1 Introduction

All living things in the earth require water to grow and reproduce. Water supplies are also vital for all production and economic activities. However, only 1% of the freshwater resources can be used by natural ecological systems and socio-economic systems(Fan et al. 2023; Zheng and Sun 2023). More than a billion people currently live in water-scarce regions, and as many as 3.5 billion may experience water scarcity by 2025 due to changing climate (International Development Association 2019). A global effort is growing to deal with water scarcity, such as China (Niu et al. 2022), France(Beaumais and Crastes dit Sourd 2024) and Australian(Zhao et al. 2024).

There are various solutions that been applied in different countries. From a technology perspective, water conservation, wastewater recycle, and other technologies have been developed to improve water use efficiency (Ehrensperger et al. 2015; Huang et al. 2016). Considering the unbalanced water resources distribution, it is essential to analyze the water resources transfer among multiple regions and explore hidden mechanisms to support future water resources management.

Physical water transfer and virtual water (VW) transfer are two major water resource transfer approaches and have attracted widespread attention (Kumar et al. 2008; Dinesh Kumar 2018). To alleviate water shortage problems in arid regions, a number of physical water transfer projects have been developed in United States (He et al. 2017), China (Sun et al. 2023), and other countries. However, the physical water transfer project is challenging in many ways, such as huge construction cost (Flyvbjerg 2014), operation (Rani and Moreira 2010), and complex physical, chemical, hydrological and biological changes to the receiving areas (Tang et al. 2014; Zeng et al. 2015). Therefore, VW transfer among multiple regions is essential to gain a better understanding of sustainable water resources management.

VW transfer, which means the embodied water transfer during the production and trade process, has gained more attention recently. Multi-region input–output (MRIO) model was regarded as a useful top–down technique to attribute pollution or resource use to final demand in a consistent framework. MRIO models were widely applied and used to link production and consumption while accounting for the direct and indirect relationships between different economic activities. For example, Zhao et al., calculated the physical and virtual water flows at the provincial level in China for the year of 2007 through an environmental extended MRIO model, which showed that VW flows accounted for 35% at the national level (Zhao et al. 2015). Jiang et al., explored the impact of virtual water flows of large river economic belts on the regional water distribution(Jiang et al. 2024). Dang et al., presented a comprehensive assessment of VW flows within the United States based on intranational food transfer empirical data (Dang et al. 2015). Ali et al., analyzed the carbon emissions and water use embodied in international trade in Italy through the MRIO table and World-Input–Output Database (Ali et al. 2018). To estimate the water footprint of each province in China, Chen et al., applied interregional input–output model and found the significant diversity of various provinces in terms of water footprints (Chen et al. 2017). Lutter et al., applied a product-level MRIO model EXIOBASE to track the distribution of water use along product supply chains within and across the EU-27 countries.

When analyzing the VW transfer network, Ecological Network Analysis (ENA) was widely-used to quantify not only the direct but also the indirect and integral flows in the transformation process (Wang and Chen 2016). Ecological Network Analysis (ENA) is a method used to study the interactions within an ecological system by representing it as a network of nodes (species, habitats, or ecosystems) connected by links (energy or matter flows). An information-based ENA model was developed and applied in Heihe River Basin in the arid region of northwest China (Fang and Chen 2015). To be specific, the network control analysis was aimed at investigating the dominant sectors and pathways for VW network, while the network utility analysis was performed to identify the mutual relationships among different sectors. Yang et al., used ENA to analyze global VW trade to identify the quantitative control or dependency relations and describe the mutual relationship between two regions (Yang et al. 2012). Besides the VW network in the entire socio-economic system, ENA could be used to in VW network in smaller scale. For example, Guo et al., analysed the VW network within China’s electric power system and found that the northern grid was always input-oriented grid and the central grid was the output-oriented grid during 2007–2012 (Guo et al. 2016).

