Predicting Hydraulic Conductivity of Compacted Lateritic Soils: A Reliability Approach

 

Charles Malachy Okechukwu Nwaiwu

Department of Civil and Water Resources Engineering,
University of Maiduguri, Maiduguri, Borno State, Nigeria
E-mail: nknwaiwu@yahoo.co.uk

Joseph O. Afolayan

and

Kolawole ‘Juwonlo Osinubi

Department of Civil Engineering , Ahmadu Bello University
Zaria, Kaduna State, Nigeria

ABSTRACT

First-order reliability method (FORM), initially developed for structural reliability applications, was employed in the analysis of hydraulic conductivity (permeability) of compacted lateritic soils as landfill liners. The analysis, which involved the use of a predictive model developed from laboratory data, was used to estimate hydraulic conductivity at various reliability levels for three compactive efforts. Results of the analysis show that, for any given compactive effort, the prediction model leads to increasing hydraulic conductivity as the reliability index augments at fixed values of degree of saturation, plasticity index and clay content. On the other hand, for a given reliability index, the hydraulic conductivity decreases when the initial degree of saturation and plasticity index increase. A methodology for selecting soil compositional and compaction variables which enhance reliable soil liner construction is proposed.

Keywords: Lateritic soils, Hydraulic conductivity, Landfill liners, Reliability analysis, Reliability index, Soil composition

INTRODUCTION

Compacted soil liners are widely used as hydraulic barriers in waste containment facilities (Benson et al., 1994), their primary purpose being to impede the flow of fluids. Judgement on the effectiveness of compacted clay liners is based on hydraulic conductivity (permeability). Compacted soil liners therefore, are required to have a hydraulic conductivity of less than or equal to 1×10–9m/s. Hydraulic conductivity is taken as the basic parameter for design and for characterizing liner performance and reliability (Bogardi et al., 1989).

Quite often, deterministic approaches are employed in the analysis and design of engineering structures. These approaches are characterized by the use of specified minimum factors of safety or specified minimum material properties. Deterministic approaches do not rigorously account for uncertainties in engineering analysis and design. In order to address uncertainty, probability theory has been widely accepted and used in engineering design in which some statistical knowledge of random variables such as their mean values and standard deviations is used to introduce them into applications (Kaymaz et al., 1998). Probabilistic methods, especially, reliability analysis, have frequently been used in structural engineering (e.g. Ang and Cornell, 1974; Rackwitz and Fiessler, 1978; Grigoriu, 1983; Afolayan, 1996, 1998) as well as in geotechnical and geoenvironmental engineering (e.g. Cornell, 1971; Halim and Tang, 1991; Christian et al., 1994; Gilbert and Tang, 1995; Rowe and Fraser, 1995; Gui et al., 2000). Reliability calculations provide a means of evaluating the combined effects of uncertainties, and a means of distinguishing between conditions where uncertainties are particularly high or low (Duncan, 2000). Moreover, reliability analysis provides a framework for establishing appropriate factors of safety and other design targets and leads to a better appreciation of the relative importance of uncertainties in different parameters (Christian and Baecher, 2001).

Due to some inherent variability, uncertainties are associated with compaction and soil compositional variables which have considerable influence on hydraulic conductivity. In this study, the first-order reliability approach was employed in the analysis of hydraulic conductivity of lateritic soils using a predictive model developed from laboratory data. The objectives of the study are: (1) to define hydraulic conductivities corresponding to different safety levels for each of the variables affecting hydraulic conductivity; and (2) to establish an acceptable combination of values of the variables that will likely result in reliable soil liner construction.

BACKGROUND

Soil liner reliability is defined as a probabilistic measure of assurance of post-construction performance characterized by hydraulic conductivity, k, and is represented as (Bogardi et al., 1989):

(1)

where ko = the specified hydraulic conductivity limit, such as 1×10–9 m/s. In this case, it will be reasonable to assume a condition of vertical porous media flow through compacted soil material with no macro pores or open flow paths (or flow channels).

Reliability assessment efforts are directed towards determining the probability that the compacted soil liner will not attain any of the known limit states likely to be violated throughout the useful life, or at least some specified design period, of the liner in the face of uncertainties as human error inputs, various environmental conditions, variation in material and engineering properties as well as prediction of future events. This probability of survival is given as:

(2)

where Ps = probability of survival and Pf = probability of failure.

