# A Simplified Model for Modular Green Roof Hydrologic Analyses and Design

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Typical Green Roof Configurations and Hydrologic Processes

#### 2.2. Governing Equations

_{s}) is equal to the height of the saturated water surface in the growth media. The water content profile and the hydraulic conductivity profile are approximated by a water retention function (θ(H)) Equation (4) from van Genuchten [14], and the hydraulic conductivity function (K(H)) Equation (5) by Mualem [15], respectively. Water content in the system is calculated by Equations (6)–(8).

^{3}/cm

^{3}) is the volumetric water content; θ

_{s}(cm

^{3}/cm

^{3}) is saturated water content; θ

_{r}(cm

^{3}/cm

^{3}) is residual water content; α, n, m (m = 1 − 1/n) are hydraulic retention curve parameters of growth media; l is the soil tortuosity parameter; H, h (cm) are height; H

_{s}is where θ = θ

_{s}; K (cm/min) is hydraulic conductivity; K

_{s}is the saturated hydraulic conductivity; A

_{o}/A

_{s}is the ratio of the drainage opening area to green roof surface area in each module; f

_{b}, and f

_{w}are sizing ratios, f

_{b}+ f

_{w}= 1.

#### 2.3. Green Roof Runoff Hydrograph Simulation

_{v}) and peak reduction (R

_{p}) were calculated as the percentage of reduced quantity over precipitation volume or maximum peak flow (Equations (9) and (10)). Peak delay (R

_{t}) was calculated as percentage of delayed time normalized by precipitation duration minus peak time (in order to compare the peak delays of precipitations with various durations) (Equation (11)).

_{v}, R

_{p}, and R

_{t}are volume reduction, peak reduction, and peak delay; D and d are precipitation depth and runoff depth (cm); P and p are precipitation peak and runoff peak rate (cm/min); T and t are precipitation peak time and runoff peak time (min); L is precipitation duration (min).

#### 2.4. Computer Simulation

_{1}, H

_{2}, f

_{a}, f

_{b}, f

_{w}; growth media hydraulic properties θ

_{r}, θ

_{s}, K

_{s}, α, n, l; ET (typically 0.6~0.8 ET

_{0}for green roofs); and precipitation time series. In this study, precipitation events were generated from Natural Resources Conservation Service (NRCS) Type I synthetic precipitations. Durations were set to 30, 60, 120, 180, 360, 720, and 1440 min and the depths of each synthetic precipitation event of each duration varied from 0 cm to 40 cm at an increment of 0.5 cm. A total of 567 precipitation events were simulated on each of 21 different green roof configurations to study the hydrologic behavior of green roofs. Outputs of this model are the hydrograph of total runoff vs. time and calculated hydrologic performance results (R

_{v}, R

_{p}, and R

_{t}).

## 3. Calibration

#### 3.1. Pilot Green Roof Experimental Data

#### 3.2. Calibration and Sensitivity of the Simplified Model

_{s}and K

_{s}are supplied by the growth media manufacturers. The growth medium used in this experiment was perlite. Typical hydraulic property parameters (θ

_{r}, α, and n) for perlite and other growth media can be found in the literature [17]. Values for all of these parameters were measured in the pilot green roof at the University of Hawaii and used to calibrate a HYDRUS-2D model as described elsewhere [16].

_{s}, α, θ

_{s}and θ

_{r}are approximately equally sensitive. θ

_{0}does not affect the results since the model quickly rebalances the water content when irrigation occurs. This calibrated model was used for extrapolation [19] to hypothetical green roofs of greater depth and to synthetic precipitation events of different durations and magnitudes as described in the following section.

## 4. Results and Discussion

#### 4.1. Simulation Results of 21 Configurations

_{e}was calculated as the amount of water retained in the green roof module without causing substantial drainage. Under the C

_{e}condition, the water retained in the module will still experience very slow but negligible drainage. The maximum capacity (C

_{m}) was calculated as the amount of water retained when the module is fully saturated less residual water content. The water retention difference between C

_{m}and C

_{e}is called transient capacity (C

_{t}) first defined by Kasmin et al. [19]. The C

_{t}is mainly determined by the growth media depth above the drainage openings. Transient water content is subject to rapid and immediate draining, and has important effects on peak flow and peak delay.

