Transport Properties of Compact Clays

Journal of Colloid and Interface Science 240, 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 Mat´eriaux Divis´es et des Interfaces, 5 boulevard Descartes, F-77454 Marne la Vall´ee Cedex 2, France; ‡PTM/LCD, BP30179, F-86962 Futuroscope, France; and § ANDRA Direction Scientifique, 1-7 rue Jean Monet, Chˆatenay-Malabry Cedex, France Received November 2, 2000; accepted May 4, 2001; published online July 12, 2001

well as the preparation of the samples. Finally, conductivity and permeability measurements are described. Results of systematic measurements of conductivity and permeability are gathered in Section 3. Data relative to powders under compaction pressures are presented and compared to data obtained with the original compact clay. Particular attention is devoted to permeability which is measured either with air or water. These data are then compared to numerical results derived by Coelho et al. (11) on packings of particles. Some concluding remarks end this first Part.

Conductivity and permeability of model and natural clays have been studied experimentally. Local properties such as porosity and zeta potentials were measured as functions of the electrolyte solutions. Whenever possible, experimental data were compared to numerical data obtained for random packings of grains of arbitrary shape, and a good agreement was found between them. ° C 2001

Academic Press

Key Words: compact clays; conductivity; permeability; zeta potential. 1. INTRODUCTION

2. MATERIALS AND MEASUREMENTS

Low permeability materials containing clay play an important role in practical life and in the natural environment. Thanks to their porous character, the transport of water through soil allows vegetal development and life. Reservoir rocks which contain oil are often confined by impermeable clay layers. In most cases, clay formations present confinement properties, such as low porosities, low permeabilities, and consequently low flow rates; these properties are highly desirable in storing nuclear wastes (1–5). Study of transport phenomena in porous media is made difficult by the complex structures of real materials. Many numerical and experimental investigations have been done about main geometrical and transport properties (6–8). However, in contrast with rocks such as sandstones and limestones, transport properties of clays have not been often studied (9, 10). Hence, a systematic analysis of these properties is both of academic and industrial interest. The first part of this series is devoted to a general presentation of the materials which have been used and to the experimental study of their basic properties, namely, permeability and conductivity. The second section is devoted to materials which are studied, namely mica, montmorillonite, and natural clay; they were characterized by various techniques including Scattering Electron Microscopy (SEM) and zeta potentials. The powders were also compacted under various pressures and the resulting porosities were systematically measured. The cells are described as 0021-9797/01 $35.00

C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.

2.1. Materials Three types of materials were used for the experiments. Two powders with different properties were employed as model materials, namely muscovite mica and sodic montmorillonite. The third material is a compact clay which has been extracted in the east part of France in a Callovo–Oxfordian formation. The muscovite mica (Comptoir de Min´eraux et Mati`eres premi`eres) is essentially composed of SiO2 (48%) and Al2 O3 (34%). This powder was analyzed by SEM; the pictures (Fig. 1) show a lamellar structure with grains whose diameters are about 4 µm; these grains tend to aggregate into larger clusters. The specific surface Ssp was measured by nitrogen adsorption and it was found to be Ssp = 2.5 × 107 m−1 .

[1]

A specific length lc can be defined as the inverse of Ssp lc = 4 × 10−8 m.

[2]

The density was measured by standard weight and volume measurements. It was deduced to be ρs = 3150 ± 150 kg/m3 .

[3]

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

498

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 bars provide the scale.

500

MAMMAR ET AL.

FIG. 3. Compact site clay: (a) the original block and two clay samples cut out of it; some powder is also shown; (b) pictures of the clay powder obtained by SEM.

water, and it can form a gel. This was also analyzed by SEM; Figure 2 shows pictures after 24 h of immersion in water; the dimensions range between grains of 2 µm diameter and clusters of 10 µm diameter. The specific surface was measured by the same technique as before, Ssp = 1.5 × 109 m−1 .

[4]

Hence, it is equal to 60 times the specific surface of mica. The corresponding characteristic length is lc = 0.67 × 10−9 m.

