ABSTRACT
Based on the confluence principle, the confluence should represent the aggregate of net rainfall confluences within each distinct basic watershed unit (BWU) of a basin. The BWUs are categorized as V-type and Horseshoe-type based on their shape characteristics and two types of time–area curves of slope convergence have been derived separately. The cascade river convergence is modeled using a lagged linear reservoir, resulting in the development of a distributed CLARK convergence model based on the BWUs of a basin (BWU-DCLARK). The key findings are as follows: (1) The BWU-DCLARK model effectively captures the runoff convergence process and has been successfully applied in the Yanduhe River basin. Modeling results demonstrate high simulation accuracy. (2) The time of slope convergence indicates that the regulatory and storage effects on runoff of BWUs cannot be overlooked. (3) The BWU-DCLARK confluence model not only enables the calculation of flow at the basin outlet but also facilitates the computation of flow at any node along the river chain which is of great significance for hydrological forecasting in un-gauged basins but the application effect will need further verification.
HIGHLIGHTS
A distributed confluence model is built.
The BWU-DCLARK model can clearly reflect the process of runoff converging.
The results show that the regulation and storage effect of basic watershed units cannot be ignored.
The confluence model can not only calculate the flow at the outlet of the basin, but also calculate the flow at any river chain node.
The research showed a new method for runoff calculating.
INTRODUCTION
As an important tool for simulating and predicting hydrological processes in river basins, basin convergence models have extensive application value in many fields such as water resources management and water disaster prevention (Wagener et al. 2001; Fathian et al. 2019; Cheng et al. 2020; Ju et al. 2020; Liu et al. 2022; Li et al. 2023a, 2023b, 2023c; Dasgupta et al. 2024). However, with the continuous impact of global climate change and human activities, basin hydrological processes have become increasingly complex and uncertain, which has brought challenges to the accuracy and reliability of basin catchment models (Madsen 2017; Xie et al. 2022; Xu et al. 2024a, 2024b). Therefore, studying a hydrological model that is more consistent with the physical process mechanism of watershed confluence has important practical significance and theoretical value (Guo et al. 2017; Bloschl et al. 2019; Cui et al. 2023; Li et al. 2023a, 2023b, 2023c).
Previous studies have mainly focused on river convergence or on the use of conceptual unit hydrograph models to address basin confluence issues (Sherman 1932; Shen et al. 2016), including widely used hydrological processing methods such as the Nash model and hydraulic treatment methods like motion wave and diffusion wave models (Lighthill & Whitham 1955; Woolhiser & Liggett 1967). With the development of computers and hydrological models, distributed confluence models have been widely studied and applied in runoff simulation (Li et al. 2023a, 2023b, 2023c; Xu et al. 2024a, 2024b). The geomorphologic instantaneous unit hydrograph routing approach, which can affect the geomorphologic characteristics on runoff routing, has been proposed and applied to un-gauged basins (Rodriguez & Valdes 1979; Gong et al. 2021).
With further research, the studies have demonstrated that the regulatory and storage effects of sub-watersheds cannot be ignored for basin confluence models (Wu et al. 2021), especially in large-scale basins (Zaitchik et al. 2010). Distributed models divide the watershed into sub-regions using approaches such as grid partitioning, natural sub-watershed division, isochronous area units, mountain slope units, and Thiessen polygons (Rui & Huang 2004). For instance, the Système Hydrologique Européen (SHE) model reflects differences in horizontal directions through grids (Abbott et al. 1986); the most widely used, the Soil and Water Assessment Tool (SWAT) model, reflects differences in hydrological response units (Mtibaa & Asano 2022; Liu et al. 2023), and the Xin'anjiang model reflects differences using the Thiessen polygon method (Zhao 1984; Zhao et al. 2023).
While the length of slope confluence is shorter than that of river confluence, the overland confluence velocity is considerably lower. Recent research indicates that, compared to surface flow, the low velocity of soil water runoff is the primary control factor in most runoff formation (Wang et al. 2019; Han et al. 2020; Bouvier et al. 2021; Zhao et al. 2022). Therefore, the regulatory and storage effects of slope confluence may possibly not be negligible, especially in small watersheds.
Currently, descriptions of slope confluence primarily rely on conceptual elements or empirical functions. The isochronous method, as outlined by Shen et al. (2016), provides a well-defined flow field and confluence propagation field, presenting a simplified distributed confluence model. In the conceptual distributed slope confluence model, ModClark comprehensively accounts for the regulation, storage, and propagation effects within the watershed. With fewer calibration parameters and relatively straightforward calculations, it is an effective simulation method for slope confluence (Clark 1945; Koppen 2020; Ogassawara et al. 2022). However, the application of watershed DEM technology in the determination of the Clark unit hydrograph still exhibits significant generalization in processing between slope and river confluences. Given the sequential order between slope flow and river flow, addressing the logical relationship between the two is of great significance for watershed confluence. Various scholars have constructed distributed confluence models for BWUs in slope confluence research, investigating the time–area curve of slope confluence based on terrain parameters (Pan & Liu 2009; Luo et al. 2012). Notably, these studies have not adequately accounted for differences in river confluence paths, specifically in the construction of slope flow and its corresponding unique river flow models.
In this work, we aim to apply the concept of BWUs in a basin, combining them with the Clark model to explore the distributed confluence of the watershed to achieve the goal of separately addressing slope and river confluence. It can clearly reflect the convergence path of net rain within each BWU. To achieve this, the BWUs are categorized as V-type and Horseshoe-type based on their shape characteristics. Meanwhile, two types of time–area area curves of slope convergence have been derived. Simultaneously, each BWU follows a fixed river confluence path, and a lagged linear reservoir is employed to calculate each fixed river confluence. This model, termed the distributed Clark model based on BWUs (BWU-DCLARK), can reflect the logical sequence of natural water flow convergence. It considers variations in slope confluence and confluence routes for each BWU, aiming to construct a clearer conceptual distributed watershed confluence model.