_{e}is usually much smaller than the C

_{m}. R

_{v}increases with increased depth of media (compare gr510, gr520, gr530 and gr540), increased drainage opening height (additional media storage) (compare gr330, gr530, gr1030 and gr1530), and added water reservoir storage (compare gr510 and gr510w; gr1020 and gr1020w; gr1530 and gr1530w). It also shows that adding water storage volume is more effective at increasing R

_{v}than increasing media depth (because the higher drainage openings take advantage only of the media porosity for additional storage whereas the reservoir is all water) (compare gr510 with gr510w and gr540; gr2010 with gr2010w and gr2040). Peak flow reductions and peak delay are greatly affected by C

_{t}.

#### 4.2. Hydrologic Performance Curves

_{e}and precipitation depth when precipitation exceeds C

_{e}(Equation (12)). Volume reduction does not seem to be affected by transient capacity since this water will slowly drain away eventually. Further, precipitation dynamics only affect volume reduction slightly as will be discussed in the following section.

_{v}is the volume reduction; C

_{e}is the effective capacity; and D is precipitation depth. (This equation tends to overestimate volume reduction by up to 10% in comparison with the model since in reality, water stored in the green roof continues to drain after the precipitation/irrigation event)

_{max}, underflow is the maximum drainage underflow rate (cm/min); A

_{o}/A

_{s}is the ratio of the drainage opening area to green roof surface area; and K

_{s}is the saturated hydraulic conductivity (cm/min).

_{m}, surface runoff occurs, and both peak reduction and peak delay decrease quickly. The initial point of rebound depends on the C

_{e}of the green roof. The rebound phase ends when surface runoff starts, and the length of the rebound depends on the draining speed, precipitation duration, and C

_{t}. Consequently, C

_{e}and C

_{t}are two controlling factors of peak reduction and peak delay.

#### 4.3. Hydrologic Performance Simulation Results and Analyses

_{v}, R

_{p}, or R

_{t}. Volume reduction (Figure 4a) of all configurations decreased sharply as the precipitation depth increased and varied only slightly due to precipitation duration (e.g., 540–30 min vs. 540–1440 min curves). Comparing various configurations, R

_{v}(Figure 4a) increased as the height of drainage opening increased (e.g., 12,710 vs. 310 curves), as the depth of growth media increased (e.g., 310 vs. 2040 curves), and when a water storage reservoir was added (e.g., 2040 vs. 2040w curves). Raising the height of the drainage opening resulting in increased R

_{v}; however, it cannot be placed so high to result in the drowning of the plants. Adding the water storage reservoir greatly increased R

_{v}as shown by gr2040w, which is far superior to the gr2040 and all other configurations. It is noted that the value of the water storage ratio used herein was 0.8 and that this is a design parameter. If smaller values of this ratio are employed, then less R

_{v}will be realized. Adding water storage is more effective than increasing media depth (without adding storage) to increase C

_{e}and consequently increase R

_{v}. It is again noted that the simulations in Figure 4 are for individual events with initial water content of 10% (dry antecedent conditions). If larger values of initial water content are employed, all of the curves would shift toward the upper right. In the worst case, the soil would be saturated at the start of the event and there would be no reduction of volume, peak or peak delay. All actual single event performances will fall between these best-case and worst-case antecedent scenarios.

_{e}of 3.18 cm and a C

_{t}of 3.18 cm. Scanning through Table 2, all of the configurations with a total thickness greater than 20 cm achieved 25% or greater on both R

_{v}and R

_{p}, except gr12720. The gr12720 is slightly low in R

_{v}(20.1%), but achieved 39% R

_{p}. The gr320, gr520, gr1020 and gr1020w would all probably have the same module cost; however the 1020w would require less growth media and have much better overall performance than the others. It would also be possible to use the gr510w configuration to meet the LEED requirements while using half as much media, which should lead to a significant cost savings compared to the gr1020w. This capacity-based approach could substitute for the performance-based standard commonly used by LEED to manage future stormwater projects with reasonable precision. This could simplify the design process and give some confidence in expected performance for a given media type.

_{p}is a non-linear function determined by the combined effects of infiltration speed, drainage speed, growth media depth, water storage, and precipitation duration. The function was derived in Li and Babcock (2015). In this study, the maximum drainage speed was the same for all modules, and infiltration speed was based on the hydraulic properties of one specific growth media that we evaluated. As shown in Figure 4b, R

_{p}was not improved much when raising drainage height (comparing gr12710 and gr510), and although increasing C

_{e}resulted in a better increase in R

_{p}as seen in configurations gr510 and gr510w or gr1030, gr2040, and gr2040w compared with raising the drainage height, increasing the media depth resulted in a superior increase in R

_{p}(comparing gr510 and gr520). The C

_{t}temporarily stored the water and slowed down the discharge. This shows the special effect of C

_{t}on R

_{p}. R

_{p}drops drastically as precipitation duration increases, which suggests that green roofs are more efficient in reducing peaks of intense events because maximum drainage speed is fixed once the system is designed.

