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Discussion of the paper by G.W. Horgan and I.M. Young
Geoderma 96 Ž2000. 277–289
Discussion of the paper by G.W. Horgan and I.M. Young 1 C.J. Moran, J.M. Kirby CSIRO Land and Water, GPO Box 1666, Canberra ACT 2601, Australia Received 11 November 2000; accepted 6 December 2000
Horgan and Young present a geometrical statistical random model of cracking patterns. They compare the simulations to cracking patterns from thin layers of homogenised soil. In this discussion, we raise four issues not explicitly dealt with by the authors which impact on how well their approach can be considered to deal with the subject of soil cracking: two other papers dealing with cracking have not been reviewed; the evolution of cracking patterns depends on the mechanics of soil drying and local heterogeneity; cracking patterns in a three-dimensional system are not necessarily well represented by two-dimensional analogues; and, finally how 2D traces might realistically be used in soil water balance modelling in swelling systems. Horgan and Young have not reviewed the work of Moran and McBratney Ž1997. nor that of Perrier et al. Ž1995. which are both relevant to this topic. Moran and McBratney presented a generalised model of soil structure including simulation of cracking patterns. Their basic approach was not dissimilar to that adopted by Horgan and Young. Moran and McBratney simulated cracks using a so-called linked distribution of points. This is a statistically constrained random
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References to this paper should include the discussion and should be cited as: Horgan, G.W., Young, I.M., 2000. An empirical stochastic model for the geometry of two-dimensional crack growth in soil Žwith Discussion.. Geoderma 96, 263–289. References to parts of the Discussion should be cited as, e.g., Moran, C.J., Kirby, J.M., 2000. In: Discussion of: Horgan, G.W., Young, I.M., 2000. An empirical stochastic model for the geometry of two-dimensional crack growth in soil. Geoderma 96, 263–289. 0016-7061r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 6 - 7 0 6 1 Ž 0 0 . 0 0 0 1 7 - 3
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walk. A directional distribution Ž Von Mises. is sampled to provide the step direction Ž du . in Horgan and Young’s terminology. A second distribution can also be invoked to determine the morphological details of the cracks. The width of cracks and the step length Ž dC . is controlled by centering an ellipse at the step. This model allows simulation of cracking patterns with variable orientation juxtapositions, e.g., blocky, polygonal and striated patterns Žsee Fig. 11 of Moran and McBratney, 1997. . Horgan and Young have introduced distance inhibition and aggregate bisecting to simulate the result of cracking. Moran and McBratney similarly require a large input specification to produce a simulation of the final structure. Both approaches suffer because they do not simulate the interaction between drying and mechanical behaviour. Perrier et al. Ž1995. presented a fractal fragmentation model based on the conceptual model of Rieu and Sposito Ž1991.. While not strictly a model of surface cracking, like the model of Moran and McBratney, it is a relevant geometrical simulation. These authors then used the cracking patterns thus generated to model the feedbacks between soil drying, hydraulic conductivity and swelling and shrinking of the soil Žthereby altering the extent of the cracks.. This remains the only paper linking the physics of water movement to the geometry of the cracking system. However, it still does not address the fundamental question of the evolution of a cracking pattern as driven by the wetting and drying sequences: the cracks of Perrier et al. Ž 1995. simply expand and contract. Observation of real soil suggests that the development of the cracking pattern is more complex and changes as wetting or drying proceeds. Surface cracking patterns often develop as the soil dries from the surface downwards. The water content depth profile plays a role in defining the tension field set up as shrinking commences. Therefore, the tension failure that we view as crack initiation is difficult to model without taking the third dimension into account. Fast drying profiles form many, small, closely spaced cracks, whereas slow drying profiles form a few, large, widely spaced cracks. Further, as cracks widen, the effective surface area available for drying increases providing feedback into the crack development process. Also not taken into account is the influence of clay mineralogy or soil mechanical properties. It is obvious that soils with different types and proportions of clay develop different cracking patterns, but the models of Horgan and Young Ž this paper. , Moran and McBratney Ž1997. and Perrier et al. Ž1995. ignore this influence. Taking the progress delivered by Horgan and Young’s simulations, we are still faced with the question of how to use such information in estimating soil water movement. Bronswijk Ž 1988. described a one-dimensional model of water balance, cracking and subsidence of clay soils. The ‘‘cracks’’ in a one-dimensional model are simply the proportion of the pore volume, as a function of depth, which is attributed to cracks rather than matrix pore space. Simulations of moisture content, surface subsidence and crack volume agreed reasonably well with ten months of measured field behaviour. Evidently, it was unnecessary to
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invoke information about crack patterns or sizes. Kirby and Ringrose-Voase Ž1999. also found reasonable agreement between a one-dimensional model simulating drying of puddled rice soils and several weeks of measured field behaviour. Again, it was unnecessary to invoke information about crack patterns or sizes. On the other hand, under saturated conditions water moves preferentially through macropores and this depends on the type of macropore and the degree to which the macropores are connected ŽHallaire and Curmi, 1994. . Ross and Bridge Ž1984. showed that cracks and their distribution patterns are crucial in determining water infiltration, and potentially, the residence times for water Ž and associated pesticides and nutrients. variously spatially distributed in a seasonally cracking soil. Characterisation of these issues probably requires representation of the soil structure and water movement in three dimensions. It is difficult to see how robust 1D or even 2D approximations can be derived without such a research model. Verification and parameterisation of such a model, as always, will prove a significant challenge. Dietrich Stoyan Institut fuer Stochastik, TU Bergakademie Freiberg, Bernhard-Õon-Cotta-Str. 2, D-09596 Freiberg, Germany The authors study and partially solve a fascinating problem — modelling the geometry of planar crack systems in soil surfaces. My own observations Ž see Figs. 9.10 and 10.2 in Stoyan et al., 1995. say that the authors are able to produce excellent simulations, which are rather similar to real crack patterns. I congratulate them on this success. My following remarks concern the problem of modelling crack systems in general and try to point to further research. Most of my ideas also appear in the paper discussed here. Modelling of crack systems and crack tessellations is a problem which is important in many fields of science and engineering, not only in soil science. I believe that it could be fruitful to collect the widely scattered material and to combine the various approaches, which seem to be developed independently by physicists, engineers and statisticians, in a unique and general theory. Clearly, the cracking materials determine speed and form of cracks and geometry of crack systems; cracks on the surface of metals or ceramics are quite different to cracks in soil. The ultimate aim of research in crack systems should be physical models with measurable material parameters, which produce the geometry as a ‘‘by-product’’. Pure geometrical models such as that given by the authors can be only a first step in modelling. The existing geometrical models can perhaps be classified as follows: The final product of planar crack processes is a planar tessellation Žsometimes with the exception of some cracks which do not join other cracks and end blindly. .
