Reliability of the liner system using the response surface method

Environmental Geotechnics

Reliability of the liner system using the response surface methodSanthosh and Sivakumar Babu

Environmental Geotechnicshttp://dx.doi.org/10.1680/envgeo.13.00002Paper 13.00002Received 14/03/2013; accepted 13/06/2013Keywords: groundwater/landfill/risk & probability analysis

ICE Publishing: All rights reserved

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Reliability of the liner system using the response surface method

NotationC contaminant concentrationC0 initial concentration of the chloride contaminantCr rate of increase in concentration with time due to the

addition of massC−1 inverse of the covariance matrix of random variablesCaq chloride concentration in the base aquiferCaqlim maximum threshold value for chloride concentration

in the aquifer (denoted as g(x) in the reliability or probability part)

D coefficient of hydrodynamic dispersion or clay diffusion coefficient of the compacted clay liner

Dg coefficient of hydrodynamic dispersion or clay diffusion coefficient of the geomembrane

e error from other sourcesF failure region in the transformed spaceHb entire thickness of the base stratahb thickness of the base aquiferHr reference head of the leachate above geomembraneKd distribution/partitioning (sorption) coefficient of a

contaminantL length of the landfill parallel to the base outflow velocity

in the aquiferM vector of mean value of random variablesn porosity of the compacted clay layer and base soil

nb porosity of the base aquiferqc volume of leachate collected per unit area of the landfill

per unit timeT transposet timeV advective velocityVb Darcy velocity or base outflow velocity in the aquiferX vector of random variablesy surface responsez depth below ground levelb reliability indexf(b ) probability of failure of landfill liner systemf(–b ) probability of no failure of landfill liner systemµ mean of a variableρ dry density of the clay soilσ standard deviation of a variable

IntroductionLandfills are one of the most significant sources of groundwater

contamination. Leachate generated due to biodegradation of

waste and infiltration of rainwater reaches groundwater through

the subsurface soil. Leachate contains dissolved organic matter,

inorganic macro components, heavy metals and so on. Among

various contaminants, chloride (Cl–) is an ion that is of great

concern, and excess amounts of chloride can make water salty

One of the most convenient ways to communicate the risk of groundwater contamination in the vicinity of a landfill

is through the use of probability of failure or reliability of the containment system of the landfill. The reliability of

the engineered containment system influences the risk of groundwater contamination. In this paper, an approach to

understanding the effectiveness of the liner system using the response surface method, taking into consideration the

variability of five parameters (initial chloride concentration, clay diffusion coefficient, soil porosity, aquifer porosity

and base outflow velocity) is presented. Comparative analysis was performed with respect to the relationship of the

reliability index and the coefficient of variation of each variable in order to understand their influence on the reliability

of the liner system. The implications of the study are presented here. It may be noted that the developed response

surface equations are only valid for the conditions considered in this analysis.

2 G.L. Sivakumar Babu ME, PhD Professor, Department of Civil Engineering, Centre for Sustainable

Technologies, Indian Institute of Science, Bangalore, Karnataka, India

1 L.G. Santhosh MTech Research Scholar, Centre for Sustainable Technologies, Indian Institute

of Science, Bangalore, Karnataka, India

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Environmental Geotechnics Reliability of the liner system using the response surface methodSanthosh and Sivakumar Babu

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and undrinkable. Not all forms of chlorine are harmful, however,

chloride-containing compounds can pose a risk to human health.

The threshold for chloride concentration in drinking water ranges

from 200 mg/l to 300 mg/l (Bureau of Indian Standards (BIS),

2005, 2009; World Health Organization (WHO), 2011). Rowe

(1991) indicates that a permissible level of 125 mg/l is used to

define the lower bound for drinking water in Canada. One of the

most convenient ways to communicate risk is through the use of

probability, or severity of risk, of chloride contamination, and the

reliability of the containment system of the landfill as an engineered

structure. Quantification and risk assessment of groundwater

contamination, as well as the reliability of the containment system,

play a decisive role in taking remedial measures to reduce adverse

affects on the environment and human health in the vicinity of

landfill sites. Pre-assessment of risk also guides engineers and

policy makers in efficient solid-waste management. Parameters

associated with groundwater contamination are variable in nature.

