A mathematical approach to evaluate the extent of groundwater contamination using polynomial approximation Open Access

Groundwater is a primary source of drinking water in India but the day-by-day increment of groundwater contamination due to various anthropogenic activities as well as geogenic sources makes life difficult especially for rural people. The main question is whether the available water in the bore wells/dug wells is drinkable or not. Mathematical modeling has been presented by researchers and scientists to predict the contamination level in the aquifer–aquitard system. Yates (1992) used the Laplace transform technique to solve the one-dimensional generalized advection–dispersion equation (ADE) in a semi-infinite domain. Logan (1996) extended the work of Yates (1992) using scale-dependent dispersion and periodic boundary conditions. Aral & Liao (1996) and Batu (1996) solved two-dimensional and three-dimensional generalized ADE respectively. Fractional ADE have been incorporated to study the impact of groundwater contamination (Benson et al. 2000; Marinca et al. 2009; Pandey et al. 2012; Singh & Chatterjee 2017). In the above-mentioned papers the contaminant concentration has been taken care of but the specific contaminant concentration for one element and the overall quality of the groundwater is not predicted. Often it will become difficult to predict the contaminant concentration for a certain zone using ADE.

According to the World Health Organization (WHO), the use of contaminated water causes around 80% of total diseases in humans. Pollution in the groundwater system is a severe threat to public health as well as to economic and social life (Milovanovic 2007). Therefore, monitoring groundwater quality and controlling the pollution level in groundwater is the need of the hour (Simeonov et al. 2003). Groundwater may have a direct impact on agriculture as most irrigation systems depend on the bore well in India. So, it is quite essential to maintain the quality of the irrigation water. Additionally, experts from all disciplines have agreed that factors such as urbanization, industrialization, poor land organization, and environmental pollution impose further stress on irrigation production. These factors have a significant effect in terms of both quantity and quality of water for irrigation.

The Water Quality Index (WQI) is regarded as one of the most efficient ways for the rating and classification of water quality for drinking and irrigation purposes (Brown et al. 1972; Ott 1978; Miller et al. 1986; Bordalo et al. 2001; Cude 2001; Hallock 2002; Agarwal et al. 2021a, 2021b). So, it is quite essential that an author can predict the WQI index for any point present in the domain of study.

Here in this present study our main goal is to approximate the contaminant concentration profile using the polynomial approximation method for the whole domain using some limited numbers of data. It is often difficult as well as costly to collect the large numbers of data which are mostly used for statistical or machine-learning-based solutions. Depending upon the degree of the polynomial we are going to choose the number of data required, and as an example one can say that we need three values to identify the two-dimensional linear curve. So, in this present paper we are not only using the polynomial approximation method to find the contaminant concentration level but also validating our work with the field data. In the literature, the polynomial approximation method has not been used for estimation of pollutants in groundwater contamination problems. This is a novel approval as it has not been utilized earlier in water quality studies and is the first of its kind to the best of the authors' knowledge. Hence, in this present study the authors utilized the polynomial approximation method to predict the contaminant concentration values in different locations. The polynomial approximation method requires a smaller data set to fit the curve, but depending upon the required accuracy the number of data may be increased although a smaller number of data also provides a good result as discussed. Convergence of the method is assured according to the Weierstrass approximation theorem. The polynomial approximation (algorithm) is very easy to understand, and one can implement this in various fields of study.

The Pindrawan tank command area was the area under study (Figure 1); it is situated within 81°45′–81°50′ E and 21°20′–21°25′ N in the upper Mahanadi River valley (southeastern part) and comes under the Raipur district of Chhattisgarh, India. A total of nine villages, namely, Pauni, Amlitalab, Khauna, Deogaon, Bangoli, Dhansuli, Kurra, Baraonda, and Nilja, falls under the study area, which has a tropical wet and dry climate. The temperature in this part of India remains moderate throughout the year. The highest temperatures in the year are observed as 48°C between March to June. The maximum rainfall recorded is around 325 mm. The average depth of the groundwater table is a minimum of 0.33 m below ground level (bgl) and a maximum of 17.14 mgbl.

The groundwater samples from open and bore wells (37 sites) are collected and the average value is used for drinking and irrigation purposes in the Prindrawan tank area as given in Table 1. The identification of the sampling points is performed using topographic sheets and GPS, and the maps are prepared using ArcGIS 10.1. Topographic sheets are utilized for the preparation of the base map and to recognize the general features of the area. The GPS technique is utilized to identify the geographic position of each sampling point. The study area is situated in an industrial zone where mining activity is very active, with coal, limestone, and quarries. Because of these, the pH parameter is selected for analysis. Along with this, as per the literature, fluoride and sulphate are very high, i.e., above permissible limits and such contaminated groundwater is not good for human beings. That is why fluoride and sulphate are selected as parameters. The collected groundwater samples are investigated for the concentration of the three parameters of pH, sulphate (SO₄), and fluoride (F) only, as per the specification of American Public Health Association (2005). The pH of the collected samples is measured in the field itself using a pH meter. The pH determination is usually done by the electrometric method, which is the most accurate one, and free from interferences. The turbidimetric method is used for the determination of sulphate ions. The sulphate ion (SO₄–) is precipitated in an acetic acid medium with barium chloride (BaCl₂) so as to form barium sulphate (BaSO₄) crystals of uniform size. Light absorbance of the BaSO₄ suspension is measured by a photometer or the scattering of light by a nephelometer. The fluoride concentration is analyzed based on the selective electrode method. When the fluoride electrode is dipped in a sample whose concentration is to be measured, a potential is established by the presence of fluoride ions by any modern pH meter having an expanded millivolt scale. The fluoride ion selective electrode can be used to measure the activity or concentration of fluoride in an aqueous sample by use of an appropriate calibration curve. However, fluoride activity depends on the total ionic strength of the sample. During the analysis, prescribed safety measures are followed. The latitude and longitude values of sampling stations range from 81°78′ E to 81°86′ E and 21°37′ N to 21°43′ N respectively.

