# Probabilistic Approach to Tank Design in Rainwater Harvesting Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Probabilistic Modeling of a Rainwater Tank

#### 2.2. Continuous Simulation

#### 2.3. Model Validation

#### 2.4. Application

^{2}and an average consumption of 20 L/day per person for toilet flushing, those values correspond to a total water demand of 15, 20 and 25 users, respectively. The water demand scenarios analyzed are related to different ratios between mean annual water demand and mean annual water supply from rainfall (Table 1).

## 3. Results and Discussion

_{1}), σ

^{2}(Δ

_{1}), µ

_{Δ,}σ

_{Δ}

^{2}and afterwards (step 4) using Equations (9) and (10) to obtain F

_{Δ(n)}(x), f

_{Δ(n)}(x), and µ

_{Δ}+ with Equation (11). At this point, the CDF of the active storage F

_{W}(w) was estimated with Equation (16). The obtained F

_{W}(w) for the defined water demands and rainfall data from Milan are shown in Figure 3.

_{W}(w). The continuous simulation was held with the operation rule in Equations (17) and (18) on a storm-event-based step with a storage capacity sufficiently large to avoid spills, obtaining the active storage after each event. The empirical distribution function from the continuous simulation of active storage is plotted on Figure 3.

^{2}and volume in m

^{3}. A volumetric reliability index (I

_{v}) was obtained as the ratio of total volume supplied and requested. The volume supplied was estimated using the operating rule in Equations (17) and (18) for the selected volumes from the entire historical long-term rainfall series. The I

_{v}obtained for the rainfall series from 1971–2017 in Milan, considering the three demands refer on this study, showed agreement with the probability obtained from the model, as can also be observed in Figure 4, assessing that the model provides accuracy comparable with continuous simulation using an analytical equation. Decisionmakers can use this methodology coupled with a cost–benefit analysis to establish the optimal level of probability of failure for each project, for example, by considering local water tariff and tank costs per m

^{3}.

_{v}obtained, as shown in Table 5. Considering the volume for T = 50 years from Table 5, volumes from Ds were, respectively, larger of +87.39%, +137.04% and 102.73%. The overestimation observed with the Ds approach can be explained by the fact that the methodology completely neglects the rainfall stochastic process and defines the volume basely solely on the longest dry period, while the probabilistic approach presented considers the alternation of inflows and outflows and the consequent deficit sub-periods or surplus sub-periods. The total storage capacity designed with the Ds approach may be frequently unused, as the available inflow is insufficient to fill the tank completely. For large-scale projects, this unused volume is translated as a considerable cost factor that does not reflect proportionally on the I

