Flow and diffusion in clays

Applied Clay Science, 4 (1989) 179-192

179

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

Flow and D i f f u s i o n in Clays R. H A S E N P A T T

I,W. D E G E N 2 and G. K A H R I

ITonmineralogisches Labor des InstitutesfiirGrundbau und Bodenmechanik, ETH-Zentrum,

CH-8092 Zi~rich (Switzerland) 21nstitut fiir Grundbau und Bodenmechanik, ETH-HiJnggerberg, CH-8093 Ziirich (Switzerland) (Received November 28, 1988; accepted February 10, 1989)

ABSTRACT

Hasenpatt, R., Degen, W. and Kahr, G., 1989. Flow and diffusion in clays. Appl. Clay Sci., 4: 179192. Generally, two transport mechanisms can be distinguishedin a clay: (I) diffusion,for which the propellingforceisthe concentration gradientof the diffusingions;and (2) flow,for which the propellingforceisthe water pressure gradient.These two transportphenomena are compared for the stationaryand transientcases. The similaritiesand differencesbetween diffusionand flow are demonstrated in an example. The mathematical solutionsfor the differentboundary conditionsare explained. Finally,the changes in a clay caused by chemically contaminated water are demonstrated in an extreme example.

INTRODUCTION

Clays are earning increased attention for their use in the isolation of radioactive and chemical wastes. In the past, only little was known about the changes in a clay brought into contact with waste materials. The clays were used mainly because of their extremely low permeability and diffusion coefficients, as an isolation to prevent the toxic agents from entering the biosphere via the groundwater. The increased awareness about and attention to environmental protection has led to a more strictly enforced control of waste deposits. In those controls, contamination of the soil underlying several waste deposits with natural and artificial seals was observed. The question arose, as to which mechanisms are at work and in what time frame the toxic compounds migrate through a barrier that was believed to be impervious. The aim of this paper is to describe the transport mechanisms which act in a clay. Generally, two mechanisms can be distinguished: (1) diffusion, for which 0169-1317/89/$03.50

© 1989 Elsevier Science Publishers B.V.

180

the propelling force is the concentration gradient of the diffusing ions; (2~ flow, for which the propelling force is the water pressure gradient. These two transport phenomena will be compared for the stationary and transient cases. The similarities and differences between diffusion and flow are demonstrated in an example. The mathematical solutions for the different boundary conditions are explained. Finally, the changes in a clay caused by chemically contaminated water are demonstrated in an extreme example. FUNDAMENTALSOF DIFFUSION AND FLOW Fundamental papers on the diffusion of ions in clays have been written by Lai and Mortland (1960). They examined the diffusion of exchangeable cations in bentonites. They measured (in 1961 ) the diffusion velocity of a variety of ions and proved the validity of Fick's law for diffusion in clays. Gast and Mortland (1971) examined the diffusion of radioactive alkylammonium ions in bentonites saturated with alkylammonium. They found out that the diffusion velocity depends on the water content and that a difference in diffusion velocity can be measured only for higher water contents for alkylammonium ions of differing charges and lengths. Crooks and Quickley (1984) compared the diffusion of salt in disturbed laboratory specimens with the in-situ diffusion below a domestic waste deposit. They found out that the coefficient of diffusion D in the one-dimensional dispersion-convection equation gives a good approximation of the salt concentrations measured in the in-situ soil. By theoretical assumptions, Gray and Weber (1984) came to the conclusion that a diffusion-controlled breakthrough in a clay barrier can happen in a relatively short frame of time. The transport of the solvent is ruled by diffusion and can, for a sufficiently large concentration gradient, even be oriented against the hydraulic gradient. They confirmed their theoretical assumptions by showing that the pollution front propagated against the hydraulic gradient below several waste deposits. Jost (1972), Crank (1975), Crank et al. (1981), Kahr et al. (1985), and others have dealt with the mathematical solution and application of Fick's second law on diffusion, considering also the influence of adsorption and advection. The constituent equations for stationary flow were already developed by Darcy (1856), while the fundamentals for the calculation of the transient flow were developed by Terzaghi and FrShlich in 1936. COMPARISONBETWEENFLOWAND DIFFUSIONUNDERSTATIONARYCONDITIONS Stationary diffusion as well as stationary flow describe a particle transfer between two locations with different energy potentials without change of con-

