# Chapter 3 : Reservoir models

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1 Chapter 3 : Reservoir models 3.1 History In earlier days, the dot graph of Kuipers was used to perform an impact assessment for combined sewer systems [Ribbius, 1951]. For a long period of rainfall, individual storm events were distinguished and for every event the storm duration and the storm volume were calculated (figure 3.1) [Berlamont, 1997]. These values were plotted on a graph with the storm duration on the horizontal axis and the storm volume on the vertical axis (figure 3.2). Figure 3.1 : Determination of a dot in the Kuipers graph. Figure 3.2 : Principle of the dot graph of Kuipers; the dots above the throughflow line will lead to an overflow event. Chapter 3 : Reservoir models 3.1

2 In this graph each storm is represented by a dot, but the distinction in individual storms is rather artificial. Although the rainfall variation is only incorporated in an approximate way, the importance of a long term approach was already stressed. In order to determine the number of storms that lead to an overflow event, the throughflow line is drawn. The throughflow is the outflow from the sewer system to a treatment plant or to a downstream catchment at the place the subcatchment is hydraulically disconnected. The throughflow line crosses the vertical axis at a volume equal to the storage in the combined sewer system and has a slope equal to the throughflow from the combined sewer system, when a constant (pumped) throughflow is assumed (figure 3.2). All dots above the throughflow line will lead to an overflow event. Based on this principle, graphs were made in which the overflow frequency is presented as a function of the maximum storage in the system and the maximum throughflow from the system (figure 3.3) [Berlamont, 199]. Figure 3.3 : Overflow frequency calculated with the dot method of Kuipers for the Uccle rainfall for the period [Berlamont, 199]. 3.2 The influence of rainfall and model simplification on combined sewer system design

3 In this figure 3.3 the constant throughflow is presented on the horizontal axis (in [m 3 /h/ha] for the upper horizontal axis and in [mm/h] for the lower horizontal axis), while the maximum storage in the system is presented on the vertical axis (in [m 3 /ha] at the left side and in [mm] at the right sight). The lines from the left top to the right bottom are the isolines for the overflow frequency (from 2 to 25 p.a.). The major disadvantage of this dot method is the simplification of both rainfall and model parameters. The method does not take into account the antecedent rainfall nor the variability of the rainfall within a storm. By linking every storm to only one storm duration (i.e. the duration of the total storm event), short overflow events for sewer systems with a smaller critical storm duration may be missed. All this will lead to an underestimation of the overflow frequency. There is also no criterion used to distinguish dependent overflow events. Furthermore, the parameters of the combined sewer system (storage and throughflow) are used as static parameters and are most often determined very roughly. It can be concluded that the dot graph of Kuipers can provide a rough estimation of the overflow frequency, but there remains a very high uncertainty on the results. The advantage of this method is its simplicity and in earlier days it was the only relevant possibility for the prediction of the overflow frequency. It is obvious that by using computer technology, better methods can be developed, while retaining the simplicity of the methodology. In the early eighties at the Hydraulics Laboratory of the University of Leuven, a single reservoir modelling system was developed [Berlamont & Smits, 1984a,b,c], which was later extended to the multiple reservoir modelling system Glas [Glas, 1986; Berlamont et al., 1987]. Glas is a linear reservoir modelling system, which means that the throughflow of the reservoir is defined as a linear function of the storage in the reservoir. Several reservoirs for subcatchments can be connected in series or parallel. With such a reservoir model the real variability of the rainfall can be included. Also the antecedent rainfall is included, because the time variation of the storage in the sewer system is taken into account. The reservoir modelling system Glas used hourly rainfall data as input for the period measured at Uccle. This might be too smoothed as rainfall input, because the rainfall variation within one hour can be high. Because of the fact that the real succession of the rainfall is taken into account for a long period (2 years) and a statistical analysis on the results is performed afterwards, the assessment of the overflow parameters will be much better than using the Kuipers graph. Using this kind of model, parameters other than the overflow frequency can be easily determined (statistically), as there are overflow volumes, discharges and durations. The remaining disadvantage of the Glas modelling system is the assumption of the linearity between storage and throughflow, which is for many applications not justified. The major bottleneck in the use of reservoir models is the calibration of the sewer system parameters. The calibration is a very important stage, which is often disregarded. Chapter 3 : Reservoir models 3.3

