## Abstract

Among the most widespread structures for successfully retaining water and checking erosion on the semi-arid portions of China's Loess Plateau, check dams retain silt at slower than projected rates, leading to flood control issues. Meanwhile, the shortage and the uneven distribution of time and space of water resources in semi-arid areas can easily cause droughts and floods, which seriously restricted the rapid development of the socioeconomic. However, some of the high-quality rain and flood resources accumulated in the check dams can be used to alleviate part of the water resources crisis instead of causing flood. With the goal of holistically maximizing a projected check dam array's water resource, ecosystem and socioeconomic benefits, a Check Dam Benefit Maximization Model (CDBMM) was first developed. The CDBMM was first applied to the Si Jiagou Basin, and the model showed the total costs represent 7.07% of the total and rather significant benefits. Water resources benefits accounting for 45.40% of the total benefits, indicating that the water resources benefits were substantial and should be considered as the main influencing factors in the basin's ecosystem-friendly design and construction. Use of the CDBMM in watershed planning will allow a more efficient use of water and soil resources and greatly alleviate water crises in the semi-arid area. It can further provide a reference for both check dam system planning and the system benefits analysis.

## HIGHLIGHTS

A Check Dam Benefit Maximization Model (CDBMM) is proposed.

CDBMM maximize the overall benefits of the check dam system and efficiently use water resources.

For the first time, CDBMM considers water resource benefits of the check dam system in the check dam construction.

Provide new ways for the sustainable development of semi-arid areas and the construction of water-saving and ecological check dam systems.

## INTRODUCTION

Located in the upper and middle reaches of China's Yellow River and stretching over parts of seven provinces, the Loess Plateau (0.64 × 10^{6} km^{2}) is the most seriously eroded region in China, having suffered from severe water and soil loss for many years. Over 60% of the land is subject to soil and water losses (Xu *et al.* 2004). Mean erosion rates are 5–10 Gg km^{−2} yr^{−1} and can even approach 20–30 Gg km^{−2} yr^{−1} in some regions. The Loess Plateau's soils have been termed the ‘most highly erodible on earth,’ a characteristic which limits local socioeconomic development and increases downstream flood risks. The main reasons are the region's unique pedology (i.e., the silty sandy loam forming most of the loess is highly erodible), landforms (i.e., high density of gullies that promote soil erosion), and dry climate (i.e., rainfall, 150–800 mm yr^{−1}; evaporation, 1,400–2,000 mm yr^{−1}) (Xu *et al.* 2004; Wang *et al.* 2011; Jin *et al.* 2012; Zhao *et al.* 2017). The control of erosion and the delivery of suspended materials to stream channels have been undertaken through a combination of engineering measures (e.g., check dam, sediment trapping dam, overflow dam, pond and structural style of dike) and planting measures (e.g., shifts in land use, planting methods, terracing). Among the most effective of engineering measures, check dam plays a key role in erosion controlling and resource utilization (Zhao *et al.* 2017). Over the last 50 years, over 110,000 check dams have been constructed to preserve the region's fragile ecosystems. These store 21 × 10^{9}m^{3} of sediments and have formed 3,340 km^{2} of dam farmlands (Laflen *et al.* 2000; China C. M. o. W. R. o. P. R. 2003; Xu *et al.* 2004; Wang *et al.* 2011; Jin *et al.* 2012). Check dams have been shown to be an effective engineering structure to control surface runoff (Xu *et al.* 2013; Shi *et al.* 2015; Tang *et al.* 2019) and soil erosion (Boix-Fayos *et al.* 2008; Ran *et al.* 2008; Shi & Wang 2015), and are deemed one of the most effective structures for rapidly reducing the volume of coarse sediment entering the Yellow River (Ran *et al.* 2008). Accordingly, in 2000, the region's check dam project was listed as one of three key engineering measures by the Chinese Ministry of Water Resources (Li *et al.* 2016).

The construction of check dams has proved effective in improving individuals' living conditions and productivity, increasing farmers' incomes, promoting regional economic development, and bringing significant social, economic and ecological benefits. In recent years, check dam construction has progressed from simply blocking silt and forming land to comprehensive and efficient utilization of river basin dam systems, and from single dam control to comprehensive dam array control (Xin Shuzhi 1982; Cui Qvpeng 2002; Yingying 2010). Some basins have pre-existing dam systems, which have played a role in flood prevention, silt blocking and increasing production (Cao *et al.* 2007). Current studies on check dams have achieved remarkable results in terms of silt deposition (Chen *et al.* 2010; Tang *et al.* 2018; Pal & Galelli 2019), farmland silting (Zhao *et al.* 2009; Wei *et al.* 2016; Xu *et al.* 2018) and the effectiveness of soil and water conservation efforts (Norman & Niraula 2016; Galicia *et al.* 2019). Besides having been widely implemented on China's Loess Plateau, check dam construction is also common abroad: e.g., Australia, Spain, Italy and Iran (Pal *et al.* 2018; Pal & Galelli 2019). Meanwhile, studies on check dam planning and structural design optimization have also progressed (Campisano *et al.* 2014; Akita *et al.* 2017; Cucchiaro *et al.* 2019), and the real-life case studies of optimization techniques on hydrologic prediction have developed rapidly (Wu & Chau 2013; Taormina & Chau 2015; Alizadeh *et al.* 2017; Chen & Chau 2019; Zavala *et al.* 2019; Feng *et al.* 2020; Kargar *et al.* 2020; Shamshirband *et al.* 2020). However, studies to promote multiple check dam arrays in a basin as a coordinated effort to maximize check dam benefits are few if any.

The Loess Plateau in Gansu, China, is a typical semi-arid region. While the construction of check dams have achieved results, severe problems still exist: (i) as few funds are allocated to construction, no flood control facilities are installed or at best simple water release pipes which fail to meet flood control standards, (ii) unified planning and design prior to check dam construction is lacking, (iii) the soil and water conservation effects of the basin's check dam array have not been established and (iv) under the premise of ensuring safety, the full check dam array's potential for efficient use of water and soil resources has not been achieved (Gao *et al*. 2016b, 2017, 2019a, 2019b).

On the Gansu Loess Plateau, water resources are very scarce and some areas are further plagued by high water mineralization, making water difficult to drink and use for irrigation. The main targets of check dam construction in traditional semi-arid basins were to block eroded matter from clogging water courses and to build farmland. However, due to the significant achievements in soil and water erosion control in the last 50 years (Hui *et al.* 2010; Gao *et al.* 2016a), the vegetation coverage rate has increased, and the amount of erosion has been greatly reduced, leading many check dam reservoirs to not fill beyond some minor silting in. Instead, a large amount of their storage capacity remains, and a considerable amount of water resources have been collected, which potentially pose a threat for downstream basins. However, the safe operation of the check dams on the Loess Plateau, along with an efficient use of the stored water resources and their attendant benefits, can play a crucial role in alleviating the water resource crisis in the semi-arid areas of Gansu province. If the benefits of existing water resources are taken into consideration in the subsequent construction or reconstruction of check dams in these semi-arid areas, the array of check dams in the basin can form an organic system that maximizes water resources benefits, and offers new tools to solve water resources shortage issues and improve the water resource use efficiency within the basin.

Most check dam models are based on the benefits of soil and water conservation, production and flood control (Parimalarenganayaki & Elango 2014; Norman & Niraula 2016; Pal *et al.* 2018; Galicia *et al.* 2019). Only a few of them consider the establishment of check dam systems in the entire basin, and few models consider the water resource benefits of the check dam system which is essential for semi-arid water-scarce areas (Xu *et al.* 2004; Cao *et al.* 2007; Pal *et al.* 2018; Pal & Galelli 2019).

