## Abstract

To study the relationship between the effective utilization coefficient of irrigation water and canal system structure including the influences of various factors in irrigation districts, the fractal dimensions of each irrigation district by Horton's law were calculated by using canal data from 20 typical irrigation districts in Heilongjiang Province. The results showed that the fractal dimensions of the three-level irrigation districts were within the general range of the Horton water law. Of the four-level irrigation districts, only the fractal dimension of the Wutong River irrigation district was 3.037, which was beyond the upper limit of approximately 1.23%. Using gray relational analysis, the correlation degrees of four factors such as the effective irrigated area, the water-saving area ratio, the complexity of the canal structure, the channel density with the effective utilization coefficient of irrigation water and the fractal dimensions were all above 0.5 in the three- and four-level irrigation districts.

## INTRODUCTION

Israelsen (1950) first proposed the term ‘effective utilization coefficient of irrigation water’ as a measure of the extent to which water diverted from a river or abstracted from a well actually reached the farmer's field and contributed to crop growth (Perry 2011), and it is the primary indicator for assessing irrigation efficiency and potential. In addition, it is the basis data for regional scientific configuration and the development of water-saving irrigation development plans, which is an important basis for government departments to make macro-decisions (Wang *et al.* 2012). Domestic and foreign scholars have conducted extensive research on the effective utilization coefficient of irrigation water. Rodríguez-Díaz *et al.* (2008) applied benchmarking and multivariate data analysis techniques to conduct a cluster analysis of nine irrigation districts in Andalusia, Spain, and found that the unit water-use cost for farmers was significantly related to the effective utilization coefficient of irrigation water. Grashey-Jansen (2014) summarized the results of a study of soil water dynamics and determined that the effective utilization coefficient of irrigation water of precision irrigation and soil-specific irrigation increased with decreasing of irrigation amounts. Liu *et al.* (2013) found that the socio-economic development level (GDP) and the effective irrigated area (the farmland area where the land is relatively flat and there is a certain amount of water sources and irrigation facilities that can be normally irrigated in the normal year) have a significant effect on the effective utilization coefficient of irrigation water in an irrigation district. However, these studies primarily investigated the effective utilization coefficient of irrigation water from the aspects of economy, resources and irrigation methods, and there have been only a few studies on the influence of canal arrangement on the effective utilization coefficient of irrigation water.

Horton's law can accurately characterize the structural characteristics of a river network. In 1945, Horton proposed a scheme to quantify the classification of water systems, which then began a new stage of the study of river system morphology (Tarboton 1996; Saha *et al.* 2009; Gupta & Mesa 2014). Many scholars have used Horton's law to study the structure of artificial irrigation districts. For example, Qu *et al.* (2015) studied the effect of the fractal dimension of the Hetao irrigation system in Inner Mongolia on canal system water use efficiency; Huang *et al.* (2017) used fractal theory to analyze the relationship between the fractal dimension and the canal system water use efficiency of the Sharsen irrigation district; and Liu *et al.* (2005) used the box counting method to calculate the fractal dimension of the canal system in the Zhanghe irrigation district and demonstrated a correlation between the fractal dimension of the canal system and the irrigation modulus. These results validate the rationality of Horton's law as it is applied to the analysis of canal structure in irrigation. Although research on the fractal dimension of canals has made some progress, the theoretical system is still weak, and there is no clear research conclusion to improve irrigation canal structure to maximize the effective utilization coefficient of irrigation water. There are few studies on the primary influencing factors of the fractal dimension and effective utilization coefficient of irrigation water or on the contribution of each factor to the effective utilization coefficient of irrigation water and the fractal dimension. In this study, Horton's law was used to calculate the fractal dimension of each irrigation district to analyze the contribution rate of each influencing factor to the fractal dimension and the effective utilization coefficient of irrigation water. Based on cited literature and to improve the performance of utilization of the coefficient of irrigation, the current study aimed to achieve the following objectives:

- (1)
Calculate canal characteristic parameters and fractal dimensions using Horton's law.

- (2)
Determine the effect of the fractal dimension of the canal system on the effective utilization coefficient of irrigation water.

- (3)
Calculate the correlation between each factor and the effective utilization coefficient of irrigation water and the fractal dimension value.

