# Behavior of a Fully-Looped Drainage Network and the Corresponding Dendritic Networks

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

**Figure 1.**Drainage networks in urban catchments: (

**a**) Mokdong area in Seoul, South Korea near the Han River; (

**b**) downtown area of Chicago, USA near Lake Michigan.

#### 2.1. Synthetic Catchment and the Behavior of a Fully Looped Network

_{1}and s

_{2}be adjacent. The transition probability from s

_{1}to s

_{2}can then be defined as follows [15]:

_{1}) is the set of trees adjacent to s

_{1}, and β is a parameter that represents the extent to which the sinuosity of the network is reflected in the generation of the new spanning tree, s

_{2}. For example, when β is equal to zero, the overall sinuosity of a network has no relationship to the transition probability and the transition probabilities are the same in all possible directions, which is identical to the uniform distribution. The maximum degree of the points in S, r, is defined as [15],

_{s}is the distance to an outlet along s from v, while ξ

_{B}is the shortest distance to the outlet from d. The actual process to generate a network with the Gibbsian model is as follows. First, start from a network, s

_{1}, generated by the Uniform model and randomly select a point, v, in the network and assign a new flow direction from v to generate the adjacent network s

_{2}. Second, check whether the new network, s

_{2}is acyclic. If not, repeat the first step. Third, draw a random probability x between zero and one and check that x is greater than e

^{−β[ΔH]}where ΔH is equal to H(s

_{2})—H(s

_{1}). If this holds, then take s

_{2}as a new network. In the next step, use s

_{2}as the starting network and repeat these steps a sufficient number of times until the resulting tree has a distribution close to the stationary Gibbs’ distribution. The Uniform model [20,21] is defined on a lattice with flow allowed in each of the four possible directions with equal probability; hence, it tends to have high sinuosity. The uniform model is equivalent to the Gibbsian model with β equal to zero. Figure 2 shows a fully looped network and also dendritic networks on the same grids generated by the Gibbsian model depending on the value of β, where a smaller β produces a more sinuous network compared with a larger β.

**Figure 2.**A synthetic rectangular catchment composed of 8 × 8 grids and dendritic networks generated by the Gibbsian model: what would behave like the fully looped network?

#### 2.2. SWMM and Rainfall-Runoff Model Runs

^{1/3}, and the roughness of pervious area was set to 0.1 s/m

^{1/3}. The invert elevation at the outlet of the drainage network is set to zero and invert elevations of other junctions were determined based on the catchment slope and the distance from the outlet. The conduits have the same length, 500 m and the maximum depth was set to 6 m. The layout of the drainage network of the SWMM model was built depending on the Gibbsian model generated for a given value of β. The duration of the rainfall is fixed to one hour and the rainfall intensities from 4 to 30 mm were tested. In terms of the infiltration model, Horton’s method was used with the same parameter values for all subcatchments. Obviously, model section and spatial variation of parameter values for the nonlinear infiltration process can affect the results of this study and finding equivalent dendritic networks for a fully looped network. We left this for a future study because the original intention of this study is focusing on the possibility of finding an equivalent dendritic network for a fully looped network.

## 3. Results and Discussion

#### 3.1. Behavior of the Fully Looped Drainage Network and a Corresponding Dendritic Network

_{c}). Figure 3 shows the direct runoff hydrographs of the fully looped network (Figure 2) depending on slope for a 1 hour storm with hourly rainfall intensity of 15 mm. As shown in Figure 3a, the peak flow of the hydrograph sharply decreases as the catchment slope decreases. Figure 3b shows the peak flows of the fully-looped network as a function of the catchment slope. The result shows that a transition exists in the relation between the catchment slope and the peak flows of a fully looped network as shown in Figure 3b; the relation between the catchment slope and peak flows is logarithmic but the slope of peak flow with respect to the catchment slope changes near the catchment slope of 9 × 10

^{−3}.

^{−4}, 10

^{−4}, and 4 × 10

^{−5}. The results clearly show that the shape of the hydrograph from a fully looped network is close to the shapes of the hydrographs from dendritic networks, which are generated by the Gibbsian model depending on the catchment slope. For example, when the catchment slope is relatively steep (4 × 10

^{−4}), the hydrograph of the fully looped network is close to that of the dendritic network with higher β (10°). In contrast, when the catchment slope is relatively mild (S = 4 × 10

^{−5}), the hydrograph of the fully looped network coincides with the dendritic network with smaller β (10

^{−4}). The behavior of the fully looped network implies that it is possible to make a pair of dendritic networks generated by the Gibbsian model and the fully looped network depending on the catchment slope.