181 centration or flow volume with time or location. In the following, "potential" refers to a specific energy level and not only, as usual in soil mechanics, to a specific pressure level. With an increasing degree of compaction of a clay layer, i.e. with a decreasing pore radius and volume, the influence of osmosis and diffusion as transportation mechanisms become larger, while the influence of flow decreases. The hydration of the counter-ions creates a film of chemically bound water on the clay surface. The smaller the pores get, the more interaction occurs between the free porewater and the hydrated layer on the surface. The free flow of water is thereby increasingly inhibited. The transition between flow and diffusion is continuous; the effects can be superposed or even act in opposite directions (Gray and Weber, 1984; Crooks and Quickley, 1984). Flow and diffusion are equalizing processes in which a particle transport through a soil, for example a clay layer, takes place under a pressure, temperature or concentration gradient. For stationary conditions the flow is described by Darcy's law, while diffusion is ruled by Fick's first law. The similarities and differences between the two laws for flow and diffusion are best shown in the formula units. According to Darcy a flow volume qh (m 3 s -~) is transported under a gradient ih (mm -1) over the distance h (m) through an area A (m 2) in a given time: (1)

qh = kh" ih'A ma m m m2 s s m _

with qh ---- flow volume (m 3 s - 1), k h = coeff, of permeability (m s - 1), ih = gradient (m m - l ) , and A = f l o w area (m2). In the diffusion process, an ion transport of concentration c (mol m -3) over a concentration gradient is (mol m -3 m - ~) takes place, transporting the quantity J s (mol s -1) in a given time across the area A (m2). The process is described using Fick's first law: Js =Ds "is .A mol s -

-

_

m 2 m o l ' m -3 s m

(2) . m

2

_

m 2 tool m2 s m4

with Js = flow volume (mol s - 1), Ds = coeff, of diffusion ( m 2 S - 1 ), is = gradient (tool m -3 m - l ) , a n d A = f l o w area (m2). kh and Ds are proportionality factors, which can be calculated by measuring qh, ih and A, or Js, is and A, respectively. The dimensions of kh and Ds cannot be compared directly, since the flow media (in the one case water, in the other

182

ions) are different, kh and D~ are determined for a clay under complete saturation for a specific bulk density or water content. COMPARISON BETWEEN FLOW AND DIFFUSION UNDER TRA N S I EN T CONDITIONS

In the case of transient flow or diffusion, the equations describing the transport are much more complicated than for the stationary case. The complexity of the problem results from the water pressure and ion concentration not being constant over time. Solving the problem leads to the well known second-order partial differential equation: dx d2x dt = const-~-~

(3)

where x, in the case of diffusion stands for the ion concentration c, and in the case of flow (consolidation), for the excess pore water pressure Au. Eq. 3 can be solved quite easily, either numerically or analytically, assuming a solving function (Gaussian distribution, error function). The main problem therefore is not solving eq. 3, but finding the right assumptions for the boundary conditions in time and space. Those boundary conditions shall be explained in the following, on two examples. For this purpose a problem in foundation engineering is compared with a diffusion laboratory test.

Foundation engineering problem A road is to be constructed on a clayey subsoil. The embankment for this purpose brings about an instant loading of Aa on the clay (see Fig. 1 ). If the Load

A U

/rn2)

Clay ~'

~0)

Iz~

Rock (undrained)

o) Geotechnica[ Profile

zO)

(m) b)

Excess porewoter pressure Au at various times

Fig. I. Foundation engineering problem.

183 clay can be assumed to be completely saturated and if the water is incompressible compared to the clay structure, an excess porewater pressure Aunt=o) = •a will built up within the soil at time t=O (see Fig. 1 ). Subsequently, this pressure will decrease according to the curves in Fig. 1. T h e decrease of pressure with time obeys eq. 3 in the following modified form: dAu d2Au dt =cv dz 2

