GAS AND LIQUID TRANSPORT IN POROUS MEDIA WITH FRACTAL GEOMETRY* O. Yu. Dinariev

UDC 532.546:515.124

The applications of fractal geometry in the theory of flow through porous media are reviewed. The last five years have been marked by the vigorous development of applications of the theory of fractals -- sets with noninteger dimensionatity - - in physics, mechanics, and chemical kinetics. The expansion of the sphere of applications has often considerably outstripped progress in the corresponding branches of mathematics. For this reason investigators, not having an adequate mathematical apparatus at their disposal, frequently obtain their results at the semi-intuitive level. This situation can also be attributed to the fact that so far the group of natural phenomena that can be described by means of fractal geometry has not been precisely defined, the class of corresponding concepts has not been completely formulated and all the related problems which arise in mathematics have not been recognized. At present, the process of accumulating general ideas, concepts and approaches is continuing. Accordingly, the new fractal models of natural objects being constructed today may generate new independent branches of science tomorrow. 1. Fractals - - sets with a noninteger Hausdorff--Besicovitch dimension -- have been known in mathematics for a long time. However, it is only comparatively recently that they began to be widely used for describing various natural objects, phenomena and processes, mainly thanks to studies [1--3]. These studies are very simple from the standpoint of the mathematical apparatus employed, but contain a large number of concrete examples taken from geography, biology, cosmology, mechanics, and physics to illustrate the use of the concept of a fractal and fractal dimensionality. The publication of studies [1--3] has led to the appearance of an enormous number of papers on the applications of fractal theory in a wide variety of branches of science, so many in fact that they are already difficult to list. An excellent review of the situation in fractal theory up to t987 is given in [4]. The concept of dimensionality arose in topology, and until recently the dimensionality of a set was understood to be precisely its topological dimensionality. To determine the topological dimensionality d r of a given set it is sufficient for there to be a topology in it, i.e., for example, a closure operation satisfying certain axioms [5]. The topological dimensionality d r can take only integer values d r = - 1, 0, 1, 2 .... , the value d r = - 1 being characteristic of the empty set. Determining the Hausdorff--Besicovitch dimension requires at least the presence of a metric structure. Let S be some compact set, and C a finite covering of the set S with open balls of different radii. For any ball B E C let r(B) be the radius of the ball B. We assume that by definition r(C)=max{r(B)!BE C}. We then determine for any non-negative number d the finite quantity

( c , d) = y~ r ~ (B)

(1.11

BEC

For any positive number e we minimize the sums (1.1) over all coverings C such that r(C)_ d . ) ,

+ ~(d
The quantity d H is called the Hausdorff Besicovitch dimension of the set S or the fractal dimensionality. For Euclidean "Based on paper read at the Seventh Congress on Theoretical and Applied Mechanics, Moscow, August i991. Presented by V. N. Nikolaevskii. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 101-109, September-October, 1992. Original article submitted January 21, 1992. 9 682

0015-4628/92/2705-0682512.50 9 1993 Plenum Publishing Corporation

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--

S2

Fig 1

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st Fig 2

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spaces and smooth manifolds d H coincides with the ordinary topological dimensionality. However, there are m,'my examples [ I - - 3 ] o f the Hausdorff--Besicovitch dimension o f a set exceeding its topological dimensionality. In this case the set is called a fractal set or simply a fractal. If, as usual in applications, the set S is embedded in a Euclidean space R D, then the rougher "practical" definition of fractal dimensionality is used:

in

(e)

d H = lira, ~-.o m ( l / e )

(1.2)

where N(e) is the minimum number o f D-cubes with edge g with which it is possible to cover the set S. In determining the fractal dimensionality it is not stipulated that the limit o f (~dH(S, g) as g--K) be finite and nonzero. If it turns out that this is in fact the case, then the passage to the limit can be used to specify the Hausdorff measure #H for S [6]. On the basis o f this measure, in accordance with a standard procedure [7], it is possible to develop a theory o f integration on S. Thus, i f f is a continuous function on S, then it is possible to determine the integral o f f from the formula ffd/z H= S

limYf(x 3~H($3, t~O

S=

L1 S~

i

where S is divided into a finite number of integrable subsets S i with diameter not exceeding e, and x i is an arbitrarily chosen point in S i. Below we present some simple examples o f fractal sets. Cantor Set (see Fig. 1). This set is constructed by means o f the following recurrence procedure. Let S o = J 0 , 1] be a segment o f unit length. F r o m the middle of this segment we cut out an open interval of length 1/3, obtaining the set S 1 =[0, 1/3] U [2/3, 1]. In the next step we cut out an interval o f length (1/3) 2 from the middle o f each o f the two segments obtained, and so on. W e obtain a decreasing sequence of closed sets So D S l D S 2 D . . . . By definition the Cantor set S can be obtained as the intersection o f the sets S n. The topological dimensionality of this set is equal to zero. In order to determine its fractaI dimensionality we cover the set S with segments o f variable length e n =(1/3)", n = 0 , l, 2 . . . . Then it is obvious that N(en) = 2 n. Substituting these relations in (1.2) and letting n tend to + 0o we obtain d H = l n 2/In 3. In this case, as it is easy to see: dz
(I .3)