**Figure 3.**Behavior of a fully looped network depending on the catchment slope (S

_{c}): (

**a**) runoff hydrographs; (

**b**) peak flows (rainfall intensity = 15 mm/h).

**Figure 4.**Comparison of the flow hydrographs from a fully looped network and dendritic networks generated by the Gibbsian model depending on the catchment slope (rainfall intensity = 15 mm/h).

^{−5}, the peak flow of the fully looped network is close to the dendritic networks generated by the Gibbsian model with β equal to 10

^{−4}. In contrast, when the catchment slope becomes higher (4 × 10

^{−3}), the closest dendritic network is generated with β higher than 10

^{1}(Figure 5d). Figure 5 shows that the corresponding dendritic network changes as the catchment slope changes. Table 1 lists the peak flows of the fully looped network and the dendritic networks generated by the Gibbsian model depending on slope and β. The results clearly show that β of the corresponding equivalent network to the fully looped network (Q

_{p}/Q

_{p,l}= 1) increases as the catchment slope increases (Figure 6). Figure 7 shows how the corresponding dendritic network (the Gibbsian model with β) changes as the catchment slope and the rainfall intensity changes. For example, the bottom left most circle indicates that, when the rainfall intensity is 30 mm/h and the catchment slope is equal to 2.93 × 10

^{−4}, the Gibbsian model with β = 6.54 × 10

^{−5}behaves like a fully looped drainage network.

**Figure 5.**Behavior of the peak flow of a fully looped network depending on the catchment slope of the order of: (

**a**) 4 × 10

^{−5}; (

**b**) 6 × 10

^{−5}; and (

**c**) 1 × 10

^{−4}; 4 × 10

^{−3}(rainfall intensity = 15 mm/h).

**Table 1.**Averaged peak flow ratio (Q

_{p}/Q

_{p,l}) of dendritic drainage networks generated with the Gibbsian model to the fully looped network depending on the catchment slope (rainfall intensity = 15 mm/h).

Catchment Slope | Gibbs Model (β) | ||||||
---|---|---|---|---|---|---|---|

10^{−4} | 10^{−3} | 10^{−2} | 10^{−1} | 10^{0} | 10^{1} | 10^{2} | |

4.00 × 10^{−5} | 1.03 | 1.06 | 1.17 | 1.48 | 1.65 | 1.66 | 1.66 |

5.00 × 10^{−5} | 0.93 | 0.96 | 1.06 | 1.37 | 1.54 | 1.54 | 1.55 |

6.00 × 10^{−5} | 0.83 | 0.86 | 0.96 | 1.26 | 1.42 | 1.43 | 1.43 |

7.00 × 10^{−5} | 0.75 | 0.78 | 0.88 | 1.16 | 1.32 | 1.33 | 1.33 |

8.00 × 10^{−5} | 0.69 | 0.72 | 0.82 | 1.09 | 1.25 | 1.26 | 1.26 |

9.00 × 10^{−5} | 0.65 | 0.67 | 0.77 | 1.04 | 1.19 | 1.20 | 1.20 |

1.00 × 10^{−5} | 0.61 | 0.63 | 0.73 | 0.99 | 1.14 | 1.15 | 1.15 |

1.10 × 10^{−5} | 0.58 | 0.60 | 0.70 | 0.95 | 1.10 | 1.11 | 1.11 |

1.20 × 10^{−5} | 0.56 | 0.58 | 0.68 | 0.93 | 1.08 | 1.09 | 1.09 |

1.30 × 10^{−5} | 0.54 | 0.56 | 0.66 | 0.91 | 1.06 | 1.07 | 1.07 |

1.40 × 10^{−5} | 0.52 | 0.54 | 0.64 | 0.89 | 1.04 | 1.05 | 1.05 |

1.50 × 10^{−5} | 0.51 | 0.53 | 0.62 | 0.87 | 1.03 | 1.04 | 1.04 |

1.60 × 10^{−5} | 0.50 | 0.51 | 0.61 | 0.86 | 1.02 | 1.02 | 1.02 |

**Figure 6.**Averaged peak flow ratio (Q

_{p}/Q

_{p,l}) of dendritic drainage networks generated with the Gibbsian model to the fully looped network depending on the catchment slope (rainfall intensity = 15 mm/h).

**Figure 7.**Behavior of the peak flow of a fully looped network depending on the catchment slope and rainfall intensity.

^{−1}, the other relationship for β greater than 10

^{−1}regardless of the rainfall intensities (Table 2). Figure 8 depicts the corresponding Gibbsian models (β) to a fully looped network depending on the catchment slope and rainfall intensity. As mentioned earlier, rainfall intensity is another factor affecting results: the corresponding dendritic network changes as rainfall intensity varies. However, as shown in Figure 8, the corresponding Gibbsian model (value of β) converges as rainfall intensity increases. The result also shows that the convergence exists for smaller rainfall intensity when the catchment slope is smaller.

