A Mathematical Model of Horizontal Averaged Groundwater Pollution Measurement with Several Substances due to Chemical Reaction

Chloride is a well-known chemical compound that is very useful in industry and agricultural. Chloride can be transformed to hypochlorite, chlorite, chlorate and perchlorate, chloride and their substances are not dangerous if they are used in the optimal level. Groundwater containing contamination chloride and their substances impacts human health, for example, drinking water contaminated chloride exceed 250 mg/L can cause heart problems and lead to high blood pressure. To avoid this problem, we used mathematical models to explain groundwater contamination with chloride and their substances. Transient groundwater flow model provides the hydraulic head of groundwater. In this model we will get the level of groundwater, next, we need to find its velocity and direction by using the result in first model put into second model. Groundwater velocity model provides xand z-direction vector in groundwater, after computation we will plugin the result into the last model to approximate the chloride concentration in groundwater. Groundwater contamination dispersion model provides chloride, hypochlorite, chlorite, chlorate and perchlorate concentration. The proposed explicit finite difference techniques are used to approximate the model solution. Explicit method was used to solved hydraulic head model. Forward space described groundwater velocity model. Forward time and central space used to predict transient groundwater contaminated models. The simulations can be used to indicate when each simulated zone becomes a hazardous zone or a protection zone.


Introduction
Nowadays, we can say that water is an important part of life, whether in daily life, agricultural and industry. The water pollution problem is created by above activity, we can found the contaminated water from the natural source, contaminated water has so much effect, it can cause many life diseases and problems. Nitrates or nitrites in water contaminates drinking water can impacts human health by decreasing blood cell ability to carry oxygen, which can be linked to blue baby syndrome [2], this is one of the effects of contaminated water. In this research we talk about the effect of chloride and their substance.
Chloride occurs naturally in groundwater but is found in greater concentrations in seawater. It generally combines with sodium, calcium, or magnesium. For example, sodium chloride (NaCl) is formed when chloride and sodium combine.
The other forms of chloride do not come from the only combination of other substances, the oxidation numbers or oxidation states is the well-know process to obtained a new form of chemical compound, for example, Chloride can be changed to hypochlorite (ClO) if the oxidation number increase by 1 or chlorite if added by 3. Several Substances due to Chemical Reaction irrigation, and give drinking water an unpleasant taste. Sodium chloride is high corrosivity,which will also damage plumbing, appliances, and water heaters, causing toxic metals to leach into your water. It can complicate existing heart problems and contribute to high blood pressure when ingested in excess [1]. There are many dangers of chloride, but we can prevent amounts of them, from exceeding standards by management based on mathematical models. Yamashita and Sugio (2000) developed a model of advection dispersion and biochemical reactions [3]. Pochai and Kraychang (2016) solved a two mathematical models for simulating water pollutant level and pollution control in a connected reservoir system [7]. Gardenas (2005) talks about a two-dimensional modeling approach to improve fertigation strategies and soil types on nitrate leaching potential [4]. The one-dimensional advection-diffusion equation with constant coefficients have been solved by Dehghan (2004) [5]. Kewalee and Pochai (2018) described the governing equation in the air quality model in three-dimensional advection-diffusion equations with time dependence [6].
In this paper, we measured groundwater that has been contaminated by chloride on a landfill. The simulation required data concerned with the groundwater flow velocity of the current points and any time in the domain. The groundwater flow velocity can be obtained by using transient twodimensional groundwater flow model. We used the transient two-dimensional advection diffusion equation to approximated the chloride concentration. The finite elements [19,20,21] and finite difference methods [16,17,18] are the most popular numerical solution techniques. Among of these, finite difference methods, including both explicit and implicit schemes, are mostly used for two-dimensional domain such as in latitude and longitude stream.
2 Chloride pollutant measure models

Transient groundwater flow model
The governing equation of a latitudinally integrated Darcy's flow in a two-dimensional advection-diffusion equation [8], is the hydraulic head(metre), S matrix of specific storage(1/metre), L x is the considered area length, L z is the depth of considered groundwater area, T is the stationary time of simulation as shown in Figure 1. The hydraulic conductivity(metre/day) component in the x, z directions are denoted by K x , K z , respectively. Assuming that the soil topography in the considered area is homogeneous, these mean that the hydraulic conductivity are constant. The initial condition is defined by an interpolation function of measured raw data as be low where h(x, z) is a given potential hydraulic head function.

The boundary conditions
The top, right and bottom boundary conditions are assumed by the average rate of change of hydraulic head around the top, right and bottom boundaries. The left boundary condition is assumed by the interpolation function of measured raw data in the considered landfill as shown in Figure 1. The boundary condition, are also assumed by

Groundwater flow velocity model
We can obtain that the groundwater flow velocity in xdirection is a decreasing rate of change of the hydraulic head x-direction, Similarly, the groundwater flow velocity in z-direction is a decreasing rate of change of the hydraulic head in z-direction,

A total chloride dispersion model
The pollutant concentration measurement of total chloride in surface water can be describe by a two-dimensional advectiondiffusion reaction equation.
∂c(x, z, t) ∂t + u ∂c(x, z, t) ∂x + w ∂c(x, z, t) ∂z where c(x, z, t) is total chloride pollutant concentration of groundwater(kg/m 3 ), D x andD z are the diffusion coefficient in xand z-directions, u(x, z, t), w(x, z, t) are the groundwater flow velocity in the xand z-directions, and R is transformed chloride rate.

