INTRODUCTION F o r m a n y y e a r s t h e t w o a u t h o r s have b e e n writing c o m p u t e r p r o g r a m s f o r d i f f e r e n t o r g a n i z a t i o n s ( G o o d w i l l et al., 197 0; Chidley, 197 0, 19 71; G o o d w i l l , 1 9 7 2 ) t o s i m u l a t e c a t c h m e n t b e h a v i o u r . F r e q u e n t l y o n l y small c o m p u t e r s h a v e b e e n available w h i c h m e a n t t h a t p r o g r a m s h a d to be e f f i c i e n t and c o m p a c t While w o r k i n g in d i f f e r e n t s i t u a t i o n s and w i t h d i f f e r e n t h y d r o l o g i s t s it was a p p a r e n t t h a t n o single m o d e l w o u l d m e e t all r e q u i r e m e n t s . T h e writers believe t h a t k n o w l e d g e o f t h e s u b j e c t is t o o sparse f o r t h e fixing o f d e v e l o p m e n t b y c o n c e i v i n g a general m o d e l w h i c h can, b y a d j u s t m e n t o f p a r a m e t e r s , be m a d e to w o r k in a n u m b e r o f basically d i f f e r e n t situations. T h e b e s t s t r a t e g y is to devise or recall a m o d e l w i t h t h e f e w e s t n u m b e r o f p a r a m e t e r s a n d simplest s t r u c t u r e t h a t will m e e t w i t h t h e r e q u i r e m e n t s . O t h e r p r o b l e m s w h i c h h a v e b e e n e n c o u n t e r e d are with d a t a i n p u t a n d outp u t . T h e d a t a available are usually d i f f e r e n t in f o r m a t a n d s c o p e f r o m j o b to job. O u t p u t r e q u i r e m e n t s vary, s o m e a n a l y s t s r e q u i r i n g d e t a i l e d statistical results, o t h e r s r e q u i r i n g to use t h e d a t a f o r f u r t h e r calculations, reservoir simulation f o r instance. I f o n e uses w a t e r s h e d m o d e l s w h i c h have b e e n p u b l i s h e d as c o m p u t e r c o d e o n e finds t h a t i n p u t a n d o u t p u t is an integral p a r t o f t h e p r o g r a m s t r u c t u r e a n d it is d i f f i c u l t to m o d i f y it w i t h o u t c o m p l e t e l y r e w r i t i n g the program. T h e s y s t e m d e s c r i b e d herein uses e i t h e r disc- or m a g n e t i c - t a p e based c o m puters. D a t a b a n k s can be b u i l t u p q u i t e i n d e p e n d e n t l y o f the w a t e r s h e d m o d elling s y s t e m . T h e d a t a is c o p i e d o n t o t h e d a t a b a n k s and results are d i s p l a y e d
156 using separate routines from those used in the modelling phase. The standard display routines for the watershed model results can be used for reporting on any o th er data in the bank. An account is given of how the program was used to study the r u n o f f from a c a t c h m e n t in Botswana. The main model was based u p o n the work of Boughton (1968) with modifications introduced by the authors to m eet the special requirements of the situation. THESYSTEM The actual model is based upon a generalized system concept. It is assumed that any hydrologic process can be broken down into one or more sub-processes or systems which transform a set of inputs to a set of outputs. A set of parameters and the specification for the transformation comprises each sub-system. The o u t p u t and input are linked by the modeller. There is a whole hierarchy of sub-systems which could be specified. This system permits only two types of sub-system, one to transform precipitation to r u n o f f and seepage to groundwater and a n o t h e r to route this r u n o f f through a channel network. The sub-systems for processing soil moisture, etc., which are lower down the hierarchy, are incorporated by the modeller into the first t y p e o f sub-system. The linking o f sub-systems is based upon the assumption that any river or drainage c a t c h m e n t upstream o f a given poi nt has a dendritic pattern. This means that there are no closed loops or bi-furcations in the stream pattern. To describe the t o p o l o g y of the resultant tree-like structure a channel ordering system has been devised. The r u n o f f f r om each c a t c h m e n t can be determined i n d ep en d en tly o f any ordering scheme, but channel routing has to be carried o u t in the o u t e r m o s t branches of the tree first, accumulating the flow as one approaches the trunk. In