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Reliability-based assessment of compacted lateritic soil liners
Computers and Geotechnics 32 (2005) 505–519 www.elsevier.com/locate/compgeo
Reliability-based assessment of compacted lateritic soil liners J.O. Afolayan a, C.M.O. Nwaiwu
b,*
a
b
Department of Civil Engineering, Ahmadu Bello University, Zaria, Kaduna State, Nigeria Department of Civil and Water Resources Engineering, University of Maiduguri, Maiduguri, Borno State, Nigeria Received 5 January 2005; received in revised form 5 August 2005; accepted 22 August 2005 Available online 2 December 2005
Abstract The first-order reliability method is employed to assess the suitability of lateritic soils as landfill liner materials based on existing models developed from laboratory and field data. The statistical characteristics of lateritic soil properties were first compared with those from which the regression models were developed. Using regression models for hydraulic conductivity, k, and established distributions for the relevant soil parameters, reliability indices are computed considering hydraulic conductivity to have a normal/ log-normal distribution. The results indicate that, for the laboratory-based model, reliability index is sensitive to variability in compactive effort and initial degree of saturation. The choice of distribution of initial degree of saturation affects reliability indices significantly and the energy of the British Standard heavy compaction has the highest reliability indices. On the basis of the reliability indices estimated from the field-based model, three variables, namely: compactor weight, initial saturation and plasticity index are identified as significantly affecting the field geometric mean hydraulic conductivity of lateritic soils. However, the hydraulic conductivity of lateritic soils can be better predicted if the compactor weight, as the most significant parameter, can be modelled as Gumbel-type distribution. 2005 Elsevier Ltd. All rights reserved. Keywords: Compaction energy; Compacted soil liners; Hydraulic conductivity; Lateritic soils; Reliability analysis; Reliability indices; Soil properties; Statistical analysis; Statistical distributions
1. Introduction The concept of reliability analysis or evaluation has been successfully applied in many aspects of geotechnical engineering designs. The most common applications of reliability theory are in the areas of slope stability [1,2], offshore structures [3] and stability of anchored sheet pile wall [4]. However, there have been applications of reliability analysis in geoenvironmental engineering in recent times. Most prominent among these analyses are for liners of landfills [5,6]. Reliability analysis has also been applied to contaminant transport in porous media [7]. Most landfills used for the containment of municipal solid wastes and hazardous wastes often employ compacted natural soils as liner and cover materials (i.e., *
Corresponding author. E-mail address: [email protected] (C.M.O. Nwaiwu).
0266-352X/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2005.08.001
hydraulic barrier layers). However, in the absence of suitable natural soils, processed clays or sand/processed clay mixtures can be used as barrier materials. In order to ensure proper functioning of barrier layers in landfills, preliminary laboratory tests are usually conducted for each selected soil type to establish appropriate ranges of water contents and dry densities needed to achieve the required hydraulic conductivity (this is usually expected to be less than 1 · 109 m/s). The procedures for defining the acceptable zone of water content and dry density on the compaction plane have been described by [8]. Developing this acceptable zone on the compaction plane in the field may be impracticable as a result of high costs. This is because different compaction plants will need to be hired and several field measurements of dry unit weights and moisture contents made for each soil to be used. Although laboratory results are usually employed, it will be noted
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that laboratory-scale compaction does not perfectly duplicate the repeated passage of compaction equipment over each lift of soil in the field. Thus, it is not necessary to completely rely on results obtained from laboratory experiments. This is because there can be differences between laboratory-scale and field-scale hydraulic conductivities. In order to assess the suitability of a given soil type for use as hydraulic barrier material, the hydraulic conductivity of such a soil may be predicted using any existing prediction model, especially one that is developed from field data as in [9]. The possibility of extending the use of this model for predicting field hydraulic conductivity to tropical soils such as lateritic soils (as the model was developed from data on temperate zone soils) may be assessed using a reliability approach. Lateritic soils which are generally abundant in many parts of the tropics constitute an important soil group that may be considered for use in waste containment. The differences in the modes and environments of formation of tropical and temperate zone soils give rise to differences in basic engineering properties and behaviour of the soils occurring in these climatic regions. For instance, lateritic soils are characterised by the presence of sesquioxides (i.e., Fe2O3 and Al2O3) which have considerable influence on their properties and behaviour [10]. According to Ola [11], the most significant differences between temperate zone soils and lateritic soils are the significant proportions of iron and aluminium oxides found in lateritic soils which tend to cement the soil particles to form a coarse-grained weakly bonded (aggregated) particulate material and the higher temperature of laterite formation which is a feature of tropical climate compared to a temperate one. Due to the immense necessity of ensuring safe environmental containment of wastes, the uncertainties associated with the variability of properties of lateritic soils (especially those that significantly affect hydraulic conductivity) should be accounted for using a suitable probabilistic approach. The objective of this study was to assess the suitability of compacted lateritic soils as hydraulic barrier materials in landfill waste containment using the first-order reliability method. Limit state functions were first established employing deterministic models in published literature that were developed from both laboratory and field data. The values of the relevant variables identified in the limit state function were quantified statistically for lateritic soils in a database. The first-order-reliability method was employed to estimate reliability indices from the predictive models for varying inputs. The results showed that, for the laboratory-based model, variations in compactive effort and initial degree of saturation have the most significant impact on the reliability of soil liner hydraulic conductivity. On the other hand, compactor weight, initial degree of saturation, and plasticity index are the most critical variables in landfill liner construction using lateritic soils based on the field model.
2. Reliability analysis 2.1. Limit state function The reliability analysis in this study for compacted lateritic soils is based on post-construction performance characterized by only hydraulic conductivity. The reliability is: R ¼ Pðk e < k 0 Þ;
ð1Þ
where k0 is the specified hydraulic conductivity limit which is 1 · 107 cm/s (or 1 · 109 m/s) and ke is expected hydraulic conductivity. The appropriate limit state function is of the form: gðxÞ ¼ ln k 0 ln k e .
ð2Þ
Failure can be said to occur when ke exceeds k0, the regulatory maximum hydraulic conductivity or some other specified value that is less than the regulatory maximum. ke is represented as follows [12]: ln k e ¼ Y ¼ X b0 þ e;
ð3Þ
where Y = a random variable describing distribution of ln ke; X = a vector containing m random variables that describe the spatial distribution of m quality-control measurements related to hydraulic conductivity such as index properties and compaction variables; b 0 = a vector of coefficients; and e = an independent mean-zero Gaussian random error term with variance, r2e . On the basis of this limit state, the first-order reliability method was employed to estimate the first-order second-moment reliability index (b) values. Given that Xi = x1, x2, . . ., xn, the reduced variate is: X 0i ¼ x01 ; x02 ; . . . ; x0n ;
ð4Þ
where X 0i ¼
X i lxi ; rxi
ð5Þ
therefore, X i ¼ rxi x0i þ lxi .
ð6Þ
A performance function and limit state is GðX Þ ¼ Gðrx1 x01 þ lx1 ; rx2 x02 þ lx2 ; . . . ; rxn x0n þ rxn Þ ¼ 0. ð7Þ The reliability index, b, associated with Eq. (7) can be calculated either using the invariant solution by Hasofer and Lind [13] or the Mean-Value First-Order Second-Moment (MFOSM) also known as the Mean-First-Order Reliability Method (MFORM). The reliability based on the FORM model is given by: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 b ¼ min ðx01 Þ þ ðx02 Þ þ þ ðx0n Þ ; ð8Þ x2F
where X 01 ; X 02 ; . . . ; X 0n are the random variables in the limit state function given by G(X) = 0.
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The minimization of Eq. (8) is performed through an optimization procedure over the failure domain F corresponding to the region G(X) 6 0. This can be accomplished using FORM5 (First-order reliability method version 5-see [14]). FORM5 has the following advantages: (1) solution to problems can be obtained by working with original, rather than previously transformed or reduced random variable space; (2) the partial derivatives of G(X) need not be provided, and (3) correlated and non-normal variables are handled easily through transformations [15]. FORM (written in FORTRAN) provides an approximation to Z P f ¼ P ðX 2 FÞ ¼ P ðGðXÞ 6 0Þ ¼ dF x ðxÞ; ð9Þ GðxÞ60
by transforming non-Gaussian (non-normal) variables into independent standard normal variables, by locating the bpoint (most likely failure point) through an optimization procedure, by linearising the limit state function in that point and by estimating the failure probability using the standard normal integral [14]. A first approximation to Pf = P(G(X) 6 0) is P f ¼ UðbÞ;
ð10Þ
where U(.) is the standard normal integral and b is the (geometric) safety index or reliability index [14]. It then follows that [16]: bf ¼ U1 ðP f Þ.
ð11Þ
2.2. Hydraulic conductivity models The limit state function used was incorporated into two FORTRAN-based programs with which the reliability analyses were performed. The first-order reliability method (FORM) was used in this study to estimate the first-order second-moment reliability index (b) values for hydraulic conductivity in both laboratory and field cases. 2.3. Regression models The regression models employed in the reliability analyses are described subsequently. These models have been developed from hydraulic conductivity measurements, either in the laboratory or field, on soils being used for compacted clay liners at landfills. 2.4. Laboratory case Benson and Trast [17] used stepwise linear regression to identify the compaction and soil compositional variables that have the greatest influence on laboratory-measured hydraulic conductivity and to develop an equation that could be used to estimate hydraulic conductivity. A ‘‘partial-F’’ statistic (which is the ratio describing the portion of the variance of the dependent variable that is described by the independent variable) was computed for each independent variable. The partial-F was required to exceed 4
507
(this value of 4 corresponds to a significance level of 5%, i.e., a = 0.05, or the probability of falsely rejecting significance) for the independent variable to be deemed significant [17]. The resulting regression equation obtained by Benson and Trast [17] is: ln k ¼ 15:0 0:087S i 0:054PI þ 0:022C þ 0:91E þ e ðR2 ¼ 0:81Þ.
ð12Þ
In Eq. (12), k is hydraulic conductivity in m/s, Si is initial saturation in percent, PI is plasticity index in percent, C is clay content (% < 2 lm), E is the compactive effort index (an integer categorical variable describing compactive effort with values assigned as follows: 1, 0 and 1 for modified, standard and reduced Proctor compactive efforts). These integer categorical values correspond to the heavy, light and ‘‘reduced light’’ British standard compactive efforts adopted in this study. Attributes of Eq. (12) are described in [17]. According to Benson and Trast [17], Eq. (12) may prove useful when considering potential borrow sources, selecting compaction machinery, or estimating the shape of an acceptable zone for compaction control. It is also the basis for the limit state function required in the laboratory case of the reliability evaluation. 2.5. Field case Benson et al. [9] also developed a regression model for estimating field hydraulic conductivities that can be obtained for different soil types and compaction conditions. The independent variables in the regression model were selected using the method of stepwise linear regression. The significance of each independent decision variable in the model was checked with the partial-F that was greater than 4 (this corresponds to a significance level of 0.05). The model developed by the stepwise linear regression is [9]: pffiffiffiffi 894 ln k e ¼ 18:85 þ 0:08PI 2:87S i þ 0:32 G W ð13Þ þ 0:02C þ eðR2 ¼ 0:78Þ; where kg is geometric mean hydraulic conductivity in cm/s; W is compactor weight in kN; PI is plasticity index in percent; Si is initial saturation (in decimal form, and estimated with a constant value of specific gravity, Gs = 2.7); G is gravel content in percent; and C is clay content in percent. The statistical attributes of Eq. (13) are described in [9]. Eq. (13) is also the basis of the limit state function employed in the reliability analysis for field application. 2.6. Database and statistical analysis A database was compiled by extracting data on lateritic soils from published literature (e.g., [18,19,11,20–24]) and laboratory tests. The data from published literature cover lateritic soils occurring within and outside Africa. The literature are those published between 1967 and 1998.
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of their gravel contents and fines contents which differed significantly. The gravel content for the lateritic soils is higher than those of the liner soils, hence the necessity to scale-down the mean value of the percent gravel content when using the equation for the field case. However, the fines contents for these soils fall within comparable ranges. The mean value of the gravel content of the liner soils in [9] was much lower than the corresponding mean value for lateritic soils while the mean of the fines content was about twice the corresponding mean value for lateritic soils (Table 1). The mean value of the initial degree of saturation for the three soil groups is essentially similar.
The statistical characteristics of the soil composition and compaction variables for lateritic soils as well as for soils investigated by Benson and Trast [17] and Benson et al. [9] are shown in Tables 1–3, respectively. The statistical characteristics of the soil composition and compaction variables for the liner soils are presented for purposes of comparison with lateritic soils. The statistical characteristics of the properties of the lateritic soils in the database were first compared with those of the soils for which the regression models were developed by [9,17]. Lateritic soils and the liner soils found in the database developed by [9,17] have comparable properties, except for the means
Table 1 Statistical characteristics of composition and compaction variables for lateritic soils Soil parameter
Range (N)
Mean
Standard deviation
Coefficient of variation (%)
Liquid limit (%) Plastic limit (%) Plasticity index (%) Specific gravity Gravel (%) Sand (%) Fines (%) Clay (%) Silt (%) Activity Initial degree of saturation (%)
23–69 (53) 13–39 (53) 4–34.9 (53) 2.55–2.90 (49) 0.5–59 (33) 9–61.7 (32) 11–100 (50) 0.5–60 (34) 1.0–99.5 (27) 0.33–1.67 (31) 69.09–121.74 (55)
41.31 24.45 16.86 2.72 25.56 37.81 41.65 23.48 21.16 0.83 92.94
11.05 6.46 8.18 0.07 17.34 12.72 20.85 15.34 23.34 0.37 12.84
26.74 26.41 48.53 2.74 67.83 33.65 50.08 65.41 110.32 44.06 13.82
N = number of data points.