_{p}, R

_{t}is also a non-linear function determined by the combined effects of infiltration speed, drainage speed, growth media depth, water storage, and precipitation duration [16]. Increasing C

_{e}increased R

_{t}before the rebound as seen in configurations gr2040 and gr540 (Figure 4c). Increasing C

_{t}drastically increased the rebound trajectory when comparing gr530 and gr540. Further, adding water storage improved the overall R

_{t}as seen in gr510, gr510w and gr2040, gr2040w. Clearly, C

_{e}and C

_{t}are two major factors affecting R

_{t}, but similar to R

_{p}, R

_{t}becomes less significant as precipitation duration increases as seen in the gr12710 simulation series (30-min to 1440-min).

#### 4.4. Long-Term Runoff Volume Reductions

_{v}of 21 green roof configurations. Evaluations of R

_{p}and R

_{t}are only relevant for individual events and were irrelevant to this analysis. The simulation used a one-year historical precipitation record during the green roof pilot test period at the University of Hawaii [16]. Reference evapotranspiration (ET

_{0}) was calculated by the Penman-Monteith method [23] and ET was 0.65ET

_{0}. The use of a crop coefficient of 0.65 to estimate ET was based on our pilot green roof operations [16]; however, in general, the value can be adjusted for any type of green roof vegetation employed. Irrigation was operated as needed at a rate of 0.5 cm/day when the water content was below 10% and when there was no precipitation. Precipitation data were acquired from a rooftop weather station. See Figure 5a.

_{v}increased with increases in the growth media depth and the drainage opening height, and with the addition of water storage. For example, gr520 reduced 7% more annual precipitation volume than gr510, and gr1020 reduced 10% more volume than gr320. R

_{v}increased dramatically for modules with water storage. Module gr510w was better than all the modules to the left of gr540 in the chart; the 20-cm module gr1020w (67.8%) was slightly better than the 40-cm module gr1540, and was 27.6% better than the same thickness module gr12720. To summarize: raising the drainage opening increased R

_{v}by an average of 4.6%; increasing the total media depth increased R

_{v}by an average of 5.3%; and installing water storage increased R

_{v}by an average of 23.5% for the configurations simulated in this study. R

_{v}varied with evapotranspiration which was the source of water consumption. The error bars denote the reduction efficiency for variations of ±10% for the ET estimate. In summary, the most effective method to increase the cumulative R

_{v}efficiency is to install water storage to increase C

_{e}.

## 5. Conclusions

_{r}, θ

_{s}, K

_{s}, α, n, l) from literature or laboratory tests; (3) obtain/prepare precipitation event (NRCS or NOAA Atlas 14) or long term data, and ET

_{0}estimations (Penman-Montieth [24] or from weather station); (4) utilize the simplified model to calculate hydrologic performance; and (5) evaluate the results and iterate the design as needed.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic layout of a green roof system and its hydrologic mechanisms. h

_{1}and h

_{2}(cm) are the thickness of lower and upper growth media; k

_{1}, k

_{2}, and k

_{s}(cm/min) are hydraulic conductivities at the drainage, at the surface and at saturation; θ, θ

_{r}, and θ

_{s}(cm

^{3}/cm

^{3}) are the media water content, residual water content, and saturated water content; and f

_{a}, f

_{b}, and f

_{w}are sizing ratios, f

_{a}= 1 and f

_{b}+ f

_{w}= 1.

**Figure 4.**Volume reduction (R

_{v}) (

**a**); peak reduction (R

_{p}) (

**b**); and peak delay (R

_{t}) (

**c**) vs. precipitation depth.

**Figure 5.**Irrigation, precipitation, and evapotranspiration from November 2012 to November 2013 (

**a**) and cumulative runoff reductions of various green roof designs; error bars denote reduction efficiency of ±10% reference evapotranspiration values (

**b**).

Variable | θ_{r} | θ_{s} | α | n | K_{s} | l |
---|---|---|---|---|---|---|

Unit | cm^{3}/cm^{3} | cm^{3}/cm^{3} | cm^{−1} | - | cm/min | - |

Lab Measurement | 0.05–0.16 | 0.55–0.82 | 0.19–0.37 | 1.71–2.17 | 8.00–10.00 | 0.50 |

Calibration Simplified Model | 0.05 | 0.60 | 0.29 | 1.80 | 10.00 | 0.50 |

**Table 2.**Simulated green roof configurations with resulting water content capacities and hydrologic performance results assuming dry antecedent conditions.

