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TRANSPORT and ROAD RESEARCH LABORATORY Department of the Environment TRRL LABORATORY REPORT 706 THE TRRL EAST AFRICAN FLOOD MODEL by D Fiddes Any views expressed in this Report are not necessarily those of the Department of the Environment Environment Division Transport Systems Department Transport and Road Research Laboratory Crowthorne, Berkshire 1976 ISSN 0305–1 293 CONTENTS Abstract 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Introduction Catchment Models 2.1 The unit hydrography 2.2 Conceptual models 2.3 Analogue models The TRRL Flood Model 3.1 Description of model 3.2 Finite difference equations for channel flow 3.3 Stability 3.4 Model calibration and proving 3.5 Error functions 3.6 Model calibrations Generalisation of Flood Model 4.1 Form of model 4.2 Initial retention (~ 4.3 Contributing area coefficient (CA) 4.4 Catchment lag times (K) 4.5 Base time Summary of Design Method Discussions and Conclusions Acknowledgements References Appendix 1. Derivation of Finite Difference Equations Appendix 2. Worked Example @CROWN COPYRIGHT 1976 Page 1 1 1 1 2 2 3 3 4 5 6 6 6 7 7 8 9 9 10 11 12 13 13 22 24 Extracts from the text may be reproduced, except for commercial purposes, provided the source is acknowledged THE TRRL EAST AFRICAN FLOOD MODEL ABSTRACT Four years ofdata from 13small representative rural catchmentsin Kenya and Uganda were analysed to develop improved methods of flood estimation forhighway bridges and culverts. Dueto the short period of record and the very quick response time of the catchments, Unit Hydrography techniques were found inappropriate. A technique which made better use of limited data, therefore, had to be developed. Rainfall and runoff were fitted to a simple three parameter conceptual catchment model. The model was then used to predict the 10 year flood using a 10 year design storm. A simple technique is then developed for predicting the peak flow and base time of design hydrography for ungauged catchments. 1. INTRODUCTION A large proportion of the total cost of building a road in East Africa is for the construction of bridges and culverts to cross streams from small catchmentsl. Wereas most of the larger rivers in East Africa have flow measuring stations, very few smaller streams are so equipped. Design methods must therefore be based on rainfall-runoff models. Very few data are available for the development of suitable flood models. In 1966 the Kenya and Uganda Governments and the UK Transport and Road Research bboratory co-operated in a project to instrument 12 representative catchrnents, results from which could be used to develop improved flood design methods. The choice of location and instrumentation of these catchments is described elsewhere 2. This report describes the programme of analysis of results and the design method that was developed. It was presented as a paper to a Hood Hydrology Symposium in Nairobi, Kenya in October 1975 which was jointly sponsored by the Economic Commission for Africa, the East African Community and the Transport and Road Research bboratory. 2. CATCHMENT MODELS A number of possible rainfall-runoff models were considered, 2.1 The unit hydrography A unit hydrography is a hydrography of unit volume resulting from a rainstorm of unit duration and specified areal pattern. Hydrography for other storms of similar areal pattern can be constructed by superimposing hydrography, suitably offset in time with ordinates proportional to the flow assumed to result from the rainfall in each unit time period. It will be seen that the unit hydrography simply distributes the runoff in time. The volume of runoff must be estimated separately and this generally involves deriving a rainfall-runoff correlation. The simplest rainfall-runoff correlation is a plot of average rainfall over the catchment and resulting runoff. Typically the relationship is slightly curved indicating a somewhat higher percentage runoff with higher rainfall totals. The scatter of the points about the regression line is often large but may be decreased by introducing additional variables such as antecedent catchment wetness and intensity of rainfall. The difference between the total rainfall and runoff is assumed to be water held on the surface of the ground or vegetation, which subsequently evaporates, or infiltration which does not appear in the stream flow until the storm runoff has effectively finished. These losses tend to be greater at the start of the storm, although to simplify the estimation of the rainfall input to the unit hydrography they are often assumed to occur at a constant rate. Their acceptance as standard practice shows that such methods have been remarkably successful, considering their simplicity, but they have two drawbacks for the present study. (a) An adequate rainfall-runoff correlation requires a large number of storm data including many producing high flows. (b) Storms that can be used to derive unit hydrography, ie high intensity storms of unit duration, are rare, particularly on small catchments where the unit time is short compared to the typical storm length. A method where more effective use can be made of the limited data that can be collected in a few years was therefore required for this study. 2.2 Conceptual models . . . 1 A water balance for a catchment maybe envisaged of the form R= P–E– AM– AG where R = runoff P = precipitation E = evaporation A M = change in soil moisture A G = recharge to deep storage If records are available for precipitation, models can be set up to compute a running sequence of values for the other terms on the right hand side of the equation thus giving a running sequence of runoff values which can be suitably distributed in time. The best known example of this type of model is the Stanford watershed mode13. Complicated computer programs and considerable data for calibration purposes are required which make this approach inappropriate for the present study. 