# Integrating Local Scale Drainage Measures in Meso Scale Catchment Modelling

^{*}

## Abstract

**:**

^{2}illustrated the applicability of the model on the regional scale.

## 1. Introduction

#### 1.1. Numerical Model Review

#### 1.2. The Scale Issue in Numerical Models

#### 1.3. The Example of Local Scale Drainage Measures and the Deficits in Numerical Models

#### 1.4. Outline

## 2. Theoretical Approach

#### 2.1. Multiscale Modelling

^{2}to 10 km

^{2}) in sub-catchments. On the reach scale, ranging from 0.5 km

^{2}to 1 km

^{2}, the preferred flow paths are defined. On the local scale the flow and retention processes of drainage measures like in LSDM are represented. The size of these drainage measures ranges from 1 m

^{2}to about 100 m

^{2}. The spatial micro scale is used to represent processes on a small size (<1 m

^{2}) like the soil water regime in different soil layers and vegetation processes. The spatial and temporal scales are independently defined. The temporal scales range from the analysis of rapid events like after local heavy rainfall events in less than an hour to seasonal impacts within 1 year to long term effects over 50 to 100 years. Physical processes, like the infiltration and exfiltration processes in the soil, are determined in even smaller time scales of seconds.

#### 2.2. Features to Integrate LSDM in Catchment Models

- (1)
- Spatial micro scale and temporal process scale features:
- (a)
- Physical process features on the micro scale: e.g., interception, infiltration, evaporation, transpiration, soil pore space storage, water retention and detention, vertical and lateral water flow in layers.
- (b)
- Interaction and feedback features: Backwater effect and exceedance flow generation in coupled layers.
- (c)
- Material features: Supporting the use of hydrological parameters of material tested in laboratories and physical model tests.

- (2)
- Spatial local scale and temporal short-term scale features:
- (a)
- Spatial features: Geographic defined local scale areas realized with GIS data import and data processing functions to enable multiple type definitions, e.g., different green roof types per meso scale sub-catchment.
- (b)
- Variable design features: Supporting a flexible setup of drainage measures with multiple layers to model new designs.
- (c)
- Rainwater harvesting features: Modelling the water withdrawal of local measures.

- (3)
- Spatial local scale and temporal seasonal scale features:
- (a)
- Vegetative features: Parameterisation of vegetation systems according to seasonal changes for different vegetation types (e.g., interception storage, root depth, crop factor).

- (4)
- Spatial reach scale and temporal short-term scale features:
- (a)
- Real-time control features: Water control and drainage functions according to local rainfall forecasts (e.g., predrainage of cisterns or retention roofs). Integrating radar-based precipitation forecasting techniques.
- (b)
- Water redistribution features: Water storage and hydrological processes depending on interactions among linked individual elements. Modelling exceedance flow control in a cascade of local measures on the reach scale.

- (5)
- Spatial meso scale and temporal short-term scale features:
- (a)
- Adoption of meso scale features: Relevant preset parameters of meso scale features are adopted for local scale measures (e.g., geological attributes defined on the sub-catchment scale).
- (b)
- Backwater effect features: Backwater effects between local scale and meso scale elements (e.g., when the capacity of retention measures is exceeded) or backwater effects derived on the meso scale by external forces (e.g., tidal effect or increased groundwater level).

- (6)
- Spatial regional scale and long-term scale features:
- (a)
- Enabling the simulation of prewetting and initial water storage conditions on the basis of continuous water balance simulations.

## 3. Methodology

#### 3.1. Data Mapping with “Overlay” Data Objects

#### 3.2. Interlinked Multiple Scale Data Objects

#### 3.3. Multiple Interlinked Micro Scale Layers

#### 3.3.1. Integration of the Concept in the Overall Computation Procedure

#### 3.3.2. The Water Balance Computation

^{2}= mm) per unit area (m

^{2}) and per time step t into the top layer (Layer i = 1) is calculated with the following equation:

^{2}at time step t (mm), t is the counter of the time steps 1 to the entity of n (-), ${\dot{V}}_{feeder}$(t) is the volumetric exceedance flux or feeding flux from linked overlay elements on the reach scale and sub-catchments on the meso scale (cp. Figure 4i,ii) in (mm/s), E(t) is the actual evaporation per time step computed with a canopy interception and evapotranspiration model in (mm). The potential specific evapotranspiration from plants depends on the seasonal defined crop factor, root depth for each land use class and the actual water content in the root zone. I(t) is the actual interception volume per time step (mm). The input parameters for the canopy interception and evapotranspiration model are defined according to land use classes on the temporal seasonal scale. Per month of an ideal year the following parameters are defined: root depth (mm), maximal canopy storage (mm) and crop factor (-). Further input parameters are the observed air temperature (°C), sunshine duration (h/day), relative humidity (%), wind speed (m/s) and precipitation (mm). Two approaches can be utilized in the evapotranspiration model: the FAO-Penman-Monteith equation or the Turc-Wendling equation (see [30]).

