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.

  • 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.

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.

The data utilized in this study are from the Yanduhe Basin, situated in the upper reaches of the Yangtze River, China (Figure 1). This basin is humid and occupies a drainage area of 601 km2. The basin experiences a temperate climate with an average annual precipitation of 1,337 mm and an approximate runoff coefficient of 0.81. Forests and farmlands account for 70 and 10% of the basin area, respectively.
Figure 1

River system of the Yanduhe Basin.

Figure 1

River system of the Yanduhe Basin.

Close modal

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.

Clark model

In 1945, Clark introduced a unit hydrograph based on an isochronous time–area curve connected to a simple linear reservoir to simulate watershed confluence (Figure 2). Here, denotes the confluence time on the slope, and K represents the storage parameter of the linear reservoir. A notable limitation of the time–area curve based on isochrones is the assumption that the velocity is uniformly distributed on the slope. This shortcoming can be remedied by connecting it with a linear reservoir.
Figure 2

Clark model.

BWU-DCLARK model

In this study, the D8 algorithm is utilized to extract BWUs and underlying surface features of the watershed based on DEM data. Both slope confluence and river confluence processes within each BWU of the basin are investigated. These confluence processes for all BWUs are linearly superimposed to obtain the BWU-DCLARK model for the entire basin. The mathematical expression for the model is presented in Equation (1):
(1)
where j is the serial number of the BWUs in the basin, n is the total number of BWUs, Aj is the area of each BWU, A is the total basin area, represents the time–area curve of each BWU, corresponds to a lagged linear reservoir, and is the lag time for each linear reservoir.
The specific steps are as follows:
  • 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.

Figure 3

Lagged Clark model.

Figure 3

Lagged Clark model.

Close modal

BWU of a basin

A basin comprises a river system and slopes. Based on the inherent composition characteristics of river systems and slopes in natural basins, the basin can be subdivided into multiple BWUs. Based on the DEM of the Yanduhe Basin, Figure 4 illustrates the river system, river chains, and BWUs that constitute the river chains in the basin. As depicted in Figure 4, there are two types of BWUs. The first is an inner chain type of BWU with a central channel and slopes on both sides. Net rainfall on the slope directly flows into the inner chain channel (e.g., No. 5 and No. 7 in Figure 4). The second type is the outer chain channel, with slopes on both sides and the outer end. Net rainfall from the slopes flows into the outer chain channel (e.g., No. 1 and No. 2 in Figure 4).
Figure 4

River system and BWUs of the Yanduhe Basin.

Figure 4

River system and BWUs of the Yanduhe Basin.

Close modal
Based on the shape characteristics of the BWUs, those with inner chains are categorized as V-type, and those with outer chains are categorized as Horseshoe-type, as illustrated in Figure 5. Here, BC represents the river chain of each BWU.
Figure 5

Simplified shape of BWUs: (a) V-type BWU and (b) Horseshoe-type BWU.

Figure 5

Simplified shape of BWUs: (a) V-type BWU and (b) Horseshoe-type BWU.

Close modal

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

For V-type BWUs, we assume that each V-type BWU consists of an inner chain and two identical rectangles. The unit hydrograph of confluence can be obtained from Equation (2):
(2)
where t is the moment; and are the areas of the two rectangular slopes, , is the total area of the V-type BWU; is the relative area of the river chain reached at time t; and and are the relative areas of the two rectangular slopes reaching the river chain at time t.
Assuming that the slope velocity is stable as V, for a rectangular slope there is:
(3)
where V is the stable slope velocity of the BWU, l is the length of the river chain, and L is the slope confluence length.
Consequently, the convergence time on slopes can be calculated using Equation (4):
(4)
From Equations (3) and (4), there is:
(5)
Thus, the slope instantaneous unit hydrograph is given as follows:
(6)
The unit hydrograph of V-type BWU is further obtained as follows:
(7)
where Δt indicates the calculation period.
  • (2)

