# Peak Flows and Stormwater Networks Design—Current and Future Management of Urban Surface Watersheds

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Numerical Modeling in Open Channel Gradually Varied Unsteady Flow

#### 2.1.1. Saint-Venant Equations Succinctly Revisited

#### 2.1.2. Overview of the Developed Numerical Integration Process

_{x(i)}, U

_{x(i)}, hm

_{x(i)}, Rh

_{x(i)}, and J between adjacent times t(j) and t(j+1) are respectively:

#### 2.1.3. Initial Conditions under Steady Flow Modelled and Implemented

#### 2.1.4. Boundary Conditions Modelled and Implemented

_{0}, at the downstream end of the reach (the sole downstream boundary condition implemented in the original implicit hydrodynamic model) [22]. That is, an approximation similar to the one that is assumed in the kinematic wave model [27,45]. These simplifications in the upstream hydrograph and in the downstream boundary condition make it possible to consider each link separately and sequentially in the downstream direction. It is then possible to articulate a design hydraulic model in steady uniform flow (that calculates, for each link, the sewer diameters and invert elevations using the peak flow of the input hydrograph in the upstream node of the link), with the simulation model in unsteady flow that calculates the input hydrographs that result from the simulation of the upstream links with sewer pipes already sized and installed. Another approximation mentioned in the literature in the case of free downstream discharge is the consideration of normal depth in the node for supercritical flows or critical depth in the node for subcritical flows [32]. For flow control structures, such as weirs or orifices, the second boundary condition is the discharge law of the hydraulic structure used.

#### 2.1.5. Application of the Implemented Hydrodynamic Model

#### 2.2. Classic Rational Formulation, and Simplified Surface Runoff Hydrographs, Revisited

_{max}(Tr), in a small drainage basin with upstream area Ad for a given frequency which is equal to the inverse of the return period, Tr, occurs when the entire area of the drainage basin is contributing to the flow; and (ii) this flow is a fraction of the average rain for the same frequency (1/Tr) with maximum average intensity I (Tr, Tc), which is constant both in space, along Ad, and in time, during Tc, where Tc is the time of concentration of the basin. Tc is originally defined as the travel time between the point kinematically more distant from the basin and the section for which the maximum flow is being calculated. The formulation may be then expressed as:

_{max}(Tr) for the same intensity of precipitation. It does not look consensual if the shape must be triangular, trapezoidal, or other, and if the peak flow reached must be a fraction of Q

_{max}(Tr) given by Tp/Tc, a fraction given by 2Tp/(Tp + Tc), as proposed by Reference [48], or any other value.

_{max}(Tr); the beginning of the recession limb after Tp; and to reduce C (for Tp ≥ Tc) by (Tp + Tc)/2Tc. Other Engineering approximations maintaining the approximated triangular shape, runoff coefficient, and slope of the rising limb are eventually possible. One conceivable solution is to consider that the duration of the recession limb may be increased and approximated to 2Tc − Tp, instead of Tc. Another alternative is to assume that the duration of the rising limb may be increased and approximated to 2TpTc/(Tp + Tc), instead of Tp, but with the total duration of the hydrograph, Tp + Tc, unchanged, reaching in this case the same peak flow as the one proposed by Reference [48], but not at Tp.

^{6})] and a, b, and c are constants determined using a suitable adjustment method for the existing records for intense rainfall of short duration in the region. It should be noted that in some correlations, b = 0 or alternatively c = 1. The intensity of precipitation used for computing the maximum or peak flow, that is, the design flow in the rational method, decreases when the time of concentration of the basin in the section of the sewer pipe considered increases, that is, in the downstream direction of the network, given that IDF curves for a given Tr diminish with Tp. The design flow in each section is thus computed for successive lower intensities of precipitation in the downstream direction. For the same area drained in one reach that does not receive any additional lateral inflow along it, the flow then decreases in the downstream direction. Applying Equations (12) and (13) for any reach in these circumstances, for a given Tr and for Tp = Tc, the ratio between the maximum flow in any section of the sewer reach, ${\mathrm{Q}}_{\mathrm{max}2}$, and the maximum flow in the upstream extremity of the reach, ${\mathrm{Q}}_{\mathrm{max}1}$, with times of concentration $\mathrm{Tc}2$ and $\mathrm{Tc}1$, respectively, may then be expressed by:

