CLAY LINER; PERMEABILITY: EVALUATION

' AND VARIATION

By Kingsley Harrop-Williams,1 M. ASCE

ABSTRACT: The primary criterion used in evaluating the suitability of hazard­ous waste landfills for containing hazardous wastes is permeability, and many regulatory agencies have adopted regulations requiring clay-lined hazardous waste landfills to have a coefficient of permeability no greater than a fixed value. However, the measurement of in-situ permeability of compacted clay is time-consuming and difficult. If used to monitor construction, it slows the construc­tion rate. Another equally important problem is that clay-liner permeability is extremely variable. Solutions to both of these problems are presented. Firstly, a relationship is developed between permeability and easily measured dry unit •Weight and moisture content. This would allow for the immediate monitoring .— of clay liners during construction. -Speerffy, an alternative is provided to t h e / J ) conventional approach in which permeability il"streated as a single-valued quan-( ly tity. A probabilistic description of the permeability of clay liners is developed from considerations of the heterogeneity of the soil. This would improve the design of clay liners by establishing confidence levels associated with possible ranges of the permeability. **)

INTRODUCTION

The containment of hazardous wastes is certainly one of the most ur­gent problems facing civil engineers today. Of the many liner materials presently in use to contain hazardous wastes, the local availability of suitable clay soils in many regions of the United States makes the use of soil liners an economically attractive design alternative. In addition, clay liners will always be used as back-up for synthetics. The assurance that the compacted liner soil satisfies an acceptance criterion is often related to its permeability. It is important then that reliable predictions of the permeability of compacted clay liners be made. Ideally, the field control of clay liner construction using permeability as an acceptance criterion should be accomplished by simply inserting a probe in the soil and directly measuring the coefficient of permeability. Unfortunately, this option is not presently available. Instead, four options might be con­sidered to test the suitability of a particular liner soil. These options in­clude:

1. Field permeability testing of field compacted soils. 2. Laboratory permeability testing of undisturbed samples of field

compacted soils. 3. Laboratory permeability testing of laboratory compacted soils. 4. Estimating permeability from more easily measured soil properties.

Of these, option 4 is particularly advantageous because it offers a basis for accepting the result of soil placement activities in the field. Options

xStaff Member, The BDM Corp., McLean, Va. 22102. Note.—Discussion open until March 1, 1986. To extend the closing date one

month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on September 4, 1984. This paper is part of the Journal of Geotechnical Engineering, Vol. I l l , No. 10, October, 1985. ©ASCE, ISSN 0733-9410/85/0010-1211/$01.00. Paper No. 20076.

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1 and 2, while providing a direct measure of compacted clay soil per­meability, slow the rate of construction. Option 3 introduces the risk that unsuitable soil layers may be buried and must be reprocessed later at an added cost. In this paper, a solution is proposed for option 4. A direct relationship is developed between permeability and the easily measured dry unit weight and moisture content of the clay liner.

As permeability controls the velocity of fluid flow through soil, and thus the volume of flow, it is the primary criterion used in evaluating the suitability of clay liners for containing hazardous wastes. This has led many regulatory agencies to adopt regulations requiring clay-lined hazardous waste landfills to have a coefficient of permeability no greater than some fixed value, typically 1 x 10~7 cm/s. The difficulty with es­tablishing specific limiting values for permeability has been outlined by Daniel (7). He observed approximately a dozen cases where the actual permeability of clay liners used to retain water were 10 to 1,000 greater than that designed for. This is not surprising, for permeability is gen­erally recognized as one of the most variable of engineering properties associated with construction materials. The range is over 10 billion times from gravel to clay and 1,000 times in clay alone (5).

Designs based on extremely variable properties are common in engi­neering. Many problems, including earthquake design, offshore design and design of wind loads, are similar. These problems have been ap­proached successfully using probability theory. In this paper, the prob­ability distribution describing the variation in clay liner permeability is identified.