[5]

The density of montmorillonite is equal to ρs = 2450 ± 180 kg/m3 .

[6]

A block of argilite has been supplied to us by ANDRA from the drilling referred to as EST 104; this cylinder extracted at a depth of 483.6 m with a diameter of 100 mm and a length of 82 mm (Fig. 3a) is labeled as EST 02364. This block has been used in different ways in order to obtain either clay powders or solid cylinders. In order to start in the simplest possible way, broken parts of the original block were crushed in an agate mortar. The resulting

501

TRANSPORT PROPERTIES OF COMPACT CLAYS, I

particles were selected with successive sieves whose smallest diameter is 40 µm. It is important to note that the remaining powder which could not be filtered by the sieves, was crushed again and again until the whole powder could pass through the smallest diameter. Pictures of the resulting powder were taken with SEM and are displayed in Fig. 3b. The structure is quite heterogeneous and many aggregates are found; because of the careful crushing process, the average grain radius ranges from 1 to 10 µm. The density ρs of the clay powder was measured by the same technique as before and was deduced to be ρs = 2660 ± 150 kg/m . 3

[7]

No analysis of the specific surface was performed on this clay. 2.2. Zeta Potentials Let us start with the definition of the double layer thickness κ −1 (15) κ −1 =

·µ

e2 ²el kT

¶X

¸−1/2 ci0 z i2

,

[8]

i

where z i is the valency of ion i, e is the charge of the electron, k is the Boltzmann constant, T is the absolute temperature, ²el is the dielectric constant of the fluid, and ci0 is the volumetric concentration of the ion i out of the double layer. For a monovalent salt such as NaCl, when c is given in mol/l, κ −1 (in nm) can be expressed as κ −1 = 3.0410−10 c−1/2 .

[9]

When charged particles are immersed into an electrolyte, and when they are subjected to an external electric field E, they move with a velocity u p . When E is small enough, u p is proportional to E and the coefficient of proportionality µe is the electrophoretic mobility µe =

up eζ = f (κa), E 6πµ

[10]

where ζ is the zeta potential, µ is the viscosity of the solution, and a is a characteristic dimension of the particle. f is a function which depends on the shape of the particle. It has been tabulated in the literature for spheres (13). In order to measure the zeta potential, a zetameter was used (14). It consists of a rectangular channel with cross-section 2 × 5 mm, which contains the electrolyte and the particles. An electric field is applied along the axis of the channel in order to create a flow. Because the flow velocity should vanish at the walls and because the total flow rate should be equal to zero, there exist two locations called the stationary planes where the flow velocity vanishes (15). In this cell with an aspect ratio of

2.5, the location of the stationary plane was calculated to be located at 306 µm from the walls. The electrophoretic mobility of the particles was measured in these stationary planes by observation through an optical microscope. Because of the range of the particle dimensions and of the solute concentration, Eq. [10] can be simplified into the Smoluchowski equation (16) µe =

eζ . 4π µ

[11]

Results for the zeta potentials in NaCl solutions of various concentrations c for the various materials are displayed in Table 1. For the sake of convenience, the Debye–H¨uckel length κ −1 was also systematically calculated with Eq. [8]. The small differences which are apparent in Table 1 are due to small temperature variations in the laboratory. The zeta potentials are seen to depend on c and on the materials. It is interesting to note that |ζ | is an increasing function of c for the clay while it is decreasing for the two other materials. Moreover, for low c, mica is much more charged than the two others; for large c, the values are equivalent. The last behavior is the usual one. The fairly high negative zeta potential obtained on the montmorillonite suggests that it is almost completely sodium saturated as would be expected. 2.3. Powder Preparation and Compaction Powders were compacted in a simple cell where the conductivity and permeability could also be measured (see Fig. 4). This cell is a circular Plexiglass cylinder of internal diameter equal to 15 mm. The powder is located between two sintered bronze plates of 4.5 mm whose pore diameters range between 40 and 90 µm. Two silver membranes of thickness 50 µm are also put between the powder and the bronze plates; the order of magnitude of the pore diameter in the membranes is equal to 0.8 µm. The compaction pressure is exerted on the bronze plate by a piston of diameter 10 mm which is related to a lever arm; weights can be placed at the other extremity of the lever arm. The upper bronze plate can move freely in the cylindrical tube. Pressures up to 105 bars can be exerted. Note that the cell has also an inlet and an outlet in order to let the fluids flow. TABLE 1 Zeta Potential ζ for the Model Materials and the Argilite in NaCl Solutions of Concentration c Muscovite mica

