# Stormwater Uptake in Sponge-Like Porous Bodies Surrounded by a Pond: A Fluid Mechanics Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modelling of the Water Uptake of the Surrounding Pond for Different Conditions

#### 2.1. The Up-Flow SPB Storage Concept

#### 2.2. Governing Equations

#### 2.2.1. Mathematical Modelling for Case 1

#### 2.2.2. Mathematical Modelling for Case 2

#### 2.2.3. Mathematical Modelling for Case 3

#### 2.2.4. Mathematical Modelling for Case 4

#### 2.3. Parametric Study

## 3. Results

^{−10}(the function relies on the fourth-order Runge–Kutta method). Unless otherwise specified, the results are primarily presented for $r=\frac{{A}_{1}}{{A}_{2}}=0.35,{\varphi =0.7,h}_{0}=0.5\mathrm{m}$ and $I=0.68\xb7{10}^{-5}$ m/s per unit area with an average duration of $T=1\mathrm{h}$.

#### 3.1. Case 1

#### 3.2. Case 2

#### 3.3. Case 3 and Case 4

#### 3.4. Summary

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Case 1

**Figure A2.**Up-take height plotted at initial time stages: (

**a**) when dynamical pressure is not included and (

**b**) when dynamical pressure is included.

#### Appendix A.2. Case 2

#### Appendix A.3. Case 3

#### Appendix A.4. Case 4

## Appendix B

## Appendix C

## References

- Lundström, T.S.; Åkerstedt, H.O.; Larsson, I.A.S.; Marsalek, J.; Viklander, M. Dynamic Distributed Storage of Stormwater in Sponge-Like Porous Bodies: Modelling Water Uptake. Water
**2020**, 12, 2080. [Google Scholar] [CrossRef] - Marsalek, J.; Schreier, H. Innovation in Stormwater Management in Canada: The Way Forward. Water Qual. Res. J.
**2009**, 44, v–x. [Google Scholar] [CrossRef] - Toronto Region Conservation Authority. Evaluation of Residential Lot Level Stormwater Practices; Toronto Region Conservation Authority (TRCA): Toronto, ON, Canada, 2013. [Google Scholar]
- Åkerstedt, H.O.; Lundström, T.S.; Larsson, I.A.S.; Marsalek, J.; Viklander, M. Modeling the Swelling of Hydrogels with Application to Storage of Stormwater. Water
**2021**, 13, 34. [Google Scholar] [CrossRef] - Zarandi, M.A.F.; Pillai, K.M.; Kimmel, A.S. Spontaneous Imbibition of Liquids in Glass-Fiber Wicks. Part I: Usefulness of a Sharp-Front Approach. AIChE J.
**2018**, 64, 294–305. [Google Scholar] [CrossRef] - Faghihi Zarandi, M.A.; Pillai, K. Spontaneous Imbibition of Liquid in Glass Fiber Wicks, Part II: Validation of a Diffuse-Front Model. AIChE J.
**2017**, 64, 306–315. [Google Scholar] [CrossRef] - Vo, H.N.; Pucci, M.F.; Corn, S.; Le Moigne, N.; Garat, W.; Drapier, S.; Liotier, P.J. Capillary Wicking in Bio-Based Reinforcements Undergoing Swelling—Dual Scale Consideration of Porous Medium. Compos. Part A Appl. Sci. Manuf.
**2020**, 134, 105893. [Google Scholar] [CrossRef] - Foudazi, R.; Zowada, R.; Manas-Zloczower, I.; Feke, D.L. Porous Hydrogels: Present Challenges and Future Opportunities. Langmuir
**2023**, 39, 2092–2111. [Google Scholar] [CrossRef] - Khayamyan, S.; Lundström, T.S.; Hellström, J.G.I.; Gren, P.; Lycksam, H. Measurements of Transitional and Turbulent Flow in a Randomly Packed Bed of Spheres with Particle Image Velocimetry. Transp. Porous Media
**2017**, 116, 413–431. [Google Scholar] [CrossRef] - Alastal, K.; Ababou, R. Moving Multi-Front (MMF): A Generalized Green-Ampt Approach for Vertical Unsaturated Flows. J. Hydrol.
**2019**, 579, 124184. [Google Scholar] [CrossRef] - Caputo, J.-G.; Stepanyants, Y.A. Front Solutions of Richards’ Equation. Transp. Porous Media
**2008**, 74, 1–20. [Google Scholar] [CrossRef] - Richards, L.A. Capillary conduction of liquids through porous mediums. Physics
**1931**, 1, 318–333. [Google Scholar] [CrossRef] - Bear, J. 7.6 Direct Integration in One-Dimensional Problems. In Dynamics of Fluids in Porous Media; Environmental Science Series; American Elsevier: New York, NY, USA, 1972; ISBN 978-0-444-00114-6. [Google Scholar]
- Bear, J. 9.4 Unsaturated Flow. In Dynamics of Fluids in Porous Media; Environmental Science Series; American Elsevier: New York, NY, USA, 1972; ISBN 978-0-444-00114-6. [Google Scholar]