STUDY AREA AND MATERIALS
A comprehensive hydro-meteorological measurement database for the Yanduhe Basin is available which was compiled from hydro-meteorological stations in China. This database includes precipitation and evaporation measurements from six stations (hourly values) as well as discharge data from the river outlet (hourly values). Thirty flood events from 1981 to 1986 were employed in the case study.
In the BWU-DCLARK model, the input variable is net rainfall, and the output is runoff at the river outlet. The net rainfall is computed using the Xin'an Jiang rainfall–runoff model (Zhao et al. 1992), a well-established approach in hydrological modeling.
Yanduhe DEM data can be downloaded from the SRTMDEM 90M of the Geospatial Data Cloud: https://www.gscloud.cn/#page1/4.
RESEARCH METHODS
Clark model

BWU-DCLARK model



Extraction of the BWUs of the watershed using DEM data through the D8 method.
Investigation of the time–area curves for each BWU within the basin.
Examination of the river confluence path for each BWU.
Use of a lagged linear reservoir to construct the river confluence module along the unique path for each BWU within the basin.
Integration of the time–area curve for each BWU with the lagged linear reservoir module to obtain the lagged Clark model, as illustrated in Figure 3.
Superimposition of the lagged Clark model of each BWU according to the area weight to obtain the distributed Clark confluence model.
BWU of a basin
Simplified shape of BWUs: (a) V-type BWU and (b) Horseshoe-type BWU.
Time–area curve
For a watershed with regular geometry, we assume that the slope within each BWU is uniform, and the slope velocity is constant. Following the concept of isochrones, the time–area curves for V-type and Horseshoe-type BWUs are derived as follows:
- (1)
Time–area curve for V-type BWUs















Lagging single linear reservoir model




BWU and river confluence velocity



BWU and river confluence length
The V-type BWU is assumed to consist of two identical rectangles; the Horseshoe-type BWU is assumed to consist of two identical rectangles and a sector, and the sector and rectangle are equal in area.

The confluence length of the river channel is obtained by analyzing the confluence path of each BWU, as shown in Section 3.4.
Performance of the BWU-DCLARK model










RESULTS AND DISCUSSION
Geomorphic features and confluence paths of each BWU
BWU no. . | Chain length (km) . | Area of each BWU (km2) . | Slope of each BWU . | Slope of each chain . | River confluence path . | River confluence length (km) . | Velocity in each BWU (m/s) . | Confluence time (rounding value) (h) . | Velocity in each river path (m/s) . | Lag time (rounding value) (h) . |
---|---|---|---|---|---|---|---|---|---|---|
1 | 3.215 | 19.051 | 0.338 | 0.031109 | 1-3-5-7-9-13-15-17-21-23-25-29-31-45-47-49-59 | 47.3000 | 0.7098 | 1 | 10.1536 | 2 |
2 | 1.554 | 13.871 | 0.582 | 0.128734 | 2-3-5-7-9-13-15-17-21-23-25-29-31-45-47-49-59 | 45.6390 | 0.7592 | 2 | 10.2098 | 2 |
3 | 6.016 | 21.942 | 0.508 | 0.160901 | 3-5-7-9-13-15-17-21-23-25-29-31-45-47-49-59 | 44.0850 | 0.7465 | 1 | 10.1942 | 2 |
4 | 4.204 | 16.661 | 0.639 | 0.16435 | 4-5-7-9-13-15-17-21-23-25-29-31-45-47-49-59 | 42.2730 | 0.7680 | 1 | 10.1963 | 2 |
5 | 2.434 | 5.044 | 0.485 | 0.041083 | 5-7-9-13-15-17-21-23-25-29-31-45-47-49-59 | 38.0690 | 0.7423 | 1 | 10.1550 | 2 |
6 | 1.080 | 8.227 | 0.321 | 0.185224 | 6-7-9-13-15-17-21-23-25-29-31-45-47-49-59 | 36.7150 | 0.7053 | 2 | 10.2474 | 1 |