Koch Curve (Fie. 2). The set is constructed by means of a recurrence procedure on a Euclidean plane. SO is a unit segment of a straight line, and S ! is obtained as a result o f cutting out from the middle of SO an open interval o f length 1/3 and replacing it with two connected segments o f length 1/3. In the next step the same operation is repeated for each of the four segments making up S1, and so on. A sequence o f polygonal lines SO, S 1.... is obtained. In the limit we have a set S with topological dimensionality d z= 1 called a Koch curve. In order to determine the fractal dimension we measure the length o f S by means of a ruler with a variable scale r n, n = 0 , 1 . . . . . Then N(en) = 4 n. From (1.2) we have rill=In 4/ln 3. In the case in question D = 2 and, as before, the inequality (1.3) holds. Sierpinski Gasket (Fig. 3). Let SO be a set o f points within an equilateral triangle with unit side. In this triangle we draw the medians. These divide SO into four equilateral triangles with sides o f length 1/2. We cut out from S O the points that fall in the inner triangle. We denote the set obtained by S1- In the next step the same operation is performed on each of the three triangles that make up S 1. As a result, we have a decreasing sequence o f closed sets in the Euclidean plane S0 D S 1D S2 D . . . . We assume that, by definition, S is the intersection o f the sets S n. Then S is a set with the topological dimensionatity d r = 1. In connection with the calculation o f the tYactal dimensionality we note that in relation (1.2) it is possible to consider coverings not only with cubes but also with spheres, simplexes, etc. - - any similar figures with characteristic dimension ~. In

683

80

sl Fig 3

az

s/ Fig 4 this case, as before, the result (1.2)will hold. We will consider coverings o f S with equilateral triangles having sides of length en=(1/2) n, n = l , 2 . . . . . It is easy to see that N(en) = 3 n. From (1.2) we obtain dH=ln 3/ln 2, and the inequality (1.3) will again be valid. Sieminski S~on~e (Fig. 4). Let SO be a unit cube. S 1 is obtained from So by cutting out square holes with sides of length 1/3. Then the same procedure is applied to the 20 cubes making up S l, and so on. S is the intersection o f the sets S n. This is a three-dimensional fractal with topological dimensionality d 7 2 and fractal dimensionality d H = l n 20/ln 3. The inequality (1.3) holds. All these fractats are obtained as a result of applying in each step a certain uniform procedure (generator), specified by a certain geometric figure. Accordingly, these fractals are called regular geometric fractals. Natural objects can rarely be described by regular geometric fractals. However, for these fractals it is possible to carry out exact calculations o f practical interest. Moreover, by modifying the fractal construction procedure (alternating two or more generators) it is possible to construct more realistic objects. We note that when inequalities (1.3) are satisfied, the fractal has a zero Lebesgue measure in the covering space. A study o f the above examples quickly shows that all these fractals are locally self-similar. In general, the local selfsimilarity property is very common among regular geometric fractals and is sometimes included in the definition o f a fractal. In the case o f random or stochastic fractals, which do not possess local self-similarity in the strict sense, it is assumed that selfsimilarity is realized in the generalized average sense. Mathematically, local self-similarity means the following. For a fractal S there exists some number X (0 < X < 1) and some everywhere dense subset SX, called the set of ),-centers, such that for any point a E S x there exists a neighborhood of the point a in 5' that is transformed into itself in the similarity mapping x~ ~.(x-

a) + ,~

For a Cantor set X= 1/3, and S x is the set of all the ends of segments. For the Koch curve X= 1/3, and S x is the set of all the vertices on the curve. For the fractats corresponding to the last two examples h = 1/3, and SX, is the set of vertices of the triangles and cubes, respectively. The set of h-centers does not exhaust the entire fractal. In [8] it was shown that the points of the fractal that are not Xcenters also form an everywhere dense set. The existence of X-centers makes it possible to define differentiation on a fractal. In fact, let f be some function on the fractal and a a ),-center and let x be some point from a fairly small neighborhood of the point a. Then for a > 0 it is possible to define the fractional derivative o f f in the direction h = x - a as the limit (if it exists) LrJ (a) = lira f (a + .a."h)

-

We introduce the notion o f the chemical dimensionality of a fractal.