    Time–area curve for Horseshoe-type BWUs

For Horseshoe-type BWUs, we assume that the rectangular slope on both sides of the outer chain BC in Figure 5(b) is equal to the area of the sector slope (i.e., ), and that the slope and confluence length are equal. The time–area curve of the Horseshoe-type BWU can be obtained from Equation (8).
(8)
where , and are the areas of the rectangular and sector slopes of the Horseshoe-type BWU, respectively, , ; ; and , respectively, represent the relative area of the rectangular and sector slopes reaching the river chain at time t.
Assuming that the slope velocity is also stable and equal, for the sector slope, we have:
(9)
where is the angle of the sector slope, and the other parameters are the same as those described above.
Similarly, if , we get:
(10)
By combining Eqs (5) and (10), the time–area curve of the Horseshoe-type BWU is as follows:
(11)
The unit hydrograph of the Horseshoe-type BWU is further obtained as follows:
(12)

Lagging single linear reservoir model

The single linear reservoir model is the Nash instantaneous unit hydrograph model of a single reservoir (Nash 1957), as shown in Equation (13):
(13)
Since the river confluence path varies for each watershed, the lag time is introduced according to the length of the river path, and the single linear reservoir model is obtained using Equation (14):
(14)
where K is storage time of the reservoir. is determined by the confluence time of the river:
(15)
where is the length of the river confluence path of the BWU, and is the average confluence velocity of the river.

BWU and river confluence velocity

The average slope confluence velocity and the average river confluence velocity are calculated according to Equations (16) and (17), respectively, proposed by Sircar in 1991 with reference to the Manning formula:
(16)
where a and b are slope confluence velocity parameters, and i is the average slope of the BWU.
(17)
where and are river confluence velocity parameters, and is the average slope of the river confluence path.

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.

If the area of each BWU is and the chain length is l, then the confluence length of the V-type BWU can be calculated using Equation (18).
(18)
The confluence length of the Horseshoe-type BWU can be obtained using Equation (19).
(19)

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

The following metrics were used to compare the observed flood processes and those estimated using the BWU-DCLARK model: NASH coefficient (CN) (Equation (20)), relative error of peak flow (REP) (Equation (21)), and relative error of total flow (RET) (Equation (22)).
(20)
(21)
(22)
where is the observed flow at time i, is the estimated flow at time i, is the average observed flow , N refers to the number of observations, is the observed peak flow , and is the estimated peak flow .
Utilizing the Yanduhe DEM, the basin is partitioned into 59 BWUs, comprising 30 Horseshoe-type BWUs and 29 V-type BWUs (Figure 4,). The total area of the basin is 601 km2, with an average BWU area of 10.1864 km2. Topographic and geomorphological data are presented in Table 1. The generalized water system diagram of the basin is presented in Figure 6 based on Figure 4, and the river confluence paths of each BWU are obtained according to both Figures 4 and 6, as shown in Table 1.
Table 1