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The elements of any tessellation are nodes, edges and cells. The process of forming a tessellation can be based on these elements. In tessellations based on nodes, the cracks appear between the nodes, so as were their endpoints prescribed before; an example of such a model is that discussed in Stoyan and Stoyan Ž1990., which was used for modeling of cracks in basaltic structures. The model presented by Horgan and Young could be interpreted as an edge-based model, since here the edges are the primary elements. The most famous models are cell-based, such as the Voronoi and Johnson–Mehl tessellation, which are created starting with cell centres. These models have been used for modelling cracks, see, e.g., Winkler et al. Ž 1992. and Schlather and Stoyan Ž 1999. . The starting point of the model given by the authors is in my opinion quite natural. To begin with random points and to have crack growth which follows somewhat and which is similar to a random walk is natural. However, already here some refining may be useful: the points could positioned more regularly than independent points Žperhaps this can make the final patterns more similar to real patterns. and the cracks could be modelled as the curÕe images in ˇ Grenander Ž1996., p. 109; by the way, Simak et al. Ž 1998. discuss a Markovchain-based fibre process model in the context of fractography, which produces cracks similar to those produced by the authors. The other rules seem to be results of observations of soil crack processes and help to produce realistic geometries. They lack any quantitative physical argument and are similar to the ad hoc rules in Stoyan and Stoyan Ž 1990. . Here still a lot of physical work has to be done which should finally lead to physical models which produce crack systems where the cracks minimize the shrinking force. Perhaps Gibbs processes will be used one day in this context. I am not entirely happy with the finite design of the model. I would prefer a Žspace-. stationary or homogeneous model, which starts with the points of a stationary point process that replaces the N0 independent points. The further rules of the model should then be formulated in such a way that a stationary planar geometrical structure is obtained. Finally, some words on the statistics of crack systems. The problem of parameter estimation for models such as that of the authors can be probably solved as described by the authors. However, the problem of testing the goodness-of-fit is another theme. It should be not only restricted to local elements of the crack system such as the authors’ summaries, but should instead consider the spatial aspect of the structures and use ideas of spatial statistics. Perhaps the experiences with tests of the hypothesis that a given tessellation is a Voronoi tessellation Žsee Stoyan et al., 1995; Hahn and Lorz, 1994. can be inspiring. I would recommend statistical methods that use point process methods as in Stoyan and Stoyan Ž 1990. . Any crack system is associated with various point patterns, for example, the nodes, cell centres, edge centres etc. These point processes could be statistically analysed and then compared with the corresponding processes of the model.
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Hans-Joerg Vogel UniÕersity of Heidelberg, Institute of EnÕironmental Physics, 69120 Heidelberg, Germany Numerous experimental studies and theoretical considerations taught us that soil structure is a key to understanding the phenomenology of processes in soil and therefore a key to formulate quantitative models. This is certainly true for the structure of cracks and the process of water flow and solute transport. However the dynamics and often the mere existence of cracks is typically ignored in modeling transport processes. The difficulty is the lack of operational tools providing a quantitative characterization of the geometry of crack networks and their dynamics in dependency of material properties and state variables. The present paper thus deals with a very important field. The approach of Horgan and Young starts from the top, the phenomenology at the macroscale, and does not explicitly consider the microscopic processes that lead to fragmentation. The starting point is the observable network of cracks and the question of how to define a stochastic process generating a similar structure. Essentially, the resulting model provides a parametric description of the crack geometry which is attractive. In contrast to purely geometrical measures, which were used in the present paper to compare the results of simulations with real crack networks, the parameters of the proposed model are expected to be more easily related to the physics of crack formation. This is because the parameters directly define the propagation of cracks at different hierarchical levels. However, the relation to the underlying physics remains to be clarified and so far, the approach at the present stage is merely descriptive. Another approach could be to start from the bottom with microscopic processes, i.e., the cohesive forces between microscopic regions. These cohesive forces may be distributed randomly to introduce a predefined heterogeneity. Modelling the course of forces during shrinkage together with the introduction of a critical stress intensity could lead to a realistic pattern of cracks. Whatever the starting point, we should aspire to arrive at models which conceptualize the underlying physical processes. Only then, there is a chance that the model parameters can be related to Ži. material properties such as the texture and the quality of mineral composition that allow the model to be extrapolated and, Ž ii. state variables of the material such as water content and water potential that provide a handle to the temporal dynamics of the system. Consequently, the next step following the approach of Horgan and Young should be the physical interpretation of the parameters — they have provided some speculations in this direction, which are reasonable — and the algorithms for crack generation. As an example, the model predicts systematic differences between branchings at the initial point of a crack and its end point, clearly because the implemented rules of crack propagation are different. This should not be difficult to verify.
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In contrast to Horgan and Young’s statement, I recommend doing this step first, before relating the model parameters to soil properties. Then, hopefully, this relation will not remain empirical, our understanding of the underlying processes will be improved and the model may be extended to three dimensions. Otherwise, we may end up with another doubtful pedotransfer function. I am looking forward to see things developing along these lines. B. Velde Ecole Normale Superieure, CNRS UMR, 24 rue Lhomond, 75231 Paris, ´ France This paper presents a model, rather ex nihilo, for producing patterns of crack growth in soil surfaces. It is a 2D model. An obvious success of the model is to create slightly curved crack segments that are often seen in natural field soil-crack patterns. A second interesting feature is the production of crack intersections with three branches, another predominant feature in natural soilcrack patterns. The crack patterns seem, a priori, to conform to those seen in soils. One defect, noted by the authors is the lack of unconnected cracks Ž dead ends., which are typical of natural crack patterns. Some drawbacks of the method, using a stochastic model, can hinder the comparison of these ‘‘synthetic’’ networks with natural ones. Natural crack networks appear to have fractal distributions in 2D space Ž Velde, 1999. . This fact excludes comparison of the distribution patterns of cracks in nature with those generated by the model. As noted by the authors, the size of the delineated zones Žaggregates in the text. is less variable than that of the ‘‘natural’’ soil given in Preston et al. Ž1997.. However it appears that changing some of the parameters in the model, splitting threshold A1 shown in Fig. 5, can give larger variations in aggregate size. It might be mentioned that these soils, quoted in this paper as an example of a natural occurrence are in fact artificial, being a homogenised soil material which was dried in the laboratory. As it turns out, the example of Preston et al. Ž 1997. is not of fractal distribution in 2D space which differentiates it from soils found in the field. It seems, overall, that the model proposed gives a rather regular distribution of aggregate sizes, crack lengths, neighbouring aggregates, and so forth. This suggests that it produces a somehow more regular pattern than the natural field soil-crack patterns. Comparison of field examples with model results could lead to a more realistic model by varying some of the parameters to fit more closely the patterns of natural field crack patterns if this is in fact the goal of the project. The project is a good one, but as the case with most idealisations of natural objects, it must be confirmed by a direct comparison with the objects in question. I apologise for a large portion of the above comments in that they refer to work that was only recently published Ž Velde, 1999. and was unavailable to the
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authors when they wrote their manuscript. This is of course unfair to the authors concerned. C.E. Mullins Department of Plant & Soil Science, UniÕersity of Aberdeen, Scotland, UK The empirical model of Horgan and Young Ž 1999. aims to describe the pattern of cracking in a drying soil. Although there is no discussion of effective stress Ž i.e., the stress field generated due to drying. and hence of the physical factors that give rise to cracking, the model makes implicit reference to these via the assumptions that underpin it Ž such as that of a perpendicular growth direction of one crack initiating along the length of another. . To the extent that this and other such models can be demonstrated to be physically based, they can be expected to give insight into those soil and other factors Ž such as rate of drying. that affect cracking pattern. Where assumptions are made Ž or modelling parameters are introduced. that do not have a demonstrable physical basis, there is a danger that the model may become an object of study in its own right. However, a descriptive model may be used to generate artificial crack distribution patterns possessing similar physical properties Ž such as gaseous diffusion. to the modelled soil. To this extent it may be of practical relevance. What the Horgan and Young model lacks is a physical basis on which to predict the characteristic size of the smallest units Ž the ‘‘stable aggregate size’’. . Despite this, the model could be used to determine the extent to which all cracking soils share a given set of organisational features in common. The simplicity and transparency of the model and the small number of parameters used in the crack pattern summaries suggests that it could be very useful for this purpose. Attempts have been made to relate cracking to soil physical properties Ž e.g., Towner, 1988. and also to explain why there is little if any cracking in hardsetting soils ŽMullins et al., 1992.. These highlight a major challenge to theories of cracking, which is to explain why different soils produce fine or coarse angular blocky structures ŽReeve and Hall, 1978. , are hardsetting, or produce a hierarchy of aggregation as in some vertisols. This represents the problem of predicting the stable aggregate size. Progress in this area probably needs at the minimum: Ži. an understanding of how K ŽCm . , the ‘‘unsaturated’’ conductivity function varies as soil dries, combined with Ž ii. knowledge of the initial distribution of elongated air-filled voids, and Ž iii. an understanding of how crack-stopping mechanisms operate at different matric potentials Ž Mullins et al., 1992.. The first of these indicates whether localised differences in effective stress can be evened out by water flow before they sufficient to cause cracking. The second indicates the distribution of sites at which crack initiation could start, and the third represents the ease with which cracks can extend.