A major problem with clay liners used for waste disposal is that the permeability may increase or decrease, depending on the nature of the chemicals it is expected to contain (4). Thus, the uncertainty associated with the permeability of the clay liners is increased by the uncertainty of the chemical nature of the waste and the uncertain effects of the chemicals on the clay. In this paper, the uncertainty in the permeability of compacted clays to water is investigated. It is expected that this will form a base from which the variation of permeability of the clays to chemical permeants can be extrapolated. The concept of using water per­meability as the base on which chemical permeants can be correlated is done conventionally in clay liner design (10).

PERMEABILITY-COMPACTION RELATIONSHIP

It is very difficult to monitor construction activities by directly mea­suring permeability in situ, or with in-situ samples tested in a labora­tory, due to the length of time required for testing. Therefore, for real time control of clay placement to achieve a desired permeability, other parameters must be measured and the permeability implied, The most frequently used technique for the control of earth placement is to mon­itor placement moisture content, w, and dry unit weight, yd, and cor­relate them to the desired property. This is typically done for strength and displacement characteristics, and to a lesser extent for permeability. The obvious advantage for such an approach is real-time monitoring. Direct monitoring can be performed as part of the compaction process. This provides all concerned parties (designer, builder, regulatory, and

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owner) with near immediate feedback (in comparison to permeability testing) regarding the acceptability of liner construction. If a criterion relating moisture content and dry unit weight to permeability is valid, then, assuming the clay type is constant throughout, all that is necessary is for the clay liner to be compacted to the design dry unit weight and moisture content. The relation of permeability to dry unit weight and moisture content of compacted clays was first investigated by Lambe (15). The general behavior of permeability with compaction is shown in Fig. 1. The permeability of compacted clays reaches a minimum value on the wet side of optimum moisture content, wopt, and at about max­imum dry unit weight, 7dmax. This suggests that for a given clay, per­meability can be minimized if the soil is compacted to the appropriate moisture content and dry unit weight.

Soil compaction is a process in which the air and, to a lesser degree, the water void volume of a soil mass is reduced. Numerous soil com­paction theories have been proposed (13) to explain the relationship be­tween dry unit weight and moisture content (Fig. 2). The simplest ex­planation involves lubrication of the soil particles by a water film evidenced by an increase in yd up to the optimum moisture content, followed by a marked decrease in yd with increasing w resulting from a replacement of soil particles by water.

In addition to the effect of moisture content on density, the amount of compactive effort can also influence the moisture density relationship. As shown in Fig. 2, as the compactive effort increases, 7dmax increases and wopt decreases. The two extreme curves in Fig. 2 represent the two most commonly used laboratory compaction test efforts and are referred to as Standard Compactive Effort (ASTM D 698-78, Ref. 1) and Modified Compactive Effort (ASTM D 1557-78, Ref. 1). These laboratory compac­tion tests are performed by dropping a weight a number of times per lift of soil (i.e., impact compaction).

Regardless of the compaction effort, as shown in Fig. 2, the soil never attains 100% saturation. This is due to its low permeability. Above the optimum moisture content additional moisture results in local shear fail­ure around the hammer. Thus, no matter how much water is added, the soil never becomes completely saturated (14). Hence, permeability of compacted clay may be analyzed from the viewpoint of flow through unsaturated porous media.

Mitchell, et al. (18), using a simple capillary model, showed that the permeability of an unsaturated capillary tube can be expressed as

k = fesatS3 (1)

where fesat is the saturated permeability of the tube and S is its degree of saturation. They noted, however, the limitations on using such a model to describe flow through fine-grained soil, and suggested that the ex­ponent may be different than 3. More intensive empirical work on flow through unsaturated media by Corey (6) showed that

-(If » where Pb is the bubbling pressure (air pressure needed to force air through

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l o " 6

, o " 6

, n ' 7

122

118

110

2

/ >

\ }

\ \

\

13 l i t

/

f f —«.

