Statistical Modeling of Municipal Solid Waste Settlement from a Lysimeter

The sanitary landfill settlement prediction is an important tool for an integrated waste management system once it allows the evaluation of the sanitary landfill useful life and assists in the development of planning actions. Several settlement models have been used as an attempt to estimate the waste volume reduction over time. However, there are few models that consider the waste biodegradation processes. The objective of this paper is to build a simple linear regression model to determine the settlements that took place over time in a lysimeter filled with municipal solid waste from the city of Campina Grande, State of Paraíba, Brazil. The lysimeter was built in the Environmental Geotechnics Group’s experimental field which is located at the Federal University of Campina Grande’s main campus. For the settlement model development, the municipal solid waste settlement from a lysimeter was monitored during 1309 days. It was observed that in the interquartile range, data exhibited a trend of linearity, so this monitoring time was selected in order to develop the model. Through the tests carried out, it can be considered that the developed model presents significant statistical parameters to estimate the waste settlement. Such model can be applied for the estimation of settlement from sanitary landfills with similar features to the ones used in this model.


Introduction
Waste settlement monitoring is one of the most important steps during the sanitary landfill's operation phase.The settlement, also known as waste body displacement, not only affects the geotechnical and environmental aspects of a sanitary landfill but also maximizes its useful life.However, during the sanitary landfill's project implementation phase, this parameter is not taken into account.Only meteorological, social and geographical parameters are usually considered.
Sanitary landfill settlement is a direct consequence of compression, particles rearrangement and biodegradation of municipal solid waste (MSW).According to Qian et al. (2002), the total settlement of a sanitary landfill may vary from 25% to 50% of its initial height.When settlement is predicted, many problems can be minimized, such as damage to the liquid and gas collection system, cracks in the soil cover layer and instability of MSW mass (Durmusoglu et al., 2005 ;Ouvry and Page, 2005;Melo, 2003).
The primary settlement occurs due to loads imposed over the waste mass such as the waste self-weight and the cover layer weight, which influence the compaction pro-cess.The secondary settlement takes place through the MSW decomposition process that can occur for many years until the organic matter biodegradation is completed (Babu & Lakshmikanthan, 2015;Hettiarachchi et al., 2009).According to Farias (2014), the study of waste settlement mechanism is complex but can it be simplified with the use of lysimeter that simulates the behavior of a full scale landfill cell.Bareither & Kwak (2015) instrumented a steel made lysimeter (8.2 m height, 2.4 m diameter) to monitor settlement at various depths in the waste column.Such system consisted of steel plates attached to steel rods that extended from the bottom to the top of the lysimeter.Rafizul et al. (2014) conducted experiments in a lysimeter made of PVC pipe with height and inner diameter of 1.8 m and 2 m, respectively.Several settlement models, as the ones developed by Sowers (1973), Zimmerman et al. (1977), Simões (2000) and Babu et al. (2010), have been used as an attempt to estimate the waste volume reduction over time.However, not all models consider the waste biodegradation processes (El-Fadel & Khoury, 2000).Hachich (2000) states that models are intended to explain reality, predict behavior and support decisions.The efficiency of a model is directly related to its purpose.To explain the reality, models must incorporate all parameters that influence the studied variable.Nonetheless, models should preferably be simple and depend on easily obtainable parameters (Boscov, 2006).
This study aims to develop a simple linear regression model in order to determine the settlement occurring over time in a lysimeter filled with municipal solid waste.

