Flash floods typically occur suddenly within hours of heavy rainfall. Accurate forecasting of flash floods in advance using the two-dimensional (2D) shallow water equations (SWEs) remains a challenge, due to the governing SWEs being difficult-to-solve partial differential equations (PDEs). Aiming at shortening the computational time and gaining more time for issuing early warnings of flash floods, constructing a new relationship between water storage and outflow in the rainfall–runoff process is attempted by assuming the catchment as a water storage system. Through numerical simulations of the diffusion wave (DW) approximation of SWEs, the water storage and discharge are found to be limited to envelope lines, and the discharge/water-depth process lines during water rising and falling showed a grid-shaped distribution. Furthermore, if a catchment is regarded as a semi-open water storage system, then there is a nonlinear relationship between the inside average water depth and the outlet water depth, namely, the water storage ratio curve, which resembles the shape of a plume. In the case of an open channel without considering spatial variability, the water storage ratio curve is limited to three values (i.e., the upper, the steady, and the lower limits), which are found to be independent of meteorological (rainfall intensity), vegetation (Manning's coefficient), and terrain (slope gradient) conditions. Meteorological, vegetation, and terrain conditions only affect the size of the plume without changing its shape. Rainfall, especially weak rain (i.e., when rainfall intensity is less than 5.0 mm h

Flood disaster is a significant global health and economic threat. Disastrous floods have caused millions of fatalities in the 20th century and billions of dollars in direct economic losses each year (Merkuryeva et al., 2015; Merz et al., 2021; Ruidas et al., 2022). According to statistics (Lee et al., 2020), from 2001 to 2018, over 2900 floods caused over 93 000 deaths and over USD 490 billion in economic damages worldwide. Based on 250 m resolution daily satellite images of 913 major flood events during the same period, the total area inundated by floods is estimated to be

Flood simulation provides an effective means of flood forecasting to reduce the loss of property and life in flood-threatened areas around the world. In particular, weather-prediction-based distributed hydrological/hydraulic models are considered to be an effective strategy for flood simulation (Ming et al., 2020). Hence, a large number of scholars are committed to improving the simulation efficiency or simulation accuracy of distributed hydrological/hydraulic models. Accordingly, they have developed many forms of hydrological models and hydrodynamic models in the past decades. Among them, the hydrological models include Stanford Watershed Model IV (SWM) (Crawford and Linsley, 1966), the SHE/MIKESHE model (Abbott et al., 1986), the TANK model (Sugawara, 1995), Soil and Water Assessment Tool (SWAT) (Arnold and Williams, 1987), and TOPMODEL (Beven and Kirkby, 1979). The hydrodynamic models include the one-dimensional (1D) Saint-Venant equations (Köhne et al., 2011), the two-dimensional (2D) shallow water equations (SWEs) (Camassa et al., 1994), and the three-dimensional (3D) integrated equations of runoff and seepage (Mori et al., 2015). In addition, a variety of hydrological–hydrodynamic coupling models have also been proposed by Kim et al. (2012), Liu et al. (2019), Hoch et al. (2019), and other scholars. In particular, SWEs are the main governing equations for simulating floods. However, flood simulation based on SWEs is a time-consuming process due to its governing equations being a hyperbolic system of first-order nonlinear partial differential equations (PDEs) (Li and Fan, 2017). Therefore, many scholars attempted to improve the efficiency and accuracy of flood simulation through computer technology, e.g., applying GPU parallel computing (Crossley et al., 2010) or advanced numerical schemes (Sanders et al., 2010). For hydrological studies, the performance of hydrological modeling is usually challenged by model calibration and uncertainty analysis during modeling exercises (Wu et al., 2021).

Efficient and stable solutions for hydrodynamic models has long been an important issue in flood forecasting. Since the SWEs are nonlinear hyperbolic PDEs, the increase in the calculation domain and the increase in the degree of discreteness will greatly increase the difficulty of solving SWEs. In addition, when using high-resolution terrain to improve model calculation accuracy, non-physical phenomena such as false high-flow velocity in steep terrain will also occur, resulting in calculation distortion and a sharp increase in calculation time. Hence, we try to ignore the complex exchange/transfer process of mass and momentum (hydrodynamic models), and we also abandon the empirical relationships (hydrological models) between the input (precipitation), the transmission (flow rate), and the output (discharge) in the catchment area. A catchment is regarded as a semi-open water storage system, and the complex problem is simplified into three megascopic variables, i.e., inflow, water storage, and outflow. For one watershed, the complex internal flow processes could be ignored if the physical mechanism between inflow, water storage, and outflow can be found under different meteorological, geographical, and geological conditions. In other words, if we can give a physical-based relationship between the three megascopic variables, then flood forecasting will become much simpler.