## MATERIALS AND METHODOLOGY

### Classification of irrigation canal

In this paper, the canal system was classified according to the order of water diversion from the source: (1) directly from the water diversion channels for the first-level systems; (2) from the first-level branch channel water diversion channels for secondary systems; (3) from the second branch channel water diversion channels for the third canal system, and so on. Grading from top to bottom from the water source, which is simple and common, is not limited by the size of the last canal system and river course of the irrigation district or by the type and scale of irrigation district. The canal system at all levels should correspond to the trunk canal, branch canal, lateral canal and field canal.

### Horton's law and characteristic parameter calculation

*R*

_{c}) reflects the richness of the drainage system in the irrigation district, and the higher the development degree of the number and length of the canal network, the more developed the canal system: where

*N*

_{0}is irrigation district rank, and

*L*

_{g}is the length of the main canal in the irrigation district, km.

### Calculation of river water system fractal dimensions

*N*) and length (

*L*) in the same watershed vary with the river level (Veltri

*et al.*1996; Schuller

*et al.*2001), that is: where

*ω*is river level serial number,

*W*is the highest level of the river,

*N*is the number of rivers of

_{ω}*ω*level,

*L*

_{1}is grade 1 river average length, m, and

*L*is the average length of the

_{ω}*ω*river, m.

*ω*as the abscissa represents the value of

*R*

_{b}and

*R*

_{l}, respectively, that is: where

*R*is the water structure parameter (

_{x}*x*= b, l), and

*k*is the slope of the

_{x}*ω*-lg

*N*and

_{ω}*ω*-lg

*L*regression line, respectively.

_{ω}*D*formula:

Tarboton *et al.* (1988) argue that the larger the water system, the more complex the water system, and vice versa. This paper uses Equation (9) to calculate the canal fractal dimensions. The flow chart of the Horton's law calculation process is shown in Figure 1.

### Gray relational analysis (GRA)

To study the uncertainty problem of poor information and small samples, gray system theory can be used (Liu & Yang 2015). Gray relational analysis is an important part of gray system theory. Compared with statistical methods, this method can solve a problem when the sample size is small and information is scarce. The existing correlation quantification model includes the Deng correlation degree, the gray absolute correlation degree, the gray similarity degree, the *T*-type correlation degree and the gray slope correlation degree (Zhang *et al.* 2015). According to the advantages and disadvantages of the model analysis, this paper uses gray absolute correlation degree for analysis and operation.

- (1)
Dimensionless data processing

_{i}_{(k)}is the original number, and is the average of the original series.

- (2)
Calculate the correlation coefficient

- (3)
Calculate the degree of correlation/correlation dimension

## STUDY AREA AND DATA SOURCES

### Study area

Heilongjiang Province is located in northeastern China, and it is the highest latitudinal and the most eastern province, latitude 43°26′–53°33′, longitude 121°11′–135°05′. It has a temperate continental monsoon climate, and the province's annual average temperature is low. It is northwest and southeast of high terrain, the northeast and southwest are relatively low, and the province's total land area is approximately 47.3 million hm^{2}. Heilongjiang Province is China's agricultural province, with an arable land area of 11.73 million hectares, accounting for 9% of the national arable land. There are 8,312 completed irrigation districts, of which 20 have large-scale irrigation and 305 have a medium-sized irrigation district (Lan 2015). According to the survey of effective utilization coefficient of irrigation water in Heilongjiang Province in 2014, the average effective utilization coefficient of irrigation water in Heilongjiang Province in 2014 was 54%, lower than the national average (Fu *et al.* 2017). Therefore, it is important to utilize comprehensive methods to improve the effective utilization coefficient of irrigation water in Heilongjiang Province to develop water-saving agriculture, improve crop yield, and develop the national economy.