Design | H_{1} | H_{2} | f_{a} | f_{b} | f_{w} | C_{e} | C_{t} | C_{m} | R_{v} * | R_{p} * | R_{t} * |
---|---|---|---|---|---|---|---|---|---|---|---|

cm | cm | - | - | - | cm | cm | cm | % | % | % | |

gr12710 | 1.27 | 10 | 1 | 0.6 | 0 | 2.12 | 3.4 | 5.52 | 12.77 | 33.52 | 1.01 |

gr12715 | 1.27 | 15 | 1 | 0.6 | 0 | 2.83 | 5.43 | 8.26 | 17.05 | 36.60 | 1.55 |

gr12720 | 1.27 | 20 | 1 | 0.6 | 0 | 3.43 | 7.58 | 11.01 | 20.08 | 40.59 | 1.97 |

gr310 | 3 | 10 | 1 | 1 | 0 | 2.61 | 2.88 | 5.49 | 18.20 | 36.12 | 2.07 |

gr320 | 3 | 20 | 1 | 1 | 0 | 4.03 | 6.95 | 10.98 | 26.76 | 40.85 | 2.13 |

gr330 | 3 | 30 | 1 | 1 | 0 | 5.03 | 11.44 | 16.47 | 31.33 | 53.24 | 2.71 |

gr340 | 3 | 40 | 1 | 1 | 0 | 5.82 | 16.14 | 21.96 | 34.18 | 65.05 | 3.54 |

gr510 | 5 | 10 | 1 | 1 | 0 | 3.14 | 2.35 | 5.49 | 23.14 | 28.19 | 1.49 |

gr510w | 5 | 10 | 1 | 0.2 | 0.8 | 5.59 | 1.70 | 7.29 | 41.46 | 65.50 | 5.96 |

gr520 | 5 | 20 | 1 | 1 | 0 | 4.69 | 6.29 | 10.98 | 33.14 | 52.38 | 2.74 |

gr530 | 5 | 30 | 1 | 1 | 0 | 5.76 | 10.71 | 16.47 | 38.41 | 70.40 | 4.56 |

gr540 | 5 | 40 | 1 | 1 | 0 | 6.58 | 15.38 | 21.96 | 41.41 | 79.13 | 6.69 |

gr1020 | 10 | 20 | 1 | 1 | 0 | 6.72 | 4.26 | 10.98 | 48.76 | 83.48 | 10.74 |

gr1020w | 10 | 20 | 1 | 0.2 | 0.8 | 11.00 | 3.59 | 14.59 | 81.93 | 94.95 | 71.12 |

gr1030 | 10 | 30 | 1 | 1 | 0 | 7.98 | 8.49 | 16.47 | 55.77 | 89.55 | 22.37 |

gr1040 | 10 | 40 | 1 | 1 | 0 | 8.90 | 13.06 | 21.96 | 59.82 | 91.88 | 30.99 |

gr1530 | 15 | 30 | 1 | 1 | 0 | 10.15 | 6.32 | 16.47 | 72.48 | 93.78 | 56.49 |

gr1530w | 15 | 30 | 1 | 0.2 | 0.8 | 16.23 | 5.65 | 21.88 | 100.00 | 100.00 | 99.98 |

gr1540 | 15 | 40 | 1 | 1 | 0 | 11.21 | 10.75 | 21.96 | 77.72 | 94.84 | 68.07 |

gr2040 | 20 | 40 | 1 | 1 | 0 | 13.47 | 8.49 | 21.96 | 93.24 | 98.27 | 89.10 |

gr2040w | 20 | 40 | 1 | 0.2 | 0.8 | 21.36 | 7.82 | 29.18 | 100.00 | 100.00 | 99.98 |

_{v}, R

_{p}, and R

_{t}are simulation results of volume reduction, peak reduction, and peak delay, respectively, for a 2-year, 24-h storm (12.7-cm depth).

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**MDPI and ACS Style**

Li, Y.; Babcock, R.W.
A Simplified Model for Modular Green Roof Hydrologic Analyses and Design. *Water* **2016**, *8*, 343.
https://doi.org/10.3390/w8080343

**AMA Style**

Li Y, Babcock RW.
A Simplified Model for Modular Green Roof Hydrologic Analyses and Design. *Water*. 2016; 8(8):343.
https://doi.org/10.3390/w8080343

**Chicago/Turabian Style**

Li, Yanling, and Roger W. Babcock.
2016. "A Simplified Model for Modular Green Roof Hydrologic Analyses and Design" *Water* 8, no. 8: 343.
https://doi.org/10.3390/w8080343