Table 2 Statistical characteristics of composition and compaction variables for compacted soil liner materials in [17] Soil parameter
Range (N = 13)
Mean
Standard deviation
Coefficient of variation (%)
Liquid limit (%) Plastic limit (%) Plasticity index (%) Specific gravity Gravel (%) Sand (%) Fines (%) Clay (%) Silt (%) Activity Initial degree of saturation (%)
24–70 12–32 11–46 2.68–2.90 0–8 6–48 52–94 16–65 25–58 0.32–1.00 NA
40.77 17 23.769 2.7839 2 16.38 81.77 37.62 44.15 0.65 NA
14.87 5.67 11.15 0.05 2.71 12.77 12.86 15.38 9.90 0.19 NA
36.48 33.36 46.92 1.92 135.40 77.94 15.73 41.31 22.42 28.84 NA
Table 3 Statistical characteristics of composition and compaction variables for compacted soil liner materials in [9] Soil parameter
Range (N)
Mean
Standard deviation
Coefficient of variation (%)
Liquid limit (%) Plastic limit (%) Plasticity index (%) Specific gravity Gravel (%) Sand (%) Fines (%) Clay (%) Silt (%) Activity Initial degree of saturation (%)
15–91(67) 8–34(67) 2–62(67) 2.7 0–22.2(65) 0–39.4(65) 44.3–100(66) 13.9–75.3(65) 16.5–65.7(65) 0.14–1.12(65) 77–104(56)
45.10 19.46 25.64 – 2.39 15.2 81.88 40.49 41.35 0.62 91.57
18.07 5.50 14.14 – 3.53 9.32 12.38 15.17 11.99 0.18 5.46
40.06 28.27 55.16 – 148.01 61.3 15.12 37.46 28.98 29.41 5.96
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Table 4 Initial input data for reliability analysis using FORM5-Laboratory case Variables
Distribution type
Maximum hydraulic conductivity, k0 Initial saturation, Si Plasticity index PI Clay content, C Compactive effort index, E
Mean Ex(i)
Log-normal (=3) Log-normal (=3) Log-normal (=3) Normal (=2) Deterministic parameter
9
1 · 10 m/s 92.9% 16.9% 23.5% (1,0,+1)
Standard deviation, Sx(i) 10
2.7 · 10 12.8 8.2 15.4 –
m/s
Coefficient of variation, CoV (%), i 27 13.8 48.5 65.5 –
2.7. Statistical distributions
3. Reliability estimates and discussion of results
A reliability-based analysis of the hydraulic conductivity of compacted soil liners requires that the type of probability distribution functions for both hydraulic conductivity, (k) and the quality control measurements that are related to hydraulic conductivity be established. Hydraulic conductivity is traditionally assumed to be log-normally distributed [2,5,25–27]. Using 30 data points Harrop-Williams [28] employed the Kolmogorov–Smirnov (K–S) test of goodness-of-fit to show that hydraulic conductivity had the following distributions in order of preference; b, c, normal and log-normal. However, only two probability distribution functions, normal and log-normal, were considered as preliminary reliability estimates did not reveal significant differences in reliability indices among these and other distributions such as two parameter c, Gumbel and Weibull distributions. A K–S goodness-of-fit test on the independent variables in Eqs. (12) and (13) revealed that initial saturation and plasticity index are log-normally distributed, clay content and gravel content are normally distributed while the compactor weight has a Weibull distribution. The compactor weights are those of sheepsfoot rollers reported in the database by [9]. These weights are globally the same and are not restricted to certain parts of the world.
3.1. Set up of numerical simulations
4
The determination of reliability indices for the laboratory case using the model in Eq. (12) was made for hydraulic conductivity, over a range of coefficients of variation from 10% to 100% for k, Si, PI, and C. Table 4 shows the initial input parameters for the laboratory case. Taking hydraulic conductivity to be normally and log-normally distributed, the results of varying the coefficients of variation for each variable in turn (while the values for other variables are kept constant) are shown in Fig. 1a through to Fig. 1d. The independent variables in the regression model in Eq. (13) are compactor weight (W), plasticity index (PI), initial saturation (Si), gravel content (G) and clay content (C). The initial input parameters in the reliability analysis using FORM5 [14] are shown in Table 5. The numbers in brackets in column 3 of Table 5 are various code numbers for the statistical distributions of the variables as used in FORM5 [14]. The regulatory maximum hydraulic conductivity for liners is 1 · 107 cm/s. The corresponding standard deviation of 2.7 · 108 cm/s is based on the minimum value of coefficient of variation for hydraulic conductivity obtained by [25] for 57 landfill sites and by [9] for 67 landfill sites.
BSH(N)
BSL(N)
RED. BSL(N)
BSH(LN)
BSL(LN)
RED. BSL(LN)
3.5
Reliability index
3 2.5 2 1.5 1
(N)-normal distribution 0.5
(LN)-log-normal distribution 0 0
20
40
60
80
100
Coefficient of variation (%)
Fig. 1a. Reliability index versus coefficient of variation for hydraulic conductivity, k.
120
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Table 5 Initial input data for reliability analysis using FORM5-Field case S/No
Variables
Distribution type
Mean Ex(i)
Standard deviation Sx(i)
Coefficient of variation Cov(%),i
1 2 3 4 5 6
Maximum hydraulic conductivity, k0 Compactor weight (sheeps foot rollers only), W Plasticity index, PI Initial degree of saturation, Si Gravel content, G Clay content, C
Log-normal (=3) Weibull (=9) Log-normal (=3) Log-normal (=9) Normal (=2) Normal (=2)
1 · 107cm/s 261.4 (165–338) 16.9 (4–34.9) 92.9 (69.09–121.74) 25.6 (0.59) 23.5 (0.5–60)
2.7 · 108cm/s 61.4 8.2 12.8 17.3 15.4
27 23.5 48.5 13.8 67.8 65.4
The values in parentheses in column (4) are the ranges for the variables. The values in column (6) are automatically calculated and used in FORM5. For sensitivity analysis, the variables were in turn made to assume values of coefficient of variation ranging between 10% and 100%. The values of the hydraulic conductivity were first varied while the values of the input parameters were kept constant for a normally – distributed hydraulic conductivity. These initial reliability estimates yielded reliability indices less than unity. The mean value of the compactor weight (W) was then scaled up to the maximum value of 338 kN (see [9]) while that of the gravel content (G) was scaled down to assume the maximum value of 10% recommended by the US Environmental Protection Agency [29,30]. Thus, for a specified compactor weight, only soils from lateritic deposits having gravel contents less than or equal to 10% can be used as hydraulic barrier materials. The only distribution types for hydraulic conductivity that were considered are normal and log-normal distributions. 3.2. Simulation results: laboratory case 3.2.1. Effect of variation of hydraulic conductivity on reliability index The range of coefficient of variation of 0.1 to 1.0 for a normally distributed K did not affect the reliability indices for a given compactive effort significantly. At the effort of the BS heavy compaction, reliability index, b, varied from 3.48 to 3.48 as coefficient of variation of hydraulic conductivity (CoV(HC)) was changed from 0.1 to 1.0. Similar trends were observed when lower compactive efforts were employed. The reliability indices decreased from 2.447 to 2.445 for BS light compaction and from 1.51 to 1.51 for reduced BS light compaction over the same range of CoV(HC). Benson [25] reported values of coefficient of variation for hydraulic conductivity ranging between 0.27 and 7.67 while Benson et al. [9] recorded values ranging between 0.27 and 2.77. At a CoV(HC) of 0.27 the estimated reliability indices are 3.48, 2.45 and 1.50 for BS heavy, BS light and reduced BS light compaction, respectively. The corresponding reliability indices at CoV(HC) of 2.77 are 3.47, 2.44 and 1.51 and at CoV(HC) of 7.67 are 3.37, 2.38 and 1.48, for BS heavy, BS light and reduced BS light compactive efforts, respectively. The decrease in reliability indices as the CoV(HC) increased was not significant for
each compactive effort considered. For log-normally distributed hydraulic conductivity b decreased from 3.48 to 3.08 for BS heavy, 2.44 to 2.08 for BS light, and from 1.51 to 1.18 for reduced BS light compaction energies, respectively as shown in Fig. 1a. The effect of the differences in compactive efforts is clearly seen in Fig. 1a. There is a consistent decrease in reliability index as the compactive efforts decreased from BS heavy to reduced BS light, and for each CoV(HC) considered. Soil specimens compacted with higher compactive efforts generally have lower hydraulic conductivity. This trend corresponds to that of decreasing reliability index values as compactive effort is decreased. 3.2.2. Effect of variation of initial saturation on reliability index The variation of reliability indices as the coefficient of variation for degree initial of saturation (CoV(Si)) increased is shown in Fig. 1b. As the CoV(Si) varied from 0.1 to 1.0, reliability index decreased non-linearly from 4.40 to 0.28, 3.13 to 0.06 and from 1.96 to 0.12 for BS heavy, BS light and reduced BS light compactive efforts, respectively. A CoV(HC) of 0.27 was maintained, while the CoV(Si) was varied. Similar trends were obtained when k assumed a log-normal distribution as shown in Fig. 1b. The rapid change in the reliability indices as the CoV(Si) varied clearly demonstrates that hydraulic conductivity is significantly affected by changes in initial degree of saturation. Thus, the initial saturation is a parameter that needs to be carefully controlled when compacting soils for use as liners and covers in waste-containment systems. Significant differences in reliability indices were also observed at lower CoV(Si) values with respect to the different compactive efforts used. These differences are smaller at higher values of CoV(Si). For each given coefficient of variation, the higher compactive efforts yielded higher reliability indices. This indicates that higher initial degrees of saturation can be obtained at higher compactive efforts. In practice, increases in initial saturation generally occur as a result of increasing moulding water content without changing compactive effort by using the same compactor and number of passes, increasing compactive effort without increasing moulding water content, or increasing both compactive effort and moulding water content [17]. These conditions are said to decrease the hydraulic conductivity of soils. The trends observed for initial saturation with
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511
5 4.5
BSH(N)
BSL(N)
RED. BSL(N)
BSH(LN)
BSL(LN)
RED. BSL(LN)
4
Reliability index
3.5 3 2.5 2
(N)-normal distribution
1.5
(LN)-log-normal distribution 1 0.5 0 -0.5 0
20
40
60
80
100
120
Coefficient of variation (%)
Fig. 1b. Reliability index versus coefficient of variation for initial degree of saturation, Si.
increasing coefficient of variation is expected as it has a high partial-F in Eq. (12). 3.2.3. Effect of variation of plasticity index on reliability index Some slight decreases in reliability indices occurred as a result of increasing the coefficient of variation for plasticity index for each of the three compactive efforts. These trends are shown in Fig. 1c for the cases when k is normally or log-normally distributed. As coefficient of variation increased from 0.1 to 1.0, reliability indices decreased from 3.81 to 3.20, 2.69 to 2.20 and from 1.68 to 1.30 for BS heavy, BS light and reduced BS light compaction, respectively when k is normally distributed. The mean value used
4
for plasticity index was 16.9. With standard deviations up to 16.9, high reliability index values can still be obtained particularly at the effort of the BS heavy compaction. However, concerns about compactibility are often the basis for restricting the value of plasticity index of soils to be used as hydraulic barriers in waste containment facilities. The minimum recommended values of plasticity index are P7–10% [31] and P12–15% [32]. EPA [29] recommends that the plasticity index of soil liner materials should be greater than 10 and discourages the use of soils with a plasticity index greater than about 30 as such soils are difficult to work with in the field (see [30]). The reliability indices were generally higher at higher compactive efforts. As with other variables, the reliability
BSH(N) BSH(LN)
BSL(N) BSL(LN)
RED. BSL(N) RED. BSL(LN)
3.5
Reliability index
3 2.5 2 1.5 1 (N)-normal distribution 0.5 (LN)-log-normal distribution 0 0
20
40
60
80
100
Coefficient of variation (%)
Fig. 1c. Reliability index versus coefficient of variation for plasticity index, PI.
120
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indices obtained with reduced BS light compaction were all lower than 2.0 for coefficients of variation within 0.1 to 1.0 for both normal and log-normal distributions of k.
lated F-value exceeds a corresponding critical F-value for a specified level of significance. The effects of coefficient of variation and compactive effort on reliability indices for each of the parameters considered are statistically significant at a = 0.05 (see Table 6). The highest F-value with respect to the effect of coefficient of variation was obtained for plasticity index (F = 62.82); followed by initial degree of saturation (F = 27.05). The effects of variations in hydraulic conductivity and clay content on reliability indices, although statistically significant, were less with F = 8.89 for hydraulic conductivity and F = 8.80 for clay content. The influence of compactive efforts on the reliability indices was more pronounced than that of coefficient of variation for hydraulic conductivity, plasticity index and clay content. Compactive effort has an over whelming influence on the reliability index values for hydraulic conductivity (F = 1485.73). The lowest Fvalue (19.77) was observed for initial degree of saturation with respect to the effect of compactive effort on reliability index values.