order to explain the philosophy of the system furt her a h y p o t h e t i c a l example o f its application is shown. Each c a t c h m e n t in Fig.1 is t h o u g h t of as a black box, the input to the box is precipitation. The only o u t p u t of interest in this case is the runoff. The r u n o f f from each c a t c h m e n t flows through a channel considered to lie within the catchment. The flow f r om the channels in the o u t e r m o s t branches of the tree is considered to run through the channel lower down in the tree. Fig.2 shows the system f o r m a t of the p r o t o t y p e c a t c h m e n t in Fig.1. The channel itself can be treated as a distributed parameter model, for instance, it could be divided up into a n u m b e r of segments with parameters to indicate geom et ry o f each segment. Flows f r om upstream channels would enter the most upstream segment of a reach, while inflow from the current c a t c h m e n t would be p r o p o r t i o n e d o u t over the length of the current channel. However, in this distributed parameter case the model would only consider what went in at the beginning o f the channel and what left at the end. Alternatively more r u n o f f models and channel connections could be used.
157
Fig.1. Schematic arrangement of a typical catchment. Numerals denote sub-catchments; letters denote main channels; dotted lines are main channels.
rcl)n
rain
ro.
runoff
of.
channel
flow
I 01 co.c,m.ot too0., channel cf
rain
ro
cf
ro[n
rain
Fig.2. Block diagram of catchment.
For a given time increment a n y b o x or circle in Fig.2 can be written functionally as:
Qi = fi (Xi, Pi, Ei) where Qi is the o u t f l o w from the ith b o x , X i are an array o f parameters and initial values associated with the ith b o x , Pi is the precipitation or inflows to the b o x as the case may be and E i is the p o t e n t i a l evaporation rate. The initial values are u p d a t e d after the f u n c t i o n fi is called. The o u t p u t at p o i n t C in the system can be expressed as:
Qi = fi (Xi,
Pi
,El )
Qc = fc { Xc,fB(XB,f2(X2,P2,E2),EB),f3(Xa,P3,Ea),Ec } The c o m p u t e r program is told in which order it will find the successive
158
catchments and channels and when a run of the model is requested it simply calls the appropriate functions which operate on the given data. Obviously the input files must have been loaded either in a previous run of part of the system or earlier in the current run. Results can be displayed as part of the current run or later. In the form used for the Botswana study the analyst modifies his program accordingly. Work is current on incorporating the system into the problem orientated language GENYSYS. THE PROGRAM
The computer program consists of a series of modules which could be subroutines or self-standing programs. As to what they are depends upon the computer system being used. One series of modules provides means of inputting the parametric and control data to the model and loads the data banks with the meteorologic data and possibly the historic runoffs. Whilst another series performs the watershed model calculations. A third series provides means of analysing results, performing parameter sensitivity analysis for instance. The fourth and final series displays data and results from the data bank. Fig.3 shows the structure of the data base used. There are two main files, one to contain r u n o f f and the other precipitation and potential evaporation data. The latter is assumed to be uniform over the basin though this need not necessarily be so. Each line {record) in the data bank contains some identifying information about the spatial and temporal location of the data in that record. This is followed by a string of data which could typically be the rainfall for each day for a particular catchment for a particular month. In order to obtain machine efficiency and minimum processor size the user normally processes a batch of records at a time for each location. For m o n t h l y data this batch would comprise a year of data. If a record contained 24 hourly Precipitation data fi1¢ r¢c no
pr¢ area I
i --
Renoff data file. re c no.
ar~Q
2
area
3
I i
!
runoff cat
I
cat
2
cat
3
ctc.
ctc
Fig.3. Layout of disc file storage.
