# Modelling the Hydraulic Behaviour of Growing Media with the Explicit Finite Volume Solution

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Model and Governing Equation

^{3}/L

^{3}), ψ is the suction head (L), K is the unsaturated hydraulic conductivity (L/T), t is the time (T) and z is the depth (L) (positive downward). The Richards equation can be derived by combining two physical laws:

- -
- The continuity equation, written in its differential form:$$\frac{\partial \text{\theta}(\text{\psi})}{\partial t}+\nabla q=0$$

- -
- the Darcy law:$$q=-K(\text{\theta}(\text{\psi}))\nabla (\text{\psi}-z)$$

^{2}/T). The main advantage of the moisture-based form is that it is expressed in terms of the conserved variable. The pressure-based form can handle discontinuities in the porous media, but may lead to significant mass balance errors. The mixed form can conserve mass, and allows for saturated flows and heterogeneity in the porous medium.

#### Unsaturated Soil Constitutive Models

_{r}(L

^{3}/L

^{3}) is the residual water content, θ

_{s}(L

^{3}/L

^{3}) is the saturated water content, K

_{s}(L/T) is the saturated hydraulic conductivity, n is a parameter which measures pore-size distribution and α (L

^{−1}) is a parameter related to the inverse of the air-entry pressure. L indicates the tortuosity and is usually assumed to be L = 0.5 [10]. S

_{e}is the effective saturation (L

^{3}/L

^{3}).

#### 2.2. Numerical Model

#### 2.2.1. Finite Volume Framework

_{Ω}of the control volume. After the control volume is discretized into a finite volume V

_{j}and the surface S

_{Ω}is discretized into N

_{ω}faces of area A

_{ω}, the finite volume scheme is obtained:

_{ω}represents the distance between two adjacent cells. The vertical 1D simplification can be written as:

_{j+1}− z

_{j}represents the distance between the two adjacent cell centers and K

_{j±}

_{1/2}represents the interface conductivity, which must be interpolated or approximated from cell center values.

#### 2.2.2. Explicit Formulation

**Figure 2.**Illustration of the Finite Volume Scheme for updating the cell values by the boundary fluxes.

^{i}

_{j+}

_{1/2}is a numerical approximation of the flux along z = z

_{j+}

_{1/2}. A fully discrete method is obtained by approximating the average flux based on the θ

^{i}values of the cells that are adjacent to the boundary in question. In the case of the upwind method, the flux through the top edge of the cell is entirely determined by the value θ

^{i}

_{j−1}in the cell above. The numerical flux for the upwind method is written as:

^{−1}). The Courant number reflects the importance of the convective flux in the unsaturated flow.

#### 2.3. Green Roof Substrate

#### 2.4. Simplified Formulation

^{2}for the two-polynomial regression is very high.

**Figure 5.**Computation of the velocity value for the determination of the mesh size as indicated in Equation (47).

#### 2.5. Model Performance Evaluation

_{0}is the initial mass in the domain (L

^{3}L

^{−3}), θ

_{n}is the final mass (L

^{3}L

^{−3}), q

_{in}is the influent flow in the domain (LT

^{−1}), q

_{ef}is the effluent (LT

^{−1}), Δz is the mesh size (L), Δt is the time step (T), N is the number of finite volume elements of the domain, and i is the time index.

_{mean}

^{obs}is the mean value of observed data. NSE coefficient ranges between −∞ and 1.0, in the case of a perfect agreement; generally, values 0 and 1.0 are considered acceptable.

## 3. Results and Discussion

^{3}m

^{−3}and 0.08 m

^{3}m

^{−3}.

Substrate Soils | θ_{r} (-) | θ_{s} (-) | α (1/cm) | n | K_{0} (cm/min) |
---|---|---|---|---|---|

Vulcaflor | 0.165 | 0.400 | 0.124 | 2.280 | 4.800 |

Hilten | 0.045 | 0.570 | 0.065 | 2.320 | 1.720 |

Sand | 0.045 | 0.430 | 0.145 | 2.680 | 0.495 |

Loamy sand | 0.065 | 0.41 | 0.075 | 1.89 | 0.074 |

_{r}is the soil residual water content; θ

_{s}is the saturated soil water content; α and n are two characteristic parameters of the MG model; and K

_{0}is the matching point at saturation.