4 3.2 Characterisation of sewer systems Concept In the past many procedures were used to determine the parameters for simplified models. This has lead to a distrust in simplified models, because the results were often not accurate. In this work, special attention is paid to the calibration of the sewer system parameters. The purpose is to establish a methodology to characterise combined sewer systems in order to find the necessary parameters for a reservoir model. In order to be generally applicable in practice, this methodology must be as simple as possible and unambiguous. This can only be obtained using a physically based approach in order to find the sewer system parameters. As the flow in any system has to comply with the continuity equation, this equation will be the base of the system characterisation. Whereas in the reservoir modelling system Glas the relationship between storage in the system and the throughflow was prescribed (assumed to be linear), in this work a methodology will be established to find the real relationships. This approach can only be used when the inflow into the system and the throughflow are known. As very limited measurement data on both these aspects at the same time are available and the accuracy of the measurements is highly influenced by factors other than the sewer system parameters themselves, it is better to base this characterisation on simulations with detailed hydrodynamic models. For this study the hydrodynamic models Spida and Hydroworks (Wallingford Software, UK) are used, as they are commonly used in Flanders. If a detailed hydrodynamic model is available for a specific sewer system, the major sewer system parameters can be easily determined using some single storm simulations. In order to obtain a good calibration, a wide range of single storm simulations must be used in order to get a complete picture of the model behaviour. In most cases a detailed hydrodynamic model is available (e.g. in the framework of the Hydronaut studies), because the sewer system was designed using detailed hydrodynamic simulations. In this case the hydrodynamic model is verified and assumed to be as close to reality as possible. Moreover, if long term hydrodynamic simulations should be performed, this verified hydrodynamic model would be used anyway. In order to calibrate the simplified model, in the first instance only the sewer system parameters are taken into account. Therefore, in the hydrodynamic simulations the runoff model is not considered in this routing calibration stage. The calibration procedure is split into three parts. First, the concentration time is determined. Secondly, the relationships between the storage in the system and the throughflow are studied. Finally, the runoff module is calibrated. In a first stage a simple catchment with one downstream throughflow and overflow is assumed to explain the procedure. Afterwards the extension to more complex systems is addressed. 3.4 The influence of rainfall and model simplification on combined sewer system design

5 3.2.2 Concentration time As was already mentioned in the previous chapters, the concentration time is a very important parameter in sewer system design. It is one of the basic parameters for the flow through a sewer system and should therefore be incorporated in the simplified model. The concentration time at a specific calculation point can be found as the time between the peak rainfall input and the maximum discharge in the calculation point (see paragraph 2.1.1). In this way it is also easy to determine the concentration time using single storm events. The composite storms are peaked, which makes it easy to determine the differences between the peak in inflow and the peak in outflow. As the concentration time is different for every pipe or node in the system, the place must be defined where the concentration time must be determined. For the impact assessment of combined sewer systems, an optimal prediction of the throughflow and the overflow discharges is preferred. Therefore, the concentration time is determined as the difference between the peak of the rainfall input and the peak of the sum of throughflow and overflow discharges, because this sum is the total outflow from the system at the downstream end. The whole catchment is assumed as one reservoir for which only the outflow (i.e. throughflow + overflow) is important. The flow inside all the individual pipes themselves is not important in this case. Therefore, only the concentration time for the global outflow is considered. As the concentration time is the time the water needs to pass through the system and the flow velocity is to some extent dependent on the magnitude of the flow, it is clear that the concentration time is not a constant value. For pipe sections with increasing (or equal) width for increasing height, the velocity is monotonously increasing with the water depth, just as the flow, but for circular conduits this is not the case [Berlamont, 1997]. However, the velocity is increasing faster than the discharge for low water depths and the increase is slowing down for larger water depths (figure 3.4). There is a limitation for the increase in velocity with increasing flow, which is more rapidly reached for closed sections. When the pipes get pressurised, the velocity is linearly varying with the flow, while for lower flows the variation of the velocity is depending on the filling ratio. For a whole network this range of variability will be phased out, such that the overall relationship between discharge and velocity can be approximated with a linear relationship. As the concentration time inversely varies with the velocity, in a first approximation, the concentration time could be assumed to vary with the inverse of the discharge (hyperbolic function). For several sewer systems the variation of the concentration time with the flow is calculated using a wide range of composite storms with frequencies between 2 p.a. till return periods of 2 years. A typical result is shown in figure 3.5. The concentration times (obtained as the peak shifts between rainfall and outflow) are presented as a function of the maximum discharge. Chapter 3 : Reservoir models 3.5

6 1.2 U/U full circular section square section Q/Q full Figure 3.4 : Relationship between velocity U and discharge Q (relative to the velocity U full and the discharge Q full in a completely filled pipe) in different types of sections. 16 concentration time (min) maximum discharge (m 3 /s) Figure 3.5 : Variation of the concentration time with the peak discharge for a small (partly steep) gravitary sewer system and a hyperbolic fit to it. The concentration time is strongly varying for this small sewer system and the variation can be very well approximated by a hyperbolic function. It can be observed in figure 3.5 that the decrease in concentration time is limited for larger discharges. As the available rainfall series has a time step of 1 minutes, an error on the 3.6 The influence of rainfall and model simplification on combined sewer system design

