The mathematical model of rainwater catchment in Wroclaw

On the basis of field studies of canalized storm water catchment of Gaj and Tarnogaj estate in Wrocław the rules of construction of hydrodynamic models in SWMM program were presented. The process of identification of hydrological and hydraulic parameters of the model, in the course of its calibration and validation were presented. To assess the quality of model the rates and statistical criteria were proposed to compare the results of stream simulation and volume of runoff of effective precipitation with the results of the measurements.


Introduction
Escalating in recent years extreme weather phenomena such as incidental or prolonged rainfall and associated with them flooding cause significant economic losses.In the existing, modernized or newly designed sewage systems the verification of hydraulic capacity of networks and objects is now recommended, including overflows from channels towards hydrodynamic modeling at different scenarios of precipitation load.
Modeling of systems reliability recommended by standard PN-EN 752:2008 is rarely applied in Poland [1] -with regard to the verification of the frequency of overflow from channels and even required by law according to the regulation of the Minister of the environment from 2014 [2] -with regard to the verification of the frequency of storm water overflows.This is mainly the result of the lack of sufficient basis of modeling methods as well as appropriate output databases with regard to monitoring of precipitation and overflows in sewage systems [3].For instance, the level of integration of sub-catchment area is as yet intuitively.Usually channels with diameter smaller than 0.5 m are omitted and the width of the hydraulic sub-catchment is designated from several different formulas.In calibration and validation of hydrodynamic models data from the short period of observation are taken into consideration (i.e.few months) and the assessment of the quality of hydrodynamic models is based on various statistical indicators for comparison of the simulation results of sewage with the measurement results.
Based on the example of local data regarding the precipitation and streams of storm water in Wrocław the principles of the construction of mathematical model for drainage the area in SWMM program were presented.
For field research the canalized catchment of Gaj and Tarnogaj estate in Wrocław were selected, with the area F = 104 hectares -giving runoff to indoor drainage collector KD1 (with diameters from 0.3 m to 1.4 m).From the total area of estates, the dimpled areas were excluded, including allotments from which the sewage runoff of precipitation taking place into the ditches.The total length of the inventoried channels was 17731 m and the number of drains 509.This gives an average spacing of drains 34.8 m, which can be considered as representative for the cities.
On the stage of identification, as a minimum diameter of channels taking dmin ≥ 0.3 m, 75 catchment area were discharged.In sub-catchments sealed areas were distinguished -not having retention Fd (roofs), sealed areas-with retention Fa (asphalt road) and Fk.b (carriageways with concrete block or cobblestone) and areas not sealed ("green" areas)with retention Fnu.Then degrees of sealed area were calculated in each sub-catchments: total sealing degree: %Fu = Fu/Fi, degree of area sealing without retention: %Fubr = Fubr/Fu and degree of area sealing with retention: %Fuzr = Fuzr/Fu.Slope of area (ip) in the sub-catchments were adopted as weighted average-from the area of the terrain, streets, squares and roofs (tab.1).As the basis to determine hydraulic width of sub-catchment (W), the basic form of the formula was taken: W = x√F (1) where: F -area of storm water catchment/sub-catchment, m 2 .
In subject literature, the formula ( 1) is most commonly used with multipliers for values: x = 1.0 [6][7][8] or x = 1.5 [9][10][11].In Wroclaw's conditions to render the parameter W it turned out that the best value was x = 1.6, which will be demonstrated later in this study.
To conduct simulation calculations in SWMM program it was necessary to create databases involving precipitation of rain in catchment and measurement of sewage streams in KD1 collector.For metering sewage streams two ultrasonic flow meters were used.Nivus type flow meter (P1 -fig. 1) was installed in drain no.18 on the section of KD1 with a diameter of 1.4 m.The second type of flow meter Teledyne (P2) was installed in the drain no.239, on the section KD1 with a diameter of 1.2 m.
The rain gauge type TRwS was a device used to register rainfall in the catchment located on the southern boundary of the catchment area of Gaj and Tarnogaj estate at a distance of about 620 m from the center of gravity of the tested catchment (fig.1).Registry of precipitation was conducted for a period of 2 years, from July 29 th 2013 to July 19 th 2015.Precipitation with duration time t > 45 min and height h > 10 mm were taken for calibration and validation of the model.Both torrential rains -those of short duration (convective K), as well as those of long-term (frontal F and lowland N), which hyetographs have a continuous character were taken into consideration.The chosen precipitation was assigned later to their frequency of occurrence in Wrocław.As a criterion to determine the frequency of precipitation occurrence was taken the probabilistic model of maximum height in Wrocław (based on the distribution of Fisher-Tippett type III) -for C  [1; 100] years [12], and for precipitation occurring more frequently than once a year (C  [0.1; 1)) a physical model was used [3].These precipitations were detailed about episodes of time duration equal to time of flow of sewage in collector KD1 to the intersection of flowmeter P1 (t ≈ 45 min) and P2 (t ≈ 15 min).The parameters of 8 precipitation selected for calibration and validation of the model are given in table 2. To assess the quality of the model selected statistical measure were used for comparing the results of measurements and calculations of the outflow (Q), such as [3-5, 13-15]:  special rate of correlation RS:  relative residual error WBR:  average value error SWS:  relative error of maximum streams ΔQmax: Indexes "p", and "o" in formulas (2-5) mean respectively measurements and calculations.The value "n" corresponds to the number of extracted compartments-averaged values Q in a registered hydrographic of runoff.Depending on the value of the rate, the model can be qualified for the specified category.Ranges of values of rates RS and WBR together with corresponding categories of models were listed in table 3.For SWS rates and ΔQmax the ranges of values for the appropriate category of model were not specified.The only known limit values appropriate for situation when the model perfectly reproduces reality: SWS = 1.0 i ΔQmax = 0.About the optimum values of parameters of the model proclaim the achieved values of indicators: RS, WBR, SWS and ∆Qmax, adopted as the statistical criteria to evaluate the quality of the model.However, the most important was the achievement of the compliance of simulated values with the values measured with respect to the balance of the volume of runoff (V), as shown in Chapter 4. In table 5 the final results of the calculation accuracy of the model for 5 precipitation calibrations and three forms of formula for the parameter W. The best results were achieved for W3.On the basis of the value of the indicators RS and WBR the calibrated model was assessed on the border of grades good and very good (according to the criteria from the table 3).
Figure 2 shows, for example, a histogram of precipitation and hydrographs of sewage streams in collector KD1 (intersection P1) for the latest lowland precipitation (N) from October 22 nd to 23 rd 2014.