_{v}. For this reason, normal Ds is used as a preliminary design tool that is then adjusted with long-term continuous simulations. However, long-term rainfall data is not often available, neither RWH projects are held by hydrology specialists. The probabilistic approach presented in this research gives a relatively simple analytical equation that can be used as a design tool but provides robustness comparable to continuous simulation.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Whitmee, S.; Haines, A.; Beyrer, C.; Boltz, F.; Capon, A.G.; De Souza Dias, B.F.; Ezeh, A.; Frumkin, H.; Gong, P.; Head, P.; et al. Safeguarding human health in the Anthropocene epoch: Report of the Rockefeller Foundation-Lancet Commission on planetary health. Lancet
**2015**, 386, 1973–2028. [Google Scholar] [CrossRef] [PubMed] - Cook, B.I.; Mankin, J.S.; Anchukaitis, K.J. Climate Change and Drought: From Past to Future. Curr. Clim. Chang. Rep.
**2018**, 4, 164–179. [Google Scholar] [CrossRef] - Monier, E.; Gao, X. Climate change impacts on extreme events in the United States: An uncertainty analysis. Clim. Chang.
**2015**, 131, 67–81. [Google Scholar] [CrossRef][Green Version] - Mukherjee, S.; Mishra, A.; Trenberth, K.E. Climate Change and Drought: A Perspective on Drought Indices. Curr. Clim. Chang. Rep.
**2018**, 4, 145–163. [Google Scholar] [CrossRef] - Dada, A.; Urich, C.; Berteni, F.; Pezzagno, M.; Piro, P.; Grossi, G. Suya Duyarlı Şehirler: Parma’da (Kuzey İtalya) Kentsel Sel Direncini Artırmak İçin Bütünleşik Bir Yaklaşım-Water sensitive cities: An integrated approach to enhance urban flood resilience in Parma (Northern Italy). Climate
**2021**, 9, 152. [Google Scholar] [CrossRef] - Jacobson, C.R. Identification and quantification of the hydrological impacts of imperviousness in urban catchments: A review. J. Environ. Manag.
**2011**, 92, 1438–1448. [Google Scholar] [CrossRef] - Miller, J.D.; Kim, H.; Kjeldsen, T.R.; Packman, J.; Grebby, S.; Dearden, R. Assessing the impact of urbanization on storm runoff in a peri-urban catchment using historical change in impervious cover. J. Hydrol.
**2014**, 515, 59–70. [Google Scholar] [CrossRef][Green Version] - Scalenghe, R.; Ajmone-Marsan, F. The anthropogenic sealing of soils in urban areas. Landsc. Urban Plan.
**2009**, 90, 1–10. [Google Scholar] [CrossRef] - Qadir, M.; Sharma, B.R.; Bruggeman, A.; Choukr-Allah, R.; Karajeh, F. Non-conventional water resources and opportunities for water augmentation to achieve food security in water scarce countries. Agric. Water Manag.
**2007**, 87, 2–22. [Google Scholar] [CrossRef] - Campisano, A.; Butler, D.; Ward, S.; Burns, M.J.; Friedler, E.; Debusk, K.; Fisher-jeffes, L.N.; Ghisi, E.; Rahman, A.; Furumai, H. Urban rainwater harvesting systems: Research, implementation and future perspectives. Water Res.
**2017**, 115, 195–209. [Google Scholar] [CrossRef] - Alim, M.A.; Rahman, A.; Tao, Z.; Samali, B.; Khan, M.M.; Shirin, S. Suitability of roof harvested rainwater for potential potable water production: A scoping review. J. Clean. Prod.
**2020**, 248, 119226. [Google Scholar] [CrossRef] - Judeh, T.; Shahrour, I.; Comair, F. Smart Rainwater Harvesting for Sustainable Potable Water Supply in Arid and Semi-Arid Areas. Sustainability
**2022**, 14, 9271. [Google Scholar] [CrossRef] - Burns, M.J.; Fletcher, T.D.; Hatt, B.E.; Anthony, R.; Walsh, C.J. Can allotment-scale rainwater harvesting manage urban flood risk and protect stream health? In Proceedings of the 7th International Conference on Sustainable Techniques and Strategies for Urban Water Management 2010, Lyon, France, 27 June–1 July 2010; pp. 1–10. [Google Scholar]
- Freni, G.; Liuzzo, L. Effectiveness of rainwater harvesting systems for flood reduction in residential urban areas. Water
