e same as the ones used in Section 2.2.
We substituted the parameters into Equation (11), and obtained the numerical relationship between E2 and ΔC0:
We plotted the functional relationship between the error E2 and the deviation ΔC0, as shown in Figure 2.
It was clear that there was a positive linear correlation between E2 and ΔC0. When ΔC0 increased, E2 also increased at the corresponding displacement x. This was because ΔC0 is the coefficient of E2 in Equation (11). Similarly, in Equation (8), C0 is the coefficient of C(x). There was also a positive linear correlation between C(x) and C0. When C0 increased, C(x) also increased at the corresponding displacement x.
On the other hand, according to the Water Quality Standards , a BOD concentration of 4.2 mg/L corresponds to Class III. For natural water, it is difficult to meet water quality requirements for this initial concentration. The concentration is officially required to decay back to 3 mg/L, which corresponds to Class I, by the self-purification of the river itself. The pollutant process satisfied Equation (8), so that at least a 75 km distance would be needed if the water pollution began to decay from the initial concentration of 4.2 mg/L. The decay processes in the river are shown in Figure 3.
From Figure 3 we can see, if “3 mg/L”, which was the limiting value for Class I in Water Quality Standards, was actually required at a distance of 40 km from a pollution source, the water quality situation at present would not meet the requirements, and the concentration could not decay from the initial concentration of 4.2 mg/L to 3 mg/L at the 40 km. That means the water quality would need to be managed effectively. Reduction of the initial concentration at the pollution source is the most common and effective way to manage water quality. We implemented a reduction of the concentration at the pollution source, such that the initial concentration would be reduced to 3.5 mg/L within 40 km. Thus “4.2 mg/L” was reduced by ΔC0 = 0.7 mg/L, and now the water quality
Figure 2. Concentration error and deviation of initial concentration.
Figure 3. Decay processes for different initial concentrations in the river.
began to change from an initial concentration of 3.5 mg/L. The concentration at the distance of 40 km could decay to the requirement of 3 mg/L though the river self-purification process, and its decay processes in the river are also shown in Figure 3. We calculated the initial concentration which could meet the requirement that concentration should decay to 3 mg/L at the 40 km distance from the pollution source according to Equation (9). The result was = 3.5687 mg/L.
2.4. Stochastic Factors
In this section the impact of stochastic factors in the water environment on the water quality model are discussed. The stochastic factors in the water environment included: External or anthropogenic factors on the water environment, uncertain measurement of parameters in the water quality model, the interaction and chemical reaction of the solutes, precipitation and drought and other climatic factors, etc. . As we knew, when the dispersion effect was not considered, Equation (2) was transformed into Equation (3). We took stochastic factors into account in the water quality model. The integrity of the stochastic factors was seen as one factor, and we mainly considered the impact of this factor on the model . Thus a stochastic factor “S” was added into the equation. Then Equation (3) was transformed:
The stochastic factor “S” was assumed to be white noise with variance of σ2 , and “S” was represented by. Then the water quality model was transferred into a first-order constant coefficient linear stochastic differential equation (12):
White noise could not be used as a stochastic process with a sample function in the usual sense, so Equation (12) was not determinate . However, because , Equation (12) was transferred into Equation (13):
We integrated both sides of Equation (13), and obtained
Equation (14) was a stochastic integral equation, and was determinate, because for each ω Î Ω, the sample function of the Brownian Motion W(x) = W(x, w) was a defined continuous function. C(x) was set in the interval I, x0 Î I, and C(x0) was the known stochastic variable. Then we named the stochastic process “C”, which satisfied Equation (14) and had continuous sample functions, and was the solution of the stochastic differential model Equation (13) in the interval I.
We used the constant variation method to solve the first-order non-homogeneous differential Equation (13). The general solution to the homogeneous equation corresponding to Equation (13) was
where a = −k/u. We set F = F(x), then
We substituted this into Equation (13), and obtained
We integrated both sides of the upper equation from x0 to x, and obtained
Because, we had . We substituted this into the upper equation and simplified the results, and obtained
This was consistent with the structure of a typical differential equation. General solution of Equation (13) was equal to general solution of the corresponding homogeneous equation coupled with a special solution of Equation (13). The difference was that C(x0) here was a stochastic variable, but not necessarily constant.
We set x0 = 0, and C(x) was the solution of Equation (13) in the interval of [0, ∞], which satisfied the initial conditions C(0) = c0. Thus we had
This was a Gaussian process. Since the expectation of the integral of the white noise was 0, the expectation of C(x) was:
The covariance of C(x) was:
Specifically, we set s = x, and we obtained the variance of C(x) which was
After theoretical analysis, we used the data for simulation to investigate the impact of stochastic factors in the water environment on the water quality model. Simulation parameters were the same as the ones used in Section 2.2. We substituted the simulation parameters into Equation (16), and obtained the solution of the water quality model with stochastic process.
Furthermore, we substituted the simulation parameters into Equations (17) and (18), and the expectation and covariance of C(x) would be obtained.
Stochastic factors were considered in the water quality model, so determinate solutions could not be given. Solutions could only be represented in the form of the stochastic process. We plotted the solutions with the stochastic process, as shown in Figure 4. In this study the stochastic process was the white noise process with variance σ2 = 0.75.
From Figure 4 we can see that the water concentration still showed an attenuated trend, while the concentration would fluctuate around the expectation centered at each corresponding displacement x. The fluctuations had a certain range, with variance of σ2 = 0.75. The impact of
Figure 4. Solutions of water quality model with stochastic process.
stochastic factors on the water quality was uncertain, and it was difficult to describe it exactly and numerically. However, we made some reasonable assumptions for stochastic factors and dealt with them using some mathematical tools. Thus we could obtain solutions with a stochastic process.
The impacts of changes of various parameters and stochastic factors on the water quality model were analyzed and discussed. Preliminary conclusions were obtained as follows.
1) The values of Δk and E1 had the opposite sign. When the deviation of the degradation coefficient Δk increased, the concentration error E1 decreased numerically at the corresponding displacement x. When Δk > 0, E1 < 0, and the value of Cs would decrease; when Δk < 0, E1 > 0, and the value of Cs would increase. A 1% change in the degradation coefficient would cause about 0.15% error in the solute concentration. The impacts of Δk on E1 decreased with increasing x.
2) When the deviation of initial concentration “ΔC0” increased, the error of the model “E2” also increased at the corresponding displacement x. There was a positive linear correlation between E2 and ΔC0. To reduce the initial concentration at the pollution source is the most common and effective way to manage water quality. By reducing the initial concentration, the requirements of water quality could be achieved through the self-purification ability of the river itself.
3) Stochastic factors did not significantly affect the attenuation trend of water quality. However, because of the stochastic interference, the concentration of the solute would fluctuate around the expectation centered at each corresponding displacement x.
This work was supported by the National Natural Science Foundation of China (No. 41071322, 71031001).