7 (implemented) concentration time of some minutes is negligible as compared with the intrinsic smoothing within the rainfall data. For larger systems the variation of the concentration time will be (relatively) smaller, because the flow is more smoothed. For more complex and non-linear systems there will often be a larger variability on the concentration time. This leads to poorer fits of the variation of the concentration time with the flow. However, from the modelling point of view it is especially important to predict correctly the lower limit for the concentration time in order to ensure that at least the shift in peak flows is sufficiently accurate. In cases where no clear peak shift in the flow can be determined (e.g. when only one single pump is giving a constant maximum throughflow over a long time), the time of the maximum flow at the end of the sewer system can be observed taking into account the maximum water level. The peak flow is then determined as the time of the maximum water level during the (longest) period that the pump is working. This is an indication of the time for the maximum discharge that occurs upstream of the pump Storage/throughflow-relationships Introduction The most important parameters of the sewer system are those which describe the relationship between the state parameter (i.e. the storage volume) and the flow parameters (i.e overflow discharge and throughflow). When the continuity equation is applied to the inflow (i.e. composite storms) and the outflow (i.e. the results of hydrodynamic simulations), the storage volume in the system can be calculated at every time step. By eliminating the time using the time dependent storage and the outflow hydrographs, the relationships between storage in the system and throughflow can be determined. This can be performed for hydrodynamic simulations using a wide range of single (composite) storm events. This relationship can be easily visualised in a storage/throughflow-graph (e.g. figure 3.6). In order to match the proposed reservoir model the rainfall input is first smoothed over the concentration time before the continuity equation is applied (see paragraph 3.3.1). As can be seen in figure 3.6 there is a clear relationship between the storage in the sewer system and the throughflow. However, there is also a clear influence of the input into the system as well. The storage volume is much larger during the rising branch than during the falling branch of the storm. This hysteresis effect can be assigned to the dynamic storage in the system. It has been checked if this variation is not dependent on the rainfall type using a wide range of rainfall events. The two extreme types of storms are a constant intensity during the storm duration corresponding to the IDF-relationships on the one hand and the peaked composite storms on the other hand. These two storm types were simulated Chapter 3 : Reservoir models 3.7

8 for a wide range of return periods in several sewer systems. Some results are shown in figure throughflow (mm/h) decreasing rainfall intensity f = 1 f = increasing rainfall intensity f = 3 f = 4 f = 5 f = 7 f = 1 f = storage (mm) Figure 3.6 : Storage/throughflow-relationship for a small gravitary (partly steep) sewer system for a range of composite storms with different frequencies f [p.a.]. 12 storage in the sewer system (mm) maximum beginning end maximum overflow discharge (mm/h) Figure 3.7 : Variation of the storage during the overflow event (at the beginning, at the end and maximum) in two sewer systems ( and ) for a wide range of rainfall events (constant intensities as well as peaked composite storms). 3.8 The influence of rainfall and model simplification on combined sewer system design

9 In figure 3.7 the storage in the system at the beginning and at the end of the storm are shown as well as the maximum storage that occurred. If these storage values are ranked as a function of the maximum overflow discharge (i.e. approximately the maximum outflow minus a constant throughflow when the overflow starts spilling), a clear trend is observed. The storage at the end of the overflow event is an almost constant value. At this moment the rainfall input into the system is most often nil or very small. In that case the system behaves as a static reservoir with a fixed volume that is completely full. The maximum storage in the system appears to be linearly varying with the maximum discharge. The difference between the maximum storage and the storage at the end of the event can be assigned to the dynamic storage, which seems to be highly correlated with the maximum discharge. The fluctuation of the storage at the beginning of the overflow event is due to the shape of the hyetograph. For peaked storms and (high) constant rainfall input over short durations the storage at the beginning of the overflow event will be near to the maximum storage, while for constant rainfall input over longer durations the storage at the beginning of the overflow event will be more an intermediate value. This indicates that the dynamic storage is highly dependent on the instantaneous inflow Static storage The maximum static storage is the storage in a sewer system when the system is filled up to the crest of the weir, when there is no throughflow and the water is stagnant (figure 3.8). Figure 3.8 : Longitudinal profile of a sewer system on which the static storage is indicated (hatched area). Chapter 3 : Reservoir models 3.9

10 Although this is an unrealistic (artificial) situation, this state is most often a very good approximation of the situation at the end of an overflow event (figure 3.7). At least the maximum static storage in a sewer system can be assessed in this way, even if there is a flow in the system which creates an extra storage (this is most often the case). In the same way the static storage at any time can be assessed as the storage during the emptying of the system when no rainfall input is present. This means that the instantaneous static storage, for instance for the case of figure 3.6, must be very close to the storage during the falling branch of the storm (slightly smaller (see figure 3.15)). In figure 3.7 it can be seen that the storage at the end of the overflow event is a good indication of the maximum static storage in the system. If only the static storage is incorporated, the storage at the beginning of the smallest overflow event can also be a good estimation for the static storage (values near to on the horizontal axis in figure 3.7). Although the maximum static storage can be easily obtained at the end of the overflow event, it is important to use the instantaneous relationship between storage and throughflow for the rising branch of the storm up to the moment the overflow starts spilling. This is necessary to obtain a good estimation of the start of the overflow. The primary objective should be to detect if an overflow occurs or not and when it starts. The emptying of the system after the overflow event only has a second order effect, because the storage in the system can have an influence on the following overflow event. This means that in the first instance the dynamic storage can be neglected. As for sewer system calculations in Flanders the composite storms are used, it is logical to use the composite storm that will just lead to an overflow event to determine this storage/throughflow-relationship in a first approach. This first approach has been applied to a number of different sewer systems and provides already a lot of information on the sewer systems behaviour. In figure 3.9 the static storage/throughflow-relationship is shown for a small sewer system with a gravitary throughflow. 3.1 The influence of rainfall and model simplification on combined sewer system design

