Journal of Colloid and Interface Science240,498–508 (2001)doi:10.1006/jcis.2001.7697, available online at http://www.idealibrary.com on

Transport Properties of Compact Clays

I. Conductivity and Permeability

N. Mammar,∗ M. Rosanne,∗ B. Prunet-Foch,† J.-F. Thovert,‡ E. Tevissen,§ and P. M. Adler∗∗IPGP, tour 24, 4, Place Jussieu, 75252 Paris Cedex 05, France;†Laboratoire de Physique des Materiaux Divises et des Interfaces,

5 boulevard Descartes, F-77454 Marne la Vallee Cedex 2, France;‡PTM/LCD, BP30179, F-86962 Futuroscope, France;and§ANDRA Direction Scientifique, 1-7 rue Jean Monet, Chatenay-Malabry Cedex, France

Received November 2, 2000; accepted May 4, 2001; published online July 12, 2001

Conductivity and permeability of model and natural clays havebeen studied experimentally. Local properties such as porosity andzeta potentials were measured as functions of the electrolyte so-lutions. Whenever possible, experimental data were compared tonumerical data obtained for random packings of grains of ar-bitrary shape, and a good agreement was found between them.C© 2001 Academic Press

Key Words: compact clays; conductivity; permeability; zeta po-tential.

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1. INTRODUCTION

Low permeability materials containing clay play an imptant role in practical life and in the natural environment. Thato their porous character, the transport of water through solows vegetal development and life. Reservoir rocks which ctain oil are often confined by impermeable clay layers. In mcases, clay formations present confinement properties, sulow porosities, low permeabilities, and consequently low flrates; these properties are highly desirable in storing nucwastes (1–5).

Study of transport phenomena in porous media is made dcult by the complex structures of real materials. Many numerand experimental investigations have been done about maiometrical and transport properties (6–8).

However, in contrast with rocks such as sandstones and lstones, transport properties of clays have not been often stu(9, 10). Hence, a systematic analysis of these properties isof academic and industrial interest.

The first part of this series is devoted to a general presentaof the materials which have been used and to the experimstudy of their basic properties, namely, permeability and cductivity.

The second section is devoted to materials which are sied, namely mica, montmorillonite, and natural clay; they wcharacterized by various techniques including Scattering Etron Microscopy (SEM) and zeta potentials. The powders walso compacted under various pressures and the resulting pties were systematically measured. The cells are describe

490021-9797/01 $35.00Copyright C© 2001 by Academic PressAll rights of reproduction in any form reserved.

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well as the preparation of the samples. Finally, conductivitypermeability measurements are described.

Results of systematic measurements of conductivity andmeability are gathered in Section 3. Data relative to powdunder compaction pressures are presented and compared tobtained with the original compact clay. Particular attentiondevoted to permeability which is measured either with airwater. These data are then compared to numerical resultrived by Coelhoet al. (11) on packings of particles.

Some concluding remarks end this first Part.

2. MATERIALS AND MEASUREMENTS

2.1. Materials

Three types of materials were used for the experiments.powders with different properties were employed as modelterials, namely muscovite mica and sodic montmorillonite. Tthird material is a compact clay which has been extracted ineast part of France in a Callovo–Oxfordian formation.

The muscovite mica (Comptoir de Min´eraux et Matierespremieres) is essentially composed of SiO2 (48%) and Al2O3

(34%). This powder was analyzed by SEM; the pictures (Figshow a lamellar structure with grains whose diameters are a4µm; these grains tend to aggregate into larger clusters.

The specific surfaceSsp was measured by nitrogen adsorptiand it was found to be

Ssp= 2.5× 107 m−1. [1]

A specific lengthlc can be defined as the inverse ofSsp

lc = 4× 10−8 m. [2]

The density was measured by standard weight and volmeasurements. It was deduced to be

ρs = 3150± 150 kg/m3. [3]

The second material is a sodic montmorillonite, i.e., a btonite (Oeno, France) whose ionic exchange capacity is eto 80 meq/100 g; it may double its volume when immersed

8

TRANSPORT PROPERTIES OF COMPACT CLAYS, I 499

FIG. 1. Pictures of muscovite mica powder obtained by SEM. The horizontal bars provide the scale.