\ a U-

15 16

X N ^

1" is i

*-s= 100%

•H 126

•a -H £ 122 JJ -H £ 1 1 8

\

/

\ V

\

\

100% 90% 80%

2 13 m 15 16 17

Water Content (%)

11 13 15. 17 19

Water Content (%)

(a) (b)

FIG. 1.—Relationship of Permeability to Dry Density and Water Content (after Ref. 16): (a) Compaction-Permeability Tests on Jamaica Sandy Clay; (b) Compaction-Permeability Tests on Siburua Clay (Note: 1 Ib/cu ft = 0.157 kN/m3)

Flocculated Fabric Dispersed

Fabric

Moisture Content, w

FIG. 2.—Moisture-Density-Soil Fabric Relationships (after Ref. 16)

an initially water-saturated sample), Pc is the capillary pressure (differ­ence between air and water pressure in the soil), X is a pore size distri­bution index and K is a constant. The constant K is related to the sat­urated permeability. Another observation made by Corey (6) is that the

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degree of saturation is also related to Pb and Pc. This relation was found to be

"©' « where S, is the effective water saturation of the clay and defined as

S-Sr

'-'max ur (4)

Here Smax is the maximum possible saturation of the clay and Sr is its residual water saturation (moisture that remains in the sample after drainage). Combining Eqs. 2, 3 and 4 leads to

k = K'(S- Sr)a (5)

where a = (2 + 3X)/\, and K' = K(Smax - Sr)a is a constant.

Eq. 5 is seen to be of a nature similar to Mitchell's equation Eq. 1. The values Sr, K' and a are unique properties of a given clay. The degree of saturation of a compacted clay can also be found in terms of the dry unit weight yd and the moisture content w from the relationship (17)

G7a> , , ,

^ = —^G ( 6 )

1 + — S

where G is the specific gravity of the solids and yw is the unit weight of water. Solving for S from Eq. 6 and substituting into Eq. 5, the expres­sion for the permeability of compacted clay in terms of its compaction variables is obtained as

/ ydwG \ k = K'(-2 Sr) (7)

PREDICTION OF PERMEABILITY FROM COMPACTION VARIABLES

The relationship developed in Eq. 7 requires the determination of the residual saturation Sr for the soil used. The constants K and a are also material properties; however, their physical determination is not nec­essary, as they can be obtained as regression constants if some perme­ability measurements are made on the compacted clay. In fact, Eq. 7 allows the logarithm of the permeability to be written as the linear regression equation

In k = A + ax (8)

where A = In K' (9)

and x = In ( - ^ S'r) (10) \Gyw -yd I

The effectiveness of Eq. 7 for predicting the permeability of a com-

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pacted clay was tested using the data in Fig. 1(b). For simplicity, the residual saturation was taken as 0%. Using yw = 62A pcf and G = 2.7 (a recommended value for clay), a regression was performed and the resulting equation is

lnk= -19.873-5.19In (11)

with a coefficient of correlation of 99.04%. For comparison, the experi­mental points and the proposed relationship Eq. 7 are plotted in Fig. 3.

The excellent prediction made by Eq. 7, considering the assumption Sr = 0, and using the degrees of saturation that existed before perme­ability testing indicates that, for clay compacted in the range of optimum moisture content, the change in saturation during the permeability test does not significantly affect the relationship. In fact any change may be reflected in the parameters K' and a.

Eq. 7 was further fitted to the permeability results at different levels of compaction to investigate its validity at higher levels of compaction. This was done using the data presented by Mitchell, et al. (18) and is

11 13 15' 17 19

Moisture Content w (%)

FIG. 3.—Fit of Suggested Curve (Data from Ftef. 16) (1 Ib/cu ft = 0.187 kN/m3)

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shown in Fig. 4. This figure also shows excellent agreement. The soil used by Mitchell was a silty clay. Indicating that the compactive effort and clay type are reflected in the empirical constants K' and a. The smaller coefficient of correlation (R) in these curves is most likely due to insuf­ficient data points.

