Fractal porous media

Transport in Porous Media 13: 41-78, 1993. @ 1993 KluwerAcademic Publishers. Printed in the Netherlands.

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Fractal Porous Media R M. ADLER and J.-E THOVERT Laboratoire d'A~rothermique du CNRS, 4, ter, Route des Gardes, 92190-Meudon, France

(Received: 24 March 1993) Abstract. The transport properties of continuousdeterministicfractals are reviewed.The methodof construction, the fractal dimension, and the major features of transport are summarized. Then the major single-phasetransports are addressed; attention is focused on the numerical results and on the analyticalargumentswhich may be used to derive these results in a simple way, wheneverit is possible. Key words: Fractal, permeability,conductivity,transports. 1. I n t r o d u c t i o n

It is certainly well known and generally now admitted that the concept of fractals is relevant to the study of real porous media, ever since the introduction of this concept by Mandelbrot [1 ]. In order to keep the size of this review within reasonable bounds, it is focused on the study of continuous deterministic fractals, i.e. of exactly self-similar sets. It may be useful for the newcomer to the field to detail what has been left out. First, the structures such as capillary networks are not addressed; the simplest example of these networks is the classical Sierpinski gasket; the major utility of these structures is to be amenable to analytical calculations, but they are not representative of the real world. Second, all the random structures are deliberately left out. The more classical example of these structures is a porous medium made of elementary solid cubes which are chosen at random with a given probability. Close to the percolation threshold, the medium is known to be fractal. Such structures present complete disorder for which many approximate tools have been devised. A third type of geometrical structure is the one of reconstructed media. Media with a given correlation function are generated by a now standard procedure. When this correlation function decays as a power law, the resulting medium can be considered as being fractal at the local scale. All these topics have been addressed in some of our previous reviews [2, 3, 4], which contain all the relevant references to the major contributions to the field. The following presentation is mostly based on our own work. It surveys all our contributions on deterministic fractals, including some recent ones which have never been published before. This presentation is organized as follows. In Section 2, the most classical continuous fractals are presented. The construc-

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P.M. ADLER AND J.-E THOVERT

tion scheme due to [5] is recalled, as well as a short introduction to the concept and to the determination of fractal dimension. Section 3 tries to give a unified approach to the description of porous media and to the analysis of transports. Porous media are viewed as materials with multiple scales. Depending upon the respective orders of magnitude of these scales, three basic situations are distinguished. The most important one occurs when the material is homogeneous at the large scale, but contains a fractal microstructure. In the second part of this section, the general properties of the transports are summarized for each basic situation. The rest of this paper deals with four particular transports of general interest: conduction, convection, Taylor dispersion, and deposition. This is always done in media which are fractal at the microscale and spatially periodic at the large scale. Our attention is focused on the presentation of the transport itself, on the major numerical results and on the derivation of renormalization arguments. This last topic is quite developed because it is the precise place where the continuous deterministic fractals are the most useful from a theoretical point of view. Conduction is addressed in Section 4 for two- and three-dimensional structures derived from the Cantor set. It is quite interesting that most macroscopic conductivities follow the classical Archie's law. i.e, a power law as a function of porosity. Simple renormalization arguments compare well with the data; for instance, the conductivity of suspension-like media can be obtained by considering them as dilute suspensions of particles in an equivalent medium at the previous generation. Convection is addressed in Section 5 for the same structures. The most interesting case from a theoretical viewpoint is one-dimensional flow in Sierpinski carpets. Numerical data, approximate renormalization arguments and exact derivation coincide. It is also the only case where permeability can be expressed as a function of porosity and fractal dimension. The situation is much worse for twoand three-dimensional flows where no satisfactory arguments could be found. Convection-diffusion processes are also known under the name of Taylor dispersion. They are addressed in Section 6 by two different means, i.e. by random walks and the method of moments. One-, two-, three-dimensional structures are analyzed. No renormalization argument can be derived. Finally, deposition is calculated in Section 7. Deposition in porous media and their subsequent plugging is a topic of very high industrial interest, since it occurs very frequently. Calculations were performed by random walks in the limit where the walls act as sinks. Only the example of the Menger sponge was worked out. It was shown that the larger pores are not plugged until their size becomes comparable with the smaller ones; then they are all plugged simultaneously. Some general remarks end up this paper in Section 8.