FIG. 2. Pictures of sodic montmorillonite powder obtained by SEM: (a) dry powder; (b) wet powder after swelling during 24 h in water. The horizontal barsprovide the scale.

500 MAMMAR ET AL.

c tained by

Mths

u

T

mt aof

eenor

FIG. 3. Compact site clay: (a) the original block and two clay samplesSEM.

water, and it can form a gel. This was also analyzed by SEFigure 2 shows pictures after 24 h of immersion in water;dimensions range between grains of 2µm diameter and clusterof 10µm diameter.

The specific surface was measured by the same techniqbefore,

Ssp= 1.5× 109 m−1. [4]

Hence, it is equal to 60 times the specific surface of mica.corresponding characteristic length is

lc = 0.67× 10−9 m. [5]

ut out of it; some powder is also shown; (b) pictures of the clay powder ob

;e

e as

he

The density of montmorillonite is equal to

ρs = 2450± 180 kg/m3. [6]

A block of argilite has been supplied to us by ANDRA frothe drilling referred to as EST 104; this cylinder extracted adepth of 483.6 m with a diameter of 100 mm and a length82 mm (Fig. 3a) is labeled as EST 02364. This block has bused in different ways in order to obtain either clay powderssolid cylinders.

In order to start in the simplest possible way, broken parts ofthe original block were crushed in an agate mortar. The resulting

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TRANSPORT PROPERT

particles were selected with successive sieves whose smdiameter is 40µm. It is important to note that the remaininpowder which could not be filtered by the sieves, was crusagain and again until the whole powder could pass throughsmallest diameter. Pictures of the resulting powder were tawith SEM and are displayed in Fig. 3b. The structure is quheterogeneous and many aggregates are found; becausecareful crushing process, the average grain radius ranges1 to 10µm.

The densityρs of the clay powder was measured by the satechnique as before and was deduced to be

ρs = 2660± 150 kg/m3. [7]

No analysis of the specific surface was performed on this c

2.2. Zeta Potentials

Let us start with the definition of the double layer thickneκ−1 (15)

κ−1 =[(

e2

εelkT

)∑i

c0i z2

i

]−1/2

, [8]

wherezi is the valency of ioni , e is the charge of the electronk is the Boltzmann constant,T is the absolute temperature,εel

is the dielectric constant of the fluid, andc0i is the volumetric

concentration of the ioni out of the double layer.For a monovalent salt such as NaCl, whenc is given in mol/l,

κ−1 (in nm) can be expressed as

κ−1 = 3.0410−10c−1/2. [9]

When charged particles are immersed into an electroand when they are subjected to an external electric fieldE,they move with a velocityup. WhenE is small enough,up isproportional toE and the coefficient of proportionalityµe isthe electrophoretic mobility

µe = up

E= eζ

6πµf (κa), [10]

whereζ is the zeta potential,µ is the viscosity of the solutionand a is a characteristic dimension of the particle.f is afunction which depends on the shape of the particle. It has btabulated in the literature for spheres (13).

In order to measure the zeta potential, a zetameter was(14). It consists of a rectangular channel with cross-sec2× 5 mm, which contains the electrolyte and the particles.electric field is applied along the axis of the channel in orto create a flow. Because the flow velocity should vanish atwalls and because the total flow rate should be equal to z

there exist two locations called the stationary planes whereflow velocity vanishes (15). In this cell with an aspect ratio

S OF COMPACT CLAYS, I 501

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2.5, the location of the stationary plane was calculated tolocated at 306µm from the walls.

The electrophoretic mobility of the particles was measuin these stationary planes by observation through an opticacroscope. Because of the range of the particle dimensionsof the solute concentration, Eq. [10] can be simplified intoSmoluchowski equation (16)

µe = eζ

4πµ. [11]

Results for the zeta potentials in NaCl solutions of vaous concentrationsc for the various materials are displayedTable 1. For the sake of convenience, the Debye–H¨uckel lengthκ−1 was also systematically calculated with Eq. [8]. The smdifferences which are apparent in Table 1 are due to small tperature variations in the laboratory.