Geomorphic features and confluence paths of each BWU

BWU no.Chain length (km)Area of each BWU (km2)Slope of each BWUSlope of each chainRiver confluence pathRiver 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)
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 10.1536 
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 10.2098 
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 10.1942 
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 10.1963 
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 10.1550 
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 10.2474 
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 10.1945 
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 10.2143 
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 10.2673 
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 10.2684 
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 10.2561 
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 10.3052 
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 10.3111 
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 10.4278 
15 1.451 1.520 0.374 0.137844 15-17-21-23-25-29-31-45-47-49-59 31.6430 0.7188 10.3975 
16 4.194 13.203 0.389 0.190746 16-17-21-23-25-29-31-45-47-49-59 34.3860 0.7223 10.4355 
17 7.587 25.899 0.726 0.064983 17-21-23-25-29-31-45-47-49-59 30.1920 0.7803 10.3866 
18 3.175 11.639 0.440 0.052908 18-20-21-23-25-29-31-45-47-49-59 27.9320 0.7334 10.3192 
19 2.638 7.242 0.658 0.040568 19-20-21-23-25-29-31-45-47-49-59 27.3950 0.7708 10.3095 
20 2.152 4.660 0.412 0.046463 20-21-23-25-29-31-45-47-49-59 24.7570 0.7274 10.3714 
21 6.256 19.904 1.038 0.143852 21-23-25-29-31-45-47-49-59 22.6050 0.8156 10.4377 
22 3.157 11.145 0.870 0.221764 22-23-25-29-31-45-47-49-59 19.5060 0.7979 10.5028 
23 2.453 4.038 0.322 0.0001 23-25-29-31-45-47-49-59 16.3490 0.7056 10.4235 
24 4.270 16.802 0.564 0.140508 24-25-29-31-45-47-49-59 18.1660 0.7562 10.5534 
25 0.616 8.526 0.666 0.162345 25-29-31-45-47-49-59 13.8960 0.7720 10.5580 
26 1.335 8.307 0.748 0.149848 26-28-29-31-45-47-49-59 19.1390 0.7831 10.5171 
27 2.310 4.507 0.670 0.129888 27-28-29-31-45-47-49-59 20.1140 0.7725 10.4989 
28 4.524 24.470 0.527 0.11206 28-29-31-45-47-49-59 17.8040 0.7499 10.5069 
29 1.736 2.456 0.594 0.11518 29-31-45-47-49-59 13.2800 0.7611 10.5386 
30 3.850 9.422 0.567 0.077913 30-31-45-47-49-59 15.3940 0.7567 10.4937 
31 4.106 6.381 0.307 0.413994 31-45-47-49-59 11.5440 0.7014 10.5773 
32 1.467 1.567 0.398 0.068149 32-34-36-40-42-44–45-47-49-59 22.6450 0.7243 10.1983 
33 3.436 4.452 0.649 0.029105 33-34-36-40-42-44–45-47-49-59 24.6140 0.7695 10.1600 
34 3.704 6.174 0.565 0.135005 34-36-40-42-44–45-47-49-59 21.1780 0.7564 10.2336 
35 2.753 6.241 0.642 0.0 29105 35-36-40-42-44–45-47-49-59 20.2270 0.7685 10.0838 
36 4.094 5.783 0.459 0.091117 36-40-42-44–45-47-49-59 17.4740 0.7372 10.1984 
37 3.075 9.422 0.690 0.065044 37-39-40-42-44–45-47-49-59 18.4100 0.7754 10.0522 
38 5.576 19.361 0.452 0.093619 38-39-40-42-44–45-47-49-59 20.9110 0.7358 10.0867 
39 1.955 3.841 0.555 0.0 650439 39-40-42-44–45-47-49-59 15.3350 0.7547 10.0827 
40 2.809 14.259 0.459 0.071211 40-42-44–45-47-49-59 13.3800 0.7372 10.2128 
41 1.046 9.919 0.561 0.191255 41-42-44–45-47-49-59 11.6170 0.7557 10.3648 
42 0.705 7.727 0.344 0.283585 42-44–45-47-49-59 10.5710 0.7114 10.2622 
43 1.984 2.949 0.326 0.050412 43-44–45-47-49-59 11.8500 0.7066 9.8339 
44 2.428 5.416 0.573 0.038303 44–45-47-49-59 9.8660 0.7577 9.8847 
45 1.892 2.848 0.571 0.0504116 45-47-49-59 7.4380 0.7574 9.9925 
46 0.956 6.659 0.478 0.296006 46-47-49-59 6.5020 0.7409 10.6282 
47 1.845 10.028 0.568 0.216852 47-49-59 5.5460 0.7569 10.2722 
48 0.891 7.023 0.528 0.12122 48-49-59 4.5920 0.7501 9.9384 
49 2.687 2.858 0.863 0.111667 49-59 3.7010 0.7971 9.6290 
50 4.105 16.331 0.501 0.146169 50-52-54-56-58-59 27.1730 0.7452 10.1744 
51 1.094 0.893 0.609 0.091411 51-52-54-56-58-59 24.1620 0.7635 10.0804 
52 2.613 13.795 0.765 0.11482 52-54-56-58-59 23.0680 0.7853 10.0775 
53 2.456 6.069 0.609 0.0914114 53-54-56-58-59 22.9110 0.7635 9.7956 
54 6.214 7.305 0.626 0.305743 54-56-58-59 20.4550 0.7661 10.0076 
55 1.395 11.473 0.428 0.071671 55-56-58-59 15.6360 0.7309 8.9117 
56 10.718 42.814 0.767 0.027991 56-58-59 14.2410 0.7856 8.1131 
57 3.643 5.669 0.794 0.0716708 57-58-59 7.1660 0.7889 5.2435 
58 2.509 8.060 0.530 0.0279913 58-59 3.5230 0.7504 5.2435 
59 1.014 1.151 1.079 0.04448 59 1.0140 0.8195 5.2435 