11 throughflow (l/s) real relationship linear 'static' approach storage (m 3 ) Figure 3.9 : Storage/throughflow-relationships up to the moment the overflow starts spilling for a small sewer system with a gravitary throughflow. It can be seen in figure 3.9 that the relationship between storage and throughflow is almost linear. In that case the sewer system is labelled as behaving linearly. As the throughflow from the system is linearly varying with the volume in the system and the volume in the system is a function of the inflow through the continuity equation, a specific input hydrograph will be smoothed, but the antecedent condition in the reservoir will have only a limited influence. The system behaviour is then driven by the flow. In figure 3.1 the static storage/throughflow-relationship is given for a small sewer system with a single pump for the throughflow. It can be seen that the relationship between storage and throughflow in this system is far from linear. If this relationship would be approximated with the best linear fit (as is shown in figure 3.1) large errors on the system behaviour are made. If a linear approximation is made based on the maximum storage and the maximum throughflow only, the error will be even larger. This clearly shows that it is not sufficient to determine the maximum storage and maximum throughflow only, but that the instantaneous relationship between storage and throughflow is equally important. When the relationship between the storage in the system and the throughflow is divergent from linearity, the system behaviour is labelled as non-linear. For this system (figure 3.1), the throughflow is even completely independent on the storage volume in the system. This means that the throughflow is hardly influenced by the immediate input into the system, but mainly by the antecedent rainfall over a longer time. The system behaviour is driven by the volume instead of the flow. The throughflow (and overflow) hydrograph can be completely different when the same input hydrograph is used, depending on the antecedent rainfall. Chapter 3 : Reservoir models 3.11

12 throughflow (l/s) bi-linear fitting linear through maxima fitted linear 1 storage (m 3 ) Figure 3.1 : Storage/throughflow-relationship for a small sewer system with a single pump for the throughflow. The two previous examples are the two most extreme possible behaviours of a sewer system. In most cases the behaviour is intermediate. In figure 3.11, the storage/throughflow-relationship is shown for a medium size sewer system with a complex throughflow relationship (pumping station with five pumps). Although there are some parts in the relationship which are non-linear, the global effect does not diverge largely from a linear relationship. 1.2 throughflow (m 3 /s) hydrodynamic linear reservoir storage (m 3 ) Figure 3.11 : Storage/throughflow-relationship for the sewer system of Dessel (with 5 pumps in series) for a composite storm which just leads to an overflow event The influence of rainfall and model simplification on combined sewer system design

13 The influence of the simplifications in the system behaviour, as shown in figure 3.1, on the effects caused by the composite storm, which has been used to derive the storage/throughflow-relationship, can be seen in the figures 3.12 and If a good linear approach is fitted, the overflow and throughflow volumes will be approximated well, but the shape of the hydrographs will be different. Consequently, the type of rainfall input can significantly influence the outflow. However, it is difficult to specify what a good linear approximation means. In this work the volume under the storage/throughflow-relationship is taken equal for both curves (e.g. figures 3.9 to 3.11), which seems to lead to good results. In that case the mean deviation from this state/flow-relationship is minimised, but it remains an arbitrary choice. If the linear approximation would only be based on the maximum storage and throughflow as is shown in figure 3.1, the overflow events will even be larger. These results amplify the conclusion that not only the maximum system parameters are important, but that the instantaneous relationship between storage and throughflow is equally important. 12 overflow discharge (l/s) hydrodynamic model linear reservoir model bi-linear reservoir model time (min) Figure 3.12 : Overflow discharge for a single composite storm that will just lead to an overflow event for a small sewer system with a single pump for the throughflow (storage/throughflow-relationships in figure 3.1). Chapter 3 : Reservoir models 3.13

14 25 throughflow volume (m 3 ) hydrodynamic model linear reservoir model bi-linear reservoir model time (min) Figure 3.13 : Throughflow volume for a single composite storm that will just lead to an overflow event in a small sewer system with a single pump for the throughflow (storage/throughflow-relationships in figure 3.1) Dynamic storage If the static storage is defined as the storage below a horizontal plane at the crest level of the overflow, the dynamic storage can be defined as the storage above this horizontal plane, which is due to the flow in the system (figure 3.14). Although this is an artificial division, these two components can be recognised in the system behaviour The influence of rainfall and model simplification on combined sewer system design

