Soil Science Society of America Journal 63:1523-1530 (1999)
© 1999 Soil Science Society of America
DIVISION S-1-SOIL PHYSICS
Tortuosity of Crack Networks in Swelling Clay Soils
V.Y. Chertkova and
I. Ravinaa
a Faculty of Agricultural Engineering, Technion, Haifa 32000, Israel
agvictor{at}techunix.technion.ac.il
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ABSTRACT
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Tortuosity of the crack network in a swelling clay soil determines the actual mean length of flow through the cracks and is an important factor influencing the hydraulic properties of the soil. In the literature there are works related only to tortuosity of the pore system of a nonswelling-soil matrix. The approach to estimate tortuosity proposed here is based on an established model of crack network geometry in swelling clay soils. Dependency of the tortuosity on the connectedness of a statistically isotropic crack network is derived from the fact that connected cracks outline fragments (peds) and are their boundaries. The second parameter that characterizes the crack network, the mean spacing between cracks, does not influence the tortuosity. The estimated two- and three-dimensional tortuosities vary from 1.5 to 2.2 and from 1.4 to 3.25, respectively, when connectedness decreases from unity to zero (for instance, with increasing soil depth). A proposed method for processing a two-dimensional image of crack networks enabled us to estimate experimental two- and three-dimensional tortuosities and connectedness of an assumedly isotropic crack network. This direct estimation from two-dimensional images is practical for sufficiently large values of connectedness. Data of seventeen different two-dimensional images (available in the literature) were used to show the applicability of the proposed dependencies of tortuosity on crack connectedness in the range 0.5 to1.
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INTRODUCTION
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HYDRAULIC PROPERTIES OF SWELLING CLAY SOILS are controlled to a large extent by the geometries of their crack networks. One characteristic of a crack network is tortuosity. It determines the actual mean length of flow through the cracks in a layer. Tortuosity of the crack network is a result of merging, branching, and curving of cracks, and has a meaning only for through-connected cracks that create continuous paths for water flow. In the literature there are a number of works discussing the tortuosity of a pore system in a nonswelling-soil matrix on the basis of a fractal approach (e.g., Tyler and Wheatcraft, 1989; Shepard, 1993; Bird et al., 1996). Recently this approach was also applied to characterize soil cracks (Preston et al., 1997), but not tortuosity. We have not found works dealing with tortuosity of the crack network in swelling clay soils. The objective of this paper is to suggest and validate a model for estimating the mean tortuosities of two- and three-dimensional crack networks. The approach is based on the model of Chertkov and Ravina (1998), which describes geometrical characteristics of the crack network as related to crack widths.
To define the mean tortuosity of a planar crack network, T2, we regard a crack network within a given square that includes a large number of cracks (Fig. 1)
. Cracks of type 2 (Fig. 1) inside the square are connected with each other and form continuous through-paths between the sides of the square. The product of the intersection number of such cracks with the lower side of the square (Fig. 1) by the height of the square, H2, gives an imaginary length of through-cracks inside the square. The imaginary length corresponds to a hypothetical situation in which tortuosity T2 is equal to unity. The actual mean tortuosity of a two-dimensional crack network, T2, is defined as a ratio of the actual length of the through-connected cracks within the square (Fig. 1) to the imaginary length. Procedures for estimating the mean tortuosity, on the basis of real two-dimensional images of crack networks, will be presented in the following.
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Fig. 1 A schematic two-dimensional crack network image. The numbers indicate different types of separate cracks: 1, isolated cracks; 2, through-connected cracks; and 3, locally connected cracks
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The mean tortuosity of a spatial crack network, T3, is similarly defined in a given cube that includes many cracks. We account only for through-connected cracks that create continuous paths between the faces of the cube. We define the product of the cube height, H3, and the total length, Ltc, of the traces of the through-connected cracks on the lower face of the cube as an imaginary surface area of the through-connected cracks inside the cube. The imaginary area corresponds to a hypothetical situation of vertical through-cracks with tortuosity equal to unity. The actual mean tortuosity of a spatial crack network, T3, is defined as the ratio of the actual surface area of the through-connected cracks within the cube to the imaginary surface area. At present we can not theoretically estimate the tortuosity of an anisotropic shrinkage-crack network. We are, therefore, grounded in the model of a statistically homogeneous and isotropic crack network (Chertkov and Ravina, 1998). Such a crack network is characterized by a dimensionless crack network connectedness, c, which is the ratio of the number of connected cracks to the total number of cracks, and by a parameter, d, which is the average spacing between crack intersections with a straight line (Chertkov, 1986, 1995a, 1995b; Chertkov and Ravina, 1998). Statistical homogeneity of a crack network means that in all parts of the volume the values of c and d are the same. One may consider a crack network in a sufficiently small volume, as well as in any of its cross-sections, as statistically homogeneous. Statistical isotropy of a crack network means that (i) the average spacing, d, between neighboring intersections of cracks with a straight line is similar at all orientations of the line; and that (ii) connectedness, c, of a planar crack network is similar at all orientations of the plane in the volume. Results of the application of the statistically homogeneous and isotropic crack network model to estimating the width and volume of vertical shrinkage cracks (Chertkov and Ravina, 1998) are suggestive and justify using the model for estimating the shrinkage-crack network tortuosity.