The effect of this variability needs to be properly analysed to

obtain rational inputs on which landfill design guidelines can be

evaluated.

Literature reviewMany researchers have been working on understanding the

process of contaminant migration in the subsurface environment

through experiments, measurements and modelling. In the analysis

of contaminant migration, chloride is normally chosen as the

contaminant ion on which to evaluate containment system designs

as it is a conservative ion. Measurements of the contamination of

groundwater sources close to landfills due to chloride have been

made in India. Kumar and Alappat (2003) analysed leachate from

the Okhla municipal landfill, New Delhi, and found chloride

concentration in excess of 16 000 mg/l. Mor et al. (2006) indicated

that the high chloride content of groundwater is likely to originate

from pollution sources such as domestic effluents, fertilisers and

septic tanks, as well as from natural sources such as rainfall and

the dissolution of fluid inclusions. Srivastava and Ramanathan

(2008) analysed leachate samples in Delhi and indicated chloride

concentration in the range 4000–4500 mg/l. Venkatesan and

Swaminathan (2009) reviewed the status of contamination of

groundwater in nearby landfill sites due to chloride and sulfate.

Jhamnani and Singh (2009) found a chloride concentration of

4000 mg/l in a leachate sample from the Bhalaswa landfill, Delhi.

Singh and Mittal (2009) conducted tests to analyse the leachate

characteristics of the municipal solid waste (MSW) landfill in

Delhi and reported that chloride concentration varied between

16 000 mg/l and 23 000 mg/l in the area under study. Kanmani and

Gandhimathi (2013) studied the physicochemical characteristics of

leachate and reported that chloride concentration ranges between

1996 mg/l and 6998 mg/l. Ramprasad (2012) characterised

leachate samples from a landfill in Chennai, India, and reported

chloride concentration in the range 1505–1667 mg/l. D’Souza and

Somashekar (2013) analysed the physicochemical characteristics of

leachate samples from various landfills in Bangalore city and found

that the concentration of chloride in the leachate varied between

1300 mg/l and 3900 mg/l.

Gillham et al. (1984) presented advection–dispersion and

advection–diffusion models for contaminant solute transport

process in heterogeneous unconsolidated geological deposits.

Rowe and Booker (1985) described a technique to analyse

contaminant transport from a landfill into a homogeneous clayey

layer in a two-dimensional space, taking into consideration the

concentration of the contaminant as it infiltrates into the soil. They

showed that there is a critical velocity in the aquifer at which the

maximum concentration at a point occurs. They also investigated

contaminant transport through soils, barriers, natural clayey soils,

two- or three-dimensional fractured networks, fractured rock, intact

shale, fractured till, layered zones and clayey silt underlain by fine

sand or silt using numerical and analytical models. Abriola (1987)

presented details on contaminant transport when in an immiscible

phase and in fractured media, along with dispersion modelling and

numerical modelling for saturated groundwater aquifer systems.

Giroud and Bonaparte (1989) modelled and analysed the leakage of

contaminant through geomembrane alone and with low permeable

soil layers assuming perfect contact and interface flow between the

layers in two- and three-dimensional space. Reddi and Scarlatos

(1996) collected actual data and modelled migration of leachate

from 11 landfills in Florida, installed with modern liner systems,

and revealed that the liners provided were effective in preventing

groundwater contamination. Foose et al. (2002) modelled transport

of cadmium and toluene through geomembrane, geosynthetic

clay liner and thick soil barrier using one- and three-dimensional

numerical models under various conditions. Peters et al. (2002)

modelled solute transport through a typical porous medium from

an engineered landfill, considering the porous medium to be

rigid material. El-Zein (2008) developed an approach to study

time-dependent contaminant migration through composite liners

with leaking geomembranes for various combinations of system

conditions, state of waste, constant mass of contaminants in the

waste, steady-state or transient flow in the geomembrane and

steady-state or transient seepage velocities in the mineral liner. Du

et al. (2008) evaluated the performance of the German standard

MSW landfill bottom liner system and compared two types of

bottom liner systems considering the one-dimensional advection–

diffusion–dispersion theory of contaminant transport process.

Varank et al. (2011) carried out field studies on four identical

pilot-level landfill reactors with different alternative composite

liners. These were operated simultaneously over a period of about

540 days to investigate and simulate the migration behaviours of

phenolic compounds and heavy metals from the landfill leachate to

the groundwater.