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P.A.C. Raats Department of Agricultural, EnÕironmental and Systems Technology, Wageningen UniÕersity, the Netherlands Horgan and Young Žthis paper. present a novel approach to the elusive problem of crack growth in soil. Rather realistic looking cracking patterns are obtained by a random walk subject to certain empirical rules. Actually this is not altogether surprising. Their rules for attraction of a growing crack by an existing crack and for initiation of a new crack at an existing crack are qualitatively in agreement with known mechanical aspects of the problem Ž see Raats, 1994 for a review of some of the early literature.. In this connection, a key feature of an appropriate mechanical model is the anisotropy of the stress relief near an existing crack, this relief being larger perpendicular to the crack than parallel to the crack. In the remainder of this commentary, I shall relate some properties of the resulting patterns given in Tables 2, 3, 4, and 5 of Horgan and Young Ž this paper. to the theory of planar tessellations. Euler showed that for any convex polyhedron the number of vertices Õ, edges e, and faces f are related by: Õ y e q f s 2.
Ž D1.
This formula also applies to any connected network in the plane, provided the region outside the network is counted as one polygon Ž Coxeter, 1973. . Let p be the number of edges and vertices per polygon and q the number of edges and polygons at each vertex. The numbers p and q are measures of the connectedness of the network, respectively, per polygon and per vertex. It can be shown that as the number of edges e goes to infinity Eq. ŽD1. is equivalent to: 1rp q (q s '2 .
Ž D2.
The three integer solutions of Ž D2. correspond to the three well-known regular tessellations of the plane by equilateral triangles wŽ p,q . s Ž3,6.x, squares wŽ p,q . s Ž4,4.x, and hexagons wŽ p,q . s Ž6,3.x. The ‘‘compactness’’ as defined by Horgan and Young Ž this paper. is 1 for circles, 0.907 for hexagons, 0.785 for squares, and 0.605 for equilateral triangles. It is interesting to note in Tables 2, 3, 4 and 5 that only one of the 11 values for the compactness is intermediate between those for squares and hexagons and the other 10 are intermediate between those for equilateral triangles and squares. For random networks the average connectedness p per polygon and q per vertex can be shown to satisfy an equation of the same form as ŽD3. ŽGray et al., 1976. : 1rp q (q s '2 .
Ž D3.
Gray et al. Ž1976. distinguish three kinds of junctions: true triple connected junctions indicated by Y, truncated triple connected junctions indicated by T,
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and four-connected junctions indicated by X. Let f Y , f T , and f X be the fractions of the junctions, respectively of the types Y, T and X. These fractions are subject to the constraint f Y q f T q f X s 1.
Ž D4.
The connectedness per vertex is 3 for the Y and T junctions, and 4 for the X junctions. Using ŽD4. it is easily shown that q s 3 q fX .
Ž D5.
Solving ŽD3. for p in terms of q and substituting Ž5. gives: p s Ž 6q2 f X . r Ž 1qf X . .
Ž D6.
After a network of polygons is taken apart, one of the three polygons which met in the intact network at a T junction will have two of its edges in line with each other. As a result, for each T junction one vertex and one edge will not be counted. The average number g of edges of the disassembled polygons can be shown to be given by: g s Ž 6 q 2 f X y 2 f T . r Ž 1 q f X . s p y 2 f Tr Ž 1 q f X . .
Ž D7.
Gray et al Ž 1976, Fig. I. present graphically iso-p and iso-g lines as functions of f Y , f T , and f X . Since X junctions are rare and Y junctions become rarer as subdivision proceeds, two special cases are of particular interest. If f X approaches zero, then ŽD5., ŽD6., and ŽD7. reduce, respectively, to q s 3,
p s 6,
g s 6 y 2 fT s p y 2 fT .
Ž D8.
If moreover f Y approaches zero, so that f T approaches unity, as will be caused by indefinite subdivision of polygons, then ŽD8. further reduces to: q s 3,
p s 6,
g s 4.
Ž D9.