BWU no.Chain length (km)Area of each BWU (km2)Slope of each BWUSlope of each chainRiver confluence pathRiver 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)
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 10.1536 
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 10.2098 
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 10.1942 
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 10.1963 
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 10.1550 
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 10.2474 
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 10.1945 
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 10.2143 
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 10.2673 
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 10.2684 
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 10.2561 
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 10.3052 
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 10.3111 
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 10.4278 
15 1.451 1.520 0.374 0.137844 15-17-21-23-25-29-31-45-47-49-59 31.6430 0.7188 10.3975 
16 4.194 13.203 0.389 0.190746 16-17-21-23-25-29-31-45-47-49-59 34.3860 0.7223 10.4355 
17 7.587 25.899 0.726 0.064983 17-21-23-25-29-31-45-47-49-59 30.1920 0.7803 10.3866 
18 3.175 11.639 0.440 0.052908 18-20-21-23-25-29-31-45-47-49-59 27.9320 0.7334 10.3192 
19 2.638 7.242 0.658 0.040568 19-20-21-23-25-29-31-45-47-49-59 27.3950 0.7708 10.3095 
20 2.152 4.660 0.412 0.046463 20-21-23-25-29-31-45-47-49-59 24.7570 0.7274 10.3714 
21 6.256 19.904 1.038 0.143852 21-23-25-29-31-45-47-49-59 22.6050 0.8156 10.4377 
22 3.157 11.145 0.870 0.221764 22-23-25-29-31-45-47-49-59 19.5060 0.7979 10.5028 
23 2.453 4.038 0.322 0.0001 23-25-29-31-45-47-49-59 16.3490 0.7056 10.4235 
24 4.270 16.802 0.564 0.140508 24-25-29-31-45-47-49-59 18.1660 0.7562 10.5534 
25 0.616 8.526 0.666 0.162345 25-29-31-45-47-49-59 13.8960 0.7720 10.5580 
26 1.335 8.307 0.748 0.149848 26-28-29-31-45-47-49-59 19.1390 0.7831 10.5171 
27 2.310 4.507 0.670 0.129888 27-28-29-31-45-47-49-59 20.1140 0.7725 10.4989 
28 4.524 24.470 0.527 0.11206 28-29-31-45-47-49-59 17.8040 0.7499 10.5069 
29 1.736 2.456 0.594 0.11518 29-31-45-47-49-59 13.2800 0.7611 10.5386 
30 3.850 9.422 0.567 0.077913 30-31-45-47-49-59 15.3940 0.7567 10.4937 
31 4.106 6.381 0.307 0.413994 31-45-47-49-59 11.5440 0.7014 10.5773 
32 1.467 1.567 0.398 0.068149 32-34-36-40-42-44–45-47-49-59 22.6450 0.7243 10.1983 
33 3.436 4.452 0.649 0.029105 33-34-36-40-42-44–45-47-49-59 24.6140 0.7695 10.1600 
34 3.704 6.174 0.565 0.135005 34-36-40-42-44–45-47-49-59 21.1780 0.7564 10.2336 
35 2.753 6.241 0.642 0.0 29105 35-36-40-42-44–45-47-49-59 20.2270 0.7685 10.0838 
36 4.094 5.783 0.459 0.091117 36-40-42-44–45-47-49-59 17.4740 0.7372 10.1984 
37 3.075 9.422 0.690 0.065044 37-39-40-42-44–45-47-49-59 18.4100 0.7754 10.0522 
38 5.576 19.361 0.452 0.093619 38-39-40-42-44–45-47-49-59 20.9110 0.7358 10.0867 
39 1.955 3.841 0.555 0.0 650439 39-40-42-44–45-47-49-59 15.3350 0.7547 10.0827 
40 2.809 14.259 0.459 0.071211 40-42-44–45-47-49-59 13.3800 0.7372 10.2128 
41 1.046 9.919 0.561 0.191255 41-42-44–45-47-49-59 11.6170 0.7557 10.3648 
42 0.705 7.727 0.344 0.283585 42-44–45-47-49-59 10.5710 0.7114 10.2622 
43 1.984 2.949 0.326 0.050412 43-44–45-47-49-59 11.8500 0.7066 9.8339 
44 2.428 5.416 0.573 0.038303 44–45-47-49-59 9.8660 0.7577 9.8847 
45 1.892 2.848 0.571 0.0504116 45-47-49-59 7.4380 0.7574 9.9925 
46 0.956 6.659 0.478 0.296006 46-47-49-59 6.5020 0.7409 10.6282 
47 1.845 10.028 0.568 0.216852 47-49-59 5.5460 0.7569 10.2722 
48 0.891 7.023 0.528 0.12122 48-49-59 4.5920 0.7501 9.9384 
49 2.687 2.858 0.863 0.111667 49-59 3.7010 0.7971 9.6290 
50 4.105 16.331 0.501 0.146169 50-52-54-56-58-59 27.1730 0.7452 10.1744 
51 1.094 0.893 0.609 0.091411 51-52-54-56-58-59 24.1620 0.7635 10.0804 
52 2.613 13.795 0.765 0.11482 52-54-56-58-59 23.0680 0.7853 10.0775 
53 2.456 6.069 0.609 0.0914114 53-54-56-58-59 22.9110 0.7635 9.7956 
54 6.214 7.305 0.626 0.305743 54-56-58-59 20.4550 0.7661 10.0076 
55 1.395 11.473 0.428 0.071671 55-56-58-59 15.6360 0.7309 8.9117 
56 10.718 42.814 0.767 0.027991 56-58-59 14.2410 0.7856 8.1131 
57 3.643 5.669 0.794 0.0716708 57-58-59 7.1660 0.7889 5.2435 
58 2.509 8.060 0.530 0.0279913 58-59 3.5230 0.7504 5.2435 
59 1.014 1.151 1.079 0.04448 59 1.0140 0.8195 5.2435 
Figure 6