15 Figure 3.14 : Longitudinal profile of a sewer system on which the static and dynamic storage is indicated using a strict definition of storage under and above the crest level. As already mentioned, the maximum dynamic storage is correlated with the maximum overflow discharge (figure 3.7). Therefore, it could be assumed that there is a relationship between the instantaneous dynamic storage and the global outflow from the system (overflow + throughflow). It is more common to link the dynamic storage to the inflow (e.g. Muskingum method [Cunge, 1969]) than to link it with the overflow discharge, although the optimum relationship will be a combination of both. If the inflow hydrograph is first smoothed in the same way as during a hydrodynamic routing, the instantaneous values of this smoothed hydrograph will be near to the sum of the instantaneous values of the overflow and the throughflow. This means that the dynamic storage can be assessed as a function of the smoothed inflow hydrograph. In this work this smoothing is obtained by an averaging of the runoff over the concentration time as will be further addressed in paragraph In figure 3.15 an example is given where the dynamic storage is included as a linear function of the smoothed inflow. The total storage can then be obtained by the summation of the static and the dynamic storage. The correlation between the relationship for the total storage obtained with a hydrodynamic simulation and those obtained using the linear relationships between the dynamic storage and inflow on the one hand and between the static storage and the throughflow on the other hand is very good for this sewer system (figure 3.15). The incorporation of the dynamic storage can certainly improve the estimation of the overflows, especially with respect to the overflow duration and the (instantaneous) overflow discharges, although with only a static storage a good result is already obtained in many cases. The incorporation of the dynamic storage not only improves the shape of the overflow events (i.e. the variation of the overflow discharge in time), it also allows a better approximation of the emptying of the system. The antecedent rainfall conditions for the next storm will be better incorporated in this way. Chapter 3 : Reservoir models 3.15

16 throughflow Q through (m 3 /s) V d = k dyn * Q in V s = k stat * Q trough V tot = V d + V s real relationship dynamic storage static storage V s total storage V tot storage (m 3 ) V d Figure 3.15 : Linear dynamic approach (with slopes k stat and k dyn ) for the storage/throughflow-relationship of a small gravitary sewer system (for the composite storm which will just lead to an overflow event). In order to obtain a good approximation of the relationship between inflow and dynamic storage, the relationship cannot be fitted only using the composite storm which will just lead to an overflow event. In that case the range for which the relationship is valid is too limited. Therefore, it is better that the fitting is performed using the whole range of composite storms. This increases the calibration efforts, but it can be easily automated. Unfortunately, the determination of a dynamic storage is not always that easy as can be seen in figure However, the advantage is that the falling branch is better described, while the rising branch of the storm is predicted equally well as in the static approach. The reason that the fitting in figure 3.16 does not seem to be optimal is due to the fact that the relationship between inflow and dynamic storage is based on a whole range of different composite storms. The overall correlation is optimal, but for each specific storm the differences may be considerable The influence of rainfall and model simplification on combined sewer system design

17 1.2 throughflow (m 3 /s) hydrodynamic model dynamic reservoir model storage (m 3 ) Figure 3.16 : Storage/throughflow-relationship for the sewer system of Dessel when the dynamic storage is included for a composite storm that will just lead to an overflow event Extra storage during the overflow event During the overflow event the water will rise above the crest level. In some cases this increasing water level can lead to significant extra backwater effects, which results in increased storage in the system. This storage above the crest level can be defined as dynamic storage as is shown in figure 3.14, but it is difficult to link this dynamic storage unambiguously to the inflow as proposed in paragraph Therefore, it is better to define an extra storage above the crest level as the storage between a horizontal plane at the crest level and a horizontal plane at the maximum water level above the crest of the overflow (figure 3.17). In that definition the dynamic storage is only the part above the horizontal plane at the maximum water level above the crest of the overflow. Again, this is an artificial division, but these three components can be recognised in the system behaviour. This extra (static) storage above the weir can be assumed to be a function of the overflow discharge. For a free overflow the relationship between discharge Q and water level h above the crest is a non-linear function [Berlamont, 1996] : (3.1) This relationship may be the case for a storage basin (figure 3.18). Chapter 3 : Reservoir models 3.17

18 Figure 3.17 : Longitudinal profile of a sewer system on which the static, dynamic and extra storage during the overflow event are indicated. 1 overflow discharge (m 3 /s) static storage (m 3 ) Figure 3.18 : Relationship between overflow discharge and volume in the system for the storage basin in the sewer system of Dessel The influence of rainfall and model simplification on combined sewer system design