(4)

where: k'ME Cv

--

Yw with Au = excess porewater pressure (kN m - 2), Cv= coefficient of consolidation (m 2 s - l ) , z = d e p t h from surface (m), ME=Young's modulus (kN m-2), k = coefficient of permeability (m s- 1), and ~w= bulk density of water (kN m-3). In the example presented, the boundary conditions are the following: (1) no water can drain away at the lower boundary of the clay layer; (2) free drainage is possible at the upper boundary; (3) at time t = 0, the load ~a is applied suddenly onto the clay and is not subsequently removed; and (4) the load is transferred on such a large area that z#u
Among other tests conducted in the clay mineralogy laboratory is the investigation of ion diffusion in clays. For this purpose, the clay is brought into the

184

testing cylinder in a dry and homogeneous state, clean from the later diffusing ions. The clay powder is then compressed within the cylindric testing mould up to the desired bulk density and then saturated with water. At time t = 0 the top part of the specimen is brought into contact with a solution with concentration cl. At t=O the concentration at the top of the specimen is therefore Ctop,t=o: c1, while Cbott. . . . t=o = 0 is still valid at the bottom of the specimen (see Fig. 2). With time, the ions diffuse to the bottom part of the sample, and the ion concentration increases according to the curves in Fig. 2. This increase of ion concentration with time again obeys eq. 3 in the following modified form: dc -D dt

d% dz 2

(5)

with D = coefficient of diffusion (m 2 s - 1), c = concentration of solvent (mol m - a ), and z = depth from top of specimen (m). The boundary conditions for this problem are comparable with those for the foundation engineering problem; they are listed below, in the same order, according to this analogy: (1) At the bottom of the specimen the start-concentration is Cbottom,t=o O. (2) At the top of the specimen the start-concentration is Ctop,t= 0 = C1. (3) At t = 0 the specimen is brought into contact with a solvent of concentration cl. This concentration remains constant throughout the whole test. (4) At time t = ~ , stationary conditions are reached, for which :

Ctop,t = ~

~--- C b o t t o m , t

= ~

~

C 1.

Here again boundary conditions ( 1 ) and (2) determine the direction of motion of the ions, while boundary conditions (3) and (4) determine the distribution of the potential under which the diffusion occurs. The identity of the form of eqs. 4 and 5, as well as the very similar boundary conditions ( 1 ) - ( 4 ), make it clear that the curves for the excess pore pressure and those for the ion concentration are completely analogous (see hatchings

C(t:O)

z (m) a)

Diffusion A p p a r a t u s

Fig. 2. Diffusion laboratory test.

b)

ConcentrQtion c at various times t

185

in Figs. I and 2 ). The principal difference between the two examples is that in the foundation engineering problem, the coefficient: C v --

k'ME Yw

is known from oedometer tests, and the left part of the equation (d~u/dt)is unknown, whereas for the diffusion laboratory test, the coefficient D is unknown, while the left part of the equation (dc/dt)is measured in the test. For this purpose, a series of diffusion tests is carried out under similar conditions. The tests are stopped each at a different time, and the distribution of the ions across the specimen is determined. Out of this, a concentration profile across the specimen for different times t emerges and in that way several values of c=f(t,z) from eq. 5 are found. The following deals with the most usual approaches to solving the problem of transient diffusion. MATHEMATICAL SOLUTIONS---DIFFUSION, ADSORPTION, ADVECTION

One possible analytical solution for eq. 5 is to assume that the concentration profile has a Gaussian distribution. To assume such a Gaussian distribution is in principle only allowed if the diffusion test is carried out with a small but well-defined amount of solvent Q, with a concentration of cl. Those conditions are not held in our example, since the solution was fed to the specimen in unlimited quantities. Nevertheless, the use of the Gaussian function may be tolerated, since for large times t, the discrepancy between the Gaussian function and the exact solution is small enough. "Small enough" means in this case smaller than the precision of the testing apparatus. The exact solution is obtained by assuming the error function for the distribution of the concentration profile; this is mathematically more difficult than the Gaussian function.

Approximation with the Gaussian distribution Assuming a Gaussian distribution, the following concentration profile can be calculated: C(t'z)

=

Q/A

. e - (z2/4Dt)

lnc = K - z2/4Dt log c = K - log e. z 2/4Dt log c = K - O . 4 3 4 3 . z 2 / 4 Dt

(6)

186 log c = K - 0.1086" z~/Dt where:

Q/A K-v/4~.Dt with c(t,z) = c o n c e n t r a t i o n at time t and location z (mol m - 3 ) , Q = a m o u n t of solution (mol), A = cross-area of specimen (m2), D = unknown coefficient of diffusion (m 2 s - ' ) , and z = distance from top-of-specimen to interesting depth

(m). If the concentration is plotted as log c against the distance squared (z2), then the slope of the straight line is proportional to the coefficient of diffusion D.