7 | 0.759 | 0.322 | 0.413 | 0.0001 | 7-9-13-15-17-21-23-25-29-31-45-47-49-59 | 35.6350 | 0.7276 | 1 | 10.1945 | 1 |
8 | 3.451 | 18.348 | 0.462 | 0.02898 | 8-9-13-15-17-21-23-25-29-31-45-47-49-59 | 38.3270 | 0.7378 | 1 | 10.2143 | 2 |
9 | 2.009 | 2.747 | 0.593 | 0.049776 | 9-13-15-17-21-23-25-29-31-45-47-49-59 | 34.8760 | 0.7610 | 1 | 10.2673 | 1 |
10 | 5.786 | 20.297 | 0.626 | 0.055305 | 10-12-13-15-17-21-23-25-29-31-45-47-49-59 | 43.3310 | 0.7661 | 1 | 10.2684 | 2 |
11 | 4.766 | 12.528 | 0.529 | 0.036297 | 11-12-13-15-17-21-23-25-29-31-45-47-49-59 | 42.3110 | 0.7503 | 1 | 10.2561 | 2 |
12 | 4.678 | 10.678 | 0.725 | 0.105182 | 12-13-15-17-21-23-25-29-31-45-47-49-59 | 37.5450 | 0.7801 | 1 | 10.3052 | 2 |
13 | 1.224 | 2.308 | 0.591 | 0.0362967 | 13-15-17-21-23-25-29-31-45-47-49-59 | 32.8670 | 0.7606 | 1 | 10.3111 | 1 |
14 | 5.293 | 14.491 | 0.752 | 0.170022 | 14-15-17-21-23-25-29-31-45-47-49-59 | 36.9360 | 0.7837 | 1 | 10.4278 | 1 |
15 | 1.451 | 1.520 | 0.374 | 0.137844 | 15-17-21-23-25-29-31-45-47-49-59 | 31.6430 | 0.7188 | 1 | 10.3975 | 1 |
16 | 4.194 | 13.203 | 0.389 | 0.190746 | 16-17-21-23-25-29-31-45-47-49-59 | 34.3860 | 0.7223 | 1 | 10.4355 | 1 |
17 | 7.587 | 25.899 | 0.726 | 0.064983 | 17-21-23-25-29-31-45-47-49-59 | 30.1920 | 0.7803 | 1 | 10.3866 | 1 |
18 | 3.175 | 11.639 | 0.440 | 0.052908 | 18-20-21-23-25-29-31-45-47-49-59 | 27.9320 | 0.7334 | 1 | 10.3192 | 1 |
19 | 2.638 | 7.242 | 0.658 | 0.040568 | 19-20-21-23-25-29-31-45-47-49-59 | 27.3950 | 0.7708 | 1 | 10.3095 | 1 |
20 | 2.152 | 4.660 | 0.412 | 0.046463 | 20-21-23-25-29-31-45-47-49-59 | 24.7570 | 0.7274 | 1 | 10.3714 | 1 |
21 | 6.256 | 19.904 | 1.038 | 0.143852 | 21-23-25-29-31-45-47-49-59 | 22.6050 | 0.8156 | 1 | 10.4377 | 1 |
22 | 3.157 | 11.145 | 0.870 | 0.221764 | 22-23-25-29-31-45-47-49-59 | 19.5060 | 0.7979 | 1 | 10.5028 | 1 |
23 | 2.453 | 4.038 | 0.322 | 0.0001 | 23-25-29-31-45-47-49-59 | 16.3490 | 0.7056 | 1 | 10.4235 | 1 |
24 | 4.270 | 16.802 | 0.564 | 0.140508 | 24-25-29-31-45-47-49-59 | 18.1660 | 0.7562 | 1 | 10.5534 | 1 |
25 | 0.616 | 8.526 | 0.666 | 0.162345 | 25-29-31-45-47-49-59 | 13.8960 | 0.7720 | 3 | 10.5580 | 1 |
26 | 1.335 | 8.307 | 0.748 | 0.149848 | 26-28-29-31-45-47-49-59 | 19.1390 | 0.7831 | 1 | 10.5171 | 1 |
27 | 2.310 | 4.507 | 0.670 | 0.129888 | 27-28-29-31-45-47-49-59 | 20.1140 | 0.7725 | 1 | 10.4989 | 1 |
28 | 4.524 | 24.470 | 0.527 | 0.11206 | 28-29-31-45-47-49-59 | 17.8040 | 0.7499 | 2 | 10.5069 | 1 |
29 | 1.736 | 2.456 | 0.594 | 0.11518 | 29-31-45-47-49-59 | 13.2800 | 0.7611 | 1 | 10.5386 | 1 |
30 | 3.850 | 9.422 | 0.567 | 0.077913 | 30-31-45-47-49-59 | 15.3940 | 0.7567 | 1 | 10.4937 | 1 |
31 | 4.106 | 6.381 | 0.307 | 0.413994 | 31-45-47-49-59 | 11.5440 | 0.7014 | 1 | 10.5773 | 1 |
32 | 1.467 | 1.567 | 0.398 | 0.068149 | 32-34-36-40-42-44–45-47-49-59 | 22.6450 | 0.7243 | 1 | 10.1983 | 1 |
33 | 3.436 | 4.452 | 0.649 | 0.029105 | 33-34-36-40-42-44–45-47-49-59 | 24.6140 | 0.7695 | 1 | 10.1600 | 1 |
34 | 3.704 | 6.174 | 0.565 | 0.135005 | 34-36-40-42-44–45-47-49-59 | 21.1780 | 0.7564 | 1 | 10.2336 | 1 |
35 | 2.753 | 6.241 | 0.642 | 0.0 29105 | 35-36-40-42-44–45-47-49-59 | 20.2270 | 0.7685 | 1 | 10.0838 | 1 |
36 | 4.094 | 5.783 | 0.459 | 0.091117 | 36-40-42-44–45-47-49-59 | 17.4740 | 0.7372 | 1 | 10.1984 | 1 |
37 | 3.075 | 9.422 | 0.690 | 0.065044 | 37-39-40-42-44–45-47-49-59 | 18.4100 | 0.7754 | 1 | 10.0522 | 1 |
38 | 5.576 | 19.361 | 0.452 | 0.093619 | 38-39-40-42-44–45-47-49-59 | 20.9110 | 0.7358 | 1 | 10.0867 | 1 |