FRACTAL POROUS MEDIA

43

2. Self-Similar Sets Because of their regular character, self-similar sets are usually easier to understand than fractals. As a tentative definition (see [1]), a body, a shape or a mathematical entity is self-similar when each of its pieces is geometrically similar to the whole. Some classical examples of self-similar sets can be found in most references such as the Cantor set, the Koch curve, the Sierpinski gasket, etc. A general and simple mode of construction was devised by [5] through the action of a set of similitudes on geometric sets. A similitude S is a geometric transformation which can be defined as the composition of a rotation, a homothety (or a contraction with a ratio r < 1) and a translation by a vector t. Several similitudes can be combined in order to generate a fractal. Suppose that one has a set of similitudes {S/, i = 1 , . . . , M } and a compact set K0; then successive sets K1, / ( 2 , . . . can be obtained by application of the recursion relationship M

KN = U Si ( K N - 1 ) .

(1)

i=l

This formula means that all the similitudes are applied to the set KN-1 and that IQv is equal to the union of all these M 'reduced' sets. The index N is called the generation number. For instance, the Sierpinski carpet can be obtained by M =8 similitudes with a contraction ratio r = 1. Starting from a square which is the initial compact set Ko, one rearranges them into a new square as indicated in Figure la; this process can be iterated. The iteration of the recursion process is shown in the right side of Figure 1. It is precisely the result of an infinite number of iterations which is called the Sierpinski carpet. Of course, this set is invariant by successive applications of the elementary similitudes. This invariance provides us with an operational definition of the fractal set constructed by means of the Hutchinson's process; the fractal set I f is the only set invariant by the set of similitudes {Si, i = 1 , . . . , M } , i.e. M

If = U Si(K).

(2)

The invariant and the unique (irrespective of the starting set K0) characters of K can be rigorously proved (cf. [5]). Another classical example, the Ben Avraham and Havlin carpet, is displayed in Figure 1. The contraction ratio is equal to ~ and M =16. Three-dimensional examples are displayed in Figure 2. The fractal foam (denoted by FF) is a three-dimensional version of the Sierpinski carpet; the parameters for this foam are equal to M = 26, r = 89 The Menger sponge can be constructed as the fractal foam, but there are two ways to proceed depending upon the location

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P.M. ADLER AND J.-E THOVERT

Fig. 1. Two-dimensional deterministic fractals at construction state N =1 and N + 2 . The whole media are supposed to be composed of cells identical to these ones. (a): SC; (b): BAH; (c): BAH after removal of the lateral connections.

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FRACTAL POROUSMEDIA

1

D

D

DI'/j!I

D

D V

L I --

Lli--

/ Z

<

h

>

{'~/'y - <

h

>

X 9.

b.

Fig. 2. Three-dimensionaldeterministic fractals at construction stage N =2. The whole media are supposed to be composed of cells identical to these ones. (a): FF, the inner cubes correspond to the solid phase; (b): both phases are continuous. MS 1 is obtained when the liquid phase corresponds to LI, and MS2 in the opposite case.

of the liquid phase; this is illustrated in Figure 2 under the names MS1 and MS2. The parameters for these two structures are

MSI" MS2"

r= 89

M=20,

r = l,

M = 7.

(3)

Usually, the concept of dimension is defined by the number of independent vectors which can be found in a given space. A very different approach is the one corresponding to the similarity dimension Dsi. In order to give an intuitive feeling for this dimension, one can start with standard objects such as segments, squares and cubes. Such objects can be covered by N(r) reduced versions of themselves, where r is the contraction ratio. For these simple objects, N(r) is obviously equal to

N(r) = (1/r) d,

(4)

where d is the usual spatial dimension. This formula can be solved for d d = Log g ( r ) Log 1/ r '

(5)

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P. M, ADLER AND J.-E THOVERT

More generally, the similarity dimension of self-similar objects constructed from a set of M similitudes with the same contraction ratio r is defined as Log M D s i - Log 1 / r ' (6) since N(r) = M.

(7)

This definition can be applied to most of the self-similar sets which were introduced until now. The Sierpinski carpet is covered by M = 8 versions of itself reduced in the ratio r = 1. Hence, the similarity dimension of the Sierpinski carpet is equal to Dsi-

Log 2 Log 3 - 1.893.

(8)

3. Transports in Fractals We are now in a situation where we can systematize both our approach to the description of porous media and our analysis to the major transports which may occur in porous media (cf. also [2-4]). 3.1. POROUS MEDIA WITH MULTIPLE STRUCTURES Since all the real porous materials are random, we shall mostly insist on the combination of structures at different length scales. Historically, porous media were first modelled as structures which were statistically homogeneous, i.e. as structures which possess a translational symmetry (as opposed to fractals which possess a dilational symmetry); this is the reason why most of the anterior interest on porous media was focused on the former structures which are quite well sketched by spatially periodic structures. Now the fractal character of geological porous media is well established and we wish to give a few examples of 'theoretical' media which clearly display several symmetries (cf. [4]). Multifractals provide the right background for these structures, but we think that it is more pedagogical to display these multiple structures in an elementary way. Consider a porous medium; in general it can be characterized by its two extreme scales I and L, as illustrated in Figure 3. I is a representative length scale of the elementary particles which compose the solid phase; L corresponds to the external boundaries of the medium. It is assumed that I 1. Since the ratio K/h 2 is almost constant