An arbitrary catchment (Fig. 1b) could be assumed to be a conceptual water tank (Fig. 1a). In this water tank, according to the law of conservation of mass, the complex confluence process of surface runoff could be neglected, and it can be described only by the relationship between input, storage, and output, which can be expressed as Eq. (1),

Conceptual schematic of the distributed runoff model (DRM) and numerical model.

In this section, attention is focused on the surface flow of runoff, so the runoff–atmosphere moisture exchange (evaporation) and runoff–soil moisture exchange (infiltration and/or exfiltration) are not considered. Zhu et al. (2020) validated the effectiveness of a diffusion wave (DW) approximation of shallow water equations by numerical simulations for simulating ground surface runoff,

To improve the computational efficiency of the hydrodynamic model, after strict mathematical derivation according to the basic hydrodynamic equation and the law of conservation of mass, Zhu et al. (2022) proposed a hydrological–hydrodynamic integrated model, i.e., distributed runoff model (DRM), as

The conceptual hydrological model takes the inside average water depth (

Water storage ratio curves.

Finally, it is found that water storage ratio curves resemble the shape of a plume. When the water outlet depth is the same, the water storage ratio (

Discharge/water-depth process lines during water rising and falling.

To obtain further insights into the causes of the formation of the water-rising limb and the water-falling limb of the water storage ratio curve, the ratio of discharge (i.e., the ratio of the total outflows (

On the other hand, the process lines of discharge and water depth during water rising and falling present a grid-shaped cross-distribution (Fig. 3a and d). Similarly, from the view of the gradient of the discharge and water depth process lines during water rising and falling, the discharge gradient curves (Fig. 3b) and the water depth gradient curves (Fig. 3e) also present a grid-shaped cross-distribution during water rising and falling, which might be the cause of the looped rating curve (Fig. 3c), i.e., higher discharges for the rising limb (

Based on the water storage ratio curve, a hydrological–hydrodynamic integrated model, namely, the distributed runoff model (DRM), is established with the governing equations in Eq. (3). To check the effectiveness and applicability of the DRM, a comparative analysis of the numerical results obtained from the DRM and the DW model is implemented. We found that the DRM quickly reproduces the calculation results of the time-consuming DW model under different rainfall intensities (Fig. 4a and b), different Manning's coefficients (Fig. 4c), and different slope gradients (Fig. 4d). meaning that the water storage ratio curve will provide new ideas for simulation and early warning of floods. In addition, due to the governing equations of the DRM being ordinary differential equations (ODEs), the computational efficiency of the DRM is much higher than the DW model, which is governed by nonlinear partial differential equations (PDEs). More attention should be paid to the determination of the nonlinear relationship of the water storage ratio curve under different geographical scenarios, which will be beneficial to the proposal of more efficient flood-forecasting methods or early-warning systems.

Comparative analyses of discharge calculated by DW and the DRM under designed rainfall.

In the above section, the simulations of DW and the DRM are based on an impermeable conceptual slope model as shown in Fig. 1c. After considering infiltration in DW and the DRM, Eqs. (2) and (3) become

Infiltration parameter sets.

A rainfall event begins with a weak precipitation intensity. When the rainfall intensity is less than the infiltration capacity, all the rainwater will infiltrate into the soil. When the rainfall intensity exceeds the soil infiltration capacity, the surface water is generated, and Horton law (Eq. 6) applies:

Outlet discharge (

After implementing a real rainfall event in the impermeable conceptual slope model (Fig. 1c), the change in the water storage ratio is calculated as shown in Fig. 6. Rainfall data were recorded from 9 August 2022 00:00 JST (Japan standard time, UTC+9) to 10 August 2022 00:00 JST in Aomori Prefecture, Japan, and from 29 August 2016 01:00 JST to 31 August 2016 09:00 JST in Nissho Pass, Japan (

The fluctuation of water storage ratio and the effectiveness of the DRM in natural rainfall events.