Montmorillonite

Clay

Particle c (mol/1)

ζ (mV)

κ −1 (nm)

ζ (mV)

κ −1 (nm)

ζ (mV)

κ −1 (nm)

10−4 10−3 10−2 10−1

−76.5 −53.93 −31.1 −23.7

30.07 9.065 3.048 1.019

−48.5 −43.2 −36.1 −22.0

30.509 9.648 3.049 0.965

−21.7 −23.7 −28.5 −34.4

30.479 9.635 3.047 0.945

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 water column 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.

The samples, the silver membranes, the two bronze plates, and the lower part of the Plexiglass cell are put in a large dish which contains the electrolyte solution; the whole set is degassed. Then a sintered plate and a silver membrane are inserted into the cell filled with degassed electrolyte; a known mass of powder is introduced. All the system is degassed again since bubbles trapped in the pores could disturb the measurements. Then the suspension settles during one day. A second sintered plate is put on the powder sediment. Particular care should be taken. Montmorillonite is allowed to swell before it is degassed. Extra ions are also dissolved from the powders. The powders are thus rinsed several times whenever necessary and this phenomenon is controled by conductivity measurements as illustrated in Fig. 5; this operation is stopped when the electrolyte conductivity corresponds to the conductivity of the initial electrolyte. This phenomenon is particularly important for montmorillonite. The powder is compacted under various pressures P in order to measure the corresponding macroscopic properties, as will be seen later. Such compaction curves are given in Fig. 6 for the three materials. For the same value of P and as long as P is smaller than 80 bars, the porosity ² p of the clay powder is smaller than the mica porosity, which is itself smaller than the montmorillonite powder. The concentration of NaCl solution is equal to 10−4 mol/l. This might be due to the less polydisperse character of the clay powder. When P is larger than 40 bars, the clay and mica porosities do not vary anymore with P. This is not true for montmorillonite whose porosity considerably decreases above 100 bars and becomes smaller than the porosity of mica.

2.4. Preparation of the Clay Samples The clay solid samples were prepared according to a simple technique which appears to give excellent results. The first step consists of cutting a cylinder with the desired diameter by means of a circular saw from the original clay blocks. The speed of rotation of the saw is very slow and equal to 200 rotations/min. When the desired cylinder length is obtained, the cylinder is cut

FIG. 5. Conductivity of supernatant liquid as a function of time when samples of montmorillonite (e) and mica (n) are prepared. At times indicated by the vertical broken lines, the samples are rinsed.

503

TRANSPORT PROPERTIES OF COMPACT CLAYS, I

The corresponding precision is estimated to be on the order of 0.5%. Finally, it is important to notice that the various powders and the clay sample are thoroughly rinsed with the aqueous solution. 2.5. Conductivity Sample conductivity was measured by a classical method; a mass m s of powder in suspension in a given electrolytic solution concentration is located between two sintered bronze plates (see Fig. 4). Porosity can be modified according to the compaction pressure and expressed as ²p = 1 −

FIG. 6. The powder porosity as a function of the compaction pressure P. Data are for C = 10−4 M of NaCl: (s) sodic montmorillonite; (h) muscovite mica; (n) clay powder.

away from the block by means of a manual saw. Then the sample faces are manually polished with abrasive papers of decreasing thicknesses. Examples are displayed in Fig. 3a. The error on the sample thickness is estimated to be equal to 0.2 mm. The sample imbibition is performed by putting it into a ring of Delrin which has the same thickness (see Fig. 7). The sample is surrounded by a Teflon film so that the junction between it and the Delrin cylinder is impermeable. In most cases, two silver membranes (pore diameters: 0.8 µm; thickness 50 µm) and two sintered glasses are used in order to maintain everything. This is put into a large beaker which contains the degassed aqueous solution. The sample is then degassed during a day by means of a vacuum trump. The weights m d and m w of the dry and imbibed sample are measured and they yield the porosity ² p of the clay sample.

