Structure of surface cracks in soil and muds

Transcription

1 Ž. Geoderma Structure of surface cracks in soil and muds B. Velde ) Laboratoire de Geologie, URA 1316 Ecole Normale Superieure, 24 rue Lhomond, Paris, France Received 9 April 1998; received in revised form 21 October 1998; received in revised form 20 May 1999; accepted 20 May 1999 Abstract Image analysis of photographs of surface cracks in cultivated soils for a variety of types Ž Vertisol, Andosol, Mollisol., and surface cracks in mud deposits shows various similar geometric characteristics. Analysis of the skeletal structure of these crack networks indicates that the relations of intersections and numbers of crack segments show a tendency to lie between those of square and hexagonal networks while the relations between particles and the number of bounding crack segments which define particle contours indicates the presence of more segments than found in a hexagonal network. Crack length distribution appears to be symmetric on log length linear frequency co-ordinates. Area distributions of pore size or crack width indicate that the cultivated soils have a regular, symmetric distribution on log width linear area percent occupied co-ordinates while muds have skewed distributions. Particle size abundance Ž defined by crack boundaries. indicates a non-linear cumulative frequency distribution on a log frequency log size analysis that suggests non-fractal relations. Measurement of the fractal dimension of crack patterns using box counting indicates that as the porosity increases the regularity of the distribution of the cracks increases. Cultivated soils show the greatest irregularity for a given porosity. Each type of crack structure Ž soils, muds. has a linear relation on a porosity fractal dimension plot. The surface area spatial characteristics of the crack network and the particles defined by them appear to be different. q 1999 Elsevier Science B.V. All rights reserved. Keywords: soils; muds; cracks; networks; fractal dimension; porosity 1. Introduction The study of the geometry of cracks in soils and muds has been undertaken in Ž various ways but rarely up until the recent past in a numerical fashion Scott et ) Fax: q ; papa@euclase.ens.fr r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. Ž. PII: S

2 102 B. VelderGeoderma al., 1986; Fies, ` 1992; Vogel, 1997, for example.. Interest in such studies lies in the potential application to modelling of crack and hence a portion of the pore structures in soils. If one can determine the real structure Žsize, connectivity, branching, etc.. one can approach the study of soil response to wetting and drying Ž Perrier, 1995, Perrier et al., An initial step in understanding the development of soil fragmentation by drying is the study of surface cracks formed by shrinking. In order to develop a viable model of the type of cracks Ž soil pores. formed at the surface one must quantify the properties of the crack networks using attributions such as the number of intersections between cracks and the size of different cracks in length and width. These are attributions of geometry, which can be observed in different soil types subjected to shrinkage due to drying. Simulations of crack structures have been made in laboratory studies where the soil was homogenised Ž sieved. and re-constructed Ž Gimenez et al., 1997; Preston et al., Laboratory simulations often give different results from those obtained in the field. This is expected because soils are highly complex, being conditioned by a large number of variables, which are often effaced in the preparation procedure of laboratory experiments. In the present study, cracking is observed in the field in the hope that we can find some unifying mechanisms which can lead to simple models of soil behaviour in the shrinking processes. The phenomenon observed here is the surface shrinkage crack network. This does not imply that the crack network formed in surface shrinkage cracks is the only aggregate-forming process in soils. Thus crack pore space should not be confused with total porosity in soils. In this study a limited number of observations on a certain range of soil types, Andosol, Vertisol, Mollisols and muds are used to initiate the systematic study of the differences of surface crack structure in soils as a function of soil type. 2. Experimental method and materials 2.1. Analysis methods The samples were observed photographically in the field. An attempt was made to choose instances where cracks were clearly apparent at the surface and evident in a normal photographic observation. The soil types were relatively varied, limited of course to those soils, which show significant shrinkage on drying. Thus Vertisol, Andosol and Mollisol soils were investigated, all having been cultivated recently using deep plowing and tilling methods, within the growing year. To compare these soil surface structures with other cases of shrinkage at the surface, two examples of mud deposits Ž very high clay content. were studied. These were one of a smectite mud puddle, which had dried, and one of freshly sedimented marine deposits formed on a tidal bank in salt marsh environment.