Construction and instrumentation of the lysimeter
The lysimeter (Fig. 1) was constructed in the Environmental Geotechnics Group's experimental field which is located at the Federal University of Campina Grande's main campus, city of Campina Grande, Brazil.The lysimeter was built with solid bricks masonry and had an internal diameter, height and volume of 2.0 m, 3.5 m and 11 m 3 , respectively.The lysimeter's cylindrical shape tends to facilitate waste arrangement and compaction, lateral pressure distribution and prevention of preferential leachate paths.Also, this shape reduces the contact between waste and inner wall's surface.The structure was built on a concrete base.The bottom and cover layer was made with a lowpermeability soil (permeability coefficient: 10 -6 m.s -1 ).The lysimeter had a drainage system composed of a 0.04 m diameter PVC pipe that was drilled and placed on the compacted soil layer.A crushed stone layer was spread over the soil to induce the leachate flow.Besides that, the lysimeter was equipped with gases drainage system, piezo-meter for liquid level measurement, magnetic and circular steel plates for depth and surface settlement measurement as well as thermocouples for temperature measurement over depth (Fig. 2).

Sampling and collection of MSW
In order to obtain a representative sample of the waste from the city of Campina Grande, a planning process was carried out through statistical inference.This statistical investigation presents a confidence level of 95,5% and a maximum error of 5%.
The statistical planning considered the city's population, area and the amount of waste produced per person based on data from the Brazilian Institute of Statistical Geography (IBGE in Portuguese abbreviation).The sample's size was determined through Eq. 1: where Z = abscissa of the standard normal distribution, s = standard deviation of population, N = population size, e = sampling error and n = sample size.
A value of Z = 2.00 was considered, which means a confidence level of (1 -a) = 0,955.For the standard deviation of population, a predetermined value was used though amplitude approximation.The amplitude represented the difference between the higher (1.5 kg) and lower (1.0 kg) waste amount produced by one person in Campina Grande, so the amounts were 532.623 and 355.082 kg, respectively.This way, the standard deviation of population was about 44.385.The population size is equivalent to the number of neighborhoods in the city, which was 50 according to IBGE's database.The admitted sampling error was 5%.
The sample size calculated in Eq. 1 corresponded to the number of neighborhoods, where the MSW would be collected.The obtained value was n = 12.Then, the city was divided into 4 different zones (north, south, east and west).After, a stratified sampling was carried out proportionally to the number of neighborhoods in each zone.Finally, the neighborhoods where randomly chosen (Table 1).
The waste samples were collected from commercial and domestic buildings and were not previously segregated.From the total collected waste amount, about 7,800 kg were destined to the lysimeter filling process and approximately 600 kg were used in the physical characterization (gravimetric composition), which was conducted based on LIPOR's (2000) methodology -adapted by Leite (2008) and Pereira et.al. (2010).

Lysimeter's filling and monitoring processes
The collected waste was homogenized and quartered (according to ABNT NBR 10007, 2004).Then, four buckets (volume of 0.006 m 3 ) were filled with waste samples and dumped into the lysimeter (Fig. 3a) followed by manual compaction (Fig. 3b) that was performed by using a weight and a number of blows that produced a compaction energy proportionally similar to the one generated by a compaction vehicle in sanitary landfill sites.The referred steps (buckets filling, waste deposition into lysimeter, and compaction) were repeated until the predetermined waste body height was achieved.
The surface waste settlement was monitored according to the procedures described by Leite (2008), Melo (2011) and Farias (2014), with necessary adaptations to this specific study.To obtain this parameter, vertical displacements of the metal plates were measured.These plates had a diameter of 0.15 m and were attached to a 0.6 m height rod (Fig. 4b).The two plates were installed after the lysimeter filling phase and were located between the top waste layer and the soil cover layer.The surface settlement measurements were carried out by stretching a nylon string, setting it on the edge of the lysimeter wall, and then measuring the vertical distance between the string and the top of the rods (Figs.4c and 4d).