Ls , where L

ms 1 , ρs S

[12]

where S is the sample surface and L its length. During the experiment, length variations due to variations of the compaction pressure were measured with a cathetometer with an accuracy of 0.1 mm. Porosities are obtained with an accuracy of 5%. An alternative voltage with a frequency of 4 kHz was imposed, between the two bronze plates, by a generator whose frequency ranges from 20 Hz to 1 MHz. The generated current intensity was measured with a Keithley 2000 multimeter. The sample resistance R can be estimated with Ohm’s law, and the resistivity ρ p or the conductivity σ p can be deduced as σp =

1 1 L = . ρp RS

[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 measurements (see Fig. 4). A steady flow was generated by means of a pressure difference 1P. The water permeability K may be determined by Darcy’s law which can be written as K =

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

Ls =

Q L µ , S 1P

[14]

where Q is the volumetric flow rate, obtained by measuring the liquid mass which flows through the sample during a given time. For permeabilities larger than 10−16 m2 , measurements were performed by applying a pressure of 0.1 bar by means of a liquid column as displayed in Fig. 4a. However, the measurement of low permeabilities necessitates the application of larger pressures which were obtained by compressed gas (see Fig. 4b). Accuracy of permeability measurements was about 2 to 3.5%. 2.6.2. Air permeability for the clay sample. This standard measurement is made under unsteady conditions where the air compressibility is taken into account. As schematized in Fig. 8a, the clay sample is located between two bronze sintered plates and is surrounded by cavities of volumes V1 and V2 . In

504

MAMMAR ET AL.

number of moles in reservoir i Pi Vi = n i RT

i = 1, 2,

[17]

where R = 8.31 J mol−1 K−1 . The gas volume Vi which leaves or enters a reservoir can be expressed as Vi = Vm

¯ K ∂P ¯¯ dn i =− dt µ ∂ x ¯xi

xi = 0 or e,

[18]

where Vm = 22.4 l under normal conditions. Under these circumstances, the solution can be expressed as P1 − P2 (t) = exp(−αt), P1

[19]

where the constant α is given by α=

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

the upstream cavity (left), a constant pressure is imposed while a pressure P2 < P1 is imposed at t = 0 in the downstream cavity; for t > 0, P2 varies and the pressure difference P1 − P2 is recorded. This standard measurement is easy to perform and it does not require any difficult gas flow rate measurement. This experiment can be schematized as indicated in Fig. 8b. Since the pressure variations are relatively small, the pressure P inside the clay sample obeys a diffusion equation 1 ∂P ∇ · (K ∇ P) = 0, − ∂t ² p µCt

[15]

where K is the permeability (m2 ), µ is the dynamic viscosity (kg m−1 s−1 ), and Ct is the isothermal compressibility coefficient (Pa −1 ). In addition, P satisfies the following boundary conditions x = 0 : P = P1 = constant x = e : P = P2

t > 0.

[16]

This must be completed by an analysis of the mass transfers from and into the two reservoirs. Let us assume that the temperatures T1 and T2 are constant and equal to T ; let n i denote the

RT K S . µV2 Vm e

[20]

This coefficient can be obtained by analyzing the experimental data. Since all the other quantities are known, α can be used to derive K . 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 compaction pressures; the experimental device described in Section 2.3 was used. The electrolyte is a NaCl solution of concentration 10−4 mol/l. For all these measurements, the samples were rinsed several times; hence, the presence of ions originating from the sample is unlikely. For instance, for the clay powder, the suspension was filtered, and the wet powder was dryed in an oven at a temperature of 100◦ C during 24 h. It is more convenient to represent the experimental results in terms of the formation factor F which is defined as the ratio between the fluid conductivity σ f and the porous medium conductivity σ p F=

