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The “Loughborough Loess” Monte Carlo model of soil structure
PII:
Computers & Geosciences Vol. 24, No. 4, pp. 345±352, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0098-3004(97)00118-0 0098-3004/98 $19.00 + 0.00
THE ``LOUGHBOROUGH LOESS'' MONTE CARLO MODEL OF SOIL STRUCTURE S. C. DIBBEN, I. F. JEFFERSON, and I. J. SMALLEY Collapsing Soils Research Group, Department of Civil and Structural Engineering, Nottingham Trent University, Nottingham NG1 4BU, U.K. (e-mail: [email protected]) (Received 20 November 1996; revised 26 July 1997) AbstractÐA major problem encountered with loess deposits is that of structural collapse when loaded and wetted, the process known as hydrocollapse. To elucidate this problem it is bene®cial to model the soil at a microstructural level. The ``Loughborough Loess'' has been developed as a Monte Carlo computer-based simulation to model soil structure. The simulation creates a random packing, representative of the initial aeolian deposition of loess. From this the in¯uence of particle shape and size on the structure and void ratio are analysed. Secondly, an archetypal soil has been manufactured from crushed sand and clay mineral. This serves to validate the computer simulation and to provide a benchmark for the comparison of worldwide loess deposits. This has been achieved by the manufacture of loess from crushed sand and clay mineral using a recently developed airfall method. Both the arti®cial and natural loess samples have shown that the computer simulation closely reproduces characteristic initial loess structures with typical void ratios. Further work is currently being undertaken to improve the computer model to take account of three dimensions. The manufactured samples are tested alongside samples of actual loess using double oedometer tests. # 1998 Elsevier Science Ltd. All rights reserved Code available from http://www.iamg.org/CGEditor/index.htm Key Words: Loess, Collapse, Structure, Particulate, Monte Carlo.
INTRODUCTION
of this process is still incomplete in certain key areas. To understand the collapse mechanisms it is necessary to model the soil on a microstructural scale. This is made dicult by the scales of analysis involved and the complexities of the system. Loess is a relatively simple soil in which the structure is dominated by 20±30 mm quartz particles of an 8:5:2 aspect ratio, tending to be blade shaped (Rogers and Smalley, 1993; Assallay, Rogers and Smalley, 1996a). To study the loess collapse phenomenon an ideal, archetypal collapsing soil has been developed called ``the Loughborough Loess''. It can be manufactured from crushed sand and clay mineral to form a material which exhibits ideal collapse properties, but it can also be modelled theoretically using Monte Carlo simulation. The aims of this paper are to discuss the computer model developed to aid understanding of the collapse mechanisms in loess. The model is evaluated and considered alongside experimental
Loess is a partially saturated, aeolian, collapsible soil. Upon deposition it forms a metastable structure consisting of an open skeleton of quartz particles, bonded by various clay and carbonate bridges and skins. In its dry state this structure has signi®cant strength and can withstand large loads. However, on saturation the bonding disintegrates and a denser structure is achieved, a mechanism known as hydrocollapse. Loess occurs as extensive sedimentary drapes in many regions of the world, including the Midwest of the U.S.A., central Europe, the Ukraine and much of China. Many of the loess regions are densely populated, a fact partly rooted in the favourable agricultural properties of the material. The loess collapse phenomenon has caused many subsidence problems, two of the best documented being the 1976 Teton Dam failure in Idaho, U.S.A. and the 1920 Kansu Province earth¯ow in China. The grave consequences of such events have led to much research into the collapse mechanisms. Unfortunately current understanding 345
346
S. C. Dibben, I. F. Jeerson and I. J. Smalley
evidence of the behaviour of loess. Future developments of the model are then considered.