39 | 1.955 | 3.841 | 0.555 | 0.0 650439 | 39-40-42-44–45-47-49-59 | 15.3350 | 0.7547 | 1 | 10.0827 | 1 |
40 | 2.809 | 14.259 | 0.459 | 0.071211 | 40-42-44–45-47-49-59 | 13.3800 | 0.7372 | 1 | 10.2128 | 1 |
41 | 1.046 | 9.919 | 0.561 | 0.191255 | 41-42-44–45-47-49-59 | 11.6170 | 0.7557 | 2 | 10.3648 | 1 |
42 | 0.705 | 7.727 | 0.344 | 0.283585 | 42-44–45-47-49-59 | 10.5710 | 0.7114 | 3 | 10.2622 | 1 |
43 | 1.984 | 2.949 | 0.326 | 0.050412 | 43-44–45-47-49-59 | 11.8500 | 0.7066 | 1 | 9.8339 | 1 |
44 | 2.428 | 5.416 | 0.573 | 0.038303 | 44–45-47-49-59 | 9.8660 | 0.7577 | 1 | 9.8847 | 1 |
45 | 1.892 | 2.848 | 0.571 | 0.0504116 | 45-47-49-59 | 7.4380 | 0.7574 | 1 | 9.9925 | 1 |
46 | 0.956 | 6.659 | 0.478 | 0.296006 | 46-47-49-59 | 6.5020 | 0.7409 | 1 | 10.6282 | 1 |
47 | 1.845 | 10.028 | 0.568 | 0.216852 | 47-49-59 | 5.5460 | 0.7569 | 1 | 10.2722 | 1 |
48 | 0.891 | 7.023 | 0.528 | 0.12122 | 48-49-59 | 4.5920 | 0.7501 | 1 | 9.9384 | 1 |
49 | 2.687 | 2.858 | 0.863 | 0.111667 | 49-59 | 3.7010 | 0.7971 | 1 | 9.6290 | 1 |
50 | 4.105 | 16.331 | 0.501 | 0.146169 | 50-52-54-56-58-59 | 27.1730 | 0.7452 | 1 | 10.1744 | 1 |
51 | 1.094 | 0.893 | 0.609 | 0.091411 | 51-52-54-56-58-59 | 24.1620 | 0.7635 | 1 | 10.0804 | 1 |
52 | 2.613 | 13.795 | 0.765 | 0.11482 | 52-54-56-58-59 | 23.0680 | 0.7853 | 1 | 10.0775 | 1 |
53 | 2.456 | 6.069 | 0.609 | 0.0914114 | 53-54-56-58-59 | 22.9110 | 0.7635 | 1 | 9.7956 | 1 |
54 | 6.214 | 7.305 | 0.626 | 0.305743 | 54-56-58-59 | 20.4550 | 0.7661 | 1 | 10.0076 | 1 |
55 | 1.395 | 11.473 | 0.428 | 0.071671 | 55-56-58-59 | 15.6360 | 0.7309 | 2 | 8.9117 | 1 |
56 | 10.718 | 42.814 | 0.767 | 0.027991 | 56-58-59 | 14.2410 | 0.7856 | 1 | 8.1131 | 1 |
57 | 3.643 | 5.669 | 0.794 | 0.0716708 | 57-58-59 | 7.1660 | 0.7889 | 1 | 5.2435 | 1 |
58 | 2.509 | 8.060 | 0.530 | 0.0279913 | 58-59 | 3.5230 | 0.7504 | 1 | 5.2435 | 1 |
59 | 1.014 | 1.151 | 1.079 | 0.04448 | 59 | 1.0140 | 0.8195 | 1 | 5.2435 | 1 |
BWU no. . | Chain length (km) . | Area of each BWU (km2) . | Slope of each BWU . | Slope of each chain . | River confluence path . | River confluence length (km) . | Velocity in each BWU (m/s) . | Confluence time (rounding value) (h) . | Velocity in each river path (m/s) . | Lag time (rounding value) (h) . |
---|---|---|---|---|---|---|---|---|---|---|
1 | 3.215 | 19.051 | 0.338 | 0.031109 | 1-3-5-7-9-13-15-17-21-23-25-29-31-45-47-49-59 | 47.3000 | 0.7098 | 1 | 10.1536 | 2 |
2 | 1.554 | 13.871 | 0.582 | 0.128734 | 2-3-5-7-9-13-15-17-21-23-25-29-31-45-47-49-59 | 45.6390 | 0.7592 | 2 | 10.2098 | 2 |
3 | 6.016 | 21.942 | 0.508 | 0.160901 | 3-5-7-9-13-15-17-21-23-25-29-31-45-47-49-59 | 44.0850 | 0.7465 | 1 | 10.1942 | 2 |
4 | 4.204 | 16.661 | 0.639 | 0.16435 | 4-5-7-9-13-15-17-21-23-25-29-31-45-47-49-59 | 42.2730 | 0.7680 | 1 | 10.1963 | 2 |
5 | 2.434 | 5.044 | 0.485 | 0.041083 | 5-7-9-13-15-17-21-23-25-29-31-45-47-49-59 | 38.0690 | 0.7423 | 1 | 10.1550 | 2 |
6 | 1.080 | 8.227 | 0.321 | 0.185224 | 6-7-9-13-15-17-21-23-25-29-31-45-47-49-59 | 36.7150 | 0.7053 | 2 | 10.2474 | 1 |
7 | 0.759 | 0.322 | 0.413 | 0.0001 | 7-9-13-15-17-21-23-25-29-31-45-47-49-59 | 35.6350 | 0.7276 | 1 | 10.1945 | 1 |
8 | 3.451 | 18.348 | 0.462 | 0.02898 | 8-9-13-15-17-21-23-25-29-31-45-47-49-59 | 38.3270 | 0.7378 | 1 | 10.2143 | 2 |
9 | 2.009 | 2.747 | 0.593 | 0.049776 | 9-13-15-17-21-23-25-29-31-45-47-49-59 | 34.8760 | 0.7610 | 1 | 10.2673 | 1 |
10 | 5.786 | 20.297 | 0.626 | 0.055305 | 10-12-13-15-17-21-23-25-29-31-45-47-49-59 | 43.3310 | 0.7661 | 1 | 10.2684 | 2 |