3 B. VelderGeoderma Colour photographs of the surface expression of the crack structure of the argillaceous materials were digitised using a document scanner at 256 Gy levels. These reproductions were thresholded, resulting in binary black and white images and these images were then treated by morphological image analysis opening and skeletisation routines Ž Moreau et al., These operations allow one to establish the crack thickness and hence determination of the percent of porosity attributable to cracks of different widths as indicated in Moreau et al. Ž and used in Velde et al. Ž in a series of thin section images from a Vertisol depth sequence. These routines permit one to establish frequency diagrams of the percent area occupied by cracks of different width. The above procedure presents some important pitfalls. It must be remembered that the simplification of a colour photograph to a grey level image, which is in turn reduced to a binary image, represents several steps of data manipulation which cause error or fluctuation in the final results. Repetition of the data reduction process used here to produce the binary image gives global results within 10% of each other in area attributed to cracks. However, when considering connectivity or crack continuity for example, the human eye is much more reliable when one wishes to identify events recorded by photographic means. This fact introduces subjective decisions into the process of data reduction. The major problem occurs when one reduces the grey level image into a binary one. Here one uses a unique intensity level to select or reject events. This is justified if the intensity level of these events is the same all over the entire image. This is rarely or in fact never true in normal photographic reproduction. Lighting angle and the resultant shadow effects from small surface irregularities on field photographs results in unwanted dark areas, which do not represent crack events. Further the grey level of a crack may be lower on one side of a photograph than the level of the same crack on another portion of the image due to uneven lighting intensity. Selecting the grey level to describe the existence of a crack becomes subjective to a certain extent. Usually, the choice of selection of a grey level of acceptance or rejection of events is sufficiently small that only some correction Ž manual. is necessary, such as elimination of over exposed zones, etc. The process of binarisation of the image is controlled here by reference to the original photograph. In other words, if one cannot see an event it does not exist. This process takes away some of the rigour from our interpretations. However, as we hope will be seen, the total results are sufficiently coherent to warrant some attention to arguments based upon them. Relations of fracture networks were also determined from the skeletonised representation of the networks. The creation of the image used to skeletonise the fracture networks was made by tracing the fractures by hand as a single width line. In this way asperities and skeletal appendages due to irregularity in crack width are avoided. This method is to a certain extent arbitrary, but the result is more easy to interpret than the interpretation of the difficult results of normal automatic skeletisation routines have proved to be where wide cracks give

4 104 B. VelderGeoderma numerous unreal skeletal figures. As fractures and pores have different widths the intersection point of a small fracture with a large one is of course subject to a certain amount of estimation. The distance between intersections in this schema was determined from the skeleton of the fracture network according to the methods developed by Sardini Ž 1996., Sardini et al. Ž The total number and size distribution of soil particles outlined by cracks was also determined using the algorithms of Moreau Ž Fig. 1 illustrates the different steps in these procedures showing Ž a. the initial grey level image, Ž b. the extracted binary Ž black and white., and finally Ž c. the hand traced median lines of the cracks where intersections are shown. Line width in Fig. 1c has been enlarged to be more easily seen. Another use of the binary 2D image of the crack structure is a determination of the fractal dimension of the pattern. This is determined by a box counting method where the total number of points Ž pixels. in the crack or pore structure present in each analysis box is counted. This is called here, according to tradition, the mass method because the totality of the event is measured in each observational area instead of the existence or not of an intersection of the image with the observational box as in the Cantor s dust method Žsee Tyler and Wheatcroft, 1992; Perrier et al., 1995; Crawford and Matsui, 1996; Preston et al., 1997; for a description of this use of the mass fractal analysis and Velde et al., 1991 for the Cantor s Dust method.. In the routine used here, a series of 200 points are selected at random in the image and boxes of different sizes are generated from each point. The total number of pixels representing cracks Ž black. in each box is counted. The averages of box content of these 200 points for the different box sizes are plotted against the box size on log log scales. The slope Ž regression of the points. is the fractal dimension. In order for this analysis method to be valid, the points must be distributed equally on either side along a regression line. If they are grouped to one side at the extremities of the line, the relation is a curve and the results indicate a non-fractal structure. One must be careful in this analysis to be sure that the box sizes are neither above nor below Ž. the largest or smallest events Velde et al., Samples studied The materials, which developed the cracks due to drying, were of several types, cultivated soils and homogeneously deposited and dried mud deposits. As a comparison published images of mud deposits created in the laboratory from soil samples and thin sections of soils in a soil profile depth sequence were analysed using the same methods. The cultivated soils photographed were observed in the following countries listed below. Italy: Andosol developed on volcanic sediments Ž Vesuvian. near Mercato, 10% clay, Ando1, 2 and 3. Soil cracks were observed in a road cut which

5 B. VelderGeoderma Fig. 1. Example of photograph of soil with cracks and steps in image treatment. Sample from Princeton, IL, USA. Surface of silt loam cultivated soil. Ž. a initial photo scanned in 256 grey levels, Ž. b binary representation of the crack structure, Ž. c schematised structure of crack elements used to determine lengths and intersections.