684

f (a)

Let ain and afin be two points on the fractal, and let Le(ain, afin) be the length of the shortest polygonal line connecting ain and afin such that the vertices o f the polygonal line belong to the fractal, while the segments of the polygonal line do not exceed e. If as g--,0 L, (ai,, a f i , ) = const el-at then the quantity d c is called the chemical dimensionality of the fractal. It is easy to see that for a Cantor set the chemical dimensionality cannot be definite, for the Koch curve d c = d t t = l n 4/ln 3, and for the other examples dc= 1. As the following lemma shows, in the definition (1.4) it makes sense to use o~<_dc. L e m m a . Let f b e a function on the fractal S for which the upper bound of the ratios If(x) - f(y) l/[Ix - y]t = is bounded by some constant. Then if ce>dc, f = c o n s t . In fact, we will construct a polygonal line with vertices a i on a fractal with segment lengths not greater than e and with extreme points a 0 = x and a n =y. Then n-I

If(x) - f ( y ) l

_< ~, I f(a~+,) - f ( @

! <- const ~~-dC

If oe > de, then from this as g--~0 we obtain f ( x ) = f ( y ) . In Euclidean spaces d c = 1 and it makes sense to consider derivatives (1.4) with c~ _< 1. For fractals with nontrivial chemical dimensionality the possibilities of differential calculus are greater. In addition to the dimensionalities d and dc, for fractals it is possible to determine many other dimensionalities. We note, for example, that f o r / 3 > 0 when we integrate over the fractal the function r ~, where r is the distance from the chosen point on the fractal, we obtain the asymptotic form: f rad/xH - const.R v+dc6}

(1.5)

r<_R

Here, d(0)=dH, and when/3 > 0 in the general case expression (1.5) defines a whole family o f dimensional[ties d(jS). 2. When fractal geometry is used to describe porous materials, in principle the following models are possible: 1) the fractaI is the pore space, 2) the fractal is the rock skeleton, 3) the fractal is the surface of the rock skeleton, 4) the fractal is a system of fractures in the porous matrix. Because the fractal has a zero Lebesgue measure in the covering space, the model 1 can be used for describing materials with almost zero porosity, and model 2 for describing materials with a porosity almost equal to unity. The equality o f Lebesgue measure of the fractal to zero often prompts the question how is it possible to use such a mathematical image to describe natural objects, when all bodies in nature have nonzero volume? Those who ask this question forget that mechanics is always concerned with models which, to some extent, reflect reality, but never coincide with it. Certainly, natural objects can be considered fractal only on a certain scale interval and subject to certain rough approximations. If we are to apply models 1--4 to real porous materials, we must be able to determine the corresponding fractal dimensional[ties. There are direct and indirect methods of determining d H. The direct methods involve the direct experimental study of the geometric structure of the object. The indirect methods involve the investigation o f the processes taking place in the object (diffusion, flow in porous media, electrical conduction, etc.), during which certain model equations with phenomenological coefficients, are derived and solved, the analytic or numerical solutions obtained are compared with the experimental data and the unknown parameters are chosen to give the best fit between theory and experiment. Extensive possibilities of determining d H by direct experiment are offered by the analysis o f the scattering of different types of radiation (see the reviews in [9, I0]). In principle, it is possible to scan natural objects with electromagnetic radiation (on different intervals of the spectrum), electrons and neutrons. This method is based on the simple fact that when a wave with wave vector k and length L is scattered on a fractal S the scattering amplitude A of the wave does not depend on the details of the fractal structure on scales much smaller than L. Therefore a given scattering amplitude will coincide with the amplitude for scattering on an object S(L) obtained from S by smoothing on scales smaller than L: A(k, S ) = A ( k , S ( L ) )

(2.[)

If we calculate the differential scattering cross section in the direction k / ! k l , then because of relation (2. i) we obtain the characteristic dependence on the wave frequency c~

685

cr~const + const J where t3 depends on d H. Thus, by measuring the cross section cr it is possible to find d H. Another method o f directly determining fractal dimensionality is based on the adsorption effect. If a given fractal surface (model 3) adsorbs molecules of different substances with different characteristic dimensions l (for example, hydrocarbons), then the number o f molecules in the adsorbed monomolecular layer will depend on l in accordance with a p o w e r law: N(/)-const l-'~ Finally, it is possible to determine the dimensionality from the pore or grain distribution [11]. Thus, let the fractal be a pore space (model 1) and let p=p(1) be the rock grain density distribution per unit volume with respect to the characteristic dimensions l. We assume that it is possible so to define the dimension l that the quantity