### Data sources

This paper is based on annual data from 107 large, medium, and small well-irrigated areas in Heilongjiang Province in 2014, and field research was conducted to classify the canal structure statistics to ensure the uniqueness of the statistical data and the reliability of the conclusions. On the basis of the difficulty of data acquisition and the uniformity of the distribution points of the sample irrigation districts, typical irrigation districts for the three-level irrigation districts were selected. In addition, ten typical irrigation districts for the four-level irrigation districts were also selected. The basic data of the irrigation districts were mainly from the farm water management center of Heilongjiang Province, and the *Heilongjiang Province Water Conservancy Construction Statistical Yearbook*. The layout of the canal system and the typical irrigation districts in Heilongjiang Province are shown in Figure 2. The measured data of effective utilization coefficient of irrigation water, effective irrigated area, water-saving area ratio (ratio of irrigated area to total irrigated area using water-saving irrigation technology) and so on in different irrigation districts are shown in Table S1 (available with the online version of this paper).

## RESULTS AND DISCUSSION

### Structural characteristics of canal system

Using Equations (3) and (4) to calculate the canal branching ratio and length ratio was difficult, so the number of channels and the average length were transformed into a logarithmic function, while putting the original geometric relationship into a linear relationship, to obtain the regression linear parameters in Table S2 (available with the online version of this paper). From Table S2, the fitting degree of the canal level and logarithmic function was good (*R*^{2} > 0.9), so the fractal dimension of the calculated irrigation district was high and could be further analyzed.

For the lg*N*-*ω* and lg*L*-*ω* fitting curves, the absolute value of the slope was taken as the antilogarithm, and the branching ratio and the length ratio were obtained. The fractal dimensions of each irrigation district were obtained according to Equation (9). The results are shown in Table S2. Because of the similarity of the canal structure in the irrigation district, the branching ratio, length ratio, and fractal dimension of the three-level irrigation districts and the four-level irrigation districts were not very different. Previous studies showed that the branching ratio of the river network was between 3 and 5, the length ratio was 1.5 to 3, and the fractal dimension was in the range of 1 to 3 (Kirchner 1993; Xie *et al.* 2007; Wang *et al.* 2012). From Table S1, it is evident that the branches of the Cangliang irrigation district, Hamatong irrigation district, Xingfu irrigation district, and Hongqi irrigation district are relatively large, which indicate that the internal canal structure of the irrigation districts is more complicated and influenced by human factors. The length ratio and fractal dimension were within a reasonable range, however, indicating that the typical irrigation districts in Heilongjiang Province are within the general range of Horton's law, which is that the distribution of the canals in each irrigation district conforms to the natural self-organization optimization structure. However, the length ratio of most irrigation districts was close to the upper bound of the general range, resulting in a smaller fractal dimension, because the proportion of ineffective channels in the irrigation district was large. Therefore, in the future, by increasing the length of the lower channel of the canal system, or shortening the length of the upper channel, the long channel can be changed to a short channel, so that the canal length ratio can be properly reduced.

### Effect of fractal dimension of canal system on effective utilization coefficient of irrigation water

The canal characteristics reflect to a certain extent whether the drainage system is reasonable. Therefore, it was important to improve the effective utilization coefficient of irrigation water. The correlation between the effective utilization coefficient of irrigation water and the fractal dimension of different irrigation districts was analyzed (Figure 3). The results exhibited that the effective utilization coefficient of irrigation water and the distribution of the fractal dimension of the three-level irrigation districts show a parabolic shape with an upward opening (*R*^{2} = 0.7833), and with an inflection point (1.790, 0.416). The effective utilization coefficient of irrigation water and distribution of fractal dimension of the four-level irrigation districts also showed a parabolic shape (*R*^{2} = 0.5893) with an inflection point (2.329, 0.405), demonstrating that with increasing fractal dimension, the effective utilization coefficient of irrigation water in both the three-level irrigation districts and four-level irrigation districts decreased first and then increased. Compared with the four-level irrigation districts, the effective utilization coefficient of irrigation water in the three-level irrigation districts was more obvious with the fractal dimension. In a certain range, when the effective utilization coefficient of irrigation water decreased with increasing branching ratio, some of the three-level irrigation districts and four-level irrigation districts were relatively ineffective in function, which is not conducive to the improvement of the effective utilization coefficient of irrigation water. Therefore, the effective utilization coefficient of irrigation water must be improved through the transformation of the irrigation district, increasing the ability of water conveyance and reducing the loss of canal water.