159
records of data one may then elect to process 30 records, or 1 month, at a time. Furthermore, while one batch of data is being processed the machine is set to search for the location on the backing store device of another batch. THE CATCHMENT
In order to illustrate the use of the system its application to a catchment study carried out by the authors is described. The catchment is the Mahalapshwe in eastern Botswana (see Fig.4). The area above the gauging site at Madiba covers an area of 727 km 2. The river drains an area that has a comparatively high population of cattle and is, therefore, over-grazed. The over-grazing causes denuding and trampling of the land; these factors help to increase overland flow causing high flood peaks. The catchment is situated in an area which shows little relief except for isolated kopjes (small rocky hills), characteristic of granite topography in this part of the world. The land slopes gently eastward towards the Limpopo. The area is underlain by granite rocks of the basement complex. The catchment is bounded in the southwest by the Shoshong hills which rise to a considerable height above the adjacent plain. The catchment has broad flat valleys in the northern and eastern regions but in the southwest has steeper incised streams which rise on the northern slopes of the Shoshong hills. The mean elevation of the catchment is approximately 1,150 m. The Mahalapshwe is typical of the rivers in eastern Botswana in having a dry sandy bed. The groundwater in the catchment is largely confined to these sands which are recharged whenever floods occur. This groundwater is depleted by pumping, evaporation, sub-surface flow downstream and downwards seepage through faults in the underlying rocks. The infiltration capacity of the sand is high, it is evident t h a t it has to be fully saturated before r u n o f f can take place. Away from the streams the ability of the ground to store water is poor due to the shallow soils. The soils consist largely of sandy loams. The natural vegetation is pasture and acacia woodland. At the end of the dry season the pasture is thin due to over-grazing and trampling by the cattle. This together with a hard soil crust reduces infiltration and increases storm runoff. The climate is semi-arid and characterized by summer rainfall. Approximately 80% of the rainfall occurs within the 5-month period from November to March. Most of the remainder fails in October and April.The rainfall is derived from the easterly air streams originating in the Indian Ocean. The annual rainfall is highly variable and the distribution t h r o u g h o u t the season is erratic. During the period October 1959--October 1969 the m a x i m u m annual rainfall was 913 mm and the minimum 229 mm. The annual coefficient of variation for this record is 33~%. Mean daffy temperatures at Mahaiapye vary from about 12°C in winter to 25°C in summer. Relative h u m i d i t y is low. Potential evaporation is high at 1,800 mm/year. The prevailing wind is from the east or northeast. It is usually light and variable but sometimes strong in winter.
160
% ~ ,
t~
o~ ~ , .
.....
~ . ~
z~o~
,
~-4~'
Fig.4. Mahalapshwe river catchment. Rain gauge network 1970--71.
AVAILABLE DATA No great a m o u n t o f data was available for the study and this illustrates the typical developing c o u n t r y application of watershed models. One reason for the model study was to e x t e n d the r u n o f f records, particularly to acquire i n fo r matio n on variability. Long term daily rainfall records exist for two
161 stations in the area, Mahalapye and Kalamare; b o t h lie outside the c a t c h m e n t b o u n d ar y . The collection o f rainfall data for the study c o m m e n c e d in D ecem ber 1968 with the installation of 15 gauges, 3 of which were o f the recording type. A fu rth er 15 recording gauges were added during the course o f the study up to late 1970. These gauges were, on the whole, reliable yielding accurate and c o n tin u o u s data. U n f o r t u n a t e l y n o t all of the recording gauge data could be used because all long t er m records are on a dally basis. Any missing data was estimated by a simple three-station arithmetic average because there was insufficient data available to use the normal ratio m et hod. R u n o f f f r o m the c a t c h m e n t is measured only at Madiba. Measurement of r u n o f f at this point started in the 1968--69 wet season with a staff gauge. The record can be e x t e n d e d back one year because it is know n t hat r u n o f f did not occur in the 1 9 67- - 68 wet season. In S e p t e m b e r 1969 a Leupold and Stevens water-level recorder with a pen trace was installed. In January 1970 construction was started on a concr e t e weir at Madiba, the work was com pl et ed the following June. The structure was designed as a sand storage device rather than a gauging weir. However, it did give a greater degree of cont rol for calibration of the water-level record. The control was calibrated during the 1970--71 wet season using an O t t cableway located a b o u t 1 km upstream of the weir. Ten daily mean values of potential evaporation for the model were c o m p u t e d by means of the Penman m e t h o d using the climatological record at Mahalapye. THE MODEL USED Several basic watershed models were tried in the study and of these the one based u p o n the work of B ought on seemed to be the most suitable, An extra sub-program was added to allow for the infiltration and r u n o f f from the sand bed o f the river. The model, in c o m m o n with most other c a t c h m e n t models, is a b o o k keeping process using, in this case, a time increment of 1 day. The precipitation is distributed over various storages, it is then routed out of the model as surface or u n d e r g r o u n d flow. The various processes are as follows.