For the first time, we propose the CDBMM (the Check Dam Benefit Maximization Model) to maximize the overall benefits of the check dam system and efficiently use water resources in the basin. Based on the presence of two pre-existing check dams and considerable water resources stored in them, the Si Jiagou Basin, situated in the semi-arid area of the Loess Plateau, was selected as a case study in building a model for maximizing the overall benefits of the basin's dam system array (Figure 1). Addressing safety concerns, the model took full account of the benefits of water resources, maximized the efficient use of water and soil resources of the check dam system, and effectively solved the problem of water resources shortages in the basin. The CDBMM promoting the formation of check dam systems in the basin, maximizing the overall ecological, water resources and socioeconomic benefits, will further promote the level of water and soil conservation engineering facilities in the semi-arid area of the Loess Plateau, and greatly improve the water resources and soil and water conservation efficiency of check dam systems in the basin.

## STUDY AREA

### Study area overview

The CDBMM, described below, was first applied to the Si Jiagou Basin, a 1,349 ha area located on the upper reaches of Jinhe River in the northeast corner of Xifeng District (35°43′N, 107°38′E) in the eastern portion of Gansu Province (Figure 1). Xifeng District is subject to a semi-arid continental climate, with the dual climate characteristics of monsoon and the Loess Plateau. The annual mean temperature is 9.6 °C, with a range of −14.5 to 34.3 °C. The maximum depth of frozen soil is 0.8 m. Rainfall occurs predominantly in the summer and autumn, with the heaviest rains occurring in July and August. In contrast, spring and winter are windy and dry, and typical of arid and dry regions. Therefore, surface water resources in the district are scarce, the natural conditions are harsh, and agricultural production is difficult (Yearbook E. B. o. G. D. 2018). Cultivated soils are mainly composed of dark loessial soils and are slightly alkaline, fertile, loose and showing strong vertical penetration. The main crops are wheat and corn, and apple cultivation is at its optimal latitude. The selection of the Si Jiagou Basin as a research area is representative and typicality because of the following reasons: (1) The area of Si Jiagou Basin was suitable for the maximum benefit problem research. (2) The two pre-existing check dams in the basin have been built for more than 20 years and have not been fully silted, which was typical for the water resources benefits study. (3) The gully shape and water catchment method in the basin were typical, and it was more representative to research the check dam construction potential and construction methods in the basin as a whole. (4) East check dam within the basin has been developed as an eco-tourism area, which is meaningful for exploring the ecological development of the basin and can also provide guidance for new development directions. In the basin, the main land uses are grassland, forest, bare land, farmland and water area. While forest, grasslands and bare lands are distributed throughout the basin, grasslands are mainly situated on sloping fields, forests in valleys or lower slopes, and bare lands mainly on the steep slopes. Farmland exists in the form of terraced and sloping farmland, while water bodies are mainly restricted to the reservoirs of the two check dams in the basin (Gao *et al.* 2016b, 2017).

### Check dam construction and flood prevention capacity in the basin

There were two pre-existing check dams (east check dam (*Ecd*) and west check dam (*Wcd*)) in the basin (Figure 1). Completed in November 1996, with a control area extending over 3.11 km^{2}, a design storage capacity of 0.502 × 10^{6} m^{3}, the 22.30 m tall *Ecd* has now long exceeded its design siltation period of 16 years. Nonetheless, at present, with the silt surface at 13.6 m from the top of the dam, its reservoir has only retained 0.132 × 10^{6} m^{3} of silt or roughly 60% of its design silt storage capacity of 0.222 × 10^{6} m^{3}. At present, the reservoir's water storage is roughly 0.116 × 10^{6} m^{3}, and the roads for flood prevention are unblocked. The dam body is well maintained, and the area around the facility has developed into a tourism and fishing resort. However, there are no flood discharge facilities (e.g., spillways), only water discharge pipes. The *Ecd* can presently resist 1-in-10-year floods or the equivalent of 0.400 × 10^{6} m^{3} in its reservoir. Based on a calculation method from the literature (Gao *et al.* 2019a), after 10 years at the current level of soil erosion control, representing a mean sedimentation rate is 5.8 × 10^{3}m^{3}yr^{−1}, the remaining storage capacity will be 0.342 × 10^{6} m^{3} and the dam will still have the capacity to resist 1-in-10-year floods. However, after 20 years at a mean sedimentation rate of 4.6 × 10^{3} m^{3} yr^{−1}, the remaining storage capacity will be 0.296 × 10^{6} m^{3}, but the dam will only have the capacity to resist 1-in-5-year floods.

Also completed in November 1996, with a control area extending over 6.50 km^{2}, a design storage capacity of 0.810 × 10^{6} m^{3}, the 21.30 m tall *Wcd* has now long exceeded its design siltation period of 10 years. Nonetheless, at present, with the silt surface at 11.6 m from the top of the dam, its reservoir has only retained 0.187 × 10^{6} m^{3} of silt or roughly 64% of its design silt storage capacity of 0.290 × 10^{6} m^{3}. However, there are no flood discharge facilities (e.g., spillways), only water discharge pipes. At present, the flood prevention roads are unblocked. The *Wcd* can presently resist 1-in-20-year floods, or the equivalent of 0.632 × 10^{6} m^{3} in its reservoir. Under current soil erosion controls, after 10 years at a mean sedimentation rate of 7.7 × 10^{3} m^{3} yr^{−1} the remaining storage capacity will be 0.546 × 10^{6}m^{3} and the dam will still have the capacity to resist 1-in-20-year floods. Likewise, after 20 years at a mean sedimentation rate is 5.1 × 10^{3}m^{3}yr^{−1}, the remaining storage capacity will be 0.505 × 10^{6}m^{3}, and the dam will retain its capacity to resist 1-in-20-year floods (Gao *et al.* 2019a).

Through the greening of the Loess Plateau and the remarkable results of soil and water conservation (Shangguan 2009; Chen *et al.* 2010; Fu *et al.* 2011; Zhao *et al.* 2013; Gao *et al.* 2016a), sediment in the runoff in Si Jiagou Basin has been greatly reduced, making it difficult to fill up the two check dams within the design period. At the current level of sedimentation, the two check dams will not be full in 10 years. At the same time, the water resources accumulated in the check dam can cause significant flood damage during the flood season. In Si Jiagou Basin and similar semi-arid areas where water resources are scarce, such high-quality water resources should be used properly and efficiently instead of causing harm.

There is no current plan for the establishment of further check dams within the basin. Check dams should be constructed in accordance with different benefit targets (e.g., water resources, water savings, ecosystems, the economy and society) and based on full consideration of check dam construction potential in the basin. Most importantly, all the benefits, especially the water resources benefits should be fully taken into consideration to make the array of check dams in the basin and organic system offering maximum benefits while ensuring safety.

## METHODS

Based on the characteristics of the semi-arid Si Jiagou Basin and the targets of intercepting sand, creating arable land through silting and efficient use of water resources, the check dam construction potential of the basin was analysed. An Analytic Hierarchy Process (AHP) method (Arabameri *et al.* 2019) was adopted to select the appropriate site for check dam construction analysis. The check dam's social, ecological, economic and water resource benefits were assessed based on their equivalent economic benefits, allowing a unified measurement of benefits. The sum of the benefits minus the cost of the check dam construction is deemed the overall benefit.

A Multi-objective Non-linear Programming Model, the CDBMM was developed to maximize the overall benefits of installing a check dam array in the basin. It employs a sequential quadratic programming (SQP) method to obtain the check dam construction parameters and system benefits when overall benefits of check dam construction in the basin are maximized (Research steps and framework are shown in Figure 2).

### Data collection

Analytical data was provided by the Gansu Provincial Meteorological Bureau, the Water Affairs Bureau of Qingyang City, the Statistics Affairs Bureau of Qingyang City, the Soil and Water Conservation Bureau of Qingyang City and the Soil and Water Conservation Bureau of Xifeng District. The data included 5 m resolution digital elevation model (DEM) data from the Shuttle Radar Topography Mission (SRTM). The datasets were provided by the Environmental and Ecological Science Data Center for West China.