6 106 B. VelderGeoderma intersected the surface horizon of the soil. Areas photographed are 1000 to cm. Vertisol near Calitri Žsee description in Righi et al., 1995a; Velde et al., % clay, two samples. Nap1 and Nap2. Areas represented are 3000 and 4000 cm 2. France: Vertic Kendzine cultivated soil developed on a Jurassic marne, La Touche, Vienne, 50%, clay LaT and LaT2 Žarea 1300 and 3000 cm 2.. Vertic Mollisols developed from polder sediments cultivated since 1912 in the Marais Poitevin near Saint-Michel-l Herme, Vendee Žsee Righi et al., 1995b for sampling sites and detailed mineralogy., 60% clay, four samples of cultivated soils Ž. 2 Dive 1, 2, 5 and 6 surface area of near 650 cm each and one sample of a Ž. Ž. 2 compacted by tractor traffic soil of the same material Triaze1 area 300 cm. USA: Cultivated silt loam Mollisol developed on glacial loess cover of till near Princeton IL, 17% clay, Princ1 and 2 Žarea 5000 and 600 cm 2.. The sedimented muds were found in France at two sites. Smectites derived from bentonites near Rochefort, Charentes Maritime, near 100% clay, Roch1 and 2 surface area 700 cm 2. Intertidal clay deposits near Saint-Michel-l Herme, Vendee, 60% clay, samples related to the Vertic Mollisols above , 2, 3, and 4 Žsurface areas and 5000 cm 2.. Other data: These surface features can be compared with soil crack and pore structures developed in a Vertisol depth sequence made using the thin sections of samples reported in Velde et al. Ž Samples 936, 954, 949, 937, and 939. Areas near 60 cm 2. Samples of mud cracks produced from natural, disaggregated clays dried in the laboratory were used for comparison Žarea 400 cm 2.. These are reported in Preston et al. Ž Another sample of crack structures in soils developed in a permafrost structure Ž perma. published by Williams and Ž. 2 Smith 1989, p. 190 area of about 1 km, is included in the data set as another comparison of materials from a very different environment. 3. Experimental results There are two major ways to assess the structures of crack networks. One way is by the relations of the network formed by the cracks, assuming a two dimensional, width-constant structure. This analysis can be obtained by a skeletonisation of the pore and crack structure, which reduces the cracks to a linear representation. It will be called here a linear analysis. The second type of observation is made on the entire crack pattern, which takes into account the widths of the cracks and their spatial distribution. It will be called here an area analysis Linear analysis of crack networks Initially, the structure of soil cracks can be described as a sequence of segments which intersect other segments. Table 1 gives the relations of the

7 B. VelderGeoderma Table 1 Relations of the skeletal representations of the crack networks. Samples NapsVertisol Žin area reported by Velde et al., 1996., Divescultivated soil in Marais Poitevin; rochsbentonite mud deposit; princ ssilt loam Princeton, IL, USA; LaT Redizine on marl, Vienne, France; Andos Andosol roadcut outcrop, near Mercato, southern Italy Ž Vesuvian region.; 1912 stidal muds Marais Poitevin; Triazescompacted soil Marais Poitevin; mudsartificial mud cracks produced in the laboratory from soil materials Ž Preston et al., 1997., permas permafrost structure figured in Williams and Smith Ž Sample Intersections Segments Particles % Dead ends Nap Nap Dive Dive Dive Dive roch roch princ princ Lat Lat Ando Ando Ando Triaize perma mud mud mud measurements on the skeletonised crack networks of the soils and muds. Values of the number of segments, the number of intersections and the number of singly connected segments Ž dead end cracks. are given. These segments define the outline of the soil fragments formed by cracking. One can imagine an orthogonal network of squares, a hexagonal network or a triangular network Ž Fig. 2.. These regular structures will give fixed relations of the number of segments related to each intersection. For the triangular net the relations SrI Žsegments per intersection. is 3, for squares the value is 2 and for hexagons the value is 1.5. One can imagine an orthogonal system where the intersections are slightly irregular, as in Fig. 2d. In this case the SrI number will be less than 2 and greater than 1.5.