## 3. Results and Discussion

#### 3.1. Comparison between the Rational Approach and the Results of Numerical Simulation in a Single Link

^{−1/3}s, for each slope and diameter tested, and the average velocities in the transversal sections for these flows, both fictitious and real, that is, h/D = 1 and h/D = 0.82 (assuming n constant). Table 1 also provides estimates of the magnitudes of the time of concentration at sewer entrances, Tc1, and of the drainage areas served, Ad, considering in this last case the abovementioned coefficients for the IDF curves (with Tr = 10 years) and average limiting values for C in Equation (12).

^{1/3}s

^{−1}and the time step and weighting factor used were Δt = 1 s and ψ = 0.55, respectively. All terms of the dynamic equation were normally considered (complete dynamic model). However, in some circumstances of supercritical flows (larger diameters, steeper conduit slopes, and larger flows, e.g., diameters of 1 and 2 m with S0 = 0.5%), local and convective accelerations were neglected in the equation of dynamics (diffusive wave model) in order to meet the convergence requirements of the iterative process by successive approximations.

_{0}= 0.5% and the larger diameters of 1 and 2 m. In both cases, they were considered constant along the pipe length, which is on the safe side, otherwise the peak flows obtained with the rational method would decrease even more.

#### 3.2. Comparison between the Rational Approach and the Results of Numerical Simulation in a Branched Network

^{1/3}s

^{−1}that was assumed to be constant with flow depth, the pipes aligned through the internal top crown, and a maximum relative water depth of 0.800 (which allows an extra vacant capacity until the maximum capacity reached about h/D = 0.94). The pluvial pipes are concrete. The IDF curves used were for Portuguese territory and had a return period of 10 years [Equation (13), with a = 290.68, b = 0, and c = 0.549].

_{max}occurring for a long time (see Figure 2c). Then, if the previously mentioned assumptions of the rational method were valid, particularly the premise that the maximum flow in a given section occurs precisely for Tp = Tc in that section, or, in other words, when the whole of the drainage basin upstream of the section is contributing to forming the flow, the method seems to be an excellent approximation when the time of precipitation is much longer than the time of concentration/entrance of the individual surface sub-basins, as occurred in the downstream section of the network presented here. However, for precipitations of lower duration than Tc of the whole of the basin upstream, and thus more intense, the response of the drainage system upstream is faster and the maximum flows reached in a given section can be clearly higher. Thus, the maximum flow can occur when only part of the upstream basin is draining for the respective section, producing larger peak flows than those computed with the classic formulation, as is extensively shown in Figure 5.

#### 3.3. Some Practical Consequences

#### 3.4. Maximum Design Flows and Simulation in Unsteady Flow

- (i)
- Calculate or estimate the time of concentration, Tc, which is the sum of the entrance time and the sewer travel time that can be estimated in steady uniform flow for a full or partially filled section, for example, and the intensity of precipitation, I, for Tp = Tc, using IDF curves. Determine the inlet surface hydrographs in each upstream node of the section or node under consideration. If the section or node is not an upstream extremity, calculate the input hydrograph resulting from the simulation in unsteady flow of all upstream links. Add the surface hydrograph of the node and identify the maximum flow reached;
- (ii)
- With the maximum design flow obtained, determine, in steady uniform flow conditions, the diameter, the invert elevations, and the slope of the downstream sewer link. Move to the next node of the sequential process and repeat steps (i) and (ii) successively until the whole of the network has been sized and routed.