Initial condition of the total chloride dispersion model
The chloride dispersion is described with conditions in the following sections, where the potential groundwater pollutant concentration in the consider area is described by where c 0 (x, z) is a averaged potential total chloride concentration in the considered area.

Boundary condition of total chloride dispersion model
The left boundary condition is assumed by the interpolation function of measured raw data at the considered landfill. The top, right and bottom boundary conditions are assumed by the averaged rates of change of pollutant concentration around the top, right and bottom boundaries. The boundary conditions are also assumed by where k 1 L x , k 2 L x are referred to as the range area of the total chloride pollutant area source, and g L (z), g T (x), g R (z) and g B (x) are the rate of change of the total chloride concentration with respect to distance around the top, the bottom and the right boundaries along the considered area, respectively.

A hypochlorite dispersion model
The total chloride is transformed to be the hypochlorite. The hypochlorite dispersion model is described by are the groundwater flow velocity in the xand z-directions and R 1 is transformed hypochlorite rate.

Initial condition of hypochlorite dispersion model
Dispersion of hypochlorite with the following conditions, if the potential hypochlorite concentration in the consider area is described by is a averaged potential hypochlorite concentration in the considered area.

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A Mathematical Model of Horizontal Averaged Groundwater Pollution Measurement with Several Substances due to Chemical Reaction

Boundary condition of hypochlorite dispersion model
The rate of change of the pollutant concentration along the domain boundaries are assumed to be: are the rate of change of the hypochlorite concentration with respect distance around the top, the bottom and the right boundaries along the considered area, respectively.

A chlorite dispersion model
The model of chlorite dispersion model can be described by , D2 x , D2 z are the diffusion coefficient in xand z-directions, u(x, z, t), w(x, z, t) are the groundwater flow velocity in the xand z-directions, R 2 is transformed chlorite rate.

Initial condition of chlorite dispersion model
Dispersion of chlorite with the following conditions, if the potential chlorite concentration in the consider area is described by where f η (x, z) is a averaged potential chlorite concentration in the considered area.

Boundary condition of chlorite dispersion model
Similarly, the rate of change are assumed by g2 L (z), g2 T (x), g2 R (z) and g2 B (x).

A chlorate dispersion model
The model of chlorate dispersion model can be described by are the groundwater flow velocity in the xand z-directions, R 3 is transformed chlorate rate.

Initial condition of chlorate dispersion model
Dispersion of chlorate with following conditions, if the potential chlorate concentration in the consider area is described by is a averaged potential chlorate concentration in the considered area.

Boundary condition of chlorate dispersion model
Similarly, the rate of change are assumed by g3 L (z), g3 T (x), g3 R (z) and g3 B (x).

A perchlorate dispersion model
The model of perchlorate dispersion model can be described by are the groundwater flow velocity in the xand z-directions, R 4 is transformed perchlorate rate.

Initial condition of perchlorate dispersion model
Dispersion of perchlorate with following conditions, if the potential perchlorate concentration in the consider area is described by where f ξ (x, z) is a averaged potential perchlorate concentration in the considered area.

Boundary condition of perchlorate dispersion model
Similarly, the rate of change are assumed by g4 L (z), g4 T (x), g4 R (z) and g4 B (x).

Numerical techniques
In this paper, we will propose finite difference methods to the transient groundwater model by using the forward time central space method. We now discretize the domain by dividing the interval where α = K x (∆t) S(∆x) 2 and β = The forward space technique is used to approximate fictitious points on the boundaries solution such as,

Explicit finite difference method for twodimensional groundwater pollution dispersion model
In this section, the considered domain is defined in a similar grid spacing as the previous. We will employ forward time central space difference scheme (FTCS) to Eqs.(2.10) The forward space technique is used to approximate fictitious points on the boundaries solution such as,  where λ 3 = D1 x ∆t (∆x) 2 , λ 4 = u n i,j ∆t 2∆x , τ 3 = D1 z ∆t (∆z) 2 . and The forward space technique is used to approximate fictitious points on the boundaries solution such as, where The forward space technique is used to approximate fictitious points on the boundaries solution such as,  where The forward space technique is used to approximate fictitious points on the boundaries solution such as, where λ 9 = D4 x ∆t (∆x) 2 , λ 10 = u n i,j ∆t 2∆x , τ 9 = D4 z ∆t (∆z) 2 . and τ 10 = w n i,j ∆t 2∆z . The forward space technique is used to approximate fictitious points on the boundaries solution such as,

