Colin Clark describes a non-linear unit hydrograph model for real time flood forecasting below a dam


A CENTRAL problem with real time flood forecasting is that the models have often not been tested under the conditions for which they were designed. This is especially true where a flood warning system is intended for a community situated below a flood detention dam. The high standard of protection afforded by the dam means that only the rare floods will be of importance. A further problem, which is even more critical, and is in many ways two-edged, is the reliability of the warning itself: a false warning will waste time and cause unnecessary anxiety, while a warning given too late will give insufficient time for those at risk to take action, and could, in a catastrophic flood, lead to a loss of life. The problem of flood forecasting is made even more difficult where there are no records of discharge to aid design. In this case use can be made of historic flood data providing they are used with care. This situation exists at Bruton in Somerset, UK, (figure 1) which has for the past 20 years been largely protected from floods by an 8m embankment dam with uncontrolled spillway (figure 2). The town has a long history of flooding dating from 1768 when the worst flood ever recorded took place. During this event the river rose very rapidly, completely destroying one of the stone bridges, with a force of water causing the breaking of house windows in the nearby village of Pitcombe (Clark, 1999). Since the dam was completed in 1984, only once has a flood warning been issued, which was not necessary since the actual level behind the dam was about 3m lower than the level recorded by the instrumentation.

Real time flood forecasting for the Brue was first proposed by the author (Clark, 1996a) but it is only recently that the idea has been taken up by the Environment Agency, following the production of a non-linear unit hydrograph model (Clark, 2004). This model has the unique feature of being largely based on the historic floods of 1968,1974, 1979, and 1982, with the great flood of 1917 being used as a rare flood to test the model.

The unit hydrograph model comes from the pioneer work of Sherman (1932) and is also described in Shaw (1994). While Jakeman et al (1990) and Young and Bevan (1994) have considered the non-linearity of the unit hydrograph response, this is the first paper to show how historic data have been used to produce and test a non-linear UH model for a flood warning system now in use for a town sited below a category A dam site in England.

The upper Brue catchment area
The river Brue drains an area of 28km2 at the Sheephouse dam site (Figure 1). The land surface is moderate to steep, reaching a height of about 260m near Alfred’s Tower. The underlying geology is mainly Oxford Clay, Fuller’s Earth, and Forest Marble. These rocks weather to give a stiff impermeable clay. The land use is mainly grassland with only 10% of the area covered with woodland. Under woodland the soil is much more permeable, although lower down the soil profile the effect of geology takes over, resulting in a slowly permeable subsoil. An important feature of the Upper Brue is the roughly circular catchment shape. This, coupled with the distribution of the drainage system, leads to a near synchronous arrival time of runoff at the dam site. The result is a rapid rise and fall of riverflow.

Model specification
Figure 3 shows the main features of the model. The data inputs to the model are:

• Telemetered rainfall from one of two sites. The Charldon Hill Research Station (CHRS) also has its own tipping bucket raingauge which is located just below the dam site.

• Soil moisture deficit (SMD) data measured directly via a weighing lysimeter (Figure 4), at CHRS. This data has been gathered since 1995 as part of a long term study of evaporation in the Upper Brue catchment area. The data have been compared with the Meteorological Office Rainfall and Evaporation Calculation System (MORECS) and also with water balance data for the river Brue at Lovington gauging station which is 11km downstream (Clark, 2002a, 2003). The results from the lysimeters are more realistic than those produced by MORECS. The lysimeters are weighed daily and the data uploaded onto a website which is accessible by the Environment Agency at Exeter.

• Saturated soil hydraulic conductivity data (Ksat), measured at 105 sites within the upper Brue catchment area. The main control of hydraulic conductivity in this catchment area is land use. Figure 5 shows the frequency distribution of this data for woodland and grassland soils, while Figure 6 shows the data transformed into percentage runoff in relation to rainfall intensity which is used in the runoff model.

• Time to peak of the unit hydrograph has been estimated from the four historic floods of 1968, 1974, 1979, and 1982, as related to the maximum hourly rainfall intensity and shown in Figure 7. In an earlier version of the model there was a log-linear relationship between these two variables, but when tested in very extreme conditions, gave unrealistic results. This led to the use of the untransformed data in the relationship.