Accordingly, disassembling any regular or random T junction network will obliterate 1r3 of the edges and vertices per polygon. In Tables 2, 3, 4, and 5 of Horgan and Young Ž this paper. the values of the properties denoted as ‘‘edges’’ and ‘‘neighbours’’ closely agree, respectively, with the values of g given by Ž D9. and of p given by Ž D8. and Ž D9. . The small deviation may be related to the fact that in the theory outlined above the region outside the network is to be counted as one polygon. The Authors’ Reply The discussants have offered some thoughtful comments for which we are grateful. To give structure to our reply, we will classify these into three types: those relating to the physics of soil cracking, those which quibble with and make
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suggestions about aspects of our model, and those which compare them with other models. We will consider these three topics in turn. Physics of cracking The patterns observed when soil dries and cracks are at the top of a hierarchy of processes, which start at the molecular level, and extend up to the macroscopic. A full understanding requires some knowledge at all levels, and we admit that our model only addresses what occurs at the top, macroscopic, level where the cracking can be seen. Our starting point was real cracks, produced within a thin layer of homogenised soil, which we examined in order to discern what geometric processes produce these cracks. Our intention is to understand cracking by working down the scale hierarchy. We ask what geometric processes can produce the patterns we see, and have concluded that random walk growth, crack attraction and aggregate splitting are essential. The next step, as we state in our paper, is then to ask what physical processes can produce these geometric effects, and how physical parameters relate to geometric parameters. To include additional parameters at this very early stage in our model would have been a mistake. Hence, some of the criticisms ŽMullins and Vogel. are inappropriate. Some parts of our geometric model are easier than others to relate to soil physics. Random walk crack growth is one. This must surely be driven by heterogeneity in the composition of the soil. In a perfectly uniform material, there would be nothing to cause a crack to deviate from a smooth path. It seems reasonable to suppose that, within limits, the greater the heterogeneity the more curved will be the cracks. However, it is not clear how this heterogeneity should be defined. It will be partly compositional and partly spatial: an even mixture of different components can be considered less heterogeneous, one which is more spatially correlated. Our model incorporates a transition from unconstrained random crack growth to cracking which splits aggregates. This transition is an abrupt change, which is surely artificial. We expect the reality to be more gradual. Our defence for our approach is that any ideas we had for a more gradual transition would have been computationally intractable to simulate. Understanding what is really happening here is important, as this aspect of the geometry is essential: there must be cracking constrained to split aggregates, and this constraint must increase as cracking progresses. This is related to another important parameter, the stable aggregate size. Moran, Kirby and Mullins point to some of the features of soil which affect these processes, and we will study this further. Weaknesses of, and improÕements to, the model Two discussants ŽStoyan and Velde. comment on the geometry of our model. Velde observes that our results are more regular than would be observed in the field. However, the degree of heterogeneity in surface crack geometry found in
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the field is far greater than that suggested by Velde. Indeed, in his most recent paper Ž Velde, 1999. some of the surface cracks presented have remarkably simple patterns. If his point is that our model must be able to account for a relatively wide range in cracking patterns, then it is well taken. Increasing crack irregularity in our model would not be difficult to incorporate. We could allow some of the parameters Žsay A2 and a . to vary with position, either deterministically or stochastically, as a Gaussian random field, for example. To do so would produce more irregular patterns than those we exhibit, without changing the essential geometry of the model. Stoyan suggests other models for the initial process of crack starting points, and would prefer these to be a space-stationary process. We accept that this would be more satisfying, and could be easily done. Our simulations are finite for computational convenience, and edge effects no worse than those in a real occurrence of cracks. The summary by Raats of relevant results from planar tessellation theory is useful. Another reason for the small deviations of our mean edge and neighbour counts from theoretical expectations is that the model is simulated on a discrete grid, although defined in the continuous plane. Cracks therefore have a nonzero thickness, and a slight overestimate of neighbours and underestimate of edges can result. The remark by Velde with regard to the fractal nature of cracks analysed in Preston et al. Ž1997., while inappropriate to our present work, is easily dealt with. No natural features are truly fractal in the mathematical sense: all such features have upper and lower bounds where the use of fractals becomes inappropriate. However, within the context of the approximations outlined in some detail by Preston et al. Ž1997. , and developed by Preston et al. Ž1999., the use of a power-law descriptor is sound. See Crawford and Young Ž 1998. , for a more detailed explanation. Indeed the work of Velde Ž 1999. does not show truly fractal cracking patterns. Comparison with other models Many other attempts have been made to model cracking processes. Stoyan gives a useful summary of the geometric approaches, and places ours in context. Our model is indeed edge-based rather than cell based, and we maintain that this is truer to physical reality. Moran and Kirby draw attention to other work which we had not reviewed. We regret this omission. The work of Moran and McBratney Ž1997. appears closest in construction to ours, and so shares some of its attractions and limitations. It aims to simulate soil pore structure in general, whereas we have concentrated on macroscopic cracks produced by drying. The model of Perrier et al. Ž1995. differs from ours in being cell-based rather than edge-based, and in being fractal, with cracks occurring at many size scales. Our model is not fractal, as the cracks we observed did not appear to be, but this is certainly a feature sometimes occurring in real soil. The model could be made
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fractal by allowing the cracking to continue to much smaller cell sizes, with crack thickness decreasing also. Other remarks We finish by saying that we hope our ideas can be seen as the start of a modelling process, rather than the end-product. The geometric ideas need to be extended in two directions: towards the more fundamental physical processes which ultimately are driving it, and towards the greater complexity which must result when the heterogeneity, at many scales, of real soil is accounted for. Only by doing so can the usefulness of the model be determined. The point we emphasise in our paper, and which is echoed by Moran and Kirby, is that the ultimate aim of our research on the genesis of cracking is to provide functional significance to the observed physical structures in soil systems.
References Bronswijk, J.J.B., 1988. Modelling of water balance, cracking and subsidence of clay soils. J. Hydrol. 97, 199–212. Coxeter, H.S.M., 1973. Regular Polytopes. 3rd edn. Dover, New York, 321 pp. Crawford, J.W., Young, I.M., 1998. The interactions between soil structure and microbial dynamics. In: Baveye, P., Parlange, J.Y., Stewart, B.A. ŽEds.., Fractals in Soil Science. CRC Press, Boca Raton, FL, pp. 233–260. Gray, N.H., Anderson, J.B., Devine, J.D., Kwasnik, J.M., 1976. Topological properties of crack networks. Math. Geol. 8, 617–626. Grenander, U., 1996. Elements of Pattern Theory. Johns Hopkins University Press, Baltimore. Hahn, U., Lorz, U., 1994. Stereological analysis of the spatial Poisson–Voronoi tessellation. J. Microsc. 175, 176–185. Hallaire, V., Curmi, P., 1994. Image analysis of pore space morphology in soil sections, in relation to water movement. In: Ringrose-Voase, A.J., Humphreys, G.S. ŽEds.., Soil Micromorphology: Studies in Management and Genesis. Proceedings of the IX International Workshop on Soil Micromorphology, Townsville, Australia, July 1992. Dev. Soil Sci. 22 Elsevier, Amsterdam, pp. 559–567. Kirby, J.M., Ringrose-Voase, A.J., 1999. Drying of puddled rice soils following surface drainage: numerical analysis using a swelling soil flow model. Soil Tillage Res., submitted for publication. Moran, C.J., McBratney, A.B., 1997. A two-dimensional fuzzy random model of soil pore structure. Math. Geol. 29, 755–777. Mullins, C.E., Cass, A., MacLeod, D.A., Hall, D.J.M., Blackwell, P.S., 1992. Strength development during drying of cultivated, flood-irrigated hardsetting soil: II. Trangie soil and comparison with theoretical predictions. Soil Tillage Res. 25, 129–147. Perrier, E., Mullon, C., Rieu, M., de Marsily, G., 1995. Computer construction of fractal soil structure: simulation of their hydraulic and shrinkage properties. Water Resour. Res. 31, 2927–2943. Preston, S., Griffiths, B.S., Young, I.M., 1997. An investigation into sources of soil crack heterogeneity using fractal geometry. Eur. J. Soil Sci. 48, 31–37. Preston, S., Griffiths, B.S., Young, I.M., 1999. Links between substrate additions, native microbes, and the structural complexity and stability of soils. Soil Biol. Biochem. 31, 1541–1547.