The performance function of a soil liner can be modelled in terms of certain basic random variables x1, x2, ... xn, which must operate within certain limits for the liner to function satisfactorily. Values of x1, x2, ... xn, outside of these limits constitute the failure domain, and that surface within the n-dimensional space of basic variables xi that divides values of these variables into the failure domain and the domain for adequate performance is called the failure surface (Ocholi, 2000). A mathematical representation of this surface is known as the limit state equation (Ditlevsen, 1981).

In the reliability analysis of compacted soil liners, failure may be defined as the event of a liner hydraulic conductivity equal to or greater than the specified regulatory maximum (that is, 1×10–9m/s), over a given period of time such as the design life. The reliability problem is then formulated in terms of a limit state function g(X), where X is a vector of random variables, and g(X) < 0 denotes the region in which the threshold value is met or exceeded. Adequate performance of the liner is established when g(X) > 0. The limit state surface is denoted as g(X) = 0. Thus, for a threshold hydraulic conductivity ko, the limit state function can be formulated as

(3)

where ke is the expected hydraulic conductivity.

If X = x1, x2, ... xn, has a joint probability distribution function (pdf) given by:

(4)

the probability of failure (that is, the probability that the threshold value is exceeded), P(ko < ke) is obtained by integrating the joint pdf in the region where g(x) < 0:

(5)

where fx(x) is the joint probability density or distribution function (pdf) of X and the n-fold integral is over the unsafe region. In practice, a direct numerical evaluation of the multifold integral is virtually impossible (Jang et al., 1994) due to a lack of full probability information. Some approximate methods have been developed for evaluating the integral, including first - and second - order reliability methods (FORM and SORM), simulation methods, and hybrid methods combining simulation with FORM and/or SORM (Jang et al., 1994).

In FORM, the integral in Eq. (5) is evaluated in standard normal space by transforming the random variables X into a set of uncorrelated standard normal variates U = U(X), having the pdf (Jang et al., 1994);

(6)

where n is the number of random variables. Eq. (5) can then be written as:

(7)

where G(u) = g (x(u)) is the limit state function in the transformed space. The limit state surface is replaced by a tangent point u* and the distance from the origin, known as the reliability index, b, is given by the inner product

(8)

where a* is the unit normal at the design point directed toward the failure region. The first-order approximation of the probability of exceeding the regulatory maximum is (Jang et al., 1994):

(9)

where j ( ) is the standard cumulative normal probability and b is also known as (geometric) safety index.

MATERIALS AND METHODS

The limit state function of the form of Eq. (3) used in this study was incorporated into a FORTRAN-based program with which the reliability analysis was performed. The expected hydraulic conductivity, ke, is represented as follows:

(10)

where ke is in m/s, Si is initial degree of saturation in percent, PI is plasticity index in percent, C is clay content. (% < 2 µm), E is the compactive effort index, and e is a random error term. The compactive effort index is an integer categorical variable describing compactive effort; E assumes the values -1, 0 and 1 for modified, standard and reduced Proctor compactive efforts, respectively. Eq. (10) is a prediction model based on laboratory test results developed by Benson and Trast (1995), using the step-wise regression procedure, for estimating hydraulic conductivity of soil liners. The statistical attributes of this model are described in Benson and Trast (1995).

Database and Statistical Analysis

A database was compiled by extracting data on lateritic soils from published literature (e.g. Meireles, 1967; Ola, 1975, 1983; Ogunsanwo, 1985a, 1985b, 1993; Osinubi, 1998a, 1998b) spanning between 1967 and 1998 and from laboratory test results. The data from published literature cover lateritic soils occurring within and outside Africa.

The statistical characteristics of the soil composition and compaction variables for lateritic soils are shown in Table 1. Benson and Trast (1995) cautioned that Eq. (10) should only be used for soils that have similar characteristics as those used in developing the equation. The statistical characteristics of these liner soil materials are therefore presented in Table 2 for purposes of comparison with lateritic soils being considered in this study. The two groups of soils are seen to have comparable ranges but slightly differing mean values, standard deviations and coefficients of variation for the corresponding soil properties.