Validation of model
Validation of tested hydrodynamic model of drainage system consists in checking the accuracy of the already calibrated model on the 3 short-lived intense convective precipitation (K-tab.2) with criterion of compatibility of the balance volume of runoff (V). Figure 3   Table 6 presents the results of a calculation parameter W for 3 validated precipitation (K).Accuracy of mapping of tested phenomena precipitation-outflow through the created model of canalized storm water catchment were presented cumulatively for 8 precipitation, i.e. 5 from calibration and 3 from validation. Figure 4 shows the correspondence with the volume balance of runoff (V).

Fig. 1 .
Fig. 1.The scheme of the analyzed storm water drainage in SWMM program.

Fig. 2 .
Fig. 2. A histogram of precipitation and hydrographs of flow into the collector KD1 (P1) for the latest lowland precipitation from October 22 nd to 23 rd 2014.
shows an example of histogram of convective precipitation from July 19 th 2015 and hydrographs of drainage (measured and simulated) in collector KD1 in the intersection of flowmeter P1.

Fig. 3 .
Fig. 3.A histogram of precipitation and hydrographs of flow into the collector KD1 (P1)for convective precipitation from July 19 th 2015.

Fig. 4 .
Fig. 4. The volume balance of runoff for precipitation from calibration and validation model.

Figure 4
Figure 4 shows that for 8 precipitations (from calibration and validation model) the volume of runoff describes the simple equation Vs ≈ Vp where R 2 = 0.981.This means high compliance of the simulated (S) and measured (P) volumes of precipitations runoff -14 results of measurements and simulation is in the range accuracy ± 10%.

Table 1 .
Summary of characteristic area and slope of sub-catchments.

Table 3 .
Categories for the classification of the models.Calibration of hydrodynamic model was based on determining the values of parameters: hydraulic -roughness rate of channels (n -to Manning formula) and roughness rate of sealed (npu) and unsealed (npnu) catchment areas,  hydrological -the height of retention on sealed (h pu) and unsealed (hpnu) catchment areas.Hydrogeological parameters of model (infiltration) were estimated by model of expert.As a result of the simulation calculations in SWMM program for 5 long-term precipitation calibration (type F and N-table 2) the empirical parameters of the model were determined.In table 4 the literature (output) and resulting values of the parameters of model of the tested drainage system were listed.

Table 4 .
The value of the calibration parameters of the model of tested drainage system.

Table 5 .
The results of the calculations of the calibration model.