Several methods of risk and reliability analysis are available in

the literature. The most widely used methods are the Monte Carlo

simulation, first-order reliability method (FORM) and second-order

reliability method (SORM). Many researchers have used these

methods to predict uncertainties involved in the parameters that

influence contaminant transport. Jim Yeh (1992) made an attempt

to understand the uncertainties in predictions and estimations of

groundwater flow and solute transport using stochastic modelling

techniques. Jang et al. (1994) applied FORM and SORM to

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model contaminant transport probabilistically in one and two

dimensions, taking into account the random spatial variability of

the input parameters, along with their spatial correlation. Adams

(1994) examined the use of the response surface method (RSM) as

a parameter estimation technique for groundwater flow modelling

at a landfill site. The author calibrated three hydraulic parameters

(porosity, transverse permeability and rate of recharge) of an

existing two-dimensional, steady-state flow model. The regression

model showed that the response had a high degree of variability

for transverse permeability. Hamed and Bedient (1999) presented

a probabilistic modelling tool based on FORM and SORM to ana-

lyse uncertainty involved in the contaminant transport process. This

methodology was also applied to analytical groundwater models to

provide a simple screening tool for the assessment of contamination

and remediation. In addition, numerical-analysis-based reliability

models were developed to account for aquifer spatial heterogeneity

and correlation structure in more complex systems. Ranade et al.

(2010) described the use of stochastic response surface technique

to estimate the uncertainty in groundwater modelling. Datta and

Kushwaha (2011) studied the importance of various hydrogeologic

parameters in the uncertainty assessment of the contaminant

concentration using stochastic response methodology.

Though measurements and modelling tools are available, there is a

need to develop a methodology to assess the integrity of the landfill

liner system.

ObjectivesThe reported case studies of contamination of groundwater in India

presented in Section 1×1 highlight the need for a methodology

for reliability analysis of liners in landfills. In this paper, an

approach to assess the risk of the landfill liner system contributing

to groundwater contamination due to chloride migration near a

landfill site is presented. The approach uses the response surface

equations obtained from RSM, which are subsequently used for

reliability analysis. It may be noted that the developed response

surface equations are only valid for the conditions considered in

this analysis.

MethodologyModeling of contaminant transport beneath landfill siteBiodegradation of MSW stored in landfills generates a huge quantity

of leachate. This degradation process in the landfill is very complex

and continues for a very long time. The contaminant transport

process is difficult to understand due to the complex matrix of

the waste mass and the heterogeneity of the subsurface soil. The

single composite liner landfill model shown in Figure 1 consists

of an upper component of a flexible-membrane liner and a lower

component of compacted clay layer (CCL) and is considered for

one-dimensional modelling in the computer program POLLUTE

v7 (SSG, 2012). Assuming uniform groundwater velocity (V) and

linear sorption of contaminant, the governing equation for one-

dimensional contaminant migration for an intact material, as given

by Rowe and Booker (1985, 1987, 1991), is

1.

2

2

ddC d C dC K dCD V

dt dz n dzdz

ρ= - -

The top boundary is assumed to have finite mass, in which case

the source concentration starts at C0, increases linearly with time

Finite mass

10·0-m-thick municipal solid waste

1·524-mm-thick geomembrane

1·0-m-thick compacted clay liner

5·0-m-thick clayey soil(attenuation layer)

3·0-m-thick alluvial aquifer

Fixed base outflow

1·0 m

Figure 1. Cross-section of the single composite liner landfill model



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at a rate of Cr and then decreases with time t as contaminant is

transported into the soil and collected by a leachate-collection

system (1) if present. Therefore, the concentration at the top

boundary is given by

2. c

0 r

r r

1( ) ( , ) ( ) .