Some researchers explored more detailed characterizations of VW through conducting VW decomposition analysis in different aspects. Three types of water resources were valued in VW analysis, including Green water, Blue water and Grey water. For example, Tian et al., provided insights into China’s water footprint and virtual water trade using three specific water through MRIO analysis at national and sectoral analysis levels (Tian et al. 2018). Tocados-Franco et al., examined the water transfer induced by energy consumption and design the water management policies (Tocados-Franco et al. 2024). Qian et al., explored the driving factors of agricultural VW trade between China and the Belt and Road Countries with a consideration of three VW. It was found that the trade with Southern Asia and Eastern Asia caused the blue and gray water losses, while gray water savings were due to the trade with Southern Asia and Eastern Europe (Qian et al. 2019). Some studies focused on one specific industry and investigated the VW embodied in the economic activities in the industry. For instance, a node-flow model was developed to investigate the VW embodied in the electricity transmission network (Zhang et al. 2017). A Water Rights trading and VW Export Compensation Model was established to analyze the relationship of these two measures and optimized to coordinate economic, environmental benefits, and food security, as well as to improve water allocation efficient (Wang et al. 2017).

However, the majority of previous studies failed to analyze the integral VW transfers through multiple path-lengths, which is essential for the water resources management. In addition, there are complex interactions among various economic sectors, which will further affect the VW transfers among various sectors and regions. The quantification of the interactive effects is also ignored by previous studies, which can provide more insights about the outcomes of different water resources allocation plans and support the realization of water stress relief in multiple regions through the most economic ways. Therefore, a detailed VW transfer through multiple path-lengths and interaction analysis is desired.

The objective of this study is to develop a Disaggregated Multi-region Virtual Water Flow and Interaction (DrWIn) model to facilitate VW transfer analysis among multiple regions and along multiple path-lengths. In detail, the DrWIn will be used to obtain the VW inflow, outflow and transfer balance for each region in the study system for identifying the hotspot of water resources management. By analyzing these flows, policymakers and water managers can develop strategies to use water more efficiently, particularly in water-scarce regions. In addition, factorial analysis will be conducted to quantify the impacts of the identified industry and region as the main factors and their interactions to further reveal the hidden mechanism. The developed model is applied to a detailed case study of China to illustrate its potential benefits. By optimizing water use and promoting sustainability, VW research contributes to achieving these SDGs (e.g., clean water sanitation (SDG 6), responsible consumption and production (SDG 12)) goals. The obtained results provide a solid scientific basis for identifying the key industries and regions across a multi-region study system and supporting industrial water resource utilization management in the future.

2 Methods

2.1 Disaggregated Multi-Region Virtual Water Flow and Interaction Model

A DrWIn model is developed to facilitate the analysis of VW inflow, outflow, and transfer balance, as well as reveal the related interactions. In a multi-region system, each region consumes freshwater to support its production activities. Meanwhile, VW embodied in the commodities from other industries and regions are consumed through intermediate utilization. Considering the variances that exist among various regions, the freshwater consumption, VW inflow and outflow of various regions are meaningful to understand comprehensive water utilization conditions. Then, the VW transfer balance can be calculated to further reveal the VW transfers among various regions. Considering the complexities of the transfer network, a factorial analysis is integrated to investigate the interactions among major impact factors, which will provide an in-depth basis for the water resources management.

The DrWIn model is based on the Leontief I-O framework (Leontief 1986; Liu et al. 2018b). In an economy with N regions and each region has K sectors, the direct VW transfer network can be calculated through Eqs. (1) and (4).

$$\mathbf{X}={[\mathbf{I}-\mathbf{A}]}^{-1}\mathbf{f}$$
(1)
$$\mathbf{F}\mathbf{W}+\varepsilon \mathbf{Z}=\varepsilon \mathbf{X}$$
(2)
$$\varepsilon =\mathbf{F}\mathbf{W}{[\mathbf{X}-\mathbf{Z}]}^{-1}$$
(3)
$$\mathbf{V}\mathbf{W}=\varepsilon \mathbf{Z}$$
(4)

where X is the total output column vector,

I is the unit vector,

A is the direct intermediate inputs coefficients matrix,

f is the final demand matrix,

FW is the industrial freshwater consumption matrix,

\({\varvec{\upvarepsilon}}\) is the embodied VW coefficient matrix,

Z is the value flow matrix in the I-O table,

VW is the VW transfer matrix.