### Model construction

Prior to developing the model, a target analysis (Feng *et al.* 2020) of the check dams construction potential in Si Jiagou Basin to achieve sediment reduction, farmland silting and efficient water use targets, led to the conclusion that the potential existed for the basin to house 10 check dams, including 2 large-scale check dams and 8 small- to medium-sized check dams (Liu Hanxi 1995; Liu *et al.* 2005; Chang *et al.* 2006; Department of Land and Resources of Shanxi Province 2006; Song 2010; Bussi *et al.* 2013). Given the two existing check dams, the Si Jiagou Basin showed the potential to house eight additional check dams. Taking an array of 10 check dams in the basin as a whole, the check dam system was arranged to provide maximum benefits. In Si Jiagou Basin, there were 20 sites suitable for the construction of check dams, so the top 8 new dam sites were selected using an AHP method.

There are many methods for decision-making of the check dam sites optimization, such as principal component analysis, decision tree, cluster analysis, fuzzy analysis research, random decision forests and AHP. Only the AHP method can divide each factor in a complex problem into related and orderly hierarchies, and quantitatively describe the comparison between the factors in the decision problem, and is a decision-making method combining quantitative and qualitative analysis. It has the advantages of being objective, comprehensive, fair and accurate (Moreno-Jiménez & Vargas 2018; Arabameri *et al.* 2019; Dos Santos *et al.* 2019; Leccese *et al.* 2020). In the process of selecting the suitable check dam sites in Si Jiagou Basin, it is necessary to identify the most suitable sites and the weight of different influencing factors. However, the AHP method formulates a hierarchy of complex decision systems to provide quantitative basis for analysis and decision-making by comparing the importance of various related factors layer by layer, which can meet the check dam site selection requirements in the basin. In the AHP's expert scoring process, eight stakeholders, including individuals in the fields of water resources science, ecology, water conservancy engineering and local administration, along with local farmers, were selected to score each site in terms of its unique socioeconomic, water-saving and ecosystem characteristics. In the selection of scoring factors, we have integrated almost all 18 elements of the 4 factors (basin geography and dam construction, socioeconomic, water resources and ecological factors) related to the check dam site location for expert scoring. In particular, we also added the scoring factors of water resources and ecological factors, starting from the selection of the check dam location, the construction of water-saving-ecological check dam system was considered. The AHP model added water resources and ecological influencing factors, and increased the weight of water resources efficient utilization factors, to construct the 4 criterion hierarchies and 18 index hierarchies model. And the scoring factors included:

- 1.
*Basin geography and dam construction factors*: channel topography, geological conditions, check dam control area and check dam building materials; - 2.
*Socioeconomic factors*: flood prevention capacity, replacement of roads with check dams and increasing cultivated land; - 3.
*Water resources factors*: water use efficiency, potential to build water storage facilities around drinking water projects for humans and livestock, water for small processing industries, irrigation water and aquaculture; - 4.
*Ecological factors*: conservation of water sources, erosion reduction and ditch fixation, slope treatment and plant growth (Hassanli*et al.*2009; Huang*et al.*2009; Hui*et al.*2010; Wang*et al.*2016).

In order to overcome the shortcomings of expert scoring in the AHP method, which are subjective, incomplete and inconsistent, PSO (Particle Swarm Optimization) method was introduced to modify the expert scoring matrix. The PSO method finds the optimal solution by initializing a group of random solutions and multiple iterations. The solution for each optimization problem is called a particle, and each particle has a position vector and a velocity vector. According to the particle's own ‘experience’ and the ‘experience’ of the best particles in the population to determine how to adjust and change the direction and speed in the next iteration, and through the stepwise iteration, the particles gradually approaching the optimal solution (Poli *et al.* 2007; Marini & Walczak 2015; Sengupta *et al.* 2018; Fernandes & Yen 2019). Due to the huge amount of data and limited space, the authors only list the original scoring matrix (Supplementary Material, Appendix A) and the revised matrix and weights (Supplementary Material, Appendix B) of expert 1 of the potential to build water storage facilities around the element in the water-saving factor. Once the AHP model was constructed, the optimal construction sites were obtained through analysis.

Based on the analysis and selection of dam sites, the CDBMM was constructed to determine check dam construction parameters such as silt storage capacity and flood control capacity. A non-linear programming model, CDBMM selects reasonable decision variables and solves objective functions by applying restrictions.

The main steps for building a non-linear programming model include (Xu *et al.* 2011; Hammad *et al.* 2016; Foroozandeh *et al.* 2017; Li *et al.* 2020):

- 1.
Program the mathematical model of check dam system planning into the non-linear planning program and debug the program;

- 2.
Assign values to the decision variables and solve them;

- 3.
Remove the check dam construction parameters with small benefits;

- 4.
Modify the system model and perform the next round of optimization according to steps 2 and 3;

- 5.
Repeat the above process until the objective function reaches the maximum and the optimization result is practically feasible.

However, in the process of constructing the check dam system's benefit function, the measures of each benefit function proved inconsistent. Therefore, all benefits were converted into economic benefits to uniformly construct the objective function, and then the cost of check dam construction was subtracted to complete the construction of the objective function. The objective function was then restricted by restrictions, and the decision variables were determined by using the constraints. Finally, the value of the decision variable that the objective function reaches at its maximum (the overall benefits of the check dam system reach a maximum) within the range of the constraint conditions was obtained.

#### Choice of the decision variables

Decision variables are parameters that can be changed during the planning process of the system but must be determined by the end of the planning. Due to the characteristics of the basin's ecosystem and economic systems and the limitations of the related mathematical planning methods, the choice of decision variables should meet the following two requirements (Zhou *et al.* 2003; Campisano *et al.* 2014; Jiang *et al.* 2014; Gharaei & Pasandideh 2016; Cucchiaro *et al.* 2019):

- 1.
Fully reflect the constraints of the environment on the subsystems, that is, the restriction of check dam system parameters by social, ecological, water resource and economic factors and

- 2.
Establish a system model with as few variables as possible, which can not only reflect the parameters of the check dam but also help to simplify and solve the model.

In the construction of check dams, in addition to storage capacity (silt storage capacity and flood control capacity), the construction parameters also include the length, width and height of the check dam, as well as the length, width, height and the maximum flow amount of the spillway. In principle, all the above parameters can be used as decision variables. However, there was a regression relationship among the storage capacity, the length, width and height of the check dam in the basin, and the maximum flow amount had a fixed regression relationship with the length, width and height of the spillway. Determine one parameter, and other parameters can be obtained by calculation according to the regression relationship. In the benefit calculation of CDBMM, the silt storage capacity can reflect the optimization and benefits of the check dam system, the flood control capacity can reflect the flood control ability and water storage benefits of the check dam system, and the maximum flow amount of the spillway can reflect the flood control capacity of the check dam. The benefits had more direct relationships with the silt storage capacity, flood control capacity and the maximum flow amount of the spillway, and choosing these three parameters made the benefits calculation more accurate. Therefore, the decision variables are preliminarily selected as the silt storage capacity, the flood control capacity and the maximum flow of the spillway. However, the flood control capacity and the maximum flow of the spillway are related through a functional relationship represented by the Gocherin (Yansheng 1985; Jisheng 1994; Ruzhao 1994; Haitao 2011; Wang & Cao 2017) formula. Accordingly, the decision variable of maximum spillway flow was removed. Finally, the two decision variables, silt storage capacity and flood control capacity remained and employed to establish the CDBMM and achieve maximum check dam system benefits.