According to definitions of the average crack spacing, statistical homogeneity, and isotropy, d parameters of planar and spatial crack networks are the same in the statistically homogeneous and isotropic case. To regard relations between connectedness in this case, c3 and c2 of spatial and planar crack networks, respectively, it is sufficient to deal with cracks in a volume that have the form of a layer of thickness 
d. It is intuitively clear that an overwhelming majority of the cracks in the layer will intersect its surfaces. Therefore, accounting for the definition of crack network connectedness (Chertkov and Ravina, 1998) one may assume that c2 
c3 
c. To estimate a loss of accuracy from the use of the model of isotropic and homogeneous cracking and the assumption c2 c3, we must compare consequences of these approximations with those of a more exact model. However, at present a model suitable for such comparison is absent. In the following, predictions based on the model of Chertkov and Ravina (1998) will be compared with experimental data.
Thus, mean tortuosities T2 and T3 can depend only on the two characteristic parameters of a crack network, c and d. By definition, tortuosities T3 and T2 are ratios of the specific surface area of through-connected cracks in a volume (per unit volume) to the specific length of their traces at a cross-sectional plane (per unit area), and of the latter value to the specific number of their intersections with a straight line (per unit length) at the cross-sectional plane, respectively. According to the physical meaning of through-connected cracks (of Type 2 in Fig. 1) in an isotropic case, if they are in a volume (irrespective of their specific number), their traces are at cross-sectional planes as are their intersections with straight lines at the planes. Therefore, in decreasing connectedness c (a ratio of the number of cracks of Types 2 and 3 in Fig. 1 to the total number of cracks) from 1 to 0, the specific surface area of through-connected cracks, the specific lengths of their traces, and the specific number of their intersections with a straight line are small values of the same order of magnitude. Hence, the tortuosities T2 and T3 must tend to finite limits when connectedness c tends to zero.
Actually, in clay soil crack-concentration decreases with increasing depth (we can speak only about local homogeneity; i.e., in a sufficiently small volume). As a result, the parameters d and c vary with soil depth (Chertkov and Ravina, 1998). These variations determine corresponding dependencies of the mean tortuosities T2 and T3 on soil depth z.
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Theory
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In computation of the mean tortuosity of two-dimensional crack network T2, we consider a square of unit area. We define L1 as the mean specific number of through-connected cracks that intersect a straight line of unit length and L2 as the mean specific length of through-connected cracks (per unit area of the network under study). Then, according to the definition of the mean tortuosity T2
 | (1) |
The key point in the approach to estimate the values of L2 and L1 is that connected cracks outline, or nearly outline, fragments (peds) and are their boundaries. Consequently, the values of L2 and L1 may be estimated from the fragment–dimension distributions. These distributions are determined by cumulative fraction f(x) of area of (planar) fragments of a dimension <x (in case of L2), or by the cumulative fraction of length f(x) of spacings <(x) between intersections of fragments with a line (in case of L1) (Chertkov, 1986, 1995a; Chertkov and Ravina, 1998), given by
 | (2) |
where I (x) is the average cracking on a plane, such that I2 (x) (i.e., the average number of cracks of a dimension x in an area of the same dimension on the plane) is equal to
 | (3) |
or the average cracking along a line, as I1 (x) (i.e., the average number of cracks of a dimension x intersecting an interval of the same dimension on the line), which is equal to
 | (4) |
K* is the critical value of the mean distance between microcracks in units of their characteristic dimension; for rocks and soils K* 5 (Zhurkov et al.,1981). The function f(x) with Eq. [3] or [4] is designated by f2(x) and f1(x), respectively.