When analyzing the VW outflow from a specific region r, the \(\mathbf{V}{\mathbf{W}}^{r}\) can be obtained through Eqs. (5) and (7).

$$\mathbf{F}{\mathbf{W}}^{r}+{\varepsilon }^{r}\mathbf{Z}={\varepsilon }^{r}\mathbf{X}$$
(5)
$${\varepsilon }^{r}=\mathbf{F}{\mathbf{W}}^{\mathbf{r}}{[\mathbf{X}-\mathbf{Z}]}^{-1}$$
(6)
$$\mathbf{V}{\mathbf{W}}^{r}={\varepsilon }^{r}\mathbf{Z}$$
(7)

where \(\mathbf{F}{\mathbf{W}}^{r}\) is the freshwater consumption matrix which only include freshwater consumption in region \(r\),

\({\varepsilon }^{r}\) is the embodied VW coefficient matrix enabled by freshwater consumption in region \(r\),

\(\mathbf{V}{\mathbf{W}}^{r}\) is the VW outflow matrix enabled by industrial freshwater consumption in region \(r\).

Based on the \(NK\times NK\) matrix \(\mathbf{V}{\mathbf{W}}^{r}\), the VW outflow from region r to industries i in region r and to region s can be calculated through Eqs. (8) and (9).

$$EV{W}^{ri}=\sum_{a}^{NK}V{W}_{a,b}^{r}$$
(8)
$$EV{W}^{rs}=\sum_{b}^{K}\sum_{a}^{NK}V{W}_{a,b}^{r}$$
(9)

where \(EV{W}^{ri}\) is the VW outflow from region r to industries i,

\(EV{W}^{rs}\) is VW outflow from region r to region s,

a represents the set of rows in VW,

\(b\) represents the set of columns in VW.

After the calculation of VW outflow from all N regions, the VW inflow to region s can be obtained through Eq. (10).

$$AV{W}^{s}=\sum_{n=1}^{N}EV{W}^{ns}$$
(10)

where \(AV{W}^{s}\) is the VW inflow to region s.

In addition, the VW transfer balance between region r and region s can be obtained through Eq. (11).

$$VW{B}^{rs}=EV{W}^{rs}-EV{W}^{sr}$$
(11)

where \(VW{B}^{rs}\) is the VW balance between region r and region s. A positive \(VW{B}^{rs}\) means there are net VW transfer from region r to region s, while a negative \(VW{B}^{rs}\) means there are net VW transfer from region s to region r.

With the built VW transfer network, ENA is further adopted to analyse the VW transfers of different path lengths among the two sectors. The integral VW transfer in the multi-region system can be calculated through Eqs. (12) to (15).

$$T{W}_{a}=V{W}_{ba}+F{W}_{a}$$
(12)
$${g}_{ab}=V{W}_{ab}/T{W}_{b}$$
(13)
$$\mathbf{N}={\left(\mathbf{G}\right)}^{0}+{\left(\mathbf{G}\right)}^{1}+{\left(\mathbf{G}\right)}^{2}+...{\left(\mathbf{G}\right)}^{\infty }={\left(\mathbf{I}-\mathbf{G}\right)}^{-1}$$
(14)
$$\mathbf{I}\mathbf{V}\mathbf{W}=diag\left(\mathbf{T}\mathbf{W}\right)\mathbf{N}$$
(15)

where \(T{W}_{a}\) is the VW that into a,

\({g}_{ab}\) is the dimensionless input-oriented intercomponent flow from a to \(b\),

\(\mathbf{N}\) is the dimensionless integral flow intensity matrix,

\(\mathbf{I}\mathbf{V}\mathbf{W}\) is the integral VW matrix.