ORIGIN OF THE MONTE CARLO APPROACH
The Monte Carlo method was invented by Ulam in the mid 1940s, at a time when computers were becoming available to tackle complex problems using mathematical processes (Ulam, 1991). However, the basic method can be used in simple applications such as the collapsing soil problem and such an approach has been promoted by Smalley since the early 1960s. The ®rst application was to the problem of the closest random packing of spheres in one dimension (Smalley, 1962) and these original experiments have recently been repeated with the 0.78 packing density con®rmed (Assallay, Rogers and Smalley, 1996a). The method allowed model drumlin ®elds to be constructed (Smalley and Unwin, 1968) and produced some initial soil structures for post-glacial marine clays (Smalley, 1978). However, the most successful application was in the determination of the modal shape of loess particles (Rogers and Smalley, 1993). Assallay, Rogers and Smalley (1996a) developed a simple structural model from randomly placed particles. Figure 1 illustrates their initial ideal structure, showing the eect that pure randomness can have. The two-dimensional structure is composed of a series of one-dimensional packings, developed sequentially and placed on top of each other to ®ll a 100 100 unit reference frame. Assallay, Rogers and Smalley (1996a) then went on to develop a
more realistic model which allowed rectangular particles to fall freely according to a random number allocation until contacting the existing structure. A major problem with much of the soil structure work to date that uses the Monte Carlo approach is that it has been conducted by hand using random number tables (Assallay, Rogers and Smalley, 1996a). The work is time consuming and thus the structures obtained are small and few. This raises questions about the repeatability of the results and the behaviour of the structure at a larger scale. The application of computing to the model allows the scale of the model to be increased signi®cantly and the time taken to produce a structure becomes negligible.
PARTICLE PACKING
Alongside the use of Monte Carlo simulation, much work has been conducted on the subject of particle packing particularly in relation to powder technology, but, as Smalley (1970) noted, it is equally applicable to silty soils. Smalley studied the work of Graton and Fraser on simple, regular packings (Graton and Fraser, 1935). Starting from a simple cubic packing, certain movements of the unit cell layers can be described so as to give a total of nine basic packings of that type. Smalley (1970) showed that, if classi®ed in terms of packing density, these regular packings in three dimensions have a range from 0.056 to 0.740. These are empirically established limits and are not proven density bounds. By studying these packing struc-
Figure 1. Ideal structure of two-dimensional particle packing (after Assallay, Rogers and Smalley, 1996a).
Monte Carlo model of soil structure
tures it may be possible to understand the mechanisms of collapse from the most open packing to a denser structure. THE ``LOUGHBOROUGH LOESS'' MODEL
The computer model An original model, written in FORTRAN 90, simulates the initial formation of the loess deposit. At this stage in the formation an unstable structure is developed which in reality will undergo some rapid adjustment to form a stronger metastable structure. The program sets up a two dimensional array of up to 1000 units wide by up to 500 units deep. The unit is an arbitrary integer value, all scaled to ®t the computer screen. Default values of 200 by 100 have been set as these appear to produce a suitable scale for the graphics on a standard PC screen and on A4 size printouts. Rectangular blocks are used to represent the quartz particles and have a dimension of one unit high by any value greater than two units wide. A default width of 3 units has been set as this best represents the aspect ratio of the quartz particles in a simpli®ed two dimensional form. A pseudo-random number generator is used to drop the particles into the array to form an open, random packing. The randomness of the generator has been tested using probability theory and found to be satisfactory. For a particle width of 3 units, a random number (r1) is generated and converted to an integer of value between 0 and the array width. A block is
347
placed at the location, x1 coinciding with r1 on the ®rst row of the empty array. Blocks are also placed at locations on either side of r1, i.e. at (x1ÿ1) and (x1+1) so that a particle 3 units wide is created. A second random number (r2) is generated and placed at location x2. If any of the locations x2, (x2ÿ1) or (x2+1) coincide with the locations x1, (x1ÿ1) or (x1+1) then the particles are overlapping and the blocks will be placed at the three locations in row 2 of the array. In this way the random, open structure is built up. Figure 2 is an example of the structure created using a particle width of 3 units. The method of particle placement used by the program means that the array formed is only pseudo-in®nite. For example, if an array of 10 10 is created and a particle of width 3 units is located at each position in turn then a de®ciency of particles occurs in columns 1 and 10 due to the nonexistence of columns 0 and 11 (Fig. 3). The decrease in the probability of these outermost columns receiving particles will increase the porosity of the structure. Thus, side boundaries are introduced to eliminate these columns from the calculations. After a number of particles equal to the width of the array divided by 10 have been located in the top row of the array, the program will stop generating random numbers. This is necessary to prevent the upper part of the array from ®lling up inde®nitely. However, errors may result from columns which are un®lled. To allow for this a boundary is located at 80% of the height of the array. This value has been obtained from studies of numerous runs of the program. Finally, the void
Figure 2. Initial structure of loess using rectangular particles.