Although impact compaction tests are routinely used for predicting the compaction behavior of soils in the field, an often overlooked con­sideration is their lack of modeling similitude in matching the conditions likely to occur in the field. As soils become more fine-grained (e.g., fine silt and clay soils), different methods of compaction produce different structures of fabric during compaction that can have a significant effect on the engineering behavior of these soils. The field method of roller compaction is assumed to be more easily modeled by kneading com­paction (18). If kneading compaction is used, Eq. 7 is evaluated using the data presented by Mitchell, et al. (18). The results are presented in Fig. 5 with those of static compaction. Again the fit is excellent and the method of compaction reflected in the constants K and a. t

The relationship of permeability and compaction variables derived in n|. Eq. 8 reflects the soil type, method and level of compaction in the regres- I sion constants A and a. If m identical specimens are subjected to the »• same conditions and each generates an equation very similar to Eq. 8, | then the predicted permeability should be a weighted average of the "li regression equations describing the specimens. As greater confidence is |!|

Ill »ll

in coefficient I I " error of estimate

tsion constants .

li ! :

f; II; !!•• ' I l l

[; J .,. II

Moisture content (%)

FIG. 4.—Effect of Compaction Effort (Data from Ref. 18) (1 Ib/cu ft = 0.157 k N / m3)

5 2

10

,„"6

5

2

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2

,n"8

5

116

108

10"t

100

92

R = S e =

A -a =

R -

4 =" a ="

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1 1 3.901 1.851

-19. i -28.8

\ et>

~A 0.961 0,88;

20.11 18.5?

0L,2£ Olb

3

01 ¥

1

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e

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= 0 . !

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= - 2 ! = -91

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ndarc

regr t

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86

.150

.128

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N X s &

_3 2

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106

Id

102

100

R -S„ „

A , a s

Knead

R = 0

S P -

A = " a = "

® Kne @Kne

A S t a

C o r r e

S t a n d .

Regr>

i n g -

. 9 8 5 0.281*

2 1 . 4 1 1 3 6 . 1 1 -

©

i d i n g i d i n g

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a t i o i

r d e i

s s i o i

_ " y «

l " x 2 . 3 . 5 " x

" x 2 . 8

c o e f : i c i e i t

r o r o : e s t : m a t e

c o n s

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°^V

B " 4 H

I."I"*I

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a n t s

^

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S t a t l R = S =

e

A - -a = -

• s =

3 . 9 8 4 1.302

t . 3 3 t

100%

K 13 15 17 19 21 23 25 27

Moisture content (%)

FIG. 5.—Effect of Method of Compaction (Data from Ref. 18) (1 Ib/cu ft = 0.157 kN/m3)

placed on the reliability of a regression curve with a smaller standard error, the calculated weight of each specimen can be taken as the inverse of its standard error.

Letting y = In k, the standard error of estimate of Eq. 8 is

1 " l V 2

Se = ln-2i

X (y* - ax< - Af (12)

where y, and xt are the observed values of y and x. The equations for the / th specimens becomes

yi n . . -> ei ya

ly»J

> = \ + \ (13)

where w; is the number of observed points and e„; is the error at the rtj th point. In a more condensed form this can be written as

M/ = WP}, + H • • (14) 1218

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The weight applied to the /th specimen is the diagonal matrix

1

si

[WJi = (15)

1

where Sej is the standard error of estimate for the ; th specimen. For m specimens the weighted average coefficients can be solved from

Eq. 14 to be

(16) {p} = (xTwxy1 (XTWY)

m

where (XTWX) = £ [X]j [W], [X]y

7 = 1

n

and (XTWY) = ^[X]J[W]j[Y]j i=i

Thus the general relationship for the logarithm of the permeability be­comes

y = A + ax (17)

where A and a are the elements of {p}, and are weighted averages of the regression constants A and a obtained for the specimens.

VARIATION IN CLAY LINER PERMEABILITY

The permeability of a soil varies from point to point within the soil mass. Thus, if a laboratory permeability test is made on a soil sample, it represents only a point estimate of the true permeability of the soil mass. However, if a sufficient number of samples are taken, one can easily plot a histogram of the results that would represent the true ran­dom behavior of the permeability, from which important statistical con­clusions can be drawn. It is generally believed that, like most soils prop­erties, there exists a probability distribution that governs the random behavior of the permeability of soils (11,22).

