Are we facing water scarcity?
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1 Nomenclatures
Variable symbols Definitions
Maximum population capacity
Population at t time
population growth rate
Industry water consumption of i sector at i time
Output in year t of the i-th sector
Output in year t i-th sector of industrial output value accounting for the proportion of
Total water consumption in year t of the i-th sector
T year industrial output
Industrial water recycling rate in year t of the i-th sector
Water consumption yuan output value in year t of the i-th sector
Industrial water utility
Economies of scale
Industrial Structure Effect
Effect of water industry strength
ρ Multiplying Water utilization ratio（depth of rainfall）
Fir Multiplying irrigation
μ Multiplying irrigation return coefficient
（storage coefficient）
Qfix Irrigation norm
SSEW Water supply from sewage
VAG Water demand from agriculture
AF Size of collecting area
VIN Water demand for industry
VTG Total water demand
R volume of runoff
S Volume of storage
AC Size of contributing area
AF Size of collecting area
Agricultural land area
time
Agricultural water and farmland area factor
Flexible sewage and time（depth of storage）
Prediction yuan output value of emissions
Prediction of industrial output
Domestic sewage
Industry sewage
Utilization rate of the recycled water
Sn Natural water supply
Ss Surface water resources
ST Total available water
SG Groundwater resources
W1 Rainfall infiltration
W2 Irrigation return flow
VAG Water demand from agriculture
VIN Water demand for industry
VTG Total water demand
S Volume of storage
AC Size of contributing area
Forecast of rainfall
depth of storage
3 Mathematical models
3.1 Analysis of water demand & supply
3.2 Model 1： Water demand model
The water demand mainly contains water use for domestic living， industry and agriculture. we have a detailed analysis of the three primary water use in the following paragraphs and then obtain the water demand model，
.
3.2.1 Water for domestic use
According to the Logistic Growth model， the population in a specific area is limited by many factors. To specifically， the growth of population increased when the Increase Rate drops.
，
Thus the amount of water demand for domestic use is
3.2.2 Water for industry use
The Logarithmic Mean Divisia Index shows that the water for industry use is determined by the repeated utilization ratio， the industrial output， industrial structure and so on. The regional industrial water use of factor decomposition model is
= ，where ，
， .
The difference of water use for industry between year 0 and year t is， ，with the LMDI decomposition formulas below.
3.2.3 Water for agriculture use
The primary driver of water using in agriculture is the changes of agricultural land， which has a downward tendency with time. The formula of water for agriculture use is
.
Thus the total water demand is .
3.3 Model 2： Water supply model
Water supply of one certain region derives from two broadly categories： natural water supply and non-natural water supply. Natural water supply mainly contains surface water and groundwater in shallow aquifers. We have a detailed analysis of the two primary supply sources in the following paragraphs with its correspond complicate mechanisms， and then obtain the water supply model，
3.3.1 Introduction of natural water resources
Natural water supply mainly contains surface water and groundwater in shallow aquifers. Surface water resources （ ） primarily incorporate reservoirs， rivers， lakes and so on. Since surface water resources are exposed to the sunlight， they evaporate more easily， and therefore the temperature （ ） is a main determined factor. According to historically statistic data about water resources， the groundwater is the principle source for natural water supply. Groundwater resources （ ） mainly contain rainfall infiltration （ ） and irrigation return flow （ ）. According to the formula， ，
we can calculate the total amount of groundwater resources in one region. Then the total available water （ ） of the region is the sum of and SG， that is . The natural water supply （ ） equals total available water （ ） multiplying the water utilization ratio （ ）， which is .
3.3.1.1 Surface water model
Since surface water exposes to the natural environment， sunlight， wind， temperatures and other natural factors can affect the amount of surface water. Therefore we assume the amount of surface water fluctuates with different temperatures （ ）， and it is also the function of year （ ）. ，where is the coefficient measuring changes of with different years.
3.3.1.2 Rainfall infiltration （ ） & irrigation return flow （ ）
Rainfall infiltration is the main component for groundwater resources. According to the formula， the rainfall infiltration （ ） equals rainfall （ ） multiplying the coefficient of permeability （ ） and control area （ ）. Specifically， varies with various type of （i=1，2，3，…）， where can be farmland， forest land， garden land， meadow land and so on. The exact formula is .