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Raats, P.A.C., 1994. Mechanics of cracking soils. In: Bouma, J., Raats, P.A.C. ŽEds.., Proceedings ISSS Symposium on Water and Solute Movement in Heavy Clay Soils. ILRI Publication 27. International Institute for Land Reclamation and Improvement, Wageningen, the Netherlands, pp. 23–38. Reeve, M.J., Hall, D.G.M., 1978. Shrinkage in clayey subsoils of contrasting structure. J. Soil Sci. 29, 315–323. Rieu, M., Sposito, G., 1991. Fractal fragmentation, soil porosity and soil water properties: 1. Theory. Soil Sci. Soc. Am. J. 55, 1231–1238. Ross, P.J., Bridge, B.J., 1984. MICCS: a model of infiltration into cracking clay soil. In: McGarity, J.W., Hoult, E.H., So, H.B. ŽEds.., The Properties and Utilization of Cracking Clay Soils, Proceedings of a Symposium held at the University of New England, Armidale, NSW, 24–28 August, 1981, Armidale, University of New England. pp. 155–163. Schlather, M., Stoyan, D., 1999. Edge systems of time-dependent incomplete Poisson Voronoi tessellations. Stochastic Models Žin press.. Sˇ imak, J., Benes, M., 1998. Simulation of fibre ˇ V., Lauschmann, H., Cejka, V., Masata, ˇ processes, application in fractography. Acta Stereol. 17, 315–320. Stoyan, D., Kendall, W.S., Mecke, J., 1995. Stochastic Geometry and its Applications. Wiley, Chichester. Stoyan, D., Stoyan, H., 1990. Exploratory data analysis for planar tessellations: structural analysis and point process methods. Appl. Stochastic Mod. Data Anal. 6, 13–25. Towner, G.D., 1988. The influence of sand-and silt-size particles on the cracking during drying of small clay-domained aggregates. J. Soil Sci. 39, 347–356. Velde, B., 1999. Structure of surface cracks in soil and muds. Geoderma 93, 101–124. Winkler, T., Bruckner-Foit, A., Riesch-Oppermann, H., 1992. Statistical characterization of ¨ random crack patterns caused by thermal fatigue. Fatigue Fract. Eng. Mater. Struct. 15, 1025–1039.
Discussion of the paper by G.W. Horgan and I.M. Young 1 C.J. Moran, J.M. Kirby CSIRO Land and Water, GPO Box 1666, Canberra ACT 2601, Australia Received 11 November 2000; accepted 6 December 2000
Horgan and Young present a geometrical statistical random model of cracking patterns. They compare the simulations to cracking patterns from thin layers of homogenised soil. In this discussion, we raise four issues not explicitly dealt with by the authors which impact on how well their approach can be considered to deal with the subject of soil cracking: two other papers dealing with cracking have not been reviewed; the evolution of cracking patterns depends on the mechanics of soil drying and local heterogeneity; cracking patterns in a three-dimensional system are not necessarily well represented by two-dimensional analogues; and, finally how 2D traces might realistically be used in soil water balance modelling in swelling systems. Horgan and Young have not reviewed the work of Moran and McBratney Ž1997. nor that of Perrier et al. Ž1995. which are both relevant to this topic. Moran and McBratney presented a generalised model of soil structure including simulation of cracking patterns. Their basic approach was not dissimilar to that adopted by Horgan and Young. Moran and McBratney simulated cracks using a so-called linked distribution of points. This is a statistically constrained random
1
References to this paper should include the discussion and should be cited as: Horgan, G.W., Young, I.M., 2000. An empirical stochastic model for the geometry of two-dimensional crack growth in soil Žwith Discussion.. Geoderma 96, 263–289. References to parts of the Discussion should be cited as, e.g., Moran, C.J., Kirby, J.M., 2000. In: Discussion of: Horgan, G.W., Young, I.M., 2000. An empirical stochastic model for the geometry of two-dimensional crack growth in soil. Geoderma 96, 263–289. 0016-7061r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 6 - 7 0 6 1 Ž 0 0 . 0 0 0 1 7 - 3
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walk. A directional distribution Ž Von Mises. is sampled to provide the step direction Ž du . in Horgan and Young’s terminology. A second distribution can also be invoked to determine the morphological details of the cracks. The width of cracks and the step length Ž dC . is controlled by centering an ellipse at the step. This model allows simulation of cracking patterns with variable orientation juxtapositions, e.g., blocky, polygonal and striated patterns Žsee Fig. 11 of Moran and McBratney, 1997. . Horgan and Young have introduced distance inhibition and aggregate bisecting to simulate the result of cracking. Moran and McBratney similarly require a large input specification to produce a simulation of the final structure. Both approaches suffer because they do not simulate the interaction between drying and mechanical behaviour. Perrier et al. Ž1995. presented a fractal fragmentation model based on the conceptual model of Rieu and Sposito Ž1991.. While not strictly a model of surface cracking, like the model of Moran and McBratney, it is a relevant geometrical simulation. These authors then used the cracking patterns thus generated to model the feedbacks between soil drying, hydraulic conductivity and swelling and shrinking of the soil Žthereby altering the extent of the cracks.. This remains the only paper linking the physics of water movement to the geometry of the cracking system. However, it still does not address the fundamental question of the evolution of a cracking pattern as driven by the wetting and drying sequences: the cracks of Perrier et al. Ž 1995. simply expand and contract. Observation of real soil suggests that the development of the cracking pattern is more complex and changes as wetting or drying proceeds. Surface cracking patterns often develop as the soil dries from the surface downwards. The water content depth profile plays a role in defining the tension field set up as shrinking commences. Therefore, the tension failure that we view as crack initiation is difficult to model without taking the third dimension into account. Fast drying profiles form many, small, closely spaced cracks, whereas slow drying profiles form a few, large, widely spaced cracks. Further, as cracks widen, the effective surface area available for drying increases providing feedback into the crack development process. Also not taken into account is the influence of clay mineralogy or soil mechanical properties. It is obvious that soils with different types and proportions of clay develop different cracking patterns, but the models of Horgan and Young Ž this paper. , Moran and McBratney Ž1997. and Perrier et al. Ž1995. ignore this influence. Taking the progress delivered by Horgan and Young’s simulations, we are still faced with the question of how to use such information in estimating soil water movement. Bronswijk Ž 1988. described a one-dimensional model of water balance, cracking and subsidence of clay soils. The ‘‘cracks’’ in a one-dimensional model are simply the proportion of the pore volume, as a function of depth, which is attributed to cracks rather than matrix pore space. Simulations of moisture content, surface subsidence and crack volume agreed reasonably well with ten months of measured field behaviour. Evidently, it was unnecessary to
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invoke information about crack patterns or sizes. Kirby and Ringrose-Voase Ž1999. also found reasonable agreement between a one-dimensional model simulating drying of puddled rice soils and several weeks of measured field behaviour. Again, it was unnecessary to invoke information about crack patterns or sizes. On the other hand, under saturated conditions water moves preferentially through macropores and this depends on the type of macropore and the degree to which the macropores are connected ŽHallaire and Curmi, 1994. . Ross and Bridge Ž1984. showed that cracks and their distribution patterns are crucial in determining water infiltration, and potentially, the residence times for water Ž and associated pesticides and nutrients. variously spatially distributed in a seasonally cracking soil. Characterisation of these issues probably requires representation of the soil structure and water movement in three dimensions. It is difficult to see how robust 1D or even 2D approximations can be derived without such a research model. Verification and parameterisation of such a model, as always, will prove a significant challenge. Dietrich Stoyan Institut fuer Stochastik, TU Bergakademie Freiberg, Bernhard-Õon-Cotta-Str. 2, D-09596 Freiberg, Germany The authors study and partially solve a fascinating problem — modelling the geometry of planar crack systems in soil surfaces. My own observations Ž see Figs. 9.10 and 10.2 in Stoyan et al., 1995. say that the authors are able to produce excellent simulations, which are rather similar to real crack patterns. I congratulate them on this success. My following remarks concern the problem of modelling crack systems in general and try to point to further research. Most of my ideas also appear in the paper discussed here. Modelling of crack systems and crack tessellations is a problem which is important in many fields of science and engineering, not only in soil science. I believe that it could be fruitful to collect the widely scattered material and to combine the various approaches, which seem to be developed independently by physicists, engineers and statisticians, in a unique and general theory. Clearly, the cracking materials determine speed and form of cracks and geometry of crack systems; cracks on the surface of metals or ceramics are quite different to cracks in soil. The ultimate aim of research in crack systems should be physical models with measurable material parameters, which produce the geometry as a ‘‘by-product’’. Pure geometrical models such as that given by the authors can be only a first step in modelling. The existing geometrical models can perhaps be classified as follows: The final product of planar crack processes is a planar tessellation Žsometimes with the exception of some cracks which do not join other cracks and end blindly. .