The zeta potentials are seen to depend onc and on the mate-rials. It is interesting to note that|ζ | is an increasing function oc for the clay while it is decreasing for the two other materiaMoreover, for lowc, mica is much more charged than the twothers; for largec, the values are equivalent. The last behaviothe usual one. The fairly high negative zeta potential obtainethe montmorillonite suggests that it is almost completely sodsaturated as would be expected.

2.3. Powder Preparation and Compaction

Powders were compacted in a simple cell where the condtivity and permeability could also be measured (see Fig. 4).

This cell is a circular Plexiglass cylinder of internal diameequal to 15 mm. The powder is located between two sintebronze plates of 4.5 mm whose pore diameters range betw40 and 90µm. Two silver membranes of thickness 50µm arealso put between the powder and the bronze plates; the ordmagnitude of the pore diameter in the membranes is equ0.8µm.

The compaction pressure is exerted on the bronze platepiston of diameter 10 mm which is related to a lever arm; weigcan be placed at the other extremity of the lever arm. The upbronze plate can move freely in the cylindrical tube. Pressuup to 105 bars can be exerted. Note that the cell has also anand an outlet in order to let the fluids flow.

TABLE 1Zeta Potential ζ for the Model Materials and the Argilite in NaCl

Solutions of Concentration c

Muscovite mica Montmorillonite ClayParticle

c (mol/1) ζ (mV) κ−1 (nm) ζ (mV) κ−1 (nm) ζ (mV) κ−1 (nm)

10−4 −76.5 30.07 −48.5 30.509 −21.7 30.47910−3 −53.93 9.065 −43.2 9.648 −23.7 9.63510−2 −31.1 3.048 −36.1 3.049 −28.5 3.047

the

of10−1 −23.7 1.019 −22.0 0.965 −34.4 0.945

ater column

502 MAMMAR ET AL.

FIG. 4. Experimental setup for conductivity and permeability measurements of a sample of compacted clay: (a) pressure exerted by means of a w

for large porosities and permeabilities; (b) pressure exerted by means of liquid nitrogen for low porosities and permeabilities; (1) compacted powder; (2) and(3) sintered bronze plates; (4) lever arm; (5) piston; (6) weights; (7) water inlet; (8) water outlet.

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The samples, the silver membranes, the two bronze platethe lower part of the Plexiglass cell are put in a large dish wcontains the electrolyte solution; the whole set is degassed.a sintered plate and a silver membrane are inserted intcell filled with degassed electrolyte; a known mass of powis introduced. All the system is degassed again since butrapped in the pores could disturb the measurements. Thesuspension settles during one day. A second sintered plateon the powder sediment.

Particular care should be taken. Montmorillonite is alloweswell before it is degassed. Extra ions are also dissolved frompowders. The powders are thus rinsed several times whenecessary and this phenomenon is controled by conducmeasurements as illustrated in Fig. 5; this operation is stowhen the electrolyte conductivity corresponds to the contivity of the initial electrolyte. This phenomenon is particulaimportant for montmorillonite.

The powder is compacted under various pressuresP in orderto measure the corresponding macroscopic properties, abe seen later. Such compaction curves are given in Fig.the three materials. For the same value ofP and as long asPis smaller than 80 bars, the porosityεp of the clay powder issmaller than the mica porosity, which is itself smaller thanmontmorillonite powder. The concentration of NaCl solutioequal to 10−4 mol/l. This might be due to the less polydispecharacter of the clay powder. WhenP is larger than 40 barsthe clay and mica porosities do not vary anymore withP. Thisis not true for montmorillonite whose porosity considerably

creases above 100 bars and becomes smaller than the porosmica.

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2.4. Preparation of the Clay Samples

The clay solid samples were prepared according to a simtechnique which appears to give excellent results. The firstconsists of cutting a cylinder with the desired diameter by meof a circular saw from the original clay blocks. The speedrotation of the saw is very slow and equal to 200 rotations/mWhen the desired cylinder length is obtained, the cylinder is

FIG. 5. Conductivity of supernatant liquid as a function of time when sa

ity ofples of montmorillonite (e) and mica (n) are prepared. At times indicated bythe vertical broken lines, the samples are rinsed.

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TRANSPORT PROPERT

FIG. 6. The powder porosity as a function of the compaction pressurData are forC = 10−4 M of NaCl: (s) sodic montmorillonite; (h) muscovitemica; (n) clay powder.

away from the block by means of a manual saw. Then the samfaces are manually polished with abrasive papers of decreathicknesses. Examples are displayed in Fig. 3a. The error osample thickness is estimated to be equal to 0.2 mm.