3.2.4. Effect of variation of clay content on reliability index Although compacted clay liners are required to have clay contents (% <2 lm)P20–25% [32], variations in clay contents affected reliability indices. The values of reliability index decreased as coefficient of variation increased from 0.1 to 1.0 (Fig. 1d). The decrease in reliability index values was more marked or pronounced at higher compactive efforts, although the reliability index remained above 3.0 for all the values of coefficient of variation at the energy of the BS heavy (BSH) compaction, and above 2.0 at the energy of the BS light (BSL) compaction. This trend was the same for both normally- and log-normally-distributed k. However, log-normally distributed k yielded lower values of reliability index. 3.2.5. Analysis of variance The results for normally-distributed hydraulic conductivity shown in Figs. 1a–1c and 1d were investigated statistically to show the level of the effects of (1) coefficient of variation, and (2) compactive effort, on the reliability indices obtained. The two-way analysis of variance (without replication) was employed at 5% level of significance (that is, a = 0.05). The results are shown in Table 5. The degrees of freedom, calculated F-values, p-values (tail probability) as well as the critical F-values obtainable from statistical tables, are shown for hydraulic conductivity (k); initial degree of saturation (Si); plasticity index (PI) and clay content (C), respectively. The calculated F-values represent ratios of the between sample variance to the within sample variance for a given set of data. The effect of a particular treatment is said to be statistically significant when a calcu-
3.2.6. Assessment of model suitability In structural reliability, reliability indices (b) ranging between 2.5 and 4.0 are considered to provide reasonable margins of safety for a given structure or structural component. The corresponding probability of failure within this region of safety index values is generally less than 6.25 · 103. In terms of acceptable probability of failure, 2.5 can be taken as a convenient lower limit for the reliability assessment of compacted lateritic soil liners. The high estimated b values indicate that the model is suitable for predicting the hydraulic conductivity of laboratorycompacted lateritic soils. On the basis of the minimum value of safety index adopted and considering the statistical values of the initial
4.5 4
BSH(N)
BSL(N)
RED. BSL(N)
BSH(LN)
BSL(LN)
RED. BSL(LN)
Reliability index
3.5 3 2.5 2 1.5 1 (N)-normal distribution 0.5 (LN)-log-normal distribution 0 0
20
40
60
80
100
Coefficient of variation (%)
Fig. 1d. Reliability index versus coefficient of variation for clay content, C.
120
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Table 6 Analysis of variance of reliability index values Variable
Source of variation
Degree of freedom
F-value (calculated)
F-value (critical)
Hydraulic conductivity, k0
Coefficient of variation Compactive effort Error (Total)
9 2 18(29)
8.89 9.7E + 07
2.46 3.55
Initial saturation, Si
Coefficient of variation Compactive effort Error (Total)
9 2 18(29)
27.05 19.77
2.46 3.55
Plasticity index, PI
Coefficient of variation Compactive effort Error/Total
9 2 18(29)
62.82 6861.57
2.46 3.55
Clay content, C
Coefficient of variation Compactive effort Error/Total
9 2 18(29)
8.80 1485.73
2.46 3.55
degree of saturation, plasticity index and clay percent, only the compaction energy of the BS heavy gave reliability or safety indices that were well above 2.5 (see Fig. 1a) for the range of coefficients of variation considered. Reliability indices obtained at the energy of the BS light compaction however, were close to the value of 2.5 and ranged between 2.45 and 2.45. The corresponding values for the reduced BS light varied from 1.51 to 1.51. These values are for hydraulic conductivity when it is normally distributed. Similar results were obtained when hydraulic conductivity assumed a log-normal distribution. Landfill liner construction is a specialized one and based on the estimated reliability indices, only samples compacted at the energy of the BS heavy compaction would yield laboratory hydraulic conductivity values suitable for design purposes. In Fig. 1b, it can be seen that the BS heavy compaction will be able to accommodate up to 21% coefficient of variation in the value of the initial saturation while up to 15% of the initial degree of saturation can be accommodated when the BS light compaction is used. Reliability indices obtained for compaction at the energy of the reduced BS light compaction were generally lower than 2.0. Similar results were obtained for log-normally distributed hydraulic conductivity, k. Fairly wide variations in plasticity index and clay percent within a lateritic soil deposit can be allowed provided compaction is carried out at the energy of the BS heavy compaction as the reliability indices ranged between 3.34 and 3.81 for plasticity index and between 3.17 and 3.83 for clay percent for normally-distributed hydraulic conductivity. However, with the energy of BS light compaction, a coefficient of variation of up to 40% for plasticity index and 53% for clay percent in terms of variations within a lateritic soil deposit, can be accommodated (see Figs. 1c and 1d). The energy of the reduced BS light compaction did not provide acceptable values of reliability index, as they are generally below 2.0 for both plasticity index and clay percent. Slightly lower values of coefficient of variation were obtained for plasticity index and clay percent when hydraulic conductivity assumed a log-normal distribution.
The estimated reliability indices show that the model used in this study is suitable for predicting the hydraulic conductivity of laboratory-compacted lateritic soils. The lowest values of hydraulic conductivity are obtained for samples compacted at the BS heavy compaction energy. Acceptable results can be obtained at the BS light compaction energy for only limited values of coefficient of variation especially for plasticity index and clay content. Lower values of the coefficient of variation for degree initial of saturation can be obtained at this compaction energy. The reliability indices also show that lateritic soils can serve as good landfill liner materials. Considering the Figs. 1a–1c and 1d, for instance, it is seen that compactive effort and the initial degree of saturation are essentially the two critical parameters that require very strict control. These observations are in good agreement with the partial-F values obtained by Benson and Trast [17] for compactive effort, E (65) and initial degree of saturation, Si (262). The plasticity index of lateritic soils also requires to be carefully controlled especially when compacting at the energy of the BS light. A partial-F value of 24 was obtained by Benson and Trast [17] for plasticity index. Plasticity index values of lateritic soils will be limited by considerations for constructability as soils with plasticity index values higher than 30% will tend to stick to the wheels of field compaction equipment. 3.2.7. Distributions of initial saturation and reliability index The K–S goodness-of-fit test indicated that the initial degree of saturation was log-normally distributed. However, due to the rapid decrease in reliability indices with increasing coefficient of variation, it was necessary to investigate the trend when other probability distribution functions were assumed. The only case considered was when hydraulic conductivity assumed a log-normal distribution. The results are shown graphically in Figs. 2a, 2b and 2c, for different compactive efforts. Similar trends of decreasing reliability indices with increasing coefficient of variation were observed in all the cases considered. However, some differences were observed
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Normal Gamma Weibull
Reliability index
5
Lognormal Gumbel
4
3
2
1
0 0
10
20
30
40
50
60
70
80
90
100
110
Coefficient of variation (%)
Fig. 2a. Reliability index versus coefficient of variation for initial degree of saturation (Si) for different distributions of Si (distribution type for k = lognormal at BSH).
4
Normal Gamma Weibull
3.5
Lognormal Gumbel
Reliability index
3 2.5 2 1.5 1 0.5 0 0
10
20
30
40
50
60
70
80
90
100
110
Coefficient of variation (%)
Fig. 2b. Reliability index versus coefficient of variation for initial degree of saturation (Si) for different distributions of Si (distribution type for k = lognormal at BSL).
for the different distributions of initial saturation as well as for the different compactive efforts. The extreme value Type I (Gumbel) gave the highest values of reliability index at 10% (0.1) coefficient of variation followed by lognormal, gamma normal and extreme value Type III (Weibull) distributions. The Weibull distribution consistently gave the lowest reliability indices at the energies of B.S. heavy and B.S. light compaction, but a different trend occurred with reduced B.S. light compaction. Moreover, at 1.0 (100%) coefficient of variation, the normal distribution gave the highest reliability indices for all the cases considered. At the B.S. heavy compactive effort, the reliability indices did not differ significantly for the five distributions when coefficient of variation was in the range of 50–70% (Fig. 2a). Similar observations were made for B.S. light
compaction at 30–50% coefficient of variation (Fig. 2b) and for reduced B.S. light compaction at 20–30% coefficient of variation (Fig. 2c). The differences in reliability index values among the different distributions were highest for reduced B.S. light compaction and lowest for B.S. heavy compaction at 100% (1.0) coefficient of variation. The results presented are for log-normally distributed hydraulic conductivity. Similar results were obtained for normally distributed hydraulic conductivity. The reliability indices for the various distributions of initial saturation at coefficients of variation of 0.1 and 1.0 and for the three compactive efforts are shown in Table 7. The reliability indices when coefficient of variation was equal to 100% were the same for gamma and Weibull distributions when the distribution types for hydraulic conductivity and
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Normal Gamma Weibull
2
515
Lognormal Gumbel
Reliability index
1.5
1
0.5
0
-0.5 0
10
20
30
40
50
60
70
80
90
100
110
Coefficient of variation (%)
Fig. 2c. Reliability index versus coefficient of variation for initial degree of saturation (Si) for different distributions of Si (distribution type for k = lognormal at RED. BSL).
Table 7 Reliability index values for different distributions of initial degree of saturation Distribution type for initial saturation, Si
Coefficient of variation
For log-normally distributed hydraulic conductivity Normal 0.1 1.0 Log-normal 0.1 1.0 Gamma 0.1 1.0 Gumbel 0.1 1.0 Weibull 0.1 1.0 Distribution type for initial saturation, Sr
Coefficient of variation
For normally distributed hydraulic conductivity Normal 0.1 1.0 Log-normal 0.1 1.0 Gamma 0.1 1.0 Gumbel 0.1 1.0 Weibull 0.1 1.0
Reliability index BSH
BSL
RED. BSL
3.83 0.44 4.35 0.27 4.18 0.15 5.65 0.32 3.07 0.18
2.83 0.33 3.08 0.06 3.00 0.02 3.81 0.19 2.41 0.02
1.84 0.21 1.91 0.22 1.89 0.11 2.19 0.05 1.71 0.11
Reliability index BSH
BSL
RED. BSL
3.86 0.44 4.40 0.28 4.23 0.18 5.72 0.33 3.10 0.18
2.86 0.33 3.13 0.06 3.04 0.03 3.88 0.19 2.44 0.03
1.87 0.22 1.96 0.12 1.93 0.11 2.25 0.06 1.73 0.11
BSH = BS heavy compaction; BSL = BS light compaction; and RED, BSL = reduced, BS light compaction.
the respective compactive efforts are considered. Reliability indices obtained when hydraulic conductivity was normally distributed were generally higher than the corresponding
values at coefficient of variation of 0.1 but generally lower than the corresponding values at coefficient of variation of 1.0 for log-normally distributed hydraulic conductivity. Moreover, the information in Table 7 tends to suggest the following probability distribution types for initial saturation, in order of preference; extreme value Type I (Gumbel), log-normal, gamma, normal and extreme value Type III (Weibull). This observation is in good agreement with K–S goodness-of-fit test performed on soil compositional and compaction variables for lateritic soils [33]. 3.2.8. Implications for field application The two main variables observed to have the most significant impact on the reliability indices associated with the model considered are compactive effort and degree of saturation. The compactive effort is the laboratory equivalent of compactor weight under field conditions. The higher the compactive effort the higher the reliability indices at any given coefficient of variation, for hydraulic conductivity as well as the independent variables, namely; initial saturation, plasticity index and clay content. For purpose of construction of soil liners with the required very low probability of failure, high compactor weights should be used. Moreover, careful construction quality control should be maintained to achieve very minimal variations in the values of initial saturation as this would result in wide variations in hydraulic conductivity values. High safety or reliability levels can best be achieved provided initial saturation values of compacted soil layers do not vary by more than 10%. Daniel [31] suggests the use of 20,000 kg compactors. Rubber-tired rollers can be as heavy as 18,000 kg and are capable of delivering tire pressures of between 450 and 1000 kPa, while sheepsfoot rollers can deliver pressures of between 1750 and 4500 kPa [34]. These rollers are suitable
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Hydraulic conductivity Compactor weight Plasticity index Initial saturation Clay content Gravel content
2.5
Reliability index
2
1.5
1
0.5
0 0
20
40
60
80
100
120
Coefficient of variation (%)
Fig. 3a. Reliability index versus coefficient of variation (distribution type for field k = normal).
3
Hydraulic conductivity Compactor weight Plasticity index
2.5
Initial saturation Clay content Gravel content
Reliability index
2
1.5
1
0.5
0 0
20
40
60
80
100
120
Coefficient of variation (%)
Fig. 3b. Reliability index versus coefficient of variation (distribution type for field k = lognormal).
for compacting clay soils of low plasticity which are usually recommended for use as soil liners [34]. 3.3. Simulation results: field case 3.3.1. Effect of variation in hydraulic conductivity When the distribution type for hydraulic conductivity was normal and its coefficient of variation ranged from 10% to 100% reliability indices, b, only decreased gradually from 1.58 to 1.57 (Fig. 3a). However, there was an appreciable decrease in reliability index, b, from 1.58 to 1.37 for a range of 10% to 100% in coefficient of variation when a log-normal distribution was assumed (Fig. 3b). This difference tends to suggest that normal distribution is superior to log-normal distribution for hydraulic conductivity of compacted lateritic soils.