2.3 Analogue models The similarity between the response of a catchment to storm rainfall and the flow through a series of reservoirs has been noted by many workers. Zoch suggested the concept of linear reservoir routing as early as 2 1934. The response of cascading flows through a number of linear reservoirs both in series and parallel was studied by Sugawara and Marayamas. The purpose of this study is to develop a model which can be applied to ungauged catchments using very few measurements of catchment characteristics such as area, slope, soil type. The model must therefore be simple and have a limited number of parameters. The lumping or averaging of certain catchment mechanisms inherent in a simple reservoir model is thus not a disadvantage. ~ls was therefore considered to be the most suitable type of model for the East Africa study. 3. THE TRRL FLOOD MODEL 3.1 Description of model The model is made up of two parts. A linear reservoir model is used for the period between the rain hitting the ground surface and the floodflow entering the stream system (the “land phase”) and a finite difference routing technique is used for the passage of the flood wave down the river to the catchment outfall. A reservoir is said to be linear if the outflow (q) is related to the water stored in the reservoir by the linear relationship q=K 1s . . . 2 where S is the reservoir storage K is the reservoir lag time The flow from a linear reservoir with zero inflow decreases exponentially. The lag time maybe conveniently thought of as the time required for the recession curve to fall to approximately + of its initial value. The reservoir storage contains the rain falling during the storm which contributes to the flood hydrography. This can either be surface runoff or rapid subsurface runoff (interflow). Runoff does not occur uniformly over a catchment. Parts of the catchment are less permeable than others due to variation in soil type, or, in low lying areas, to slower drying out after previous rain. A uniform runoff coefficient can therefore be misleading and recently the concept of catchment contributing area (CA) has been substituted. Between storms, drying of the soil takes place due to the combined effects of evaporation and plant transpiration. The drying takes place principally in the surface layers. This accounts for the high infiltration rate at the start of a storm and also for the lack of subsurface runoff until this loss has been made good by infiltration. This is modelled by a storage called initial retention (~, which must be filled before flood runoff occurs. The simple land phase model may therefore be summarised as: (a) firly rain fills the initial retention (~. Runoff at this stage is zero. (b) Subsequent rain falling on the parts of the catchment from which runoff will occur (CA) enters the reservoir storage. 3 (c) Runoff is given by equation (2). This simple model, and derivations from it, were compared with the unit hydrography using data from a small catchment 7. The results were very promising. With small catchments the attenuation of the flood hydrography during travel down the stream system is ne~igible. For larger catchments it can be considerable. It has been suggested that these translation effects can be allowed for by varying the reservoir lag time (K)s. This was attempted when data from the larger catchments were analysed but poor results were obtained. In addition, if the value of the lag time is dependent on catchment size, values obtained for small catchments are difficult to apply to large catchments. The approach finally adopted was to divide the catchment into a number of sub-catchments, the runoff from which was simulated using the land phase model. The translation of this runoff down the stream system to the catchment outfall was modelled using a modification of the finite difference technique developed by Morgali and Linsley9. 3.2 Finite difference equations for channel flow h exact mathematical solution of the equations describing the generation of a flood hydrography in a stream system is not possible. Finite difference techniques can be used to get approximate solutions. The principle by which they operate is that very simple equations are adequate to describe the flow over very short distances and times. It follows that accurate solutions by this method forwhole catchments would involve many repetitions of the calculations. This would be very tedious if done by hand but is quite feasible by digital computer. ~agrammatic view of a reach of the stream: SECTION OF STREW (2) (1) Lines 1 and 2 represent the water surface at times separated by a time increment At. L, M and R are subscripts which define three stations on line 1. P is the subscript of the middle station on line 2. 4 If the depth and velocities at the stations on line 1 are all known together with the lateral inflow from the linear reservoir, by moving progressively down the stream, the depths and velocities along line 2 can be computed using the momentum and continuity equations. The derivation of the appropriate finite difference equations is given in Appendix 1. From the diagram it will be seen that values for the extreme upstream and downstream stations are not calculated and have to be estimated. For the uppermost reach the upstream velocities and depths are assumed zero. For lower reaches the upstream values are assumed to be the same as the downstream values for the previous reach. At junctions the following equations apply A1+A2=A3 Q1+Q, =Q3 where A = cross sectional area Q = flOW and subscripts 1, 2 and 3 refer to the three reaches forming the junction. The values for the downstream station of each reach are calculated by extrapolation of the values for the two immediately upstream stations. A flow diagram for the computer program is shown in Fig 1. 