11 | 4.766 | 12.528 | 0.529 | 0.036297 | 11-12-13-15-17-21-23-25-29-31-45-47-49-59 | 42.3110 | 0.7503 | 1 | 10.2561 | 2 |
12 | 4.678 | 10.678 | 0.725 | 0.105182 | 12-13-15-17-21-23-25-29-31-45-47-49-59 | 37.5450 | 0.7801 | 1 | 10.3052 | 2 |
13 | 1.224 | 2.308 | 0.591 | 0.0362967 | 13-15-17-21-23-25-29-31-45-47-49-59 | 32.8670 | 0.7606 | 1 | 10.3111 | 1 |
14 | 5.293 | 14.491 | 0.752 | 0.170022 | 14-15-17-21-23-25-29-31-45-47-49-59 | 36.9360 | 0.7837 | 1 | 10.4278 | 1 |
15 | 1.451 | 1.520 | 0.374 | 0.137844 | 15-17-21-23-25-29-31-45-47-49-59 | 31.6430 | 0.7188 | 1 | 10.3975 | 1 |
16 | 4.194 | 13.203 | 0.389 | 0.190746 | 16-17-21-23-25-29-31-45-47-49-59 | 34.3860 | 0.7223 | 1 | 10.4355 | 1 |
17 | 7.587 | 25.899 | 0.726 | 0.064983 | 17-21-23-25-29-31-45-47-49-59 | 30.1920 | 0.7803 | 1 | 10.3866 | 1 |
18 | 3.175 | 11.639 | 0.440 | 0.052908 | 18-20-21-23-25-29-31-45-47-49-59 | 27.9320 | 0.7334 | 1 | 10.3192 | 1 |
19 | 2.638 | 7.242 | 0.658 | 0.040568 | 19-20-21-23-25-29-31-45-47-49-59 | 27.3950 | 0.7708 | 1 | 10.3095 | 1 |
20 | 2.152 | 4.660 | 0.412 | 0.046463 | 20-21-23-25-29-31-45-47-49-59 | 24.7570 | 0.7274 | 1 | 10.3714 | 1 |
21 | 6.256 | 19.904 | 1.038 | 0.143852 | 21-23-25-29-31-45-47-49-59 | 22.6050 | 0.8156 | 1 | 10.4377 | 1 |
22 | 3.157 | 11.145 | 0.870 | 0.221764 | 22-23-25-29-31-45-47-49-59 | 19.5060 | 0.7979 | 1 | 10.5028 | 1 |
23 | 2.453 | 4.038 | 0.322 | 0.0001 | 23-25-29-31-45-47-49-59 | 16.3490 | 0.7056 | 1 | 10.4235 | 1 |
24 | 4.270 | 16.802 | 0.564 | 0.140508 | 24-25-29-31-45-47-49-59 | 18.1660 | 0.7562 | 1 | 10.5534 | 1 |
25 | 0.616 | 8.526 | 0.666 | 0.162345 | 25-29-31-45-47-49-59 | 13.8960 | 0.7720 | 3 | 10.5580 | 1 |
26 | 1.335 | 8.307 | 0.748 | 0.149848 | 26-28-29-31-45-47-49-59 | 19.1390 | 0.7831 | 1 | 10.5171 | 1 |
27 | 2.310 | 4.507 | 0.670 | 0.129888 | 27-28-29-31-45-47-49-59 | 20.1140 | 0.7725 | 1 | 10.4989 | 1 |
28 | 4.524 | 24.470 | 0.527 | 0.11206 | 28-29-31-45-47-49-59 | 17.8040 | 0.7499 | 2 | 10.5069 | 1 |
29 | 1.736 | 2.456 | 0.594 | 0.11518 | 29-31-45-47-49-59 | 13.2800 | 0.7611 | 1 | 10.5386 | 1 |
30 | 3.850 | 9.422 | 0.567 | 0.077913 | 30-31-45-47-49-59 | 15.3940 | 0.7567 | 1 | 10.4937 | 1 |
31 | 4.106 | 6.381 | 0.307 | 0.413994 | 31-45-47-49-59 | 11.5440 | 0.7014 | 1 | 10.5773 | 1 |
32 | 1.467 | 1.567 | 0.398 | 0.068149 | 32-34-36-40-42-44–45-47-49-59 | 22.6450 | 0.7243 | 1 | 10.1983 | 1 |
33 | 3.436 | 4.452 | 0.649 | 0.029105 | 33-34-36-40-42-44–45-47-49-59 | 24.6140 | 0.7695 | 1 | 10.1600 | 1 |
34 | 3.704 | 6.174 | 0.565 | 0.135005 | 34-36-40-42-44–45-47-49-59 | 21.1780 | 0.7564 | 1 | 10.2336 | 1 |
35 | 2.753 | 6.241 | 0.642 | 0.0 29105 | 35-36-40-42-44–45-47-49-59 | 20.2270 | 0.7685 | 1 | 10.0838 | 1 |
36 | 4.094 | 5.783 | 0.459 | 0.091117 | 36-40-42-44–45-47-49-59 | 17.4740 | 0.7372 | 1 | 10.1984 | 1 |
37 | 3.075 | 9.422 | 0.690 | 0.065044 | 37-39-40-42-44–45-47-49-59 | 18.4100 | 0.7754 | 1 | 10.0522 | 1 |
38 | 5.576 | 19.361 | 0.452 | 0.093619 | 38-39-40-42-44–45-47-49-59 | 20.9110 | 0.7358 | 1 | 10.0867 | 1 |
39 | 1.955 | 3.841 | 0.555 | 0.0 650439 | 39-40-42-44–45-47-49-59 | 15.3350 | 0.7547 | 1 | 10.0827 | 1 |
40 | 2.809 | 14.259 | 0.459 | 0.071211 | 40-42-44–45-47-49-59 | 13.3800 | 0.7372 | 1 | 10.2128 | 1 |
41 | 1.046 | 9.919 | 0.561 | 0.191255 | 41-42-44–45-47-49-59 | 11.6170 | 0.7557 | 2 | 10.3648 | 1 |
42 | 0.705 | 7.727 | 0.344 | 0.283585 | 42-44–45-47-49-59 | 10.5710 | 0.7114 | 3 | 10.2622 | 1 |
