# A Coupled Hydrologic–Hydraulic Model (XAJ–HiPIMS) for Flood Simulation

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}, while hydraulic models are more advantageous for applications on smaller scales. In order to take advantages of these two types of models, this paper coupled a hydrologic model, the Xinanjing model (XAJ), with a hydraulic model, the Graphics Processing Unit (GPU)-accelerated high-performance integrated hydraulic modelling system (HiPIMS). The study was completed in the Misai basin (797 km

^{2}), located in Zhejiang Province, China. The coupled XAJ–HiPIMS model was validated against observed flood events. The simulated results agree well with the data observed at the basin outlet. The study proves that a coupled hydrologic and hydraulic model is capable of providing flood information on a small scale for a large basin and shows the potential of the research.

## 1. Introduction

^{2}, while hydraulic models are more advantageous for applications on smaller scales. In order to take advantage of these two types of models, this study attempts to couple a hydrologic model with a hydraulic model.

^{2}), located in Zhejiang Province, China, is taken as the study area. For the hydrologic model, XAJ was selected as it has been widely used in China for a long time [34,35,36]. For the hydraulic model, the high-performance integrated hydraulic modelling system (HiPIMS) was chosen because of its high-speed calculations and wide application [23,24,25]. This paper introduces how the XAJ–HiPIMS is coupled from XAJ and HiPIMS and validates the new model against typical floods. Section 2 outlines the materials and methods, Section 3 presents results and discussion, and Section 4 gives conclusions.

## 2. Materials and Methods

#### 2.1. XAJ

#### 2.2. HiPIMS

_{b}, and S

_{f}denote the source terms of rainfall, bed slope, and friction, respectively. In Equation (2), h is the water depth (h = η − z

_{b}, where η is the water surface elevation and z

_{b}is the bed elevation); u and v are the depth-averaged velocity components in the x- and y-directions, respectively; g is the acceleration of gravity; ρ is the water density; r is the generated surface runoff (which is the ${r}_{i,j}^{n}$ in Equation (2)); ∂b / ∂x and ∂b / ∂y denote the bed slope in the x- and y-directions; τ

_{bx}and τ

_{by}are the bed friction stresses, calculated by the following equations:

_{f}( = gn

^{2}/ h

^{1/3}) stands for the bed roughness coefficient; and n is the Manning’s coefficient.

_{i}stands for the area of cell i; F

_{k}(q) presents the fluxes that are normal to the cell edges; k is the index of the cell edges (k = 1–4); l

_{k}is the corresponding cell edge length; and n = (n

_{x},n

_{y}) denotes the unit vector of the outward normal direction. In HiPIMS, the flux terms and bed slope term are calculated by an explicit scheme. The interface fluxes are calculated by the HLLC (Harten–Lax–van Leer-Contact) approximate Riemann solver. The local Riemann problems at the cell interfaces are solved by implementing the surface reconstruction method [39]. The friction term is solved by an implicit scheme [40], which is capable of maintaining the numerical stability when dealing with very small water depth. The adaptive time step method is implemented and controlled by the Courant–Friedrichs–Lewy (CFL) criterion. The high-performance of this CPU/GPU-integrating model is achieved by adopting the OpenCL programming framework [38,41].

#### 2.3. Coupling Framework

_{XAJ}denotes the surface runoff calculated by the XAJ; $\Delta {t}^{n}$ is the time-step of HiPIMS; n means the presently considered time-step; and ${r}_{i,j}^{*}$ stands for the downscaled surface runoff. In Equation (6), ${r}_{i,j}^{n}$ means the surface runoff in terms of the source term in the mass conservation equation of HiPIMS and m is the amount of the computing grid cell.

#### 2.4. Statistical Method

_{s}) stands for the simulated peak discharge and MAX(Q

_{o}) denotes the observed peak discharge. ARED∈ (−∞,+∞),

_{MAX(Qs)}is the arrival time of simulated peak discharge by XAJ and XAJ–HiPIMS and T

_{MAX(Qo)}is the arrival time of the observed peak discharge.

_{1}is the Nash–Sutcliffe efficiency coefficient computed by the XAJ simulation and observation; NSE

_{2}is the Nash–Sutcliffe efficiency coefficient computed by the XAJ–HiPIMS simulation and observation; Q

_{o}is the observed discharge; Q

_{1}is the simulated discharge of XAJ; Q

_{2}is the simulated discharge of XAJ-HiPIMS; and $\overline{{\text{}\mathrm{Q}}_{\mathrm{o}}}$ is the averaged observed discharge.