The sample imbibition is performed by putting it into a rinof Delrin which has the same thickness (see Fig. 7).

The sample is surrounded by a Teflon film so that the jution between it and the Delrin cylinder is impermeable. In mcases, two silver membranes (pore diameters: 0.8µm; thickness50µm) and two sintered glasses are used in order to maineverything. This is put into a large beaker which containsdegassed aqueous solution. The sample is then degasseda day by means of a vacuum trump.

The weightsmd andmw of the dry and imbibed sample ameasured and they yield the porosityεp of the clay sample

FIG. 7. Imbibition of the clay solid sample. (1) sintered plates; (2) Ag/AgCmembranes; (3) Delrin ring; (4) clay solid sample.

S OF COMPACT CLAYS, I 503

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The corresponding precision is estimated to be on the orde0.5%.

Finally, it is important to notice that the various powders athe clay sample are thoroughly rinsed with the aqueous solut

2.5. Conductivity

Sample conductivity was measured by a classical methomassms of powder in suspension in a given electrolytic solutioconcentration is located between two sintered bronze platesFig. 4). Porosity can be modified according to the compactpressure and expressed as

εp = 1− Ls

L, where Ls = ms

ρs

1

S, [12]

whereS is the sample surface andL its length.During the experiment, length variations due to variatio

of the compaction pressure were measured with a cathetoter with an accuracy of 0.1 mm. Porosities are obtained wan accuracy of 5%. An alternative voltage with a frequency4 kHz was imposed, between the two bronze plates, by a gentor whose frequency ranges from 20 Hz to 1 MHz. The generacurrent intensity was measured with a Keithley 2000 multimeThe sample resistanceR can be estimated with Ohm’s law, anthe resistivityρp or the conductivityσp can be deduced as

σp = 1

ρp= 1

R

L

S. [13]

Accuracy is estimated to be about 2%.

2.6. Permeability

2.6.1. Water permeability.Water permeability was measured with the same experimental setup as for conductivity msurements (see Fig. 4). A steady flow was generated by mof a pressure difference1P. The water permeabilityK may bedetermined by Darcy’s law which can be written as

K = Q

Sµ

L

1P, [14]

whereQ is the volumetric flow rate, obtained by measuring tliquid mass which flows through the sample during a given tim

For permeabilities larger than 10−16 m2, measurements werperformed by applying a pressure of 0.1 bar by means of a liqcolumn as displayed in Fig. 4a. However, the measuremenlow permeabilities necessitates the application of larger psures which were obtained by compressed gas (see Fig. 4b

Accuracy of permeability measurements was about 2 to 3.

2.6.2. Air permeability for the clay sample.This stan-dard measurement is made under unsteady conditions wthe air compressibility is taken into account. As schematized

lplates and is surrounded by cavities of volumesV1 andV2. In

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504 MAMMA

FIG. 8. (a) Schematic of the measurement of air permeability of thesolid sample; (1) and (2) upstream and downstream reservoirs; (3) clayple (see Fig. 7); (4) air supply; (5) pressure gauge; (6) water manom(b) Schematic of the model measurement of air permeability; (1) and (2stream and downstream reservoirs.

the upstream cavity (left), a constant pressure is imposed wa pressureP2 < P1 is imposed att = 0 in the downstream cavity; for t > 0, P2 varies and the pressure differenceP1− P2 isrecorded. This standard measurement is easy to perform adoes not require any difficult gas flow rate measurement.