- Kuraz, M.; Holub, J.; Jerabek, J. Numerical Solution of the Richards Equation Based Catchment Runoff Model with Dd-Adaptivity Algorithm and Boussinesq Equation Estimator. Pollack Period.
**2017**, 12, 29–44. [Google Scholar] [CrossRef] - Piikki, K.; Söderström, M. Digital Soil Mapping of Arable Land in Sweden—Validation of Performance at Multiple Scales. Geoderma
**2019**, 352, 342–350. [Google Scholar] [CrossRef] - Detmann, B. Linear Elastic Wave Propagation in Unsaturated Sands, Silts, Loams and Clays. Transp. Porous Media
**2011**, 86, 537–557. [Google Scholar] [CrossRef] - Haghighatafshar, S.; Nordlöf, B.; Roldin, M.; Gustafsson, L.-G.; la Cour Jansen, J.; Jönsson, K. Efficiency of Blue-Green Stormwater Retrofits for Flood Mitigation—Conclusions Drawn from a Case Study in Malmö, Sweden. J. Environ. Manag.
**2018**, 207, 60–69. [Google Scholar] [CrossRef] - MSE-MyWebPages Melbourne School of Engineering. Available online: https://people.eng.unimelb.edu.au/stsy/geomechanics_text/Ch5_Flow.pdf (accessed on 11 March 2023).
- Woessner, W.W.; Eileen, P.P. Hydrogeologic Properties of Earth Materials and Principles of Groundwater Flow; The Groundwater Project: Guelph, ON, Canada, 2020. [Google Scholar]
- Yu, C.; Cheng, J.J.; Jones, L.G.; Wang, Y.Y.; Faillace, E.; Loureiro, C.; Chia, Y.P. Data Collection Handbook to Support Modeling the Impacts of Radioactive Material in Soil; USA, 1993; p. 152. Available online: https://resrad.evs.anl.gov/docs/data_collection_1993.pdf (accessed on 5 April 2023).
- Olsson, J.; Södling, J.; Berg, P.; Wern, L.; Eronn, A. Short-duration rainfall extremes in Sweden: A regional analysis. Hydrol. Res.
**2019**, 50, 945–960. [Google Scholar] [CrossRef] - Berggren, K. Urban Stormwater Systems in Future Climates—Assessment and Management of Hydraulic Overloading. Ph.D. Thesis, Luleå University of Technology, Luleå, Sweden, 2014. [Google Scholar]
- Zhao, G.; Wan, Y.; Lei, Z.; Liang, R.; Li, K.; Pu, X. Effect of Urban Underlying Surface Change on Stormwater Runoff Process Based on the SWMM and Green-Ampt Infiltration Model. Water Supply
**2021**, 21, 4301–4315. [Google Scholar] [CrossRef] - Vodák, R.; Fürst, T.; Šír, M.; Kmec, J. The Difference between Semi-Continuum Model and Richards’ Equation for Unsaturated Porous Media Flow. Sci. Rep.
**2022**, 12, 7650. [Google Scholar] [CrossRef] - Chali, A.K.N.; Hashemi, S.R.; Akbarpour, A. Numerical Solution of the Richards Equation in Unsaturated Soil Using the Meshless Petrov–Galerkin Method. Appl. Water Sci.
**2023**, 13, 119. [Google Scholar] [CrossRef] - Xiao, Y.; Zhu, Y. Study of the Water Vertical Infiltration Path in Unsaturated Soil Based on a Variational Method: Application of Power Function Distribution of D (θ). Hydrol. Sci. J.
**2022**, 67, 2254–2261. [Google Scholar] [CrossRef] - Wei, L.; Yang, M.; Li, Z.; Shao, J.; Li, L.; Chen, P.; Li, S.; Zhao, R. Experimental Investigation of Relationship between Infiltration Rate and Soil Moisture under Rainfall Conditions. Water
**2022**, 14, 1347. [Google Scholar] [CrossRef] - Timsina, R.C.; Khanal, H.; Ludu, A.; Uprety, K.N. Numerical Solution of Water Flow in Unsaturated Soil with Evapotraspiration. Nepali Math. Sci. Rep.
**2021**, 38, 35–45. [Google Scholar] [CrossRef]