In a multi-region VW transfer systems, the consumption of each industry can be changed to different levels, which will further affect the entire VW transfers among various sectors and regions. The impacts of single sector management can be obtained through a sensitivity analysis. But when planning the water utilization of multiple industries, the relationship between the single sector water utilization and total VW transfer are nonlinear. Thus, it is essential to further investigate the impacts of single factors and their interactions by integrating a factorial analysis. Factorial analysis is a powerful way to study the interactions of different levels of multiple factors and their impacts on system performance (Zhou et al. 2016a, b; Liu et al. 2018a). Therefore, a factorial analysis is introduced to the model to investigate the system performance under different scenarios. In the DrWIn model, a \({2}^{k}\) factorial design is adopted to quantify the complexity of the VW transfer system. Considering that the inputs of the model and the developed model are deterministic, replicates are not necessary in this study. Due to the limitation of the degree of freedom, the error item is considered as the highest order of interaction. The factorial analysis was implemented in the software Designer Expert (Zhou et al. 2016a, b; Liu et al. 2018a).

2.2 Case Study and Data Sources

Severe water shortages, growing population, and rapid economic development all affect the water resources of China. The overall water resources in China (i.e., 2,000 m3 per person) are above the level where water stress starts (i.e., 1700 m3 per pserson) (Jiang 2009). However, 80% of the water is in southern China, leading to eight northern provinces suffering from acute water scarcity, four from scarcity, and a further two (i.e., Xinjiang and Inner Mongolia) are largely desert (Yang and Zehnder 2001). Moreover, these 12 provinces account for 38% of China’s agriculture, 46% of its industry, 50% of its power generation, and 41% of its population (Yang et al. 2003). Considering the unbalanced water resources and water demand, China was chosen as the case study.

The total amount of water resources in Guangdong Province (i.e., 202.7 billion m3) ranks at the top of all provinces in China. It is of particular interest to investigate how to relieve water stresses in northern provinces through future water utilization in Guangdong Province. Guangdong Province is also the leader in stimulating the Chinese economy, with its population and entire GDP both ranked first in China (Zhai et al. 2018). Thus, it is essential to analyze the VW transfer enabled by Guangdong Province and explore economic efficient pathways to relieve water stresses in China. In this study, Guangdong Province is investigated as in detail.

In this study, data collection is mainly from two parts, including the Chinese MRIO tables and the industrial freshwater consumption of different industries in different provinces. The freshwater consumption of different industries in the year of 2012 were used, which is taken from China Statistical Yearbook (National Bureau of Statistics 2012). To make the data consistent, the MRIO table in the year of 2012 used in this study was compiled by Mi's research (Mi et al. 2017), which includes 30 provinces (excluding Tibet, Hong Kong, Macau, and Taiwan provinces due to limited data) and 30 economic sectors for each province, as shown in Tables 1 and 2. Meanwhile, due to the data limitation, the freshwater consumption data of industrial freshwater consumption (i.e., I2-I23) is considered in this study.

Table 1 Province definition
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Table 2 Sector definition
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3 Results and Discussion

3.1 Virtual Water Outflow from Guangdong Province

Using the DrWIn model developed in this study, the VW outflow from Guangdong Province is analyzed first, as shown in Figs. 1 and 2. The freshwater consumption by 22 industries in Guangdong Province, VW outflow to 30 provinces, and VW outflow to 30 sectors in Guangdong are revealed in Fig. 1. The integral VW outflow to the other 29 provinces are shown in Fig. 2.

Fig. 1
figure 1

VW outflow from Guangdong Province

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Fig. 2
figure 2

Integral VW outflow from Guangdong Province

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As shown in the left part of Fig. 1, the freshwater consumption of 22 industries varies significantly, which reflects the different roles of various industries in water utilization. It can be seen that the freshwater consumption of I22 (i.e., 3221 Mt) is much higher than other industries, which is in line with the fact that the Electricity and hot water production and supply industry take the majority of freshwater from nature. Besides, I19, I10, I7, and I12 also rank at the top of the 22 industries, with their freshwater consumption being more than 1200 Mt. Among them, I19 and I12 consume considerable freshwater due to their considerable economic production activities, which also brings relatively high GDP. In contrast, the freshwater consumption of I10 and I7 are mainly caused by high unit water consumption. The freshwater consumption in the rest of the industries are all lower than 1000 Mt and the total freshwater consumption in these 17 industries only accounts for 39 percent of the total freshwater consumption, indicating that the freshwater consumption in Guangdong Province is highly concentrated in the afore-mentioned 5 industries. Therefore, I22, I19, I10, I7, and I12 will be emphasized in the later factorial analysis.