t t

o o

qC t C C t f C t dt C t dt

H H

= + - -ò ò

The bottom boundary is assumed to have a fixed outflow to

represent a base aquifer, where the concentration varies with time

as contaminant mass is transported into the aquifer from the landfill

and transported out of the aquifer with velocity Vb. Consideration of

the conservation of mass gives the base concentration as

3. b b b

b

b b b0

( , , ( , )( , )

t

f t z H C V C t z HC t z H dt

n h n L

ì ü= == = -í ýî þ

ò

To model the one-dimensional contaminant migration process

through the clayey subsurface soil mass, a landfill with a length

of 700 m and a width of 1 m is considered as the source of

contamination. The longitudinal dimension of the landfill is parallel

to the groundwater table. The Resource Conservation and Recovery

Act, Subtitle D (USC, 1976) Landfill consists of a composite liner

and leachate-collection system. The contaminant source type is

assumed to be a finite mass with a waste thickness of 10 m. The unit

weight of MSW varies between 5 kN/m3 and 10×4 kN/m

3 (Zekkos

et al., 2006). Kavazanjian et al. (1995) studied the variation of waste

properties in landfills and indicated that the unit weight of MSW

varied from 6 kN/m3 at the surface to about 13 kN/m

3 at depths of

45 m or greater. Therefore, in this study, the unit weight of the waste

is taken as 6 kN/m3. Moo-Young et al. (2004) reviewed the literature

to characterise infiltration rates from landfill liners and indicated that

average infiltration rates through the CCL range 1×6 × 10−9

cm/s to

1×2 × 10−6

cm/s, with an average of 3×1 × 10−7

cm/s. Therefore, in

this study, an infiltration rate of 4×75 × 10−8

cm/s is assumed. Rowe

(1995) reported that chloride constitutes about 0×07–0×21% of the in

situ mass of waste with 0×18% as the most probable value and 0×2%

as a reasonable value for leachable chloride mass. Based on this,

the mass of the chloride contaminant is taken as 0×2% of the total

mass of the waste. In India, leachate-collection systems are neither

effectively installed nor efficiently working to remove the stagnant

leachate from the bottom of the waste matrix. Hence the leachate-

collection system is not considered for modelling of the contaminant

transport process in the present study. Based on the available

literature, an initial chloride concentration of 1500 mg/l to 4000 mg/l

with a standard deviation of 622 mg/l is considered for analysis.

The hydraulic head of the leachate above the geomembrane is

taken as 1 m. Geomembrane placed above the 1-m-thick clay

liner has a thickness of 1×52 mm. The diffusion coefficient for

the geomembrane is assumed to be 3 × 10–5

m2/year. Leakage

through the defects in a composite liner system is computed

using the equations given by Rowe et al. (2004). Porosity of the

clayey soil varied between 0×4 and 0×6 and is in accordance

with the study by Adams (1994). Giroud and Bonaparte (1989)

indicated that three to five defects per hectare can be expected

even when the geomembrane is installed well above the CCL.

The number of defects in the geomembrane per hectare is taken

as 2×5, with radius, transmissivity and conductivity of 5×64 mm,

1 × 10–10

m2/s and 0×1 mm/s respectively (Rowe et al., 2004). The

bulk density and hydraulic conductivity for 1-m thick CCL below

the geomembrane and 5-m thick clayey soil mass (attenuation

layer) are taken as 19 kN/m3 and 1 × 10

–9 m/s respectively. In

clayey soils, the coefficient of hydrodynamic dispersion is often

controlled by the diffusion coefficient, and the coefficient of

mechanical dispersion is negligible. Therefore, the study is carried

out considering the clay diffusion coefficient D values in the

range 0×018–0×020 m2/year as determined by Rowe and Booker

(1987). With the chloride ion being a conservative contaminant

(no sorption), the value of Kd is taken as zero for both the CCL and

the attenuation layer (Rowe, 1995). The thickness of the alluvial

aquifer is considered as 3 m with porosity nb in the range 0×2–0×4

(Adams, 1994; Rowe and Booker 1985). A summary of the input

values for each variable is presented in Table 1. It is difficult to

determine the values of the base flow velocity (Rowe, 1995),

Landfill component Value

Source Landfill length 700.0 mLandfill width 1.0 mSource type Finite mass

Waste thickness 10.0 mWaste density 6 kN/m3

Per cent of mass 0.2Hydraulic head Leachate head on

primary landfill liner1.0 m

Groundwater level relative to top of the

aquifer

0.0 m

Geomembrane Thickness 1.52 mmDiffusion coefficient 3 × 10–5 m2/year

Leakage Hole frequency 2.5/haHole radius 5.64 mm

Wrinkle radius 5.64 mmTransmissivity 1 × 10–10 m2/sConductivity 0.1 mm/s

Distribution coefficient 0 ml/gClay liner Thickness 1.0 m

Density 19 kN/m3

Conductivity 1 × 10–9 m/sAquitard (attenuation layer)

Thickness 5.0 mDensity 19 kN/m3

Conductivity 1 × 10–7 m/sAquifer Thickness 3.0 m

Table 1. Summary of input values considered for modeling



5

and the velocity of the water is very slow in the alluvial aquifer.

