Outflow adjustment coefficient for the design of storage facilities using the rain envelope method applied to Brazilian state capitals

Detention devices are often used as alternative measures for stormwater control. The Envelope Curve Method is widely used in Brazil to estimate detention device volumes. This method estimates the storage volume based on inlet and outlet balance, where the inlet is obtained by the Rational Method and the outlet by orifice bottom discharge. Usually, the outlet flow is adopted as a constant and equivalent to the maximum allowed, and this procedure can cause reservoir undersizing. This paper evaluates detention control measures’ hydraulic behavior for the Envelope Curve Method and proposes the inclusion of an outflow adjustment coefficient (Cout), seeking to compensate for the adoption of constant outlet flow simplification. Values for this coefficient were estimated for several Brazilian state capitals, ranging from 0.62 up to 0.65. The undersizing hypothesis due to the adoption of constant outlet flow was confirmed, as the simulations showed the need for an increase between 8.4% to 16.8% in the device size. This undersizing may be compensated for by applying the outflow adjustment coefficient (Cout).


INTRODUCTION
The urbanization process modifies the physical characteristics of the watershed, increasing impervious surfaces and changing the hydrologic cycle. These changes cause an increase in stormwater runoff volume and peak flow and decrease flow time (Chen et al., 2015;Guan et al., 2016;Zhou, 2014).
Their impacts on the hydrologic cycle result in an increase in the frequency of water-related disasters such as floods and, as a consequence, there are stormwater control measures (SCMs) that seek to compensate for the effects of urban occupation. Among the SCMs adopted, storage measures stand out, which restrict the inlet hydrograph, providing temporary storage of the runoff. Several municipalities in Brazil, such as Porto Alegre, Brasília, and São José do Rio Preto, have adopted these measures for new construction in order to control hydrological impacts, with the requirement that reservoir construction include an outflow restriction of flow to the public drainage system (Brasília, 2018;Porto Alegre, 2014;São José do Rio Preto, 2008) There are several methods for design or preliminary design of detention facilities, among them the rain envelope method, which is often used in Brazil. Butler et al. (2018) presents the envelope method as a preliminary sizing of reservoirs, and Azzout et al. (1994) presents it as a simplified sizing method. Baptista et al. (2011) andMiguez et al. (2015) call the methodology the rainfall method of envelope curves, and present it as a simplified form of structure design. Some examples of use from the rain envelope method can be seen in Lucas et al. (2013), in the design of a filter-swale-trench system; Angelini Sobrinha et al. (2012), in the design of an infiltration well; and in Cadore et al. (2016), for bioretention areas. In addition, Silveira and Goldenfum (2007) presented a generalized methodology for preliminary sizing of SCMs using the envelope curve method.
The envelope curve method is characterized as being simple and straightforward, with several simplifications, among which we highlight the adoption of the outflow rate as a constant equal to the rate of outflow when the storage facility is full and the outlet is under maximum head, which may cause under-dimensioning in the structures. These simplifications are the core of criticism of adopting the sizing results of this method needing further sizing checks. On the other hand, there is no approach to redress the undersizing effect on the results by envelope curve use.
This article evaluates the hydraulic behavior of stormwater control detention measures designed by the rain envelope method, proposes a change in this methodology with the creation of an outflow adjustment coefficient (Cout) in order to correct the underestimation caused by using the outflow rate as a constant value.