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Moore, T.L.; Rodak, C.M.; Vogel, J.R. Urban stormwater characterization, control, and treatment. Water Environ. Res.
**2017**, 89, 1876–1927. [Google Scholar] [CrossRef] [PubMed] - Zhou, Z.; Smith, J.A.; Yang, L.; Baeck, M.L.; Chaney, M.; Veldhuis, M.-C.T.; Deng, H.; Liu, S. The complexities of urban flood response: Flood frequency analyses for the Charlotte metropolitan region. Water Resour. Res.
**2017**, 53, 7401–7425. [Google Scholar] [CrossRef] [Green Version] - Guan, M.; Sillanpää, N.; Koivusalo, H. Modelling and assessment of hydrological changes in a developing urban catchment. Hydrol. Process.
**2015**, 29, 2880–2894. [Google Scholar] [CrossRef] [Green Version] - Lyu, H.M.; Sun, W.J.; Shen, S.L.; Arulrajah, A. Flood risk assessment in metro systems of mega-cities using a GIS-based modeling approach. Sci. Total Environ.
**2018**, 626, 1012–1025. [Google Scholar] [CrossRef] [PubMed] - Lyu, H.M.; Shen, S.L.; Zhou, A.; Yang, J. Perspectives for flood risk assessment and management for mega-city metro system. Tunnel. Undergr. Space Technol.
**2019**, 84, 31–44. [Google Scholar] [CrossRef] - Stewart, R.D.; Lee, J.G.; Shuster, W.D.; Darner, R.A. Modelling hydrological response to a fully-monitored urban bioretention cell. Hydrol. Process.
**2017**, 31, 4626–4638. [Google Scholar] [CrossRef] - Winston, R.J.; Dorsey, J.D.; Hunt, W.F. Quantifying volume reduction and peak flow mitigation for three bioretention cells in clay soils in northeast Ohio. Sci. Total Environ.
**2016**, 553, 83–95. [Google Scholar] [CrossRef] - Zhang, S.; Guo, Y. Stormwater capture efficiency of bioretention systems. Water Resources Management.
**2014**, 28, 149–168. [Google Scholar] [CrossRef] - Hixon, L.F.; Dymond, R.L. Comparison of stormwater management strategies with an urban watershed model. J. Hydrol. Eng.
**2015**, 20, 04014091. [Google Scholar] [CrossRef] - Campisano, A.; Modica, C. Rainwater harvesting as source control option to reduce roof runoff peaks to downstream drainage systems. J. Hydroinform.
**2016**, 18, 23–32. [Google Scholar] [CrossRef] - Wella-Hewage, C.S.; Hewa, G.A.; Pezzaniti, D. Can water sensitive urban design systems help to preserve natural channel-forming flow regimes in an urbanised catchment? Water Sci. Technol.
**2016**, 73, 78–87. [Google Scholar] [CrossRef] [PubMed] - Chenevey, B. Development and Its Impact on the Water Balance of an Urban Watershed. Master’s Thesis, University of Cincinnati, Cincinnati, OH, USA, 2013. [Google Scholar]
- Stovin, V.; Vesuviano, G.; Kasmin, H. The hydrological performance of a green roof test bed under UK climatic conditions. J. Hydrol.
**2012**, 414–415, 148–161. [Google Scholar] [CrossRef] - Jefferson, A.J.; Bhaskar, A.S.; Hopkins, K.G.; Fanelli, R.; Avellaneda, P.M.; McMillan, S.K. Stormwater management network effectiveness and implications for urban watershed function: A critical review. Hydrol. Process.
**2017**, 31, 4056–4080. [Google Scholar] [CrossRef] - Diogo, A.F.; Oliveira, M.C. A simplified approach for the computation of steady two-phase flow in inverted siphons. J. Environ. Manag.
**2016**, 166, 294–308. [Google Scholar] [CrossRef] [PubMed] - Diogo, A.F.; Barros, L.T.; Santos, J.; Temido, J.S. An effective and comprehensive model for optimal rehabilitation of separate sanitary sewer systems. Sci. Total Environ.
**2018**, 612, 1042–1057. [Google Scholar] [CrossRef] [PubMed] - ASCE; WPCF. Design and construction of sanitary and storm sewers. In ASCE—Manuals and Reports on Engineering Practice; No. 37; ASCE: Washington, DC, USA, 1969. [Google Scholar]
- ASCE; WEF. Design and construction of urban stormwater management systems. In ASCE—Manuals and Reports on Engineering Practice; No. 77; ASCE: Virginia, NV, USA, 1992. [Google Scholar]