The middle part of Fig. 1 shows the VW outflow from Guangdong Province to all provinces. It is illustrated that the VW within Guangdong Province is the largest, which is almost 10 times of the VW outflow to the second largest province. When comparing the VW outflow to the other 29 provinces, it can be seen that the VW outflow to P10 is remarkable, with the VW outflow being 1055 Mt. Upon further investigation, P10 imported large quantities of commodities from I18, I19, and I12 in Guangdong Province, since the economic structure of these two provinces are similar. In addition, I18, I19, and I12 are all water consumption-intensive industries in Guangdong Province, resulting in the large VW outflow from Guangdong Province to P10. A similar phenomenon can be observed in P11, which ranks at third of the 30 provinces. There are other four provinces that have more than 400 Mt VW outflow from Guangdong Province, including P16, P18, P9, and P20. Among them, P18 and P9 have a similar industrial structure with Guangdong Province, which facilitated imports from light industries. P18 and P20 are driven mainly by heavy industry, such as Metallurgy, Machinery, and Electricity Industry. Although the industrial structures of P18 and P20 are different from Guangdong Province, they are located closely to each other, which facilitates their VW transfers through various industries.

When disaggregating the VW in Guangdong Province into sectors, the VW outflow to the detailed 30 sectors are shown in the right part of Fig. 1. It can be seen that the VW outflow to I11 is the highest, with the VW outflow being 1342 Mt. According to the industrial freshwater consumption analysis, the freshwater consumption of I11 is limited. This result demonstrates that the majority VW of I11 is obtained through the consumption of intermediate inputs from other sectors, including I12, I 26, and I17. It is also worth noting that I17 and I26 all rank at the top of the VW outflow from Guangdong. The close relationships among these sectors are another reason for their relatively high received VW outflow. This finding further reveals the VW transfers from Guangdong Province to P18 and P20, since the main imports from Guangdong Province are I11, I13, I17, and I26.

Figure 2 shows the integral VW outflow from Guangdong Province. The integral VW outflow to Guangdong Province (i.e., 205201 Mt) is still the largest. Due to the wide difference between this number and the integral VW outflow to other provinces, only the 29 integral VW outflow to other provinces are reflected in Fig. 2. When comparing the ranking between VW and integral VW outflow to different provinces, it can be seen that most provinces with higher VW outflow also obtain higher integral VW outflow, such as P10, P11, P16, and P9. Besides these provinces, P1, P6 and P23 are observed to obtain relatively high integral VW outflow from Guangdong Province, which indicates that the VW of different path lengths may be significant. It is worth mentioning that the standard deviation of integral VW outflow to different provinces is much smaller than the standard deviation of VW outflow to different provinces. This result illustrates that the VW transfers among various provinces are intensified by the multiple path-lengths transfers, which further proves the necessity of considering integral VW transfer.

3.2 Virtual Water Inflow to Guangdong Province

Both the VW and integral VW inflow to Guangdong Province are shown in Fig. 3. In each subfigure, the various provinces are marked with different colors. The trajectories, which have the same color with the source provinces, represent the VW transfers from source provinces to Guangdong Province. The numbers and percentages shown in the edges of the figures represent the amount and proportion of VW inflow from different provinces. In addition, the VW transfer amounts can be seen more intuitively through the width of the trajectories.