Substrate Soils | Θ_{fc} (-) | θ_{fc} (-) |
---|---|---|

Vulcaflor | 0.056 | 0.178 |

Hilten | 0.066 | 0.079 |

Sand | 0.057 | 0.067 |

Loamy Sand | 0.106 | 0.094 |

_{fc}is the field capacity value in terms of effective saturation; θ

_{fc}is the field capacity in terms of volumetric water content.

Rainfall Event | h (mm) | i_{max} (mm/h) | D (min) |
---|---|---|---|

11 November 2013 00:27 | 64.2 | 96.0 | 1599 |

22 November 2013 23:31 | 69.0 | 60.0 | 3116 |

29 November 2013 22:19 | 48.0 | 24.0 | 3301 |

26 December 2013 09:58 | 26.2 | 12.0 | 1648 |

20 January 2014 17:39 | 48.2 | 48.0 | 2710 |

31 January 2014 18:54 | 33.8 | 24.0 | 2448 |

02 February 2014 19:43 | 27.8 | 24.0 | 1944 |

11 February 2014 22:58 | 29.6 | 24.0 | 2039 |

23 March 2014 21:55 | 40.4 | 36.0 | 2357 |

27 March 2014 02:27 | 31.8 | 36.0 | 2507 |

_{max}: max intensity during the rainfall event (mm/h); D: duration of the event (min).

#### 3.1. Results

Substrate Soils | λ (-) | ω (-) | V_{m} (cm/min) | Δz (cm) | N_{100} (-) | N_{400} (-) |
---|---|---|---|---|---|---|

Vulcaflor | 10.3 | 0.196 | 0.218 | 10.3 | 1 | 4 |

Hilten | 8.36 | 0.52 | 0.084 | 8.4 | 2 | 5 |

Sand | 3.84 | 2.16 | 0.036 | 3.9 | 3 | 10 |

Loamy sand | 5.81 | 2.02 | 0.011 | 4.7 | 2 | 7 |

_{m}is the mean value of the velocity computed by assuming that the saturation range does not exceed 50%; Δz is the mesh size calculated by using Equation (30); N

_{100}and N

_{400}are the numbers of subdivision elements for the two selected substrate depths.

Soil | NSE_{100} (-) | NSE_{400} (-) | PBias_{100} (%) | PBias_{400} (%) | RSR_{100} (-) | RSR_{400} (-) |
---|---|---|---|---|---|---|

Vulcaflor | 0.95 ± 0.04 | 0.99 ± 0.01 | 0.45 ± 0.39 | −1.17 ± 0.45 | 0.23 ± 0.07 | 0.10 ± 0.01 |

Hilten | 0.97 ± 0.01 | 0.89 ± 0.06 | −0.03 ± 0.24 | −1.96 ± 0.69 | 0.16 ± 0.03 | 0.33 ± 0.09 |

Sand | 0.98 ± 0.01 | 0.95 ± 0.03 | 0.52 ± 0.18 | 1.74 ± 0.53 | 0.12 ± 0.03 | 0.23 ± 0.06 |

Loamy sand | 0.98 ± 0.01 | 0.95 ± 0.02 | 0.78 ± 0.28 | 2.17 ± 1.09 | 0.13 ± 0.03 | 0.22 ± 0.06 |

**Figure 7.**Graphical and statistical evaluations of comparing the proposed model soil outflow rates to those produced by Hydrus-1D, for the event of 29 November 2013 22:19:00 and the Vulcaflor soil.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Carbone, M.; Brunetti, G.; Piro, P.
Modelling the Hydraulic Behaviour of Growing Media with the Explicit Finite Volume Solution. *Water* **2015**, *7*, 568-591.
https://doi.org/10.3390/w7020568

**AMA Style**

Carbone M, Brunetti G, Piro P.
Modelling the Hydraulic Behaviour of Growing Media with the Explicit Finite Volume Solution. *Water*. 2015; 7(2):568-591.
https://doi.org/10.3390/w7020568

**Chicago/Turabian Style**

Carbone, Marco, Giuseppe Brunetti, and Patrizia Piro.
2015. "Modelling the Hydraulic Behaviour of Growing Media with the Explicit Finite Volume Solution" *Water* 7, no. 2: 568-591.
https://doi.org/10.3390/w7020568