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The elements of any tessellation are nodes, edges and cells. The process of forming a tessellation can be based on these elements. In tessellations based on nodes, the cracks appear between the nodes, so as were their endpoints prescribed before; an example of such a model is that discussed in Stoyan and Stoyan Ž1990., which was used for modeling of cracks in basaltic structures. The model presented by Horgan and Young could be interpreted as an edge-based model, since here the edges are the primary elements. The most famous models are cell-based, such as the Voronoi and Johnson–Mehl tessellation, which are created starting with cell centres. These models have been used for modelling cracks, see, e.g., Winkler et al. Ž 1992. and Schlather and Stoyan Ž 1999. . The starting point of the model given by the authors is in my opinion quite natural. To begin with random points and to have crack growth which follows somewhat and which is similar to a random walk is natural. However, already here some refining may be useful: the points could positioned more regularly than independent points Žperhaps this can make the final patterns more similar to real patterns. and the cracks could be modelled as the curÕe images in ˇ Grenander Ž1996., p. 109; by the way, Simak et al. Ž 1998. discuss a Markovchain-based fibre process model in the context of fractography, which produces cracks similar to those produced by the authors. The other rules seem to be results of observations of soil crack processes and help to produce realistic geometries. They lack any quantitative physical argument and are similar to the ad hoc rules in Stoyan and Stoyan Ž 1990. . Here still a lot of physical work has to be done which should finally lead to physical models which produce crack systems where the cracks minimize the shrinking force. Perhaps Gibbs processes will be used one day in this context. I am not entirely happy with the finite design of the model. I would prefer a Žspace-. stationary or homogeneous model, which starts with the points of a stationary point process that replaces the N0 independent points. The further rules of the model should then be formulated in such a way that a stationary planar geometrical structure is obtained. Finally, some words on the statistics of crack systems. The problem of parameter estimation for models such as that of the authors can be probably solved as described by the authors. However, the problem of testing the goodness-of-fit is another theme. It should be not only restricted to local elements of the crack system such as the authors’ summaries, but should instead consider the spatial aspect of the structures and use ideas of spatial statistics. Perhaps the experiences with tests of the hypothesis that a given tessellation is a Voronoi tessellation Žsee Stoyan et al., 1995; Hahn and Lorz, 1994. can be inspiring. I would recommend statistical methods that use point process methods as in Stoyan and Stoyan Ž 1990. . Any crack system is associated with various point patterns, for example, the nodes, cell centres, edge centres etc. These point processes could be statistically analysed and then compared with the corresponding processes of the model.
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Hans-Joerg Vogel UniÕersity of Heidelberg, Institute of EnÕironmental Physics, 69120 Heidelberg, Germany Numerous experimental studies and theoretical considerations taught us that soil structure is a key to understanding the phenomenology of processes in soil and therefore a key to formulate quantitative models. This is certainly true for the structure of cracks and the process of water flow and solute transport. However the dynamics and often the mere existence of cracks is typically ignored in modeling transport processes. The difficulty is the lack of operational tools providing a quantitative characterization of the geometry of crack networks and their dynamics in dependency of material properties and state variables. The present paper thus deals with a very important field. The approach of Horgan and Young starts from the top, the phenomenology at the macroscale, and does not explicitly consider the microscopic processes that lead to fragmentation. The starting point is the observable network of cracks and the question of how to define a stochastic process generating a similar structure. Essentially, the resulting model provides a parametric description of the crack geometry which is attractive. In contrast to purely geometrical measures, which were used in the present paper to compare the results of simulations with real crack networks, the parameters of the proposed model are expected to be more easily related to the physics of crack formation. This is because the parameters directly define the propagation of cracks at different hierarchical levels. However, the relation to the underlying physics remains to be clarified and so far, the approach at the present stage is merely descriptive. Another approach could be to start from the bottom with microscopic processes, i.e., the cohesive forces between microscopic regions. These cohesive forces may be distributed randomly to introduce a predefined heterogeneity. Modelling the course of forces during shrinkage together with the introduction of a critical stress intensity could lead to a realistic pattern of cracks. Whatever the starting point, we should aspire to arrive at models which conceptualize the underlying physical processes. Only then, there is a chance that the model parameters can be related to Ži. material properties such as the texture and the quality of mineral composition that allow the model to be extrapolated and, Ž ii. state variables of the material such as water content and water potential that provide a handle to the temporal dynamics of the system. Consequently, the next step following the approach of Horgan and Young should be the physical interpretation of the parameters — they have provided some speculations in this direction, which are reasonable — and the algorithms for crack generation. As an example, the model predicts systematic differences between branchings at the initial point of a crack and its end point, clearly because the implemented rules of crack propagation are different. This should not be difficult to verify.