Table 1. Statistical characteristics of 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)
13-39.5 (53)
4-34.9 (53)
2.55-2.9 (49)
0-59 (33)
9-61.7 (32)
11-100 (50)
1-99.5(27)
0.5-60 (34)
0.33-1.67 (31)
69.09-121.74 (55)
41.305
24.447
16.862
2.718
25.563
37.811
41.646
23.48
21.162
0.829
92.94
11.045
6.456
8.184
0.075
17.339
12.723
20.855
15.339
23.345
0.365
12.842
26.739
26.409
48.533
 2.744
67.828
33.650
50.076
65.413
110.316
44.060
13.818

Statistical Distributions of Variables

Hydraulic conductivity is usually assumed to be lognormally distributed (Harrop-Williams, 1985; Bogardi et al., 1989; Benson, 1993; Benson and Daniel, 1994a, 1994b; Gui et al., 2000). Although alternative distributions have been proposed by Harrop-Williams (1985) and Benson (1993), the two parameter lognormal distribution type was adopted here for hydraulic conductivity.

A K-S goodness-of-fit test on lateritic soil material properties indicates that both plasticity index and initial degree of saturation are lognormally distributed while clay content is normally distributed. These soil composition and compaction variables can be described by some alternative probability distribution types; however, only the ones stated above were used in this study.

Set-up of Numerical Experiments

The initial input parameters in the reliability analysis using FORM5 (Gollwitzer et al., 1988) are shown in Table 3. The numbers in brackets in column 3 of Table 3 are the various code numbers for the statistical distributions of the variables as used in FORM 5. The regulatory maximum hydraulic conductivity of 1×10–9 m/s for soil liners was set as the upper limit of hydraulic conductivity values and 1×10–12 m/s as the lower limit. The standard deviation of 2.7×10–10 m/s used here is based on values in literature (see Benson, 1993; Benson et al., 1994). Six values of target reliability index (bt) ranging from 0.5 to 3.0 were used. For each bt value selected, hydraulic conductivity values were estimated at various mean values of the independent variables, in turn, and for each index of the compactive efforts. The following procedure was utilized in the estimation of hydraulic conductivites: (1) Mean values, standard deviations and the probability distribution function for each variable were established (FORM 5 automatically calculates the coefficient of variation); a target reliability index was also selected from the range given above to obtain a set of hydraulic conductivities. (2) For each run of FORM 5, a mean value of a variable was selected and its coefficient of variation varied while the coefficients of variation of other variable were kept constant. When the range of coefficients of variation was exhausted, a new target reliability index was chosen until the full set of b values was completely utilized.

Table 2. Statistical characteristics of variables for liner soils (Benson and Trast, 1995)
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
40.767
17.00 
23.769
 2.784
 2.000
16.385
81.769
37.615
14.154
 0.651
NA
14.873
 5.672 
11.152
 0.053
 2.708
12.771
12.859
15.378
 9.898
 0.188
NA
 36.48 
 33.362
 46.916
  1.921
135.401
77.943
15.726
41.131
22.412
28.840
NA

The assumed mean values were selected as follows; initial degree of saturation (70% - 120%) plasticity index (5% - 70%) and clay content (0.5% - 80%). These ranges cover those for liner soil materials and lateritic soils in the database.

RESULTS AND DISCUSSION

Variations in Soil Composition and Compaction Variables

Initial Degree of Saturation

The variations in hydraulic conductivity as degree of saturation increases are shown in Fig.1 (a) for modified Proctor; Fig. 1 (b) for standard Proctor and Fig. 1 (c) for reduced Proctor efforts. At any given value of bt the hydraulic conductivity decreases as initial degree of saturation increases; lower values of hydraulic conductivity being obtained at higher compactive efforts for corresponding values of ßt and initial degree of saturation. For instance, at a ßt value of 1.5 and for values of initial degree of saturation ranging between 95% and 120%, hydraulic conductivity decreases from 1.47×10–10 m/s to 2.4×10–11 m/s for modified Proctor effort, from 3.67×10–10 m/s to 6.3×10–11 m/s for standard Proctor effort and from 9.11×10–10 m/s 1.154×10–10 m/s.


Figure 1a. Hydraulic conductivity versus degree of saturation (modified Proctor)

 


Figure 1b. Hydraulic conductivity versus degree of initial saturation (standard Proctor)

 


Figure 1c. Hydraulic conductivity versus degree of initial saturation (reduced Proctor)

 

In order to achieve acceptable hydraulic conductivities at high bt values higher compactive efforts or higher degrees of initial saturation or both are required. This is consistent with the findings of Mitchell et al (1965), Harrop-Williams (1985), Benson et al (1994, 1999) and Benson and Trast (1995).