Input Depth of daily precipitation weighed by the Thiessen m e t h o d and potential evaporation were the only time series data used. Snow does n o t occur on the c a t c h m e n t so is not included in this model.
Storages Interception store Part of the precipitation reaching the ground is intercepted by the vegetation and retained there. For modelling purposes this storage has to be full before
162 precipitation can reach the land surface. Evaporation from this store is at the potential rate. The true limit of interception storage for any given type of vegetal cover is n o t constant but largely depends on the time of year and the intensity and duration of rainfall. In the model it is assumed that interception has a constant value.
Upper-soil store Any excess precipitation that has passed through the interception store enters the upper-soil store. Boughton assumes that this store can only be depleted by evapotranspiration. Because evaporation and transpiration account for a high proportion of the water lost from a catchment, their accurate evaluation is important. Most of the loss occurs as transpiration from vegetation. Although it is now relatively easy to compute potential evaporation and transpiration rates, it is n o t easy to compute rates under conditions other than potential. For this model Boughton uses a simple m e t h o d based on the work of Denmead and Shaw (1962) and Slatyer and Denmead {1963). They have shown that the ratio of actual to potential transpiration depends n o t only on the soil-moisture level but also on the prevailing potential rate.
Depression store This store is called the drainage store by Boughton. When the precipitation reaches the land surface it collects in surface depressions as soon as the uppersoil store has been filled. When the depression store is full, the remaining water will either run off as overland flow or, alternatively, infiltrate into the subsoil store. Boughton modified the exponential equation of Horton (1939) and proposed that infiltration could be obtained from:
f= fi* tanh (P/fi*) where P is the excess moisture in the drainage store, f is the infiltration rate and fi* is the infiltration capacity according to Horton, which is a function of the soil-moisture c o n t e n t of the soil at the given time. Using this equation it is seen that the r u n o f f Q is the remainder of the excess moisture after infiltration has been satisfied.
Subsoil store Moisture enters the subsoil store by infiltration from the depression store. The rate of infiltration is controlled entirely by the moisture level in the subsoil store and the available moisture in the depression store. The subsoil can lose moisture by drainage to the groundwater and by evaporation. The drainage c o m p o n e n t is estimated by assuming that the subsoil store drains as a linear reservoir. The evaporation loss is computed in exactly the same manner as from the upper-soil store.