**Figure 1.**Theoretical concepts: (

**a**) down-flow sponge-like porous body (SPB) storage; (

**b**) up-flow SPB storage with the pre-installed vertical structures expanding horizontally; and (

**c**) up-flow SPB storage with new vertical structures growing up from the ground when absorbing water. The figure is directly copied from [1].

**Figure 2.**Illustrative representations of the four scenarios showing water uptake of the up-flow storage device when surrounded by a pond: (

**a**) case 1: impervious ground surface without precipitation; (

**b**) case 2: impervious ground surface with precipitation; (

**c**) case 3: permeable ground surface without precipitation; (

**d**) case 4: permeable ground surface with precipitation. Note that the ground color denotes permeability: green indicates permeable ground surface, and grey is impermeable. ${A}_{1}$ denotes the inlet surface area of the up-flow model and ${A}_{2}$ denotes the surface area of the surrounding ground. $h\left(t\right)$ denotes the pond height ($h\left(t=0\right)={h}_{0}$), $z\left(t\right)$ water uptake height within the model, $H\left(t\right)$ the depth of the infiltrated water within the soil, and $I$ the uniform rain intensity. $\delta $ denotes the gap. Arrows represent the water movement.

**Figure 3.**Illustration of the domain of the soil extending infinitely with an infinite number of cells. Each cell contains the model.

**Figure 4.**(

**a**) Water column and pond heights and (

**b**) their corresponding volumes for case 1 are plotted against time for various initial height values (Table 3); (

**c**) water column and pond heights and (

**d**) their corresponding volumes for case 1 are plotted against time for various porosity values (Table 3).

**Figure 5.**Time taken to absorb the pond is plotted as a function of the initial pond height value and ratio: (

**a**) plots follow the analytical expressions: red plot contains, and the blue does not contain the effect of dynamical pressure; (

**b**) the graph is plotted numerically; (

**c**) the graph is plotted following the analytical expression (the effect of the dynamical pressure is neglected); (

**d**) the graph is plotted numerically.

**Figure 6.**(

**a**) Water column and pond heights and (

**b**) their corresponding volumes for case 2 are plotted against time for various initial height values (Table 3); (

**c**) water column and pond heights and (

**d**) their corresponding volumes for case 2 are plotted against time for various porosity values (Table 3).

**Figure 7.**(

**a**) Water column and pond heights and (

**b**) their corresponding volumes for case 2 are plotted against time for various rain intensity values (Table 2); (

**c**) time taken to absorb the pond is plotted as a function of initial pond height value after precipitation; (

**d**) time taken to absorb the pond is plotted as a function of area ratio.

**Figure 8.**The time taken to absorb the pond is plotted as a function of the duration of rain for different rain intensity values (Table 2).

**Figure 9.**(

**a**) Water column, pond, and depth of the waterfront within the soil and (

**b**) their corresponding volumes for case 3 are plotted against time for different types of soil (Table 1).

**Figure 10.**(

**a**) Water column, pond, and depth of the water within the soil and (

**b**) their corresponding volumes for case 3 are plotted against time for various initial height values (Table 3); (

**c**) water column, pond, and depth of the waterfront within the soil and (

**d**) their corresponding volumes for case 3 are plotted against time for various porosity values (Table 3).

**Figure 11.**(

**a**) Water column, pond, and depth of the water within the soil and (

**b**) their corresponding volumes for case 4 are plotted against time for various initial height values (Table 3); (

**c**) water column, pond, and depth of the waterfront within the soil and (

**d**) their corresponding volumes for case 4 are plotted against time for various porosity values (Table 3).

**Figure 12.**(

**a**) Water column, pond depth of the waterfront within the soil, and (

**b**) their corresponding volumes for case 4 are plotted against time for various rain intensity values (Table 2).

Soil Type | $\mathbf{Permeability}\mathbf{K}\left({\mathbf{m}}^{2}\right)$ | $\mathbf{Porosity}{\mathit{\Phi}}_{2}$ (%) |
---|---|---|