#### Objective functions

The check dam has social, ecological, economic and water resources benefits. The benefits of water storage, mud blocking, water savings and irrigation can be converted into economic and ecological benefits for unified measurement, and the objective function can then be constructed. The social benefit accrues from the impact of the completion of the check dam system on transportation, engineering construction, agricultural production, etc. The economic benefits (i.e., value) mainly accrues from the agricultural production income arising from the installation of the check dam, and the ecological benefits are that the silt and water are stored behind the check dam, altering the channel's ecosystem by providing good soil and water conditions for plant growth. The ecological benefits also included the amount of mud retained and water storage, so the economic benefits of the check dam system reflect both ecological and water-saving benefits. If the amount of mud held back by the check dam system is considered in the restrictions, the economic benefits can be used as the objective function, which can reflect the overall benefits of the check dam system. In semi-arid areas where water resources are scarce, when constructing water-saving ecological check dam systems, the water-saving factors of dam systems must be considered. The water-saving factors can also be converted into economic benefits to uniformly build the objective function.

*F*is the total benefit, is the planting benefit, is the aquaculture benefit, is the irrigation benefit, is the flood control benefit, is the silt blocking benefit, is the check dam construction cost and is the spillway construction cost. The benefits are all expressed in RMB (RenMinBi).

*n*subscript indicates total benefits/costs of the

*n*check dams in the basin, in RMB.

##### Benefit calculations for the check dam system

The economic benefits of check dams include direct economic benefits and indirect economic benefits. Direct economic benefits are mainly arable land for planting benefits, aquaculture benefits, irrigation benefits and flood protection benefits, while indirect benefits are mainly silt blocking benefits and other social benefits. When considering the water resources factor, the benefits of water resources utilization are converted into economic benefits of check dam construction in order to quantify them. The main methods involved are as follows (Wu Yongchang 1991; Zhou *et al.* 2003; Qin & Li 2010; Wantu 2015; Zhang *et al.* 2016).

**(i) Production benefits for newly cultivated land**

*et al.*2013; Wang

*et al.*2016): where

*i*is a counter of check dams in the basin , is the unit price of grain crops (RMB kg

^{−1}), is the unit price of non-grain crops (RMB kg

^{−1}), is the unit area yield of grain crops of the

*i*th check dam (kg ha

^{−1}), is the unit area yield of non-grain crops of the

*i*th check dam (kg ha

^{−1}), is the decision variable, which represents the silt storage capacity of the

*i*th check dam (10

^{4}m

^{3}),

*R*is the guaranteed harvest probability of the check dam (%), is the calculation period of check dam system (year), is the proportion of land area for grain crops of the

*i*th check dam (%), is the proportion of land area for non-grain crops of the

*i*th check dam (%),

*η*is the land utilization ratio, is the index to convert check dam silt storage capacity to check dam area for the

*i*th dam and is the coefficient to convert check dam silt storage capacity to check dam area for the

*i*th dam.

In the production benefits calculation, the unit price of crops listed grain and non-grain, however, there were other crops in the basin. Grain and non-grain crops were taken to represent the average crop planting situation and average benefit level in the basin. The average value could be used as a reference for benefit calculation and simplify the calculation.

**(ii) Check dam water resources aquaculture benefits**

A check dam's maximum water storage capacity was determined based on the design silt storage capacity. As the silt continues to accumulate, the water storage capacity in the dam gradually decreases. After reaching the design siltation age, the water storage function is lost. The water stored in the check dam becomes seepage that contributes to increasing the amount of groundwater recharge. This also has the effect of storing turbid water and allowing clean water to drain. Considering human and livestock drinking water, irrigation water, water to meet ecological water needs and other water consumption factors, many experts believe that the water resource utilization rate of check dams in semi-arid areas can be calculated to be 30%. Silt dams can effectively store precipitation, help alleviate the problem of drinking water for rural people and livestock in hilly areas, increase soil moisture storage, reduce flood peak flow, increase dry season flow and enhance drought resistance ability (Bureau U. a. M. Y. R. M. 2004; Liu *et al.* 2010; Zhang *et al.* 2016).

*i*th check dam (RMB kg

^{−1}),

is the unit yield of aquaculture of the *i*th check dam (kg ha^{−1}),

is the decision variable, which represents the flood control capacity of the *i*th check dam (10^{4}m^{3}) and

is the coefficient for calculating the aquaculture water surface area from flood control capacity of the *i*th check dam (Bureau U. a. M. Y. R. M. 2004).

In the water resources aquaculture benefits calculation, the benefits were calculated according to the average aquaculture unit price, and not the individual conditions of each dam. However, in the benefits calculation of the entire check dam system, it was reasonable to use the assumed average value in the basin.

**(iii) Irrigation benefits accruing from new check dam-generated water resources**

*i*th check dam (RMB kg

^{−1}), is the unit increase in crop yield achieved by irrigating with water retained by the

*i*th check dam (kg ha

^{−1}), is the index for converting flood control capacity of the

*i*th check dam to water resources available for irrigation and is the coefficient for converting flood control capacity of the

*i*th check dam to water resources available for irrigation.

In the irrigation benefits calculation, the main irrigation crop was grain; however, there were a small number of vegetables and fruit trees. In Si Jiagou Basin, most of the crops for supplementary irrigation were still grains. Therefore, it was reasonable to assume that all irrigated crops are grains. The impact on the calculation results of irrigation benefits was small for the whole basin.

**(iv) Flood control benefits**

*i*th check dam (RMB kg

^{−1}), is the unit crop yield protected by the

*i*th check dam (kg ha

^{−1}), is the index for converting the

*i*th dam's flood control capacity to the area of land protected and for converting the

*i*th dam's flood control capacity to the area of land protected.

**(v) Silt blocking benefits**

^{−1}(Hanxiong 1994; Bureau U. a. M. Y. R. M. 2004): where is the unit price of dredging silt blocking the

*i*th check dam (RMB Mg

^{−1}) and is the coefficient for converting the

*i*th check dam's silt storage capacity to the quantity of silt blocked by the dam (RMB Mg

^{−1}).

##### Cost calculation of check dam system

**(i) Check dam engineering cost**

*et al.*2007): where

*a*,

*b*,

*c*,

*d*,

*j*,

*k*are the coefficients and indices, which obtained through regression analysis,

*et al.*2014; Wantu 2015): where is the unit price of check dam earthwork volume (RMB m

^{−3}), is a coefficient that converts the storage capacity of the

*i*th check dam to particular engineering quantities and is a coefficient that converts the storage capacity of the

*i*th check dam to particular engineering quantities.

**(ii) Spillway engineering costs**

*et al.*1999; Haitao 2011; Parsaie

*et al.*2015; Wang & Cao 2017). Finally, the spillway engineering cost is calculated through a regression equation between flow and engineering cost. For a flood discharge upstream of the check dam, the spillway flow downstream of the check dam is given as follows (Zhou

*et al.*2003; Bureau U. a. M. Y. R. M. 2004; Qin & Li 2010): where is the flood peak flow of frequency

*p*within the check dam control area (m

^{3}s

^{−1}), is the maximum spillway flow of the immediately upstream check dam (m

^{3}s

^{−1}),

*Q*is the maximum flow in the spillway downstream from the check dam (m

^{3}s

^{−1}), is one of the decision variables, representing the flood control capacity of the check dam (10

^{4}m

^{3}), is the total flood amount in the check dam control area at a frequency of

*p*(10

^{4}m

^{3}) and is the total flood discharge of the immediately upstream check dam before the check dam spillway flow reaches its maximum flow (10

^{4}m

^{3}).

In Equation (12), and are constants determined by hydrological conditions, while is a decision variable; accordingly, the key to calculating the spillway flow is to calculate and . To do so, four steps must be completed (Liu Hanxi 1995):

- 1.
identify immediately upstream check dam(s),

- 2.
calculate the discharge flow, , of the immediately upstream check dam(s) through its spillway. For the sake of simplicity, the maximum value of spillway flow of the immediately upstream check dam(s) is selected as .