A (planar) fragment of a dimension x is assumed, on the average, to be of a square shape with a side of length x. Then, according to the above definition of f2(x), as the cumulative fraction of the area of planar fragments of a dimension <x, the differential fraction of the area of the fragments of dimensions between x and x + dx is a differential, f2'(x)dx, and the total length of their sides per unit area is
. Because any connected crack is the side of two fragments, the above calculation takes into account the length of a connected crack twice. Thus, the mean specific length (per unit area) of connected cracks of dimensions between
. The range of variations of x in the two-dimensional case (planar fragments) is
(Chertkov, 1995a; Chertkov and Ravina, 1998), where xm is the maximum dimension of the fragments, and d is the average spacing between crack intersections with a straight line. Hence, the mean specific length of the connected cracks at a plane, L2, is
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According to Eq. [2] and [3], Eq. [5] may be rewritten as
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where
 | (7) |
Similarly, in the one-dimensional case the mean number of spacings between intersections of planar fragments, of dimensions between x and x + dx, with a unit length line is (1/x) f1'(x)dx (where, by definition, f1(x) is the cumulative fraction of length of spacings <x between intersections of fragments with a line). Simultaneously, this is the mean specific number of connected cracks of dimensions between x and x + dx (per unit length of the line). In the one-dimensional case, the maximum value of the spacings,
, is found as the maximum point of the function I1(x) in Eq. [4] and similarly,
in the two-dimensional case (Chertkov, 1995a; Chertkov and Ravina, 1998). As a result, the mean specific number of connected cracks intersecting a line, L1, is
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Accounting for Eq. [2] and [4], Eq. [8] may be rewritten as
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where
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Eq. [1], [6], and [9] determine the mean tortuosity of a two-dimensional crack network
 | (11) |
To calculate mean tortuosity T3 of a three-dimensional crack network we consider a cube of a unit volume. In this case we define L2 as the mean specific length of the traces of the through-connected cracks (per unit area of the lower face of the cube) and L3 as the mean specific surface area of the through-connected cracks (per unit volume). Then, according to the definition of tortuosity T3
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The mean specific surface area, L3, is calculated similarly to L2 and L1. The fragment–dimension distribution is determined by the cumulative fraction, f3(x), of volume of fragments of a dimension <x, in the form of Eq. [2], where the average cracking, I3(x), (the average number of cracks of a dimension x in a volume of a similar dimension) is
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A three-dimensional fragment is assumed to be a cube of a dimension x. Then, according to the definition of the cumulative function f3(x), the volume fraction of fragments of dimensions between x and x + dx is
, and the total surface area of these fragments per unit volume is
. Consequently, the mean specific surface area (per unit volume) of the connected cracks of dimensions between x and
. The maximum dimension of three-dimensional fragments is
(Chertkov, 1986; Chertkov and Ravina, 1998). Thus, the mean specific surface area of the spatial connected cracks, L3, is
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Keeping in mind Eq. [2] and [13] we may rewrite Eq. [14] in the form
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where
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Eq. [12], [6], and [15] determine the mean tortuosity of a three-dimensional crack network
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According to Eq. [17] and [11], tortuosity T3 as well as tortuosity T2, do not depend on the scale parameter d (the mean spacing between cracks), but only on connectedness c of the crack network. Integrals in right parts of Eq. [7], [10], and [16] can be calculated numerically for any given set of physically possible values 0 < c 
1 (and for K*).