8 108 B. VelderGeoderma Fig. 2. Examples of types of regular geometrical networks based upon Ž. a squares, Ž. b hexagons, Ž. d triangles and Ž. c irregular orthogonal structure which reduces the number of intersecting segments per intersection. For the 22 samples studied the relations seem to fall between the SrI values of 2 and 1.5 Ž Fig. 3.. There does not seem to be any direct relation between soil type and SrI value. Looking at the inverse of the cracks, considering that they define the boundaries of soil fragments, the relation of fragments to the number of bounding segments necessary to define can be plotted as in Fig. 4. The values Fig. 3. Relations between the numbers of segments and intersections in the crack networks of the soils and muds investigated schematized in a skeletal structure, i.e., without the thickness of the cracks indicated. Segmentrintersection ratio is 2 for a network of squares and 1.5 for a hexagonal network. The natural networks lie between these values for the most part.

9 B. VelderGeoderma Fig. 4. Relations between the number of particles defined by the cracks as boundaries and the number of bounding crack segments for the soil and mud samples investigated. The values for a square network, hexagons and a ratio of 4 are indicated on the diagram. The natural networks lie in the range of a hexagonal network Ž segmentsrparticless3. and a greater value which indicates that there are a certain number of segments which are not connected to form a complete network Ž dead end segments.. lie between a segmentrparticle ratio of 3 to 4 or more which is equal or greater than found for a hexagonal net and more than the square or triangular networks. Assuming that the problem of representativity is not skewed due to the relatively small scale of the observations it seems that there are extra segments present in most of the crack networks which do not define particle boundaries. These are cracks that end in the center of a polygon. These are given in Table 1 in percent of the total segments. Another parameter in the linear representation of the crack pattern is the length of the fractures between intersections. Such measurement is difficult if the cracks and pores are of greatly differing widths Žsee the Roch1 mud crack example in Fig. 5 for example.. In such cases it is not possible to determine the point of intersection of a small crack with a larger one. However, in cases where the crack widths are of small variance, as is the case of the soils on marne Ž clayey limestone. or the inter-tidal mud cracks, one can estimate the size distribution of the cracks between intersections with reasonable confidence. In these cases it was found that the crack length distribution is symmetrical on log linear plots Ž Fig. 5.. There is always a decrease in the abundance of cracks of small size Crack area analysis If we look now at the total binarised pattern of the crack structures, one can measure the porosity due to the cracks and their different sizes. It should be remembered that the crack sizes are those expressed on the two dimensional

10 110 B. VelderGeoderma Fig. 5. Crack segment length frequency distributions for a Vertisol and dried mud cracks. Nap1s Andosol; tidal mud flat, Marais Poitvin. Ž. surface of the soil profiles. It is known Moreau, 1997 that the 2D expression of 3D pore structures of crack shape in soils can be misleading. However, the pattern, which interests us here, is the expression of the soil cracks at the interface of a three-dimensionally physically-constrained soil or mud block and the unconstrained atmosphere. Examples of the different types of fracture types, in cultivated soils and muds are given in Fig Porosity size distribution These measurements were made on a more limited number of samples used than in the linear skeletonisation study. It should be noted that the pores

11 B. VelderGeoderma Fig. 6. Examples of the binary crack structures with crack thickness shown. Tidal mud Ž30=150 cm., bentonite mud Ž 30=24 cm., Vertisol Ž 60=50 cm., Andosol Ž 35=50 cm. and silt loam Ž 60=90. examples are shown. observed here are all well above the 100 mm width, and hence should all be considered as macropores. Table 2 gives the porosities Žcrack area in percent of the image. of the samples which include examples of the soils or mud samples of Table 1 as well as values for five images taken from a soil depth sequence from the same site as the Italian Vertisol of Tables 1 and 2. The sequence was