- 3.
calculate the period of time of the check dam spillway discharge flow from its onset to its maximum,

- 4.calculate the discharge flow of each immediately upstream check dam during this period and sum them up as . The engineering cost of the entire spillway is then calculated as (Liu Hanxi 1995): where is the unit price of spillway lining (RMB m
^{−3}), is the unit price of spillway earthwork (RMB m^{−3}), is the spillway critical water depth of the*i*th check dam*i*(m), is the flood peak flow for the*i*th check dam in the check dam control area at a frequency*p*(m^{3}s^{−1}), is the sum of maximum spillway flow of check dams immediately upstream from the*i*th check dam (m^{3}s^{−1}), is the spillway width of the*i*th check dam (m), is the spillway length of the*i*th check dam (m), is the total flood amount of frequency*p*in the check dam control area of the*i*th check dam (10^{4}m^{3}), is the total discharge flood amount of immediately upstream check dams before the spillway flow of the*i*th check dam reaches its maximum flow (10^{4}m^{3}), is the coefficient used to convert check dam spillway flow to necessary spillway earthworks for the*i*th check dam and is the index to convert check dam spillway flow to necessary spillway earthworks for the*i*th check dam.

##### Objective functions

#### Restrictions

The restrictions should be compatible with the objective function and should reflect the requirements for water use, economic, social and ecological benefits of the check dam system arising from the check dam array's silt blocking, silting and flood detention capacities. Within a certain limit, these benefits are largest when a lot of mud blocking, a large silted in the area and a large detention reservoir capacity are achieved. Therefore, it was not necessary to limit the four benefits, if the restrictions on silt storage capacity, flood storage capacity and silt area of the check dam system were limited. The treatment not only reflected the requirements for the four benefits but also avoided the difficulty of calculating the four benefits. Besides, in addition to restrictions on the four benefits, the height of the check dam must be limited, which can be achieved by limiting the silt storage capacity or flood storage capacity of the check dam. Accordingly, the restrictions were as follows (Wu Yongchang 1991; Hanxiong 1994; Liu *et al.* 2010; Qin & Li 2010; Wantu 2015).

##### Non-negative variables

##### Terrain

*i*th check dam site (10

^{4}m

^{3}).

##### Check dam silting area

Silt storage was one of the ecological benefits, and the check dam silting area was one of the main sources of economic benefits. In order to ensure ecological and economic benefits, the minimum check dam silting area was often restricted. The minimum check dam silting area was determined by many factors. This study mainly considered six aspects.

**(i) Check dam system relatively stable**

Only when the ratio of the check dam silting area to the corresponding basin area reaches a certain value can relative check dam system stability be achieved. According to the results of the optimized simulation planning, when the ratio of the check dam silting area to the basin area was between 11 and 12, which was optimal, the check dam silting area could be determined accordingly.

**(ii) Silting thickness**

The silt deposition on the check dam was beneficial to crop growth, but an excessive thickness may flood the crops, and the thickness of silt deposition should not exceed 0.2 m yr^{−1} (Bureau U. a. M. Y. R. M. 2004, 2005; Wang *et al.* 2013). Therefore, the minimum check dam silting area can be determined based on the erosion modulus in the basin.

**(iii) Water storage depth**

*et al.*2013). The check dam silting area was calculated based on the above. All the check dams in the Si Jiagou Basin would be provided with spillways, and the height of the spillway inlet would be flush with the check dam ground, so the water storage depth could no longer be considered. One must comprehensively consider the requirements of the above aspects and the check dam silting area of many production check dams in the basin to determine the minimum check dam silting area. Then, the total check dam silting area of the check dam system should satisfy (Wu Yongchang 1991; Hanxiong 1994; Qin Xiangyang 1994; Qin & Li 2010): where is the coefficient for converting storage capacity dam into silting area for the

*i*th check dam, is the minimum check dam silting area (km

^{2}) and is the index of converting storage capacity of the check dam to the silting area for the

*i*th check dam.

##### Flood control capacity

^{4}m

^{3}).

##### Silting time

^{−1}a

^{−1}in the basin, which may cause the siltation period to be very long and the silt storage capacity of the check dam was further increased. Therefore, according to the designed siltation period and the calculation time and benefits of the entire check dam system in the basin, the siltation constraint period was 30 years and is determined as follows (Wu Yongchang 1991; Hanxiong 1994; Qin Xiangyang 1994; Qin & Li 2010): where is the

*i*th check dam's control area (ha),

*M*is the erosion modulus in the basin (Mg km

^{−2}a

^{−1}) and

*γ*is the silt dry volume-weight and has a value of 1.3 Mg m

^{−3}.

##### Water–sand balance

### Solving model

The CDBMM is a multi-constrained non-linear optimization model that seeks an optimal solution. There already exist several excellent algorithms for solving optimization problems (e.g., trust-region algorithm, conjugate gradient method, quasi-Newton method, etc.) (Raja *et al.* 2016; Zhengqing *et al*. 2019; Kim *et al.* 2020). These methods are often only applicable to linear convex optimization problems, whose local optimal solution can be considered as a global optimal solution. However, for multi-constrained non-linear models, the results obtained by classical optimization methods are often local optimums rather than global optimums. Therefore, a global optimization algorithm needed to be selected.

Global optimization algorithms were mainly divided into two categories: uncertainty algorithms and deterministic algorithms. Uncertainty algorithms mainly refer to evolutionary categories, such as genetic algorithm, PSO, simulated annealing algorithm and ant colony algorithm. Due to the randomness of the calculation process, in order to achieve the optimal solution, uncertainty algorithms need to adjust the parameters and increase the number of iteration steps, causing the calculation efficiency to be low. The deterministic algorithm, such as the SQP method, was more efficient than the evolutionary algorithm. However, for complex problems, such as multi-objective optimization, deterministic algorithms were often not suitable.

*et al.*2020). These advantages mainly arise because the limitation in the variable domain allows the optimization algorithm to make better decisions after considering the search direction and step length. Therefore, SQP was selected as the solution method, and the basic method can be expressed as follows (Boggs & Tolle 2000; Lee & Leyffer 2012; Huang

*et al.*2019): where is a symmetric matrix and .

*et al.*2006; Huang

*et al.*2019): where is the Lagrange multiplier. At

*x*, the Kuhn–Tucker (K–T) condition of Equation (23) is:

By repeating the process for more iterations, the optimization solution can be obtained. There were many optimized programs, such as open-source optimization toolboxes based on Fortran, C Programming Language and Python, etc. As an excellent software, after years of improvement and feedback from a large number of researchers, the simulation results of calculations were very reliable. In short, compared to other open-source software and programs, MATLAB's numerical calculation results were more reliable. Therefore, the solution process used the MATLAB 2018b global optimization toolbox running environment. Before performing the steps to obtain the model's optimal solution, the parameters of the model were first determined according to the conditions of the basin and the technical specifications.

## RESULTS AND DISCUSSION

### Determining the model's parameters

Before determining the parameters of the CDBMM, the proposed check dam sites were first located on the satellite map (Figure 3). According to the location information for each check dam site, ground contour data was downloaded. Based on the contour data, the check dam control area, check dam length, height and other information could be estimated for the different sites. Other parameters required for the CDBMM were obtained through regression analysis and investigation analysis, etc. (Figure 4).

The objective function of the silt storage and flood control capacity was in the form of a function (Equation (14)), whose parameters were determined (Table 1). Based on Gao *et al.*, the check dam construction potential for the entire Si Jiagou Basin was 10, so in the optimization problem, the value of *n* was 10. The CDBMM parameter values for the 10 check dams in the basin were consistent (Figure 5). For all key check dams (>20 m in height and >0.5 × 10^{6} m^{2} in earthwork volume) in Qingyang City, parameters such as dam control area, dam height, silt storage capacity and flood control capacity were determined (Tables 1 and 2).