163 The sand river
The surface r u n o f f from Boughton's model is passed directly into the sand o f the river bed. After various attempts at modelling the i n f l o w / o u t f l o w process of such a river, the following m e t h o d was devised. The r u n o f f in mm over the whole c a t c h m e n t is converted to an equivalent r u n o f f in millimetres over an assumed area of sand-river bed. This r u n o f f is then added to the soilmoisture level in the sand, if the uppe r limit of this is exceeded then the excess water is added to the surface r u n o f f reservoir. The actual daily r u n o f f is computed by assuming t hat the surface r u n o f f reservoir depletes linearly in proportion to its contents. A small c o m p u t a t i o n a l p oi nt here is t hat when as in this case one has long sequences with no input to the r u n o f f reservoir the c o m p u t e r will go on reducing the contents of the reservoir until it finally underflows the machine's capacity to register a number. Evaporation from the surface r u n o f f reservoir is considered to take place at the potential rate. Evaporation from water stored in the sand is large when the water level is near the surface. Experiments by Wipplinger (1958), in southwest Africa suggest th a t the evaporation is very high when the water level is less than 600 mm from the surface and below 750 m m it almost ceases. If there is no recharge or pumping then, according to Wipplinger, the water level will drop to this level within a m ont h. Experiments in Rhodesia confirm t ha t evaporation losses cease when the water level approaches 1 m below the surface. In Rhodesia it was found that the time taken to reach this level was m uc h longer than that found by Wipplinger. Personal observations by H y d e (1969), confirm t hat the Rhodesian times are more likely to be applicable in Botswana. In view of these experiments evaporation from the sand was assumed to decrease linearly as the water level d r o p p e d below the sand surface. At a certain pre-specified level below the surface evaporation is cut off, this level is one of the parameters that requires evaluation in the calibration process. In the model evaporation does n o t take place from the sand until the surface r u n o f f reservoir is em pt y. THE CALIBRATION PROCESS The p o wer of the system structure was utilized by dividing the c a t c h m e n t into five sub-catchments. The B ought on model was used for the r u n o f f model and the special sand river routing was used to link each catchment. Rainfall for each sub-catchment was determined by the Thiessen m e t h o d . This highlights one o f the major problems associated with this work. The combined r u n o f f and stream routing model has 16 parameters that need calibration, with 5 sub-catchments this needs 80 parameters in all. Because only one hydrograph existed for the c a t c h m e n t and this was of p o o r quality it was decided t h a t the equivalent parameter values in each sub-catchment would be kept equal. If any hydrographs had existed within the c a t c h m e n t then these
164 TABLE I Finally calibrated parameter values Parameter symbol
CEPMX USMAX DRMAX SSMAX SDRMX FO FC AK AAC H PCUS DR SS CEP US AC RST SK PCT R SR MAXEE RFFP
Sub-catchment No. 1 2
3
4
5
2.5 7.5 2.0 133.0 75.0 20.0 10.0 0.03 0.95 8.9 0.75 0.0 0.0 0.0 0.0 206.80 400.0 0.45 0.00004 100.0 0.0 150.0 0.99
2.5 10.0 2.0 133.0 75.0 20.0 10.0 0.03 0.95 8.9 0.75 0.0 0.0 0.0 0.0 164.00 750.0 0.60 0.0001 100.0 0.0 500.0 0.99
2.5 12.0 2.0 133.0 75.0 20.0 10.0 0.03 0.95 8.9 0.75 0.0 0.0 0.0 0.0 65.30 750.0 0.75 0.0006 175.0 0.0 500.0 0.99
2.5 12.0 2.0 133.0 75.0 20.0 10.0 0.03 0.95 8.9 0.75 0.0 0.0 0.0 0.0 122.60 1200.0 0.75 0.0004 175.0 0.0 950.0 0.99
2.5 7.5 2.0 133.0 75.0 20.0 10.0 0.03 0.95 8.9 0.75 0.0 0.0 0.0 0.0 168.50 400.0 0.45 0.00004 100.0 0.0 150.0 0.99