The concentration of the parameters is compared with the acceptable limits prescribed by WHO (2012) and (BIS (IS 10500) 2012). The observed water quality parameters are finally integrated into a combined single parameter referred to as WQI. The WQI in the present study was calculated using the weighted arithmetic index method (Brown et al. 1972).

Based on calculated WQI values, the quality of the groundwater for drinking purposes can be classified in to five categories, i.e., excellent water quality (WQI: 0–50), good water quality (WQI: 50–100), poor water quality (WQI: 100–200), very poor water quality (200–300), unfit for drinking (>300).

Finally, the outcomes from the analytical study are taken to the GIS for the preparation of spatial distribution maps. The IDW interpolation method was used from the geospatial analysis tools.

In this present study Weierstrass's polynomial approximation theorem in a finite domain is adopted (Schep 2007). For example, the contaminant concentration data for wells, and bore wells at different points in a certain finite region, was found. Let us assume that we are working for a specific latitude and longitude in the very beginning of the process and shift the origin to that specific point. Let be a continuous function in a finite domain for a fixed time ⁠, then can be approximated by a polynomial in the domain ⁠, where ⁠.

For the finite set of values we determined linear equations with n unknowns. By solving these equations, the polynomial is obtained. So, for a certain domain of interest (i.e., the specified region) it can be assumed that various parameters of the groundwater contaminant can be obtained mathematically in the form of a polynomial. So, by using a much smaller number of data along with longitudinal and latitudinal values it is possible to predict the amount of the harmful elements throughout the region from the latitudinal and longitudinal values. The present problem has been solved by using the MATLAB software. In this present scenario the authors considered that there is only one aquifer that is used throughout the specific region.

The concentration, distribution, and impact of different physicochemical parameters observed from the water samples collected from the Pindarwan tank area are discussed in this section. The details of the apparatus used for measuring the parameters are provided in Table 2. The ranges of concentrations observed for various parameters and the percentages of total samples exceeding the prescribed limits are presented in Table 3, along with their undesirable effects on groundwater quality and human physiology. This section provides an overview of the spatial distribution of the physicochemical parameters of pH, fluoride and sulphate that were measured in the Pindarwan tank area as shown in Figures 2–4. Out of 37 samples, 32.43% of the samples had excellent water quality, 43.24% of the samples had good water quality, 21.62% of the samples had poor water quality, and 2.71% of the samples had very poor water quality. None of the samples fell under the unfit for drinking category. The areas corresponding to these WQI values are presented in Figure 5.

For the mathematical model, the authors used the following data set to obtain the graphical representation of the pH, WQI, fluoride and sulphate distribution throughout the domain as given in Table 4.

According to these considerations the origin is shifted to (81,21) and then the polynomial is approximated. In Table 1 the other rows indicate the concentration of the respective physicochemical parameters for the respective samples. The obtained results or the polynomial for pH, WQI, fluoride and sulphate are shown by the graphs in Figures 6-9 and in the next section the result will be validated with additional data and the error will be calculated.

In this section the authors want to verify the obtained result with unused data. The authors are going to validate the result with one of the presented contaminants and if our results match the real data then we can say that the used methodology is better to use. To validate the result the authors consider pH level and fluoride as shown in Tables 5 and 6 respectively.

The average relative error is 0.07 (pH) and 0.04 (fluoride), hence the percentage error is 7% (pH) and 40% (fluoride). So, it is clear that the methodology we have used is accurate. Hence, polynomial approximation can be one of the replacements for this type of problem.

The process of WQI estimation is often associated with handling large quantities of identical data. This can create significant confusion during the calculation process and make decision-making difficult. Using this simple but powerful polynomial approximation methodology, groundwater quality prediction can be done more accurately. The two major benefits of this method are: (1) it is easy to understand (anyone can use the method); (2) less data is required (a small set of data is enough to determine the polynomial). Using the polynomial approximation method one can easily obtain the WQI index of a position and be able to decide whether the stakeholder may consume the water or not. The error is found to be 7% approximately, which is significantly small. So, the proposed method can be very effective and useful. This polynomial approximation method can also be used as the substitute of inverse modeling to determine the location of the source in the three-dimensional system. Depending upon the availability of data, in future analysis a greater number of parameters will be utilized under different weather conditions.

The authors would like to sincerely thank the Indian Institute of Technology (Indian School of Mines) Dhanbad, for extending their support and facilities. The authors would also acknowledge the data support received from the Water Resources Department, Govt. of Chhattisgarh, and Chhattisgarh Infotech Promotion Society (CHiPS), Raipur, Chhattisgarh, in carrying out this research work.

Data cannot be made publicly available; readers should contact the corresponding author for details. Data will be made available on request.

The authors declare there is no conflict.