Approximation with errorfunction A more accurate solution for the test conditions described is found by using the error function. Ogata and Banks (1961) obtain the following solution:

C=Co"[ 1 - e r f ( ~ D ~

)

(7)

where erf( ) = e r r o r function. For the error function, the following approximation can be found after Abramowitz and Stegun (1965): erf(z) = 1 - (B'A1 +B2"A2 +B3"A3)'e( - z 2) where: B-

1 (1 + 0.47047z)

A1 = 0.3480242 A2 = - 0.0958798 A3 = 0.7478556 This approach has the advantage that the boundary conditions ( 1 ) - (4), in contrast to eq. 6, are exactly fulfilled. It is therefore advised to calculate the coefficient of diffusion via eq. 7 rather than by eq. 6.

Adsorption It should be mentioned first that for the phenomena described in the following, no analogy with flow can be made. One of the p h e n o m e n a that occur during diffusion is the adsorption of ions

187

at the surface of the clay particles. With the amount of ions adsorbed (S), and under the assumption that there is no significant diffusion in the adsorbed phase on the surface of the clay particles, and that, furthermore, adsorption and desorption are fast processes compared with diffusion, the following differential equation results: dc

dS

d2c

-~ +--d-[= D - ~ in which, under the assumption that the number of ions adsorbed {S) is proportional to the concentration of the solution (c), it follows:

S=Lc dc d(Lc) _d2c dt ~ dt = u~-~ dc _ dc dc d2c ~--~+ L~-~= (1 +L)~-~=D~-~ dc D dt-l+L

L>0

d2c dz 2

The factor 1 + L causes a reduction of the speed of diffusion. This retention factor is defined as R: dc

D d2c

d t - R dz 2 d2c = DaPp" dz 2

The coefficient of diffusion D, reduced by the factor R = I + L , is called "apparent" coefficient of diffusion, Dap p (Kahr et al., 1985). Some authors base their considerations about diffusion on a pore model, in which they use the constructivity 5 and tortuosity 32 as geometrical factors (Nagra, 1985). Under consideration of those geometry factors, they try to derive the effective coefficient of diffusion from the apparent coefficient of diffusion. It has to be considered in this context that until today even the apparent coefficient of diffusion could only be measured for the transient case and those results cannot be transferred easily on the transient case (Kahr et al., 1985; Nagra, 1985 ).

Advection If the transient diffusion is superposed with a simultaneous flow of velocity v, the phenomenon is called advection. Such a flow can be concurrent with or

188 opposite to the direction of diffusion (Gray and Weber, 1984). Advection is accounted for in the differential equation as follows: dc

d2c d2c = D~(z~,t)2 - E~

with:

E

D - (1 - vt/z) 2 = coefficient of advection.

v = velocity of pore fluid. CLAY AS A SEALING MATERIAL

Structure of clay minerals The structure of the common clay minerals can be divided into two basic structural units, a tetrahedral and an octahedral unit (Grim, 1968). In the tetrahedral unit, a silicon ion is tetrahedrally coordinated by four oxygens. These tetrahedra are linked in a hexagonal network by their three base vertices. The tetrahedral sheet structure is stabilized by an octahedral sheet structure. In this layer, divalent or trivalent ions such as aluminium, magnesium or iron are octahedrally coordinated with six oxygens or hydroxides. Cations with ionic radii which do not differ by more than 15% from the central ions, can substitute for the latter. The exchange of the four-valent silicon ion in the tetrahedral sheet or of the trivalent aluminium ion in the octahedral sheet causes a negative charge on the surface, wich is compensated by hydrated cations like calcium, magnesium, sodium or potassium. These cations are not integrated in the crystal structure of the unit layers, but rather accumulate on the surface of those sheet layers.