The permeability of clay soils to chemical wastes depends on many factors, including the availability of pore spaces within the soil, the sur­face charge density of the soil and the chemical nature of the waste. The latter two, however, are chemical rather than physical in nature. It is the intention of this task to isolate the physical factors from the chemical factors and thereby develop a model for the variation of the permeability of the soil due to physical attributes alone. This model would be more applicable if the permeant was a neutral agent (e.g., water).

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A randomly chosen point in a soil is either a void or a solid with the probability of it being a void equal to its porosity (n) (11). It follows that the number of pore areas (m) in a cross subsection of soil capable of having N pore areas is binomially distributed. Its distribution can be stated as

Pm(m) = ( N j n'" (1 - n)N~'"; m = 0 ,1 , . . . N (18)

Further, if mk is the number of pore areas required to have a coefficient of permeability of value k, then because N is large, mk can be shown to have a Poisson distribution with parameter vk = Nn (11), or

(vk)mte-vk

Kk(mk) = K ' ; m = 0 ,1 , 2 (19) mk\

If m„ denotes the actual number of pore areas in a unit cross subsection, then the probability that the actual permeability of the sample, ka, is greater than any value k is equal to the probability that mk is less than ma; or

P[ka >k]= P[mk <ma] (20)

Using this equation in combination with Eq. 13 it can be shown (12) that the probability density function of permeability becomes

UK) = r f^T k'r1 e"*, 0 < ka < °o (21)

where T( ) is the gamma function. The mean value and variance of the permeability, evaluated from Eq.

21 are

tn„ k = — (22)

v

and var (k) = -$ (23) v

respectively. Equivalently, the parameters ma and v can be written in terms of the mean value and variance of k as

k2

ma = — (24) var (k)

k v = (25)

var(fc) V '

Eqs. 24 and 25 indicate that although the probability density function of permeability Eq. 21 was derived from a microscopic point of view, its parameters ma and v can be determined from macroscopically measured values of the mean and variance.

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On examination, one observes that 1/v is the permeability of one pore area, and can be written as (3):

1 7 - = C„-J& (26) V [ l

where 7 is the unit weight of the permeant (water), ]x is the viscosity of the permeant, RH is the hydraulic radius of the pore opening, and Cs is a shape factor reflecting the shape of the pore opening.

The second parameter, ma, can also be written as

total pore area nirRr

area of a pore ITRH

where RT is the total hydraulic radius of the soil sample and n is the porosity. This is equal to (23):

RT = CHDave (28)

where e is the void ratio, Dav is the average particle diameter, and CH is the shape factor.

The probability density function derived above is the gamma distri­bution. In general it is skewed similar to the lognormal distribution, the latter having been previously recommended to model soil permeability (22).

Goodness of fit of the proposed distribution for compacted clay per­meability was evaluated using 30 data points obtained from laboratory tests taken from a field compacted liner. This data is shown in Table 1. A Komologorov-Smirnof test of goodness of fit ranked this data as Beta, gamma, normal and lognormal in order of preference as to its distri­bution. This illustrates that although the gamma is a better fit than the normal or lognormal, it is weaker than the beta for this data set. The reason for this is the maximum value observed for the permeability. The­oretically, the maximum value possible for permeability is infinite, as for breakthrough conditions, a possibility modeled by the gamma distri­bution. However, if a maximum permeability value k,„ is specified, the conditional distribution based on this maximum can be determined.