12 112 B. VelderGeoderma Table 2 Total crack area of images and fractal dimension and porosity. Abbreviations as in Table 1 and numbered samples Ž svertisol soil profile determined from thin sections given in Velde et al. Ž 1996.; 85si 1100sissurface Vertisol soil blocks from southern Italy Vertisol site Sample Fractal D Porosity % Nap Nap Dive Dive roch princ Ando Ando Triaze si si si si si taken from thin sections of soil blocks ranging in depth from the surface to 1.5 m Ž see Velde et al., The frequency data distribution is obviously non-gaussian. Therefore, another representation has been used to compare the data sets. Fig. 7 shows the transformation of fracture surface-fracture width data from a linear plot to one of log linear co-ordinates. The results, plotted on log linear co-ordinates for the samples studied can be characterised in three ways, those with a near symmetrical distribution on log linear co-ordinates those skewed to low values on such plots and those skewed to high values. The use of analysis procedures to determine lognormal distributions and lognormality is carefully explained by Parkin and Robinson Ž These authors give several numerical tests to estimate approach to ideal distributions. Our use here of log linear co-ordinate representation is less rigorous, the main interest is to test whether or not a symmetric curve shape is evident when the X-axis is shown in log and the Y on linear units. Given the small number of samples observed, it is not felt that an exhaustive statistical analysis of each data set is warranted. Fig. 8 shows the results for the different samples where the variation of crack width was

13 B. VelderGeoderma Fig. 7. Crack size distributions shown in the case of a symmetric distribution in a linear linear plot of surface area vs. crack width Ž. a and plotted as log width linear surface area plot showing the symmetrical distribution of the crack widths in such a plot. sufficient to make a width analysis. These figures show that the Vertisol Ž 8a., and Andosol Ž 8d. have a rather symmetric distribution on such plots, Triaze Žthe compacted Vertic Mollisol. has a skew to high values and polder soils and the muds Ž tidal mud and bentonite. tend to have a distribution skewed to the low fracture width side Particle size distribution An important measure in the description of soils is of course the size distribution of the soil particles themselves ŽOades and Warters, 1991; Rieu and

14 114 B. VelderGeoderma Sposito, In the present study one can consider that the cracks at the surface of soils and muds delimit particles, though they are rather large, several centimetres in diameter in many cases. One can analyse the particle size distribution as a cumulative frequency and plot the log of the frequency against the log of the particle size. If the size relations are fractal, a linear relation will be found Ž Tyler and Wheatcroft, Fig. 9 shows these relations for the two Fig. 8. Plots of the different soil and mud types crack width distributions on log linear coordinates. The soils Ž Vertisol, Andosol, silt loam. are nearly symmetrical on such plots while the muds are skewed to the small crack sizes or to the large crack sizes. Laboratory mud cracks Ž Preston et al., are shown for comparison.

15 B. VelderGeoderma Ž. Fig. 8 continued. samples studied where a sufficiently large number of particles was observed. It is evident that no linear relation is shown Fractal dimensions The regularity of the crack distribution in the surface 2D plane was measured using the fractal analysis mass method of box counting of the totality of events Ž pixels. found in a given analysis area, the mass of the events. Fractal dimension Ž D. can range from 1 to 2 for a two dimensional image. For the samples investigated here the fractal number varied from 1.2 to 1.8, a very wide

16 116 B. VelderGeoderma Ž. Fig. 8 continued. range Ž Table 2.. The fractal dimension does not show a relation to the total area of the image observed. The areas cover a range of more than two orders of magnitude. It is important to note that the synthetic dried mud laboratory experiments reported by Preston et al. Ž do not give a strictly linear relation on log log plots of analytical unit vs. number of events. In fact the plots show a decided curvature on such graphs. This indicates that these laboratory experiments give results that are not truly fractal from a spatial analysis point of view. As the analysis of cracks in natural soils and muds gave linear relations on log log

17 B. VelderGeoderma Ž. Fig. 8 continued. plots, indicating a fractal crack network structure, one can conclude that the laboratory experiments do not faithfully simulate field conditions. The difference between the two sets of conditions is seen in the structure produced in the laboratory. It is of almost the same crack width over the surface observed which is not the case for natural soils and muds. Further, and possibly more importantly, the distances between the different fractures are very similar in the laboratory experiments. In fact the structure produced in the laboratory is very regular. Such structures approach a non-fractal value of 2 Žan integer value and hence not fractal. which is difficult to observe in a spatial analysis because the range of variation between the units of the structure are very closely spaced. There is little variation in the spacing between events and the effect of changing