3.3.2. Effect of variations in the independent variables Of all the independent variables in Eq. (13), compactor weight (W) exhibited the most significant change in reliability indices. For a normally distributed hydraulic conductivity, k, (Fig. 3a) reliability indices changed from 2.54 to 0.16 when the coefficient of variation was increased from 10% to 100% for W. The corresponding change in reliability indices when k was made to assume a log-normal distribution (see Fig. 3b) was from 2.49 to 0.15 for the same range of coefficient of variation. The trend in the decrease of b values for W should be expected since in the model in Eq. (13), W had the highest partial-F of 59.7. It has been shown by [9] that for given plasticity index, clay content; initial saturation and gravel content, increasing the compactor weight results in lower hydraulic conductivity. Thus, in order to obtain liners and covers that have desirable hydraulic con-
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ductivities (that is, k values that are much less than 1 · 107cm/s or 1 · 109 m/s) higher compactor weights are generally required. However, the study carried out by [9] did not involve a reliability analysis, hence no reliability index value was indicated in their analysis. For a given landfill liner construction, it is not necessary to employ compactors whose weights differ significantly from each other. The value of 10% coefficient of variation seems to be a convenient maximum for variations in compactor weight, W. It is best to use compactors of the same weight for a given soil liner construction. The second variable that showed significant decrease in reliability indices as the coefficient of variation increased from 10% to 100% is the initial saturation, Si (or initial degree of saturation). Reliability indices decreased from 1.60 to 0.78 for normally – distributed k and from 1.58 to 0.75 for log-normally distributed k (Figs. 3a and 3b) respectively. Results of tests by [17] on compacted clays show that trend of decreasing hydraulic conductivity with increasing initial saturation exists. For the soils studied by [17] hydraulic conductivities less than 1 · 107cm/s, the common regulatory maximum hydraulic conductivity in the United States, can be achieved by compacting to an initial saturation in excess of 85%. However, Benson and Trast [17] cautioned that increasing initial saturation might not always result in lower hydraulic conductivity and that higher initial saturation should be obtained without decreasing compactive effort (that is, compactor weight and number of passes). Plasticity index is another variable that exhibited appreciable decrease in b values when the coefficient of variation increased from 10% to 100%. The reliability indices decreased from 1.700 to 1.39 as displayed in Fig. 3a and from 1.68 to 1.36 in Fig. 3b. The model in Eq. (13) showed that a partial-F of 50.1 was obtained for plasticity index (PI). According to [9], hydraulic conductivity decreases quickly as the PI increases from 10 and 30, but then
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decreases more slowly as the PI is increased further. However, plasticity indices in the range 10 to 30 are recommended by [29]. It is envisaged that soils with PI values greater than 30 may give field construction problems. A minimum PI value of 7 has been suggested by [9] for liner soil materials. With a mean value of 10% and a corresponding standard deviation (scaled) of 6.76% for the gravel contents the b values decreased from 1.60 to 1.55 (Fig. 3a) and 1.59 to 1.53 (Fig. 3b) for the assumed range of coefficients of variation. It can be seen that variations in gravel content within the limits recommended by [29] do not significantly affect the reliability of lateritic soil liners. Similarly increases in clay contents did not significantly affect the estimated reliability indices. The b values decreased from 1.60 to 1.55 (Fig. 3a) and from 1.58 to 1.53 (Fig. 3b) within the assumed coefficient of variation. Proper compaction of lateritic soils having appropriate combination of soil grading and plasticity characteristics will ensure reliable soil liner construction provided that compaction is carried out at suitable moisture content on the wet side of optimum moisture content. In this case, only lateritic soils from deposits with percent gravel contents not exceeding 10% will be acceptable. 3.3.3. Assessment of model suitability for lateritic soils The model (Eq. (13)) developed by [9] explains only 78% of the variance in hydraulic conductivity. Moreover, a constant value of 2.7 was adopted for the specific gravity of the soils. In practice lateritic as well as other soils do not assume a constant value of specific gravity. Nevertheless, it is still possible to employ Eq. (13) for predicting the field hydraulic conductivity of lateritic soils particularly when higher compactor weights are used. In this study, only the compactor weights for sheepsfoot rollers were considered. Sheepsfoot compactors generally have higher weights than rubber-tired compactors. Increasing the weight of the
3
Normal distribution Lognormal distribution Gamma distribution
2.5
Gumbel distribution
Reliability index
Weibull distribution 2
1.5
1
0.5
0 0
20
40
60
80
100
120
Coefficient of variation (%)
Fig. 4a. Reliability index versus coefficient of variation for compactor weight, W (distribution type for field k = normal).
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Normal distribution Lognormal distribution Gamma distribution Gumbel distribution Weibull distribution
Reliability index
2.5
2
1.5
1
0.5
0 0
10
20
30
40
50
60
70
80
90
100
110
Coefficient of variation (%)
Fig. 4b. Reliability index versus coefficient of variation for compactor weight, W (distribution type for field k = lognormal).
compactor generally results in greater compactive efforts, more shear deformation, more smaller uniform pores, and lower hydraulic conductivity [9]. Using box plots it was shown [9] that the median hydraulic conductivity for compactors classified as sheepsfoot was k = 1.4 · 108 cm/s which is approximately 4 times lower than the median hydraulic conductivity of compactors classified as rubber-tired for which k = 6.0 · 108 cm/s. Thus, it is appropriate to use sheepsfoot compactors for the field compaction of lateritic soil liners. Although the b values at 10% coefficient of variation for some of the model parameters were fairly greater than 1.5, Eq. (13) can be used for prediction of lateritic soil hydraulic conductivity. However, b values close to 2.5 were obtained when the coefficient of variation for the compactor weight was assumed to be 10%. Thus, Eq. (12) can be used to predict the hydraulic conductivity of properly compacted lateritic soils to be used for waste containment purposes. Under field conditions, the three most important parameters that require careful control are compactor weight, initial saturation and plasticity index. Eq. (13) suggests that increasing compactor weight, PI or initial saturation will result in lower geometric mean hydraulic conductivity [9]. Even when a higher compactor weight is used and there is need to use more than one compactor it is necessary to select compactors whose weights do not differ from each other by more than 10%. The trends in Figs. 3a and 3b indicate the need for this caution. 3.3.4. Probability distributions for compactor weight Trends in Figs. 4a and 4b showed that compactor weight, W, has a tremendous influence on geometric mean field hydraulic conductivity, kg. The influence of different probability distribution types for W on the reliability estimates for cases when the hydraulic conductivity assumes normal and lognormal distributions was studied. The up-scaled compactor weight was used in this case
Table 8 Reliability indices for different distributions of compactor weight Distribution type
Gumbel Log-normal Gamma Normal Weibull
Reliability indices Normal distribution
Log-normal distribution
CoV = 10%
CoV = 100%
CoV = 10%
CoV = 100%
2.67 2.63 2.63 2.62 2.54
0.31 0.25 0.16 0.42 0.16
2.61 2.58 2.57 2.57 2.49
0.30 0.24 0.15 0.42 0.15
CoV = coefficient of variation (%).
(mean = 338 kN; standard deviation = 79.4 kN). Five alternative distributions, namely; normal, log-normal, gamma, Gumbel and Weibull, respectively were considered. The reliability indices for 10% and 100% coefficient of variation for each distribution type and for the cases when hydraulic conductivity, k, has normal and lognormal distributions, respectively, are shown in Table 8. Trends of decreasing reliability indices with increasing values of coefficient of variation are shown in Fig. 4a for normally-distributed k and in Fig. 4b for log-normal distributed k. It is evident from Table 8 and from Figs. 4a and 4b that, in decreasing order of superiority, compactor weight assumes the following distribution types: Gumbel, log-normal, gamma, normal and Weibull. This order holds regardless of the distribution type for hydraulic conductivity, k. 4. Conclusions Statistical characteristics of lateritic soil properties were established and compared with those of land fill liner soils. The distributional forms for relevant soil properties were employed to accomplish a reliability-based assessment of the suitability of lateritic soils as hydraulic barriers in waste containment facilities using existing hydraulic conductivity
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models developed from laboratory and field data considering variations in soil parameters. Variations in compactive effort and initial degree of saturation were observed to have the most significant impact on the reliability of soil liner hydraulic conductivity based on the laboratory model. Sheepsfoot rollers can provide, under field conditions, the equivalent of laboratory B.S. heavy compactive effort which generally gave high reliability indices. An additional assessment of the suitability of lateritic soils for use as compacted liners and covers in landfills was also made using a first-order reliability approach. The assessment was based on a regression model developed from field data (for 67 actual landfill sites) adopted for predicting the geometric mean hydraulic conductivity of lateritic soils. The values of the reliability index, b, obtained as coefficient of variation of the various variables increased suggest that this regression model is suitable for predicting the hydraulic conductivity of lateritic soils. The trends in the variation of reliability indices as coefficient of variation of the various variables increased also show that the compactor weight has most significant effect on the geometric mean hydraulic conductivity of lateritic soils, followed by initial saturation and plasticity index, respectively. The results also show the importance of the probability distribution of the compactor weight in the reliability estimate. References [1] Christian JT, Ladd CC, Baecher GB. Reliability and probability in stability analysis. Proceedings of Conference on Stability and Performance of Slopes and Embankments – II, vol. 2. New York, NY: ASCE; 1992. p. 1071–111. [2] Gui S, Zhang R, Turner JP, Xu X. Probabilistic slope stability analysis with stochastic soil hydraulic conductivity. J Geotech Geoenviron Eng, ASCE 2000;126(1):1–9. [3] Ronold KO, Bysveen S. Probabilistic stability analysis for deep-water foundation. J Geotech Eng, ASCE 1992;118(3):394–405. [4] Cherubini C, Garrasi A, Petrolla C. The reliability of an anchored sheet-pile wall embedded in cohesionless soil. Can Geotech J 1992;29(3):426–35. [5] Bogardi I, Kelly WE, Bardossy A. Reliability model for soil liners: Initial design. J Geotech Eng ASCE 1989;115(5):658–69. [6] Rowe RK, Fraser MJ. Effect of uncertainty in the assessment of the potential impact of waste disposal facilities. Geoenvironment 2000, Geotech. Spec. Publ. 46, ASCE, New York, 1995. p. 270–84. [7] Jang YS, Sitar N, Der Kiureghian A. Reliability analysis of contaminant transport in saturated porous media. Water Resour Res 1994;30(8):2435–48. [8] Daniel DE, Benson CH. Water content-density criteria for compacted soil liners. J Geotech Eng, ASCE 1990;116(12):1811–30. [9] Benson CH, Zhai H, Wang X. Estimating hydraulic conductivity of compacted clay liners. J Geotech Eng, ASCE 1994;120(20):366–87. [10] Townsend FC, Manke GP, Parcher JV. The influence of sesquioxides on laterite soils properties. Highway Res Board Record 1971;374: 80–92.
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[11] Ola SA. Geotechnical properties and behaviour of some Nigerian lateritic soils. In: Ola SA, editor. Tropical soils of Nigeria in engineering practice. Rotterdam: A.A. Balkema; 1983. p. 61–84. [12] Benson CH, Zhai H, Rashad SM. Statistical sample size for construction of soil liners, ASCE. J Geotech Eng 1994;120(101): 1704–24. [13] Hasofer AM, Lind NC. An exact and invariant first-order reliability format. J Eng Mech Div, ASCE 1974;100(1):111–21. [14] Gollwitzer S, Abdo T, Rackwitz R. First order reliability method manual, v. 5. Munich, 1988, 47p. [15] Juang CH, Rosowsky DV, Tang WH. Reliability-based method for assessing liquefaction potential of soils. J Geotech Geoenviron Eng, ASCE 1999;125(8):684–9. [16] Thoft-Christensen P, Baker MJ. Structural reliability theory and its applications. Berlin, Heidelberg: Springer Verlag; 1982. [17] Benson CH, Trast JM. Hydraulic conductivity of thirteen compacted clays. Clays Clay Min 1995;45(6):669–81. [18] Meireles JMF. Trial pavements of laterite base course. In: Proceedings of the Regional Conference for Africa on Soil Mechanics and Foundation Engineering, 4th, vol. 1, Cape Town, 1967. p. 71–6. [19] Ola, S.A., Stabilisation of Nigerian lateritic soils with cement, bitumen and lime. In: Proceedings of the Regional Conference for Africa on Soil Mechanics and Foundation Engineering, 6th, vol. 1, Durban, 1975. p. 145–52. [20] Ogunsanwo O. Variability in the shear strength characteristics of an amphibolite derived laterite soil. Bull IAEG 1985;32:111–5. [21] Ogunsanwo O. The micro-fabric and permeability of a laterite clay. In: Proceedings of the First International Conference on Geomechanics of Tropical Lateritic and Saprolitic Soils, Brasilia, vol. 1, 1985. p. 397–402. [22] Ogunsanwo O. CBR and shear strengths of compacted laterite soils from southwestern Nigeria. Quart J Eng Geol 1989;22:317–28. [23] Ogunsanwo O. Some geotechnical properties of two laterite soils compacted at different energies. Eng Geol 1989;26:261–9. [24] Ogunsanwo O. Influence of sample preparation and mode of testing on the shear strength characteristics of laterite soils from southwestern Nigeria. Bull Int Assoc Eng Geol 1993;47:141–4. [25] Benson CH. Probability distributions for hydraulic conductivity of compacted soil liners. J Geotech Eng, ASCE 1993;119(3):471–86. [26] Benson CH, Daniel DE. Minimum thickness of compacted soil liners: I. Stochastic models. J Geotech Eng, ASCE 1994;120(1):129–52. [27] Benson CH, Daniel DE. Minimum thickness of compacted soil liners; II. Analysis and case histories. J Geotech Eng, ASCE 1994;120(1): 153–72. [28] Harrop-Williams K. Clay liner permeability: Evaluation and variation. J Geotech Eng, ASCE 1985;111(10):1211–25. [29] Environmental Protection Agency (EPA), Requirements for hazardous waste landfill: Design, construction and closure. Publ. No. EPA 625/4 - 89 - 022, Cincinnati, OH, 1989. [30] Das BM. Principles of geotechnical engineering. 4th ed. Boston: PWS Publishing Company; 1998. 712p.. [31] Daniel DE. Clay liners. In: Daniel DE, editor. Geotechnical practice for waste disposal. Chapman and Hall; 1993. pp. 137–63. [32] Benson CH, Daniel DE, Boutwell GP. Field performance of compacted clay liners. J Geotech Geoenviron Eng, ASCE 1999;125(5): 390–403. [33] Nwaiwu CMO. Evaluation of compacted lateritic soils as hydraulic barriers in municipal solid waste containment systems. Ph.D. dissertation, Ahmadu Bello University, Zaria, 2004. 457p. [34] Holtz RD, Kovacs WD. An introduction to geotechnical engineering. Englewood Cliffs, NJ: Prentice-Hall, Inc.; 1981. 733p.