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Materials and methods
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For confirmation of the model in this work we are interested in independent findings of measured connectedness c and tortuosities T2 and T3 from two-dimensional images of crack networks. The images may be (i) at the (undisturbed) soil surface; (ii) at different cross-sections of soil cores or blocks; and (iii) at the surface of specially prepared (i.e., disturbed) soil samples. A number of such images are available in the literature (Guidi et al., 1978; Ringrose-Voase and Bullock, 1984; Scott et al., 1986; Ringrose-Voase, 1987; Scott et al., 1988; Ringrose-Voase and Nys, 1990; Velde et al., 1996) and were used for validation of the model. These are shown in Fig. 2
in a smaller scale than the originals. They are numbered according to decreasing connectedness. Images 1 to 4, 6, 9, 11, 13 (from Fig. 2 of Guidi et al., 1978) are photographs of eight replicates of artificially prepared samples of the 1- to 2-mm fractions of an Italian Fluvisol, >48% clay (Images 1–4), and a Regosol, >46% clay (Images 6, 9, 11, 13). Image 5 from Ringrose-Voase and Bullock (1984)(Fig. 4c) is a photograph of a horizontal cross-section of a Windsor soil at depth 46 cm. Images 7 and 12 are Fig. 7
and the lower part of Fig. 1, respectively, of Scott et al. (1988); they are photographs of horizontal cross-sections of clayey subsoils of the Swanwick series at a depth of 46 cm and of the Windsor series at a depth of 35 cm, respectively. Images 8 and 16 are Fig. 1 
and Fig. 1 
, respectively, of Ringrose-Voase and Nys (1990); they are photographs of 12.7 cm by 10.16 cm horizontal cross-sections at a depth of 7 cm of blocky and prismatic soil samples, respectively, from North Wales. Image 10 from Ringrose-Voase (1987)(Fig. 1I) is a cross-section of a soil from Wales. Images 14 and 15 are two parts of Fig. 1 of Scott et al. (1986); the photographs are of 4.8 cm by 7.2 cm soil thin sections cut at random angles of both elevation and azimuth from a profile of the Windsor-series-of-London clay in Essex at depth 35 cm. Image 17 is the right part of Fig. 4b
of Velde et al. (1996), which is a photograph of a 5 cm by 10 cm soil vertical thin section from a vertisol profile (>65% clay) in Southern Italy at a depth of 155 cm.
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Fig. 2 Two-dimensional images of crack networks in clay soils used for the experimental estimates of c, T2, and T3 (arranged in the order of decreasing connectedness c)
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Fig. 7 Model predictions of the mean two- and three-dimensional tortuosities, T2 and T3, of an isotropic crack network in a swelling clay soil as functions of depth (for zm/zO = 10)
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Fig. 4 Predicted mean planar tortuosity, T2, as a function of the mean crack network connectedness, c, and experimental data (numbers correspond to images in Fig. 2)
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The procedure of preparing the images shown in Fig. 2 was as follows: (i) the original image was magnified to a linear dimension of 20 cm; (ii) a strip of the boundary of the magnified image, 1.5 cm wide, was excluded if it was distorted to allow to get a working window square of a side length of 14 to 23 cm; (iii) a grid with a spacing of 1 cm was drawn on the working window (we designated the total number of the horizontal and vertical lines by i0).
A line section, within the working window, is considered to be the trace of a spatial crack if its length > 2 mm and the ratio of the length to width > 3. Any straight section is considered to be a separate crack (1, 2, and 3 in Fig. 1). A separate crack whose two ends are connected with other separate cracks is referred to as a connected crack (2 and 3 in Fig. 1); otherwise, it is an isolated crack (1 in Fig. 1). In a series of connected cracks that cross the entire working window, in any direction, each crack is referred to as a through-connected crack (2 in Fig. 1); otherwise, it is referred to as a locally connected crack (3 in Fig. 1).
The measured mean connectedness of the crack network (in the working window) is determined according to its definition (Chertkov and Ravina, 1998) as the ratio of the number of connected cracks, namely the total number of through-connected cracks, Ntc, and locally connected cracks, Nlc, to the total number of cracks, including the isolated ones, Ni, as well:
 | (18) |
The measured mean two-dimensional tortuosity, T2, for a given image is estimated by
 | (19) |
where Nc is the mean number of the intersections of through-connected cracks with a line of length Lw, which is the dimension of window (first, we calculated the mean numbers of the intersections of through-connected cracks with each of the i0 lines of the grid, then Nc was found by averaging these i0 mean numbers). Ltc is the total length of the through-connected cracks that were measured within the window.