19 Also for combined sewer systems the extra storage (V) available can be limited (as in figure 3.18), which leads to a similar non-linear relationship (curving in the same direction as equation (3.1) : Q - V n, with n > 1). On the other hand, also the overflow discharge Q may be restricted (e.g. by the transport capacity of the pipes from the overflow structure to the receiving water). In that case a non-linear relationship with a curving in the other direction (Q - V n, with n < 1) is obtained (figure 3.19). 1.4 overflow discharge (m 3 /s) T = 1 year T = 2 year T = 5 year T = 1 year T = 2 year bi-linear fit static storage (m 3 ) Figure 3.19 : Relationship between storage and overflow discharge for the overflow in the sewer system of Dessel for design storms with different return periods (T). Figure 3.19 shows that the extra storage during the overflow event can be large. In this case this extra storage during the overflow event also has a major effect on the shape of the overflow hydrograph. In figure 3.2 the effect on the overflow events is shown for several situations : i.e. caused by a single storm using a static reservoir model only, using a dynamic reservoir model (i.e. reservoir model for which the dynamic storage is included as a function of the inflow) and incorporating the extra storage during the overflow event. The influence on the total volume of the overflow event is very small, but the influence on the shape (discharge, duration) is significant. In figure 3.21, the effect on some large overflow events is shown for the same sewer system, showing that the peak discharges and overflow durations will be predicted well for a wide range of storms. The differences in shape between the overflow events of both models (figure 3.21) are due to the hysteresis of the extra storage during the overflow event (figure 3.19). Chapter 3 : Reservoir models 3.19

20 1.6 overflow discharge (m 3 /s) hydrodynamic model static reservoir model dynamic reservoir model + storage during overflow time (min) Figure 3.2 : Overflow discharges for the combined sewer system of Dessel for different models and a composite storm which will just lead to an overflow event overflow discharge (m 3 /s) hydrodynamic model optimal reservoir model time (min) Figure 3.21 : Overflow discharges for the sewer system of Dessel for return periods between 1 and 2 times p.a. obtained using the hydrodynamic model and the reservoir model with extra storage above the weir (see figure 3.19). 3.2 The influence of rainfall and model simplification on combined sewer system design

21 3.2.4 Runoff models After the calibration of the routing through the sewer system against a verified detailed hydrodynamic model, the runoff model can be calibrated separately. As the variability of the rainfall and consequently of the water input into the sewer system is high, the calibration of the runoff model must be carried out using a wide range of measurements. The calibration must be performed in such a way that the input volumes are similar in the model and in the measurements. Most often it is not possible to also obtain similar instantaneous discharges in model and measurements, because of the high uncertainty on the parameters. There are significant uncertainties on the rainfall measurements, on the runoff coefficients, on the losses and on the contributing area. If the cumulative input is corresponding well in the model and measurements, the errors on the instantaneous discharges are only of second order importance. An example of the calibration of a runoff model is given in paragraph 6.5. A wide range of runoff models can be used. From the modelling point of view the most extreme runoff models are on the one hand a fixed runoff coefficient and on the other hand a fixed depression storage (i.e. the storage in depressions on the surface). Using a fixed runoff coefficient leads to rainfall losses which are a fixed percentage of the rainfall input. This is a model with a linear behaviour, because the output is linearly varying with the input. The losses are independent of the antecedent rainfall. Using a fixed depression storage on the other hand, leads to a non-linear relationship between in- and outflow. If the depression storage is empty, it will be filled during the first part of the storm and the shape of the inlet hydrograph might be changed. If the depression storage is already filled by antecedent rainfall, it will have no influence on the storm. For this kind of model the antecedent rainfall and the intrinsic variability of the rainfall are very important. For continuous simulations the runoff models should not be event based, even if they are calibrated using single measured events. For that reason a depression storage model cannot simply be used to subtract the first volume of a storm as is often proposed in event based models [Van den Herik & Van Luytelaar, 199; WS, 1996], because of the subjective definition of a storm. The depression storage must be written as a function of instantaneous parameters. This can for instance be performed using a simple reservoir model for which the outflow is linearly varying with the inflow and with the depression storage (figure 3.22). The depression storage S is thus limited to S max. If this relationship is incorporated in the continuity equation and this differential equation is solved, an exponential filling of the depression storage is obtained : S Vin = S max 1 exp (3.2) Smax In equation (3.2), V in is the cumulative inflow into the reservoir. Chapter 3 : Reservoir models 3.21

22 Figure 3.22 : Depression storage model. Using this equation (3.2) for the depression storage model simplifies continuous simulations, because it is not event based. Moreover, this continuous relationship is physically more realistic than the storage of the first rainfall volume. However, the exponential filling has most often no significant influence on the rainfall runoff as compared for instance with a fixed initial depression storage at the beginning of a storm. Using this exponential filling of the depression storage, an inflowing rainfall volume equal to S max will be retained for 63 %. This kind of non-linear runoff model responds very different from a runoff model using a fixed runoff coefficient. This can be shown when for the depression storage model the instantaneous runoff coefficients are calculated. In figure 3.23, the instantaneous runoff coefficients for a depression storage model are calculated in every (1 minute) time step of the historical rainfall series of 27 years assuming a maximum depression storage of 1.5 mm. These instantaneous runoff coefficients are ranked as a function of the rainfall intensity. For the rainfall intensities that occur several times, the mean value for the instantaneous runoff coefficients is presented in figure 3.23 by the markers according to the primary (left) vertical axis. The number of times that a certain rainfall intensity occurs is presented by the line according to the secondary (right) vertical axis. Only the mean values per rainfall intensity are shown, but for the low intensities there is a large variation (using the whole range from to 1). An extra parameter p (between and 1) could be incorporated to delay the filling of the depression storage (i = rainfall intensity; Q r = runoff discharge) : ( 1 ) Qr = p i + p S S max i (3.3) S S p V in = max 1 exp (3.4) Smax 3.22 The influence of rainfall and model simplification on combined sewer system design