Hence, the value of the base flow velocity is considered to be in

the range 3×5–4×5 m/year.

Response surface methodThe RSM is a collection of mathematical and statistical techniques

useful in a situation where a greater number of variables are

potentially influencing or controlling the outcome response of the

process. It can be found from the literature that many researchers

have applied RSM in various fields to develop, improve and optimise

the performance measure or the quality characteristic of the process

that involves uncertainties in the variables. In this study, using the

RSM, an approximate explicit functional relationship between the

output response – that is, chloride contaminant concentration in the

aquifer (Caq) – and input variables x1, x2, x3, x4 and x5 for initial

chloride concentration C0 (mg/l), clay diffusion coefficient D (m2/

year), soil porosity n, aquifer porosity nb and base outflow velocity

Vb (m/year), respectively, is established using regression analysis

with least-squares error approach. This relationship is termed as a

response surface equation and is given as

4. 1 2 3 4 5( , , , , )y f x x x x x e= +

A normal probability plot that follows a straight line defines the

adequacy of the fitted model and gives a good approximation of

the true system (Babu and Srivastava, 2007). The coefficient of

multiple determination R2 is a measure of the amount of reduction

in the variability of output response y obtained by incorporating the

uncertain variables (Myers and Montgomery, 2002).

In RSM, a single replicate 2n (n is the total number of input variables)

factorial design is used to fit a first-order linear regression model.

For developing a response surface equation, 32 sample points are

required. The upper and lower limits related to the mean (µ) and

standard deviation (σ) can be defined using relationships xu = m +

1×65σ and xl = m – 1×65σ, respectively. The RSM equations are

developed after computing the rate of contamination in the aquifer

(Caq) for each set of sample points. Regression analysis is also

performed to check the adequacy model.

Reliability analysisIt is known that parameters involved in the contamination

migration are random variables and hence the use of reliability

analysis is desirable. For landfills, though many guidelines have

been formulated, there is a need to examine to what extent these

guidelines correspond to reliability-based guidelines, using the

reliability index. The second-moment reliability index b as defined

by Hasofer and Lind (1974) is generally considered to be a better

approach for reliability-based designs in civil engineering (Low and

Tang, 1997). The reliability index b is calculated on the basis of

the matrix formulation for the second moment and Hasofer–Lind

reliability index (b) defined as

5. 1min ( ) ( )

T

X F

X M C X Mβ -Î

= - -

The reliability index b in the present study is defined in terms of

f(b ) exceeding the permissible level of concentration in ground-

water, given that the influencing factors such as C0, D, n, nb and Vb

are random variables.

Variables Mean Standard deviation CoV: % Minimum Maximum Distribution

Initial chloride concentration, C0: mg/l 2750 622 22.6 1723.7 3776.3 NormalClay diffusion coefficient, D: m2/year 0.0190 0.001 5.3 0.0174 0.0207 NormalPorosity, n 0.33 0.013 4.0 0.3036 0.3465 NormalAquifer porosity, nb 0.275 0.0413 15.0 0.2069 0.3431 NormalBase outflow velocity, Vb: m/year 4.00 0.2 5.0 3.67 4.33 Normal

Table 2. Statistics of variables for case I

Variables Mean Standard deviation CoV: % Minimum Maximum Distribution

Initial chloride concentration, C0: mg/l 2750.0 137.5 5.0 2523.13 2976.88 NormalClay diffusion coefficient, D: m2/year 0.019 0.00095 5.0 0.0174 0.0206 NormalPorosity, n 0.325 0.01625 5.0 0.298 0.351 NormalAquifer porosity, nb 0.275 0.01375 5.0 0.252 0.297 NormalBase outflow velocity, Vb: m/year 4.00 0.2 5.0 3.67 4.33 Normal

Table 3. Statistics of variables for case II



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Results and discussionThis study has been carried out for two cases. In one case, referred

to as case I, different coefficient of variation (CoV) values based

on the data available in the literature are considered and represent

realistic scenarios. In the second case, referred to as case II, all the

variables are assumed to have a CoV of 5% and is a typical analysis

in reliability calculations. This is done to examine the variation

of regression coefficients in the two cases and the corresponding

effect on reliability analysis results. Statistics of the five variables

for both cases are presented in Tables 2 and 3.