Fig. 3
figure 3

(a)VW and (b) Integral VW inflow to Guangdong Province from other 29 provinces

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As shown in Fig. 3(a), the VW inflow to Guangdong Province from P10 is the largest, which is also the only province that provided more than 1000 Mt VW transfer to Guangdong Province. Based on the analysis in Sect. 3.1, it can be seen that the VW transfers between Guangdong Province and P10 are frequent from either direction. Following P10, the VW inflow from P13, P18, and P20 are also noteworthy. This result further demonstrates that the VW transfers among provinces that are located nearby are more likely to happen. Figure 3(b) shows the integral VW inflow to Guangdong Province from different provinces. It can be seen that the basic trends of these two sub-figures are similar. For example, the trajectories from P10 to Guangdong Province are the most striking and the trajectories on the upper-right part (e.g., P4 and P5) are the narrowest. On the other hand, there are detailed differences among the VW transfer and integral VW transfer. Except P10, the provinces that rank on the top 5 of these two analyses are all different. To be specific, P9, P17, P14, P12 rank second to fourth respectively when analyzing the integral VW inflow to Guangdong Province. The above four provinces are not geographically close to Guangdong Province and they did not obtain high VW from Guangdong Province, illustrating that Guangdong Province imports abundant commodities from these provinces. This result is also related to the economic structures. For instance, P17 is the transition region of northern wheat growing area to the southern wheat growing area, leading to its considerably high outputs from the agriculture sector. The outputs of the agriculture sector in P17 can then be transferred to Guangdong Province to be processed, which is one potential VW transfer approach.

There are provinces that provided limited VW inflow to Guangdong Province. Both VW and integral VW inflow to Guangdong Province from P1, P2, P28, and P29 are the lowest. Specially, the VW inflow from P28 to Guangdong Province is only 9 Mt. Considering water resources availability, these four provinces are all suffering severe water shortage. Thus, the low VW inflow from these provinces to Guangdong Province are understandable and acceptable.

3.3 Virtual Water Transfer Balance

According to the above VW inflow and outflow results of Guangdong Province, the VW transfer balances between Guangdong Province and other provinces are calculated from both direct and integral perspectives, as shown in Fig. 4(a) and (b). In these two sub-figures, the warm colors represent positive values, which means that the province obtained net VW transfer from Guangdong Province and cold colors represent negative values, which means that the province provided net VW transfer to Guangdong Province. In addition, Fig. 4(c) shows the total water resources in different provinces in 2012, which can reflect the water resources availability and water stress.

Fig. 4
figure 4

The VW balance between Guangdong Province and other provinces and the total water resources in different provinces. (a) shows the direct VW balance between Guangdong Province and other provinces, (b) shows the integral VW balance between Guangdong Province and other provinces, (c) shows the total water resources in different provinces

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When comparing Fig. 4(a) and (b), we can see that there are more provinces marked with cold colors in Fig. 4(a), indicating that Guangdong Province provided more net VW transfers to other provinces through the integral VW network. To be specific, eleven provinces provided net VW flow to Guangdong Province with consideration of direct VW transfers, while five provinces provided net VW flow to Guangdong Province when taking integral VW transfers into account. Therefore, the integral VW transfers among Guangdong Province with other provinces should be emphasized to release the water stresses in China, since it can provide more VW flow to other provinces.

P11, P1, P23, P15, and P6 rank at the top five for the direct VW balance. According to Fig. 4(c), only P1 and P15 are water-deficient provinces. Other provinces that received net VW flow from Guangdong Province are rich in water resources. A different phenomenon can be observed in the integral VW balance analysis. Guangdong Province provides high net integral VW flow to four water-deficient provinces, including P1, P2, P21, and P4. For the provinces that have a negative VW balance with Guangdong Province, the majority of them are water-rich provinces, as shown in Fig. 4(a) and (b).

However, P9 provide a net VW transfer to Guangdong Province when considering direct VW and integral VW transfers. P9 is one of the most serious water-deficient provinces, with its annual water resources being 3390 Mt. Upon further investigation, the other provinces that suffer severe water shortage (i.e., P3 and P16) also receive limited VW from Guangdong Province. Since the water resources availability is high in Guangdong Province, the VW transfer from Guangdong Province to water-deficient provinces should be improved in future planning. Therefore, the integral VW transfer from Guangdong Province to P3, P9, and P16 will be further investigated through factorial analysis.

3.4 Water Consumption Interaction Analysis

According to the above analysis, I22, I19, I10, I7 and I12 are the largest consumers of freshwater in Guangdong Province and the water consumption in these five industries has great impacts on the VW outflow from Guangdong Province. Thus, the freshwater consumption of these five industries are chosen as the investigated factors in this study. The integral VW from Guangdong Province to P3, P9, and P16 are chosen as the response in the factorial analysis.