#### 2.5. Study Area

#### 2.6. Flood Processes

^{3}/s), medium (M: 500 m

^{3}/s < peak discharge < 1000 m

^{3}/s), and small (S: peak discharge < 500 m

^{3}/s) floods, as shown in Table 2.

#### 2.7. Modelling Set

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The framework of the Xinanjing model (XAJ) [37].

**Figure 6.**(

**a**) FP(S): Snapshot of inundated extent at the peak moment (left) and map of maximum water depth (right); (

**b**) FP(M): Snapshot of inundated extent at the peak moment (left) and map of maximum water depth (right); (

**c**) FP(B): Snapshot of inundated extent at the peak moment (left) and map of maximum water depth (right).

No. | Land Use Type | Area Proportion (%) | Manning’s n ^{1} |
---|---|---|---|

L1 | Forest | 45.5 | 0.139 |

L2 | Heavy brush | 37.3 | 0.098 |

L3 | Cultivated land | 2.2 | 0.041 |

L4 | Grass land | 7.3 | 0.031 |

L5 | Pond/river | 5.8 | 0.021 |

L6 | Bare land | 0.7 | 0.026 |

L7 | Urban land | 1.2 | 0.013 |

Name | Start Time | End Time | Rainfall Height | Peak Flow |
---|---|---|---|---|

FP(S) | 1987/5/26 8:00 | 1987/5/27 20:00 | 65 mm | 202 m^{3}/s |

FP(M) | 1985/5/5 8:00 | 1985/5/7 12:00 | 106 mm | 708 m^{3}/s |

FP(B) | 1983/5/29 8:00 | 1983/5/30 13:00 | 222 mm | 1820 m^{3}/s |

Module | Parameters | Physical Meaning | Value |
---|---|---|---|

Evapotranspiration | WUM | Averaged soil moisture storage capacity of the upper layer | 14 |

WLM | Averaged soil moisture storage capacity of the lower layer | 86 | |

WDM | Averaged soil moisture storage capacity of the deep layer | 33 | |

K | Conversion coefficient of evaporation | 1 | |

C | Coefficient of the deep layer | 0.126 | |

Runoff generation | B | Exponential of the distribution to tension water capacity | 0.375 |

IMP | Percentage of impervious and saturated areas in the catchment | 10 | |

Runoff source partition | SM | Areal mean free water capacity of the surface soil layer | 97 |

EX | Exponent of the free water capacity curve influencing the development of the saturated area | 1.03 | |

KG | Outflow coefficients of the free water storage to groundwater relationships | 0.459 | |

KSS | Outflow coefficients of the free water storage to interflow relationships | 0.07 | |

Runoff routing | KKG | Recession constants of the groundwater storage | 0.997 |

KKSS | Recession constants of the lower interflow storage | 0.747 |

FP No. | Peak discharge (m^{3}/s) | ARED (%) | DPAT (hour) | NSE | |||||
---|---|---|---|---|---|---|---|---|---|

O | H | C | H | C | H | C | NSE_{1} | NSE_{2} | |

FP(S) | 202 | 291 | 250 | 0.44 | 0.24 | 0 | 1 | 0.6392 | 0.8459 |

FP(M) | 708 | 900 | 822 | 0.27 | 0.16 | 0 | 0 | 0.7111 | 0.8524 |

FP(B) | 1820 | 1620 | 1817 | 0.11 | 0.002 | −1 | −1 | 0.9757 | 0.9422 |

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**MDPI and ACS Style**

Wang, Y.; Yang, X.
A Coupled Hydrologic–Hydraulic Model (XAJ–HiPIMS) for Flood Simulation. *Water* **2020**, *12*, 1288.
https://doi.org/10.3390/w12051288

**AMA Style**

Wang Y, Yang X.
A Coupled Hydrologic–Hydraulic Model (XAJ–HiPIMS) for Flood Simulation. *Water*. 2020; 12(5):1288.
https://doi.org/10.3390/w12051288

**Chicago/Turabian Style**

Wang, Yueling, and Xiaoliu Yang.
2020. "A Coupled Hydrologic–Hydraulic Model (XAJ–HiPIMS) for Flood Simulation" *Water* 12, no. 5: 1288.
https://doi.org/10.3390/w12051288