Reliability-based assessment of compacted lateritic soil liners J.O. Afolayan a, C.M.O. Nwaiwu
b,*
a
b
Department of Civil Engineering, Ahmadu Bello University, Zaria, Kaduna State, Nigeria Department of Civil and Water Resources Engineering, University of Maiduguri, Maiduguri, Borno State, Nigeria Received 5 January 2005; received in revised form 5 August 2005; accepted 22 August 2005 Available online 2 December 2005
Abstract The first-order reliability method is employed to assess the suitability of lateritic soils as landfill liner materials based on existing models developed from laboratory and field data. The statistical characteristics of lateritic soil properties were first compared with those from which the regression models were developed. Using regression models for hydraulic conductivity, k, and established distributions for the relevant soil parameters, reliability indices are computed considering hydraulic conductivity to have a normal/ log-normal distribution. The results indicate that, for the laboratory-based model, reliability index is sensitive to variability in compactive effort and initial degree of saturation. The choice of distribution of initial degree of saturation affects reliability indices significantly and the energy of the British Standard heavy compaction has the highest reliability indices. On the basis of the reliability indices estimated from the field-based model, three variables, namely: compactor weight, initial saturation and plasticity index are identified as significantly affecting the field geometric mean hydraulic conductivity of lateritic soils. However, the hydraulic conductivity of lateritic soils can be better predicted if the compactor weight, as the most significant parameter, can be modelled as Gumbel-type distribution. 2005 Elsevier Ltd. All rights reserved. Keywords: Compaction energy; Compacted soil liners; Hydraulic conductivity; Lateritic soils; Reliability analysis; Reliability indices; Soil properties; Statistical analysis; Statistical distributions
1. Introduction The concept of reliability analysis or evaluation has been successfully applied in many aspects of geotechnical engineering designs. The most common applications of reliability theory are in the areas of slope stability [1,2], offshore structures [3] and stability of anchored sheet pile wall [4]. However, there have been applications of reliability analysis in geoenvironmental engineering in recent times. Most prominent among these analyses are for liners of landfills [5,6]. Reliability analysis has also been applied to contaminant transport in porous media [7]. Most landfills used for the containment of municipal solid wastes and hazardous wastes often employ compacted natural soils as liner and cover materials (i.e., *
Corresponding author. E-mail address: [email protected] (C.M.O. Nwaiwu).
0266-352X/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2005.08.001
hydraulic barrier layers). However, in the absence of suitable natural soils, processed clays or sand/processed clay mixtures can be used as barrier materials. In order to ensure proper functioning of barrier layers in landfills, preliminary laboratory tests are usually conducted for each selected soil type to establish appropriate ranges of water contents and dry densities needed to achieve the required hydraulic conductivity (this is usually expected to be less than 1 · 109 m/s). The procedures for defining the acceptable zone of water content and dry density on the compaction plane have been described by [8]. Developing this acceptable zone on the compaction plane in the field may be impracticable as a result of high costs. This is because different compaction plants will need to be hired and several field measurements of dry unit weights and moisture contents made for each soil to be used. Although laboratory results are usually employed, it will be noted
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that laboratory-scale compaction does not perfectly duplicate the repeated passage of compaction equipment over each lift of soil in the field. Thus, it is not necessary to completely rely on results obtained from laboratory experiments. This is because there can be differences between laboratory-scale and field-scale hydraulic conductivities. In order to assess the suitability of a given soil type for use as hydraulic barrier material, the hydraulic conductivity of such a soil may be predicted using any existing prediction model, especially one that is developed from field data as in [9]. The possibility of extending the use of this model for predicting field hydraulic conductivity to tropical soils such as lateritic soils (as the model was developed from data on temperate zone soils) may be assessed using a reliability approach. Lateritic soils which are generally abundant in many parts of the tropics constitute an important soil group that may be considered for use in waste containment. The differences in the modes and environments of formation of tropical and temperate zone soils give rise to differences in basic engineering properties and behaviour of the soils occurring in these climatic regions. For instance, lateritic soils are characterised by the presence of sesquioxides (i.e., Fe2O3 and Al2O3) which have considerable influence on their properties and behaviour [10]. According to Ola [11], the most significant differences between temperate zone soils and lateritic soils are the significant proportions of iron and aluminium oxides found in lateritic soils which tend to cement the soil particles to form a coarse-grained weakly bonded (aggregated) particulate material and the higher temperature of laterite formation which is a feature of tropical climate compared to a temperate one. Due to the immense necessity of ensuring safe environmental containment of wastes, the uncertainties associated with the variability of properties of lateritic soils (especially those that significantly affect hydraulic conductivity) should be accounted for using a suitable probabilistic approach. The objective of this study was to assess the suitability of compacted lateritic soils as hydraulic barrier materials in landfill waste containment using the first-order reliability method. Limit state functions were first established employing deterministic models in published literature that were developed from both laboratory and field data. The values of the relevant variables identified in the limit state function were quantified statistically for lateritic soils in a database. The first-order-reliability method was employed to estimate reliability indices from the predictive models for varying inputs. The results showed that, for the laboratory-based model, variations in compactive effort and initial degree of saturation have the most significant impact on the reliability of soil liner hydraulic conductivity. On the other hand, compactor weight, initial degree of saturation, and plasticity index are the most critical variables in landfill liner construction using lateritic soils based on the field model.
2. Reliability analysis 2.1. Limit state function The reliability analysis in this study for compacted lateritic soils is based on post-construction performance characterized by only hydraulic conductivity. The reliability is: R ¼ Pðk e < k 0 Þ;
ð1Þ
where k0 is the specified hydraulic conductivity limit which is 1 · 107 cm/s (or 1 · 109 m/s) and ke is expected hydraulic conductivity. The appropriate limit state function is of the form: gðxÞ ¼ ln k 0 ln k e .
ð2Þ
Failure can be said to occur when ke exceeds k0, the regulatory maximum hydraulic conductivity or some other specified value that is less than the regulatory maximum. ke is represented as follows [12]: ln k e ¼ Y ¼ X b0 þ e;
ð3Þ
where Y = a random variable describing distribution of ln ke; X = a vector containing m random variables that describe the spatial distribution of m quality-control measurements related to hydraulic conductivity such as index properties and compaction variables; b 0 = a vector of coefficients; and e = an independent mean-zero Gaussian random error term with variance, r2e . On the basis of this limit state, the first-order reliability method was employed to estimate the first-order second-moment reliability index (b) values. Given that Xi = x1, x2, . . ., xn, the reduced variate is: X 0i ¼ x01 ; x02 ; . . . ; x0n ;
ð4Þ
where X 0i ¼
X i lxi ; rxi
ð5Þ
therefore, X i ¼ rxi x0i þ lxi .
ð6Þ
A performance function and limit state is GðX Þ ¼ Gðrx1 x01 þ lx1 ; rx2 x02 þ lx2 ; . . . ; rxn x0n þ rxn Þ ¼ 0. ð7Þ The reliability index, b, associated with Eq. (7) can be calculated either using the invariant solution by Hasofer and Lind [13] or the Mean-Value First-Order Second-Moment (MFOSM) also known as the Mean-First-Order Reliability Method (MFORM). The reliability based on the FORM model is given by: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 b ¼ min ðx01 Þ þ ðx02 Þ þ þ ðx0n Þ ; ð8Þ x2F
where X 01 ; X 02 ; . . . ; X 0n are the random variables in the limit state function given by G(X) = 0.
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The minimization of Eq. (8) is performed through an optimization procedure over the failure domain F corresponding to the region G(X) 6 0. This can be accomplished using FORM5 (First-order reliability method version 5-see [14]). FORM5 has the following advantages: (1) solution to problems can be obtained by working with original, rather than previously transformed or reduced random variable space; (2) the partial derivatives of G(X) need not be provided, and (3) correlated and non-normal variables are handled easily through transformations [15]. FORM (written in FORTRAN) provides an approximation to Z P f ¼ P ðX 2 FÞ ¼ P ðGðXÞ 6 0Þ ¼ dF x ðxÞ; ð9Þ GðxÞ60
by transforming non-Gaussian (non-normal) variables into independent standard normal variables, by locating the bpoint (most likely failure point) through an optimization procedure, by linearising the limit state function in that point and by estimating the failure probability using the standard normal integral [14]. A first approximation to Pf = P(G(X) 6 0) is P f ¼ UðbÞ;
ð10Þ
where U(.) is the standard normal integral and b is the (geometric) safety index or reliability index [14]. It then follows that [16]: bf ¼ U1 ðP f Þ.
ð11Þ
2.2. Hydraulic conductivity models The limit state function used was incorporated into two FORTRAN-based programs with which the reliability analyses were performed. The first-order reliability method (FORM) was used in this study to estimate the first-order second-moment reliability index (b) values for hydraulic conductivity in both laboratory and field cases. 2.3. Regression models The regression models employed in the reliability analyses are described subsequently. These models have been developed from hydraulic conductivity measurements, either in the laboratory or field, on soils being used for compacted clay liners at landfills. 2.4. Laboratory case Benson and Trast [17] used stepwise linear regression to identify the compaction and soil compositional variables that have the greatest influence on laboratory-measured hydraulic conductivity and to develop an equation that could be used to estimate hydraulic conductivity. A ‘‘partial-F’’ statistic (which is the ratio describing the portion of the variance of the dependent variable that is described by the independent variable) was computed for each independent variable. The partial-F was required to exceed 4
507
(this value of 4 corresponds to a significance level of 5%, i.e., a = 0.05, or the probability of falsely rejecting significance) for the independent variable to be deemed significant [17]. The resulting regression equation obtained by Benson and Trast [17] is: ln k ¼ 15:0 0:087S i 0:054PI þ 0:022C þ 0:91E þ e ðR2 ¼ 0:81Þ.
ð12Þ
In Eq. (12), k is hydraulic conductivity in m/s, Si is initial saturation in percent, PI is plasticity index in percent, C is clay content (% < 2 lm), E is the compactive effort index (an integer categorical variable describing compactive effort with values assigned as follows: 1, 0 and 1 for modified, standard and reduced Proctor compactive efforts). These integer categorical values correspond to the heavy, light and ‘‘reduced light’’ British standard compactive efforts adopted in this study. Attributes of Eq. (12) are described in [17]. According to Benson and Trast [17], Eq. (12) may prove useful when considering potential borrow sources, selecting compaction machinery, or estimating the shape of an acceptable zone for compaction control. It is also the basis for the limit state function required in the laboratory case of the reliability evaluation. 2.5. Field case Benson et al. [9] also developed a regression model for estimating field hydraulic conductivities that can be obtained for different soil types and compaction conditions. The independent variables in the regression model were selected using the method of stepwise linear regression. The significance of each independent decision variable in the model was checked with the partial-F that was greater than 4 (this corresponds to a significance level of 0.05). The model developed by the stepwise linear regression is [9]: pffiffiffiffi 894 ln k e ¼ 18:85 þ 0:08PI 2:87S i þ 0:32 G W ð13Þ þ 0:02C þ eðR2 ¼ 0:78Þ; where kg is geometric mean hydraulic conductivity in cm/s; W is compactor weight in kN; PI is plasticity index in percent; Si is initial saturation (in decimal form, and estimated with a constant value of specific gravity, Gs = 2.7); G is gravel content in percent; and C is clay content in percent. The statistical attributes of Eq. (13) are described in [9]. Eq. (13) is also the basis of the limit state function employed in the reliability analysis for field application. 2.6. Database and statistical analysis A database was compiled by extracting data on lateritic soils from published literature (e.g., [18,19,11,20–24]) and laboratory tests. The data from published literature cover lateritic soils occurring within and outside Africa. The literature are those published between 1967 and 1998.
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of their gravel contents and fines contents which differed significantly. The gravel content for the lateritic soils is higher than those of the liner soils, hence the necessity to scale-down the mean value of the percent gravel content when using the equation for the field case. However, the fines contents for these soils fall within comparable ranges. The mean value of the gravel content of the liner soils in [9] was much lower than the corresponding mean value for lateritic soils while the mean of the fines content was about twice the corresponding mean value for lateritic soils (Table 1). The mean value of the initial degree of saturation for the three soil groups is essentially similar.
The statistical characteristics of the soil composition and compaction variables for lateritic soils as well as for soils investigated by Benson and Trast [17] and Benson et al. [9] are shown in Tables 1–3, respectively. The statistical characteristics of the soil composition and compaction variables for the liner soils are presented for purposes of comparison with lateritic soils. The statistical characteristics of the properties of the lateritic soils in the database were first compared with those of the soils for which the regression models were developed by [9,17]. Lateritic soils and the liner soils found in the database developed by [9,17] have comparable properties, except for the means
Table 1 Statistical characteristics of composition and compaction variables for lateritic soils Soil parameter
Range (N)
Mean
Standard deviation
Coefficient of variation (%)
Liquid limit (%) Plastic limit (%) Plasticity index (%) Specific gravity Gravel (%) Sand (%) Fines (%) Clay (%) Silt (%) Activity Initial degree of saturation (%)
23–69 (53) 13–39 (53) 4–34.9 (53) 2.55–2.90 (49) 0.5–59 (33) 9–61.7 (32) 11–100 (50) 0.5–60 (34) 1.0–99.5 (27) 0.33–1.67 (31) 69.09–121.74 (55)
41.31 24.45 16.86 2.72 25.56 37.81 41.65 23.48 21.16 0.83 92.94
11.05 6.46 8.18 0.07 17.34 12.72 20.85 15.34 23.34 0.37 12.84
26.74 26.41 48.53 2.74 67.83 33.65 50.08 65.41 110.32 44.06 13.82
N = number of data points.