Example of the change of a pure bentonite by diffusion of cationic detergents The changes in the soil-mechanical properties of a pure clay after diffusion and intercalation of chemical compounds were examined experimentally. It has to be mentioned that this experiment was carried out in the laboratory under extreme conditions, in order to be on the safe side in evaluating the changes taking place in a clay barrier. With this example, the long-time dangers, which can occur by using clay as a sealing material will be demonstrated. In a diffusion apparatus, a calcium-bentonite specimen was exposed, from one side, to a 0.3 mol diammoniumdodecane solution, a waste product of the soap industry. The chemical formula for this compound is (NHa)2C12H24 and will be called C12 in the following. The sample was removed from the appa-

189

Fig. 3. Shrinkage cracks in a bentonite specimen after diffusion of diammoniumdodecane.

ratus after a time of diffusion of 16 days. In the water-saturated state, it showed the clearly visible cracks shown in Fig. 3. After diffusion, the specimen showed a water content of 59% and a bulk density of 7= 1.61 Mg m -3. The apparent coefficient of diffusion was determined for the boundary conditions mentioned above (p.184), from the concentration distribution of the diammoniumdodecane according to eq. 6. It was found to be D,pp = 5.3-10 -s m 2 s -1. In further tests, the stationary condition was simulated, a condition which would be reached after the ions have migrated through the barrier. In order to simulate those conditions, the Ca-bentonite was placed several times in a concentrated diammoniumdodecane solution and all electrolytes were washed away. After a complete transformation of the bentonite, which would correspond to a complete diffusion of the clay barrier, mineralogical and soil-mechanical investigations were carried out on the transformed bentonite and compared with the results for the untreated bentonite (see Table I). After ultrasonic treatment in a calgon solution, the grain-size distribution was determined in a pipette apparatus. The percentage of the clay fraction smaller than 0.002 mm decreased by 50% for the chemically treated bentonite. The basal distance, determined in the X-ray diffractometer, decreased to 1.3 nm and no subsequent increase of this basal distance could be achieved by addition of water. The hydraulic permeability, determined in oedometer tests, increased for the same bulk density and load increment from 10 -9 m s -1 in the untreated bentonite to 10 -6 m s-1 in the bentonite treated with diammoniumdodecane. According to the plasticity criteria (Atterberg limits) the bentonite showed a change from a highly active to an inactive material after chemical treatment.

Interpretation The increase in permeability as well as the change in soil-mechanical parameters can be explained as follows.

190 TABLE I

Mineralogical and soil-mechanical parameters of the pure bentonite and after transformation with diammoniumdodecane Grain-size distribution (in % )

Untreated C12

> 63

63-20

20- 2

< 2/zm

3.0 2.8

8.4 8.2

16.9 52.9

71.7 36.2

Basal distance d,,,,~ (in n m )

Untreated C12

humid

dry

1.99 1.32

1.49 1.30

Atterberg limits

Untreated C12

Untreated C12

Wr,(% )

WI.(% )

lr

]A

38.3 65.1

137.0 83.1

98.7 18.0

1.38 0.50

Bulk density

Permeability

~,~= 12.2 k N m -:~ }'~l= 12.5 k N m -:~

k=2.8"10-~m s- 1 k = 1.5.10 -6 m s -

In a water-saturated calcium bentonite, the counter-ions are surrounded by hydration water, and thereby enlarge the basal distance from 1.0 nm to 2.0 nm. The positively charged carbon hydroxyl compounds are intercalated in the interlayer space and expel the hydrated calcium ions. Besides the electrical interaction of the two ammonium groups with the negatively charged clay surface, there are Van der Waals forces acting between the carbon hydroxyl groups and the clay surface. The intercalation of the carbon hydroxyl groups causes the clay surface to become hydrophobic and the basal distance to shrink from 2.0 nm to 1.3 nm. As a result of this contraction, the pore volume increases and the cracks shown in Fig. 3 develop. Whether these cracks can also occur in real barriers depends, beside other factors, on the state of stress within the clay barrier. The carbon hydroxyl group is not only intercalated into the interlayer space but also on the outer surface of the minerals, causing an aggregation between neighbouring particles which is strong enough to resist separation by ultrasonic treatment {Table I). The cation exchange therefore changes the clay into