Sand | ${10}^{-11}$ | $42$ |

Silty | ${10}^{-13}$ | $44$ |

Clay | ${10}^{-17}$ | $46$ |

**Table 2.**Direct inflow of rainwater into storage (Table is copied from [1]).

Event | $\mathbf{Direct}\mathbf{Rainwater}\mathbf{Inflow}\mathbf{Velocity}\mathit{I}\mathbf{m}/\mathbf{s}$$\mathbf{per}\mathbf{Unit}\mathbf{Area}\left[{\mathbf{m}}^{2}\right]$ | |
---|---|---|

Southwest (SW) | North (N) | |

Uniform intensity 60 min duration, with return period 1:10 years | $0.68\xb7{10}^{-5}$ | $0.53\xb7{10}^{-5}$ |

60 min duration event with a high-intensity burst of 5 min, return period 1:10 years, Berggren [23] | $3.47\xb7{10}^{-5}$ Preceding rainfall (during the first 27.5 min) = 7 mm | $2.73\xb7{10}^{-5}$ Preceding rainfall (during the first 27.5 min) = 5.4 mm |

$\mathbf{Initial}\mathbf{Pond}\mathbf{Height}{\mathit{h}}_{0}$[m] | $\mathbf{Porosity}{\mathit{\Phi}}_{1}$ (%) |
---|---|

$0.3$ | $60$ |

$0.5$ | $70$ |

$1$ | $80$ |

**Table 4.**Absorption time values (${T}_{ap}$) extracted from the plots for different porosity, initial pond height, and rain intensity scenarios in all four cases.

${\mathit{\varphi}}_{1}=0.6$ | ${\mathit{\varphi}}_{1}=0.7$ | ${\mathit{\varphi}}_{1}=0.8$ | ${\mathit{h}}_{0}=0.3\mathbf{m}$ | ${\mathit{h}}_{0}=0.5\mathbf{m}$ | ${\mathit{h}}_{0}=1\mathbf{m}$ | $\mathit{I}=0.53\xb7{10}^{-5}\frac{\mathbf{m}}{\mathbf{s}}$ per Unit Area | $\mathit{I}=0.68\xb7{10}^{-5}\frac{\mathbf{m}}{\mathbf{s}}$ per Unit Area | $\mathit{I}=3.47\xb7{10}^{-5}\frac{\mathbf{m}}{\mathbf{s}}$ per Unit Area | |
---|---|---|---|---|---|---|---|---|---|

${\mathit{T}}_{\mathit{a}\mathit{p}}\left[\mathbf{s}\right]$for case 1 | indefinite | 2523 | 1458 | 407 | 2522 | indefinite | no rain | no rain | no rain |

${\mathit{T}}_{\mathit{a}\mathit{p}}\left[\mathbf{s}\right]$for case 2 | indefinite | 9221 | 1649 | 423 | 9220 | indefinite | 4630 | 9221 | indefinite |

${\mathit{T}}_{\mathit{a}\mathit{p}}\left[\mathbf{s}\right]$for case 3 | 461 | 410 | 366 | 148 | 410 | 1481 | no rain | no rain | no rain |

${\mathit{T}}_{\mathit{a}\mathit{p}}\left[\mathbf{s}\right]$for case 4 | 472 | 420 | 370 | 153 | 420 | 1540 | 417 | 419 | 458 |

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**MDPI and ACS Style**

Barcot, A.; Åkerstedt, H.O.; Larsson, I.A.S.; Lundström, T.S.
Stormwater Uptake in Sponge-Like Porous Bodies Surrounded by a Pond: A Fluid Mechanics Analysis. *Water* **2023**, *15*, 3209.
https://doi.org/10.3390/w15183209

**AMA Style**

Barcot A, Åkerstedt HO, Larsson IAS, Lundström TS.
Stormwater Uptake in Sponge-Like Porous Bodies Surrounded by a Pond: A Fluid Mechanics Analysis. *Water*. 2023; 15(18):3209.
https://doi.org/10.3390/w15183209

**Chicago/Turabian Style**

Barcot, Ana, Hans O. Åkerstedt, I. A. Sofia Larsson, and T. Staffan Lundström.
2023. "Stormwater Uptake in Sponge-Like Porous Bodies Surrounded by a Pond: A Fluid Mechanics Analysis" *Water* 15, no. 18: 3209.
https://doi.org/10.3390/w15183209