Table 2 Statistical characteristics of composition and compaction variables for compacted soil liner materials in [17] Soil parameter
Range (N = 13)
Mean
Standard deviation
Coefficient of variation (%)
Liquid limit (%) Plastic limit (%) Plasticity index (%) Specific gravity Gravel (%) Sand (%) Fines (%) Clay (%) Silt (%) Activity Initial degree of saturation (%)
24–70 12–32 11–46 2.68–2.90 0–8 6–48 52–94 16–65 25–58 0.32–1.00 NA
40.77 17 23.769 2.7839 2 16.38 81.77 37.62 44.15 0.65 NA
14.87 5.67 11.15 0.05 2.71 12.77 12.86 15.38 9.90 0.19 NA
36.48 33.36 46.92 1.92 135.40 77.94 15.73 41.31 22.42 28.84 NA
Table 3 Statistical characteristics of composition and compaction variables for compacted soil liner materials in [9] Soil parameter
Range (N)
Mean
Standard deviation
Coefficient of variation (%)
Liquid limit (%) Plastic limit (%) Plasticity index (%) Specific gravity Gravel (%) Sand (%) Fines (%) Clay (%) Silt (%) Activity Initial degree of saturation (%)
15–91(67) 8–34(67) 2–62(67) 2.7 0–22.2(65) 0–39.4(65) 44.3–100(66) 13.9–75.3(65) 16.5–65.7(65) 0.14–1.12(65) 77–104(56)
45.10 19.46 25.64 – 2.39 15.2 81.88 40.49 41.35 0.62 91.57
18.07 5.50 14.14 – 3.53 9.32 12.38 15.17 11.99 0.18 5.46
40.06 28.27 55.16 – 148.01 61.3 15.12 37.46 28.98 29.41 5.96
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Table 4 Initial input data for reliability analysis using FORM5-Laboratory case Variables
Distribution type
Maximum hydraulic conductivity, k0 Initial saturation, Si Plasticity index PI Clay content, C Compactive effort index, E
Mean Ex(i)
Log-normal (=3) Log-normal (=3) Log-normal (=3) Normal (=2) Deterministic parameter
9
1 · 10 m/s 92.9% 16.9% 23.5% (1,0,+1)
Standard deviation, Sx(i) 10
2.7 · 10 12.8 8.2 15.4 –
m/s
Coefficient of variation, CoV (%), i 27 13.8 48.5 65.5 –
2.7. Statistical distributions
3. Reliability estimates and discussion of results
A reliability-based analysis of the hydraulic conductivity of compacted soil liners requires that the type of probability distribution functions for both hydraulic conductivity, (k) and the quality control measurements that are related to hydraulic conductivity be established. Hydraulic conductivity is traditionally assumed to be log-normally distributed [2,5,25–27]. Using 30 data points Harrop-Williams [28] employed the Kolmogorov–Smirnov (K–S) test of goodness-of-fit to show that hydraulic conductivity had the following distributions in order of preference; b, c, normal and log-normal. However, only two probability distribution functions, normal and log-normal, were considered as preliminary reliability estimates did not reveal significant differences in reliability indices among these and other distributions such as two parameter c, Gumbel and Weibull distributions. A K–S goodness-of-fit test on the independent variables in Eqs. (12) and (13) revealed that initial saturation and plasticity index are log-normally distributed, clay content and gravel content are normally distributed while the compactor weight has a Weibull distribution. The compactor weights are those of sheepsfoot rollers reported in the database by [9]. These weights are globally the same and are not restricted to certain parts of the world.
3.1. Set up of numerical simulations
4
The determination of reliability indices for the laboratory case using the model in Eq. (12) was made for hydraulic conductivity, over a range of coefficients of variation from 10% to 100% for k, Si, PI, and C. Table 4 shows the initial input parameters for the laboratory case. Taking hydraulic conductivity to be normally and log-normally distributed, the results of varying the coefficients of variation for each variable in turn (while the values for other variables are kept constant) are shown in Fig. 1a through to Fig. 1d. The independent variables in the regression model in Eq. (13) are compactor weight (W), plasticity index (PI), initial saturation (Si), gravel content (G) and clay content (C). The initial input parameters in the reliability analysis using FORM5 [14] are shown in Table 5. The numbers in brackets in column 3 of Table 5 are various code numbers for the statistical distributions of the variables as used in FORM5 [14]. The regulatory maximum hydraulic conductivity for liners is 1 · 107 cm/s. The corresponding standard deviation of 2.7 · 108 cm/s is based on the minimum value of coefficient of variation for hydraulic conductivity obtained by [25] for 57 landfill sites and by [9] for 67 landfill sites.
BSH(N)
BSL(N)
RED. BSL(N)
BSH(LN)
BSL(LN)
RED. BSL(LN)
3.5
Reliability index
3 2.5 2 1.5 1
(N)-normal distribution 0.5
(LN)-log-normal distribution 0 0
20
40
60
80
100
Coefficient of variation (%)
Fig. 1a. Reliability index versus coefficient of variation for hydraulic conductivity, k.
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Table 5 Initial input data for reliability analysis using FORM5-Field case S/No
Variables
Distribution type
Mean Ex(i)
Standard deviation Sx(i)
Coefficient of variation Cov(%),i
1 2 3 4 5 6
Maximum hydraulic conductivity, k0 Compactor weight (sheeps foot rollers only), W Plasticity index, PI Initial degree of saturation, Si Gravel content, G Clay content, C
Log-normal (=3) Weibull (=9) Log-normal (=3) Log-normal (=9) Normal (=2) Normal (=2)
1 · 107cm/s 261.4 (165–338) 16.9 (4–34.9) 92.9 (69.09–121.74) 25.6 (0.59) 23.5 (0.5–60)
2.7 · 108cm/s 61.4 8.2 12.8 17.3 15.4
27 23.5 48.5 13.8 67.8 65.4
The values in parentheses in column (4) are the ranges for the variables. The values in column (6) are automatically calculated and used in FORM5. For sensitivity analysis, the variables were in turn made to assume values of coefficient of variation ranging between 10% and 100%. The values of the hydraulic conductivity were first varied while the values of the input parameters were kept constant for a normally – distributed hydraulic conductivity. These initial reliability estimates yielded reliability indices less than unity. The mean value of the compactor weight (W) was then scaled up to the maximum value of 338 kN (see [9]) while that of the gravel content (G) was scaled down to assume the maximum value of 10% recommended by the US Environmental Protection Agency [29,30]. Thus, for a specified compactor weight, only soils from lateritic deposits having gravel contents less than or equal to 10% can be used as hydraulic barrier materials. The only distribution types for hydraulic conductivity that were considered are normal and log-normal distributions. 3.2. Simulation results: laboratory case 3.2.1. Effect of variation of hydraulic conductivity on reliability index The range of coefficient of variation of 0.1 to 1.0 for a normally distributed K did not affect the reliability indices for a given compactive effort significantly. At the effort of the BS heavy compaction, reliability index, b, varied from 3.48 to 3.48 as coefficient of variation of hydraulic conductivity (CoV(HC)) was changed from 0.1 to 1.0. Similar trends were observed when lower compactive efforts were employed. The reliability indices decreased from 2.447 to 2.445 for BS light compaction and from 1.51 to 1.51 for reduced BS light compaction over the same range of CoV(HC). Benson [25] reported values of coefficient of variation for hydraulic conductivity ranging between 0.27 and 7.67 while Benson et al. [9] recorded values ranging between 0.27 and 2.77. At a CoV(HC) of 0.27 the estimated reliability indices are 3.48, 2.45 and 1.50 for BS heavy, BS light and reduced BS light compaction, respectively. The corresponding reliability indices at CoV(HC) of 2.77 are 3.47, 2.44 and 1.51 and at CoV(HC) of 7.67 are 3.37, 2.38 and 1.48, for BS heavy, BS light and reduced BS light compactive efforts, respectively. The decrease in reliability indices as the CoV(HC) increased was not significant for
each compactive effort considered. For log-normally distributed hydraulic conductivity b decreased from 3.48 to 3.08 for BS heavy, 2.44 to 2.08 for BS light, and from 1.51 to 1.18 for reduced BS light compaction energies, respectively as shown in Fig. 1a. The effect of the differences in compactive efforts is clearly seen in Fig. 1a. There is a consistent decrease in reliability index as the compactive efforts decreased from BS heavy to reduced BS light, and for each CoV(HC) considered. Soil specimens compacted with higher compactive efforts generally have lower hydraulic conductivity. This trend corresponds to that of decreasing reliability index values as compactive effort is decreased. 3.2.2. Effect of variation of initial saturation on reliability index The variation of reliability indices as the coefficient of variation for degree initial of saturation (CoV(Si)) increased is shown in Fig. 1b. As the CoV(Si) varied from 0.1 to 1.0, reliability index decreased non-linearly from 4.40 to 0.28, 3.13 to 0.06 and from 1.96 to 0.12 for BS heavy, BS light and reduced BS light compactive efforts, respectively. A CoV(HC) of 0.27 was maintained, while the CoV(Si) was varied. Similar trends were obtained when k assumed a log-normal distribution as shown in Fig. 1b. The rapid change in the reliability indices as the CoV(Si) varied clearly demonstrates that hydraulic conductivity is significantly affected by changes in initial degree of saturation. Thus, the initial saturation is a parameter that needs to be carefully controlled when compacting soils for use as liners and covers in waste-containment systems. Significant differences in reliability indices were also observed at lower CoV(Si) values with respect to the different compactive efforts used. These differences are smaller at higher values of CoV(Si). For each given coefficient of variation, the higher compactive efforts yielded higher reliability indices. This indicates that higher initial degrees of saturation can be obtained at higher compactive efforts. In practice, increases in initial saturation generally occur as a result of increasing moulding water content without changing compactive effort by using the same compactor and number of passes, increasing compactive effort without increasing moulding water content, or increasing both compactive effort and moulding water content [17]. These conditions are said to decrease the hydraulic conductivity of soils. The trends observed for initial saturation with
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511
5 4.5
BSH(N)
BSL(N)
RED. BSL(N)
BSH(LN)
BSL(LN)
RED. BSL(LN)
4
Reliability index
3.5 3 2.5 2
(N)-normal distribution
1.5
(LN)-log-normal distribution 1 0.5 0 -0.5 0
20
40
60
80
100
120
Coefficient of variation (%)
Fig. 1b. Reliability index versus coefficient of variation for initial degree of saturation, Si.
increasing coefficient of variation is expected as it has a high partial-F in Eq. (12). 3.2.3. Effect of variation of plasticity index on reliability index Some slight decreases in reliability indices occurred as a result of increasing the coefficient of variation for plasticity index for each of the three compactive efforts. These trends are shown in Fig. 1c for the cases when k is normally or log-normally distributed. As coefficient of variation increased from 0.1 to 1.0, reliability indices decreased from 3.81 to 3.20, 2.69 to 2.20 and from 1.68 to 1.30 for BS heavy, BS light and reduced BS light compaction, respectively when k is normally distributed. The mean value used
4
for plasticity index was 16.9. With standard deviations up to 16.9, high reliability index values can still be obtained particularly at the effort of the BS heavy compaction. However, concerns about compactibility are often the basis for restricting the value of plasticity index of soils to be used as hydraulic barriers in waste containment facilities. The minimum recommended values of plasticity index are P7–10% [31] and P12–15% [32]. EPA [29] recommends that the plasticity index of soil liner materials should be greater than 10 and discourages the use of soils with a plasticity index greater than about 30 as such soils are difficult to work with in the field (see [30]). The reliability indices were generally higher at higher compactive efforts. As with other variables, the reliability
BSH(N) BSH(LN)
BSL(N) BSL(LN)
RED. BSL(N) RED. BSL(LN)
3.5
Reliability index
3 2.5 2 1.5 1 (N)-normal distribution 0.5 (LN)-log-normal distribution 0 0
20
40
60
80
100
Coefficient of variation (%)
Fig. 1c. Reliability index versus coefficient of variation for plasticity index, PI.
120
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indices obtained with reduced BS light compaction were all lower than 2.0 for coefficients of variation within 0.1 to 1.0 for both normal and log-normal distributions of k.
lated F-value exceeds a corresponding critical F-value for a specified level of significance. The effects of coefficient of variation and compactive effort on reliability indices for each of the parameters considered are statistically significant at a = 0.05 (see Table 6). The highest F-value with respect to the effect of coefficient of variation was obtained for plasticity index (F = 62.82); followed by initial degree of saturation (F = 27.05). The effects of variations in hydraulic conductivity and clay content on reliability indices, although statistically significant, were less with F = 8.89 for hydraulic conductivity and F = 8.80 for clay content. The influence of compactive efforts on the reliability indices was more pronounced than that of coefficient of variation for hydraulic conductivity, plasticity index and clay content. Compactive effort has an over whelming influence on the reliability index values for hydraulic conductivity (F = 1485.73). The lowest Fvalue (19.77) was observed for initial degree of saturation with respect to the effect of compactive effort on reliability index values.