Generalized river system and BWU of the Yanduhe Basin.

Figure 6

Generalized river system and BWU of the Yanduhe Basin.

Close modal

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.

Table 2

BUW-DCLARK model calibration results of nine flood events

Flood eventsMeasured 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 0.9242 
810,714 97.1716 88.0852 −9.3509 611.6 589.0 3.84 0.9759 
810,810 127.2974 131.4548 3.2658 641.8 628.0 2.20 0.8674 
810,824 82.9152 76.6022 −7.6139 534.4 509.4 4.90 0.8672 
820,716 331.9663 321.7267 −3.0845 967.2 1,040.0 −7.00 0.9367 
820,820 171.2696 168.8709 −1.4005 602.7 572.5 5.28 0.8963 
820,908 83.7343 77.9819 −6.8698 512.0 661.0 −22.54 0.9169 
830,623 213.3171 203.1236 −4.7786 993.7 1520.0 −34.63 0.8529 
830,721 136.2846 122.0411 −10.4513 765.9 896.0 −14.52 0.9198 
Mean values   −4.1989   −0.0758 0.9064 
Flood eventsMeasured 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 0.9242 
810,714 97.1716 88.0852 −9.3509 611.6 589.0 3.84 0.9759 
810,810 127.2974 131.4548 3.2658 641.8 628.0 2.20 0.8674 
810,824 82.9152 76.6022 −7.6139 534.4 509.4 4.90 0.8672 
820,716 331.9663 321.7267 −3.0845 967.2 1,040.0 −7.00 0.9367 
820,820 171.2696 168.8709 −1.4005 602.7 572.5 5.28 0.8963 
820,908 83.7343 77.9819 −6.8698 512.0 661.0 −22.54 0.9169 
830,623 213.3171 203.1236 −4.7786 993.7 1520.0 −34.63 0.8529 
830,721 136.2846 122.0411 −10.4513 765.9 896.0 −14.52 0.9198 
Mean values   −4.1989   −0.0758 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.