M = f Pp (l) dl

(2.2)

0

is equal to the relative volume occupied by the rock. By assumption M = 1. For small I let it be possible to derive the asymptotic form:

p (l) ~ pol-"

(2.3)

For the integral (2.2) to converge it is necessary that the inequality ~ < 4 be satisfied. The volume corresponding to the pore space, if it is covered by standard figures with dimensions 6, is: V(6)=po64-~/(4-oO. F r o m this we find N(6) =p0~31-cq(4-a) and then, from (1.2), we obtain d H = a - 1 . Strictly speaking, the real grain distribution cannot have the asymptotic form (2.3) with c~> I. For fairly small l the function p(/) will be equal to zero. Therefore, it is possible to speak of the asymptotic form (2.3) and, consequently, the fractal geometry of the pore space only on a certain scale interval: /min'~l,~/max, where /rain is the characteristic dimension of the smallest grain, and/max is the characteristic dimension of the largest grain. It is possible to speak of the fractal geometry of the pore space with greatest justification when by origin the pore space is of the fractured type (model 4), since there is good reason to assume that fracture and destructive processes lead to fractal structures [12]. Now let the fractal be the rock skeleton (model 2). It is then possible to consider the pore distribution with respect to characteristic dimensions p =p(/) and formally repeat the previous reasoning. The end result is again the expression dtt=c~- 1. Thus, in principle, it is possible to determine the fractal dimensionality by direct experiment. For problems of flow through porous media it is not enough to know the dimensionality and additional phenomenological parameters characterizing the finer properties of the porous material must be introduced. 3. In studying gas and liquid transport in porous materials with fractal geometry it is possible to anticipate especially interesting results when the space through which the fluid moves possesses a fractal structure (models 1 and 4 or model 3 in the case of film flow). In this case two different approaches are possible. Firstly, it is possible to investigate the motion of the fluids in the fractal capillary networks [13]. In this way a number of exact results for regular geometric fractals can be obtained. A shortcoming of this approach is the fact that regular fractal capillary networks differ very sharply from real pore channels. Secondly, it is possible to construct phenomenological models, starting from general information concerning the fractal structure of the pore space, on scales considerably exceeding the dimensions of the detailed features of the fractal structure [ 14]. In this case it is natural to base the theory on the law of mass conservation for the fraetal d

f rnpdlz, = - f L d ~ z ?7 -v Z

(3.1)

Here, O is the mass density of the fluid, V is the part of the fractal bounded by the surface E (on the fractal), n is the normal to I2, Jn is the mass flux across E, c/~H is the Hausdorff measure, d/z1; is the induced measure on E, and m is an analog of the porosity. In order tbr the quantity

f mPd~s v

to have the dimensionality of mass, taking into account the fact that [d~H] =L dH, [p] =ML -3, it is necessary for the quantity m to have the dimensionality [m] = L ~, where a = D - d H is the space defect. When a = 0 the quantity m goes over into the usual porosity. 686

How we should establish the expression for j, i.e., how we should generalize Darcy's law for a fractal? If gravitation is neglected, it seems reasonable to postulate

], = _ k pD~zp

(3 m2~

H e r e . . is the shear viscosity, and k is a quantity_ analogous to the permeability, which in order to ensure the correct dtmenmonahty Of Jn must have the dlmenslonahty [k] = L . When d c = 1 expression (3.2) leads to Darcy's law in its usual 9

.

"

.

.

.

.

"

9

9

~

+

d

c

+

'

l

form. We will consider cylindrically symmetric ( D = 2 ) or spherically symmetric ( D = 3 ) flow, when all the functions depend on and r, where r is the distance from the center o f symmetry. Strictly speaking, fractal porous media do not allow a continuous group of symmetries; however, it is possible to anticipate the existence of p and p fields symmetric in a certain average sense on fairly large scales. From Eq. (3.1) there follows

f o <~r<--rl

r=r 0

f rmr I

which, when (3.2) is taken into account, leads to equation

m Op = r_(eH_o 0 k/~_e~p 0-7 0-7 (if~r )

(3.3)

For this equation it is possible to obtain a number of analytic solutions, which can be compared with experiment. Thus, for constant k and an isothermal process (p =PO),/z=/z(p)) from (3.3) it is possible to find the steady-state solution implicit in O P = C,r,_an+a~ + C2,