This experiment can be schematized as indicated in FigSince the pressure variations are relatively small, the presP inside the clay sample obeys a diffusion equation

∂P

∂t− 1

εpµCt∇ · (K∇P) = 0, [15]

whereK is the permeability (m2), µ is the dynamic viscosity(kg m−1 s−1), andCt is the isothermal compressibility coefficient (Pa−1). In addition, P satisfies the following boundarconditions

x = 0 : P = P1 = constant

x = e : P = P2 t > 0.[16]

This must be completed by an analysis of the mass tran

from and into the two reservoirs. Let us assume that the tempaturesT1 andT2 are constant and equal toT ; let ni denote the

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up-

hile

nd it

8b.ure

-

fers

number of moles in reservoir i

Pi Vi = ni RT i = 1, 2, [17]

whereR= 8.31 J mol−1 K−1.The gas volumeVi which leaves or enters a reservoir can b

expressed as

Vi = Vmdni

dt= −K

µ

∂P

∂x

∣∣∣∣xi

xi = 0 ore, [18]

whereVm = 22.4 l under normal conditions.Under these circumstances, the solution can be expresse

P1− P2(t)

P1= exp(−αt), [19]

where the constantα is given by

α = RT K

µV2Vm

S

e. [20]

This coefficient can be obtained by analyzing the experimental data. Since all the other quantities are known,α can beused to deriveK .

3. RESULTS AND DISCUSSION

3.1. Conductivity Measurements

3.1.1. Powders. Conductivity measurements were performed on the three powders for various values of the compactpressures; the experimental device described in Sectionwas used. The electrolyte is a NaCl solution of concentrati10−4 mol/l. For all these measurements, the samples were rinseveral times; hence, the presence of ions originating fromsample is unlikely. For instance, for the clay powder, the supension was filtered, and the wet powder was dryed in an ovat a temperature of 100◦C during 24 h.

It is more convenient to represent the experimental resuin terms of the formation factorF which is defined as the ra-tio between the fluid conductivityσ f and the porous mediumconductivityσp

F = σ f

σp. [21]

The results obtained for the three powders are displayedFig. 9. It appears that for a given value of the porosity,F islarger for clay than for montmorillonite and mica. It seems ththis dependence parallels the zeta potentials as it was obsein Section 2.2.

The results were also systematically compared to Archielaw

er-

F = kε−mp , [22]

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TRANSPORT PROPERTI

FIG. 9. Formation factorsF as functions of porosity. Data are for: powde((s) clay powder compacted when wet, (e) montmorillonite, (∗) mica); originalclay sample (n); clay powder compacted when dry (h).

wherek andmare constants. For the three powders, the measvalues are gathered in Table 2.

3.1.2. Clay samples.Conductivity measurements were alperformed with the same NaCl solution on a clay samplethickness 3 mm and diameter 30 mm. The results are displain Fig. 9. For the same porosity, the formation factor of the csample equal to 22.5 is slightly lower than the formation facof the compacted powder equal to 33. Hence, the corresponelectric resistance is slightly lower.

The influence of the saturation mode was studied by perfoing measurements on a sample of clay powder recompacteder dry conditions. Some powder was compacted under a psure of 105 bars in order to get a sample whose porosity is cto that of the original compact clay. Then this sample was weby means of a flow of an NaCl solution whose salt concentrawas 10−2 mol/l. The conductivity result is displayed in Fig. 9it is slightly smaller than the formation factor obtained for poders for the same porosity. However, when the porosity variais taken into account, there is no significant difference withoriginal clay sample.

It is interesting to note that the difference between the pder and the clay sample cannot be explained by an incompimbibition of the samples; such a phenomenon would necesily imply a larger formation factor. Moreover, clay may havbeen partially cemented, but again this effect would impllarger formation factor. Thus, it is likely that the packings strutures are different in the various samples; for instance, iwell known that the void structure is different in dry or wpacking (17).

3.1.3. Comparison with numerical results.Finally, the pre-

vious measurements can be compared to the numerical resobtained on various types of random packings by (11).

S OF COMPACT CLAYS, I 505

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c-ist

These random packings were obtained by successive ations of particles randomly located and oriented. The ceof gravity of each particle was lowered by elementary tralations and rotations compatible with the particles which walready deposited. When the packed bed is completed, spadiscretized and the formation factor is determined by solvthe Laplace equation through the pore space; the particlesassumed to be insulating. Such calculations were performemonodisperse particles of various shapes such as spheres,soids and parallelepipeds. Results are displayed in Fig. 10;can be gathered by an Archie law

F = ε−2p for 0.1≤ εp ≤ 0.75. [23]

The experimental data were reported in the same figurshould be emphasized that this comparison is direct in the sthat there is no parameter which is fitted whatsoever. Hencecept maybe for the largest porosities above 0.6 (obtained onlmica and montmorillonite), the agreement between the numcal and experimental results is very good. It is interesting to nthat for large porosities, the data for clay are in better agreemwith the numerical data than the two other powders; moreothe results obtained for the smallest porosities are well aligwith the numerical results.