43 | 1.984 | 2.949 | 0.326 | 0.050412 | 43-44–45-47-49-59 | 11.8500 | 0.7066 | 1 | 9.8339 | 1 |
44 | 2.428 | 5.416 | 0.573 | 0.038303 | 44–45-47-49-59 | 9.8660 | 0.7577 | 1 | 9.8847 | 1 |
45 | 1.892 | 2.848 | 0.571 | 0.0504116 | 45-47-49-59 | 7.4380 | 0.7574 | 1 | 9.9925 | 1 |
46 | 0.956 | 6.659 | 0.478 | 0.296006 | 46-47-49-59 | 6.5020 | 0.7409 | 1 | 10.6282 | 1 |
47 | 1.845 | 10.028 | 0.568 | 0.216852 | 47-49-59 | 5.5460 | 0.7569 | 1 | 10.2722 | 1 |
48 | 0.891 | 7.023 | 0.528 | 0.12122 | 48-49-59 | 4.5920 | 0.7501 | 1 | 9.9384 | 1 |
49 | 2.687 | 2.858 | 0.863 | 0.111667 | 49-59 | 3.7010 | 0.7971 | 1 | 9.6290 | 1 |
50 | 4.105 | 16.331 | 0.501 | 0.146169 | 50-52-54-56-58-59 | 27.1730 | 0.7452 | 1 | 10.1744 | 1 |
51 | 1.094 | 0.893 | 0.609 | 0.091411 | 51-52-54-56-58-59 | 24.1620 | 0.7635 | 1 | 10.0804 | 1 |
52 | 2.613 | 13.795 | 0.765 | 0.11482 | 52-54-56-58-59 | 23.0680 | 0.7853 | 1 | 10.0775 | 1 |
53 | 2.456 | 6.069 | 0.609 | 0.0914114 | 53-54-56-58-59 | 22.9110 | 0.7635 | 1 | 9.7956 | 1 |
54 | 6.214 | 7.305 | 0.626 | 0.305743 | 54-56-58-59 | 20.4550 | 0.7661 | 1 | 10.0076 | 1 |
55 | 1.395 | 11.473 | 0.428 | 0.071671 | 55-56-58-59 | 15.6360 | 0.7309 | 2 | 8.9117 | 1 |
56 | 10.718 | 42.814 | 0.767 | 0.027991 | 56-58-59 | 14.2410 | 0.7856 | 1 | 8.1131 | 1 |
57 | 3.643 | 5.669 | 0.794 | 0.0716708 | 57-58-59 | 7.1660 | 0.7889 | 1 | 5.2435 | 1 |
58 | 2.509 | 8.060 | 0.530 | 0.0279913 | 58-59 | 3.5230 | 0.7504 | 1 | 5.2435 | 1 |
59 | 1.014 | 1.151 | 1.079 | 0.04448 | 59 | 1.0140 | 0.8195 | 1 | 5.2435 | 1 |
Thirty flood events from 1981 to 1987 were selected for studying the application of the BWU-DCLARK model. Among them, nine flood events were used for model parameter calibration, and the remaining 21 flood events were used for model validation.
Using the Generalized Likelihood Uncertainty Estimation (GLUE) optimization method, the BWU-DCLARK model parameters were obtained as follows: a = 0.8118, a′ = 12.7041, b = 0.1238, b′ = 0.0961, and K = 3.6228 m/s. The average slope confluence velocity of each BWU and the average confluence velocity of the corresponding river path are detailed in Table 1.
As shown in Table 1, the slope flow velocity remains relatively stable, ranging from a maximum of 0.8195 m/s to a minimum of 0.7014 m/s, with an overall basin average slope confluence velocity of 0.7545 m/s. Except for minor variations in the confluence velocity of individual river channels, the average confluence velocity across other river basins also remains relatively stable, ranging from a maximum of 10.6282 m/s and a minimum of 5.2435 m/s. The average confluence velocity of all river channels in the entire basin is 9.9362 m/s, indicating that the confluence velocity in downstream river channels is low and the velocity differences among upstream river channels is minimal. Notably, the slope confluence velocity is considerably lower than the river confluence velocity.
As observed in Table 1, the slope confluence time is similar to the river channel confluence lag time. This suggests that although each BWU is relatively small (only approximately 10 km2), the regulatory and storage effects of each BWU cannot be ignored.
Table 2 presents the model calibration results for the nine flood events. The maximum value of the Nash coefficient is 0.9759, the minimum value is 0.8529, and the average value is 0.9064. These results guide the determination of the model parameters in accordance with the precision requirements of the model calculations.