#### 3.3.3. The Dynamic Time Step Size Module

_{free}) per time step is at least 10 times larger than the influx ($\dot{V}$

_{in}) per time step size ∆t′ in the short-term and long-term simulation (cp. Figure 6, Corr. 1). This adjustment proved to be valid within different case studies in recent years (see. Supplementary Materials). If open space storage layers are defined as top layer(s), this calculation is distinctive and the next upper substrate layer is used to compute the internal time step size.

_{r}is the CFL criterion (-), ∆t is the time step size (s), u is the magnitude of velocity (mm/s), ∆x is the spatial distance (mm), and the constant C

_{max}is equal to 1 for explicit calculation (see [32]).

_{c}is the required time step size to fulfil the CFL criterion (s), i is the layer index (from 1 … n), n is the index of the last soil layer above the groundwater level, ∆z

_{i}is the thickness of the actual soil layer i (mm) and k

_{i}is the saturated hydraulic conductivity of the layer i (mm/s).

_{∆t}is calculated to test the validity of the CFL criterion. If the time step size computed with the adaptation factor is smaller than the time step size computed with Corr. 1, the internal step size is computed with the adaptation factor and the actual input flux is corrected respectively (see Figure 6, Corr. 2).

_{∆t}is the adaptation factor with f

_{∆t}≥ 1, ∆t is the predefined model time step size (s), ∆t

_{c}is the required time step size to fulfil the CFL criterion (s), ∆t′

_{j}is the adapted internal time step size (s) of Corr 1., ∆t′ is the final adapted internal time step size (s) and $\lceil \rceil $ is the mathematical notation of the ceiling function. $\dot{V}$

_{in}(t) is the actual influx in the predefined time step size (mm/s) and $\dot{V}$

_{inf,i}(t′) is the adapted influx within the internal time step size (mm/s).

#### 3.3.4. Computation Procedure for Multiple Interlinked Micro Scale Layers

_{in}(t′). The influx in the deeper soil layers $\dot{V}$

_{inf,i}(t′) depends on the percolation of water from the layer above $\dot{V}$

_{perc,i}(t′). The actual flux into the layer $\dot{V}$

_{inf,i}(t′) and the actual outflow of the layer $\dot{V}$

_{perc,i}(t′) depend on the potential infiltration $\dot{V}$

_{inf,pot,i}(t′) and potential exfiltration $\dot{V}$

_{perc,pot,i}(t′) calculated with the infiltration capacity (c

_{in}) and exfiltration capacity (c

_{ex}) (cp. Equations (5)–(10)).

_{in,i}is the infiltration capacity (mm/s), k

_{i}is the hydraulic conductivity (mm/s), V

_{max,i}is the maximal storage volume per unit area (mm), V

_{WP,i}is the volume of water defining the wilting point per unit area (mm), F

_{c,in,i}is the calibration factor of the infiltration capacity (-). $\dot{V}$

_{inf,pot,i}(t′) is the potential infiltration flux (mm/s), V

_{i}(t′) is the actual water volume per unit area (mm). $\dot{V}$

_{inf,i}(t′) is the actual infiltration flux in the soil layer i (mm/s), $\dot{V}$

_{in,}

_{1}(t′) is the effective influx in the top soil layer (mm/s), $\dot{V}$

_{perc,i}

_{−1}(t′) is the actual percolation flux from the layer above (mm/s). c

_{ex,i}is the exfiltration capacity (mm/s), V

_{FC,i}is the water volume defining the field capacity per unit area (mm), F

_{c,ex}

_{,i}is the calibration factor of the exfiltration capacity (-). $\dot{V}$

_{perc,pot,i}(t′) is the potential percolation flux according to soil parameters (mm/s). $\dot{V}$

_{perc,i}(t′) is the actual percolation flux (mm/s), V

_{free,i}(t′) is the actual drainable water volume (mm), $\dot{V}$

_{inf,pot,i}

_{+1}(t′) is the potential infiltration flux into the layer below (mm/s).