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In contrast to Horgan and Young’s statement, I recommend doing this step first, before relating the model parameters to soil properties. Then, hopefully, this relation will not remain empirical, our understanding of the underlying processes will be improved and the model may be extended to three dimensions. Otherwise, we may end up with another doubtful pedotransfer function. I am looking forward to see things developing along these lines. B. Velde Ecole Normale Superieure, CNRS UMR, 24 rue Lhomond, 75231 Paris, ´ France This paper presents a model, rather ex nihilo, for producing patterns of crack growth in soil surfaces. It is a 2D model. An obvious success of the model is to create slightly curved crack segments that are often seen in natural field soil-crack patterns. A second interesting feature is the production of crack intersections with three branches, another predominant feature in natural soilcrack patterns. The crack patterns seem, a priori, to conform to those seen in soils. One defect, noted by the authors is the lack of unconnected cracks Ž dead ends., which are typical of natural crack patterns. Some drawbacks of the method, using a stochastic model, can hinder the comparison of these ‘‘synthetic’’ networks with natural ones. Natural crack networks appear to have fractal distributions in 2D space Ž Velde, 1999. . This fact excludes comparison of the distribution patterns of cracks in nature with those generated by the model. As noted by the authors, the size of the delineated zones Žaggregates in the text. is less variable than that of the ‘‘natural’’ soil given in Preston et al. Ž1997.. However it appears that changing some of the parameters in the model, splitting threshold A1 shown in Fig. 5, can give larger variations in aggregate size. It might be mentioned that these soils, quoted in this paper as an example of a natural occurrence are in fact artificial, being a homogenised soil material which was dried in the laboratory. As it turns out, the example of Preston et al. Ž 1997. is not of fractal distribution in 2D space which differentiates it from soils found in the field. It seems, overall, that the model proposed gives a rather regular distribution of aggregate sizes, crack lengths, neighbouring aggregates, and so forth. This suggests that it produces a somehow more regular pattern than the natural field soil-crack patterns. Comparison of field examples with model results could lead to a more realistic model by varying some of the parameters to fit more closely the patterns of natural field crack patterns if this is in fact the goal of the project. The project is a good one, but as the case with most idealisations of natural objects, it must be confirmed by a direct comparison with the objects in question. I apologise for a large portion of the above comments in that they refer to work that was only recently published Ž Velde, 1999. and was unavailable to the
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authors when they wrote their manuscript. This is of course unfair to the authors concerned. C.E. Mullins Department of Plant & Soil Science, UniÕersity of Aberdeen, Scotland, UK The empirical model of Horgan and Young Ž 1999. aims to describe the pattern of cracking in a drying soil. Although there is no discussion of effective stress Ž i.e., the stress field generated due to drying. and hence of the physical factors that give rise to cracking, the model makes implicit reference to these via the assumptions that underpin it Ž such as that of a perpendicular growth direction of one crack initiating along the length of another. . To the extent that this and other such models can be demonstrated to be physically based, they can be expected to give insight into those soil and other factors Ž such as rate of drying. that affect cracking pattern. Where assumptions are made Ž or modelling parameters are introduced. that do not have a demonstrable physical basis, there is a danger that the model may become an object of study in its own right. However, a descriptive model may be used to generate artificial crack distribution patterns possessing similar physical properties Ž such as gaseous diffusion. to the modelled soil. To this extent it may be of practical relevance. What the Horgan and Young model lacks is a physical basis on which to predict the characteristic size of the smallest units Ž the ‘‘stable aggregate size’’. . Despite this, the model could be used to determine the extent to which all cracking soils share a given set of organisational features in common. The simplicity and transparency of the model and the small number of parameters used in the crack pattern summaries suggests that it could be very useful for this purpose. Attempts have been made to relate cracking to soil physical properties Ž e.g., Towner, 1988. and also to explain why there is little if any cracking in hardsetting soils ŽMullins et al., 1992.. These highlight a major challenge to theories of cracking, which is to explain why different soils produce fine or coarse angular blocky structures ŽReeve and Hall, 1978. , are hardsetting, or produce a hierarchy of aggregation as in some vertisols. This represents the problem of predicting the stable aggregate size. Progress in this area probably needs at the minimum: Ži. an understanding of how K ŽCm . , the ‘‘unsaturated’’ conductivity function varies as soil dries, combined with Ž ii. knowledge of the initial distribution of elongated air-filled voids, and Ž iii. an understanding of how crack-stopping mechanisms operate at different matric potentials Ž Mullins et al., 1992.. The first of these indicates whether localised differences in effective stress can be evened out by water flow before they sufficient to cause cracking. The second indicates the distribution of sites at which crack initiation could start, and the third represents the ease with which cracks can extend.
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P.A.C. Raats Department of Agricultural, EnÕironmental and Systems Technology, Wageningen UniÕersity, the Netherlands Horgan and Young Žthis paper. present a novel approach to the elusive problem of crack growth in soil. Rather realistic looking cracking patterns are obtained by a random walk subject to certain empirical rules. Actually this is not altogether surprising. Their rules for attraction of a growing crack by an existing crack and for initiation of a new crack at an existing crack are qualitatively in agreement with known mechanical aspects of the problem Ž see Raats, 1994 for a review of some of the early literature.. In this connection, a key feature of an appropriate mechanical model is the anisotropy of the stress relief near an existing crack, this relief being larger perpendicular to the crack than parallel to the crack. In the remainder of this commentary, I shall relate some properties of the resulting patterns given in Tables 2, 3, 4, and 5 of Horgan and Young Ž this paper. to the theory of planar tessellations. Euler showed that for any convex polyhedron the number of vertices Õ, edges e, and faces f are related by: Õ y e q f s 2.
Ž D1.
This formula also applies to any connected network in the plane, provided the region outside the network is counted as one polygon Ž Coxeter, 1973. . Let p be the number of edges and vertices per polygon and q the number of edges and polygons at each vertex. The numbers p and q are measures of the connectedness of the network, respectively, per polygon and per vertex. It can be shown that as the number of edges e goes to infinity Eq. ŽD1. is equivalent to: 1rp q (q s '2 .
Ž D2.
The three integer solutions of Ž D2. correspond to the three well-known regular tessellations of the plane by equilateral triangles wŽ p,q . s Ž3,6.x, squares wŽ p,q . s Ž4,4.x, and hexagons wŽ p,q . s Ž6,3.x. The ‘‘compactness’’ as defined by Horgan and Young Ž this paper. is 1 for circles, 0.907 for hexagons, 0.785 for squares, and 0.605 for equilateral triangles. It is interesting to note in Tables 2, 3, 4 and 5 that only one of the 11 values for the compactness is intermediate between those for squares and hexagons and the other 10 are intermediate between those for equilateral triangles and squares. For random networks the average connectedness p per polygon and q per vertex can be shown to satisfy an equation of the same form as ŽD3. ŽGray et al., 1976. : 1rp q (q s '2 .
Ž D3.
Gray et al. Ž1976. distinguish three kinds of junctions: true triple connected junctions indicated by Y, truncated triple connected junctions indicated by T,
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and four-connected junctions indicated by X. Let f Y , f T , and f X be the fractions of the junctions, respectively of the types Y, T and X. These fractions are subject to the constraint f Y q f T q f X s 1.
Ž D4.