BUW-DCLARK model calibration results of nine flood events
Flood events . | Measured total flow (mm) . | Estimated total flow (mm) . | RET (%) . | Measured peak flow (m3/s) . | Estimated peak flow (m3/s) . | REP (%) . | Lag time of peak flow (h) . | Nash coefficient . |
---|---|---|---|---|---|---|---|---|
810,623 | 128.7119 | 131.9217 | 2.4938 | 1,065.3 | 1,130.0 | −5.73 | 1 | 0.9242 |
810,714 | 97.1716 | 88.0852 | −9.3509 | 611.6 | 589.0 | 3.84 | 1 | 0.9759 |
810,810 | 127.2974 | 131.4548 | 3.2658 | 641.8 | 628.0 | 2.20 | 2 | 0.8674 |
810,824 | 82.9152 | 76.6022 | −7.6139 | 534.4 | 509.4 | 4.90 | 0 | 0.8672 |
820,716 | 331.9663 | 321.7267 | −3.0845 | 967.2 | 1,040.0 | −7.00 | 1 | 0.9367 |
820,820 | 171.2696 | 168.8709 | −1.4005 | 602.7 | 572.5 | 5.28 | 0 | 0.8963 |
820,908 | 83.7343 | 77.9819 | −6.8698 | 512.0 | 661.0 | −22.54 | 1 | 0.9169 |
830,623 | 213.3171 | 203.1236 | −4.7786 | 993.7 | 1520.0 | −34.63 | 1 | 0.8529 |
830,721 | 136.2846 | 122.0411 | −10.4513 | 765.9 | 896.0 | −14.52 | 2 | 0.9198 |
Mean values | −4.1989 | −0.0758 | 1 | 0.9064 |
Flood events . | Measured total flow (mm) . | Estimated total flow (mm) . | RET (%) . | Measured peak flow (m3/s) . | Estimated peak flow (m3/s) . | REP (%) . | Lag time of peak flow (h) . | Nash coefficient . |
---|---|---|---|---|---|---|---|---|
810,623 | 128.7119 | 131.9217 | 2.4938 | 1,065.3 | 1,130.0 | −5.73 | 1 | 0.9242 |
810,714 | 97.1716 | 88.0852 | −9.3509 | 611.6 | 589.0 | 3.84 | 1 | 0.9759 |
810,810 | 127.2974 | 131.4548 | 3.2658 | 641.8 | 628.0 | 2.20 | 2 | 0.8674 |
810,824 | 82.9152 | 76.6022 | −7.6139 | 534.4 | 509.4 | 4.90 | 0 | 0.8672 |
820,716 | 331.9663 | 321.7267 | −3.0845 | 967.2 | 1,040.0 | −7.00 | 1 | 0.9367 |
820,820 | 171.2696 | 168.8709 | −1.4005 | 602.7 | 572.5 | 5.28 | 0 | 0.8963 |
820,908 | 83.7343 | 77.9819 | −6.8698 | 512.0 | 661.0 | −22.54 | 1 | 0.9169 |
830,623 | 213.3171 | 203.1236 | −4.7786 | 993.7 | 1520.0 | −34.63 | 1 | 0.8529 |
830,721 | 136.2846 | 122.0411 | −10.4513 | 765.9 | 896.0 | −14.52 | 2 | 0.9198 |
Mean values | −4.1989 | −0.0758 | 1 | 0.9064 |
The model verification results for the remaining 21 flood events are detailed in Table 3. The maximum value of the Nash coefficient is 0.9721, the minimum value is 0.7533, and the average value is 0.8999. The consistently high accuracy of the simulation results indicates that the BWU-DCLARK model can be used for flood simulation and prediction.
BUW-DCLARK model verification results of another 21 flood events
Flood events . | Measured total flow (mm) . | Estimated total flow (mm) . | RET (%) . | Measured peak flow (m3/s) . | Estimated peak flow (m3/s) . | REP (%) . | Lag time of peak flow (h) . | NASH coefficient . |
---|---|---|---|---|---|---|---|---|
830,822 | 70.5262 | 56.3863 | −20.0492 | 512.0 | 374.2 | −26.91 | 2 | 0.9243 |
830,906 | 262.3300 | 258.0910 | −1.6159 | 896.7 | 925.0 | 3.16 | 1 | 0.9619 |
830,922 | 57.7514 | 46.6503 | −19.2223 | 407.3 | 382.1 | −6.19 | 2 | 0.8723 |
831,004 | 176.7057 | 174.1784 | −1.4302 | 575.4 | 661.6 | 14.99 | −24 | 0.8934 |
831,017 | 114.6680 | 116.0373 | 1.1941 | 377.9 | 393.2 | 4.05 | −1 | 0.8847 |
840,612 | 134.8214 | 132.4066 | −1.7911 | 632.0 | 701.1 | 10.93 | 0 | 0.7533 |
840,703 | 126.6472 | 116.3503 | −8.1304 | 541.1 | 506.0 | −6.49 | 0 | 0.9665 |
840,723 | 144.3533 | 152.2364 | 5.4610 | 1,060.0 | 1009.2 | −4.79 | 1 | 0.9460 |
840,909 | 73.8589 | 76.9750 | 4.2190 | 355.9 | 317.2 | −10.87 | 0 | 0.9282 |
850,424 | 80.0629 | 78.5504 | −1.8891 | 347.4 | 363.0 | 4.50 | 0 | 0.8876 |
850,603 | 67.2568 | 68.8686 | 2.3965 | 235.0 | 242.9 | 3.34 | −3 | 0.8732 |
850,621 | 100.7181 | 106.6952 | 5.9345 | 475.8 | 497.8 | 4.63 | 0 | 0.9720 |
860,615 | 66.4281 | 68.6293 | 3.3136 | 482.3 | 446.6 | −7.41 | 1 | 0.9500 |
860,714 | 35.0487 | 35.6772 | 1.7931 | 226.9 | 189.1 | −16.63 | 1 | 0.8396 |
860,909 | 181.8900 | 184.1387 | 1.2363 | 844.0 | 791.7 | −6.20 | 1 | 0.9721 |
870,511 | 45.5293 | 46.0235 | 1.0856 | 341.1 | 243.9 | −28.51 | 0 | 0.9142 |
870,622 | 33.8167 | 32.6393 | −3.4816 | 316.0 | 304.7 | −3.58 | 1 | 0.8959 |
870,627 | 65.4857 | 68.0055 | 3.8479 | 366.9 | 293.5 | −20.01 | 1 | 0.9064 |
870,719 | 98.0883 | 93.8050 | −4.3668 | 819.0 | 977.8 | 19.39 | 2 | 0.8568 |
870,821 | 94.0498 | 91.4995 | −2.7117 | 555.8 | 459.4 | −17.34 | 1 | 0.8337 |
870,827 | 90.8083 | 78.9158 | −13.0963 | 671.5 | 425.7 | −36.60 | 3 | 0.8660 |
Mean values | −2.2525 | −0.0603 | −0.5238 | 0.8999 |
Flood events . | Measured total flow (mm) . | Estimated total flow (mm) . | RET (%) . | Measured peak flow (m3/s) . | Estimated peak flow (m3/s) . | REP (%) . | Lag time of peak flow (h) . | NASH coefficient . |
---|---|---|---|---|---|---|---|---|
830,822 | 70.5262 | 56.3863 | −20.0492 | 512.0 | 374.2 | −26.91 | 2 | 0.9243 |
830,906 | 262.3300 | 258.0910 | −1.6159 | 896.7 | 925.0 | 3.16 | 1 | 0.9619 |
830,922 | 57.7514 | 46.6503 | −19.2223 | 407.3 | 382.1 | −6.19 | 2 | 0.8723 |
831,004 | 176.7057 | 174.1784 | −1.4302 | 575.4 | 661.6 | 14.99 | −24 | 0.8934 |