3.3 Stability Finite difference equations are only approximations and the accuracy of the solutions is dependent on the incremental length and times chosen. Morgali and Llnsleyg show that the equations will become unstable and errors will be introduced unless the following equation is satisfied. where V = Y= g= velocity depth acceleration due to gravity 3 Ax and At are the chosen distance and time increments. Typical values for the initial incremental distance and times were 200m and 30s. The instability takes the form of oscillations or waves in the recession part of the predicted hydrography. In the discussion of Morgali and tinsley’s paper 10 several people pointed out that satisfying equ (3) did not automatically ensure that the equation would be stable. This was found to be correct. In the analysis, instability did occur on occasions, even if equ (3) were satisfied. This was particularly so for very steep catchments or for catchments with very large reservoir lag (K) values. Tests were therefore incorporated in the program to establish when instability was occtiing. The run was terminated and repeated with the time increment At reduced by a factor of two. 3.4 Model calibration and proving To run the program for a given storm the required input is: (a) The recorded rainfall for each 15 minute period (b) The recorded streamflow hydrography ordinates (c) For each reach - area, length, slope, K, Y, CA, Manning’s “n”, Ax, At. The model works on At time intervals but assumes that the rainfall intensity is constant over a 15 minute period. For a singe gauge, rainfall can be measured to a finer time scale, but when several raingauges are being averaged this is unjustified. 3.5 Error functions For each combination of values of the parameters, a hydrography is predicted. The ordinates of this hydrography are compared with the ordinates of the recorded hydrography and an error function calculated. A small error function indicates close agreement between predicted and recorded hydrography. The error function adopted was the usual sum of the squares of the ordinate differences. ERF = ~ (y – YO)2 where y is a predicted ordinate and y. is a recorded ordinate. To avoid undue significance being given to the larger storms a second error function was calculated, This compared the mean weighted ordinate error to the mean recorded ordinate. Per cent ordinate error = m ,,,, 70 where n = number of ordinates To. = mean of recorded ordinates 3.6 Model calibration Approximately 4 years of data were available for each catchment. For most catchments these included a number of large storms, but, inevitably with such short periods of research, on some catchments only relatively small storms were recorded. The model was run for each large storm on a catchment and for a variety of values of the parameters K and CA. The combinations giving the best agreement between recorded and predicted floods are listed in Table 1. 4. GENERALISATION OF FLOOD MODEL 4.1 Form of model The flood model had now been calibrated for each catchment. To develop a general flood model the differences in catchment response to rainfall shown by the individual catchments was next examined. As the recorded storms varied in severity it was necessary to use the model for each catchment to simulate a flood of a known recurrence interval before a comparison could take place. A 10 year flood was selected for comparison. ~s was simulated by using a 10 year storm profile (see ref 11) and appropriate values for the model parameters K, CA and Y (see 4.2 – 4.4). The results appear in Table 2. Hydrography, recorded and predicted for the largest storms and for 10 year floods are shown for each catchment in Figs 2 – 13. Once estimates of the parameters Y, CA, and K have been made for a catchment a design flood could be estimated by routing a design storm through the computer program. This can be a time consuming process and for many purposes a simpler technique is required. From CA and Y the volume of runoff from any given design storm can be calculated and if the hydrography shape can be related to the catchment lag time (K), the peak flow can dso be estimated. Many research workers have published “dimensionless hydrography” and it has been shown17 that in the United States these approximate closely to the equation . . . 4 L 4 where Q = discharge at time T after start of rise Qm = peak discharge Tm = time to peak The most widely used dimensionless hydrography is that of the US Soil Conservation,? Service l’. For arid areas Hickok et alls suggest a somewhat more peaked shape. For all of these the ratio of time to peak to base time are very similar. This was not found to be true for the East African catchments studied. For consistency the base time was assumed to be the time from 1 per cent of peak flow on the rising limb to 10 per cent of peak flow on the falling limb of the hydrography. Mfined this way, the ratio base time : time to peak is approximately 3.0 for the US hydrography. For the East African catchments it varied between 2.7 and 11.0. The use of a sin#e hydrography based on time to peak was therefore not appropriate. A much more stable ratio was found to be the peak flow (Q) divided by the average flow measured over the base time (~) (Peak How Factor F) 7 . . . 