In order to develop a multi-linear response surface model between

input and output variables, 25 = 32 sample points are used. Regression

analysis with the least-squares error method is adopted to fit the

response surface model. Coefficients of multiple determination R2

and adjusted coefficients of multiple determination 2

adjR defining

the adequacy of the fitted linear model are evaluated. Residual

analysis is also performed to check the normality assumption.

Normal probability plots of residuals for case I and case II are

shown in Figures 2 and 3, respectively. These plots approximately

follow a straight line, indicating that the assumption of normality

is satisfactory.

The statistics of the results that show the Caq (at a depth Z = 6 m, on

the top layer of the aquifer) at a time period of 100 years, obtained

from 32 sample points for each of the cases mentioned above, are

tabulated in Table 4.

Accordingly, response surface functions for case I and case II are

formulated using the results of regression analysis for 32 sample

points and are given in Equations 6 and 7, respectively.

6.

Caq = 769×73 + 0×0942C0 + 12 878×79D – 2291×08n

– 670×095nb – 17×55Vb

7.

Caq = 760×78 + 0×0929C0 + 12 806×61D – 2266×69n

– 670×495nb – 16×592Vb

Coefficients of multiple determination R2 and adjusted coefficients

of multiple determination 2

adjR obtained from regression analysis are

presented in Table 5. Statistics of the regression analysis showed

that standard error from the RSM model for case I is greater than

that from the RSM model of case II. Nevertheless, the values of

R2 in both cases are greater than 0×94 and are considered to be

satisfactory.

Reliability analysis is carried out using a spreadsheet algorithm

for FORM in a Microsoft spreadsheet developed by Low and Tang

(2007), solving Equation 5 using the solver tool of the Microsoft

spreadsheet. In order to optimise the reliability index b, the

constraint g(x) = Caqlim = 200 mg/l is considered for both cases.

To identify the influencing variables, further analysis is performed

and variation of the reliability index b with parameters C0, D, n, nb

1009080706050403020100–2 –1·5 –1 –0·5 0 1 2 30·5 1·5 2·5

Residuals

Nor

mal

pro

babi

lity:

%

Figure 3. Normal probability plot of residuals for case II

1009080706050403020100–1·5 –0·5–1 0 1 1·5 2 32·50·5

Nor

mal

pro

babi

lity:

%

Residuals

Figure 2. Normal probability plot of residuals for case I

Regression statistics Case I Case II

Multiple R 0.9762 0.9957R2 0.9529 0.9915Adjusted R2 0.9439 0.9899Standard error 29.5417 7.1211Sample points 32.0 32.0

Table 5. Regression statistics from regression analysis for both cases

Statistics Case I Case II

Mean 273.60 272.30Standard deviation 124.70 70.71CoV: % 45.50 26.00

Table 4. Statistics for contaminant concentration in the aquifer (Caq at Z = 6 m) at a time period of 100 years in both cases



7

and Vb is obtained. The response pattern in Figure 4 indicates that

an increase in the source concentration of the chloride contaminant

leads to lower values of the reliability index b, indicating higher

risk of groundwater or aquifer contamination.

Diffusion is one of the most prominent mechanisms of the

contaminant transport process. It dominates advection and

dispersion in clayey soil mass. Figure 5 shows that an increase in the

diffusivity of the contaminant through clayey soil will eventually

increase the risk of groundwater contamination.

Figures 6 and 7 show that an increase in soil and aquifer porosity

reduces the risk of contamination. Figure 8 indicates that an

increase in the base outflow velocity has no significant influence

on the reliability index b or risk of contamination. Furthermore,

the choices of permissible chloride concentration can influence

the reliability index b, the results of which are presented in Table 6

and Figure 9.