- Matos, M.R. Métodos de Análise e de Cálculo de Caudais Pluviais em Sistemas de Drenagem Urbana. Tese para Obtenção do Grau de Especialista, LNEC, Lisboa. 1987. Available online: http://livraria.lnec.pt/eng/php/livro_ficha.php?cod_produc_tirag=5398956 (accessed on 10 April 2019).
- Ben-Zvi, A. Toward A New Rational Method. J. Hydraul. Eng.
**1989**, 115, 1241–1255. [Google Scholar] [CrossRef] - MOPTC. Regulamento Geral dos Sistemas Públicos e Prediais de Distribuição de Água e de Drenagem de Águas Residuais. Decreto Regulamentar no. 23/95 de 23 de Agosto, Diário da República—I Série-B. 1995, pp. 5284–5319. Available online: https://dre.pt/application/file/a/431921 (accessed on 10 April 2019).
- Diogo, A.F. Optimização Tridimensional de Sistemas Urbanos de Drenagem. Ph.D. Thesis, Faculdade de Ciências e Tecnologia da Universidade de Coimbra, Coimbra, Portugal, 1996. [Google Scholar]
- Dhakal, N.; Fang, X.; Cleveland, T.; Thompson, D. Revisiting modified rational method. In Proceedings of the World Environmental and Water Resources Congress 2011: Bearing Knowledge for Sustainability, Palm Springs, CA, USA, 22–26 May 2011; pp. 751–762. [Google Scholar]
- NYC Environmental Protection (New York City Department of Environmental Protection) in Consultation with the New York City Department of Buildings. Guidelines for the Design and Construction of Stormwater Management Systems. 2012. Available online: http://www.nyc.gov/html/dep/pdf/green_infrastructure/stormwater_guidelines_2012_final.pdf (accessed on 25 November 2017).
- NC DEQ (North Carolina Department of Environmental Quality). Stormwater Design Manual—B. Stormwater Calculations; 2017. Available online: https://deq.nc.gov/sw-bmp-manual (accessed on 11 November 2017).
- Amein, M.; Chu, H.-L. Implicit numerical modeling of unsteady flows. J. Hydraul. Divis.
**1975**, 101, 717–731. [Google Scholar] - Froise, S.; Burges, S.J. Least-cost design of urban-drainage networks. J. Water Resour. Plan. Manag. Divis.
**1978**, 104, 75–92. [Google Scholar] - Sousa, E.R. Técnicas de simulação em sistemas de drenagem de águas pluviais. In Seminário 290; LNEC: Lisboa, Portugal, 1983; Volume 2, pp. 91–134. [Google Scholar]
- Almeida, A.B. Análise hidráulica de colectores de águas pluviais. In Seminário 290; LNEC: Lisboa, Portugal, 1983; Volume 2, pp. 27–89. [Google Scholar]
- Chow, V.T.; Maidment, D.R.; Mays, L.W. Applied Hydrology; McGraw-Hill: New York, NY, USA, 1988. [Google Scholar]
- Diogo, A.F.; Sousa, E.R.; Graveto, V.M.; Santos, F.S. Modelação hidráulica em colectores de sistemas urbanos de drenagem. In VII Encontro Nacional de Saneamento Básico; DEC/FCTUC: Coimbra, Portugal, 1996; Volume 1, pp. 177–186. [Google Scholar]
- Ji, Z. General hydrodynamic model for sewer/channel network systems. J. Hydraul. Eng.
**1998**, 124, 307–315. [Google Scholar] [CrossRef] - Carmo, J.S.A. Modelação em Hidráulica Fluvial e Ambiente, 2nd ed.; Imprensa da Universidade de Coimbra: Coimbra, Portugal, 2009. [Google Scholar]
- Rossman, L.A. Storm Water Management Model, Quality Assurance Report: Dynamic Wave Flow Routing; National Risk Management Research Laboratory, Office of Research and Development, U.S. Environmental Protection Agency (US EPA): Washington, DC, USA, 2006.