σf . σp

[21]

The results obtained for the three powders are displayed in Fig. 9. It appears that for a given value of the porosity, F is larger for clay than for montmorillonite and mica. It seems that this dependence parallels the zeta potentials as it was observed in Section 2.2. The results were also systematically compared to Archie’s law F = k² −m p ,

[22]

505

TRANSPORT PROPERTIES OF COMPACT CLAYS, I

These random packings were obtained by successive additions of particles randomly located and oriented. The center of gravity of each particle was lowered by elementary translations and rotations compatible with the particles which were already deposited. When the packed bed is completed, space is discretized and the formation factor is determined by solving the Laplace equation through the pore space; the particles are assumed to be insulating. Such calculations were performed for monodisperse particles of various shapes such as spheres, ellipsoids and parallelepipeds. Results are displayed in Fig. 10; they can be gathered by an Archie law F = ² −2 p

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

where k and m are constants. For the three powders, the measured values are gathered in Table 2. 3.1.2. Clay samples. Conductivity measurements were also performed with the same NaCl solution on a clay sample of thickness 3 mm and diameter 30 mm. The results are displayed in Fig. 9. For the same porosity, the formation factor of the clay sample equal to 22.5 is slightly lower than the formation factor of the compacted powder equal to 33. Hence, the corresponding electric resistance is slightly lower. The influence of the saturation mode was studied by performing measurements on a sample of clay powder recompacted under dry conditions. Some powder was compacted under a pressure of 105 bars in order to get a sample whose porosity is close to that of the original compact clay. Then this sample was wetted by means of a flow of an NaCl solution whose salt concentration was 10−2 mol/l. The conductivity result is displayed in Fig. 9; it is slightly smaller than the formation factor obtained for powders for the same porosity. However, when the porosity variation is taken into account, there is no significant difference with the original clay sample. It is interesting to note that the difference between the powder and the clay sample cannot be explained by an incomplete imbibition of the samples; such a phenomenon would necessarily imply a larger formation factor. Moreover, clay may have been partially cemented, but again this effect would imply a larger formation factor. Thus, it is likely that the packings structures are different in the various samples; for instance, it is well known that the void structure is different in dry or wet packing (17). 3.1.3. Comparison with numerical results. Finally, the previous measurements can be compared to the numerical results obtained on various types of random packings by (11).

for

0.1 ≤ ² p ≤ 0.75.

[23]

The experimental data were reported in the same figure. It should be emphasized that this comparison is direct in the sense that there is no parameter which is fitted whatsoever. Hence, except maybe for the largest porosities above 0.6 (obtained only for mica and montmorillonite), the agreement between the numerical and experimental results is very good. It is interesting to note that for large porosities, the data for clay are in better agreement with the numerical data than the two other powders; moreover, the results obtained for the smallest porosities are well aligned with the numerical results. 3.2. Permeability Measurements 3.2.1. Powders. Permeability measurements were performed on the powders with pressure gradients which can be modified according to the compaction pressure. Moreover, as for conductivity measurements, the concentration of the NaCl solution is equal to 10−4 mol/l.

σ

FIG. 10. Dimensionless conductivity σ0p . Data are for: powders ((s) clay powder compacted when wet, (e) montmorillonite, (∗) mica); original clay sample (n); clay powder compacted when dry (h). Numerical results (n) obtained for various types of random packings (11); experimental results for milar (×) and mica (+) (18).

506

MAMMAR ET AL.

air as described in Section 2.6. Three series of measurements were performed and they are displayed in Fig. 13. First, a cylindrical sample of diameter 14.4 mm and thickness 3 mm was submitted to three initial pressure differences equal to 12, 20, and 30 mbars. According to the theoretical law, the resulting pressure variations P1 /(P1 − P2 (t)) should not depend on the initial pressure difference. This is seen to be indeed the case in Fig. 13a. The corresponding permeability is very close to the water estimate and is equal to K = 5 × 10−18 m2 .