P = P (tg) = f

d~

Self-similar solutions o f Eq. (3.3) can be found by standard means. We now turn to the model of a fractured porous medium with fractal fracture geometry [14]. In this case Eq. (3.t) must be replaced by the system

d f m, p,d/.t~ = - f j, dtz z + f qd/.zn, -d7

d f m~o2dct~ = - f qd/z~ -dr

V

(3.4)

v

Here, Pl is the fluid density in the fractures, P2 is the fluid density in the matrix, m ! is a parameter with the dimensionality L ~ characterizing the fractal fracture system, m2_ is the porosity of the matrix, and q is the mass flux from the porous blocks to the fractures. We will consider the case of symmetric flow Pi =Pi( r, r), i = 1, 2. On going over from integral equations (3.4) to differential equations it is necessary to specify, the expression for q. It is natural to give this expression the form:

q = -~l (PzPz - PIPl) re where .~ is a phenomenological coefficient, Pl and P2 are the pressures in the fractures and the matrix, respectively, gl =#(Pt) is the shear viscosity, and I/ is a dimensionless parameter characterizing the details of the fractal structure. Then, for constant parameters mi, m2_, and k from system (3.4) we obtain the system of differential equations

Op, _ k r_(aH_o O ( tzT,ra~_d~p, Op, ~ __if__ at m, 0-7 Or / +/x,m[ (PaPa - P , P , ) re

(3.5) Op~

~ -

- pap2)

=

,(

d /

When "/=0, 6 = 0 , and de= 1 the model (3.5) reduces to the classical model of a fractured porous medium [15, 16]. In [14] the case in which 3,= - e , dc= 1 was investigated. In a number of simple situations the system (3.5) admits exact solutions, which make it possible to determine the phenomenologicai parameters of the model experimentally [14]. In particular, the solution of the problem of the pressure recovery curve for a liquid in the model with T = - e , dc= 1 gives the asymptotic expression

687

Ap -- C , f - Cz + C3t2~- ', T = (2 - dH)/(4 - d,r Therefore nonstationary well investigations could make it possible to find the fractal dimensionality o f the fractures in the near-wall zone. Equations (3.3) and (3.5) make it possible to solve substantive problems of the theory o f flow through porous media with allowance for the fractal geometry (models 1 and 4). Various improvements and modifications o f the theory proposed are possible, since the entire theory o f fractals in general continues to be developed. A certain time will be required to recognize all the consequences o f the fractal concept, compare them with experiment, and determine the values o f the phenomenological parameters. There is no doubt that this activity is o f great scientific and practical importance. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8, 9. 10. 11. 12. 13. 14. 15. 16.

688

B.B. Mandelbrot, Les Objers Fracrals: Fmvne, Hasard et Dimension, Flammarion, Paris (1975). B.B. Mandelbrot, Fracmls: Form, Chance and Dimension, Freeman, San Francisco (1977). B.B. Mandelbrot, The Fracml Geometly of Nature, Freeman, New York (1983). E. Feder, Fractals [Russian translation[, Mir, Moscow (1991). K. Kuratowski, Topology, Vol. 1, Academic Press, New York (1966). C.A. Rogers, HausdolffMeasures, Cambridge University Press, Cambridge (1970). N. Bourbaki, Etemenrs of Mathematics, Vol. 6. Measures and Integration of Measures, Addison-Wesley, New York (1965)o O. Yu. Dinariev and A. B. Mosolov, ~The self-similarity property for subsets Rn, ~ in: Modern Analysis and its Applications [in Russian], Naukova Dumka, Kiev (1989), p. 44. L. Pietronero and E. Tosatti (eds.), Fracrals in Physics, North-Holland (1986). B.M. Smirnov, Physics of Fractal Clusters [in Russianl, Nauka, Moscow (1991). A.B. Mosolov and O. Yu. Dinariev, ~Fractals, scales and the geometry of porous materials;" Zh. Tekh. Fiz., 58, No. 2, 233 (1988). A.B. Mosolov and O. Yu. Dinariev, "Self-similarity and the t'ractal geometry of fracture," Prob. Prochn., No. 1, 3 (1988). P.M. Adler, ~Transport processes in fractals," Inr. J. Multiphase Flow, 11, No. 1, 91, No. 2, 213,241 (1985). O. Yu. Dinariev, "Flow in a fractured medium with fraetal fracture geometry," Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 5, 66 (1990). G.I. Bareublatt and Yu. P. Zhelmov, "Basic equations of the flow of homogeneous fluids in fractured rocks," Dokl. Akad. Nauk SSSR, 132, 545 (1960). G.I. Barenblatt, Yu. P. Zhehnov, and I. N. Kochina, ~Basic notions of the theory of flow of homogeneous fluids in fractured rocks," P1qkl. Mat. Mekh., 24, 852 (1960).