Numerical simulations
A two-dimensional hydraulic head model provides hydraulic head. The computed hydraulic head is transformed to be the groundwater flow velocity by using the second model. The results from the second model will be input into the chloride dispersion models that provide chloride, hypochlorite, chlorite, chlorate and perchlorate concentration

Numerical simulation of transient groundwater flow velocity
A chloride compound dispersion model that provide the concentration of their substance, this computation need to using the groundwater flow velocity from the first model.
The two-dimensional of groundwater flow model, considering with area domain 1.0 × 0.5 km. Assume that the specific storage is 1 m −1 and the hydraulic conductivity in xand zdirection are 15 (m/day) and don't have source term. The grid spacing is ∆x = ∆z = 10 m, time step ∆t = 1 day at time T = 3650 day. The initial h = 0(m) and boundary conditions h T = h R = h B = 0 and h L = 0.06z + 10. Consider the hydraulic head of groundwater by using the explicit method, we get the approximated groundwater pollutant concentration as shown in Table 1 and Figures 3-4. Next, input the hydraulic head into the second model to approximate groundwater velocity in (2.8) and (2.9) by using central space. Then, we get the approximated groundwater flow velocity as shown in Tables 2-3 and Figure 5.

Numerical simulation of chloride dispersion
In the last model, we input the groundwater flow velocity to approximate chloride dispersion. The chemical compound of chloride can be transformed into hypochlorite, then into chlorite, chlorate, and lastly into perchlorate. At the beginning of the time, we assume that the amount of chemical compound in considered area is equal to zero, i.e., c 0 = 0 m and boundaries in considered domain suppose that there is no rate of change, then, g L = g T = g R = g B = 0. The diffusion coefficient in xand zdirection in each models are different due to the mass of compound, so, we assume that the diffusion coefficient D x = D z = 1.5. The transformed rate are important factor to concentration of pollutant, its valuable between 0 to 1, then, we set R = 0.001. The approximate solution is shown in Table 4 and Fig 6-7.
We set the initial and boundary conditions as the same with previous model, considering diffusion coefficient of hypochlorite D1 x = D1 z = 2.0 and rate of hypochlorite dispersion R 1 = 0.25. The approximate solution is shown in Table 5 and    We set the initial and boundary conditions as the same with previous model, considering diffusion coefficient of chlorite D2 x = D2 z = 0.5 and rate of chlorite dispersion R 2 = 0.01. The approximate solution as shown in Table 6   We set the initial and boundary conditions as the same with previous model, considering diffusion coefficient of chlorate D3 x = D3 z = 2.5 and rate of chlorate dispersion R 3 = 0.4. The approximate solution as shown in Table 7 and Fig 12-13.
We set the initial and boundary conditions as the same with previous model, considering diffusion coefficient of perchlorate D4 x = D4 z = 1 and rate of perchlorate dispersion R 4 = 0.005. The approximate solution as shown in Table 8 and Fig 14-15.

Discussion
In our simulation, we assume that the hydraulic head drive the groundwater flow from the higher hydraulic head level zone to the lower zone, the result of simulation for 10 years have been shown in Table 1 and Figure 3-4. The figures shown that the hydraulic head at the surface area is higher than the deep area. The hydraulic head is transformed to be the groundwater flow direction as shown in Fig 5. The direction has shown that groundwater flow from high to lower hydraulic head. The result has been plug into the five chloride compound dispersion models. We can measure the total chloride, hypochloride, chlorite, chlorate and perchlorate pollutant levels at 10 years as shown Table 4-8 and Fig 6-16. The figure shows that the amount of groundwater changes directly,over time the substance has been less than reactant. The approximated chloride compound is compared in Fig 16-20. In Fig 16-20, we can see that the simulation for 2 to 4 years tell us the trend of graph are the same and a little bit increasing, after that, for 4 to 6 years, they are slightly increase. Finally, in 6 to 10 years we obtained that the graph is stable.

Conclusion
A Mathematical model of horizontal average groundwater pollutant measurement with several substances due to chemical reaction is introduced. The first model is the two-dimensional transient groundwater flow model which provides the hydraulic head. The second is the groundwater flow velocity model which provides the groundwater flow in xand z-directions. The third model is the two-dimensional horizontal averaged contaminated chloride dispersion model which provides the concentration of chloride and their substances. A method to set up the initial and boundary conditions of transient groundwater flow model is proposed. The computed hydraulic head is transformed to be the groundwater flow velocity by using the second model. The results in second model will be input into the last model as a field data. The concentration of chloride and their substances are obtained by the third model. The hydraulic head of the first model is approximated by an explicit finite difference method. An explicit finite difference technique is used to obtain the groundwater flow velocity of the second model. A forward time-centered space finite difference technique is used to approximate the concentration of chloride and their substances. The groundwater quality is affected by the chloride release by the landfill. The proposed simulations show that the different levels of hydraulic head have a small effect on the overall groundwater quality level. In our simulation.It is found that the main groundwater quality factor are pollutant concentration level around the landfill and transform rate.