[25]

Second, the area S of the clay sample was varied. According to Eq. [19], the following quantity should not depend on S A= FIG. 11. Permeability measurements as functions of porosity. Data are for: powders ((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. Montmorillonite permeability is significantly smaller than the permeability of the two others; clay and mica powders are relatively close one to another. It is also interesting to note that all the curves present a change of slope for a porosity equal to 40%; this value might not be totally fortuitous since it corresponds to the minimal porosity obtained numerically for random packings of monodisperse spheres. 3.2.2. Clay samples. As already stated, air and water permeability measurements were performed on the compact clay samples. Let us start with water measurements. As explained in Section 2.6, the sample is first degassed and then imbibition starts. The cylindrical samples have a diameter of 30 mm and a thickness of 3 mm. The electrolyte is a NaCl solution of concentration 10−4 mol/l. For the sake of completeness, various pressure differences were applied on the samples. Examples of results are displayed in Fig. 12. Permeability slightly varies as a function of pressure 0.910−17 m2 ≤ K ≤ 10−17 m2 .

ln

P1 P1 −P2 (t)

S

.

[26]

It is seen in Fig. 13b that the results obtained with a sample of diameter 27.5 mm are very close to the previous results. The corresponding permeability is equal to K = 4 × 10−18 m2 .

[27]

Finally, the sample thickness was varied, i.e., e = 15 and 28 mm; the diameter is equal to 27.5 mm (Fig. 13c). According to Eq. [19], the following quantity should not depend on e B = e ln

P1 . P1 − P2 (t)

[28]

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

[24]

It is obvious that these variations cannot be explained by inertial effects since the flow rates are very small. It seems to be more likely that they are due to changes in the local structure such as microfissures. It is important to note that the permeability obtained for a compacted clay with a porosity of 0.2, is equal to 10−17 m2 as displayed in Fig. 11. Hence, the two measurements are in perfect agreement. However, because of the importance of permeability for practical storage purposes, it was decided to measure it again with

FIG. 12. Variations of the permeability K of a compact clay sample as a function of the pressure difference 1P (water measurements).

507

TRANSPORT PROPERTIES OF COMPACT CLAYS, I

FIG. 13. Air permeability measurements. (a) Variations of P1 /(P1 − P2 (t)) as functions of P1 − 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, S = 1.61 cm2 (×).

thus important to wait for complete equilibrium before starting new measurements. The corresponding value of K is equal to −18

K = 2.5 × 10

m. 2

[29]

Finally, all the results can be tentatively superposed by using the following dimensionless representation, i.e., by plotting C as a function of t 0 1/3

C= t0 =

eV2 S

ln

RT K 2/3

µV2 Vm

P1 P1 − P2 (t) t.

[30]

All the data are displayed in Fig. 13d. A regression analysis yields 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 measurements, it can be concluded that the solid clay permeability ranges between 2.5 × 10−18 and 10−17 m2 . This is not significantly different from the powder permeability for the same porosity as it can be seen in Fig. 11. Hence, the cutting and polishing operations necessary to obtain the samples do not seem to have a strong influence on the properties of the sample. 3.2.3. Comparison with numerical results. Again these permeability measurements can be compared with the numerical

508

MAMMAR ET AL.

results obtained by (11) on the same random packings as those mentioned in Section 3.1.3. The permeability of these packings was determined by solving the three-dimensional Stokes equation in the discretized packings. The comparison is not as direct as it was for conductivity since a length scale must be provided. Let R˜ be the equivalent radius of the particles in a given powder. R˜ is determined by obtaining the best fit between the numerical results and the experimental data. The comparison is displayed in Fig. 14 and the slope of the data is seen to be quite good. It should be noticed that the range of (11) was extended to lower porosities by surrounding particles in the actual packing by layers of constant thickness; the corresponding lowest porosities are thus equal to 0.15. As shown in (12), the permeability of sandstones can be well approximated by the power law K ∝ ²n

with

n = 4.15

for

0.1 ≤ ² ≤ 0.25.

