{'fraction_non_alphanumeric': 0.02546554193856438, 'fraction_numerical': 0.00668470475887315, 'mean_word_length': 5.234126984126984, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Study on contaminant transport in one-layered media with variable diffusion coefficient

With considering the distance-dependent diffusion coefficient, a one-dimensional contaminant diffusion model was developed. An analytical solution was obtained by using the orthogonal expansion method. The corresponding program was compiled on the basis of the proposed solution, and the effect of the relevant parameters on contaminant transport was analyzed. The results indicate that the diffusion coefficient has complex effects on the transport of contaminants. On one hand, the initial diffusion coefficient and heterogeneity parameter will affect the time of contaminant concentration reaches steady state, and on the other hand, the heterogeneity parameter affects the concentration of steady state. In addition, the influence of related parameters on the bottom flux of contaminant was discussed. The proposed analytical solution can be used for verification of complicated numerical models and experimental data fitting.


INTRODUCTION
Contaminant transport analysis is a hot research problem when considering geo-environmental problems such as the evaluation and design of the barrier systems for waste disposal and the remediation of existing contaminated sites. Studying the contaminant transport can effectively control the pollution problems caused by the leakage of pollutants. Mathematical model is a powerful tool for researching the contaminant transport, and is widely applied to the contaminant transport analysis (Singh, 2011).
Contaminant transport in the soil medium is influenced by many factors , such as advection, diffusion, adsorption, degradation, etc. The Chinese standard code for liner systems consisted of compacted clay requires the hydraulic conductivity less than 10 -7 cm/s. In previous studies, the hydraulic conductivity is considered as the main process of contaminant transport in clay liners. But research showed that careful attention to construction procedures results in high quality clay liners having a hydraulic conductivity less than 10 -7 cm/s (Edil, 2003). For instance, clay liners in Wisconsin typically have very low field hydraulic conductivities, on the order of 10 -8 cm/s. (Gordon ea al, 1990;Benson et al, 1999 Research has also shown that contaminant diffusion coefficient for different types, and obtained the analytical solution. The above variable diffusion coefficient models are relatively complicated, and difficult for practical application. This study will simplify the type of diffusion coefficient and simulate the contaminant transport.
In this paper, the diffusion coefficient is assumed to be a function of depth, and a one-dimensional contaminant diffusion model is proposed. The analytical solution satisfying all relevant conditions is derived by using orthogonal expansion method. According to the proposed solution, contaminant transport is analyzed and the effects of the related parameters are discussed. The method is relatively simple to be applied and can be used for evaluating experimental results, and for verifying more complex numerical models.

MATHEMATICAL MODEL
The major assumptions for the model are as follows: (1) contaminant diffusion model is one-dimensional; (2) the diffusion coefficient is a linear function of depth; (3) the retardation factor is constant; (4) adsorption is the linear type.
The governing equation for one-dimensional contaminant diffusion problem can be described as: According to Liu et al (1998), the boundary conditions for the problem considered can be expressed as follows: : ( , ) 0 where H is the thickness of the soil medium.

ANALYTICAL SOLUTION
According to the superposition principle, to convert the inhomogeneous boundary condition into a homogeneous one, the solution can be expressed as: where the solutions of () uz and ( , ) zt  are solved as follows.

Solution for () uz
The governing equation for () uz is as follows: The boundary conditions for this problem are: : The solution to Eq.6 is: where P and Q can be obtained using the boundary conditions (Eqs.7,8).

Solution for ( , )
zt  The governing equation for ( , ) zt  can be written as follows: The boundary conditions for this problem are: The initial condition is: According to the solution to homogeneous heat conduction problem, the solution of ( , ) zt  can be expressed: Flux of contaminant can be obtained as follows: where 1 J and 1 Y are first order Bessel functions of the first kind and second kind.

ANALYSIS RESULTS AND DISCUSSION
On the basis of the obtained analytical solution, the corresponding program is compiled and the contaminant transport in soil medium is illustrated. The values of relevant parameters are shown in Table 1.  The effects of the initial diffusion coefficient and heterogeneity parameter on contaminant flux at the bottom are shown in Fig. 3 and Fig. 4. As shown in figures, the bottom contaminant flux has a great difference under different conditions. With the relevant parameters increasing, the contaminant flux also increases. The parameters not only affect the required time for reaching steady-state, but also have an impact on the stable value.

CONCLUSIONS
This paper presents an analytical solution for onedimensional contaminant diffusion with distancedependent diffusion coefficient. The analytical solution is obtained by using the orthogonal expansion method and the program is written through MATLAB.
Through the parameters study, it is found that the contaminant concentration is drastically reduced with the increase of depth, and the relevant parameters have an important impact on the calculation results. The contaminant concentration increases as the initial diffusion coefficient increases, and the initial diffusion coefficient only affect the time for contaminant concentration reaching steady-state. The required time of reaching steady-state is shorter and the contaminant concentration value of steady-state is smaller when the heterogeneity parameter is larger. So an accurate value of the diffusion coefficient is important to improve the prediction accuracy. The proposed model is established on the basis of linear adsorption, therefore it is necessary to make further research for considering non-linear adsorption and decay.