23 instantaneous runoff coefficient 1.9 number of occurence instantaneous runoff coefficient number of occurrence rainfall intensity (mm/h) Figure 3.23 : Instantaneous runoff coefficients (mean values for every rainfall intensity) for a depression storage model with a maximum storage of 1.5 mm (markers) and the corresponding number of occurrence of the rainfall intensities (line). In order to make the runoff model continuous, the depression storage must also be emptied. This can for instance be implemented by applying evaporation. The evaporation is varying over the year (equation 1.1). The influence of this variation on the total evaporation volume is however much smaller than the large variation of the evaporation capacity itself, because of the limited availability of water in the depressions on the surface. Figure 3.24 shows the monthly evaporated volumes obtained with a continuous long term simulation of 27 years using different constant values for the evaporation capacity (full lines) and the effect (bold dashed line) using a variable evaporation capacity (thin dashed line). The variation of the real evaporation (i.e. based on the water availability) is much smaller than the evaporation capacity. For this reason a constant evaporation capacity could be used as an approximation without having too much effect on the prediction of the overflow events. In the winter period the evaporation is overestimated if a constant evaporation capacity of.1 mm/h is used, while in the summer this is a good approximation. The overestimation of the evaporation during the winter is about 1 mm/month. The overflow volumes in winter could be slightly underestimated in this way. However, most (and most severe) combined sewer overflows occur in the summer period. Moreover, the evaporation only influences the antecedent conditions and thus has only a second order effect on the overflow emissions. Chapter 3 : Reservoir models 3.23

24 evaporation volume (mm/month) evaporation capacity variable evaporation month of the year constant evaporation (mm/h) :.12.2 Figure 3.24 : Influence of a constant (full lines) versus a variable evaporation (bold dashed line) over the year. If not only impervious areas are modelled, but also pervious areas, then the losses can also be due to infiltration. Although infiltration is a complex process where many parameters are involved, a basic model for the infiltration capacity has been proposed by Horton [194] : f = fmin + fmax fmin exp k t (3.5) ( ) ( ) In this formula, f is the infiltration capacity (in [mm/h]), f min and f max respectively the minimum and maximum infiltration capacity, k the decay factor (in [/h]) and t the time. The physical meaning of this model is that the infiltration capacity will exponentially decrease from an initial infiltration capacity f max to a minimum infiltration capacity f min as the surface becomes wet. Also the restoration of the infiltration capacity due to the drying-up of the surface can be assumed to occur exponentially : f = fmax fmax fmin exp k t (3.6) ( ) ( ) In equation 3.6 a different decay factor k can be used than in equation (3.5). As for pervious areas the depression storage will be emptied by infiltration, this can be incorporated in the formula for the depression storage : Vin Fin S = Smax 1 exp (3.7) Smax In equation (3.7) F in is the cumulative infiltration. Only if enough water is available in the depression storage, will the infiltration be equal to the infiltration capacity The influence of rainfall and model simplification on combined sewer system design

25 Other more detailed runoff models could be used, but the calibration of the parameters requires large amounts of high quality data. Although a detailed runoff model for the pervious areas might be necessary for the modelling of runoff to brooks and rivers (e.g. [Clarke, 1994; DHI, 1993]), the influence of pervious areas and certainly the influence of the detail of the infiltration description will be small for urban drainage. If it is necessary to incorporate the infiltration, in most cases the rough description by Horton (equation 3.5) will suffice. Next to the losses, also the smoothing by the overland flow should be considered. Many complex models can be used for this, as was described in paragraph However, if this is already taken into account in the single storm simulations that are used for the calibration of the routing model, the smoothing is already incorporated in the routing part of the simplified model Complex systems In the previous paragraphs, the sewer system characterisation was applied to simple sewer systems with one overflow. In reality, sewer systems are more complex, but it is not always necessary to incorporate this complexity into the model. Before a model is built, the modeller has to decide which output he wants to obtain. Then, the model has to be built in such a way that the required output is achieved with the desired accuracy. The first step in the transformation of a combined sewer system into a reservoir model is to separate the hydraulically independent subcatchments. This can be done e.g. at pumping stations, internal weirs, places where the critical water depth always occurs,... The number of places where these conditions occur may be limited. However, other places can be used to disconnect different subcatchments as long as no significant backwater effects occur at these places. Whether a certain place is suitable for disconnecting subcatchments, can be seen in the storage/throughflow-relationships. If the storage volume in the subcatchment cannot be linked unambiguously to the inflow and outflow, the place is not suitable for disconnection of subcatchments. In all other cases the disconnection can be applied at that place. As a reservoir model is a conceptual model, the outflows of the reservoir do not necessarily have to agree with the flow in a single pipe or at a single overflow. The throughflow can be the sum of different pipes which connect two subcatchments and the overflow can be the sum of two overflows which spill into the same brook or river. This gives more possibilities to separate subcatchments, but also to simplify the models. Chapter 3 : Reservoir models 3.25