**2019**, 11, 1389. [Google Scholar] [CrossRef][Green Version] - Peng, J.; Zhang, X.M.; Zhang, Y.H. Study on combining flood control with rainwater utilization of airports in China. IOP Conf. Ser. Earth Environ. Sci.
**2018**, 191, 012133. [Google Scholar] [CrossRef] - Schuetze, T.; Chelleri, L. Integrating decentralized rainwater management in urban planning and design: Flood resilient and sustainable water management using the example of coastal cities in The Netherlands and Taiwan. Water
**2013**, 5, 593–616. [Google Scholar] [CrossRef][Green Version] - Hamel, P.; Fletcher, T.D. The impact of stormwater source-control strategies on the (low) flow regime of urban catchments. Water Sci. Technol.
**2014**, 69, 739–745. [Google Scholar] [CrossRef] - Teston, A.; Piccinini Scolaro, T.; Kuntz Maykot, J.; Ghisi, E. Comprehensive Environmental Assessment of Rainwater Harvesting Systems: A Literature Review. Water
**2022**, 14, 2716. [Google Scholar] [CrossRef] - Campisano, A.; Modica, C. Optimal sizing of storage tanks for domestic rainwater harvesting in Sicily. Resour. Conserv. Recycl.
**2012**, 63, 9–16. [Google Scholar] [CrossRef] - Londra, P.A.; Kotsatos, I.E.; Theotokatos, N.; Theocharis, A.T.; Dercas, N. Reliability analysis of rainwater harvesting tanks for irrigation use in greenhouse agriculture. Hydrology
**2021**, 8, 132. [Google Scholar] [CrossRef] - Palla, A.; Gnecco, I.; Lanza, L.G. Non-dimensional design parameters and performance assessment of rainwater harvesting systems. J. Hydrol.
**2011**, 401, 65–76. [Google Scholar] [CrossRef] - Palla, A.; Gnecco, I. On the Effectiveness of Domestic Rainwater Harvesting Systems to Support Urban Flood Resilience. Water Resour. Manag.
**2022**, 36, 5897–5914. [Google Scholar] [CrossRef] - Ward, S.; Memon, F.A.; Butler, D. Rainwater harvesting: Model-based design evaluation. Water Sci. Technol.
**2010**, 61, 85–96. [Google Scholar] [CrossRef] [PubMed] - Ghisi, E. Parameters Influencing the Sizing of Rainwater Tanks. Water Resour. Manag.
**2010**, 24, 2381–2403. [Google Scholar] [CrossRef] - Fewkes, A.; Butler, D. Simulating the performance of rainwater collection and reuse systems using behavioural models. Build. Serv. Eng. Res. Technol.
**2000**, 21, 99–106. [Google Scholar] [CrossRef] - Fewkes, A.; Warm, P. Method of modelling the performance of rainwater collection systems in the United Kingdom. Build. Serv. Eng. Res. Technol.
**2000**, 21, 257–265. [Google Scholar] [CrossRef] - Lopes VA, R.; Marques, G.F.; Dornelles, F.; Medellin-azuara, J. Performance of Rainwater Harvesting Systems under Scenarios of Non-Potable Water Demand and Roof area Typologies using a Stochastic Approach. J. Clean. Prod.
**2017**, 148, 304–313. [Google Scholar] [CrossRef] - Mitchell, V.G.; Mccarthy, D.T.; Deletic, A.; Fletcher, T.D. Urban stormwater harvesting e sensitivity of a storage behaviour model. Environ. Model. Softw.
**2008**, 23, 782–793. [Google Scholar] [CrossRef] - Bacchi, B.; Balistrocchi, M.; Grossi, G. Proposal of a semi-probabilistic approach for storage facility design. Urban Water J.
**2008**, 5, 195–208. [Google Scholar] [CrossRef] - Becciu, G.; Raimondi, A.; Dresti, C. Semi-probabilistic design of rainwater tanks: A case study in Northern Italy. Urban Water J.
**2018**, 15, 192–199. [Google Scholar] [CrossRef] - Becciu, G.; Raimondi, A. An analytical probabilistic approach for sizing rainwater tanks. In Acqua e Città 2011: Pianificazione, Protezione e Gestione; La Loggia, G., Paoletti, P., Beccin , G., Eds.; Centro Studi Idraulica Urbana: Brescia, Italy, 2011; pp. 1–13. [Google Scholar]
- Cheng, G.; Huang, G.; Guo, Y.; Baetz, B.W.; Dong, C. Stochastic Rainwater Harvesting System Modeling Under Random Rainfall Features and Variable Water Demands. Water Resour. Res.