- Rossman, L.A. Storm Water Management Model User's Manual, Version 5.0; National Risk Management Research Laboratory, Office of Research and Development, U.S. Environmental Protection Agency (US EPA): Washington, DC, USA, 2010.
- Liu, H.; Wang, H.; Liu, S.; Hu, C.; Ding, Y.; Zhang, J. Lattice Boltzmann method for the Saint-Venant equations. J. Hydrol.
**2015**, 524, 411–416. [Google Scholar] [CrossRef] - EPA (US Environmental Protection Agency). Storm Water Management Model User's Manual Version 5.1; EPA: Washington, DC, USA, 2015.
- Fread, D.L. Technique for implicit dynamic routing in rivers with tributaries. Water Resour. Res.
**1973**, 9, 918–926. [Google Scholar] [CrossRef] - Ponce, V.M.; Simons, D.B.; Indlekofer, H. Convergence of four-point implicit water wave models. J. Hydraul. Divis.
**1978**, 104, 947–958. [Google Scholar] - Quintela, A.C. Hidráulica, 8th ed.; Fundação Calouste Gulbenkian: Lisboa, Portugal, 2002. [Google Scholar]
- Sen, D.J.; Garg, N.K. Efficient algorithm for gradually varied flows in channel networks. J. Irrigation Drain. Eng.
**2002**, 128, 351–357. [Google Scholar] [CrossRef] - Islam, A.; Raghuwanshi, N.S.; Singh, R.; Sen, D.J. Comparison of gradually varied flow computation algorithms for open–channel network. J. Irrigation Drain. Eng.
**2005**, 131, 457–465. [Google Scholar] [CrossRef] - Zhu, D.; Chen, Y.; Wang, Z.; Liu, Z. Simple, robust, and efficient algorithm for gradually varied subcritical flow simulation in general channel networks. J. Hydraul. Eng.
**2011**, 137, 766–774. [Google Scholar] [CrossRef] - Djordjević, S.; Prodanović, D.; Walters, G.A. Simulation of transcritical flow in pipe/channel networks. J. Hydraul. Eng.
**2004**, 130, 1167–1178. [Google Scholar] [CrossRef] - Cembrowicz, R.G.; Krauter, G.E. Design of cost optimal sewer networks. In Proceedings of the Fourth International Conference on Urban Storm Drainage, Lausanne, Switzerland, 31 August–4 September 1987; pp. 367–372. [Google Scholar]
- Costa, P.C. O método racional generalizado. Princípios conceptuais, domínio de aplicação e resultados. In Seminário 290; LNEC: Lisboa, Portugal, 1983; Volume 1, pp. 129–160. [Google Scholar]
- Guo, J.C.Y. Rational Hydrograph Method for Small Urban Watersheds. J. Hydrol. Eng.
**2001**, 6, 352–356. [Google Scholar] [CrossRef] - Chien, J.S.; Saigal, K.K. Urban Runoff by Linearized Subhydrograph Method. J. Hydraul. Divis.
**1974**, 100, 1141–1157. [Google Scholar] - Bennis, S.; Crobeddu, E. New Runoff Simulation Model for Small Urban Catchments. J. Hydrol. Eng.
**2007**, 12, 540–544. [Google Scholar] [CrossRef] - Guo, J.C.Y. Storm Hydrographs from Small Urban Catchments. IWRA Int. J.
**2000**, 25, 481–487. [Google Scholar] - Walesh, S.G. Discussion of Urban Runoff by Linearized Subhydrograph Method by Chien and Saigal 1974. J. Hydraul. Divis.
**1975**, 101, 1447–1449. [Google Scholar] - Dhakal, N.; Fang, X.; Thompson, D.B.; Cleveland, T.G. Modified rational unit hydrograph method and applications. Proc. Inst. Civ. Eng. Water Manag.
**2014**, 167, 381–393. [Google Scholar] [CrossRef] - Diogo, A.F.; Walters, G.A.; Sousa, E.R.; Graveto, V.M. Three-Dimensional Optimization of Urban Drainage Systems. Comput.-Aided Civ. Infrastruct. Eng.
**2000**, 15, 409–425. [Google Scholar] [CrossRef]