TABLE 1.—Permeability Values from Compacted Liner (in Centimeters per Sec­ond)

Test (1) 1 2 3 4 5 6 7 8 9 10

k x lO"8

(2) 1.7

48.0 5.9

26.0 9.9 3.9 1.4 1.1 4.9 9.8

Test (3)

11 12 13 14 15 16 17 18 19 20

k x 10"8

(4)

1.1 6.7 19.0 19.0 1.4 0.59 7.3

24.0 12.0 0.36

Test (5) 21 22 23 24 25 26 27 28 29 30

k X 10"8

(6)

16.0 10.0 0.62 1.1 0.37 2.1 33.0 24.0 2.6 1.3

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The number of pore areas mk per unit cross-sectional areas that are required for soil to have a permeability k was shown to be Poisson-dis-tributed with parameter v (Eq. 21). Hence the increase in permeability caused by an additional pore area is exponentially distributed. This is due to the fact that the sum of independently exponentially distributed random variables is gamma distributed (21). The distribution of this in­crease caused by an additional pore area is

/at(AJfc) = ve~M 0 < Afc < oo . . . . ; (29)

where Afc is the increase in permeability. The value ks for ms pore areas is the sum of ms such increases. Letting Xa = vks, the probability dis­tribution of Xi is

fxAXi) = 7^—^r1 e~Xl 0 < ^ < » (30) T(ms)

If the maximum permeability value km of a soil is known it would imply a maximum number bf pore areas mm . This means that the difference k,„ - ks is also a sum of these independently exponentially distributed increases in permeability. Letting X2

= v{km — ks) and adding the in­creases between these mm - ms + 1 particles the probability distribution of X2 is found to be (2)

k(X2)=~ l — ~ ^ e ~ * 0<X 2 <=o (31) T(mm -ms + l)

The variables x1 and x2 are independent, thus their joint probability dis­tribution becomes

W * l . *2) = ~ ^ - \ X"_1 Xr' e'(X1+X2) W r(v!)r(v2)

here vj = ms and v2 = mm — ms + 1. Eq. 32 can be integrated to obtain the marginal distribution of y = (x-^Xi

+ x2) = ks\k,„, which is (24): r(vi + Vo)

f^ = rTTTT^ Y £ ~ Y) ' 0 s Y a l (33) r > i ) i > 2 )

This indicates that Y = kjkm is beta-distributed with parameters vx and v2. If the minimum value (kL) of the permeability is also specified, the distribution of k becomes (from Eq. 33) the general Beta distribution:

T{v, + v2)( 1 \",+V2-1

/*(*.) = F K T T ^ T—r (k. - hr~i (km - ksr~\ r(v!)r(v2) \km-kj

kL^kss k,„ (34) The data presented in Table 1 has specified minimum and maximum

values. Hence, it confirms to the beta distribution.

CONCLUSION

Mitchell and Younger (19) identified one of the reasons for experi­mentally induced uncertainty in laboratory-determined permeability val-

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ues of fine-grained soils as the unnaturally large hydraulic gradients ap­plied to these soils. These large gradients are needed to decrease the time in obtaining results, but, as a result, they may violate Darcy's law in local areas and cause particle migration in the sample. Olson and Daniel (20) later identified several factors that may also cause this uncertainty. Among them are smear zones formed during trimming, voids formed in sample preparation, wrong temperature, air in samples, and others. Later work by Daniel (7) suggested that other very important errors are selecting samples without a representative distribution of dessication cracks, fissures and slickensides, and for laboratory compacted samples wrongly modeling the compacted effort applied in the field. It is for these reasons that field permeability determination is preferred.

Time constraints imposed by in-situ permeability tests makes it highly impractical for clay liner construction monitoring. Spatial variability of permeability would also require many such tests to be made to ade­quately assess the overall performance of the liner. The determination of compaction characteristics, such as dry density and water content, however, is made immediately and repeatedly by such in-situ devices as the nuclear densometer. An equation like that developed in Eq. 7 allows in-situ permeabilities to be assessed immediately throughout the liner. At any point in the liner, abnormalities such as dessication cracks, fissures and slickensides are portrayed in yd and w and hence reflected in k. The procedure recommended first requires the determination of the regression constants A and a from a few laboratory permeability tests on field-compacted samples, as these parameters are functions of the soil type, compaction effort and type of compaction. If more than one similar relationships are developed, Eq. 16 provides a weighted average of the constants A and a.