CDBMM parameter . | Units . | Equation . | Check dam consensus value . |
---|---|---|---|

RMB m^{−3} | 11 | 16.5 | |

RMB m^{−3} | 13 | 555.6 | |

RMB m^{−3} | 13 | 73.3 | |

m | 13 | 2.0 | |

17 | 0.3339 | ||

7 | 0.75 | ||

RMB kg^{−1} | 4 | 15 | |

RMB kg^{−1} | 5 | 6 | |

RMB kg^{−1} | 6 | 4 | |

Mg ha^{−1} | 3 | 6.00 | |

Mg ha^{−1} | 4 | 2.75 | |

Mg ha^{−1} | 5 | 2.00 | |

Mg ha^{−1} | 6 | 8.00 | |

RMB kg^{−1} | 7 | 11.32 | |

m^{3} s^{−1} | 13 | 64.73 | |

ha | 17 | 46.12 | |

Mg km^{−1} a^{−1} | 19 | 3,000 | |

% | 3 | 80 | |

years | 3 | 50 | |

% | 18 | 72.58 | |

11 | 265.44 | ||

% | 3 | 30 | |

% | 3 | 70 | |

11 | 1.7563 | ||

Mg m^{−3} | 19 | 1.3 | |

6 | 0.30 | ||

13 | 1.9564 | ||

13 | 2.3132 | ||

% | 3 | 60 | |

20 | 0.769 | ||

5 | 0.15 | ||

3 | 0.8141 | ||

17 | 0.8141 | ||

3 | 0.339 | ||

4 | 27.09 | ||

6 | 1.00 |

CDBMM parameter . | Units . | Equation . | Check dam consensus value . |
---|---|---|---|

RMB m^{−3} | 11 | 16.5 | |

RMB m^{−3} | 13 | 555.6 | |

RMB m^{−3} | 13 | 73.3 | |

m | 13 | 2.0 | |

17 | 0.3339 | ||

7 | 0.75 | ||

RMB kg^{−1} | 4 | 15 | |

RMB kg^{−1} | 5 | 6 | |

RMB kg^{−1} | 6 | 4 | |

Mg ha^{−1} | 3 | 6.00 | |

Mg ha^{−1} | 4 | 2.75 | |

Mg ha^{−1} | 5 | 2.00 | |

Mg ha^{−1} | 6 | 8.00 | |

RMB kg^{−1} | 7 | 11.32 | |

m^{3} s^{−1} | 13 | 64.73 | |

ha | 17 | 46.12 | |

Mg km^{−1} a^{−1} | 19 | 3,000 | |

% | 3 | 80 | |

years | 3 | 50 | |

% | 18 | 72.58 | |

11 | 265.44 | ||

% | 3 | 30 | |

% | 3 | 70 | |

11 | 1.7563 | ||

Mg m^{−3} | 19 | 1.3 | |

6 | 0.30 | ||

13 | 1.9564 | ||

13 | 2.3132 | ||

% | 3 | 60 | |

20 | 0.769 | ||

5 | 0.15 | ||

3 | 0.8141 | ||

17 | 0.8141 | ||

3 | 0.339 | ||

4 | 27.09 | ||

6 | 1.00 |

Definition of the parameter can be found with the equation listed (Wu Yongchang 1991; Qin Xiangyang 1994; Niu Ping 2004; Qin & Li 2010).

Check dam No. . | Check dam control area ( Equation (19), km^{2})
. | 1-in-100-year flood volume (, Equation (13), 10^{5} m^{3})
. | Maximum Spillway length (, Equation (13), m) . | Spillway width (, Equation (13), m) . | Max spillway flow of upstream check dams (, Equation (13), m^{3} s^{−1})
. | Maximum storage capacity (, Equation (16), 10^{4} m^{3})
. |
---|---|---|---|---|---|---|

1 | 3.13 | 3.152 | 35 | 1.5 | 2.0 | 48.89 |

2 | 4.06 | 4.092 | 35 | 1.5 | 2.0 | 44.35 |

3 | 2.44 | 2.460 | 50 | 1.5 | 2.0 | 48.55 |

4 | 4.77 | 4.808 | 35 | 1.5 | 2.0 | 75.77 |

5 | 0.93 | 0.937 | 35 | 1.5 | 2.0 | 29.06 |

6 | 1.92 | 1.935 | 50 | 1.5 | 2.0 | 40.99 |

7 | 10.9 | 10.987 | 80 | 2.0 | 3.0 | 182.45 |

8 | 0.87 | 0.877 | 80 | 2.0 | 3.0 | 123.31 |

Wcd | 6.50 | 6.552 | 80 | 2.0 | 3.0 | 128.52 |

Ecd | 3.09 | 3.115 | 80 | 2.0 | 3.0 | 52.92 |

Check dam No. . | Check dam control area ( Equation (19), km^{2})
. | 1-in-100-year flood volume (, Equation (13), 10^{5} m^{3})
. | Maximum Spillway length (, Equation (13), m) . | Spillway width (, Equation (13), m) . | Max spillway flow of upstream check dams (, Equation (13), m^{3} s^{−1})
. | Maximum storage capacity (, Equation (16), 10^{4} m^{3})
. |
---|---|---|---|---|---|---|

1 | 3.13 | 3.152 | 35 | 1.5 | 2.0 | 48.89 |

2 | 4.06 | 4.092 | 35 | 1.5 | 2.0 | 44.35 |

3 | 2.44 | 2.460 | 50 | 1.5 | 2.0 | 48.55 |

4 | 4.77 | 4.808 | 35 | 1.5 | 2.0 | 75.77 |

5 | 0.93 | 0.937 | 35 | 1.5 | 2.0 | 29.06 |

6 | 1.92 | 1.935 | 50 | 1.5 | 2.0 | 40.99 |

7 | 10.9 | 10.987 | 80 | 2.0 | 3.0 | 182.45 |

8 | 0.87 | 0.877 | 80 | 2.0 | 3.0 | 123.31 |

Wcd | 6.50 | 6.552 | 80 | 2.0 | 3.0 | 128.52 |

Ecd | 3.09 | 3.115 | 80 | 2.0 | 3.0 | 52.92 |

Maximum spillway length, spillway width, maximum spillway flow of upstream check dams, and check dam storage capacity for each projected and pre-existing check dam in the Si Jiagou Basin.

There were many ways to find the relationship between various parameters, such as regression analysis, correlation analysis, analysis of variance, principal component analysis, artificial neural networks, and various methods in machine learning. However, the regression analysis method can show the significant relationship between the parameters and the degree of regression fitting. The algorithm of regression analysis is simple, the calculation speed is fast, and the calculation accuracy can also meet the requirements of the check dam parameters in Si Jiagou Basin (Schroeder *et al.* 1986; Shi *et al.* 2002; Daoud 2017; Gkioulekas & Papageorgiou 2019; Graudal *et al.* 2019; Jiang *et al.* 2019a; Abadie *et al.* 2020; Jiang & Liao 2020; Zhang *et al.* 2020). Therefore, the regression analysis with higher calculation efficiency was adopted between the silt storage capacity and dam silting area to obtain the Si Jiagou Basin's dam system parameters: and (Table 1). Obtaining similar consistency among the 10 check dams, similar regression methods determined other parameters. Based on topographic map-derived channel flood volume and the check dam control area of the basin's check dam sites, the total volume of a once-in-a-century flood in the Si Jiagou Basin was calculated for each check dam. Calculated values of is the total flood volume for a 1-in-100-year flood (i.e., *p* = 0.01) in the check dam control area of the *i*th check dam (10^{4} m^{3}) is shown in Table 2. In addition, is the total discharge flood volume of immediately upstream check dams before the spillway flow of the *i*th check dam reaches its maximum flow (10^{4} m^{3}) and had the same value as in this study. The maximum storage capacity of topographical allowance measured and calculated based on the basin topographic map, for the *i*th check dam ( 10^{4} m^{3}) is also given in Table 2.