Plasticity index

Soil of varying plasticity indices have often been used in the construction of clay liners and covers. The following ranges of values of plasticity index have been reported in literature: 2 - 62 (Benson et al 1994); 11 - 46 (Benson and Trast 1995); and 7 - 71 (Benson et al., 1999). In order to cover as much as possible these ranges, mean values of plasticity index were varied between 5% and 70%.

In Figs. 2 (a - c), the hydraulic conductivity of the lateritic soils is shown to decrease as plasticity index values increase for each compactive effort and at any given value of bt. Again, lower values of hydraulic conductivity are obtained at higher compactive effort for any given bt value. For instance, as plasticity index increased from 20 to 70% and at bt value of 1.5, the hydraulic conductivity decreased from 1.52×10–10 to 2.9×10–11 m/s for modified Proctor effort; from 3.78×10–10 to 7.2×10–11 m/s for standard Proctor effort and from 9.39×10–10 to 1.79×10–10 m/s for the reduced Proctor effort. Values of hydraulic conductivity less than 1.0×10–9 m/s could not be obtained for a bt value of 1.5 at plasticity index values less than 20%. (see Fig. 2c).


Figure 2a. Hydraulic conductivity versus plasticity index (modified Proctor)

 


Figure 2b. Hydraulic conductivity versus plasticity index (standard Proctor)

 


Figure 2c. Hydraulic conductivity versus plasticity index (reduced Proctor)

Several recommendations are made for minimum values of plasticity index for soil liners. These include: 7% (Benson et al., 1994); 20 - 25% (Benson et al., 1999), 10% (EPA, 1989; Das, 1998). The US Environmental Protection Agency recommends a maximum of 30% as higher values of plasticity index may give field construction problems (see Das, 1998). Considering the plasticity index range of 10 to 30%, it is possible to achieve low hydraulic conductivity values at reliability levels greater than 1.5 using the modified Proctor and standard Proctor compactive efforts but not the reduced Proctor effort.

Clay content

In Figs. 3 (a) - (c), the hydraulic conductivities are seen to increase with increasing percent clay content for each of the compactive efforts and bt values. At corresponding clay contents and bt values, higher compactive efforts resulted in lower hydraulic conductivities. The trend in the variation of hydraulic conductivity as clay content increases is expected as other variables were held constant when estimates were being made using FORM 5. Increasing clay content while maintaining the same plasticity index suggests that the clay fraction is composed of less active minerals, which generally corresponds to higher hydraulic conductivity (Benson and Trast, 1995). Kaolinites are less active as clay minerals and form the major group of clay minerals found in lateritic soils.


Figure 3a. Hydraulic conductivity versus clay content (modified Proctor)

 


Figure 3b. Hydraulic conductivity versus clay content (standard Proctor)

 


Figure 3c. Hydraulic conductivity versus clay content (reduced Proctor)

Benson et al (1994) suggest a minimum clay content of 15% in order to achieve hydraulic conductivities lower than 1×10–9 m/s in the field. Most soils used as compacted clay liners have clay contents in the range of 13.9% to 75.3%. However, at clay content of 70% and at bt value of 1.5, the energy of the modified Proctor compaction will be required to achieve hydraulic conductivity values less than 1×10-9 m/s. Acceptable hydraulic conductivities can only be achieved for lower bt values at higher clay contents (see Figs. 3 (a) - (c)).

Selection of Compactive Efforts and Material Properties

The apparent effects of the different compaction energies on the estimated hydraulic conductivity values, taking into consideration the various values of the soil composition and compaction variables, are clearly depicted in Figs. 4 (a) - (c). These trends are shown for a reliability index of 1.5. In structural reliability and for the serviceability limit state, reliability indices (b) ranging between 1.0 and 2.0 (see NKB Report No. 36, 1978) are considered to provide reasonable margins of safety for a given structure or structural component. The value of 1.0 may be taken as a convenient lower limit for the reliability assessment of compacted lateritic soil liners. The value of 1.5 for b, as an average value, is taken to illustrate a methodology for selecting pertinent variables in liner construction. This b value corresponds to a probability of failure of 6.68×10–2.