**Figure 1.**Initial conditions in the links in steady flow, and downstream boundary conditions in subcritical flow controlled downstream with hout > hc (a, b), and free discharge, hout < hc (c), or alternatively hout < hc/1.4 (d).

**Figure 2.**Classic rational formulation and simplified entrance hydrographs in urban stormwater networks. (A constant minimum/residual volumetric flow rate, Qbase, is assumed in all entrance hydrographs).

**Figure 3.**Simulation in open channel gradually varied unsteady flow versus rational method along a pipe sewer for known input symmetric triangular hydrographs in the upstream extremity for sewer slopes of 0.3 and 0.5%. (

**a**) Diameters of 200, 400, and 600 mm and different downstream boundary conditions. (

**b**) Diameters of 1 and 2 m, different downstream boundary conditions, and two criteria of average velocity in the rational method. (

**c**) Diameter of 1 m and different values of the exponent c of IDF curves in the rational method.

**Figure 5.**Comparison of simulation in open channel gradually varied unsteady flow for several times of precipitation with the rational method along a dendritic network.

**Figure 6.**Upstream and downstream hydrographs in the network downstream link simulated along a dendritic network for several times of precipitation. (

**a**) Base criteria. (

**b**) With a Tc increase in one branch.

**Figure 7.**Comparison of the flow hydrographs in the nodes and in the downstream end of the links for the dendritic network tested with respect to (i) the implicit simulation model considering each link separately and the normal depth as the link downstream boundary condition; (ii) the SWMM explicit model for the network as a whole considering a free discharge and the conduits aligned by the internal crown, SWMM–G1; and (iii) aligned by the invert elevation, SWMM–G2. (

**a**) Precipitation of 7.5 min. (

**b**) Precipitation of 12.5 min. Graph axes: horizontal—time (s); vertical—volumetric flow rate (L/s).

**Table 1.**Volumetric flow rates at full section (peak flow of the symmetric triangular hydrograph of base 2Tc1), average velocities in the transversal section, and estimates of the magnitude of times of concentration and drainage basin areas served for several pipe diameters and sewer slopes of 0.3 and 0.5% with the Manning coefficient n = 1/75 m

^{−1/3}s.

D (mm) | S0 = 0.3% | S0 = 0.5% | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Q (L/s) h/D = 1 | U (m/s) h/D = 0.82 | U (m/s) h/D = 1 | Tc1 (min) | Ad (ha) | Q (L/s) h/D = 1 | U (m/s) h/D = 0.82 | U (m/s) h/D = 1 | Tc1 (min) | Ad (ha) | |

200 | 17.5 | 0.636 | 0.558 | 10 | 0.1–0.15 | 22.6 | 0.820 | 0.720 | 10 | 0.1–0.2 |

400 | 111.2 | 1.009 | 0.885 | 15 | 0.6–1.2 | 143.6 | 1.302 | 1.143 | 15 | 1.0–1.5 |

600 | 327.9 | 1.322 | 1.160 | 20 | 2–4 | 423.3 | 1.707 | 1.497 | 20 | 3–5 |

1000 | 1280 | 1.858 | 1.630 | 30 | 10–20 | 1653 | 2.400 | 2.105 | 30 | 15–25 |

2000 | 8130 | 2.950 | 2.588 | 60 | 120–240 | 10496 | 3.808 | 3.341 | 60 | 150–300 |

**Table 2.**Sewers and partial drainage basins data, maximum volumetric flow rates, and base flows of the entrance hydrographs; and rational method maximum volumetric flow rates and maximum average velocities for the simplified dendritic network analyzed.

Sewers | Drainage Basins—Entrance Hydrographs | Rational Method | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Link | Length (m) | D (mm) | S0 (%) | Tc (min) | Useful Area (m ^{2}) | Entrance Maximum Flows (L/s)—Tp (min) | Base Flow (L/s) | Max. Flow (L/s) | Max. Vel. (m/s) | |||||||

Tp 7.5 | Tp 9.0 | Tp 10 | Tp 12.5 | Tp 20.4 | Tp 25.8 | Base | (a) | Base | (a) | |||||||