could have been utilized to individually calibrate the parameters of the subcatchment. In any hydrologic study using models, groundwater or surface water, the investigator should at the outset of the study formulate a possible model. Data collection procedures could then be designed to accumulate the right data. T h e M a h a l a p s h w e c a t c h m e n t is r e l a t i v e l y h o m o g e n e o u s , t h e v e g e t a t i o n a n d t o p o g r a p h y a r e u n i f o r m t h r o u g h o u t . T h e g e o l o g y is u n i f o r m , t h e w h o l e a r e a b e i n g u n d e r l a i n w i t h b a s e m e n t c o m p l e x g r a n i t i c r o c k s . T h e soils a r e n o w h e r e t h i c k , in t h e u p p e r p a r t s o f t h e c a t c h m e n t r o c k o u t c r o p s a r e c o m m o n . F o r t h i s r e a s o n t h e u p p e r - s o i l s t o r a g e w a s d e c r e a s e d in s u b - c a t c h m e n t s 1 a n d 2 t o 7 . 5 m m , a n d in s u b - c a t c h m e n t 3 t o 1 0 . 0 0 m m . In s u b - c a t c h m e n t s 4 a n d 5 i t w a s 1 2 m m . T h i s r e d u c t i o n in u p p e r - s o i l s t o r a g e w a s n e c e s s a r y in t h e u p p e r sub-catchments to increase the surface runoff. The stream channels are not homogeneous, the channel widths vary from a f e w m in t h e u p p e r r e a c h e s t o a l m o s t 50 m a t t h e d o w n s t r e a m e n d o f t h e c a t c h m e n t . T h e d e p t h o f s a n d in t h e s t r e a m b e d s v a r i e s f r o m 1 . 0 - - 1 . 5 m in t h e u p s t r e a m r e a c h e s t o a p p r o x i m a t e l y 5.0 m a t M a h a l a p y e .
165
The parameter values are shown in Table I. From this table it can be seen that the calibrated parameter values for each sub-catchment overland flow phase are the same except for the upper-soil storage which has been discussed earlier. The parameters for the channel phase are different for each channel because of the different physical dimensions of the sand beds. The meanings of the various symbols in the table are given in the appendix. RESULTS OF THE CATCHMENT STUDY
The c o m p u t e d and measured hydrographs for 1 9 6 9 - - 7 0 and 1 9 7 0 - - 7 1 are given in Figs.5 and 6. It will be seen from these that calibration could n o t be described as good. Various anomalies occur, sometimes the c o m p u t e d runoff is t o o high and sometimes t o o low. The most likely cause of error is the runoff record at Madiba. Errors in rainfall data are normally unlikely to cause very large errors in the runoff data because the weighting process and the model will usually damp them out. Although in cases where one polygon covers a large part of the
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Fig.5. A n n u a l d i s c h a r g e h y d r o g r a p h at Madiba. River M a h a l a p s h w e
JUN
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1969--70.
AUG
166
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JUN
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Fig.6. Annual discharge hydrograph at Madiba. River Mahalapshwe 1970--71.
catchment then this damping effect will not be so pronounced. When a high error was observed the rainfall data were re-checked and found to be accurate. Rainfall over the catchment often occurs in intense local storms, these are sometimes large enough to produce r u n o f f but, because of their local nature, may be inadequately measured by the rain gauge network used. For a developing c o u n t r y the network in this catchment is good and is typical of what faces the analyst in these circumstances. A further possibility for error is the time increment used for computing runoff, in this case a day. Some of the floods in the Mahalapshwe catchment start and finish in less than 24 h. For the period of calibration rainfall and r u n o f f data are available as continuous records, therefore it would be quite feasible to model the catchment using a much smaller time increment still using this system. However, long term records were on a daily basis and one of the reasons for the study was to extend r u n o f f records using the long term rainfall data, therefore calibration had to be fixed on a daffy basis. Using a