18 118 B. VelderGeoderma Fig. 9. Cumulative frequency distributions for particles Ž determined by segment boundaries. and particle surface areas for Ž. a tidal mud flat and Ž. b Vertisol samples. the size of the analysis area is quickly rendered redundant, giving a constant value for larger or smaller values than the range of variation in the fracture set Ž see Velde et al., 1991 for an explanation of this problem.. For this reason the fractal dimensions are not recorded in Table 2 for the series entitled mud Discussion of the results The objects observed here are of two main types, cultivated soils from varying origins Ž Vertisol, Andosol, Mollisols. and dried mud deposits. The choice of samples allows one to assess the crack structure differences for a range

19 B. VelderGeoderma of clay content Ž near 100% to 17%.. As clay is the major agent in soil shrinkage, such variations present a large variety of cases Crack networks length intersection relations If soils of different clay content behave in the same manner as far as the geometric relations of drying cracks is concerned, a general model for the network of cracks could suffice for all soil types. The observations here for a large range of clay contents in soils indicate a strong similarity in certain geometric aspects. The soil and mud crack structures investigated here have between 1.5 and 2 segments per intersection. There seems to be no great difference between the crack networks developed in the various cultivated soils of different types and the muds. The relations of segment to intersection lie between 1.5 and 2 even when changing the scales of the areas from hundreds of square centimetres to kilometres. This type of network connection could be modelled using a slightly deformed rectangular pattern where rectangles are of different sizes Ž see Fig. 1.. In all cases the structure of the crack pattern is quite similar in the definition of the number of sides defining a particle and the number of segments per intersection. However even though there seems to be a great similarity in the observations based upon materials of very different origins, given the small number of examples used here, further work is necessary to confirm these generalisations Pore size distribution In the following discussion one should remember that the smallest pore observed is well above 100 mm in width and hence these are all macropores. In pore size distributions, it appears that cultivated soils tend to have symmetric distributions of pore size area on log width linear percent surface area plots. Muds have more abundant surface areas for small crack sizes. The greater complexity of soil and mud shrinkage cracks suggests that different mud types Ž clay species or clay content. determine the relationship between the abundance of large cracks relative to smaller ones. The difference in the patterns of pore size distributions between cultivated soils and dried muds suggests that the cultivation process tends to affect the shrinkage process of the soil materials in their response to drying Fractal dimension of pore distributions in 2D space The use of fractal analysis to characterise natural phenomena has been in vogue for several decades now. However the significance of such methods is difficult to assess. The real possibility of multifractal phenomena Ži.e., different spatial relations in at different scales of observations in the same pattern, see Stanley and Meakin, 1988 for example. makes use of fractal analysis all the

20 120 B. VelderGeoderma more hazardous to apply to real, complex and unknown phenomena. In the present study, the use of fractal analysis of the distribution of pores in a 2D plane is only one of characterisation and comparison. No study of phenomenology is intended. There is no attempt to investigate the scaling effect in a single sample Ž relations on different scales of observation. due to the limits of photographic resolution of the crack structures. It can be remarked that the crack pore sizes varied by a factor an order of magnitude in each sample and a factor of 20 between the different samples studied. This range of observation does not cover several orders of magnitude as is frequently, but not always, the case for studies of fractality. However the total surface area of the different crack networks investigated varied by two orders of magnitude. Despite these limitations, one can compare the fractal dimensions Ž D. of the different pore patterns, which show features varying by two orders of magnitude on each of the images. If one looks at the values of D, they cover a rather large range from 1.2 to 1.8. Values of 1.2 indicate highly irregular structures while those of 1.8 indicate rather regular ones Ž Crawford and Matsui, For purposes of characterisation of natural objects this is a very good situation. If different soils show large variations in a measurement parameter, it is highly likely that his parameter can be used as a diagnostic tool. If one now looks at the porosity of the soil images as a function of the fractal dimension Ž D. as is shown in Fig. 10, it is evident that there is a regular relation between porosity and the distribution of the pores in 2D space according to the type of material Fig. 10. Plot of the porosity fractal dimension Ž D. relations for the samples investigated. The surface soils Ž for the most part cultivates soils. are grouped on a trend showing a high degree of disorder Ž low fractal dimension D. at low total crack surface areas, the muds are less so and the deep soil profiles show a minimal disorder Ž low fractal dimension. for low areas. These relations are independent of the surface area of the sample observed Ž Table 2..