3.2.4. Effect of variation of clay content on reliability index Although compacted clay liners are required to have clay contents (% <2 lm)P20–25% [32], variations in clay contents affected reliability indices. The values of reliability index decreased as coefficient of variation increased from 0.1 to 1.0 (Fig. 1d). The decrease in reliability index values was more marked or pronounced at higher compactive efforts, although the reliability index remained above 3.0 for all the values of coefficient of variation at the energy of the BS heavy (BSH) compaction, and above 2.0 at the energy of the BS light (BSL) compaction. This trend was the same for both normally- and log-normally-distributed k. However, log-normally distributed k yielded lower values of reliability index. 3.2.5. Analysis of variance The results for normally-distributed hydraulic conductivity shown in Figs. 1a–1c and 1d were investigated statistically to show the level of the effects of (1) coefficient of variation, and (2) compactive effort, on the reliability indices obtained. The two-way analysis of variance (without replication) was employed at 5% level of significance (that is, a = 0.05). The results are shown in Table 5. The degrees of freedom, calculated F-values, p-values (tail probability) as well as the critical F-values obtainable from statistical tables, are shown for hydraulic conductivity (k); initial degree of saturation (Si); plasticity index (PI) and clay content (C), respectively. The calculated F-values represent ratios of the between sample variance to the within sample variance for a given set of data. The effect of a particular treatment is said to be statistically significant when a calcu-
3.2.6. Assessment of model suitability In structural reliability, reliability indices (b) ranging between 2.5 and 4.0 are considered to provide reasonable margins of safety for a given structure or structural component. The corresponding probability of failure within this region of safety index values is generally less than 6.25 · 103. In terms of acceptable probability of failure, 2.5 can be taken as a convenient lower limit for the reliability assessment of compacted lateritic soil liners. The high estimated b values indicate that the model is suitable for predicting the hydraulic conductivity of laboratorycompacted lateritic soils. On the basis of the minimum value of safety index adopted and considering the statistical values of the initial
4.5 4
BSH(N)
BSL(N)
RED. BSL(N)
BSH(LN)
BSL(LN)
RED. BSL(LN)
Reliability index
3.5 3 2.5 2 1.5 1 (N)-normal distribution 0.5 (LN)-log-normal distribution 0 0
20
40
60
80
100
Coefficient of variation (%)
Fig. 1d. Reliability index versus coefficient of variation for clay content, C.
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Table 6 Analysis of variance of reliability index values Variable
Source of variation
Degree of freedom
F-value (calculated)
F-value (critical)
Hydraulic conductivity, k0
Coefficient of variation Compactive effort Error (Total)
9 2 18(29)
8.89 9.7E + 07
2.46 3.55
Initial saturation, Si
Coefficient of variation Compactive effort Error (Total)
9 2 18(29)
27.05 19.77
2.46 3.55
Plasticity index, PI
Coefficient of variation Compactive effort Error/Total
9 2 18(29)
62.82 6861.57
2.46 3.55
Clay content, C
Coefficient of variation Compactive effort Error/Total
9 2 18(29)
8.80 1485.73
2.46 3.55
degree of saturation, plasticity index and clay percent, only the compaction energy of the BS heavy gave reliability or safety indices that were well above 2.5 (see Fig. 1a) for the range of coefficients of variation considered. Reliability indices obtained at the energy of the BS light compaction however, were close to the value of 2.5 and ranged between 2.45 and 2.45. The corresponding values for the reduced BS light varied from 1.51 to 1.51. These values are for hydraulic conductivity when it is normally distributed. Similar results were obtained when hydraulic conductivity assumed a log-normal distribution. Landfill liner construction is a specialized one and based on the estimated reliability indices, only samples compacted at the energy of the BS heavy compaction would yield laboratory hydraulic conductivity values suitable for design purposes. In Fig. 1b, it can be seen that the BS heavy compaction will be able to accommodate up to 21% coefficient of variation in the value of the initial saturation while up to 15% of the initial degree of saturation can be accommodated when the BS light compaction is used. Reliability indices obtained for compaction at the energy of the reduced BS light compaction were generally lower than 2.0. Similar results were obtained for log-normally distributed hydraulic conductivity, k. Fairly wide variations in plasticity index and clay percent within a lateritic soil deposit can be allowed provided compaction is carried out at the energy of the BS heavy compaction as the reliability indices ranged between 3.34 and 3.81 for plasticity index and between 3.17 and 3.83 for clay percent for normally-distributed hydraulic conductivity. However, with the energy of BS light compaction, a coefficient of variation of up to 40% for plasticity index and 53% for clay percent in terms of variations within a lateritic soil deposit, can be accommodated (see Figs. 1c and 1d). The energy of the reduced BS light compaction did not provide acceptable values of reliability index, as they are generally below 2.0 for both plasticity index and clay percent. Slightly lower values of coefficient of variation were obtained for plasticity index and clay percent when hydraulic conductivity assumed a log-normal distribution.
The estimated reliability indices show that the model used in this study is suitable for predicting the hydraulic conductivity of laboratory-compacted lateritic soils. The lowest values of hydraulic conductivity are obtained for samples compacted at the BS heavy compaction energy. Acceptable results can be obtained at the BS light compaction energy for only limited values of coefficient of variation especially for plasticity index and clay content. Lower values of the coefficient of variation for degree initial of saturation can be obtained at this compaction energy. The reliability indices also show that lateritic soils can serve as good landfill liner materials. Considering the Figs. 1a–1c and 1d, for instance, it is seen that compactive effort and the initial degree of saturation are essentially the two critical parameters that require very strict control. These observations are in good agreement with the partial-F values obtained by Benson and Trast [17] for compactive effort, E (65) and initial degree of saturation, Si (262). The plasticity index of lateritic soils also requires to be carefully controlled especially when compacting at the energy of the BS light. A partial-F value of 24 was obtained by Benson and Trast [17] for plasticity index. Plasticity index values of lateritic soils will be limited by considerations for constructability as soils with plasticity index values higher than 30% will tend to stick to the wheels of field compaction equipment. 3.2.7. Distributions of initial saturation and reliability index The K–S goodness-of-fit test indicated that the initial degree of saturation was log-normally distributed. However, due to the rapid decrease in reliability indices with increasing coefficient of variation, it was necessary to investigate the trend when other probability distribution functions were assumed. The only case considered was when hydraulic conductivity assumed a log-normal distribution. The results are shown graphically in Figs. 2a, 2b and 2c, for different compactive efforts. Similar trends of decreasing reliability indices with increasing coefficient of variation were observed in all the cases considered. However, some differences were observed
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Normal Gamma Weibull
Reliability index
5
Lognormal Gumbel
4
3
2
1
0 0
10
20
30
40
50
60
70
80
90
100
110
Coefficient of variation (%)
Fig. 2a. Reliability index versus coefficient of variation for initial degree of saturation (Si) for different distributions of Si (distribution type for k = lognormal at BSH).
4
Normal Gamma Weibull
3.5
Lognormal Gumbel
Reliability index
3 2.5 2 1.5 1 0.5 0 0
10
20
30
40
50
60
70
80
90
100
110
Coefficient of variation (%)
Fig. 2b. Reliability index versus coefficient of variation for initial degree of saturation (Si) for different distributions of Si (distribution type for k = lognormal at BSL).
for the different distributions of initial saturation as well as for the different compactive efforts. The extreme value Type I (Gumbel) gave the highest values of reliability index at 10% (0.1) coefficient of variation followed by lognormal, gamma normal and extreme value Type III (Weibull) distributions. The Weibull distribution consistently gave the lowest reliability indices at the energies of B.S. heavy and B.S. light compaction, but a different trend occurred with reduced B.S. light compaction. Moreover, at 1.0 (100%) coefficient of variation, the normal distribution gave the highest reliability indices for all the cases considered. At the B.S. heavy compactive effort, the reliability indices did not differ significantly for the five distributions when coefficient of variation was in the range of 50–70% (Fig. 2a). Similar observations were made for B.S. light
compaction at 30–50% coefficient of variation (Fig. 2b) and for reduced B.S. light compaction at 20–30% coefficient of variation (Fig. 2c). The differences in reliability index values among the different distributions were highest for reduced B.S. light compaction and lowest for B.S. heavy compaction at 100% (1.0) coefficient of variation. The results presented are for log-normally distributed hydraulic conductivity. Similar results were obtained for normally distributed hydraulic conductivity. The reliability indices for the various distributions of initial saturation at coefficients of variation of 0.1 and 1.0 and for the three compactive efforts are shown in Table 7. The reliability indices when coefficient of variation was equal to 100% were the same for gamma and Weibull distributions when the distribution types for hydraulic conductivity and
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Normal Gamma Weibull
2
515
Lognormal Gumbel
Reliability index
1.5
1
0.5
0
-0.5 0
10
20
30
40
50
60
70
80
90
100
110
Coefficient of variation (%)
Fig. 2c. Reliability index versus coefficient of variation for initial degree of saturation (Si) for different distributions of Si (distribution type for k = lognormal at RED. BSL).
Table 7 Reliability index values for different distributions of initial degree of saturation Distribution type for initial saturation, Si
Coefficient of variation
For log-normally distributed hydraulic conductivity Normal 0.1 1.0 Log-normal 0.1 1.0 Gamma 0.1 1.0 Gumbel 0.1 1.0 Weibull 0.1 1.0 Distribution type for initial saturation, Sr
Coefficient of variation
For normally distributed hydraulic conductivity Normal 0.1 1.0 Log-normal 0.1 1.0 Gamma 0.1 1.0 Gumbel 0.1 1.0 Weibull 0.1 1.0
Reliability index BSH
BSL
RED. BSL
3.83 0.44 4.35 0.27 4.18 0.15 5.65 0.32 3.07 0.18
2.83 0.33 3.08 0.06 3.00 0.02 3.81 0.19 2.41 0.02
1.84 0.21 1.91 0.22 1.89 0.11 2.19 0.05 1.71 0.11
Reliability index BSH
BSL
RED. BSL
3.86 0.44 4.40 0.28 4.23 0.18 5.72 0.33 3.10 0.18
2.86 0.33 3.13 0.06 3.04 0.03 3.88 0.19 2.44 0.03
1.87 0.22 1.96 0.12 1.93 0.11 2.25 0.06 1.73 0.11
BSH = BS heavy compaction; BSL = BS light compaction; and RED, BSL = reduced, BS light compaction.
the respective compactive efforts are considered. Reliability indices obtained when hydraulic conductivity was normally distributed were generally higher than the corresponding
values at coefficient of variation of 0.1 but generally lower than the corresponding values at coefficient of variation of 1.0 for log-normally distributed hydraulic conductivity. Moreover, the information in Table 7 tends to suggest the following probability distribution types for initial saturation, in order of preference; extreme value Type I (Gumbel), log-normal, gamma, normal and extreme value Type III (Weibull). This observation is in good agreement with K–S goodness-of-fit test performed on soil compositional and compaction variables for lateritic soils [33]. 3.2.8. Implications for field application The two main variables observed to have the most significant impact on the reliability indices associated with the model considered are compactive effort and degree of saturation. The compactive effort is the laboratory equivalent of compactor weight under field conditions. The higher the compactive effort the higher the reliability indices at any given coefficient of variation, for hydraulic conductivity as well as the independent variables, namely; initial saturation, plasticity index and clay content. For purpose of construction of soil liners with the required very low probability of failure, high compactor weights should be used. Moreover, careful construction quality control should be maintained to achieve very minimal variations in the values of initial saturation as this would result in wide variations in hydraulic conductivity values. High safety or reliability levels can best be achieved provided initial saturation values of compacted soil layers do not vary by more than 10%. Daniel [31] suggests the use of 20,000 kg compactors. Rubber-tired rollers can be as heavy as 18,000 kg and are capable of delivering tire pressures of between 450 and 1000 kPa, while sheepsfoot rollers can deliver pressures of between 1750 and 4500 kPa [34]. These rollers are suitable
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Hydraulic conductivity Compactor weight Plasticity index Initial saturation Clay content Gravel content
2.5
Reliability index
2
1.5
1
0.5
0 0
20
40
60
80
100
120
Coefficient of variation (%)
Fig. 3a. Reliability index versus coefficient of variation (distribution type for field k = normal).
3
Hydraulic conductivity Compactor weight Plasticity index
2.5
Initial saturation Clay content Gravel content
Reliability index
2
1.5
1
0.5
0 0
20
40
60
80
100
120
Coefficient of variation (%)
Fig. 3b. Reliability index versus coefficient of variation (distribution type for field k = lognormal).
for compacting clay soils of low plasticity which are usually recommended for use as soil liners [34]. 3.3. Simulation results: field case 3.3.1. Effect of variation in hydraulic conductivity When the distribution type for hydraulic conductivity was normal and its coefficient of variation ranged from 10% to 100% reliability indices, b, only decreased gradually from 1.58 to 1.57 (Fig. 3a). However, there was an appreciable decrease in reliability index, b, from 1.58 to 1.37 for a range of 10% to 100% in coefficient of variation when a log-normal distribution was assumed (Fig. 3b). This difference tends to suggest that normal distribution is superior to log-normal distribution for hydraulic conductivity of compacted lateritic soils.