SEP
167
T A B L E II List o f i n p u t variables u s e d b y p r o g r a m t o g e t h e r w i t h u n i t s Variable
Unit
NY NM NT KN NYS MST MEND NCAT N NI KC NRM NIM NOPT NC NDUR MCURS CEPMX 2 USMAX 3 DRMAX 4 SSMAX 5 SDRMX 6 FO 7 FC 8 AK 9 AAC
mm mm mm mm mm mm/day mm/day
10 11 12 13 14 15 16 2c 3c 4c 5c 6c 7c
mm/day
H PCUS DR* SS* CEP* US* AC RST SK PCT R* SR* MAXEE
8c R F F P
mm mm mm mm km: mm
mm mm mm
Description N u m b e r o f years t o b e c o m p u t e d ( i n c l u d e past years) N u m b e r o f m o n t h s in a y e a r ( r e c o r d s per b a t c h ) N u m b e r o f t i m e s t e p s / m o n t h ( t i m e steps p e r r e c o r d ) Number of time steps/evaporation measurement Number of starting year Number of starting month / o n l y r e l e v a n t to N u m b e r o f finishing m o n t h monthly time steps Number of catchments Total number of catchment model parameters (including s t a r t i n g values) N u m b e r o f o p t i m i z a b l e p a r a m e t e r s ( n o t used in t h e p r e s e n t program) Number of channels Number of channel parameters Number of optimizable channel parameters If N O P T > 0 s u m o f s q u a r e s o f t h e errors will be c o m p u t e d a n d printed If NC equals 1 t h e n t h e daily results o f t h e o v e r l a n d a n d c h a n n e l flow phases are p r i n t e d T h e n u m b e r of discharge categories for a d u r a t i o n curve; if N D U R equals 0 n o d u r a t i o n curves will be p r o d u c e d If t h e value o f M C U R S is greater t h a n zero t h e n mass curves will be p r o d u c e d for t h o s e c h a n n e l s specified b y I C H A N The capacity of the interception store T h e c a p a c i t y o f t h e upper-soil s t o r e Capacity of the drainage store C a p a c i t y of t h e subsoil s t o r e Drainage c o m p o n e n t o f t h e subsoil s t o r e I n f i l t r a t i o n rate w h e n t h e subsoil m o i s t u r e level is zero M i n i m u m daily i n f i l t r a t i o n rate E x p o n e n t k in i n f i l t r a t i o n e q u a t i o n Subsoil d e p l e t i o n factor, it o p e r a t e s as a linear reservoir o n t h e subsoil M a x i m u m e v a p o t r a n s p i r a t i o n rate P e r c e n t a g e o f e v a p o r a t i o n loss f r o m t h e upper-soil s t o r e M o i s t u r e level o f t h e drainage s t o r e M o i s t u r e level o f t h e subsoil s t o r e M o i s t u r e level o f t h e i n t e r c e p t i o n s t o r e M o i s t u r e level o f t h e upper-soil s t o r e Area of the catchment (or sub-catchment) C a p a c i t y o f t h e s a n d river s t o r e S u r f a c e reservoir d e p l e t i o n f a c t o r P e r c e n t a g e o f c a t c h m e n t area t h a t is c o v e r e d b y s a n d rivers M o i s t u r e level o f t h e s a n d in t h e river M o i s t u r e level in t h e r u n o f f s t o r e T h e m i n i m u m w a t e r level in t h e s a n d at w h i c h e v a p o r a t i o n can take place A f a c t o r to s i m u l a t e p u m p i n g f r o m t h e s a n d
168 TABLE II (continued) Variable
Unit
NORDER NOUT IN XRM XCM RT EVAP ISWO
mm mm
ICHAN
ND
Description An array of values to describe the ordering of the sub-catchments Identification number of the outlet channel for each sub-catchment A two-dimensional array which indicates the identification numbers of the channels entering a sub-catchment An array into which are packed the overland flow phase parameters for all sub-catchments An array into which are packed the channel flow phase parameters for all channels An array for precipitation data An array for evaporation data If ISWO = 1 the precipitation data are on cards If ISWO = 2 then these data are on a sequential file An array of values, one for each channel; if the value for a particular channel is 1 then the outflow from that channel will have duration and mass curves produced if NDUR and MCURS have the appropriate values; if ICHAN is less than 1 then the curves will not be produced whatever the values of NDUR and MCURS The number of historic flow values input in any given year