_{WP,i}corresponds to the water volume that is held by capillary and hydroscopic forces and is not available for plants or drainage features of the layer. The field capacity V

_{FC,i}is the water volume remaining in the soil layer after gravitational drainage is ceased. It is the water volume held by capillary forces and is available for plants. The potential evapotranspiration from plants per layer depends on the overall depth of the roots and the thicknesses of the soil layers. For each soil layer, the effective root mass is calculated and used to define the potential fraction of transpiration. This calculation is distinctive if open space storage layers are defined above the soil layers. The actual transpiration is computed on the micro scale on the basis of the potential transpiration, the fractions of rooted soil layers and the available soil water above the wilting point of the specific soil layer. The thickness of rooted substrate is computed over several layers till the root depth is reached. A query is checking if the top layers are defined as substrate or free storage layers.

_{WP,i}) of the substrate. The actual stored water (V

_{i}) in the layer is calculated with the following balance equation:

_{i}(t′) is the actual water volume per unit area in that internal time step t′ (mm), V

_{i}(t′ − 1) is the water volume of the previous time step (mm), $\dot{V}$

_{inf,i}(t′) is the actual infiltration flux in the soil (mm/s), $\dot{V}$

_{ET,i}(t′) is the actual evapotranspiration per unit area (mm/s), $\dot{V}$

_{perc,i}(t′) is the actual percolation flux (mm/s).

_{ov}). The effective flow through the overflow pipe is the minimal discharge calculated with four approaches: (1) the flow over a crest height into the pipe using the Poleni approach (see [34]); (2) the maximal pipe capacity according to the Darcy-Weisbach approach with an assumed full-flowing pipe diameter; (3) the flow through a retention layer according to a prolonged flow path L

_{drain,i}; and (4) the flow through substrate computed with the Darcy’s law through porous media.

_{drain,i}(t′) is the outflow (mm

^{3}/s), D

_{outlet,i}is the diameter of the outlet (mm), μ is the overflow coefficient (-) according to [34], g = 9.81 × 10

^{3}(mm/s

^{2}) is the standard acceleration due to gravity, h

_{w,i}(t′) is the actual water level in the layer above the overflow crest height (mm), λ is the friction coefficient (-), L

_{drain,i}is the longest flow path in the drainage layer (mm), k

_{ret,drain,i}is the retention coefficient in the drainage layer (s), A

_{drain,i}is the drained area per outlet (mm

^{2}), k

_{i}is the saturated hydraulic conductivity (mm/s), I

_{drain,eff,i}is the effective gradient taking into account the actual water level and the gradient of the construction (-), W

_{drain,i}is the width of the drainage area (mm), h

_{ov,i}is the overflow crest height (mm), R

_{drain,i}is the roughness of the drainage layer (mm), Re is the Reynolds number (-), v

_{drain,i}is the velocity of flow in the layer calculated according to the Darcy-Weisbach equation (mm/s), D

_{drain,i}is the diameter of the drainage flow media (mm), I

_{drain,i}is the gradient of the drainage layer (-).

_{Sat.state}) (cp. Figure 7, 3rd layer loop). This saturation state varies according to the design of the drainage measures and is defined as calibration parameter. The flow curve through the drainage layer is computed with the retention coefficient of the drainage system (k

_{ret,drain,i}) and a unit hydrograph computation. The developed mathematical approach enables the modelling of upcoming new technologies to increase the retention time in LSDM, where drainage constructions are designed, e.g., with prolonged flow paths L

_{drain,i}.

#### 3.3.5. Design Examples of Local Scale Drainage Measures

## 4. Implementation

## 5. Validation of the Method of Multiple Interlinked Micro Scale Layers

#### 5.1. Laboratory Physical Model Setup

^{2}. The maximum fall height is currently 2.75 m and drops with an average fall velocity of 1.8 to 2.6 m/s are generated. The size of the drops can be varied between 0.4 and 0.65 mm by adjusting different meshes. The general characteristics of the rainfall simulator are described in [35,36].

#### 5.2. Numerical Model Setup and Input Parameters

^{2}.

_{max}) = 58.3 mm, hydraulic conductivity (k) = 0.115 mm/s. The initial soil water (40 mm) is gained from the experimental measurements 24 h after full saturation.

#### 5.3. Calibration Procedure and Results

_{Sat,state}) for the drainage layer. For a specific gradient of the layer, this index defines the relative filling degree of the layer before water exceeds the lower reach. It defines the point in time of backwater and exceedance flow generation between the linked layers. Further calibration parameters are the factor of the infiltration capacity (F

_{c,in}) and the factor of the exfiltration capacity (F

_{c,ex}) in the layers.

_{c,in}= 1). Likewise, for the factor of exfiltration capacity of the substrate medium no calibration was required (F

_{c,ex}= 1). But an adaptation of the exfiltration capacity of the first free storage layer is done to assure the characteristic of an empty medium. The factor of the exfiltration capacity from the top free storage layer is increased to simulate the fast exfiltration from a free storage volume: F

_{c,ex}is set to 100 for the free top layer (L1). The calibration results are presented in Figure 11.