**Table 2.**Two-step regression equation between catchment slopes (S

_{c}) and the Gibbsian model (β) (rainfall intensity = 15 mm/h).

Range | Equation | a | b | R^{2} |
---|---|---|---|---|

10^{−4} < β <10^{−1} | S_{c} = a·exp(b·β) | 5.00 × 10^{−5} | 6.5356 | 0.991 |

10^{−1} < β < 10^{1} | S_{c} = a·exp(b·β) | 9.00 × 10^{−5} | 0.6217 | 0.989 |

**Figure 8.**Corresponding Gibbsian models (β) to a fully looped network depending on the catchment slope and rainfall intensity.

#### 3.2. Effect of the Catchment Imperviousness Ratio and Rainfall Duration on the Equivalence

^{−4}. Also, as shown in Table 3, the total rainfall amount falling on the test catchment is the same as 15 mm. Figure 9 depicts the equivalent Gibbsian networks depending on test cases and the catchment imperviousness ratio (r

_{imperv}). The result is interesting in that compared with the previous results with r

_{imperv}equal to 25%, the equivalent Gibbsian network shows high sensitivity to catchment imperviousness ratio. In contrast, rainfall duration shows little difference in terms of equivalent networks, which is dissimilar to the results from rainfall intensity (Figure 8). The results indicate the equivalence still can be found and the association between a fully looped network and a dendritic network can be established in different rainfall durations and imperviousness ratios. However, the equivalence can be greatly affected by the rainfall intensity and catchment characteristics.

^{−1}, the other relationship for β greater than 10

^{−1}. This relationship can be explicitly utilized to obtain the value of β depending on the catchment slopes.

Test Cases | Precipitation (mm) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

0 min | 10 min | 20 min | 30 min | 40 min | 50 min | 60 min | 70 min | 80 min | 90 min | |

P1 | - | - | - | - | 15 | - | - | - | - | - |

P2 | - | - | - | 5 | 5 | 5 | - | - | - | - |

P3 | - | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | - | - | - |

P4 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 |

P5 | - | - | - | 2.5 | 10 | 2.5 | - | - | - | - |

**Figure 9.**Corresponding Gibbsian models (β) to a fully looped network (S

_{c}= 10

^{−4}) depending on the imperviousness ratio and temporal rainfall variation.

^{−4}), the hydrograph of the fully looped network is close to that of the dendritic network with higher β (10

^{0}). In contrast, when the catchment slope is relatively mild (S = 1 × 10

^{−4}), the hydrograph of the fully looped network is close to the dendritic network with smaller β (10

^{−4}), which is consistent regardless of the model (GIUH or SWMM). The behavior of the fully looped network implies that it is possible to set a pair of dendritic networks generated by the Gibbsian model and the fully looped network depending on the catchment slope. Seo and Schmidt [22] showed that the network property in urban catchments represented by the parameter (β) of the Gibbsian model shows greater variability compared with that of natural river networks, of which β is typically greater than 10°. Conversely, it can be inferred that the Gibbsian model is appropriate to represent the network properties of the drainage network in urban catchments compared to other stochastic network models, such as the Scheidegger model [18,19] and Uniform model [20,21]. Combined with the results from Seo and Schmidt [22,23,24], this study strongly suggests the application of the Gibbsian model to a hydrologic modeling approach in urban catchments, especially in the case of data unavailability.

**Figure 10.**Comparison of the flow hydrographs from a fully looped network using Storm Water Management Model (SWMM) and a dendritic networks generated by the Gibbsian model using Geomorphologic Instantaneous Unit Hydrograph (GIUH) and SWMM (rainfall intensity = 15 mm/h).

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Seo, Y.; Seo, Y.-H.; Kim, Y.-O. Behavior of a Fully-Looped Drainage Network and the Corresponding Dendritic Networks. *Water* **2015**, *7*, 1291-1305.
https://doi.org/10.3390/w7031291

**AMA Style**

Seo Y, Seo Y-H, Kim Y-O. Behavior of a Fully-Looped Drainage Network and the Corresponding Dendritic Networks. *Water*. 2015; 7(3):1291-1305.
https://doi.org/10.3390/w7031291

**Chicago/Turabian Style**

Seo, Yongwon, Young-Ho Seo, and Young-Oh Kim. 2015. "Behavior of a Fully-Looped Drainage Network and the Corresponding Dendritic Networks" *Water* 7, no. 3: 1291-1305.
https://doi.org/10.3390/w7031291