The unit hydrograph model uses the time to peak, percentage runoff, and a conversion of the effective rainfall into runoff at each time step of the storm, the so called ordinates of the unit hydrograph. The standard UH method (Houghton-Carr, 1999) calculates a linear transformation of rainfall into runoff, but this has not worked well for the upper Brue. Therefore a logistic function borrowed from biology (Street and Opik, 1970) has been adapted:

Log x/(a-x) = k(t-t1)

Where t and t1 are the time intervals; k = constant; a = maximum value; x = a given value. For the model, this equation can be rewritten for the rising limb of the hydrograph and optimised for the Bruton dam site:

Y = [INVLOG 2(t – 0.7 Tp)/1 + INVLOG 2(t – Tp)] Qp

Where Tp = time to peak; t = time; Qp = peak runoff rate per mm net rainfall = (330/Tp) Area/1000.

For the falling stage:
Y = [INVLOG (t1 – 0.85{TB – Tp})/ 1 + INVLOG(t1 – 0.85{TB – Tp})] Qp

Where TB = time base (Tp x 2.52), t1 = TB – t

Figure 8 shows how these ordinates vary with both time and rainfall intensity.

It has been observed that runoff from the upper Brue can take place even when there is a soil moisture deficit. This has been parameterised by the simple function Sin


rainfall, where


= mean catchment slope. The rainfall then passes into the Ksat:rainfall intensity relationship. The remaining rainfall becomes delayed flow, which for the upper Brue has a two hour time lag and a time to peak of two hours:

Y = 0.3896 (x/2) for the rising stage of delayed flow

Y = -0.01096x + 0.3896 for the falling stage.

Where y = ordinate; x = time in hours.

At each stage of the storm event the output hydrograph is calculated. Any soil moisture deficit is gradually eliminated at each time step. Before this is completed any runoff is slope runoff. On attaining zero soil moisture deficit the Ksat and delayed flow relationships come into operation.

Figure 9 shows the simulated hydrograph for the flood of 30 May 1979 during which over 36mm of rainfall was recorded at North Brewham. The key feature of this result is that both the peak discharge and timing of the flood event as a whole are well represented. Although this flood would not have caused a flood downstream if the dam were in place, it is important to show that the model faithfully represents floods over a large range of magnitudes. Three years later in 1982 a much more severe flood took place on 12 July 1982. Figure 10 shows the results: apart from the peak being about 0.25 hour different for the observed time of peak, the event as a whole has been well simulated. During this flood event nearly 100mm rainfall was measured at North Brewham, mostly within three hours. This flood has an estimated return period of about 70 years (Clark, 1997) and would have caused a flood with the flood detention dam in place.

As a storm progresses, it is possible to model each stage of the event. This has been done for the 1982 flood and is shown in Figure 11. During a storm which will occur in the future, it is not possible to tell when the rainfall will cease; storms collapse, move away, or just decline. Therefore at each time step the flood resulting from the rainfall already measured can be forecast. Referring to Figure 11, by noon on the 12 July, assuming the dam to be in place, the three hour hydrograph would predict a flood at Bruton by 14.30 hours. This is a warning time of 2.5hr. This is far superior to the previous system of issuing a warning based on the rate of rise of the water level behind the dam – giving at most about 90 minutes warning. In a very severe storm under the old system, this could be reduced to about one hour.

Whilst the model has given some confidence in the case of floods which have occurred in recent years (Table 1), it is important to know how well it performs in more extreme conditions.

On 28 June 1917, Bruton experienced one of the worst ever recorded storms in the UK (British Rainfall, 1917). A total of 243mm in about eight hours was recorded at Bruton, with an unofficial measurement of 228mm at Brewham. The profile of rainfall intensity is fairly well established (NERC, 1975) and together with an estimation of the antecedent SMD at the time of 42mm, the flood hydrograph was simulated (Figure 12). The peak discharge was estimated in the field from a scour mark (Figure 13). This concurs well with eye-witness reports and the timing of the duration of the flood which came from a newspaper report. Combined with the wrack mark surveys of the 1979 and 1982 floods an estimate of 178m3s-1 compares well with that produced by the model. However, the 1917 flood is less than half the estimated probable maximum flood (PMF) (Clark, 1996, 1997), of about 500 m3s-1.