**2021**, 57, e2021WR029731. [Google Scholar] [CrossRef] - Raimondi, A.; Becciu, G. On pre-filling probability of flood control detention facilities. Urban Water J.
**2015**, 12, 344–351. [Google Scholar] [CrossRef] - Raimondi, A.; Becciu, G. Performance of Green Roofs for Rainwater Control. Water Resour. Manag.
**2021**, 35, 99–111. [Google Scholar] [CrossRef] - Raimondi, A.; Marchioni, M.; Sanfilippo, U.; Becciu, G. Infiltration-exfiltration systems design under hydrological uncertainty. WIT Trans. Built Environ.
**2020**, 194, 143–154. [Google Scholar] [CrossRef] - Raimondi, A.; Marchioni, M.; Sanfilippo, U.; Becciu, G. Vegetation survival in green roofs without irrigation. Water
**2021**, 13, 136. [Google Scholar] [CrossRef] - Raimondi, A.; Marchioni, M.; Sanfilippo, U.; Stroppa, F.F.; Becciu, G. Probabilistic Estimation of Runoff From Green Roofs. Int. J. Comput. Methods Exp. Meas.
**2022**, 10, 13–25. [Google Scholar] [CrossRef] - Wang, J.; Guo, Y. Proper Sizing of Infiltration Trenches Using Closed-Form Analytical Equations. Water Resour. Manag.
**2020**, 34, 3809–3821. [Google Scholar] [CrossRef] - Zhang, S.; Guo, Y. Analytical Equation for Estimating the Stormwater Capture Efficiency of Permeable Pavement Systems. J. Irrig. Drain. Eng.
**2015**, 141, 06014004. [Google Scholar] [CrossRef] - Raimondi, A.; Di Chiano, M.G.; Marchioni, M.; Sanfilippo, U.; Becciu, G. Probabilistic Modelling of Sustainable Urban Drainage Systems. Urban Ecosyst.
**2022**. [Google Scholar] [CrossRef] - Guo, Y.; Baetz, B.W. Sizing of Rainwater Storage Units for Green Building Applications. J. Hydrol. Eng.
**2007**, 12, 197–205. [Google Scholar] [CrossRef] - Guo, R.; Guo, Y. Stochastic modelling of the hydrologic operation of rainwater harvesting systems. J. Hydrol.
**2018**, 562, 30–39. [Google Scholar] [CrossRef] - García, V.J.; García-Bartual, R.; Cabrera, E.; Arregui, F.; Garca-Serra, J. Stochastic Model to Evaluate Residential Water Demands. J. Water Resour. Plan. Manag.
**2004**, 130, 386–394. [Google Scholar] [CrossRef] - Barbosa, L.R.; Almeida, C.D.N.; Coelho, V.H.R.; Freitas, E.D.S.; Galvão, C.D.O.; Araújo, J.C.D. Sub-hourly rainfall patterns by hyetograph type under distinct climate conditions in Northeast of Brazil: A comparative inference of their key properties. Rbrh
**2018**, 23, 1–14. [Google Scholar] [CrossRef][Green Version] - Brasil, J.B.; Guerreiro, M.S.; de Andrade, E.M.; de Queiroz Palácio, H.A.; Medeiros, P.H.A.; Ribeiro Filho, J.C. Minimum Rainfall Inter-Event Time to Separate Rainfall Events in a Low Latitude Semi-Arid Environment. Sustainability
**2022**, 14, 1721. [Google Scholar] [CrossRef] - Dunkerley, D. Identifying individual rain events from pluviograph records: A review with analysis of data from an Australian dryland site. Hydrol. Process. Int. J.
**2008**, 22, 5024–5036. [Google Scholar] [CrossRef] - Freitas, E.D.S.; Coelho, V.H.R.; Xuan, Y.; de CD Melo, D.; Gadelha, A.N.; Santos, E.A.; Galvão, C.D.O.; Ramos Filho, G.M.; Barbosa, L.R.; Huffman, G.J.; et al. The performance of the IMERG satellite-based product in identifying sub-daily rainfall events and their properties. J. Hydrol.
**2020**, 589, 125128. [Google Scholar] [CrossRef] - Sattari, M.T.; Rezazadeh-Joudi, A.; Kusiak, A. Assessment of different methods for estimation of missing data in precipitation studies. Hydrol. Res.
**2017**, 48, 1032–1044. [Google Scholar] [CrossRef] - Balistrocchi, M.; Bacchi, B. Modelling the statistical dependence of rainfall event variables through copula functions. Hydrol. Earth Syst. Sci.
**2011**, 15, 1959–1977. [Google Scholar] [CrossRef][Green Version]