The results of sensitivity analysis expressed in terms of percentage

change in reliability index b as a result of percentage change

2·5

2·0

1·5

1·0

0·5

0·0

Relia

bilit

y in

dex

β

0·0175 0·018 0·019 0·02 0·02050·0185 0·0195

Case I

Case II

Clay diffusion coefficient, D: m2/year

Figure 5. Comparison between the reliability index b and the clay diffusion coefficient D for both cases

2·5

2·0

1·5

1·0

0·5

0·00·305 0·31 0·32 0·33 0·340·315 0·325 0·355 0·345

Soil porosity, n

Case I

Case II

Relia

bilit

y in

dex

β

Figure 6. Comparison between the reliability index b and the soil porosity n for both cases

2·01·81·61·41·21·01·80·60·40·20·0

Relia

bilit

y in

dex

β

3·8 3·9 4 4·1 4·2 4·3 4·4 4·5Base outflow velocity, Vb: m/year

Case II

Case I

Figure 8. Comparison between the reliability index b and the base outflow velocity Vb for both cases

2·5

2·0

1·5

1·0

0·5

0·00·2 0·22 0·24 0·26 0·28 0·3 0·32 0·34 0·36

Aquifer porosity, nb

Relia

bilit

y in

dex

β

Case I

Case II

Figure 7. Comparison between the reliability index b and the aquifer porosity nb for both cases

Cases g(x): mg/l b f(b ): % f(–b ): %

Case I 200 1.014 84.48 15.52150 1.702 95.57 4.43125 2.047 97.97 2.03100 2.319 99.17 0.83

Case II 200 1.7180 95.71 4.28150 2.908 99.82 0.18125 3.503 99.97 0.03100 4.099 99.99 0.01

Table 6. Variation of reliability index with threshold values g(x)

5·0

4·0

3·0

2·0

1·0

0·00 500 1000 1500 2000 2500 3000 3500 4000

Initial chloride concentration, C0: mg/l

Case I

Case II

Relia

bilit

y in

dex

β

Figure 4. Comparison between the reliability index b and the initial chloride concentration C0 for both cases



8

(1%) in each of the variables are presented in Table 7. It should

be noted that porosity n (7×14%) is the critical variable, followed

by C0 (4×26%), D (2×8%), nb (1×41%) and Vb (0×6%). Hence it is

important that these parameters, as variables, need to be estimated

properly to assess the risk of failure of the liner system and

subsequent groundwater contamination.

Summary and conclusionsThis paper provides an approach to assess the risk of the landfill liner

system contributing to contamination of groundwater due to chloride

migration near a landfill site using RSM. Properties of MSW, primary

liner systems and attenuation layer were considered for the analysis.

Among various parameters, initial chloride concentration C0, clay

diffusion coefficient D, soil porosity n, aquifer porosity nb and base

outflow velocity Vb were considered as variables.

Using the RSM, approximate explicit functional relationships

between the output response — that is, chloride contamination

in the aquifer (Caq) — and uncertain input variables C0, D, n, nb

and Vb, were established using regression analysis. RSM models

were developed for two cases. Case I was for varied CoV values

of each variable and case II was for 5% of CoV values for all five

variables. Reliability analyses show a low probability of failure of

the liner system in case I compared to case II and also indicate the

low probability of groundwater contamination in case I.

To determine the influence of the selected variables on groundwater

contamination, the variation of reliability index b and CoV of variables

C0, D, n, nb and Vb, for both cases were studied. Sensitivity analysis

showed that porosity n is the critical variable followed by C0, D, nb

and Vb. Furthermore, the choice of permissible chloride concentration

influenced the reliability index b. It may be noted that the developed

response surface equations are only valid for the conditions considered

in this analysis. It is possible to assess better the risk of the landfill

liner system contributing to the groundwater contamination near

a landfill site using the proposed approach and considering other

variables that have a high degree of variability. Computational effort

also significantly increases as the number of sample points increase,

along with an increase in the number of variables.

AcknowledgementsThe work presented in this paper is part of the research work in the

project entitled ‘Development of probabilistic design and analysis

procedures in radioactive waste disposal in NSDF and design of

NSDF closure’ supported by the Board of Research in Nuclear

Sciences, Department of Atomic Energy, Government of India,

Mumbai. Their financial support is gratefully acknowledged.

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Reliability of the liner system using the response surface method

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