5 Q This is the factor used by Rodier and Auvraylg in West Africa. For very short lag times (K ~ 0.2h) F was 2.8 * 10%. For all lag times greater than 1 hour, F was 2.3 * 10%. These figures hold true for the catchment results and were confirmed by a simulation exercise in which area, slope, lag time and contributing area coefficient were systematically varied. The peak flow can therefore be simply estimated if the average flow during the base time of the hydrographycan be calculated. The total volume of runoff is given by: RO = where P = Y= CA = A= (p-~cA.A.lo3(m3) storm rainfall (mm) during time period equal to the base time initial retention (mm) contributing area coefficient catchment area (kmz ) . . . 6 If the hydrography base time is measured to a point on the recession curve at which the flow is ~ th of the peak flow then the volume under the hydrography is approximately 7 per cent less than the total runoff given by equ (6). The average flow (c) is therefore given by: Q= 0.93 . RO 3600 . TB . . . 7 where TB = hydrography base time (hrs) Estimates of Y and CA are required to calculate RO and lag time K to calculate TB. These will now be discussed in turn. 4.2 Initial retention (Y) For the model fitting to the storms listed in Table 1 an appropriate initial retention was calculated from the balance of evapotranspiration and rainfall since the last storm to give significant runoff. This procedure could not be applied to the Mudanda catchment. Here a value of approximately 5mm was found appropriate. This is typical of arid zone catchments12. The probability of the soil on a catchment anywhere in East Africa being at field capacity has been studied by Huddart and Woodward 13. For convenience their results are reproduced in this paper (Table 3, Figs 14 and 15). It will be seen that in the wet zones the 7 day antecedent rainfall easily exceeds the potential evapotranspiration during the same period. In the dry zones the figures are much closer but only in Western Uganda is there a high probability that the surface layers will be below field capacity. Here as in the semi arid zone, a 5mm initial retention is recommended for design purposes. Elsewhere assume zero initial retention. 8 4.3 Contributing area coefficient (CA) Wen the catchrnent surface is very dry, runoff is small and only occurs from areas very close to the stream system. For storms following a wet period a larger area contributes and larger volumes of runoff occur. If the catchment were sufficiently wet, the whole area would contribute and the value of CA would approach unity. However, except on very small solid clay or rock catchments there is a practical upper limit to CA which is well below unity. Evidence for this from the USA has already been referred to6 and there is further confirmation in a recent TRRL study 14. For simplicity it is assumed that the contributing area coefficient varies linearly with soil moisture recharge until the soil reaches field capacity when the limiting value of CA is attained. Similar assumptions are made in the recent UK Flood Studies Report15. Four factors influence the size of the contributing area coefficient. These are soil type, slope, type of vegetation or Ianduse (particularly in the valley bottoms) and catchment wetness. The network of catchments had been selected to cover the range of these factors to be found in East Africa. The results could therefore be used to give indications of their effect on CA. The effect of slope and sod type was studied by comparing the results of the catchments with grass cover and the storms falling on soil at field capacity. The effect of antecedent wetness was studied by comparing the runoff volumes resulting from storms. occurring at different stages of the rainy season. The reduction in value of CA was assumed to vary linearly with the soil moisture deficit. Using Tables 1 and 3, appropriate values of the reduction in CA for design conditions for the various zones were arrived at. The effect of landuse was calculated by comparing the recorded volumes of runoff with those that would have occurred with a standard grassed catchment. The design value of the contributing area coefficient is therefore given by: CA ‘es. cw. cL . . . 8 where Cs = the standard value of contributing area coefficient for a grassed catchment at field capacity Cw = the catchment wetness factor CL = the land use factor Tables for estimating these factors are given in Tables 4,5 and 6. 4.4 Catchment lag times (K) In Table 2 it will be seen that there is a very large range of lag times (K). Attempts were made to obtain correlation of K with various catchment characteristics such as overland slope, contributing area and drainage density, but the only factor found to show a strong relationship was vegetation cover. The same conclusion was drawn in a similar study by Bell and Om Kar, involving results from 47 small catchments located throughout the United States16. 9 The appropriate value of lag time can be estimated by reference to Table 7. In assessing which category to place a given catchment it should be remembered that generally only small areas either side of the stream are contributing to the flood hydrography. It is these areas, therefore, which must be assessed. 4.5 Base time The method of estimating the base time was derived from the study referred to in section 4.1. It is made up of three parts: (a) The rainfall time (b) The recession time for the surface flow (c) The attenuation of the flood wave in the stream system. The rainfall time (Tp) is the time during which the rainfall intensity remains at high level. This can be approximated by the time during which 60 per cent of the total storm rainfall occurs. Using the general East African depth duration equation. I= a . . . 9 (T t ;)n where I = intensity T = duration a and n are constants The time to give 60 per cent of the total storm rainfall is given by solving the equation 0.6 = ; ( 2433 )n T t 0.33 . . . 