3.2. Permeability Measurements

3.2.1. Powders. Permeability measurements were perfomed on the powders with pressure gradients which can be mified according to the compaction pressure. Moreover, asconductivity measurements, the concentration of the NaCllution is equal to 10−4 mol/l.

FIG. 10. Dimensionless conductivityσpσ0

. Data are for: powders ((s) claypowder compacted when wet, (e) montmorillonite, (∗) mica); original clay sam-ple (n); clay powder compacted when dry (h). Numerical results (n) obtained

ultsfor various types of random packings (11); experimental results for milar (×)and mica (+) (18).

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nor

c

ait

nau

t

r

r

ents

essualthe

theose

g

pleThe

ing

is

506 MAMMAR

FIG. 11. Permeability measurements as functions of porosity. Data arepowders ((s) clay powder compacted when wet, (e) montmorillonite, (∗) mica);original clay sample (n); clay powder compacted when dry (h).

Dimensional results are displayed in Fig. 11. Montmorillonpermeability is significantly smaller than the permeability of ttwo others; clay and mica powders are relatively close onanother. It is also interesting to note that all the curves presechange of slope for a porosity equal to 40%; this value mightbe totally fortuitous since it corresponds to the minimal pority obtained numerically for random packings of monodispespheres.

3.2.2. Clay samples.As already stated, air and water pemeability measurements were performed on the compactsamples.

Let us start with water measurements. As explained in Stion 2.6, the sample is first degassed and then imbibition stThe cylindrical samples have a diameter of 30 mm and a thness of 3 mm. The electrolyte is a NaCl solution of concentra10−4 mol/l.

For the sake of completeness, various pressure differewere applied on the samples. Examples of results are displin Fig. 12. Permeability slightly varies as a function of press

0.910−17 m2 ≤ K ≤ 10−17 m2. [24]

It is obvious that these variations cannot be explained byertial effects since the flow rates are very small. It seems tomore likely that they are due to changes in the local strucsuch as microfissures.

It is important to note that the permeability obtained focompacted clay with a porosity of 0.2, is equal to 10−17 m2 asdisplayed in Fig. 11. Hence, the two measurements are in peagreement.

However, because of the importance of permeability for pratical storage purposes, it was decided to measure it again w

ET AL.

for:

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nt aots-se

r-lay

ec-rts.ck-ion

cesyedre

in-be

ure

a

fect

air as described in Section 2.6. Three series of measuremwere performed and they are displayed in Fig. 13.

First, a cylindrical sample of diameter 14.4 mm and thickn3 mm was submitted to three initial pressure differences eqto 12, 20, and 30 mbars. According to the theoretical law,resulting pressure variationsP1/(P1− P2(t)) should not dependon the initial pressure difference. This is seen to be indeedcase in Fig. 13a. The corresponding permeability is very clto the water estimate and is equal to

K = 5× 10−18 m2. [25]

Second, the areaSof the clay sample was varied. Accordinto Eq. [19], the following quantity should not depend onS

A =ln P1

P1−P2(t)

S. [26]

It is seen in Fig. 13b that the results obtained with a samof diameter 27.5 mm are very close to the previous results.corresponding permeability is equal to

K = 4× 10−18 m2. [27]

Finally, the sample thickness was varied, i.e.,e= 15 and28 mm; the diameter is equal to 27.5 mm (Fig. 13c). Accordto Eq. [19], the following quantity should not depend one

B = e lnP1

P1− P2(t). [28]

When the sample thickness is large (as fore= 28 mm),the final equilibrium is reached only after 15 to 24 h. It

c-ith

FIG. 12. Variations of the permeabilityK of a compact clay sample as afunction of the pressure difference1P (water measurements).