26 When a resulting subcatchment after separation still contains more than one overflow, the modelling can be performed in several stages. For instance, for a subcatchment with two overflows two stages are necessary. In the first stage the different overflows are modelled as one overflow. In the second stage the overflow which starts spilling first, is modelled together with the throughflow so that only the overflow events of the second overflow are obtained. The overflow events of the first overflow can then be obtained by subtracting both overflow series from each other. In figure 3.1 the storage/throughflow-relationship is presented for a catchment with two overflows and a single pump as throughflow, where the two overflows are modelled together. In figure 3.25 for the same catchment the storage/throughflow-relationship is given where the first spilling overflow is modelled as throughflow in order to obtain the overflow events from the second overflow. This kind of methodology can be extended to more than two overflows per subcatchment. However, this is a labourious methodology. It would be easier if these different stages would be implemented in the reservoir modelling system itself for which a methodology is proposed in paragraph throughflow (mm/h) f = 1 f = 2 f = 3 f = 4 f = 5 f = 6 f = 7 f = storage (mm) Figure 3.25 : Storage/throughflow-relationship for the second spilling overflow in a small sewer system with one pump for the throughflow for a range of composite storms with different frequencies f (the discharges of the first spilling overflow are modelled as throughflow) The influence of rainfall and model simplification on combined sewer system design

27 3.3 The reservoir modelling system Remuli Concept of the modelling system Based on the basic parameters for combined sewer systems that were identified in this work, a physically based conceptual modelling system was developed. This modelling system contains four main parts which are connected in series (figure 3.26) : a runoff module, a smoothing module, a module for the dynamic storage and a module for the static storage. The long term simulations can be performed with rainfall with a time step of 1 minutes at Uccle for the period Figure 3.26 : Schematic representation of the reservoir modelling system Remuli. The runoff module incorporates the rainfall losses and consequently the rainfall input into the system. In the runoff module a distinction can be made between impervious and pervious area. For the impervious areas, a fixed runoff coefficient can be specified as well as a depression storage (i.e. surface storage). The depression storage is filled exponentially in time as described in paragraph 3.2.4, also including an extra parameter to delay the filling up of the depression storage (equation 3.4). For the pervious areas, Chapter 3 : Reservoir models 3.27

28 the same parameters can be specified as for the impervious areas and as an extra possibility the infiltration can be modelled using the Horton equation as described in paragraph The decay factors for surface wetting (equation 3.5) and drying (equation 3.6) may be chosen differently. The depression storage on the impervious areas can be emptied during dry weather using a constant evaporation. The possible evaporation is strongly influenced by other climatic parameters as for instance the temperature. However, the actual evaporation from impervious areas varies much less, because of the limited amounts of water available in the depression storage (figure 3.24). The evaporation has only a small effect on the losses and does not influence the prediction of overflows directly. It only slightly influences the antecedent conditions. Therefore, it is acceptable to use a constant evaporation for emission predictions at combined sewer overflows. For the pervious areas the depression storage can also be emptied by infiltration. In the reservoir modelling system Glas, hourly rainfall was used which already incorporated some smoothing. Furthermore, a time shift of one or more hours (multiple of the time step) could be applied on the throughflow. This is sufficient if no better accuracy than hourly based results is necessary and for sewer systems which have concentration times of maximum a few hours. Larger concentration times seldom occured, because in earlier times only small systems were modelled with a single reservoir. Since 1 minute rainfall data are available now, it is possible to predict the combined sewer overflows with higher accuracy. In order to obtain the smoothing of the hydrograph and the time delay of the peak flow, it is necessary to add an extra smoothing module in the reservoir modelling system. This smoothing is applied to the runoff. The smoothing module contains an averaging of the runoff over the concentration time. In this way the peak shift between inflow and outflow are well determined and part of the smoothing due to the flow in the system is already incorporated. This smoothing is necessary, because a single reservoir model has an instantaneous (exponential) response. This means that the peak shift cannot be predicted well in the reservoir model, when no extra module for this is included. For the same reason (i.e. exponential response) a single linear reservoir is not a good tool for this smoothing. If the concentration time is used as a variable parameter (as a (hyperbolic) function of the runoff), discontinuities may occur for low concentration times using only multiples of the 1 minute time step. Therefore the concentration time is incorporated with a higher accuracy of 1 minute, however without reducing the simulation time step. This also allows a more accurate smoothing for small systems using a constant concentration time. The use of the relationship between peak shift and maximum runoff as described in paragraph can be generalised for the smoothing of all runoff discharges (not just only the peaks) in order to obtain continuous smoothing. This is physically acceptable and has so far led to good results. However, already good results are often obtained with a constant concentration time which is equal to the peak shift between rainfall and outflow for the (composite) storm that will just lead to an overflow event The influence of rainfall and model simplification on combined sewer system design

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