10 Values for the various rainfall zones of East Africa are given in Table 8. The time for the outflow from a linear reservoir to fall to ~th of its initial value is 2.3K where K is the reservoir lag time. The recession time for surface flow is therefore 2.3K. In the simulation study, values for base time were calculated for various areas, slopes, lag times and contributing area coefficients. Knowing the rainfall time and the surface flow recession time, the additional time for flood wave attenuation (TA) can be found by difference. It was found that this could be estimated from the equation TA = 0.028 L . . . 11 Q% ~% 10 where L = length of main stream (km) o = average flow during basetime (m3 /s) S = average slope along mainstream The base time is therefore estimated from the equation TB = Tp t 2.3K t TA . . . 12 The average flow (~) can be estimated. It will be noted that ~ appears in equ (11) so an iterative or trial and error solution is required. If initially TA is assumed zero, two iterations should be adequate. Knowing ~ and F the peak flow is calculated using equ (5). 5. SUMMARY OF DESIGN METHOD The steps involved in estimating the peak flow for a design storm are as follows: (a) ,’ (b) (c) (d) (e) (0 (g) (h) (i) 6) (k) hcate catchment on a large scale map and measure catchment area, land slope and channel slope. The land slope is estimated by superimposing a grid over the catchment and measuring the minimum distance between contours at each grid point. From these slopes are calculated and averaged to give the mean catchment value. The channel slope is the average slope from the bridge site to the uppermost part of the stream. Where information is sparse this may be taken as 85 per cent of the distance to the watershed. From site inspection establish catchment type in Table 7 and hence lag time (K). From site inspection or using Fig 15 establish soil type and with land slope estimate the standard contributing area coefficient (Cs) in Table 4 Using Fig 14 fix antecedent rainfall zone. Check in Table 3 to see if zone is wet, dry or semi-arid. Estimate catchment wetness factor from Table 5. From site inspection decide on type of vegetative cover, paying particular attention to areas close to the stream. Using table 6 estimate land use factor (Cd. Contributing area coefficient (CA) is given by: CA= CS. CW. CL If antecedent rainfall zone in (d) above is semi-arid or West Uganda, initial retention (~ is 5mm. For all other zones, Y = O. Using Fig 16 and Table 8, estimate rainfall time (Tp). Using methods outlined inref(11) calculate the design storm rainfall to be allowed for during time interval TB hours (P mm). The volume of runoff is given by equ (6). RO = CA. (p–~. A.103(m3) 11 ~) The average flow is given by ~ = 0.93 . RO 3600 . TB . . . 13 (m) Recalculate base time TB = Tp + 2.3K + TA 0.028 L where TA=— ~%~% (n) Repeat steps Q) to (m) until C is within 5 per cent of previous estimate. (o) Design peak flow (Q) is given by Q=F. ~ where peak flood factor (F) is: F = 2.8 K less than 0.5 hour F = 2.3 K more than 1 hour Table 9 indicates the accuracy achieved in using the method for the network catchments. A worked example is given in Appendix 2. 6. DISCUSSION AND CONCLUSIONS A reservoir analogue model has been developed which can be used to improve the usefulness of limited catchment data. This has been used to derive a simple method of flood prediction for small rural catchments in which the most critical factors are landuse and soil type. The catchments used to develop the method covered a range of landuse and soil type but inevitably could not cover all the combinations to be found throughout East Africa. In areas not already covered, very simple on site measurements by engineers would greatly improve the accuracy and usefulness of the design technique. If a note is kept of how many hours a flood resulting from a notable rainstorm lasts, an estimate can be made of the appropriate lag time to apply using equation 12. This is the factor that has the greatest influence on peak flow. The most commonly specified recurrence intervals for small hydraulic structures are 5 or 10 years. The method has been designed to provide these figures easily. Sometimes, for larger structures, longer recurrence intervals are specified. Using the method to estimate these introduces two elements of uncertainty. (a) The shape and volume of the appropriate storm profile. 12 (b) The catchment response to exceptional storms. Of these the second is the more problematical. There are a number of other problems in designing structures for very unusual floods. These include: – (a) If the total flow is channeled through a bridge opening very high scouring velocities occur which can endanger the pier foundations. (b) Very high flows uproot trees or islands of papyrus. Very often structure failure is due to spans being blocked by such debris rather than to the flow of the water itself. For the size of catchment being considered here (up to 200 km2 ) high flood flows only last for a few hours. If the structure itself can be safeguarded, designing for flows of greater than 10 year recurrence interval could we~ be uneconomic. From the above considerations it is recommended that where possible designs are standardised at a 10 year recurrence interval with provision for larger flows to bypass the structure with safety. An example would be to provide approach embankments at a level lower than the bridge deck, thus giving a safe flood spillway and effectively limiting the velocity of flow through the bridge opening. 