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P.M. ADLER AND J.-E THOVERT

Fig. 10. Successive stages of the deposition processes in the Menger sponge with Nc =9. In (a-e), Pe= 1. The initial solid is in white and the deposit in grey. In (e), only the deposit is displayed, with, in chronological order; dark grey; dotted dark grey; medium grey; light grey. In (f), Pe=100. The final pore space is white and the deposit with the same colour convention.

f o r any N (cf. [14]), this m e a n s t h a t - D ~ / D d e p e n d s o n l y u p o n a m o d i f i e d P6clet

FRACTALPOROUS MEDIA

75

number P e l = EPe = ~ ,

(81)

where the interstitial velocity ~* is replaced by the seepage velocity ~r The curves relative to FF (Figure 8) are very similar to the ones relative to SC (Figure 7). For large P6clet numbers, they seem to diverge with increasing N. However, the sense of variation with N is a consequence of the choice of the length scale ! used to define Pe. If the size of the smallest solid inclusion A/3N was used in the P6clet number instead of that of the largest one h/3, this sense would be reversed. This is an illustration of the impossibility to define a proper length scale in fractal media, which precisely present an infinite spectrum of scales.

7. Deposition 7.1. GENERAL Let us briefly sketch the deposition studies which have been performed by [21]. The deposition process is calculated by means of random walks. The main lines of the procedure are as follows. The velocity field is first determined by solving the Stokes equations in the current state of the pore space. Then the trajectories of the numerous tracers are followed individually. Each time a particle hits a solid wall, it is assumed to stick there and the amount of such particles is recorded for every elementary area of the solid/liquid interface, i.e. for every face of every elementary cube lying on this interface. A liquid cube in contact with the solid phase may present several such faces. Whenever the cumulated quantity of matter deposited on the faces of a given cube reaches a prescribed number, this cube is converted to solid. The geometry is modified accordingly and the process is started again. The deposition rate in a given geometry is proportional to c and otherwise depends only upon the P6clet number. Hence, it is desirable to define a dimensionless time t*, which eliminates the numerical parameters and allows the comparison of various results. All the numerical data are presented in terms of this dimensionless time t* (cf. [21] for details). The real time t is expressed by t -

f L2

Cv D

t ~.

(82)

ev is the actual volumetric concentration, f is the solid concentration in the filled elementary cubes; for instance, in cubes filled by random packings of particles, f is equal to 0.64. L is a characteristic length of the porous medium; it is also involved in the calculation of t*. The calculations were performed on a Menger sponge MS2 which is composed of a bundle of interconnected channels of various sizes (Figure 2b). In view of the considerable amount of computation required for the simulations, only sponges with N = 2 could be fully investigated. Nevertheless, capillaries with very different

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E M. ADLER AND J.-E THOVERT

k/ko 1

....

k/ko 1

i

t

10

0 1

10-2 i

0

|

I

0.02

I

QO/-,tl

f,

|

0.2

i

0.6

i

1

b

Fig. 11. The normalized permeability as a function of the dimensionless time t* (a) and porosity (b) in the Menger sponge: (o), P e = l ; (e), Pe=10.

cross-sections are already present in this medium, and it should be sufficient to answer the basic question for this type of media, i.e. which of the smaller or larger channels are plugged first. Several intermediate stages of deposition in a Menger sponge with N = 2 are displayed in Figure 10 (a-c). The P~clet number, based on the width of the main channels, is Pc-- 1. The initial geometry is displayed in (a); in (b, c, d), the deposited matter is displayed at three successive times. Depositiorl first occurs in the central cavity, then on the walls of the three main channels. Again, this is typical of a diffusion-limited deposition which takes place mainly around the largest particle reservoirs. The smaller capillaries are plugged only at the end. The ultimate state of the medium, after complete plugging, is shown in (e). Figure 10f is the result of a similar simulation with Pc= 100. The deposit is in colour with the same convention. Deposition follows exactly the same pattern as with Pc= 1. This was expected since an increase of the P6clet number corresponds to an increase of the renewal of the brine in the regions where flow is the most intense, i.e. in the larger capillaries. This is confirmed by the evolution of permeability displayed in Figure 11. For both P6clet numbers, a strong permeability decrease is caused at first by a small amount of deposit. This corresponds to the covering of the internal surface of the main channels, until their cross-section becomes comparable to that of the smaller ones. Then all the capillaries are plugged at random. The following general conclusions can be drawn from the study of these deter-

FRACTALPOROUSMEDIA

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ministic configurations. At low P6clet numbers, deposition is limited by diffusion and takes place preferentially on the walls which limit the largest pores, which are reservoirs of particles. The narrower parts are preserved until the final steps of the process and the permeability diminishes relatively slowly. At high P~clet numbers, deposition occurs where convection is able to provide fresh brine. In the Menger sponge, this occurs in large channels. In any case, it occurs along the preferential paths of the convecting fluid. Thus, the permeability is significantly diminished by a small amount of deposited matter.