Figure 4. Hydraulic conductivity versus degree of saturation, plasticity index and clay content at reliability index of 1.5: (a) modified Proctor, (b) standard Proctor and (c) reduced Proctor

Suggested methodology is as follows:

Statistical Analysis

In this analysis, hydraulic conductivity was taken to be lognormally distributed. In a Kolmogorov-Smirnov (K-S) test of goodness-of-fit on 30 data points, Harrop-Williams (1985) showed that hydraulic conductivity had the following distributions in order of preference: beta, gamma, normal and lognormal. This suggests that the normal probability distribution function better describes hydraulic conductivity than the lognormal function.

Table 3. Input data for reliability analysis using FORM 5
Variables
(1)
Distribution type
(2)
Mean, E(x)
(3)
Standard deviation
S(x)
(4)
Coefficient of variation (%)
(5)
1. Maximum hydraulic conductivity, ko
2. Initial degree of saturation, Si
3. Plasticity index, PI
4. Clay content, C
5. Compactive effort index, E
Lognormal (=3)
Lognormal (=3)
Lognormal (=3)
Normal (=2)
Deterministic parameter
1x10–9m/s
92.9
16.9
23.5
(-1,0,+1)
2.x10–10m/s
12.8
8.2
15.4
-
27
13.8
48.53
65.53
-

 

Table 4. Hydraulic conductivity data used in statistical analysis (bt = 1.5)
Clay content (%) Hydraulic conductivity (m/s):
Normal distribution
Hydraulic conductivity (m/s):
Lognormal distribution
0.5
5.0
10.0
20.0
5.63×10–10
6.25×10–10
7.06×10–10
9.24×10–10
5.84×10–10
6.47×10–10
7.31×10–10
9.57×10–10
Hydraulic conductivity values are those obtained at reduced Proctor compactive effort.

In order to test whether the differences in hydraulic conductivity values between the normal distribution and lognormal distribution functions were statistically significant, two statistical tests, the “F – test” and the “t-test” for two samples assuming equal variances, were carried out. The hydraulic conductivity values used for these tests are those obtained for clay contents (0.5% to 20%) at a bt value of 1.5 and using the reduced compactive effort. The results are shown in Table 4; the differences between the two sets of hydraulic conductivity data are statistically not significant as the calculated F value of 1.224 is less than critical F value of 9.277 at 5% level of significance. The t-test yielded the following 95% confidence interval for the true mean difference (µ1 - µ2) in hydraulic conductivity between the two distributions:

As zero is contained in this interval, it is clear that, at 95% confidence, there is no significant difference in hydraulic conductivity between the two distributions.

Moreover, the calculated t-statistic is 0.1383. This is less than the critical bt value of 1.9432 for one-tailed test and less than the critical bt value of 2.4497 for a two-tailed test, signifying that the differences in hydraulic conductivities of the two distributions are not statistically significant.

REMARKS ON LATERITIC SOIL LINER DESIGN

The b value of 1.5 has been used to illustrate a methodology for reliability design of lateritic soil liners. As laboratory test results, especially for hydraulic conductivity, are more highly controlled than in field situations and gravel was not included in the model used, a higher safety margin may be specified. The values of 2.0 to 2.5 corresponding to failure probabilities of between 2.28×10–2 and 6.2×10–3 may be considered. If the suggested ranges of soil compositional and compaction variables are adopted for lateritic soils, high b values in the range of 2.0 to 2.5 will require that field compaction equipment with compactor weights equivalent to a minimum of the standard Proctor effort be used for compacting soil liners in the field.

The use of lower compactor weights will result in lower safety margins of 1.0 or less. However, even when high b values are targeted, careful control of construction procedures are required in order to ensure reliable compacted soil liners.

CONCLUSIONS

The first-order second-moment approach employing FORM 5 has been used to estimate hydraulic conductivity values corresponding to targeted reliability indices (bt) ranging between 0.5 and 3.0 for lateritic soils. The analysis was based on a model for predicting hydraulic conductivity developed from laboratory data. The results show that, for any given bt value hydraulic conductivities decrease as initial degree of saturation increases and as plasticity index increases while increases in clay content result in increased hydraulic conductivity. For any given value of initial degree of saturation, plasticity index and clay content, hydraulic conductivity increases as bt value increases. A methodology is suggested for selecting lateritic soils for use as liners based on some specified minimum and maximum values of soil compositional and compaction variables.

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