Application of Backward Probability Method in Pollutant Source Tracking in Non-Uniform Flow Rivers

Application of Backward Probability Method in Pollutant Source Tracking in Non-Uniform Flow Rivers

Alireza Ghane
M.Sc. Student of Water Structures, Tarbiat Modares University, Tehran
Mehdi Mazaheri 
Assistant Prof., Department of Water Structures, Tarbiat Modares University, Tehran
Jamal Mohammad Vali Samani
Professor, Department of Water Structures, Tarbiat Modares University, Tehran
Introduction
Rivers are very vulnerable to the chemical pollutions that were released from industries and agriculture. Contaminants are suddenly interned into rivers. So the location of contaminant that was released to the river should be identified quickly to control and decrease the pollution and determine responsibility, If the location or the release time of contaminants are unknown, in order to determine the location or the release time, it needs to use the backward model in both location and time. Backward probability method is one of the backward models that able to determine the location and the release time of the contaminant. Identification of contaminant sources is faced to the two parameter, the first one is the location of contaminant and the second one is the release time. Accordingly, the identification problem is illustrated with two kind of probability concept. Backward location and backward travel time probability are the two different way to identify the location and the release time of the contaminant source. Backward location probability determines the location of the contaminant source based on the assumption that the release time is known. In contrast, backward travel time probability gives the information about the release time based on the assumption that the source location is known.
The ability of this model has been proven in groundwater, but it used less in surface water. Therefore, the main goal of this study is application of this model to identify the location and the release time in non-uniform and steady state rivers.

Governing Equations of Backward Probability Method
Governing equation for mass transport in non-uniform and unsteady rivers is advection-dispersion equation (ADE). Upstream boundary condition in the ADE is the third type and downstream boundary will be the first type.
Adjoint analysis is an efficient approach for sensitive analysis. In common, sensitive analysis approach we need to run the model more and more, but adjoint equation is solved once and the results are used for sensitive analysis. This equation is used as Backward Probability Method. So, the governing equation for Backward Probability Method is adjoint equation. Adjoint equation is similar to the ADE, and it will be:
(1)


is the adjoint state, is the backward time and is the source term. Source term is different in both kind of Backward Probabilities. Dirac Delta Function approximates source term. It can be utilized as an initial condition. Boundary condition for adjoint equation is shown in equation 2:

(2)
Adjoint equation governs on both Travel Time and Location Probability, but the source term make a difference between Backward Travel Time Probability and Backward Location Probability. The source term is determined based on the type of the probability. After one time simulation both Backward Probabilities can be obtained.

Backward Simulation
A numerical code was developed to compute the backward probabilities based on adjoint equation. The control volume method based on explicit scheme is utilized for the numerical code.
In this study first, we utilized the backward code with exist analytical solution for a uniform canal. Therefore, we reached an accommodation between analytical solution and Backward Probabilities Solution. After verifying the model, it was used for a hypothetical non-uniform river. To compute hydraulic parameters of the river we need a hydrodynamic model. So a standard step method was used to compute the hydraulic parameters. So, the backward model inputs are outputs of the hydrodynamic model with a modification on flow field. For using hydrodynamic results in the backward model the flow field is reversed. Three release points were assumed in non-uniform. These are 500 m, 10 km and 20 km from observation point. We used a forward explicit method to compute the arrival time of the contaminants to the observation point at x=0. Then, it will identify the release time and the location of the contaminant by using the backward probability model. The model was ran for 40000 s. we show both PDF and CDF figure for the backward location and the travel time probability.

Results
The model was verified with analytical solution. It was applied for a rectangular canal with constant velocity. The length of the canal was 8 km. the contaminant was released at 5km from detection point. It took 10000 s to arrive to the detection point. We verified both the Backward Travel Time Probability and the Backward Location Probability.

Figure1: Verification of Backward Travel Time Probability


The model predicted the release time of the contaminant very well. The most percentage of the backward location probability is at the 1000 point. So the release time of contaminant is 1000 s.
The model has been tested in a non-uniform river. The contaminant was released contaminant from 3 different points, to test the ability of the model. Here we just show the second point (10 km).


Figure2: Pollutant Released from 10km

Contaminant arrive to detection point in 4.2 hr.


Figure3: Backward Location Probability in 4.2 hr

The model predict the source of the contaminant very well. Therefore, the most percentage of backward probability is at the 10 km. So, the source of contaminant is 1000 s.

Conclusion
In the past researches, backward probability method has been used less in surface water. But in the present paper, the backward probability method was used in non-uniform and Steady State River. According to the results backward probability method is able to apply for the non-uniform and steady state rivers. This method is able to identify the location and the release time only by one simulation. The accuracy of the model depends on the condition of rivers. The accuracy is high in uniform rivers, but it decreases in non-uniform rivers. So, this method is fast, since it does not need to run several times. Finally, it is suggested that this model test in non-uniform and unsteady rivers.

Key Word: Identification of Contaminant Sources, Backward Probability Model, Adjoint- Analysis, Non-uniform Rivers