Noticed from experimental studies is that the minimum permeability occurs a little above optimum water content. Hence clay liners should be designed for this minimum. It is in this region that Eq. 7 is most applicable, as permeability testing of clays above optimum water content does not saturate the sample further. Results of Mitchell, et al. (18) have shown that on the very dry side of optimum, further saturation by per­meability tests occurs.

Further improvement of the prediction of permeability from yd and w requires the evaluation of the maximum possible saturation and the re­sidual saturation. Although it is possible to remove practically all the wetting fluid by evaporation, Corey (6) suggests that the residual sat­uration be taken as the saturation at some arbitrarily large value of the capillary pressure, where the capillary pressure is defined as the differ­ence between the air and water pressures in the soil. Also, to avoid the regression building of Eq. 7, the values of parameters K' and a should be experimentally determined for specific soil types, level and type of compaction. This would involve determination of the maximum possible saturation Smax, the bubbling pressure Pb, capillary pressure Pc and grain size distribution index A. explained in Eq. 2. The maximum saturation may be inferred from the decreasing portion of the compaction curve. Investigations on the pore size distribution on permeability has already been suggested by Garcia-Bengochea, et al. (8). For more information on Pb and Pc, one is referred to Corey (6).

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In this paper, the permeability distribution of clay liner permeability is also established as the gamma distribution. This was shown to reduce to the beta distribution if maximum and minimum values of k are spec­ified. However, it is unlikely that the true extremes are ever observed, as undetected fissures and clay clods are invariably present (7,9). For this reason, the gamma distribution is a better model.

Conventional design procedures for clay-lined hazardous waste land­fills generally require that their coefficients of permeability not exceed some fixed value. This is usually a difficult condition to fulfill; a random sampling of the permeability of the constructed clay liner often produces values both above and below this value. By modeling the randomness associated with the coefficient of permeability, an alternative to the con­ventional design approach in which clay-liner permeability is treated as a single-valued quantity can be provided. An an example, consider the case where a number of samples taken from the completed clay liner produced a mean value for the coefficient of permeability of k = 9.838 x 10"8 cm/s and a standard deviation of 11.641 x 1CT8 cm/s. If the de­sign permeability is specified as 2.4 x 10~7 cm/s, then the probability of reaching this requirement can be obtained by integrating Eq. 21, and is found to be 0.9. Similarly, the permeability value, k95, at which one is 95% certain that any randomly selected point in the liner will have a lower coefficient of permeability, can be found from the following equa­tion:

J"*93

fk(K)dks = 0.95 (35) o

where fk(ks) is given by Eq. 21. The solution to Eq. 35 yields k95 = 3.3 X 10~7 cm/s.

As is done for strength tests of steel and concrete, this approach would allow for the development of guidelines as to the percentage of per­meability values randomly sampled from the completed clay liner that must fall within a certain permeability range to provide a given confi­dence in the required design permeability.

In conclusion, a method is presented to predict the immediate and repeated determination of clay liner permeability based on the easily de­termined in-situ dry density and moisture content. Further, the proba­bility distribution of this permeability is developed and is justified when compared with actual, though limited, data. The variation in perme­ability requires that regulatory agencies specify confidence limits that the permeability must satisfy, rather than absolute limits. These can more easily be reached by clay liner designers who now are able to repair their liners during construction as they observe the field dry density and moisture contents.

APPENDIX.—REFERENCES

1. Annual Book of ASTM Standards—Part 19; Soil and Rock; Building Stone, Amer­ican Society for Testing and Materials, Philadelphia, Pa., 1982, pp. 202-208, 228-284.

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2. Benjamin, J. R., and Cornell, C. A., Probability Statistics and Decisions for Civil Engineers, McGraw-Hill, Inc., New York, N.Y., 1970.