Figure 5 shows the t-value of main factors, second and third order interactions for integral VW transfer from Guangdong Province to P3, P9, and P16. In addition, all of the factors and interactions are ranked according to their significance. It should be noted that the significance levels of main effects are much higher than the significance levels of interactions. Also, the impacts of main factors are all positive, which means that the increase of the main effects will result in an increase of the responses in the three factorial analyses. These results further illustrate that the increase of the freshwater consumption in the five identified industries will provide more VW transfers to the three water-deficient provinces. When comparing the results of the three provinces, it is found that the significance level of one effect varies. To be specific, the increase of the freshwater consumption in I22 will bring more benefits to P3, while P9 will obtain the largest benefits when the freshwater consumption in I10 and I19 are increased.

Fig. 5
figure 5

The t-value of significant factors for integral VW from Guangdong Province to (a) P3, (b) P9, and (c) P16

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The interactions, especially high order interactions, are not as obvious as the main effects. But in real-world cases, a small disturbance may bring a serious consequence. In this study, all of the second order interactions have negative effects and all of the third order interactions have positive effects. To fully represent the differences among various interactions, the contributions of main effects, second and third order interactions in terms of the integral VW from Guangdong Province to P3, P9, and P16 are listed in Table 3. It can be seen that the interaction between I10 and I22 are important for the integral VW received by P3, as well as the interaction between I19 and I22. In contrast, the integral VW received by P9 is more sensitive to the interaction between I19 and I22. To further explain the interaction effects, Fig. 6 presents the interaction plot of I19 and I22, with the integral VW from Guangdong Province to P9 being taken as the response. When I22 is at high level, the integral VW achieved by P9 will increase along with I19 at a lower rate. When I22 is at low level, the increase rate of the integral VW from Guangdong Province to P9 is relatively high. This result illustrates that the effect of I19 will change when I22 is at different levels, which further demonstrates why the effect of their interaction is negative. For future water utilization planning, the interactions among various industries should be considered to increase the VW transfer from water-rich provinces to water-deficient provinces effectively.

Table 3 The contribution of main effects, second and third order interactions
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Fig. 6
figure 6

The interaction between I19 and I22 in terms of the integral VW achieved by P9

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4 Conclusion

In this study, a Disaggregated Multi-Region Virtual Water Flow and Interaction (DrWIn) model has been developed to facilitate VW transfer analysis among multiple regions. In addition, a factorial analysis about the identified industry and region have been conducted to quantify the impacts of main factors and their interactions. The application to China has illustrated the applicability and superiority of the DrWIn model.

It is found that the frequent direct VW transfer can be enabled by two different approaches. The most common approach is among different regions that share a similar economic or industrial structure. Another approach is through the neighboring geographical locations. However, the total water resources in a certain region do not vary significantly, which means that the second VW transfer approach cannot relieve the water stresses effectively. The integral VW flow among regions that are not located closely is much higher than the direct VW flow through the multiple path-lengths transfers. In addition, the VW transfer balances among different provinces are more even when considering integral VW transfers. Specially, the water-deficient provinces receive more VW through commodity imports and exports with other provinces. Therefore, more trades among water-rich and water-deficient provinces should be encouraged to enable the VW transfers.

Another major finding of this study is the interaction effects of the freshwater consumption of different industries. The results highlighted the importance of the interaction effects, since they can affect the performances of different strategies and policies. Specially, the interaction effects of any two industries are negative, indicating that the high freshwater consumption in two industries are not the best choice. In China, water resources are limited and precious. The increase of freshwater consumption in one industry will bring more costs to the industry, as well as higher indirect costs in other industries and the entire economy. Therefore, the significant negative interaction effects must be taken into account when optimizing the future water resources allocation.

In future studies, both direct and indirect costs of freshwater consumption will be considered to reflect the economic impacts of different water resources utilization strategies. Also, due to the data limitation, only industrial freshwater consumption is considered in this study. More up-to-date analysis will be conducted based on this model when the data is available.