**Figure 3.**Empirical CDF from continuous simulation and modeled CDF of the active storage W of the tank for different water demand (D = 1.2 mm/day; D = 1.6 mm/day; D = 2.0 mm/day).

**Figure 4.**Active storage CDF obtained with the probabilistic model for different demands and volumetric reliability index (I

_{v}).

**Table 1.**Mean annual water demand A

_{d}as a percentage of the mean annual water supply A

_{s}for the different demand scenarios considered.

Case | D (mm/day) | A_{d} ^{1} (mm) | n° of Persons ^{2} | A_{s} ^{3} (mm) | A_{d}/A_{s} (%) |
---|---|---|---|---|---|

(a) | 1.2 | 438 | 15 | 929.7 | 47 |

(b) | 1.6 | 584 | 20 | 929.7 | 63 |

(c) | 2.0 | 730 | 25 | 929.7 | 79 |

^{1}Average annual water demand.

^{2}Assuming an average need for toilet flushing of 20 L/day per user.

^{3}Average annual water supply (A

_{s}= φ$\xb7P$ where P is the mean annual precipitation).

**Table 2.**Average values and the standard deviations, per event, of the three rainfall variables used in the modeling (rainfall depth h, rainfall duration θ, and interevent time d).

Milan (1971–2017) ^{1} | |
---|---|

μ_{h} [mm] | 7.17 |

μ_{θ} [days] | 0.14 |

μ_{d} [days] | 2.29 |

σ_{h} [mm] | 11.98 |

σ_{θ} [days] | 0.20 |

σ_{d} [days] | 4.34 |

^{1}Rainfall time series from Milano–Monviso gauge station.

**Table 3.**Correlation index among rainfall depth and rainfall duration (ρ

_{h,θ}), rainfall duration and interevent time (ρ

_{θ,d}) and rainfall depth and interevent time (ρ

_{h,d}) in Milan.

Correlation Index | |
---|---|

ρ_{h,θ} (-) | 0.714 |

ρ_{θ,d} (-) | −0.005 |

ρ_{h,d} (-) | 0.018 |

D (mm/day) | Parameter n (-) |
---|---|

1.2 | 1 |

1.6 | 1 |

2 | 2 |

**Table 5.**Volumetric reliability indexes (I

_{v}) as function of water demand and return period T obtained with probabilistic model and the “demand-side” approach (Ds).

D = 1.2 (mm/day) | D = 1.6 (mm/day) | D = 2 (mm/day) | |||||
---|---|---|---|---|---|---|---|

T (years) | F (-) | W (m^{3}) | I_{v} (-) | W (m^{3}) | I_{v} (-) | W (m^{3}) | I_{v} (-) |

2 | 0.5 | 1.42 | 0.53 | 1.58 | 0.47 | 2.48 | 0.50 |

5 | 0.8 | 3.26 | 0.74 | 3.60 | 0.66 | 5.40 | 0.67 |

10 | 0.9 | 4.73 | 0.82 | 5.18 | 0.74 | 7.88 | 0.75 |

20 | 0.95 | 6.08 | 0.86 | 6.75 | 0.80 | 10.13 | 0.80 |

50 | 0.98 | 8.33 | 0.91 | 8.78 | 0.85 | 12.83 | 0.84 |

Ds | 15.60 | 0.97 | 20.80 | 0.95 | 26.00 | 0.92 |

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**MDPI and ACS Style**

Di Chiano, M.G.; Marchioni, M.; Raimondi, A.; Sanfilippo, U.; Becciu, G.
Probabilistic Approach to Tank Design in Rainwater Harvesting Systems. *Hydrology* **2023**, *10*, 59.
https://doi.org/10.3390/hydrology10030059

**AMA Style**

Di Chiano MG, Marchioni M, Raimondi A, Sanfilippo U, Becciu G.
Probabilistic Approach to Tank Design in Rainwater Harvesting Systems. *Hydrology*. 2023; 10(3):59.
https://doi.org/10.3390/hydrology10030059

**Chicago/Turabian Style**

Di Chiano, Maria Gloria, Mariana Marchioni, Anita Raimondi, Umberto Sanfilippo, and Gianfranco Becciu.
2023. "Probabilistic Approach to Tank Design in Rainwater Harvesting Systems" *Hydrology* 10, no. 3: 59.
https://doi.org/10.3390/hydrology10030059