2-1 | 330.2 | 600 | 0.55 | 7.5 | 2400 | 64.1 | 58.0 | 54.7 | 48.4 | 37.0 | 32.5 | 2.0 | 435 | 376 | 1.79 | 1.76 |

3-2 | 245.2 | 600 | 0.34 | 9.0 | 11200 | 249.3 | 270.7 | 255.5 | 226.0 | 172.7 | 151.8 | 1.0 | 340 | 286 | 1.40 | 1.38 |

4-3 | 240.4 | 400 | 0.56 | 7.5 | 2800 | 74.8 | 67.7 | 63.9 | 56.5 | 43.2 | 38.0 | 5.0 | 148 | 121 | 1.37 | 1.34 |

5-4 | 280.2 | 400 | 0.42 | 7.5 | 4200 | 112.2 | 101.5 | 95.8 | 84.75 | 64.8 | 56.9 | 5.0 | 112 | 84.8 | 1.17 | 1.11 |

6-2 | 250.6 | 400 | 0.46 | 6.0 | 1600 | 42.7 | 38.7 | 36.5 | 32.3 | 24.7 | 21.7 | 1.0 | 100 | 100 | 1.19 | 1.19 |

7-6 | 310.2 | 300 | 0.88 | 10.0 | 3600 | 72.1 | 78.3 | 82.1 | 72.65 | 55.5 | 48.8 | 1.0 | 82.1 | 82.1 | 1.42 | 1.42 |

**Table 3.**Comparison of the maximum volumetric flow rates obtained in each link for the test network and for periods of precipitation of 7.5 and 12.5 min, applying the simulation model considering each link separately and the normal depth as the link downstream boundary condition, and those obtained using SWMM for the network as a whole considering a free discharge and the conduits aligned by the internal crown, SWMM–G1, and by the invert elevation, SWMM–G2.

Q Max Upstream (L/s) | ||||||||||||

Tp = 7.5 Min. | Tp = 12.5 Min. | |||||||||||

Links | 5-4 | 4-3 | 3-2 | 7-6 | 6-2 | 2-1 | 5-4 | 4-3 | 3-2 | 7-6 | 6-2 | 2-1 |

Sim. Model | 112.20 | 151.40 | 323.88 | 72.10 | 91.88 | 430.35 | 84.75 | 140.61 | 357.29 | 72.65 | 100.90 | 471.91 |

SWMM–G1 | 112.20 | 166.17 | 334.04 | 72.10 | 85.31 | 435.08 | 84.75 | 145.93 | 354.83 | 72.65 | 95.05 | 466.56 |

SWMM–G2 | 112.20 | 166.03 | 359.11 | 72.10 | 100.88 | 440.09 | 84.75 | 143.43 | 365.32 | 72.65 | 103.68 | 475.85 |

Relative Differences to Implicit Simulation Model (%) | ||||||||||||

Tp = 7.5 Min. | Tp = 12.5 Min. | |||||||||||

Links | 5-4 | 4-3 | 3-2 | 7-6 | 6-2 | 2-1 | 5-4 | 4-3 | 3-2 | 7-6 | 6-2 | 2-1 |

SWMM–G1 | – | 9.8 | 3.1 | – | −7.2 | 1.1 | – | 3.8 | −0.7 | – | −5.8 | −1.1 |

SWMM–G2 | – | 9.7 | 10.9 | – | 9.8 | 2.3 | – | 2.0 | 2.2 | – | 2.8 | 0.8 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Freire Diogo, A.; Antunes do Carmo, J.
Peak Flows and Stormwater Networks Design—Current and Future Management of Urban Surface Watersheds. *Water* **2019**, *11*, 759.
https://doi.org/10.3390/w11040759

**AMA Style**

Freire Diogo A, Antunes do Carmo J.
Peak Flows and Stormwater Networks Design—Current and Future Management of Urban Surface Watersheds. *Water*. 2019; 11(4):759.
https://doi.org/10.3390/w11040759

**Chicago/Turabian Style**

Freire Diogo, António, and José Antunes do Carmo.
2019. "Peak Flows and Stormwater Networks Design—Current and Future Management of Urban Surface Watersheds" *Water* 11, no. 4: 759.
https://doi.org/10.3390/w11040759