Statistical treatment of data and settlement model elaboration
The settlement monitoring lasted four years.After this time, the lysimeter's settlement became practically stabilized.Thus, the model considered the database from the monitoring period.
The statistical modeling process consisted of a linear model development using simple linear regression.To achieve this, the waste settlement behavior was observed over time by means of a scatter diagram.Because of data variability, only data in the interquartile range was selected for the model development.According to Paiva (2009), interquartile range represents dispersion measures that are insensitive to disturbing observations.This range is used for outlier's identification and is determined by the Eqs. 1 and 2.   with where L i = lower critical limit; L s = upper critical limit, Q 1 = first quartile, Q 3 = third quartile and A int = interquartile distance.
The normality of the dependent variable (settlement) was verified by the Kolmogorov-Smirnov test, proposed by Massey Jr. (1951).The assumption of data normality is essential for performing statistical inferences and when there is a small sample, it is necessary to carry out tests to check normality.
The correlation coefficient, proposed by Pearson (1896), was determined to check the degree of association between variables of settlement and time.According to Borges (2003), perfect correlation means a result equal to 1; very strong correlation is for values between 0.99 and 0.80; strong correlation is for values between 0.79 and 0.60; weak correlation ranges from 0.59 to 0.40; very weak correlation has values from 0:39 to 0:20; valueless correlation is for 0.19 to 0.001 values, and zero correlation is when the result is equal to 0. To determine if a particular correlation occurred due to either a random sampling error or because of chance, a Student's t-test, developed by Gosset (1908), was applied.
To establish a simple model that describes the relationship between two variables normally distributed (settlement and time), a simple linear regression was performed as described by Galton (1886).Thus, it was used a simple linear regression equation (Eq.5), in which for each monitoring time, a respective settlement value could be calculated.
After choosing the equations to model the behavior of waste settlement over time, the coefficient of determination (R 2 ) was analyzed.Such coefficient measures the relationship between two quantitative variables and ranges from 0% to 100%.The higher the value for R 2 , the greater is the dependence of a variable in relation to another or the better the model fit in relation to the monitored parameters.However, errors may occur and variables may not show any relationship of cause and effect between them.
The residual analysis was performed using the tests of Durbin & Watson (1950) to verify the independence, and Breusch & Pagan (1979) to verify the residual homoscedasticity.According to Fernandes et al. (2014), if any of these assumptions is not met, the model is not suitable and  the obtained deviation has to be corrected or considered in the model.

Gravimetric composition of MSW
Figure 5 shows that 47% of the MSW is composed of putrescible materials.This high amount of organic matter (OM) can influence waste mass settlement when the environment provides favorable conditions for the biodegradation of OM.The percentage of recyclable materials, such as plastic, papers and cardboards, metals and glass, in the total waste composition is shown as well.