Rain envelope method or rainfall method
The rain envelope method or rainfall method, presented by Urbonas and Stahre (1993), is a simple method for determination of detention volume based on mass balance in a detention facility. It uses curves of cumulative runoff volume and cumulative volume of outflow at 3 Outflow adjustment coefficient for the design of … Rev. Ambient. Água vol. 16 n. 6, e2707 -Taubaté 2021 different durations of rainfalls, where the largest difference between the two curves expresses the storage volume. The method does not consider the time of concentration of the basin and uses the rational method to estimate the flow contribution to the detention facility, being indicated to areas smaller than around 80 ha (ASCE, 2017).
The first step in this method is to calculate the cumulative runoff volume for a range of storm durations. This is done by incrementally increasing the storm duration and, for each duration, the volume is calculated from Equation 1, using the runoff coefficient (C), the ration between the runoff and the respective precipitation, and the average rainfall intensity determined by an intensity-duration-frequency (IDF) curve. = . . .
The volume of the outflow curve varies according to design characteristics. For detention facilities with nozzles or orifices as outlets, the outflow can be determined, considering free discharge at atmospheric pressure, using Equation 2. The volume of outflow is commonly estimated by Equation 3, using the simplification that the structure empties at a constant rate equal to the maximum possible outlet outflow (considering the device full and outlet under maximum head). As already shown by Urbonas and Stahre (1993), with the exception of special flow regulators the outflow is not constant and varies with the depth of water as the structure fills and empties. ( Where: = discharge coefficient ; = area of the opening (m²); = gravitational acceleration (m.s -2 ); ℎ = effective head seen by the orifice (m); = cumulative outflow volume (m³); = maximum outflow rate (m³.s -1 ); = duration (s).
To provide the controlled release of flows when the detention facility's water exceeds its storage volume, commonly spillways are used. The discharge flow of the spillway can be determined by Equation 4.
The storage volume obtained by the method is the largest difference between the two curves obtained by Equations 1 and 3, defined by Equation 5 and illustrated in Figure 1.
The simplification of using, during all rainfall periods, the outflow rate equal to the maximum possible outlet flow causes a cumulative outflow volume curve with values greater than what they are supposed to be, resulting in lower volumes of reservation, reducing the safety of the project.

Curve envelope method verification
In order to identify the impact of the simplification of the reservoirs designed by the envelope method, the hydraulic behavior of the structure storage volume obtained by the envelope curve method was simulated using one of the most traditional models to simulate reservoir routing; it was developed by Puls (Nascimento and Baptista, 2009).
The IDF curve used to obtain the rainfall used for the envelope curve and Puls method was the recommended IDF curve for the 8 th District of Porto Alegre (Bemfica et al., 2000) with a return period of 10 years (Equation 6). For the envelope curve method, the characteristics of the tributary site were: tributary area (A) of 1000 m² and runoff coefficient (C) of 0.9. The outlet was considered as an orifice with diameter of 25 mm, and the input for the outflow rate was considered a constant obtained by Equation 3, with a maximum depth of water of 1 m and a discharge coefficient of 0.94, according to experimental values found for this outlet diameter and water depth (Drummond, 2014). From this information, with Equation 5, the storage volume was obtained.
These characteristics were chosen for to be smaller than the Porto Alegre outflow restriction flow to the public drainage system (20.8 l.s -1 .ha -1 or 2.08 l.s -1 for this tributary area) (Porto Alegre, 2014).
For the Puls method, the same IDF curve was used, with a time distribution obtained by the alternating blocks hyetograph method (Chow et al., 1988), centralized peak and rain discretization of 1 minute in order to obtain a hyetograph.
The inflow hydrograph of the reservoir was determined from the hyetograph multiplied by the same values of runoff coefficient and tributary area for each time interval of the hyetograph, thus obtaining the inflow hydrograph. The outflow rate was defined by Equation 3, with the same characteristics of outlet for envelope curve, but considering the outflow rate variable in time, function of water depth. In case of exceeding storage volume during the simulation, a spillway was designed to remove excess water, with the discharge flow obtained by Equation

Reservoir hydraulic behavior with traditional envelope curve method storage volume
For the rain and tributary characteristics described in the method verification, the envelope curve is shown in Figure 2. The constant outflow rate of the method (Qlim) resulted in 2.04 l.s -1 and the storage volume was 40.2 m³, occurring with a rainfall duration of 102 minutes.  Figure 3 presents the reservoir outflow hydrographs for the Puls method, with simulations for rainfall durations from 10 to 120 minutes for Porto Alegre by the alternating blocks method. The storage volume of the reservoirs used in all simulations were the ones found using the envelope method, 40.2 m³. The results show that the design volume was not adequate to withstand the runoff volumes of rainfall durations above 60 minutes; therefore, the 102 minutes of rainfall duration, duration from the envelope method, also had water depth above 1 m, causing the spillway to remove the excess water and the outflow rate to peak above 2.04 l.s -1 . The difference between the outflow estimated by the verification method (Puls) and the constant outflow for the envelope method is shown in Figure 4.a, simulation of rainfall duration of 102 minutes for Porto Alegre, where the green area of the figure represents the outflow volume difference between the methods.