Table 3

BUW-DCLARK model verification results of another 21 flood events

Flood eventsMeasured 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 0.9243 
830,906 262.3300 258.0910 −1.6159 896.7 925.0 3.16 0.9619 
830,922 57.7514 46.6503 −19.2223 407.3 382.1 −6.19 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 0.7533 
840,703 126.6472 116.3503 −8.1304 541.1 506.0 −6.49 0.9665 
840,723 144.3533 152.2364 5.4610 1,060.0 1009.2 −4.79 0.9460 
840,909 73.8589 76.9750 4.2190 355.9 317.2 −10.87 0.9282 
850,424 80.0629 78.5504 −1.8891 347.4 363.0 4.50 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 0.9720 
860,615 66.4281 68.6293 3.3136 482.3 446.6 −7.41 0.9500 
860,714 35.0487 35.6772 1.7931 226.9 189.1 −16.63 0.8396 
860,909 181.8900 184.1387 1.2363 844.0 791.7 −6.20 0.9721 
870,511 45.5293 46.0235 1.0856 341.1 243.9 −28.51 0.9142 
870,622 33.8167 32.6393 −3.4816 316.0 304.7 −3.58 0.8959 
870,627 65.4857 68.0055 3.8479 366.9 293.5 −20.01 0.9064 
870,719 98.0883 93.8050 −4.3668 819.0 977.8 19.39 0.8568 
870,821 94.0498 91.4995 −2.7117 555.8 459.4 −17.34 0.8337 
870,827 90.8083 78.9158 −13.0963 671.5 425.7 −36.60 0.8660 
Mean values   −2.2525   −0.0603 −0.5238 0.8999 
Flood eventsMeasured 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 0.9243 
830,906 262.3300 258.0910 −1.6159 896.7 925.0 3.16 0.9619 
830,922 57.7514 46.6503 −19.2223 407.3 382.1 −6.19 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 0.7533 
840,703 126.6472 116.3503 −8.1304 541.1 506.0 −6.49 0.9665 
840,723 144.3533 152.2364 5.4610 1,060.0 1009.2 −4.79 0.9460 
840,909 73.8589 76.9750 4.2190 355.9 317.2 −10.87 0.9282 
850,424 80.0629 78.5504 −1.8891 347.4 363.0 4.50 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 0.9720 
860,615 66.4281 68.6293 3.3136 482.3 446.6 −7.41 0.9500 
860,714 35.0487 35.6772 1.7931 226.9 189.1 −16.63 0.8396 
860,909 181.8900 184.1387 1.2363 844.0 791.7 −6.20 0.9721 
870,511 45.5293 46.0235 1.0856 341.1 243.9 −28.51 0.9142 
870,622 33.8167 32.6393 −3.4816 316.0 304.7 −3.58 0.8959 
870,627 65.4857 68.0055 3.8479 366.9 293.5 −20.01 0.9064 
870,719 98.0883 93.8050 −4.3668 819.0 977.8 19.39 0.8568 
870,821 94.0498 91.4995 −2.7117 555.8 459.4 −17.34 0.8337 
870,827 90.8083 78.9158 −13.0963 671.5 425.7 −36.60 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.

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.

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.

All relevant data are available from https://www.gscloud.cn/#page1/4.

The authors declare there is no conflict.

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