[33]

It is seen in Fig. 14 that the measurements could be well approximated by such a power law, but with an exponent n = 4.91. Finally, let us comment on the values obtained for the equivalent radii R˜ obtained for the three materials. These values are gathered in Table 2, where they are compared to the radii obtained by SEM. It is interesting that the two sets of numbers can be ordered similarly and that they roughly differ by one order of magnitude. This comparison which is quite primitive in character, is very satisfactory; it should be recalled that the numerical calculations were performed for monodisperse particles, a feature which is far from being verified in the real powders; moreover, the fines which play a small role in determining conductivity, may be quite influential on permeability.

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

k

m

lc (µm)

Rm (µm)

R˜ (µm)

Montmorillonite Mica Clay

0.86 0.79 1

−2.2 −1.8 −2.2

0.67 × 10−3 4 × 10−2 —

1 2 5

0.1 0.4 0.45

Note. Characteristic length lc (see Eq. [2]); geometric radii Rm measured by SEM; equivalent radii R˜ obtained by permeability measurements.

4. CONCLUDING REMARKS

In this first paper, we have essentially focused our attention on the conduction and convection properties of model and natural clays. These properties have been studied as functions of porosity. Whenever possible, these experimental data have been compared with numerical ones. We have found in some cases fundamental differences between natural and model clays. For a given value of porosity, the formation factor is larger for natural clay than for model clays. These differences can be due to the differences in local properties such as surface charge, which is less important for natural clay than for models. It has been also pointed out that |ζ | increases with concentration for natural clay and decreases for model clays.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

15. 16. FIG. 14. Normalized permeability as a function of porosity. Data are for: clay powder (s), montmorillonite (e), mica (∗). Numerical results obtained for various packings (h) (see (11)). The dashed line represents the fitting by Eq. [33] with n = 4.91.

17. 18.

Lee, S. Y., and Tank, R. W., Appl. Clay Sci. 1, 145 (1985). Allen, C. C., and Wood, M. Y., Appl. Clay Sci. 3, 11 (1988). Bucher, F., and M¨onler-Vonmoos, M., Appl. Clay Sci. 4, 157 (1989). Weiss, A., Appl. Clay Sci. 4, 193 (1989). Felix, B., Lebon, P., Miguez, R., and Plas, F., Eng. Geology 41, 30 (1996). Adler, P. M., “Porous Media: Geometry and Transport.” Butterworth– Heinemann, Stoneham, 1992. Dullien, F. A. L., “Porous Media: Fluid Transport and Pore Structure.” Academic Press, New York, 1992. Brace, W. F., Int. J. Rock. Mech. Miner. Sci. Geomech. [Abstract] 17, 241 (1980). Jacquin, C. G., Rev. Inst. Franc¸ais P´etrole 20, 1475 (1965). Dewhurst, D. N., Yang, Y., and Aplin, A. C., Geol. Soc. London 158, 23 (1999). Coelho, D., Thovert, J.-F., and Adler, P. M., Phys. Rev. E 55, 1956 (1997). Jacquin, C. G., Rev. Inst. Franc¸ais P´etrole 19, 921 (1964). Abramson, H. A., Moyer, L. S., and Gorin, M. H., “Electrophoresis in Proteins.” Reinhold; New York, 1942. Sen´ee, J., “Influence des macromol´ecules et des particules sur la stabilit´e des mousses de solutions hydroalcooliques complexes: Application aux vins de Champagne.” Th`ese de Doctorat de INPL, Lorraine, 1996. Hunter, R. J., “Zeta Potential in Colloid Science.” Academic Press, New York, 1988. Probstein, R. F., “Physicochemical Hydrodynamics: An Introduction,” 2nd ed. Wiley–Interscience, New York, 1994. Onoda, G. Y., and Liniger, E. G., Phys. Rev. Lett. 64, 2727 (1990). Kim, J. H., Ochoa, J. A., and Whitaker, S., Transport Porous Media 2, 327 (1987).