191 a silty material. This transformation from a clay into a silt is best shown in the increased permeability of the chemically treated material. EVALUATIONOF THE FUNCTIONALITY OF A CLAYBARRIER Under extreme conditions the soil-mechanical properties of clays and especially of clays with a high swelling potential can change when these are brought into contact with chemical compounds and they can lose those properties which made them appropriate for use as a barrier material in the first place. Besides the almost complete loss of swelling potential, as was shown in the example, also large increases in permeability may be expected after migration of the ions through the barrier. This can lead to a permeability value which no longer meets the retention requirements. Shrinkage cracks such as those in Fig. 3 can occur as a result of the irreversible contraction of clay layers and the loss of swelling potential. A pure bentonite was used for the specimen shown in Fig. 3. The tests were carried out with diluted solutions which were fed to the top of the specimen in ion concentrations which are common in toxic waste filtrates. The example demonstrates that clay should not be used as a barrier material without a deeper knowledge of its soil-mechanical properties and their possible changes. One possible concept is to design multi-layer clay barriers in which every layer selectively removes a specific part of the toxic filtrates. For the evaluation of a clay as a barrier material, the following clay-mineralogical and soil-mechanical parameters have to be determined: (a) type of clay minerals and their quantitative distribution; (b) cation distribution and cation exchange capacity; (c) permeability of the chemically untreated material and of the same material after transformation with a filtrate which is selected according to the expected filtrate in the waste deposit; (d) swelling and shrinking behaviour; (e)the apparent coefficient of diffusion for pure diffusion and for diffusion with dispersion; and (f) shear-parameters such as undrained shear-strength, cohesion and friction-angle. It is obvious that a closer cooperation between geotechnical engineers and clay mineralogists for the evaluation of a barrier material and the design of multibarrier concepts is not only useful but necessary.

REFERENCES Abramowitz,M. and Stegun, I.A., 1965. Handbookof Mathematical Functions. Dover, New York. Baer, J. and Verruijt, A., 1987. ModelingGroundwaterFlow and Pollution. IGWMC, Delft. Crank, J., 1975. The Mathematics of Diffusion. Clarendon Press, Oxford. Crank, J., McFarlane, N.R., Newby,J.C., Paterson, G.D. and Pedley,J.B., 1981. Diffusion Processes in Environmental Systems. The MacmillanPress, London.

] 92 Crooks, V.E. and Quickley, R.M., 1984. Saline leachate migration through clay: A comparative laboratory and field investigation. Can. Geotech. J., 21: 349-362. Darcy, H., 1856. Les fontaines publiques de la ville de Dijon. Dalmont, Paris. Gast, R.G. and Mortland, M.M., 1971. Self diffusion of alkylammonium ions in montmorillonite. J. Colloid Interface Sci., 37: 80-92. Gray, D.H. and Weber, W.J., 1984. Diffusional transport of hazardous waste teachate across clay barriers. 7th Annu. Madison Waste Conf. Municipal and Industrial Waste, pp. 11-12. Grim, R.E., 1968. Clay Mineralogy. McGraw Hill, New York, N.Y. Jost, W. and Hauffe, K., 1972. Diffusion - - Fortschritte der physikalischen Chemie. D. Steinkopf Verlag, Darmstadt. Kahr, G., Hasenpatt, R. and Mtiller-Vonmoos, M., 1985. Ionendiffusion in hochverdichtetem Bentonit. Nagra Tech. Ber. Baden, 85-23. Lagaly, G. and Weiss, A., 1969. Zur Van der Waals-Wechselwirkung in n-Dodecylammonium Schichtsilikaten. Z. Naturforsch., 24b, 8: 1057-1058. Lai, T.M. and Mortland, M.M., 1960. Self-diffusion of exchangeable cations in bentonite. 9th Natl. Conf. Clays and Clay Miner., pp. 229-247. Lai, T.M. and Mortland, M.M., 1961. Diffusion of ions in bentonite and vermiculite. Soil Sci. Soc. Am. Proc., 35: 353-357. Nagra, 1985. Projekt Gew~ihr 1985. Endlager ftir hochaktive Abf'~ille:Das System der Sicherheitsbarrieren. Projektbericht Baden, NGB, 85-04. Ogata, A. and Banks, R.B., 1961. A solution of the differential equation of longitudinal dispersion in porous media. USGS Prof. Pap., 411-A. Terzaghi, K. and FrShlich, O.K., 1936. Theorie der Setzungen yon Tonschichten. Deuticke, Leipzig und Wien.