### Effects of different factors on effective utilization coefficient of irrigation water and fractal dimension

The correlation degree and rank of each factor and effective utilization coefficient of irrigation water and fractal dimension are shown in Tables S3 and S4 (available online). The correlation between the factors and the effective utilization coefficient of irrigation water is above 0.5 in Table S3, which demonstrates that the effective utilization coefficient of irrigation water was closely related to the effective irrigation district, the water-saving area ratio, the complexity of canal structure, and the channel density. In the three-level irrigation districts, the correlation coefficient between the canal density and effective utilization coefficient of irrigation water was the highest (0.700), followed by that of the effective irrigation district and the complexity of the canal structure (0.695 and 0.606, respectively), and then the water-saving area ratio (0.556). In the four-level irrigation districts, the correlation coefficient between the water-saving area ratio and the effective utilization coefficient of irrigation water was the largest (0.730), followed by that of the effective irrigation district and channel density (0.700 and 0.675, respectively), and, finally, by the complexity of the canal structure (0.605). Based on the ranking of the correlation degree, to improve the effective utilization coefficient of irrigation water, the appropriate effective irrigation district should be selected; then the channel density and the water-saving area ratio should be considered; and, finally, the appropriate drainage structure complexity should be selected.

From Table S4, the correlation degree between each factor and fractal dimension was above 0.5, which indicated that the fractal dimension was closely related to the effective irrigated area, the water-saving area ratio, the complexity of channel structure, and the channel density. For the three-level irrigation districts, the correlation degree between canal structure complexity and fractal dimension value was the largest (0.711), followed by that of the channel density and water-saving area ratio (0.613 and 0.585, respectively), and the correlation between the effective irrigation district and the fractal dimension was the smallest (0.575). For the four-level irrigation districts, the correlation coefficient between the effective irrigation district and the fractal dimension was the largest (0.763), followed by the water-saving area ratio and the complexity of the canal system (0.602 and 0.583, respectively), and the correlation between channel density and fractal dimension was the smallest (0.546). According to the order of the correlation degree, the complexity of the canal structure should be considered, the appropriate water-saving area ratio and effective irrigation district can be selected, and then the channel density can be considered.

## CONCLUSION

In the 20 typical irrigation districts of Heilongjiang Province, the canal structures of the three-level and four-level irrigation districts are in line with the requirements of the Horton water law, indicating that the irrigation canal system somewhat conforms to self-organization structure. However, problems with the channel level and channel length of the existing irrigation still exist, and canal reconstruction may be appropriate in the future to adjust the numbers of the branch canals and lateral canals, the length of the main canal and increase or decrease the level of the canal system, so that the branching ratio and length ratio are within a reasonable range.

At present, there are some redundant channels in both the three-level and four-level irrigation districts, and the function is low in the process of water supply and distribution, which leads to the small effective utilization coefficient of irrigation water. There is much room for improvement; the length of the lateral canal can be reduced, while also increasing the number of branch canals to reduce the fractal dimension and improve the effective utilization coefficient of irrigation water.

For the three-level irrigation districts, the rationality of the canal system structure is the key to water supply and distribution. For the four-level irrigation districts, water-saving measures are important factors in improving the effective utilization coefficient of irrigation water. Therefore, they should be reformed according to the characteristics of different levels of the irrigation districts, so as to effectively improve the effective utilization coefficient of irrigation water in irrigation districts in Heilongjiang Province. But the mechanism and factors influencing the effective utilization coefficient of irrigation water are more complex. Future research should consider additional factors, and more in-depth studies of effective utilization coefficient of irrigation water should be conducted to maximize the effectiveness of canal irrigation water distribution systems.

## CONFLICT OF INTEREST

The authors declare that they have no conflicts of interest.

## ACKNOWLEDGEMENTS

The authors are grateful for the basic study data provided by the Jiansanjiang Administration of Heilongjiang Province in China.

## FUNDING

This study is supported by the National Natural Science Foundation of China (No. 51579044, No. 41071053, No. 51479032), National Key R&D Program of China (No. 2017YFC0406002), Natural Science Foundation of Heilongjiang Province (No. E2017007), Science and Technology Program of Water Conservancy of Heilongjiang Province (No. 201319, No. 201501, No. 201503).