7. ACKNOWLEDGEMENTS The work described in this report forms part of the programme of the Environment Division (Head – Mr L H Watkins) in the Transport Systems Department of the UK Transport and Road Research bboratory and was carried out on behalf of the UK Ministry of Overseas Development. The equipment in the experimental catchments was installed and operated for TRRL by the Water Development Departments of the Kenya and Uganda Governments. The help and assistance of the Director and Commissioner are gratefully acknowledged. 1. 2. 3. 4. 5. 6. 8. REFERENCES TRURAN, R J. The needs of the consulting engineer. ECA Hood Hydrology Symposium, Nairobi 1975. FIDDES, D and J A FORSGATE. Representative rural catchments in Kenya and Uganda. Department of the Environment, TRRL Report LR 318. Crowthorne, 1970 (Transport and Road Research Laboratory). CRAWFORD, N H and R K LINSLEY. The synthesis of continuous streamflow hydrography on a digital computer. Stanford University Technical Report No. 12.1962. ZOCH, R T. On the relation between rainfall and streamflow. Monthly Weather Review. Vol 62. pp 31 5–322. 1934. SUGAWARA, M and F MARAYAMA. A method of prevision of the River Discharge by means of a rainfa~ model. Publ No 42, IASH Symposium Darcy. Vol 3. pp 71–76. 1956. BETSON, R P. What is watershed runoffl Journal Geophysical Research, Vol 69. April 1964. 13 7. FIDDES; D. A reservoir model alternative to the unit hydrography for flood estimation. Department of the Environment, TRRL Report LR554. Crowthorne, 1973. (Transport and Road Research bboratory). 8. LAURENSON, EM. A catchment storage model for runoff routing. Journal of Hydrology. Vol 2. pp 144–163. 1964. 9. MORGALI, J R and R K LINSLEY. Computer analysis of overland flow. Proc ASCE (Hyd Div). HY3. pp 81–100. May, 1965. 10. Discussion of Ref (9). Proc ASCE (Hyd Div) HY6 pp 224–238. Nov 1965. 11. FIDDES, D. Design storms for East Africa ECA Flood Hydrology Symposium. Nairobi 1975. 12. FOGEL, M M and L DUCKSTEIN. Prediction of convective storm runoff in semi-arid regions. Symposium on Representative and experimental watersheds. IASH Publ No 96. pp 465–478. New Zealand 1970. 13. HUDDART, L and P G WOODWARD. The prediction of antecedent catchment wetness. ECA Flood Hydrology Symposium. Nairobi 1975. 14. FORD, W G. The adaptation of the RRL Hydrography method for tropical conditions. ECA Flood Hydrology Symposium. Nairobi 1975. 15. NATIONAL ENVIRONMENTAL RESEARCH COUNCIL. Flood Studies Report. London 1975. 16. BELL, F C and S OM KAR. Characteristic response times in design flood estimation. Journal of Hydrology. Vol VIII. No 2. pp 173–196. 1969. 17. LINSLEY, R K, M A KOHLER and J L PAULHUS. Hydrology for engineers. McGraw-Hill. 1958. 18. HICKOK R B, R U KEPPEL and B R RAFFERTY. Hydrography synthesis for small arid watersheds. Agricultural Engineering. pp 608–615. October 1959. 19. RODIER, J A and C AUVRAY. Preliminary general studies of floods on experimental and representative catchment areas in tropical Africa. IASH Symposium of Budapest. 1965. Vol 1. pp 22–38. 14 TABLE 1 Details of storms and summary of analysis Catchment TIWI MUDANDA MI GWANI KAJIADO ESERET KIAM BU I KIAMBU 11 SAOSA BARABILI MUNYERE SUB RUBAARE LUGULA Date of Storm 21/4/70 9/5/72 11/5/72 6/7/12 Z6/11/69 14/4/71 1/12/71 2/12/71 4/1 1/72 11/11/72 19/11/72 i4/1 1/69 i3/11/71 2/12/71 !0/12/71 15/11/72 2/6/10 1/6/71 !2/12/71 !5/12/71 23/4/70 16/2/73 19/2/73 22/2/73 3/12/68 1/5/71 16/5/71 3/5/72 11/5/72 3/6/72 16/3/58 20/5/60 4/7/60 12/5/61 !9/12/62 1 6/8/66 19/8/70 28/8/71 18/7/71 6/10/71 2/11/71 8/11/71 4/3/72 25/4/70 26/2/72 1/11/69 27/3/70 1/2/72 ~ainfa mm 104.7 58.0 98.9 27.4 14.9 27.5 17.2 20.5 15.8 10.8 22.1 60.4 96.7 66.2 35.4 53.9 24.2 26.4 71.5 64.6 58.2 43.5 67.0 20.6 63.2 45.0 37.7 16.1 12.1 53.6 116.6 25.0 32.2 28.1 45.4 49.5 74.8 01.8 41.1 15.5 24.2 52.3 17.3 89.6 05.2 28.7 42.9 30.8 Peak flow n3 see-l 0.289 1.428 5.836 0.354 0.680 0.810 0.370 0.710 0.540 0.340 0.370 59.770 41.930 59.770 75.640 39.320 0.771 0.431 2.011 4.181 2.560 0.960 3.260 0.940 0.097 0.018 0.038 0.053 0.068 0.435 0.642 0.140 0.250 0.156 0.140 0.171 0,235 0.751 0.310 0.190 0.300 0.680 0.110 3.200 8.300 0.364 0.767 0.116 Rainfall in prel 30 Days mm Record incom Negligible Rai 74.3 277.5 11.2 108.8 6.2 48.9 89.6 20.8 66.8 74.9 129.2 12.0 155.4 238.4 247.4 85.0 88.5 74.7 140.1 113.3 33.8 104.9 215.1 338.0 140.4 76.5 111.2 102.8 0.0 0.0 0.0 0.0 31.5 mm 0.0 199.9 200.4 8.4 111.5 i 36.0 158.2 76.2 195.9 113.2 155.4 241.7 124.8 3US 7 Days mm ete but all 38.8 242.3 11.2 69,8 1.5 19.6 60.3 14.4 10.7 0.9 89.8 9.0 65.3 31.9 75.5 41.4 14.9 37.0 90.9 Two storms on 23/4 0.0 25.6 100.8 189.5 100.7 117.2 67.2 49.2 18.9 0.5 0.0 0.0 0.0 0.0 n 28/12/62 0.0 63,9 56.6 8.4 32.7 48.8 29.8 19.7 88.2 11.9 51.8 37.1 22.8 hsume, Initial tetentio mm 95 25 0 15 3 7 5 8 5 4 7 3 30 3 3 3 15 15 32 0 0 5 0 5 0 0 0 0 0 35 0 0 0 0 0 0 0 0 15 2 8 10 9 0 15 0 0 0 Lag Time { Minf 95 80 70 100 3 3 3 3 3 3 3 30 20 20 20 20 140 140 35 120 5 5 5 5 500 500 400 200 175 150 450 500 375 450 450 450 600 400 7 7 7 7 7 50 50 450 500 600 optimum Values for Modelling CA 0.050 0.050 0.090 0.050 0.081 0.080 0.047 0.105 0.067 0.067 0.037 0.280 0.430 0.380 0.250 0.350 0.380 0.130 0.100 0.380 0.065 0.029 0.059 0.062 0.035 0.006 0.016 0.017 0.022 0.080 0.030 0.035 0.035 0.030 0.022 0.014 9.030 ).065 ).047 ).052 ).061 ).120 ).036 ).02 1 ).038 ).140 ).280 ).060 Error ‘unctior 0.0359 0.9842 7.1360 0.2061 0.2000 0.1800 0.0200 0.1500 0.0800 0.0400 0.0200 I 1944.9 !6713.7 !4021.6 2692.1 5659.9 0.1846 0.0799 0.9810 4.5280 4.3300 0.2300 1.1800 0.3100 0.0033 0.0008 0.0017 0.0024 0.0019 0.1807 0.6090 0.0122 0.0124 0.0556 0.0056 0.0535 0.2855 0.3620 0.0100 0.0000 0.0200 0.1900 0.0000 8.1900 1.4000 0.2717 0.8892 0.0304 % Age Ordinate Error 50.4 69.6 98.7 47.5 75.5 79.2 59.7 84.4 85,6 90.2 54.2 67.1 57.0 50.2 51.6 33.5 22.3 30.8 39.6 17.0 114.6 56,6 36.7 81.1 12.3 43.0 34.0 26.6 20.1 35.2 34.2 16.3 11.9 32.8 9.5 39.5 45.3 20.6 35.3 44.0 73.5 65.4 38.6 52.5 