In the CDBMM, the parameter settings were divided into two categories. The first category was the model calculation parameters, such as the Check Dam Control Area, Maximum Spillway length and Maximum storage capacity, which were all calculated by measurement, formula according to the socioeconomic conditions and the regression analysis results (Tables 1 and 2 and formulas (3)–(26)). The second category was the numerical calculation model parameters. For the calculation of the SQP algorithm, the optimization toolbox in MATLAB was used. The model was more sensitive to the maximum number of iteration steps and termination error, which were respectively set to 1,000 and 10^{−6}. The number of iteration steps set to 3,000 was not much different from 1,000, which was just an increase in calculation time, and the setting of the termination error determined the model accuracy and calculation time. Other parameters in the algorithm, such as search step and search direction, were set as the default parameters of the toolbox. Under the default parameters, the errors of multiple calculation results were less than 0.1%, which can achieve the desired results.

### Analysis of CDBMM running results

After establishing the CDBMM, the selection of model solution methods and the determination of model parameters, which note the full array of check dams offers maximum benefits at minimum cost. Results of the CDBMM are presented in Figures 6–11.

#### Determination of check dam construction parameters

According to CDBMM, the eight new proposed check dams and the two built check dams in Si Jiagou Basin should maximize production, aquaculture, irrigation, flood control, silt blocking and minimize the check dam's construction costs. The results of the CDBMM for silt storage capacity and flood control capacity are shown in Figure 6. The total silt storage capacity was 2.478 × 10^{6} m^{3} and the total flood control capacity was 3.557 × 10^{6} m^{3} across the entire Si Jiagou Basin. The silt storage capacities of the proposed check dams 1–8 were 185, 216, 257, 64, 133, 169, 755 and 60 × 10^{3} m^{3}, and the flood control capacities were 201, 227, 328, 182, 238, 276, 984 and 175 × 10^{3} m^{3}. The top three check dams in terms of silt storage capacity and of flood control capacity were No. 7, 3 and 6; moreover, they all distributed in the midstream and downstream basin, which indicates that the silt storage capacity and the flood control capacity were consistent. The silt storage and flood control capacity of the *Ecd* check dam were 188.0 and 217.0 × 10^{3}m^{3}, respectively, compared to the original design values of 222.0 and 280.0 × 10^{3}m^{3}, i.e., a reduction of 34.0 (18.09%) and 63.0 (22.50%) × 10^{3} m^{3}. The silt storage and flood control capacity of the *Wcd* check dam were 450.0 and 730.0 × 10^{3} m^{3}, respectively, compared to the original design values of 290.0 and 520.0 × 10^{3}m^{3}, i.e., increases of 160.0 (35.56%) and 210.0 (40.38%) × 10^{4}m^{3}. The results of CDBMM for the *Ecd* check dam decreased by 20% compared with the original design value, and *Wcd* check dam increased by 40%, indicating that the original design was poor. The design of the original east and west check dams needed to be modified (Figure 7).

Meanwhile, the check dam construction had improved the flood detention and sediment retention capacity of the entire basin and provided a guarantee for the high-quality, safe and rapid development of the basin. More importantly, the high-quality rain and flood resources stored in the basin could further alleviate the water crisis, increase food production amount and economic benefits. If this part of the stored water resources was supplemented for irrigation during the critical water demand period of the crop, it could increase the grain output by 1/5 compared to the same non-irrigated land. The reasonable construction of the check dam laid a solid water resource foundation for people's production and living standards improving and ecological civilization building (Wang *et al.* 2011; Abedini *et al.* 2012; Parimalarenganayaki & Elango 2016; Malek Hosayni *et al.* 2017).

#### Benefit and cost comparison

With the benefits of new check dams in the Si Jiagou Basin set to reach their peak within 50 years, the sub-benefits for each of the 10 check dams and their construction costs are shown in Figure 8. Taking years as the calculation period, the total benefits of the 10 check dams in the basin was 338.315 million RMB and the total cost was 23.955 million RMB (the check dam construction cost was 17.236 million RMB and the spillway cost was 6.719 million RMB). The total cost presented 7.07% of the total benefits, indicating that the overall benefit of installing the check dam array was significant. Among the 10 check dams, the benefits of No. 7 (89.108 million RMB) were the greatest, with those of *Wcd* (60.781 million RMB) and No. 3 (34.082 million RMB) being the second and third, respectively. Together these accounted for 54.30% of the total basin benefits. Among all five benefits, the silt blocking benefits (105.175 million RMB) were the largest, and the irrigation benefits (320.334 million RMB) were the smallest. The silt blocking benefit of No. 7 check dam was 32.033 million RMB, that of *Wcd* (19.013 million RMB) and that of No. 3 (10.912 million RMB) were the second and third, respectively, and together accounted for 18.31% of the overall benefits. This distribution order was consistent with that for the overall benefit.

As shown in Figure 9, over the 50-year calculation period, the total benefits and total costs were significantly different, the latter being 7.07% of the former. The check dam system in Si Jiagou Basin had significant benefits. Among the proposed check dams, the lowest cost–benefit ratio was No. 4 (1.24%), the highest was No. 7 (10.66%) and the average was 5.72%. The larger the scale of the check dam, the greater the cost–benefit ratio, mainly because, as the construction scale of the check dam increased, the cost of check dam construction also increased. However, apart from check dam No. 7, the cost–benefit ratio of the remaining check dams did not exceed 8.00%. In addition, with the reduction of the check dam scale, the cost–benefit ratio was significantly reduced. Therefore, under the condition of safe operation, the construction of an array of check dams in the basin can provide huge benefits. The benefits can be divided into tangible and intangible benefits. If the accumulated water and soil resources were reasonably allocated and efficiently used, greater net benefits would be produced, which had huge potential for utilization. The check dam system scale was determined according to the results of the model, and in the long-term could obtain significant benefits and achieved the target of sustainable and efficient management of the basin (Xu *et al.* 2004; Agoramoorthy *et al.* 2016; Parimalarenganayaki & Elango 2016; Baba & Hack 2019; Dashora *et al.* 2019).

#### Benefit analysis

Figure 10 showed the net benefits and the percentages attributable to each check dam in Si Jiagou Basin for a 50-year calculation period. The top three were No. 7, *Wcd* and No. 3, showing net benefits of, respectively, 79.61, 56.72 and 31.76 million RMB. The net benefits of No. 7 check dam accounted for 25.28% of the significant net benefits. The top three check dams account for 53.38% of the basin's total net benefits and were all distributed in the downstream portion of the basin (Figure 11). They had the characteristic that the larger the check dam construction scale (silt storage capacity and flood control capacity), the better the net benefits. The last three ranked check dams in terms of net benefits were No. 8, No. 4 and No. 5 (79.61, 56.72 and 31.76 million RMB, respectively), and accounted for 15.91% of the total net benefits; all were distributed in the upper and middle reaches of the basin. At the same time, the net benefits of the original *Wcd* and *Ecd* check dams were 80.19 million RMB, and accounted for 25.46% of the total net benefits. This shows the necessity of building large- and middle-scale check dams in the basin, which can greatly increase the net benefits, and have significant effects on improving water-saving, ecology and soil and water conservation in the basin. The check dam construction transformed the arable land into high-yield farmland and high-quality orchards and had great development potential and social significance, which will better serve the basin in an ecofriendly manner (Abedini *et al.* 2012; Wolka *et al.* 2018; Jiang *et al.* 2019b).