21 B. VelderGeoderma concerned. In the figure, one sees different trends for cultivated soils and mud cracks. One finds a different relation for cracks revealed in thin sections of a depth sequence in a soil profile reported in Velde et al. Ž Two sequences Ž surface to 1.5 metres. are from profiles in a prairie soil at the Italian Vertisol site reported in the data set for surface cracks in cultivated Vertisols. The crack structure at depth in a 2D representation can be compared directly to the crack network in the cultivated surface soils. The figure shows that the soils at the surface form a sequence of porosity fractal dimension values as do the muds. The depth sequence forms a third series of relations. All seem to converge at high total crack areas. For the samples studied, the irregularity of the structures is greatest in the soils at low porosity and for the same porosity the irregularity decreases as one goes from soils to muds to soils at depth. The cultivated and surface soils show the greatest change in fractality or irregularity as a function of reduction in porosity. The consequences of these relations as they concern hydraulic conductivity and water retention must be determined by a modelling process. If one extrapolates the different trends, it appears that at 60% porosity all shrinkage crack networks reach a rather regular structure Žfractal dimension of It is highly probable that the loss of fractality is due to the analysis method more than a true absolute regularity indicated by a value of Particle sizes The distribution of soil particles has been used to describe the type of organisation of particles and by extension the distribution of crack structures Ž Rieu and Sposito, 1991 for example.. One assumes a relation between the cracks and the particles since one forms the other. In our images the cracks are macropores and hence the soil particles defined by them are macro-in size. The division of soil surfaces into soil particles by drying and crack formation observed here does not seem to produce a fractal distribution of the particle sizes outlined at the surface. The cumulative frequency of particle sizes encountered does not become linear on a log frequency log particle size plot as would be expected for a fractal distribution. This suggests that the particles revealed by surface cracking do not follow a fractal type distribution Žsee Young and Crawford, 1991, for a discussion of the theoretical aspects of differences between fractal properties of pores and particles.. It is possible that further fragmentation, into more fundamental, smaller particles will give a fractal distribution as has been observed in studies of dried soil particle distribution divisions Ž see Oades and Warters, 1991; Rieu and Sposito, Geometrical effect of shrinkage The range in fractal dimension related to porosity is important. If one assumes that the initial, low porosity D values for cultivated soils represent the

22 122 B. VelderGeoderma early stages of drying and that the high porosity values represent the more ultimate terms, one would expect that the fractal dimension of the soil aggregate analysis would be the same for each of the soil series ŽVertisol, Andosol, Mollisol. if the mechanism of the formation of aggregates is the same process Ž Rieu and Sposito, 1991; Young and Crawford, However, it is clear here that the fractal dimension increases as porosity increases, for samples of very different clay content Ž shrinkage properties., suggesting that the fragmentation process of the soil surface operative is not the same at different stages of crack development in the different soils and muds investigated. Otherwise increasing fragmentation would give the same fractal dimension and it would be scale independent. The measurements of the fractal dimension of the soil and mud crack patterns by the mass method do give perfectly fractal Žlinear plot on log log scales of number of events vs. the size of the analysis surface. relations for each sample, and they cover a large range of fractal dimension Ž Whatever the origin of the differences in D, the measurement of the fractal dimension of the crack structure of soils and muds by the mass method appears to be a useful tool to describe different types of material Crack pore space and particle distribution Observations on the particle size distributions Ž Section 4.4. and the fractal dimension of the pore space Ž Section 4.3. indicate that the pore space is fractal but the size distribution of the soil fragments produced by cracking are not. Hence there is a difference in the distribution of the pore space and the fragments of the soil in 2D space. This has been observed by Oleschko et al. Ž in thin sections of Andosol, Vertisol and Acrisols when measuring the linear fractal dimension of the different elements. Similar conclusions have been reached by Crawford and Matsui Ž from a more theoretical standpoint. It seems that, at least for surface shrinkage cracks, the pore space and the fragments determined by it are not directly complimentary. 5. Conclusions Several general relations seem to be revealed in the survey of a wide variety of soil and clay materials subject to shrinkage. Ž 1. Crack networks: The relations of crack segment frequency and intersection frequency found in all crack networks observed give a useful tool in formulating simple models of cracks in soils. The symmetrical distribution of crack lengths on a log linear plot can also be used to construct more realistic models of crack networks. Ž 2. Crack dimensions: The distribution of crack widths for the different materials Ž soils and dried muds. indicates that the distributions are not Gaussian,