Besides, the fluctuations of the water storage ratio can be divided into three modes: Mode I identified as the inverse S-shape type during the rainfall beginning stage (Fig. 7a), Mode II identified as a wave type during the weak rainfall duration stage (Fig. 7b), and Mode III identified as a checkmark type during the rainfall end stage (Fig. 7c). Among them, Mode I describes how water storage ratio drops from the upper limit to the steady limit in an inverse S-shape. Mode II represents the water storage fluctuations around the steady limit. Mode III happens when the water storage ratio first drops from the steady limit to the lower limit and then rises to the upper limit. This means that the certainty of the fluctuation modes will provide the possibility for quantitative analysis of the fluctuation of the water storage ratio induced by the change in the rainfall intensity.

Three kinds of water storage ratio fluctuation modes in natural rainfall events.

Figure 8a and b show the simulation results of discharge calculated by the DRM and DW model using the rainfall data recorded in Aomori Prefecture and Nissho Pass, Japan, respectively. Results suggest that after the determination of the water storage ratio fluctuations, the calculation results of the DRM are in good agreement with those of the DW model, meaning that the DRM provides a new and more effective theoretical scheme for flood prediction.

Time-dependent discharge calculated by the DRM and DW model.

Based on a conceptual slope model, numerical simulations of the rainfall–runoff process are performed by using the diffusion wave (DW) approximation of SWEs. A plume-shaped nonlinear relationship between water storage and outflow, defined as the water storage ratio, is found between the inside average water depth and the outlet water depth in a catchment. The water storage ratio is controlled by three limits, namely, upper limit, steady limit, and lower limit with values of approximately 1.0, 0.625, and 0.4125, respectively. Under the control of the three limits, meteorological, vegetation, and terrain conditions only affect the size of the plume without changing its shape. The regular curve shape of the water storage ratio provides the possibility to construct a correlation between the water storage in the catchment area and the outlet discharge.

Based on the water storage ratio, a hydrological–hydrodynamic integrated model (DRM) is established, which shows high calculation accuracy and computational efficiency. This is because the governing equations of the DRM are ordinary differential equations (ODEs), which are much easier to solve than nonlinear partial differential equations (PDEs). However, the calculations of the DRM and DW only involve the confluence part of surface water and infiltration, while the interbasin groundwater flow as inputs to the watershed (exfiltration) and evaporation are not considered. This is inconsistent with the real rainfall–runoff process in the watershed and may lead to deviations in the calculation results. Therefore, the flow exchange between surface water and groundwater during the existence and extinction of runoff also needs to be further realized by establishing a dynamic coupling model of surface water and groundwater.

In addition, the water storage and discharge are limited to envelope lines, and the discharge/water-depth process lines during water rising and falling showed a grid-shaped distribution, which might be the cause of the looped rating curve, i.e., higher discharges for the rising limb than for the recession limb at the same stage. Rainfall, especially weak rainfall (i.e., rainfall intensity less than 5.0 mm h

The findings in this study provide a key to establishing a simpler prediction model for flash floods. The water storage ratio has been proven to be effective in improving the effectiveness and efficiency of flood forecasting. Therefore, the determination of the nonlinear relationship of the water storage ratio curve under different geographical scenarios will provide new ideas for simulation and early warning of flash floods.

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

YUZ: conceptualization, methodology, software, validation, formal analysis, investigation, data curation, writing-original draft, writing (review and editing). YAZ: methodology, validation, investigation, resources, data curation. XX: methodology, investigation, data curation. CM: validation, investigation, data curation. YW: conceptualization, methodology, writing (original draft), writing (review and editing), supervision, project administration, funding acquisition.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This study was supported by the National Natural Science Fund of China (52279064, 52209087) and the Fundamental Research Funds for the Central Universities of China (2024MS069, 2024MS068).

This research has been supported by the National Natural Science Fund of China (grant nos. 52279064, 52209087) and the Fundamental Research Funds for the Central Universities of China (grant nos. 2024MS069, 2024MS068).

This paper was edited by Roberto Greco and reviewed by Marco Peli and one anonymous referee.