TRANSPORT PROPERTIES OF COMPACT CLAYS, I 507

FIG. 13. Air permeability measurements. (a) Variations ofP1/(P1 − P2(t)) as functions ofP1 − P2: (s) 30 mbars; (n) 20 mbars; (e) 12 mbars.(b) Variations of A for two different sections:S= 1.61 cm2 (×); S= 5.94 cm2 (n). (c) Variations of B for two different thicknesses:e= 15 mm (e); e=

28 mm (n). (d) Variations of C as function of time:e= 15 mm,S= 5.94 cm2 (e); e= 28 mm,S= 5.94 cm2 (n); e= 15 mm,S= 5.94 cm2 (s); e= 15 mm,

r sis

ea-abil-

epol-eem

S= 1.61 cm2 (×).

thus important to wait for complete equilibrium before staing new measurements. The corresponding value ofK isequal to

K = 2.5× 10−18 m2. [29]

Finally, all the results can be tentatively superposed by usthe following dimensionless representation, i.e., by plottingCas a function oft ′

C = eV1/32

Sln

P1

P1− P2(t)[30]

t ′ = RT K

µV2/32 Vm

t.

t-

ing

All the data are displayed in Fig. 13d. A regression analyyields a value of

K = 2.6× 10−18 m2 if e≥ 15 mm [31]

K = 4.4× 10−18 m2 if e= 3 mm. [32]

Hence, from all this series of water and air permeability msurements, it can be concluded that the solid clay permeity ranges between 2.5× 10−18 and 10−17 m2. This is not sig-nificantly different from the powder permeability for the samporosity as it can be seen in Fig. 11. Hence, the cutting andishing operations necessary to obtain the samples do not sto have a strong influence on the properties of the sample.

3.2.3. Comparison with numerical results.Again these per-meability measurements can be compared with the numerical

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508 MAMMAR

results obtained by (11) on the same random packings as tmentioned in Section 3.1.3. The permeability of these packiwas determined by solving the three-dimensional Stokes eqtion in the discretized packings.

The comparison is not as direct as it was for conductivity sina length scale must be provided. LetR be the equivalent radiusof the particles in a given powder.R is determined by obtainingthe best fit between the numerical results and the experimedata. The comparison is displayed in Fig. 14 and the slope ofdata is seen to be quite good.

It should be noticed that the range of (11) was extendedlower porosities by surrounding particles in the actual packinglayers of constant thickness; the corresponding lowest porosare thus equal to 0.15.

As shown in (12), the permeability of sandstones can be wapproximated by the power law

K ∝ εn with n = 4.15 for 0.1≤ ε ≤ 0.25. [33]

It is seen in Fig. 14 that the measurements could be wellproximated by such a power law, but with an exponentn = 4.91.

Finally, let us comment on the values obtained for the equalent radiiR obtained for the three materials. These valuesgathered in Table 2, where they are compared to the radiitained by SEM. It is interesting that the two sets of numbecan be ordered similarly and that they roughly differ by oneder of magnitude. This comparison which is quite primitivecharacter, is very satisfactory; it should be recalled that themerical calculations were performed for monodisperse partica feature which is far from being verified in the real powdemoreover, the fines which play a small role in determining coductivity, may be quite influential on permeability.

FIG. 14. Normalized permeability as a function of porosity. Data are foclay powder (s), montmorillonite (e), mica (∗). Numerical results obtained

for various packings (h) (see (11)). The dashed line represents the fitting bEq. [33] withn = 4.91.

ET AL.

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ce

ntalthe

tobyties

ell

ap-

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ob-rsr-

innu-es,s;n-

r:

TABLE 2Coefficients in Archie’s Law (k,m)

Material k m lc (µm) Rm (µm) R (µm)

Montmorillonite 0.86 −2.2 0.67× 10−3 1 0.1Mica 0.79 −1.8 4× 10−2 2 0.4Clay 1 −2.2 — 5 0.45

Note.Characteristic lengthlc (see Eq. [2]); geometric radiiRm measured bySEM; equivalent radiiR obtained by permeability measurements.

4. CONCLUDING REMARKS

In this first paper, we have essentially focused our attenon the conduction and convection properties of model andural clays. These properties have been studied as functionporosity. Whenever possible, these experimental data havecompared with numerical ones. We have found in some cafundamental differences between natural and model clays. Fgiven value of porosity, the formation factor is larger for natuclay than for model clays. These differences can be due todifferences in local properties such as surface charge, whicless important for natural clay than for models. It has been apointed out that|ζ | increases with concentration for natural claand decreases for model clays.

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