23 B. VelderGeoderma but for the most part not ideally lognormal either. This indicates that the shrinkage in different soils and muds is controlled by the materials and their structures. Ž 3. Crack distributions: Fractal analysis of the crack patterns suggests a relationship between total crack area and the fractal dimension of the materials. This relationship is different for soil cracks investigated and for the muds. It is possible that the area-fractal dimension relations could be used to identify shrinkage processes. In general, morphological measurements of drying cracks in soil and muds appear to give relations which at times define the types of materials Žsoils, muds, deep soils. and at times show generally respected relations of crack structures valid for all of the crack networks observed here. Further, more systematic research should give more precise information concerning these relations and hence one will be able to construct more credible models of soil behaviour. References Crawford, J.W., Matsui, N., Heterogeneity of the pore volume of soil: distinguishing a fractal space from its non-fractal complement. Geoderma 73, Fies, ` J.C., Analysis of soil textural porosity relative to skeleton particle size, using mercury porosimitry. Soil. Sci. Am. J. 56, Giminez, D., Allmaras, R.R., Nafer, E.A., Huggins, D.R., Fractal dimensions for volume and surface interaggregate pore scale effects. Geoderma 77, Moreau, E., Etude de la Micromorphologie et de la Topologie 2D et 3D d un sol Argileux par Analyse d Image. Univ Poitiers, 328 pp. Moreau, E., Sardini, P., Touchard, G., Velde, B., D and 3D morphological and topological analysis of a clay soil. Microsc. Microanal. Microstruct. 7, Oades, J.M., Warters, A.G., Aggregate hierachry in soils. Aust. J. Soil. Res. 29, Oleschko, K., Fuentes, C., Brambila, F., Alvarez, R., Linear fractal analysis of three Mexican soils in different management systems. Soil Technology 10, Parkin, T.B., Robinson, J.A., Analysis of lognormal data. In: Stewart, B.A. Ž Ed.., Advances in Soil Science. Springer-Verlag, Heidelberg. Perrier, E., Structure Geometrique et Fonctionnement Hydrique des sols: Simulations Exploratoires. ORSTOM Editons, Paris, 258 pp. Perrier, E., Mullon, C., Rieu, M., Computer construction of fractal soil structures: simulation of their hydraulic and shrinkage properties. Water Resour. Res. 31, Preston, S., Griffiths, B.S., Young, I.M., An investigation into sources of soil crack heterogeneity using fractal geometry. Eur. J. Soil Sci. 48, Rieu, M., Sposito, G., Fractal fragmentation, soil porosity and soil water properties: II. Application. Soil Sci. Am. J. 55, Righi, D., Terrible, F., Petit, S., 1995a. Low-charge beidellite conversion in a Vertisol from southern Italy. Clays Clay Miner. 43, Righi, D., Velde, B., Meunier, A., 1995b. Clay stability in clay-dominated soil systems. Clay Miner. 30, Sardini, P., Micropermeabilite du Granite de Soultz-sous-Forets, ˆ Traitment d Images et Experimentation. Univ. Poitiers, 320 pp.

24 124 B. VelderGeoderma Sardini, P., Moreau, E., Sahel, A., Badri, A., Touchard, G., Meunier, A., Beaufort, D., Two-dimensional fracture and porous network geometry: detection of dead-ends using morphological operations. Acta Sterologica 15. Scott, G.J.T., Webster, R., Northcliff, S., An analysis of crack pattern in clay soil: its density and orientation. J. Soil Sci. 37, Stanley, H.E., Meakin, P., Multifractal phenomena in physics and chemistry. Nature 335, Tyler, S.W., Wheatcroft, S.W., Fractal scaling of soil particle size distribution: analysis and limitations. Soil Sci. Soc. Am. J. 56, Velde, B., Dubois, J., Moore, D., Touchard, G., Fractal patterns of fractures in granites. Earth Planet. Sci. Letters 104, Velde, B., Moreau, E., Terribile, F., Pore networks in an Italian Vertisol: quantative characterisation by two dimensional image analysis. Geoderma 72, Vogel, H.J., Morphological determination of pore connectivity as a function of pore size using serial sections. Eur. J. Soil Sci. 48, Williams, P.J., Smith, M.W., The Frozen Earth. Cambridge Univ. Press, 306 pp. Young, I.M., Crawford, J.W., The fractal structure of soil aggregates: its measurement and interpretation. J. Soil Sci. 42,

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