To obtain data for estimating mean three-dimensional tortuosity T3 of a statistically homogeneous and isotropic spatial crack network, one can use any two-dimensional image of this crack network. Considering cracks in a thin layer of thickness h, one may assume that (i) the spatial crack network in the layer is statistically homogeneous, and that (ii) each straight section of length l of an image of the crack network at the face of the layer corresponds to a plane of a crack in it. Designating by 
the angle between the normal to the layer and a crack plane the surface area of the crack in the layer is equal to lh/cos. The total surface area Stc of all through, spatially connected cracks in the layer is
 | (20) |
where the i indicates the separate, straight crack trace in the two-dimensional image. According to the definition, the measured tortuosity of the statistically homogeneous spatial crack network is (actually it is the three-dimensional tortuosity in the vicinity of the cross-sectional image under consideration)
 | (21) |
For any arbitrary ith connected crack in the layer the corresponding angle i has a random value from the range
 | (22) |
where m is the maximum possible value of i. Herewith m < 
/2 because the crack must intersect the cross-sectional plane. Before estimating the angle m it is worth noting that in the definition of the fragment–dimension distribution and the specific fragment surface, a fragment was characterized by its largest dimension, x, only. However, the boundary value m of the random values of i is determined by geometrical considerations of the mean fragment shape. To define the three dimensions of a fragment of an arbitrary convex shape we consider the two dimensions of its largest face and the dimension normal to it, and we designate them in the order of decreasing values as x, y, z. The shape of the fragment is characterized by the ratios y/x and z/x. The mean fragment shape, that is the mean shape of a set of fragments (outlined by connected cracks), is characterized by the ensemble averages,
and
. Defining
, if the fragment adjoins the cross-sectional surface from below by its (x, y) side (Fig. 3)
, then the angle i between the normal to the cross-section, n, and the fragment face (crack) in the layer, can not surpass arctan(x/z), or it must be, on the average, less than arctan(1/a). If the adjoining side is (y, z) or (z, x), then the limit value of the corresponding angle will be even smaller. Thus,
 | (23) |
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Fig. 3 Sketch of a fragment (ped) of an arbitrary convex shape, adjoining a cross-sectional surface from below. x > y > z are the dimensions of a fragment, and i is the random angle between the normal to the cross-section and the plane of an ith crack
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Data of observations of Repin (1978), strengthened by theoretical estimates of Chertkov (1991), show that for natural rock blocks
. This value may be applied to drying clay soils, as shown by Chertkov (1995a). Because of the experimental character of the value a 0.5, one ought to replace a in Eq. [23] by a - 2
, where is the standard deviation of a (at a P = 0.95 confidence level). Because the fragment dimensions were measured to an accuracy of 5 to 7%, the accurancy of their ratio was 10 to 15%. Hence, a - 2 0.35 and
 | (24) |
The procedure for determining the random intersection angle i in Eq. [21] is as follows. Considering the isotropy of the crack-network, it is assumed that the values of the random angle i for any i are uniformly distributed in the range given by Eq. [22]. Moduli of the random azimuths 
i of the crack traces on the cross-sectional plane are also uniformly distributed in the range 0 |i| /2. So, we may take as a random value of i (if the number of cracks is large)
 | (25) |
In the numerical estimation of the tortuosity T3, the angles i were measured relatively to the horizontal side of the working window.
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Results and discussion
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The line in Fig. 4 shows the dependency T2(c) (from Eq. [11], [7], and [10]) and the line in Fig. 5
shows that of T3(c) (from Eq. [17], [7], and [16]) for K* = 5. As can be seen, the tortuosity T2 increases from 1.5 to 2.2 and the tortuosity T3 from 1.4 to 3.25 when connectedness c decreases from one to zero. Finite values of tortuosities in the tendency of connectedness c to zero are in agreement with general considerations mentioned in the introductory section. The line in Fig. 6
shows the dependency T3(T2) that results from the dependencies T2(c) and T3(c) for an isotropic and homogeneous crack system.
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Fig. 6 The mean three-dimensional tortuosity, T3, as a function of the two-dimensional tortuosity T2 (solid line) of an isotropic crack network and experimental data (points) from crack networks of clay soils (the numbers refer to soils shown in Fig. 2)
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To obtain the variation of the tortuosities with depth we use the dependency of the connectedness, c, on soil depth, z, (Chertkov and Ravina, 1998; Eq. [18] and [23]). Figure 7 shows the resulting dependencies T2(c(z)) and T3(c(z)) for a case: zm/zO = 10, where zm is the maximum crack depth and zO is the thickness of the upper layer of intensive cracking (Chertkov and Ravina, 1998).
Estimates of c, T2, and T3 for the crack networks of the images in Fig. 2 are shown by the numbered points in Fig. 4 through 6 along with the theoretical lines. The numbers correspond to numbers of the images in Fig. 2.