Figure 11(a) and 11(b) shows the distribution of total benefits in Si Jiagou Basin when the entire check dam system has yielded its maximum benefits over the 50-year calculation period. The overall benefit for the entire Si Jiagou Basin was 338.83 million RMB. The top three of five benefits were silt blocking benefits, production benefits and aquaculture benefits, which respectively accounted for 105.17, 79.81 and 77.03 million RMB (i.e., 31.04, 23.55 and 22.73% of the total). The sum of the first three benefits was 262.02 million RMB and accounted for 77.33% of the total benefits. The flood control benefits, and irrigation benefits came last, at 44.80 and 32.01million RMB, respectively, accounting for 13.22 and 9.45% of the total benefits. The construction costs (17.23 million RMB) of the check dam system were relatively small, and the construction costs of the spillway were 6.72 million RMB. The construction costs of the check dam system accounted for 7.07% of the total benefits. The check dam system benefits in the basin were large and maximized the ecological, social, economic and water-saving benefits of the whole basin. At the same time, the water resources benefits (sum of aquaculture, irrigation and flood control benefits) of the basin were 153.84 million RMB, accounting for 45.40% of the total benefits, indicating that the water resources benefits of the check dam system were large and should be considered as the main income and influencing factors in the ecological construction of the basin (Figure 11(b)). Production benefits (184.99 million RMB, 55.60%) still accounted for more than half, and they were still the primary income, which should be considered as a key factor. However, especially in the semi-arid area in Gansu, the benefits of water resources of check dams should not be neglected. It is necessary to fully consider the water resources benefits in the construction and formation of the check dam system to alleviate the water shortage crisis in the semi-arid area, as it will provide new ways to solve the water resources problems in the basin, and also provide new methods for the sustainable development of the basin and the construction of water-saving and ecological check dam arrays under the premise of ensuring safety. The construction of check dams and the formation of a system based on the CDBMM operation result will greatly improve the flood control capacity of the basin. At the same time, the comprehensive benefits of sand blocking, silting land and irrigation will be significantly improved, which can further resist drought, increase grain yield and farmers' income, and promote the rapid development of the socioeconomic benefits. The intercepted rainwater resources can solve the problem of drinking water for local people and animals, develop small irrigated land, improve local agricultural production and farmers' living conditions, improve the ecological environment and bring considerable social, economic and ecological benefits (Xu *et al.* 2004; Wang *et al.* 2011; Parimalarenganayaki & Elango 2016; Malek Hosayni *et al.* 2017; Vema *et al.* 2018).

## CONCLUSIONS

In this study, we propose the CDBMM to maximize the overall benefits of the check dam system and efficiently use water resources in the basin. For the first time, CDBMM considers water resource benefits of the check dam in the construction of the check dam systems and can obtain optimal check dam construction parameters (silt storage capacity and flood control capacity), benefits (production, aquaculture, irrigation, flood control and silt blocking) and costs (check dam engineering cost and spillway engineering cost) when the system benefits are maximized. All social, ecological and water resources benefits were converted into economic benefits to serve as a unified measure of benefits in CDBMM. The CDBMM was first applied to Si Jiagou Basin, with two pre-existing check dams, to study system construction parameters and benefits and find how overall benefits could be maximized. The main results were as follows:

The overall benefit of the entire Si Jiagou Basin was 338.83 million RMB, and the top three of the five benefits were the silt blocking, production and the aquaculture benefits. The flood control benefits and irrigation benefits came last. The construction costs were equivalent to 7.07% of the total benefits, and the check dam system benefits were very significant. Most importantly, the water resources benefits accounting for 45.40% of the total benefits, indicating that water resources benefits were significant and should be considered as the main income and influencing factors in the ecological construction. Production benefits still accounted for more than half and should also be considered as a key factor. The water resources benefits of the check dam system should not be neglected especially in a semi-arid area.

The total silt storage capacity was 247.8 × 10

^{4}m^{3}and the total flood control capacity was 355.7 10^{4}m^{3}. The top three check dams in terms of silt storage capacity were Nos. 7, 3 and 6, and the order of the top three rankings of flood control capacity was the same, indicating that the silt storage capacity and flood control capacity were consistent, and all distributed in the midstream and downstream basin. The original design was not very reasonable, and the design of the original*Ecd*and*Wcd*check dams needed to be modified.The total cost of the 10 check dam array represented 7.07% of the total benefits, though the overall benefit of the check dam system was significant. The larger the scale of the check dam, the greater the cost–benefit ratio. With the reduction of the check dam scale, the cost–benefit ratio was significantly reduced. Under conditions of safe operation, the construction of check dams in the basin can provide huge benefits.

The top three net benefits were obtained for dams No. 7,

*Wcd*and No. 3 which were all distributed in the downstream portion of the basin. The net benefits were significant. As the construction scale increased, the net benefit became more significant. The last three ranked check dams in terms of benefits were No. 8, No. 4 and No. 5 which were mainly distributed in the upper and middle reaches. Thus, it was necessary to build large- and middle-scale check dams in the basin to significantly increase the net benefits.In Si Jiagou Basin, there were 2 check dams and a 10 check dam construction potential. The design siltation period of the two pre-existing dams has been exceeded and they are still not full. At the current level of sedimentation, the two check dams would not be full in 10 years. In Si Jiagou Basin and other semi-arid areas, water resources are scarce, so the portion of water resources of greater quality should be used properly and efficiently instead of causing harm.

Water resources benefits were significant and important in terms of overall benefits and the ecological state of the basin, especially in this semi-arid area where it is necessary to fully consider the water resources benefits in the construction and formation of a check dam array. The CDBMM added water resource benefits to the analysis and used water and soil resources more efficiently, thereby potentially alleviating the water crisis in a semi-arid area under conditions of safe operation. The CDBMM method will provide ways for the sustainable development of a basin and the construction of water-saving and ecological check dam systems, and can further provide a reference for both check dam system planning and the system benefits analysis. One limitation of this study that may deserve further research was the construction time of the eight proposed check dams. Due to the different construction times of the upstream and downstream check dams, the construction of upstream check dams will reduce the amount of incoming water and sediment, and increase the siltation period of the downstream check dams, which will lead to changes in benefits. Therefore, the equations and restrictions for calculating the benefits must be adjusted accordingly. Studying the suitable construction time and the benefits of the eight check dams during the calculation period that maximize the benefits of the entire basin will be done next, using the new model with new objective functions and restrictions.

## AUTHORS CONTRIBUTIONS

Conceptualization: Y.G., X.Z. (Xinmin Zhang) and X.Z. (Xiaoyou Zhang); Data acquisition: J.T. and Y.G.; Funding acquisition: Y.G. and X.Z. (Xinmin Zhang); Formal analysis: D.L. and M.Y.; Investigation: Y.G., M.Y. and X.Z. (Xinmin Zhang); Methodology: X.Z. (Xinmin Zhang) and X.Z. (Xiaoyou Zhang); Visualization: R.H. and Y.G.

## FUNDING

This work was financially supported by the National Key Research and Development Plan (2017YFC0504704), the Young Creative Talents Support Program in Longyuan ([2014]93), the Water Resources Foundation Support Project of Gansu Provincial Water Resources Department ([2017]293), the Technology Promotion Projects of Water Conservation Research in Gansu (2014]223, [2015]55), Gansu Provincial Key Research and Development Plan (18YF1NA031) and 2019 Provincial Key Talent Project ([2019]39).

## ACKNOWLEDGEMENTS

The authors would like to thank Gansu Provincial Meteorological Bureau, Water Affairs Bureau of Qingyang City, Statistics Affairs Bureau of Qingyang City, the Soil and Water Conservation Bureau of Qingyang City and the Soil and Water Conservation Bureau of Xifeng District for providing climatic and water resources data.

## CONFLICTS OF INTEREST

The authors declare no conflict of interest.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.