Note. V a r . l - - 1 6 are repeated NCAT times and are contained in XRM. Var.2c--8c are repeated KC times and are contained in XCM. * Indicates an initial value. small time increment allows the effects of rainfall intensity to be taken into a c c o u n t . T h i s is o f p a r t i c u l a r i m p o r t a n c e f o r t h e s u c c e s s f u l m o d e l l i n g o f f l a s h f l o o d s . A f u r t h e r r e l a t e d a d v a n t a g e o f u s i n g a s m a l l t i m e i n c r e m e n t is t h a t i f r u n o f f t a k e s p l a c e f o r o n l y p a r t o f a d a y o r a t t h e e n d o f a d a y a n d is t h e n carried over into the next day it can be better simulated. CONCLUSIONS The main conclusions to be derived from this piece of work are that: ( 1 ) I t is p o s s i b l e t o d e r i v e a d i s t r i b u t e d p a r a m e t e r w a t e r s h e d m o d e l t o r u n on a small computer (8K X 16 bit word plus one disc transport, minimum). (2) The system architecture can make available a wide variety of watershed model and river routing techniques. (3) Separation of input/output functions and generalization of parameters renders simple modification and extension of the system. (4) While 2-years data and a handful of flood events were probably too few for adequate calibration of the model in this case, sufficient success was obt a i n e d t o give e n c o u r a g e m e n t in d e v e l o p i n g t h e p a r t i c u l a r s t u d y as m o r e d a t a become available. (5) Problems with multi-parameter models are inevitable and must be faced i f p r o g r e s s in t h i s f i e l d o f h y d r o l o g y is t o b e m a d e . T h e m o d e l s t r u c t u r e sug-
1 69
gested here gives the analyst a powerful investigatory tool which could be incorporated into a language such as H Y D R O or GENYSYS. For those interested a listing of the program used for the Botswana study is available. AC KN OWLE DGEM ENTS
The writers wish to acknowledge the Permanent Secretary, Ministry of Mineral Resources, Republic of Botswana and the F o o d and Agriculture Organization of the U.N. for giving permission to publish this paper and also to the Officer in Charge of the F.A.O./U.N. Special Fund Project in Botswana, Mr. L.W. H y d e and the Government Senior Hydrological Engineer, Mr. B.H. Wilson, for their advice and support.
REFERENCES Boughton, W.C., 1968. A mathematical catchment model for estimating runoff. N.Z.J. Hydrol., 17(2): 75--100, Chidley, T.R.E., 1970. Report on the analysis of hydrological data in the Ganges tidal cell. Rep. Raikes Partners UNFAO Proj. 244. Chidley, T.R.E., 1971. The utilization of computers within the Botswana project. Rep. UNFAO/SF Proj. 359. Denmead, O.T. and Shaw, R.H., 1962. Availability of soil water to plants as affected by soil moisture content and meteorological conditions. J. Agron., 54: 385--390. Goodwill, I.M., 1972. The Mahalapshwe catchment model. Rep. Raikes Partners UNFAO/ SF Proj. 359, Tech. Note 29, 50 pp. Goodwill, I.M., Underhill, H. and Schenkefeld, M., 1970. Trials of a mathematical watershed model for runoff simulation. UNFAO FAO/SF: 166/GRE. Horton, R.E., 1940. Approach toward a physical interpretation of infiltration capacity. Proc. Soil Sci. Soc. Am., 5: 399--417. Hyde, L.W., 1969. Report of mission to UNDP/SF Proj. 359, Tech. Note 1, 20 pp. Slatyer, R.O. and Denmead, O.T., 1963. Water movement through the soil--plant--atmosphere system. Proc. Nat. Symp. Water Resour. Use Manage. Melbourne Univ. Press, Melbourne, 250 pp. Wipplinger, O., 1958. Storage of Water in Sand. S.W. Afr. Adm., Water Aff. Branch, Windhoek, S.W. Afr., 25 pp.