The connectedness per vertex is 3 for the Y and T junctions, and 4 for the X junctions. Using ŽD4. it is easily shown that q s 3 q fX .
Ž D5.
Solving ŽD3. for p in terms of q and substituting Ž5. gives: p s Ž 6q2 f X . r Ž 1qf X . .
Ž D6.
After a network of polygons is taken apart, one of the three polygons which met in the intact network at a T junction will have two of its edges in line with each other. As a result, for each T junction one vertex and one edge will not be counted. The average number g of edges of the disassembled polygons can be shown to be given by: g s Ž 6 q 2 f X y 2 f T . r Ž 1 q f X . s p y 2 f Tr Ž 1 q f X . .
Ž D7.
Gray et al Ž 1976, Fig. I. present graphically iso-p and iso-g lines as functions of f Y , f T , and f X . Since X junctions are rare and Y junctions become rarer as subdivision proceeds, two special cases are of particular interest. If f X approaches zero, then ŽD5., ŽD6., and ŽD7. reduce, respectively, to q s 3,
p s 6,
g s 6 y 2 fT s p y 2 fT .
Ž D8.
If moreover f Y approaches zero, so that f T approaches unity, as will be caused by indefinite subdivision of polygons, then ŽD8. further reduces to: q s 3,
p s 6,
g s 4.
Ž D9.
Accordingly, disassembling any regular or random T junction network will obliterate 1r3 of the edges and vertices per polygon. In Tables 2, 3, 4, and 5 of Horgan and Young Ž this paper. the values of the properties denoted as ‘‘edges’’ and ‘‘neighbours’’ closely agree, respectively, with the values of g given by Ž D9. and of p given by Ž D8. and Ž D9. . The small deviation may be related to the fact that in the theory outlined above the region outside the network is to be counted as one polygon. The Authors’ Reply The discussants have offered some thoughtful comments for which we are grateful. To give structure to our reply, we will classify these into three types: those relating to the physics of soil cracking, those which quibble with and make
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suggestions about aspects of our model, and those which compare them with other models. We will consider these three topics in turn. Physics of cracking The patterns observed when soil dries and cracks are at the top of a hierarchy of processes, which start at the molecular level, and extend up to the macroscopic. A full understanding requires some knowledge at all levels, and we admit that our model only addresses what occurs at the top, macroscopic, level where the cracking can be seen. Our starting point was real cracks, produced within a thin layer of homogenised soil, which we examined in order to discern what geometric processes produce these cracks. Our intention is to understand cracking by working down the scale hierarchy. We ask what geometric processes can produce the patterns we see, and have concluded that random walk growth, crack attraction and aggregate splitting are essential. The next step, as we state in our paper, is then to ask what physical processes can produce these geometric effects, and how physical parameters relate to geometric parameters. To include additional parameters at this very early stage in our model would have been a mistake. Hence, some of the criticisms ŽMullins and Vogel. are inappropriate. Some parts of our geometric model are easier than others to relate to soil physics. Random walk crack growth is one. This must surely be driven by heterogeneity in the composition of the soil. In a perfectly uniform material, there would be nothing to cause a crack to deviate from a smooth path. It seems reasonable to suppose that, within limits, the greater the heterogeneity the more curved will be the cracks. However, it is not clear how this heterogeneity should be defined. It will be partly compositional and partly spatial: an even mixture of different components can be considered less heterogeneous, one which is more spatially correlated. Our model incorporates a transition from unconstrained random crack growth to cracking which splits aggregates. This transition is an abrupt change, which is surely artificial. We expect the reality to be more gradual. Our defence for our approach is that any ideas we had for a more gradual transition would have been computationally intractable to simulate. Understanding what is really happening here is important, as this aspect of the geometry is essential: there must be cracking constrained to split aggregates, and this constraint must increase as cracking progresses. This is related to another important parameter, the stable aggregate size. Moran, Kirby and Mullins point to some of the features of soil which affect these processes, and we will study this further. Weaknesses of, and improÕements to, the model Two discussants ŽStoyan and Velde. comment on the geometry of our model. Velde observes that our results are more regular than would be observed in the field. However, the degree of heterogeneity in surface crack geometry found in
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the field is far greater than that suggested by Velde. Indeed, in his most recent paper Ž Velde, 1999. some of the surface cracks presented have remarkably simple patterns. If his point is that our model must be able to account for a relatively wide range in cracking patterns, then it is well taken. Increasing crack irregularity in our model would not be difficult to incorporate. We could allow some of the parameters Žsay A2 and a . to vary with position, either deterministically or stochastically, as a Gaussian random field, for example. To do so would produce more irregular patterns than those we exhibit, without changing the essential geometry of the model. Stoyan suggests other models for the initial process of crack starting points, and would prefer these to be a space-stationary process. We accept that this would be more satisfying, and could be easily done. Our simulations are finite for computational convenience, and edge effects no worse than those in a real occurrence of cracks. The summary by Raats of relevant results from planar tessellation theory is useful. Another reason for the small deviations of our mean edge and neighbour counts from theoretical expectations is that the model is simulated on a discrete grid, although defined in the continuous plane. Cracks therefore have a nonzero thickness, and a slight overestimate of neighbours and underestimate of edges can result. The remark by Velde with regard to the fractal nature of cracks analysed in Preston et al. Ž1997., while inappropriate to our present work, is easily dealt with. No natural features are truly fractal in the mathematical sense: all such features have upper and lower bounds where the use of fractals becomes inappropriate. However, within the context of the approximations outlined in some detail by Preston et al. Ž1997. , and developed by Preston et al. Ž1999., the use of a power-law descriptor is sound. See Crawford and Young Ž 1998. , for a more detailed explanation. Indeed the work of Velde Ž 1999. does not show truly fractal cracking patterns. Comparison with other models Many other attempts have been made to model cracking processes. Stoyan gives a useful summary of the geometric approaches, and places ours in context. Our model is indeed edge-based rather than cell based, and we maintain that this is truer to physical reality. Moran and Kirby draw attention to other work which we had not reviewed. We regret this omission. The work of Moran and McBratney Ž1997. appears closest in construction to ours, and so shares some of its attractions and limitations. It aims to simulate soil pore structure in general, whereas we have concentrated on macroscopic cracks produced by drying. The model of Perrier et al. Ž1995. differs from ours in being cell-based rather than edge-based, and in being fractal, with cracks occurring at many size scales. Our model is not fractal, as the cracks we observed did not appear to be, but this is certainly a feature sometimes occurring in real soil. The model could be made
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fractal by allowing the cracking to continue to much smaller cell sizes, with crack thickness decreasing also. Other remarks We finish by saying that we hope our ideas can be seen as the start of a modelling process, rather than the end-product. The geometric ideas need to be extended in two directions: towards the more fundamental physical processes which ultimately are driving it, and towards the greater complexity which must result when the heterogeneity, at many scales, of real soil is accounted for. Only by doing so can the usefulness of the model be determined. The point we emphasise in our paper, and which is echoed by Moran and Kirby, is that the ultimate aim of our research on the genesis of cracking is to provide functional significance to the observed physical structures in soil systems.
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