56.5 37.1 24.0 27.9 Predicted Peak FIOW m3 see-l 0.286 1.263 5.717 0.363 0.680 0.810 0.370 0.700 0.540 0.330 0.380 163.190 233.440 158.230 72.590 135.480 0.767 0.439 2.034 4.074 2.400 0.950 3.580 0.950 0.100 0.012 0.031 0.055 0.071 0.383 0.643 0.144 0.247 0.167 0.135 0.151 0.203 0.815 0.300 0.190 0.290 0.670 0.110 3.160 8.150 0.363 0.803 0.137 15 TABLE 2 10 year flood details Time to Peak flow peak (h) (m’ /s) TP Q Average flow (m3 /s) Q Retension hg time (mm) (rein) Y K Contributing Area Coef Ca Base time (hr) TB Catchment Rainfall (mm) Tiwi I 102.5 0 70 5 5 0 50 0 120 0 5 0 500 0.09 0.20 6.00 1.60 1.35 I 5.66 2.70 2.10 Mudanda I 88.4 0.60 I 6.79 2.41 2.82 Migwani I 108.5 0.38 0.50 5.50 7.40 1.50 I 377.03 143.77 2.62 Kajiado 1.35 I 9.03 4.19 2.16 Eseret 0.06 3.00 0.85 I 2.95 1.04 2.84 Kiambu I I 80.3 0.04 22.75 2.25 I 0.14 0.06 1.63 2.25 2.18 * o 175 0 450 0 500- 0.15 9.75 1.65 I 3.55 0.04 0.07 20.50 2.00 I 0.63 0.27 2.34 22.00 2.00 I 0.67 0.28 2.42 Munyere sub I 81.4 5 17 0.08 0.03 1.50 4.18 + 0.43 1.81 2.56 5 2.56 I 50 0 I 500 0.30 21.75 2.00 2.27 0.95 2.39 Note: Base time is measured from 0.01 Q on rising limb to .,...-,,,-. — — — — 0 m m — 0 0 0 m. w m 0 m 0 0 F w — — 0 — Catchment slope Very Flat <1 .0% Moderate 1-4% Rolling 4-1 o% Hilly 10-20% Mountainous > 20% TABLE 4 Standard contributing area coefficients (Wet zone catchment, short grass cover) Well drained 0.09 0.10 0.11 0.12 Soil type Slightly impeded drainage 0.15 0.38 0.45 0.50 Impeded drainage 0.30 0.40 0.50 Note: The soil types are as in Fig 16 and are based on the soils map contained in the Handbook of Natural Resources of East Africa (see ref 13). 18 TABLE 5 Catchment wetness factor fiinfall zone Wet zones Semi arid zone Dry zones (except West Uganda) West Uganda Catchmetit wetness factor (Cw) Perennial streams I Ephemeral streams 1.0 1.0 1.0 1.0 0.75 0.50 0.60 0.30 TABLE 6 bnd use factors (Cd (Base assumes short grass cover) brgely bare soil Intense cultivation (Particularly in valleys) Grass cover Dense vegetation (particularly in valleys) Ephemeral stream, sand filled valley Swamp filled valley Forest 1.50 1.50 1.00 0.50 0.50 0.33 0.33 19 TABLE 7 Catchment lag times Catchment type I hg time (K) hrs Arid Very steep small catchments (slopes > 20%) Semi arid scrub (large bare soil patches) Poor pasture Good pasture Cultivated land (down to river bank) Forest, overgrown valley bottom Papyrus swamp in valley bottom 0.1 0.1 0.3 0.5 1.5 3.0 8.0 20.0 TABLE 8 fiinfall time (Tp) for East African 10 year storms Zone Index “n” Winfall time (Tp) (h) Inland zone 0.96 0.75 Coastal zone 0.76 4.0 Kenya-Aberdare Uuguru Zone 0.85 2.0 20 TABLE 9 Comparison between predicted 10 year floods using computer and short methods Assumed Catchment Parameters 10 YEAR PEAK FLOWS m Storm rainfall (mm) Catchment Area Lnd Channel CL Cw y(mm) k(hr) (km2 ) sl~ slope % Channel length (km) Ca Tiwi 122 0.09 0.75 I 1.00 I o I 1.0 I 6.7 2.2 I 1.2 5.08 2.67 5.7 I 6.6 Mudanda 89 105 0.10 0.38 1.50 / 1.00 I 5.0 \ 0.3 I 1.7 I 8.2 I 5.0 6.8 I 8.0 Migwani Kajiado 1.00 ] 1.00 I O I 1.0 I 83.5 I 3.0 I 1.3 19.05 68 0.50 1.00 1.00 0 1.5 3.6 8.8 2.7 0.50 1.00 0 0.1 3.2 22.0 9.2 3.43 Eseret 57 112 0.12 0.10 0.10 3.70 3.0 I 3.1 0.14 I Kiambu I 0.50 0.75 0 8.0 2.0 4.0 3.6 0.27 1.50 1.00 0 3.0 5.2 6.6 2.8 2.75 Kiambu 11 112 6.03 2.03 + Barabili 96 0.15 0.50 0.75 0 8.0 3.9 0.7 0.3 Munyere sub 1.00 0.60 5.0 0.1 0.5 27.0 23.5 Rubaare Lugula Saosa 54 0.11 0.83 + 1.11 1.12 4.6 4.3 2.27 1.74 65 0.10 1.00 I 0.30 I 5.0 I 1.0 I 13.7 I 6.0 I 4.9 6.99 103 103 0.45 0.11 0.50 I 0.75 I O I 8.0 I 3.1 I 9.0 I 0.7 2.29 0.33 I 0.75 I O I 8.0 I 6.8 I 12.0 .1 3.4 4.54 0.63 0.55 N APPENDIX 1 Derivation of finite difference equations u m 1 SECTION The equations are developed from the continuity and momentum equations. Continuity equation: A8V+VdA+aA=q ax ~ at (q = inflow per unit length) For simplicity the cros~section is assumed triangular. Therefore A = ~y2 ay Therefore a(y2v) t 2y ax = ~ ax Momentum equation Symbols y, v, x, t, g have their usual meaning, ~ is the depth to the centroid of area measured from the water surface. So is the channel bed slope . . . 1 . . . 2 Sf is the energy line slope. 22 In finite difference form av KM=/ ay -/ax M = av / = ZP ay -/at P = q= Substituting in Equ (1) YR2 VR – YL2 VL VR – VL 2Ax YR – YL 2AX VP – VM At YP-YM At Qin Ax + (YP ——. t YM) (YP - YM) Qin _ ~ Simplifying YP= From Equ 2 2Ax At mAx - ~(vR–vL) + VP–W + +x (YR - YL) = g(So - So 2Ax At . . . 3 . . . 4 From the Manning equation Vp2 n2 sf=— RP4 /3 mYP and RP = 24X where n = Manning’s ‘n”. Substituting in (4) and simplifying: m2@tu–~t#x(VR– VL) t &(YR-YL)–g So=O . ..5 4/3 At At Equations (3) and (5) are the equations used in the computer program. 23 APPENDIX 2 Worked Example A 10 year flood design is required for a catchment having the following details: (a) Area: 10 km2 (b) hnd slope: 6% (c) Channel slope: 3% (d) Channel length: 4.0 km (e) Grid reference 5°S 35°E (~ Catchment type: poor pasture From Table 7, lag time (K) = 0.5 h From Fig 16 and Table 4, standard contributing area coefficient Cs = 0.45. From Table 5, catchment wetness factor (Cw) = 0.50. From Table 6, land use factor (Co = 1.0. Therefore, design value for CA = 0.23. Initial retention (~ = O. From Table 8, Tp = 0.75 hrs. Using equ(11) with TA = O. TB = 0.75 t 2.3 . 0.5 = 1.15 hrs Rainfall during base time is given by: TB 24.33 )n . Rl” 124 (seeref(11)) ‘TB = ~ ‘TB t 0.33 where Rlo 124 = 10 year daily rainfall and n = 0.96 (see Table 8). 24 Using rainfall maps inref(11), 2 yr daily point rainfall = 63 mm 10:2 yr ratio = 1.49 10 yr daily point rainfa~ = 94 mm 1.15 24.33 0.96 R1.15 = ~ (m) . 