3.3.2. Effect of variations in the independent variables Of all the independent variables in Eq. (13), compactor weight (W) exhibited the most significant change in reliability indices. For a normally distributed hydraulic conductivity, k, (Fig. 3a) reliability indices changed from 2.54 to 0.16 when the coefficient of variation was increased from 10% to 100% for W. The corresponding change in reliability indices when k was made to assume a log-normal distribution (see Fig. 3b) was from 2.49 to 0.15 for the same range of coefficient of variation. The trend in the decrease of b values for W should be expected since in the model in Eq. (13), W had the highest partial-F of 59.7. It has been shown by [9] that for given plasticity index, clay content; initial saturation and gravel content, increasing the compactor weight results in lower hydraulic conductivity. Thus, in order to obtain liners and covers that have desirable hydraulic con-
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ductivities (that is, k values that are much less than 1 · 107cm/s or 1 · 109 m/s) higher compactor weights are generally required. However, the study carried out by [9] did not involve a reliability analysis, hence no reliability index value was indicated in their analysis. For a given landfill liner construction, it is not necessary to employ compactors whose weights differ significantly from each other. The value of 10% coefficient of variation seems to be a convenient maximum for variations in compactor weight, W. It is best to use compactors of the same weight for a given soil liner construction. The second variable that showed significant decrease in reliability indices as the coefficient of variation increased from 10% to 100% is the initial saturation, Si (or initial degree of saturation). Reliability indices decreased from 1.60 to 0.78 for normally – distributed k and from 1.58 to 0.75 for log-normally distributed k (Figs. 3a and 3b) respectively. Results of tests by [17] on compacted clays show that trend of decreasing hydraulic conductivity with increasing initial saturation exists. For the soils studied by [17] hydraulic conductivities less than 1 · 107cm/s, the common regulatory maximum hydraulic conductivity in the United States, can be achieved by compacting to an initial saturation in excess of 85%. However, Benson and Trast [17] cautioned that increasing initial saturation might not always result in lower hydraulic conductivity and that higher initial saturation should be obtained without decreasing compactive effort (that is, compactor weight and number of passes). Plasticity index is another variable that exhibited appreciable decrease in b values when the coefficient of variation increased from 10% to 100%. The reliability indices decreased from 1.700 to 1.39 as displayed in Fig. 3a and from 1.68 to 1.36 in Fig. 3b. The model in Eq. (13) showed that a partial-F of 50.1 was obtained for plasticity index (PI). According to [9], hydraulic conductivity decreases quickly as the PI increases from 10 and 30, but then
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decreases more slowly as the PI is increased further. However, plasticity indices in the range 10 to 30 are recommended by [29]. It is envisaged that soils with PI values greater than 30 may give field construction problems. A minimum PI value of 7 has been suggested by [9] for liner soil materials. With a mean value of 10% and a corresponding standard deviation (scaled) of 6.76% for the gravel contents the b values decreased from 1.60 to 1.55 (Fig. 3a) and 1.59 to 1.53 (Fig. 3b) for the assumed range of coefficients of variation. It can be seen that variations in gravel content within the limits recommended by [29] do not significantly affect the reliability of lateritic soil liners. Similarly increases in clay contents did not significantly affect the estimated reliability indices. The b values decreased from 1.60 to 1.55 (Fig. 3a) and from 1.58 to 1.53 (Fig. 3b) within the assumed coefficient of variation. Proper compaction of lateritic soils having appropriate combination of soil grading and plasticity characteristics will ensure reliable soil liner construction provided that compaction is carried out at suitable moisture content on the wet side of optimum moisture content. In this case, only lateritic soils from deposits with percent gravel contents not exceeding 10% will be acceptable. 3.3.3. Assessment of model suitability for lateritic soils The model (Eq. (13)) developed by [9] explains only 78% of the variance in hydraulic conductivity. Moreover, a constant value of 2.7 was adopted for the specific gravity of the soils. In practice lateritic as well as other soils do not assume a constant value of specific gravity. Nevertheless, it is still possible to employ Eq. (13) for predicting the field hydraulic conductivity of lateritic soils particularly when higher compactor weights are used. In this study, only the compactor weights for sheepsfoot rollers were considered. Sheepsfoot compactors generally have higher weights than rubber-tired compactors. Increasing the weight of the
3
Normal distribution Lognormal distribution Gamma distribution
2.5
Gumbel distribution
Reliability index
Weibull distribution 2
1.5
1
0.5
0 0
20
40
60
80
100
120
Coefficient of variation (%)
Fig. 4a. Reliability index versus coefficient of variation for compactor weight, W (distribution type for field k = normal).
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Normal distribution Lognormal distribution Gamma distribution Gumbel distribution Weibull distribution
Reliability index
2.5
2
1.5
1
0.5
0 0
10
20
30
40
50
60
70
80
90
100
110
Coefficient of variation (%)
Fig. 4b. Reliability index versus coefficient of variation for compactor weight, W (distribution type for field k = lognormal).
compactor generally results in greater compactive efforts, more shear deformation, more smaller uniform pores, and lower hydraulic conductivity [9]. Using box plots it was shown [9] that the median hydraulic conductivity for compactors classified as sheepsfoot was k = 1.4 · 108 cm/s which is approximately 4 times lower than the median hydraulic conductivity of compactors classified as rubber-tired for which k = 6.0 · 108 cm/s. Thus, it is appropriate to use sheepsfoot compactors for the field compaction of lateritic soil liners. Although the b values at 10% coefficient of variation for some of the model parameters were fairly greater than 1.5, Eq. (13) can be used for prediction of lateritic soil hydraulic conductivity. However, b values close to 2.5 were obtained when the coefficient of variation for the compactor weight was assumed to be 10%. Thus, Eq. (12) can be used to predict the hydraulic conductivity of properly compacted lateritic soils to be used for waste containment purposes. Under field conditions, the three most important parameters that require careful control are compactor weight, initial saturation and plasticity index. Eq. (13) suggests that increasing compactor weight, PI or initial saturation will result in lower geometric mean hydraulic conductivity [9]. Even when a higher compactor weight is used and there is need to use more than one compactor it is necessary to select compactors whose weights do not differ from each other by more than 10%. The trends in Figs. 3a and 3b indicate the need for this caution. 3.3.4. Probability distributions for compactor weight Trends in Figs. 4a and 4b showed that compactor weight, W, has a tremendous influence on geometric mean field hydraulic conductivity, kg. The influence of different probability distribution types for W on the reliability estimates for cases when the hydraulic conductivity assumes normal and lognormal distributions was studied. The up-scaled compactor weight was used in this case
Table 8 Reliability indices for different distributions of compactor weight Distribution type
Gumbel Log-normal Gamma Normal Weibull
Reliability indices Normal distribution
Log-normal distribution
CoV = 10%
CoV = 100%
CoV = 10%
CoV = 100%
2.67 2.63 2.63 2.62 2.54
0.31 0.25 0.16 0.42 0.16
2.61 2.58 2.57 2.57 2.49
0.30 0.24 0.15 0.42 0.15
CoV = coefficient of variation (%).
(mean = 338 kN; standard deviation = 79.4 kN). Five alternative distributions, namely; normal, log-normal, gamma, Gumbel and Weibull, respectively were considered. The reliability indices for 10% and 100% coefficient of variation for each distribution type and for the cases when hydraulic conductivity, k, has normal and lognormal distributions, respectively, are shown in Table 8. Trends of decreasing reliability indices with increasing values of coefficient of variation are shown in Fig. 4a for normally-distributed k and in Fig. 4b for log-normal distributed k. It is evident from Table 8 and from Figs. 4a and 4b that, in decreasing order of superiority, compactor weight assumes the following distribution types: Gumbel, log-normal, gamma, normal and Weibull. This order holds regardless of the distribution type for hydraulic conductivity, k. 4. Conclusions Statistical characteristics of lateritic soil properties were established and compared with those of land fill liner soils. The distributional forms for relevant soil properties were employed to accomplish a reliability-based assessment of the suitability of lateritic soils as hydraulic barriers in waste containment facilities using existing hydraulic conductivity
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models developed from laboratory and field data considering variations in soil parameters. Variations in compactive effort and initial degree of saturation were observed to have the most significant impact on the reliability of soil liner hydraulic conductivity based on the laboratory model. Sheepsfoot rollers can provide, under field conditions, the equivalent of laboratory B.S. heavy compactive effort which generally gave high reliability indices. An additional assessment of the suitability of lateritic soils for use as compacted liners and covers in landfills was also made using a first-order reliability approach. The assessment was based on a regression model developed from field data (for 67 actual landfill sites) adopted for predicting the geometric mean hydraulic conductivity of lateritic soils. The values of the reliability index, b, obtained as coefficient of variation of the various variables increased suggest that this regression model is suitable for predicting the hydraulic conductivity of lateritic soils. The trends in the variation of reliability indices as coefficient of variation of the various variables increased also show that the compactor weight has most significant effect on the geometric mean hydraulic conductivity of lateritic soils, followed by initial saturation and plasticity index, respectively. The results also show the importance of the probability distribution of the compactor weight in the reliability estimate. References [1] Christian JT, Ladd CC, Baecher GB. Reliability and probability in stability analysis. Proceedings of Conference on Stability and Performance of Slopes and Embankments – II, vol. 2. New York, NY: ASCE; 1992. p. 1071–111. [2] Gui S, Zhang R, Turner JP, Xu X. Probabilistic slope stability analysis with stochastic soil hydraulic conductivity. J Geotech Geoenviron Eng, ASCE 2000;126(1):1–9. [3] Ronold KO, Bysveen S. Probabilistic stability analysis for deep-water foundation. J Geotech Eng, ASCE 1992;118(3):394–405. [4] Cherubini C, Garrasi A, Petrolla C. The reliability of an anchored sheet-pile wall embedded in cohesionless soil. Can Geotech J 1992;29(3):426–35. [5] Bogardi I, Kelly WE, Bardossy A. Reliability model for soil liners: Initial design. J Geotech Eng ASCE 1989;115(5):658–69. [6] Rowe RK, Fraser MJ. Effect of uncertainty in the assessment of the potential impact of waste disposal facilities. Geoenvironment 2000, Geotech. Spec. Publ. 46, ASCE, New York, 1995. p. 270–84. [7] Jang YS, Sitar N, Der Kiureghian A. Reliability analysis of contaminant transport in saturated porous media. Water Resour Res 1994;30(8):2435–48. [8] Daniel DE, Benson CH. Water content-density criteria for compacted soil liners. J Geotech Eng, ASCE 1990;116(12):1811–30. [9] Benson CH, Zhai H, Wang X. Estimating hydraulic conductivity of compacted clay liners. J Geotech Eng, ASCE 1994;120(20):366–87. [10] Townsend FC, Manke GP, Parcher JV. The influence of sesquioxides on laterite soils properties. Highway Res Board Record 1971;374: 80–92.
519
[11] Ola SA. Geotechnical properties and behaviour of some Nigerian lateritic soils. In: Ola SA, editor. Tropical soils of Nigeria in engineering practice. Rotterdam: A.A. Balkema; 1983. p. 61–84. [12] Benson CH, Zhai H, Rashad SM. Statistical sample size for construction of soil liners, ASCE. J Geotech Eng 1994;120(101): 1704–24. [13] Hasofer AM, Lind NC. An exact and invariant first-order reliability format. J Eng Mech Div, ASCE 1974;100(1):111–21. [14] Gollwitzer S, Abdo T, Rackwitz R. First order reliability method manual, v. 5. Munich, 1988, 47p. [15] Juang CH, Rosowsky DV, Tang WH. Reliability-based method for assessing liquefaction potential of soils. J Geotech Geoenviron Eng, ASCE 1999;125(8):684–9. [16] Thoft-Christensen P, Baker MJ. Structural reliability theory and its applications. Berlin, Heidelberg: Springer Verlag; 1982. [17] Benson CH, Trast JM. Hydraulic conductivity of thirteen compacted clays. Clays Clay Min 1995;45(6):669–81. [18] Meireles JMF. Trial pavements of laterite base course. In: Proceedings of the Regional Conference for Africa on Soil Mechanics and Foundation Engineering, 4th, vol. 1, Cape Town, 1967. p. 71–6. [19] Ola, S.A., Stabilisation of Nigerian lateritic soils with cement, bitumen and lime. In: Proceedings of the Regional Conference for Africa on Soil Mechanics and Foundation Engineering, 6th, vol. 1, Durban, 1975. p. 145–52. [20] Ogunsanwo O. Variability in the shear strength characteristics of an amphibolite derived laterite soil. Bull IAEG 1985;32:111–5. [21] Ogunsanwo O. The micro-fabric and permeability of a laterite clay. In: Proceedings of the First International Conference on Geomechanics of Tropical Lateritic and Saprolitic Soils, Brasilia, vol. 1, 1985. p. 397–402. [22] Ogunsanwo O. CBR and shear strengths of compacted laterite soils from southwestern Nigeria. Quart J Eng Geol 1989;22:317–28. [23] Ogunsanwo O. Some geotechnical properties of two laterite soils compacted at different energies. Eng Geol 1989;26:261–9. [24] Ogunsanwo O. Influence of sample preparation and mode of testing on the shear strength characteristics of laterite soils from southwestern Nigeria. Bull Int Assoc Eng Geol 1993;47:141–4. [25] Benson CH. Probability distributions for hydraulic conductivity of compacted soil liners. J Geotech Eng, ASCE 1993;119(3):471–86. [26] Benson CH, Daniel DE. Minimum thickness of compacted soil liners: I. Stochastic models. J Geotech Eng, ASCE 1994;120(1):129–52. [27] Benson CH, Daniel DE. Minimum thickness of compacted soil liners; II. Analysis and case histories. J Geotech Eng, ASCE 1994;120(1): 153–72. [28] Harrop-Williams K. Clay liner permeability: Evaluation and variation. J Geotech Eng, ASCE 1985;111(10):1211–25. [29] Environmental Protection Agency (EPA), Requirements for hazardous waste landfill: Design, construction and closure. Publ. No. EPA 625/4 - 89 - 022, Cincinnati, OH, 1989. [30] Das BM. Principles of geotechnical engineering. 4th ed. Boston: PWS Publishing Company; 1998. 712p.. [31] Daniel DE. Clay liners. In: Daniel DE, editor. Geotechnical practice for waste disposal. Chapman and Hall; 1993. pp. 137–63. [32] Benson CH, Daniel DE, Boutwell GP. Field performance of compacted clay liners. J Geotech Geoenviron Eng, ASCE 1999;125(5): 390–403. [33] Nwaiwu CMO. Evaluation of compacted lateritic soils as hydraulic barriers in municipal solid waste containment systems. Ph.D. dissertation, Ahmadu Bello University, Zaria, 2004. 457p. [34] Holtz RD, Kovacs WD. An introduction to geotechnical engineering. Englewood Cliffs, NJ: Prentice-Hall, Inc.; 1981. 733p.