In view of this situation, the model was tested under a range of both rainfall and soil moisture deficit conditions. This is an example of a sensitivity test of the model and is an important part of the modelling process (Bergstrom, 1991).

The critical rainfall duration for the upper Brue is four hours, which includes the reservoir lag of about one hour or less in the event of the PMF. Figure 14 shows the results of the sensitivity tests wherein the dramatic effect of the higher rainfall depths on peak discharge is apparent. The effect of SMD is also important making it imperative to have reliable data for soil moisture. In the author’s experience, measurement of SMD using microlysimeters offers a much more realistic and safer data input than does using equations dependent on measurements of wind speed, radiation and relative humidity as used in the MORECS system. During the summer, when flood producing convective storms occur, values of SMD as shown by the lysimeters are typically 50mm lower than those produced by MORECS. Referring to Figure 14, a rainfall intensity of 90mm in four hours and an estimated SMD of 70mm according to MORECS, would not be expected to produce overflow from the Bruton dam, where the critical discharge is about 60m3s-1 However, if the correct SMD was only 20mm then a completely different outcome would be expected. This type of scenario is the more common situation as shown by the slope of the lines of equal discharge where SMD is most critical.

The probable maximum precipitation of four hours (Clark, 1995, 2002b) of 270mm gives a peak discharge of just over 500m3s-1. This is about the same as previous estimates based on historic flood frequency analysis (Clark, 1996).

Reservoir flood routing
In order to produce a flood forecast the flood has to be routed through the flood detention reservoir. A revised stage volume relationship was produced in 1996 (Clark, 1996), and largely confirmed by a later survey (Babtie, Brown and Root 2003). Figure 15 shows the stage-storage relationship which is based on a cross valley survey of 13 sections.

The culvert which passes through the dam is 1.635m diameter and was designed to pass 20m3s-1 when under full head conditions. Although the trashrack has been designed according the standard guidelines (Babtie, Brown and Root, 2003) it is not true to say that there is no impedance to flow. Considerable blockage of the trash rack has been observed but no quantitative data has been collected to clarify the situation. As a result, on 6 February 2004 the author measured the flow just downstream of the dam. The result was 4.79m3s-1. The head of water behind the dam was 1.01m above the soffit. This is associated with a discharge of 6.9m3s-1 which gives a 30% blockage for that particular situation:

% blockage = 100 [(Expected flow – actual flow)/ Expected flow]

During a major flood the extent of blockage could increase so that further measurements will have to be made in the near future. However, for the purposes of reservoir routing a second set of flow measurements were made in order to establish a rating equation for the culvert:

Q = 2.202 H0.81909

Where Q = discharge m3s-1; H = head above base of culvert m.

Routing procedure
For a given time interval ∆t there is a balance between the rate of inflow, and the sum of outflow and change in storage:

I∆t = O∆t+ ∆S

Where I = inflow, O = outflow, ∆S = change of storage.

Reservoir routing involved calculating the averages of these parameters for each time interval. Thus if the subscripts 1 and 2 are used to show the start and finish of each time interval, the equation can be rewritten:

[(I1 + I2)/2] = [(O1 + O2)/2] + [(S – S1)/∆t]

This can be rearranged to enable the unknown value of the outflow to be calculated:

[S2/∆t + O2/2] = [S1/∆t – O1/2] + [I1 + I2]/2

The first two expressions in brackets are related to the rate of outflow discharge namely the rating equation of the culvert above. The height of water in the reservoir is associated with a storage volume and an outflow discharge. Table 2 shows these variables for the Bruton detention reservoir at a time interval of 30 minutes.

Flow over the spillway will take place when the head exceeds the retention level. This is calculated using the normal weir equation for use in a high head situation:

Q = CB H1.5

Where H = head (m), B = width (m), C = 1.7 in this case. Note here that Babtie used a value of B = 50m which is too big by 2.5m.

Figure 16 shows the relationship between the storage outflow functions and the rate of outflow.