94 = 59.Omm Area reduction factor is given by ARF = 1 – 0.04 T-3 A; = 0.88 Average rainfall (P) = 59.0x 0.88 = 51.9 mm RO=CA. (P–~. A. 103 Q= 0.93 . RO = 26.23 m3 /see 3600 . TB TA = 0.028 L — = 0.29 hrs ~;~; TB (2nd approximation) = 1.15 + 0.29 = 1.44 hrs 1.44 ~24.33)0.96 ..94 = 69.9 mm R1.44 = ~ — 1.77 ARF = 0.89 Therefore, P = 62.4 mm Q = 25.11 m3/sec TA = 0.29 hrs (no change) Therefore, Q = F . 0 F = 2.8 Therefore, Q = 2.8 . 25.11 = 70.3 m3 /see 25 I t 1 I Read recorded hydrography and rainfall ordinates I t ) I I Label stream network I t For each channel read: contributing area, length and slop of channel, Ax, At I Read values for K, Ca, Y, n I t I First channel I I I t - . I I I I Compute surface hvdrograph using Linear Reservoir Model I I Compute upstream depth and velocitv I t , I Route flow through channel using Finite Difference Equations t I Calculate goodness of fit I I t 1 L Print output I & Fig. 1 dRead data for next channel I Combine flows I I FLOW DIAGRAM FOR RURAL FLOOD MODEL 30 20 10 0 Design storm rainfall t 1 20 F 1 Observed storm rainfall 10 - 0 - d - 1.0 0.8 0.6 0.4 0.2 0 . . -—— o 1 2 3 4 5 6 7 8 Time (h) Fig. 2 TIWI CATCHMENT HYDROGRAPHY 40 r 30 - Design storm rainfall 20 - 10 - 0 20 r Observed storm rainfal I 10 - 6 5 I 4 3 2 i Q . I —— Observed hydrograph ---- Predicted hydrograph I I — Design hydrography I 1 2 3 4 5 Time (h) Fig. 3 MUDANDA CATCHMENT HYDROGRAPHY -.bu r L 40 30 20 10 0 20 r Observed storm rainfall 10 - 0 - 6 5 4 3 2 7 0 0 1 2 3 4 5 6 7 8 9 10 Time (h) Fig. 4 MIGWANI CATCHMENT HYDROGRAPHY 30- 20 - Design storm rainfall 10 - 0 J 20 r Observed storm rainfall 2.5 2.0 1.5 1.0 0.5 0 I I 1 I I I I I o 1 2 3 4 5 6 7 8 9 10 Time (h) Fig. 5 KAJIADO CATCHMENT HYDROGRAPHY 30 20 10 Design storm rainfall o ;=ormrainfa” 1.2 1.0 0.8 . 0.6 0.4 0.2 c ,0 1 2 3 4 5 Time (h) Fig.6 ESERET CATCHMENT HYDROGRAPHY 30 L Design storm rainfall 20 E q~ 10 ~ — x o .5 m a 10 - Observed storm rainfal I o u.14 0.12 0.10 0.08 0.06 0.04 9.02 0 0 5 10 15 20 Time (h) Fig. 7 KIAMBU ‘1 CATCHMENT HYDROGRAPHY 30 20 10 0 L 1 Design storm rainfall L 10 Observed storm rainfall o 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1 — — Observed hydrography ––– - Predicted hydrography — Design hydrograph o 5 10 15 20 Time (h) Fig.8 KIAMBU II CATCHMENT HYDROGRAPHY 40 r 30 20 10 0 30 20 10 0 k 0.10 0.08 0.06 0.04 0.02 0 Desiqn storm rainfall L Observed storm rainfall s I : / / / / y / I I I 1 I 1 I I o 2 4 6 8 10 12 14 16 18 20 Time (h) Fig.9 SAOSA CATCHMENT HYDROGRAPHY 40 30 20 k Design storm rainfall 10 0 30 20 L Observed storm rainfall 10 0 — — Observed hydrography -—-- Predicted hydrography Design hydrography o 5 10 15 20 Time (h) Fig.10 BARABILI CATCHMENT HYDROGRAPHY 40 30 20 10 0 20 10 1 Design storm rainfall ~observed‘termainfa” 3.( 2.5 2.0 1.5 1.0 0.5 0 0 I I —— Observed hydrograph ---- Predicted hydrograph — Design hydrography L I In\ \ 1\ \ \ I \\ \ I \ \ I I I I -—— I \ ———— \ ------ 1 3 Time (h) 4 5 6 Fig. 11 MUNYERE CATCHMENT HYDROGRAPHY 40 30 20 lC I Design storm rainfall 40- 30 - 20 - Observed storm rainfall ~ 10 - 0 ~ 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 il i I !1 “\ \ o 1 2 3 4 5 Time (h) Fig.12 RUBAARE CATCHMENT HYDROGRAPHY 30 20 L Design storm rainfall 10 0, 30 20 F Observed storm rainfal I 0.8, 0.7 0.6” 0.5” 0.4 0.3 0.2 0. i o o 5 10 15 20 Time (h) Fig.13 LUGULA CATCHMENT HYDROGRAPHY 30° E Northern Uganda \ \ \ ~e~ter””aanda -~4Centre}\.=<~-= “–—- / Ursanda / ~—-—.~–- ---- Central Tanzania \ 30° E 35° E j“E 4 North Eastern Kenya —-— I . . — \/ m Nyanza I ~ ~ 1 Nairobi ,~’ \\ / -J / \\ /’ I ,’ \\/ \ ) \ \ Kitui / I f ‘E 5°! o 50 100 150 200 km 8 Fig. 14 AREAS FOR CALCULATION OF RAINFALL, AND 2 AND 7 DAY ANTECEDENT 30 DAY SMR 30° E Afr” F , I \ 30° E 35° E 40° E o 200 km ~ Fig. 15 SOIL ZONES 50 0 50 ld 30% 35°E 40°E .\ \ \ $ 1 { \ \\ \ \ \ \\ \ ‘\. Kampala INLAND ZONE Tp = 0.?5 /7 /( CA I \ \ Nairo ‘. KENYA- / I U;:;:RU/ I Tp=2.0 I I [ , ‘\ ~ )/\/ ‘ ornbasa y /C ATAL ! ZONE Ii Tp=4. O 1. !. r es Salaam Morogoro ( { ‘-l 4 \ ON ,0 0s !Os Fig. 16 RAINFALL TIME (Tp) ZONES (453) Dd209410 2M 4/76 HPLtd So’ton G191s PRINTED IN ENGLAND ABSTRACT The TRRL East African flood model: D FIDDES: Department of the Environment, TRRL Laboratory Report 706: Crowthorne, 1976 (Transport and Road Research Laboratory). Four years of data from 13 small representative rural catchments in Kenya and Uganda were analysed to develop improved methods of flood estimation for highway bridges and culverts. Due to the short period of record and the very quick response time of the catchments, Unit Hydro~aph techniques were found inappropriate. A technique which made better use of limited data, therefore, had to be developed. Rainfall and runoff data were fitted to a simple three parameter conceptual catchment model. The model was then used to predict the 10 year flood using a 10 year design storm. A simple technique is then developed for predicting the peak flow and base time of design hydrography for ungauged catchments. ISSN 0305– 1293 ABSTRACT The TR R L East African flood model: D FIDDES: Department of the Environment, TRRL Laboratory Report 706: Crowthorne, 1976 (Transport and Road Research Laboratory). Four years of data from 13 small representative rural catchments in Kenya and Uganda were analysed to develop improved methods of flood estimation for highway bridges and culverts. Due to the short period of record and the very quick response time of the catchments, Unit Hydrography techniques were found inappropriate. A technique which made better use of limited data, therefore, had to be developed. Rainfall and runoff data were fitted to a simple three parameter conceptual catchment model. The model was then used to predict the 10 year flood using a 10 year design storm. A simple technique is then developed for predicting the peak flow and base time of design hydrography for ungauged catchments. ISSN 0305– 1293