3. Bowles, J. E., Physical and Geotechnical Properties of Soils, McGraw-Hill, Inc., New York, N.Y., 1979.

4. Brown, K. W., and Anderson, D., "Effects of Organic Chemicals on Clay Liner Permeability—A Review of the Literature," Disposal of Hazardous Waste, D. Shultz, Ed., 6th Annual Research Symposium, U.S. EPA, EPA-60019-80-101, Cincinnati, Ohio, 1980, pp. 123-124.

5. Cedergren, H. R., Seepage, Drainage and Flow Nets, J. Wiley & Sons, Inc., New York, N.Y., 1965, p. 23.

6. Corey, A. T., "Mechanics of Heterogeneous Fluids in Porous Media," Water Resources Publications, P.O. Box 303, Ft. Collins, Colo., 1977.

7. Daniel, D. E., "Predicting Hydraulic Conductivity of Clay Liners," Journal of Geotechnical Engineering, ASCE, Vol. 110, No. GT2, 1984, pp. 288-300.

8. Garcia-Bengochea, I., Lovell, C. W., and Altschaeffle, A. G., "Pore Distri­bution and Permeability of Silty Clays," Journal of the Geotechnical Engineering Division, ASCE, Vol. 105, No. GT7, 1979, pp. 839-856.

9. Gau, F. L., and Olson, R. E., "Uniformity of Specimens of a Compacted Clay," Journal of Materials, ASTM, Vol. 6, No. 4, 1971, pp. 874-888.

10. Green, W. T., Lee, G. F., and Jones, R. A., "Impact of Organic Solvents on the Integrity of Clay Liners for Industrial Waste Disposal Pits: Implications for Groundwater Contamination," Final Report to U.S. EPA, Robert S. Kerr Environmental Research Laboratory, Ada, Okla., June, 1979.

11. Harr, M. E., Mechanics of Particulate Media—A Probabilistic Approach, McGraw-Hill, Inc., New York, N.Y., 1977.

12. Harrop-Williams, K., "Clay Liner Permeability—A Probabilistic Approach," Hazardous Waste Management of the 80's, T. L. Sweeney, et al., Eds., Ann Ar­bor Science, 1982, pp. 307-313.

13. Hilf, J. W., "Compacted Fill," Foundation Engineering Handbook, H. F. Win-terkorn and H. Y. Fang, Eds., Van Nostrand Reinhold Co., New York, N.Y., 1975, pp. 244-256.

14. Holtz, R. D., and Kovacs, W. D., An Introduction to Geotechnical Engineering, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1981.

15. Lambe, T. W., "The Permeability of Fine-Grained Soils," Symposium on Per­meability of Soils, ASTM STP 163, 1954, pp. 56-57.

16. Lambe, T. W., "The Engineering Behavior of Compacted Clay," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 84, No. SM2, Part 1, 1958, pp. 1655-1 to 35.

17. Lambe, T. W., and Whitman, R. V., Soil Mechanics, J. Wiley & Sons, Inc., New York, N.Y., 1969.

18. Mitchell, J. K., Hooper, D. R., and Campanella, R. G., "Permeability of Com­pacted Clay," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 91, No. SM4, 1965, pp. 41-65.

19. Mitchell, J. K., and Younger, T. S., "Abnormalities in Hydraulic Flow Through Fine Grained Soils," Permeability and Capillary of Soils, ASTM STP H7, Amer­ican Society for Testing and Materials, 1967, pp. 106-139.

20. Olson, R. E., and Daniel, D. E., "Measurement of the Hydraulic Conductiv­ity of Fine-Grained Soils," ASTM STP 746, 1981, pp. 18-64.

21. Parzen, E., "Stochastic Processes," Holden-Day, San Francisco, Calif., 1962. 22. Rogowski, A. S., "Watershed Physics: Soil Variability Criteria," Water Re­

sources Research, Vol. 8, No. 4, 1972, pp. 1015-1023. 23. Taylor, D. W., "Fundamentals of Soil Mechanics," J. Wiley & Sons, Inc.,

New York, N.Y., 1948. 24. Wilks, S. S., Mathematical Statistics, Wiley, New York, N.Y., 1962.

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