When the 1917 flood is routed through the reservoir, flooding takes place at Bruton about five hours later than if the dam was not present. The peak flow is reduced by about 20m3s-1 and the duration of the flood reduced by about seven hours. Figure 12 shows the inflow and outflow hydrographs. The outflow hydrograph does not, however, include the additional flows between the dam and Bruton, a catchment area of about 3km2.

This paper has shown the importance of both local catchment data and historic flood information in the production of a real time flood forecasting model. Details of the warning issued and the human response to that warning are described elsewhere (Clark, 2004).

There are several points arising from the present approach to flood forecasting. First, the use of measured soils data has to be superior to the HOST (Boorman, Hollis, and Lily, 1995) soil classification which is used in the Flood Estimation Handbook (IOH, 1999). Unless rainfall intensity data are an integral part of the estimation of percentage runoff, then it is hard to see how flood forecasting can be expected to achieve reasonable results under extreme conditions in the future. Very rare floods have been shown to produce a percentage runoff beyond the range of the FEH (Clark, In Press). In some studies only one value of Ksat is used for an entire catchment (Mertens et al, 2002). Given the complexity and variety of soils in the UK this can hardly be realistic, especially when distributed modelling is performed.

Second, the time to peak has been estimated from historic storms. Rapid runoff means a short time to peak and higher peak discharges all else being equal. Although ICE (1996) has advocated a reduction of the standard time to peak by one third for estimating the PMF, hard evidence for this is wanting (Lowing Pers Com.) although reductions during high intensity storms do occur (Ashfaq and Webster, 2000). In the present paper the time to peak does reduce in high intensity storms, but the rate of reduction slows down as the intensity becomes very high.

Third, there is a need for accurate and representative rainfall data. Well known for its variability in time and space, rainfall can be measured by radar, but its accuracy is not very good, especially during more intense storms (Clark, 2000) and where false reflections from the blades of wind turbines make the job of calibration more difficult. Even without this added complication, it was concluded, from the upper Brue HYREX study, that the comparison between radar and raingauge based estimates exclude cases where the radar performs very badly (Wood et al 2000). Therefore the results of the present forecasting model will always be dependent on the quality of the rainfall data.

Fourth, the effect of a partial blockage of the trash rack on the effectiveness of the flood detention reservoir should always be measured. In a severe flood, conditions may arise which have not been witnessed in the past, leading to a more rapid filling of the reservoir and a higher flood risk downstream.

A fundamental weakness of many modelling approaches described by Bevan (2000) is the reliance on long established equations which may not suit the catchment under study. The objective of flood warnings is to save people’s lives; this demands the use of real world data and the determination to gather such information where none exists.

Implications for dam safety
At present there is concern that some dam spillways may be underdesigned (Millmore, 2003). For many it is not clear how seriously this concern should be taken. Application of a realistic flow model in which the percentage runoff and time to peak are both based on locally gathered data eliminates a great uncertainty in estimating the PMF. The only remaining uncertainty in estimating the PMF is the PMP. Recently, the August 2004 Boscastle flood in Cornwall, UK, showed that the FEH (IOH, 1999) rainfall model has severe limitations. The peak rainfall depth during the Bocastle storm was 200.3mm at Otterham. The FEH gave a return period in excess of 1900 years for 200mm in 24 hours. A more realistic result comes from the four-parameter model of Clark (1991), of just over 200 years. Since there was a similar flood, though lacking detailed rainfall records, which took place in June 1957, it appears that the lower estimate of the Boscastle 2004 storm is the more realistic.

By using the old estimates of PMP (NERC, 1975) the peak discharge for the upper Brue dam site is 340m3s-1. This is well in excess of the spillway design flood, but only a little in excess of the Great Flood of 1768. Therefore, in the present case, whichever way the problem of dam safety is viewed, the use of either the standard FEH UH package, and/or the old PMP values, gives results which are at odds with the historic record.

Author Info:

Dr Colin Clark, Director, Charldon Hill Research Station, Bruton, Somerset BS10 0BJ, UK. Email:

The author would like to thank Adrian Wynn of the Environment Agency SW region for useful discussions


Table 1
Table 2