831,017 | 114.6680 | 116.0373 | 1.1941 | 377.9 | 393.2 | 4.05 | −1 | 0.8847 |
840,612 | 134.8214 | 132.4066 | −1.7911 | 632.0 | 701.1 | 10.93 | 0 | 0.7533 |
840,703 | 126.6472 | 116.3503 | −8.1304 | 541.1 | 506.0 | −6.49 | 0 | 0.9665 |
840,723 | 144.3533 | 152.2364 | 5.4610 | 1,060.0 | 1009.2 | −4.79 | 1 | 0.9460 |
840,909 | 73.8589 | 76.9750 | 4.2190 | 355.9 | 317.2 | −10.87 | 0 | 0.9282 |
850,424 | 80.0629 | 78.5504 | −1.8891 | 347.4 | 363.0 | 4.50 | 0 | 0.8876 |
850,603 | 67.2568 | 68.8686 | 2.3965 | 235.0 | 242.9 | 3.34 | −3 | 0.8732 |
850,621 | 100.7181 | 106.6952 | 5.9345 | 475.8 | 497.8 | 4.63 | 0 | 0.9720 |
860,615 | 66.4281 | 68.6293 | 3.3136 | 482.3 | 446.6 | −7.41 | 1 | 0.9500 |
860,714 | 35.0487 | 35.6772 | 1.7931 | 226.9 | 189.1 | −16.63 | 1 | 0.8396 |
860,909 | 181.8900 | 184.1387 | 1.2363 | 844.0 | 791.7 | −6.20 | 1 | 0.9721 |
870,511 | 45.5293 | 46.0235 | 1.0856 | 341.1 | 243.9 | −28.51 | 0 | 0.9142 |
870,622 | 33.8167 | 32.6393 | −3.4816 | 316.0 | 304.7 | −3.58 | 1 | 0.8959 |
870,627 | 65.4857 | 68.0055 | 3.8479 | 366.9 | 293.5 | −20.01 | 1 | 0.9064 |
870,719 | 98.0883 | 93.8050 | −4.3668 | 819.0 | 977.8 | 19.39 | 2 | 0.8568 |
870,821 | 94.0498 | 91.4995 | −2.7117 | 555.8 | 459.4 | −17.34 | 1 | 0.8337 |
870,827 | 90.8083 | 78.9158 | −13.0963 | 671.5 | 425.7 | −36.60 | 3 | 0.8660 |
Mean values | −2.2525 | −0.0603 | −0.5238 | 0.8999 |
Moreover, the BWU-DCLARK model facilitates the calculation not only of the flow process at the outlet but also at any specific node. For example, for the No. 31 river chain node (junction of the 31st and 32nd river chains), it is only necessary to recalculate the channel path and length of the 1st to 31st BWUs, and the confluence process of the river chain junction can be calculated by the BWU-DCLARK model.
CONCLUSIONS
In this study, the BWUs of the basin are delineated based on DEM data. These BWUs are simplified into V-type for inner chains and Horseshoe-type for outer chains. Utilizing the concept of isochrones, time–area curves for Horseshoe-type and V-type BWUs are derived. After the pure rainfall reaches the corresponding river chain through slope confluence in each BWU of a basin, it reaches the basin outlet in the form of river confluence via the unique path of each BWU. The river confluence path for each BWU is analyzed based on the location of the BWU, and a lagged linear reservoir confluence module is constructed for each BWU considering path length and average slope. Our main findings are as follows.
The time–area curves for Horseshoe-type and V-type BWUs which are categorized based on their shape characteristics are derived on basis of physical mechanism of slope convergence.
This yields the distributed Clark confluence model based on the BWUs (BWU-DCLARK), which includes five parameters which reflect the confluence velocities of the slope, river, and single linear reservoir.
The Yanduhe Basin of the Yangtze River, with a total area of 601 km2, serves as a case study. Utilizing DEM data, the basin is divided into 30 Horseshoe-type and 29 V-type BWUs, with the average area of the BWUs being 10.1864 km2. The BWU-DCLARK model for the Yanduhe Basin is obtained, and the simulation results show that the high simulation accuracy of the model meets prediction requirements.
Simultaneously, considering the slope confluence time and river confluence time of the BWUs in the Yanduhe Basin, although each BWU is relatively small (only approximately 10 km2), the regulatory and storage effects on runoff of each BWU cannot be ignored.
The BWU-DCLARK model for basin confluence not only facilitates the computation of discharge at the basin outlet but also allows for the forecasting of discharge at any river chain node which will be of great significance for hydrological forecasting in un-gauged basins, while the simulation accuracy has not been validated as the limitation of hydrological data in Yanduhe basin.
Overall, this study proposed a discharge calculation based on the mechanism of net rainfall convergence for improving the process-based hydrological model. While this study is limited to the hydrological data to determine the model parameters, and due to data limitations, there is a lack of discharge forecasting accuracy in un-gauged basins. Future research will be carried out in more basins which have more observed hydrological data to test the effectiveness of the model.
ACKNOWLEDGEMENTS
This research is supported by the ‘Fundamental Research Funds for the Central Universities’ at South-Central Minzu University (Grant No. CZQ18022) and the Innovation and Entrepreneurship Training Program funded by South-Central Minzu University (Grant No. 20230100017). The authors would like to thank all the reviewers who participated in the review, as well as MJEditor (www.mjeditor.com) for providing English editing services during the preparation of this manuscript.
DATA AVAILABILITY STATEMENT
All relevant data are available from https://www.gscloud.cn/#page1/4.
CONFLICT OF INTEREST
The authors declare there is no conflict.