Irrigation return flow （ ） is a point source which generally returns to the irrigation center after a period of about three to four weeks. Due to this， irrigation return flows can affect the water quality because some organisms in farmlands have entered the groundwater with the flow. Irrigation return flow （ ） equals irrigated area （ ） multiplying irrigation return coefficient （ ） and irrigation norm （ ）.The specific formula is .
Thus， the total groundwater resources model is
.
Thus the natural water supply mode is
with specific formula as follows.
， ，
，
3.3.2.1 Choose Beijing as the certain region for empirical analysis
We have collected and compiled Beijing’s historical statistic data of rainfall， size of farmland， size of forest land， size of effective irrigation land， irrigation norm of various crops， temperatures， total water supply， total groundwater， total surface water， from Nation Bureau of Statistics， from 2001 to 2014. We use these statistical data in the model 2 to for empirical analysis.
3.3.2.1.1 Covariance analysis
In order to construct the model using Ordinary Least Squares， we first obtain a covariance matrix to examine variable’s correlation with the application of Eviews. According to the covariance matrix， the covariance between every two variables is not strong， with the efficient maximum correlation coefficient being -0.001006， and thus it is reasonable to use Ordinary Least Squares to get the multiple linear regression model.
3.3.2.1.2 multiple linear regression model
Regress SG on W11 （=R*K1*F1）， W12 （=R*K2*F2）， W2 （=FIR*U*QFIX） and Tem， based on Beijing’s historical data， obtaining the multiple linear regression model as follows：
，where
， ， ， =0.28
（29.54440） （.000186） （.000766） （.00000179） （2.072147）
= .559215， Adjusted =0.363310， F=2.854527， n=14.
3.3.2.1.3 Heteroskedasticity test & Joint significance
We first examine whether this model has heteroskedasticity. Base on the Breusch-Pagan test， obtaining F=1.364179 < F（4，9）=3.63 with Eviews， we fail to reject H0 that the model has homoskedasticity. According to the test outcome， we draw the conclusion that we do not consider heteroskedasticity in this regression model.
Then we discover that the coefficient of （= ） is very small， even though the t-value is statistically significant. In order to test whether this variable is economically significant， we process the Wald Test. According to the Eviews test outcome， we find that variable （= ） and W2 （= ） are jointly significant， so we had better not omit the variable W2 （= ）.
3.3.2.1.4 Empirical analysis for Beijing’s surface water supply
We assume that Beijing’s surface water is relatively constant under different environmental conditions except temperatures. Using Beijing’s statistic data of total surface water supply， we regress Ss on t to depict the trend of surface water supply （Ss） with various years （t）.
，obtaining .
（120.3882） （.059969）
Then， using the same Beijing’s surface water supply data and temperature data from 2001 to 2014 with the application of Eviews， we obtain the following surface water supply regression model：
（122.5174） （.061425）（.577683）
= .508344， Adjusted =.418951， F=5.686673， n=14.
The Eviews outcome implies that the t-value of temperature （Tem） is -.805017， being not statistically significant. However， before omitting this variable， we test the joint significance using Wald Test. The Wald test outcome shows that the F-value is 5.686673 with df=（2，9）， so temperature and year are jointly significant， meaning that variable temperature can not be omitted.
3.3.2.1.5 Empirical analysis for Beijing’s total available water （ ） & the natural water supply （ ）
，
3.3.3 Non-natural water supply
3.3.3.1 The water resources from sewage
Sewage treatment is a process that translate sewage water to reuse water， this technology is wildly used in our daily life. The two mainly forms of sewage water are industry and domestic sewage， the analysis are as follows. Domestic sewage discharge growth model：
Choose Beijing as the certain region for empirical analysis
Industry sewage discharge growth model：
Empirical analysis we can conclude from the sheet 1：
，
So
3.3.3.2 Potential for new or alternate water resources
We mainly discuss two kinds of alternate water resources， the seawater desalination and the water harvesting techniques. The analysis are at the part of Intervention plan.