In Fig. 4 and 5 all the points are for values of c > 0.5. According to estimates of Chertkov and Ravina (1998) at depths zO (i.e., within the intensive-cracking layer) connectedness is in the interval 0.36 c 1. This is why images from these depths (such as in Fig. 2), regardless of their dimensions, do not provide information of the dependency T(c) at c 0.4. To estimate c and T at great depths by this method, one needs images of large dimensions because of the decreasing concentration of cracks with depth.
Guidi et al. (1978) noted that because of a large content of kaolinite in the Regosol samples (Fig. 2, Images 6, 9, 11, and 13) their shrinkage was different than that of the Fluvisol samples (Fig. 2, Images 1–4). Unlike the Fluvisol samples that were well divided by cracks, the fragments of the Regosol samples were interconnected. In our model this means that the mean value of c for the Regosol must be appreciably smaller than that of the Fluvisol, as indeed is found when comparing the c values of Points 1 to 4 with those of Points 6, 9, 11, and 13 in Fig. 4 and 5.
The experimental errors of c, T2, and T3 can be estimated as follows. Images 1, 2, 3, and 4 in Fig. 2 are from the same soil and conditions (Guidi et al. 1978). This is true also for Images 6, 9, 11, and 13 (Guidi et al. 1978) and for Images 14 and 15 (Scott et al., 1986). Hence, it is possible to assess the errors of c, T2, and T3 for each group of images. The standard deviations Dc, DT2, DT3 found for Images 1 through 4 are Dc 0.03, DT2 0.03, DT3 0.03; for Images 6, 9, 11, and 13 they are: Dc 0.04, DT2 0.05, DT3 0.09; and for Images 14 and 15 they are: Dc 0.004, DT2 0.03, DT3 0.4.
It is seen that standard deviations of connectedness, Dc, for above three groups of images are appreciably smaller than the connectedness variation, 
c 0.5 in the available range 0.5 < c 1
 | (26) |
Hence, the available range of experimental c values is large enough to consider discrepancies between experimental and theoretical T2 and T3 values in Fig. 4 through 6 (for 17 experimental c values) as having statistical meaning. There are three possible sources for these discrepancies: (i) the assumptions of a square shape for planar fragments and a cubic shape for three-dimensional fragments in the calculations of the mean tortuosities, T2 and T3, respectively; (ii) the assumption of a statistically isotropic crack network (equal values of connectedness for the two- and three-dimensional cases and the calculation of T3 from a two-dimensional image); (iii) the procedure for counting the number of cracks of different types within the working window (the ratio of crack length to width > 3 and crack length > 2 mm).
One should note with regard to the choice of the fragment shapes that additional testing of two alternative simple fragment shapes, triangle and tetrahedron or circle and sphere, results in much larger (more than three standard deviations) discrepancies (not shown in Fig. 4–6) between theoretical curves of T2(c), T3(c), T3(T2), and experimental points than between those in Fig. 4–6. The estimates of the experimental errors of T2 and T3 and the distribution of the points around the curves in Fig. 4 through 6 show that the discrepancies do not, as a rule, surpass two standard deviations. Hence, one may say that, in spite of the approximations used, the model estimates and the values from published images of crack networks in clay soils do agree.
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Conclusion
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The tortuosity of the crack network together with crack concentration, depths, widths, and volume determine the hydraulic properties of a swelling clay soils. The theoretical estimation of tortuosities of planar and spatial crack networks were based on the relations between cracks and fragments (fragment's faces are connected cracks). Assuming a statistically isotropic crack network, the two- and three-dimensional tortuosities, T2 and T3, depend only on the connectedness, c, of the crack network and are related to each other. Analytical expressions and graphic presentations were found for the functions T2(c), T3(c), and T3(T2). Based on the dependence of connectedness on depth, c(z) (Chertkov and Ravina, 1998) dependencies of the tortuosities on soil depth, T2(c(z)) and T3(c(z)) were also found. A method for processing two-dimensional images of crack networks is proposed for estimating the values of c, T2, and T3 (assuming crack network isotropy). Results obtained from 17 two-dimensional images that were published in seven works do not contradict the predicted dependencies T2(c), T3(c), and T3(T2) (in the available range 0.5 < c 1).
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ACKNOWLEDGMENTS
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The research is supported in part by the Israel Ministry of Science and Arts and BARD Foundation.
Received for publication March 23, 1998.
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V. Y. Chertkov
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ABSTRACT
INTRODUCTION