Settlement model
Figure 6 presents the dispersion diagram of the studied variables, which shows an association between the values of settlement and time.Initially, there is no linear relationship between the variables.However, from day 19, the monitored data presented a positive linear trend.In other words, as time increased, the values of settlement concomitantly increased.Before day 19, the settlement had a nonlinear trend, which is mainly due to volume reduction caused by mechanical processes (rearrangement of the particles and compaction due to waste's self-weight) that occur in the waste mass during the lysimeter filling process.
According to Hettiarachchi et al. (2009) and Dixon & Jones (2005), waste settlement takes place due to mechanical and biological processes.The initial settlement phase, which is usually 30 days long, is characterized by the predominance of compressibility and mechanical processes.The final settlement phase is mostly due to biological processes.These features may reduce or increase the degree of linear dependence between the variables of time and settlement.
When settlement occurs due to the predominance of mechanical processes, there is a quick waste mass deformation and a non-linear settlement behavior over time.However, when biodegradation processes are present, waste settlement tends to be linear over time, which is probably due to the organic matter biostabilization constant.The interquartile range and its critical limits (Table 2) were determined with the purpose of removing outliers and selecting a range in which the settlements values present a linear behavior over time.
As the interquartile range indicates, the model had to be developed with settlement values above 0.3068 m and less than 0.7368 m.However, before modeling, the normal-  ity of settlement data was analyzed through the Kolmogorov -Smirnov test (Table 3).Considering a significance level (a) of 0.05 and a number of observations (n) of 67, it can be seen in Table 3 that D calculated is less than the D tabulated .Thus, there was no evidence to reject the normality assumption.Also, research data followed a normal distribution since the p-value is greater than 0.05 (null hypothesis).Tables 2 and 3 only analyzed the response variable (settlement) because time is an independent variable, not requiring such studies since its variation does not depend on settlement.
The selected range for determining the linear regression model presented a correlation coefficient of 0.98, which indicates that settlement and time are directly related.In other words, there is a strong linear correlation between these variables, but this correlation may have occurred by chance.Because of that, a Student's t-test was carried out to verify if the association between the variables is real.With a significance level of 0.05 and a degree of freedom of 65, the t calculated = 39.4 was higher than the t critical = 1.10, meaning that the correlation is statistically significant.
The linear regression was determined with R 2 = 0.97, indicating that 97% of the settlement variability can be explained by the model during the considered time range.Through Table 4, it is verified that p-value was lower than 0.05, so the regression has statistical significance.Also, as indicated by the angular coefficient of the linear regression equation, for each additional day, it is expected that the waste mass presents a settlement of 0.00055 m.
Equation 6 was used to perform predictions about the settlement that occurred in the lysimeter over the time range between 19 and 809 days.
where y = settlement (mm) and x = time (day).According to Eq. 6, it is expected to occur a deformation of 0.805 m in the waste mass until day 809.Because of that, there will be a gain of available spaces that might be used to storage more waste.This way, such space saving could maximize the sanitary landfill useful life.
According to Table 4, the upper and lower equations can be determined, with 95% of confidence (Eqs.7 and 8), from the developed model, which means that the observed data is within this range.where: y = settlement (mm) and x = time (day).
Figure 7 shows the model's trend lines and the settlement values from the lysimeter monitoring.
To investigate the suitability of the regression model and validate it, a statistical residual analysis was conducted through the regression diagnosis.The Goldfeld-Quandt test (Table 5) was used to test the null hypothesis in which the errors variances are equal (homoscedasticity) and the alternative hypothesis in which the errors variances are a multiplicative function of the variable.
As it is seen in Table 5, the statistical residual has homoscedasticity since the p-value is greater than the significance level of 0.05.Thus, the standard errors of the estimators obtained by the method of least squares are consistent and statistical inference is valid, so it is possible to conclude that the estimators of the linear regression equation are appropriate.
To verify if the residual is independent, the Durbin-Watson test (Table 6) was performed.Such test had the statistical residual dependence as null hypothesis and statistical residual independence as alternative hypothesis.
According to Table 6, 0 £ DW £ DL and the p-value was less than 0.05, which means that the null hypothesis can be rejected.Therefore, it is possible to affirm, with a confidence level of 95%, that the residuals are independent and have a positive correlation.

Conclusion
The developed model considered significant statistical parameters to estimate waste settlement.Such model can be applied for estimating settlement values from sanitary landfills with similar features to the ones used in this model.
The lysimeter's total waste settlement was about 1.1 m during a monitoring time of 1300 days.The settlement prediction through the developed model is only indicated for an amount of time longer than 19 days and less than 809 days.However, this fact does not disqualify the proposed model since adjustment difficulties for models like this one are typical during the initial period.
To obtain a model that reflects the initial monitoring time, other variables such as waste density, cover layer and 56 Soils and Rocks, São Paulo, 40(1): 51-59, January-April, 2017.

Figure 2 -
Figure 2 -Schematic view of the lysimeter.

Figure 3
Figure 3 -a-waste disposal into the lysimeter and d -Waste mass compaction.

Figure 4
Figure 4 -a-Nylon string; b -Metal surface plates; c and d -Surface settlement-measurement.

Figure 7 -
Figure 7 -Dispersion of the predicted settlement values adjusted by the models.

Table 1 -
Amount of MSW collected in the selected neighborhoods from Campina Grande.