^{2}) illustrate the correlation between the data sets. The RMSE illustrates the deviation between observed and simulated results.

^{2}), a close approach to the 1:1 Line and a low RMSE value for the drainage system flux results and the exceedance flux results. In comparison to a conventional roof, the time delay to reach the peak flux is about 16 min, which demonstrates good retention potential to mitigate runoff peaks e.g., from urban catchments.

#### 5.4. Validation Results

#### 5.5. Summary of Calibration and Validation Results

## 6. Application Studies of the Catchment Model

^{2}) in Hamburg, Germany. This case study was analysed in detail within the German Research Project KLIMZUG-NORD. Three urban growth and adaptation scenarios for Hamburg were used to model the effectiveness of local scale drainage measures (e.g., green roofs and larger scale retention areas) to reduce the peak flow rates and flood prone areas. The results of the application study are published in Hellmers et al., 2015 [38].

## 7. Discussion and Conclusions

^{2}for flood peak mitigation. It is concluded that the presented and implemented methods improve the integration of local scale drainage measures in catchment modelling.

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Definition of scales: (

**a**) the heterogeneity in spatial scales; and (

**b**) the variability in temporal scales.

**Figure 2.**Data mapping with the approach of overlay data objects: meso scale sub-catchment data objects are intersected with local scale drainage measures (LSDM) data objects.

**Figure 4.**Data objects defined on respective spatial scales and new developed interconnections: (

**i**) meso via reach scale; (

**ii**) on reach scale; and (

**iii**) on micro scale.

**Figure 7.**Soil water and drainage module: Water balance computation procedure via four layer stages on the spatial and temporal micro scale with new developed drainage functions.

**Figure 8.**Design examples of LSDM made up of multiple layers (L = layer): (

**a**) green roof with overflow and down pipe outlet; (

**b**) swale-filter-drain system with coupled layers; (

**c**) swale; (

**d**) cistern with rainwater harvesting function.

**Figure 9.**Laboratory physical model setup with the Rainfall-Simulator of the Hamburg University of Technology (RS-TUHH) and two green roof test installations.

**Figure 11.**Results of the calibration run: precipitation of 1.8 mm/min for a duration of 15 min. Comparison between observed and simulated flux of the layered green roof model in hydrographs (left side) and scatter plots (right side).

**Figure 12.**Results of the validation run 1: precipitation intensity P(t) of 1.0 mm/min for a duration of 45 min. Comparison between observed and simulated flux of the layered green roof model.

**Figure 13.**Results of the validation run 2: precipitation intensity P(t) of 0.6 mm/min for a duration of 90 min. Comparison between observed and simulated flux of the layered green roof model.

Cali- and Validation Runs ^{1} | Drainage System Criteria | Exceedance System Criteria | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

(1) | (2) | (3) | (4) | (5) | (1) | (2) | (3) | (4) | (5) | |

Δ in Time Delay (min) | Δ in Time to Peak Flux (min) | Δ in Peak Flux (%) | Δ in Volume (%) | RMSE (mm/min) | Δ in Time Delay (min) | Δ in Time to Peak Flux (min) | Δ in Peak Flux (%) | Δ in Volume (%) | RMSE (mm/min) | |

Cal. 1 | <1 min | <1 min | 7.5% | <10% | 0.0422 | <1 min | <1 min | 4.5% | <10% | 0.0399 |

Val. 1 | <1 min | <1 min | 10.0% | <10% | 0.0236 | <1 min | <1 min | 3.0% | <10% | 0.0387 |

Val. 2 | <1 min | <1 min | 3.5% | <10% | 0.0257 | ~3 min | <1 min | 10.0% | <10% | 0.0243 |

^{1}Calibration Run 1: P(t) = 1.8 mm/min, D = 15 min; Validation Run 1: P(t) = 1.0 mm/min, D = 45 min; Validation Run 2: P(t) = 0.6 mm/min, D = 90 min.

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**MDPI and ACS Style**

Hellmers, S.; Fröhle, P. Integrating Local Scale Drainage Measures in Meso Scale Catchment Modelling. *Water* **2017**, *9*, 71.
https://doi.org/10.3390/w9020071

**AMA Style**

Hellmers S, Fröhle P. Integrating Local Scale Drainage Measures in Meso Scale Catchment Modelling. *Water*. 2017; 9(2):71.
https://doi.org/10.3390/w9020071

**Chicago/Turabian Style**

Hellmers, Sandra, and Peter Fröhle. 2017. "Integrating Local Scale Drainage Measures in Meso Scale Catchment Modelling" *Water* 9, no. 2: 71.
https://doi.org/10.3390/w9020071