Proposed methodology to compensate for the constant outflow simplification
An outflow adjustment coefficient (Cout) was created in order to compensate for the simplification of constant outflow, adopting the ratio between the accumulated volume obtained by the Puls methodology (an area under the red line in Figure 4.b up to reach the blue line for constant outflow) and the constant outflow (a rectangular area in Figure 4.c under blue line until to intercept the red line). The coefficient was obtained using the same rainfall duration as found for the storage volume in the envelope curve, called "critical duration time" (Tcrit), and gradually increasing the storage volume until the minimum detention volume that would not cause the water depths above the maximum, i.e., spillway unused (Equation 7).
From the Cout, the incremental volume, denominated as , is determined by Equation 8. The incremental volume is then added to the storage volume determined by the envelope curve method, resulting in the adjusted storage volume (Equation 9).

Sensitivity of Cout to input parameters
The sensitivity of Cout was evaluated by simulating different runoff coefficients and sizes of orifices as outlets, modifying the inflow and outflow of the methods. The same methodology of the previous verification was used. The sensitivity of Cout as a function of the Runoff coefficient C and Qlim, constant outflow as a function of orifice diameter (D), for a return period of 10 years is shown in Table 1. The value of C coefficient varied from 0.2 to 1.0, and the maximum flow capacity from 1.31 l.s -1 to 8.18 l.s -1 , respectively, for commercial diameters of orifices of 20 mm and 50 mm.
The results showed that the value of the Cout increases when the value of the C reduces, and that Cout is also higher when the diameter of the orifice is higher. Therefore, the combination of higher C and smaller diameter results in a smaller coefficient, which is the less favorable scenario, since the smaller the Cout the greater the incremental volume .
In order to observe the impact of a higher period return on the coefficient, the process was repeated for a period return of 50 years. The coefficients found are shown in Table 2 and were similar to those with a period return of 10 years.

Cout applied to Brazilian state capitals
According to the results of Tables 1 and 2, the least favorable condition for the coefficient Cout, the smallest value, occurs with a higher C and smaller orifice diameter. Cout values were determined for several Brazilian state capitals, using Equation 7 with the same methodology previously described. The adopted conditions were the same as used in the curve envelope method verification: coefficient C equal to 0.9; outlet considered as an orifice as 25 mm of diameter; discharge coefficient of 0.94; maximum water depth of 1 m; tributary area of 1000 m². The rain parameters for each state capital are presented in Table 3, following the format of Equation 10. Figure 5 shows the studied cities locations in Brazil.
Where: , , are adjusted parameters for each city. Even though there is a great variability in the pluviometric characteristics of the Brazilian state capitals presented, there is only a small variation in the values of Cout. Table 5 presents the storage volume ( ) needed for each city for a return period of 10 years using the envelope curve method and the adjusted storage volume ( ) when considering the coefficient Cout.
The results showed a Cout coefficient ranging from 0.62 to 0.65 and an increase in storage   Bemfica et al. (2000); 10 Bertoni and Tucci (1993); 11 Wilken (1978).  The results are similar to those found by Guo (1999), 0.64 to 0.75 for an outflow reduction coefficient and he also concludes that not using the adjustment factor can result in approximately 20% underestimation of detention volume.

DISCUSSION AND CONCLUSIONS
This paper evaluated the performance of storage facilities designed by the envelope curve method and showed undersizing in the storage volume when considering a simplification of the outflow rate as a constant value equal to the maximum discharge capacity of the outlet. The hydraulic verification results showed that the reservoirs were insufficient to store the inflow volumes. Thus, an outflow adjustment coefficient Cout was proposed in order to compensate for the considered simplification. The application of the proposed methodology was applied to several Brazilian state capitals and indicated there is little variation in the value of Cout, ranging from 0.62 to 0.65, even though there is great variability in the pluviometric characteristics of these cities. The results also showed that, for the studied cities, the adjusted methodology caused an increase in the storage volume from 8.4% to 16.8%.
Finally, the parameters adopted (lot area; runoff coefficient; return period; orifice diameter; IDF of 17 Brazilian States Capital) for the simulations were the most frequent values for the urban lots, so this criteria can indeed be representative for several areas and even for the whole country after analysis of results for Cout, as it varies only within a narrow range.