8. Concluding Remarks The fractal approach to real porous media has been proved useful in the last 10 years and this structure cannot be ignored anymore. Many data have been accumulated either experimentally or numerically. Many of them can be reasonably justified by approximate renormalization arguments similar to some presented here. A significant effort should be devoted to strengthen the theoretical aspect of these arguments which are very often derived on an ad hoc basis. The continuous deterministic fractals offer a good opportunity to the development of this theoretical analysis. M a n y other routes are also opened to further investigations, such as the analysis of random structures.

References 1. Mandelbrot,B. B., 1982, The Fractal Geometry of Nature, Freeman, San Francisco. 2. Adler,P. M. 1989, Flow in porous media, in D. Avnir. (ed.), The FractaIApproach to Heterogeneous Chemistry, J. Wiley, New York, pp. 341-359. 3. Adler,P. M., 1990, Fractal porous media, in J. Bear and M. Y. Corapcioglu (eds.), Transports Processes in Porous Media, Kluwer Acad. Pubs., Dordrecht, pp. 723-743. 4. Adler,P. M., 1992, Porous Media: Geometry and Transports, Butterworth/Heinemann,Stoneham, MA. 5. Hutchinson, J. E., 1981, Fractals and self-similarity,Indiana Univ. Math. J. 30, 713-747. 6. Vignes-Adler, M., Adler, P. M. and Gougat, P., 1987, Transport processes along fractals. The Cantor-Taylorbrush, Phys. Chem. Hydrodynamics 8,401-422. 7. Adler,P. M., 1987, Hydrodynamic properties of fractal flocs, Faraday Discussion Chem. Soc. 83, 145-153. 8. Ben Avraham, D. and Havlin, S., 1983, Exact fractals with adjustable fractal and fracton dimensionality,J. Phys. A16, L559-L563. 9. Thovert, J.-F., Wary, E, and Adler, P. M., 1990, Thermal conductivity of random media and regular fractals, J. AppL Phys., 68, 3872-3883. 10. Miller,M. N., 1969, Bounds of effective electrical, thermal and magnetic properties of heterogeneous materials, J. Math. Phys. 10, 1988-2004. 11. Milton, G. W., 1982, Bounds of the elastic and transport properties of two-component composites, J. Mech. Phys. Solids 30, 177-191. 12. Adler,P. M., 1986, Transport processes in fractals. VI, Stokes flow through Sierpinski carpets, Phys. Fluids, 29, 15-22. 13. Adler, P. M., 1988, Fractal porous media, III, Transversal Stokes flow through random and Sierpinski carpets, Transport in Porous Media 3, 185-198.

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14. LemaRre, R. and Adler, P. M., 1990, Fractal porous media, IV, Three dimensional Stokes flow through random media and regular fractals, Transport in Porous Media 5, 325-340. 15. Gefen, Y., Mandelbrot, B. B. and Aharony, A., 1980, Critical phenomena on fractal lattices, Phys. Rev. Lett. 45, 855-858. 16. Gefen, Y., Meir, Y., Mandelbrot, B. B. and Aharony, A., 1983, Geometric implementation of hypercubic lattices with non-integer dimensionafity by use of low lacunarity fractal lattices, Phys. Rev. Lett. 50, 145-148. 17. Jacquin, C. G. and Adler, E M., 1987, Fractal porous media II, Geometry of porous geological structures, Transport in Porous Media 2, 571-596. 18. Delannay, R., 1990, Dispersion de Taylor en milieux poreuxfractals, PhD thesis, Universit6 de Paris-VI. 19. Salles, J., Prevors, L., Delannay, R., Thovert, J.-E, Auriault, J.-L and Adler, E M., 1993, Taylor dispersion in random media and regular fractals, Phys. Fluids, in press. 20. Brenner, H., 1980, Dispersion resulting from flow through spatially periodic porous media, PhiL Trans. Roy. Soc., London, A297, 81-133. 21. Salles, J., Thovert, J. E, Adler, E M., 1993, Deposition and clogging, Chem. Eng. Sci., 48, 2839-2858.