348
S. C. Dibben, I. F. Jeerson and I. J. Smalley
Figure 3. De®ciency of particles in columns 1 and 10 for 10 10 array and particle width of 3 units.
ratio is calculated for the area within these boundaries and the two dimensional structure is displayed on screen. The ®rst alteration to the initial model has been to change the particle shape from a rectangle to an ellipse. This involved converting the vertical axis of the array into real numbers. Only ellipses landing directly on top of each other can gain the same height in the array as is obtained using rectangles. By using the geometry of an ellipse it has been possible to calculate the points of intersection of ellipses contacting from adjacent columns. This
procedure gives three times the number of vertical locations for the particles compared to the number available when using rectangles. The use of elliptical particles has two main eects: ®rst, it changes the appearance of the structure (Fig. 4) and, second, it changes the void ratio (Table 1). Experimental work A computer simulation is only of use if it matches the behaviour of the natural material being modelled. Therefore, an extensive pro-
Figure 4. Initial structure of loess using elliptical particles.
Monte Carlo model of soil structure
349
Table 1. Void ratios for dierent particle widths and shapes Rectangular particles
Elliptical particles
Particle width
average void ratio
deviation from mean
average void ratio
deviation from mean
2 3 4 5 6 7 9 10
0.9507 1.1895 1.2762 1.3443 1.3993 1.4096 1.4593 1.4740
20.0379 20.0470 20.0760 20.0788 20.0610 20.1307 20.1182 20.1094
0.8993 1.1361 1.2038 1.2884 1.3163 0.3456 1.3721 1.3892
20.0382 20.0455 20.0693 20.0751 20.0790 20.0873 20.1174 20.1183
gramme of experimental testing is being conducted in order to test the model. This will be used in connection with the ®ndings of other researchers who have undertaken similar experimental work on loess. Oedometer tests are being conducted on both undisturbed and manufactured samples of loess from several sources. Manufactured samples of ``Loughborough Loess'' are being created from crushed sand and clay mineral in order to obtain an ideal collapsible material allowing both repeatable and reproducible results. This material may be used as a benchmark against which loess deposits from around the world can be compared. Collapse is typically measured in an oedometer. Samples are either carefully cut blocks of undisturbed material or are manufactured samples. The latter are prepared by direct airfall sedimentation into an oedometer specimen ring in the laboratory. Using the ``Loughborough Loess'', a whole range of variables may be tested, including the quantity and type of clay mineral present, the particle shape
and size, the void ratio and the moisture content. This ideal soil may be directly compared with natural, undisturbed loess and the computer generated simulation. Using the airfall sedimentation method some collapse occurs almost instantaneously. This relates to the collapse of the largest pores shown in Figure 2, indicating the formation of a metastable structure. Applying a degree of moisture to the sample and then oven drying allows clay and carbonate bondings to form, a characteristic feature of the metastable structure. Once formed, the samples are tested in the oedometer rig. The samples are stressed axially and the subsequent deformation is monitored. Use of a double oedometer test demonstrates the eect of saturation on the sample. This test involves simultaneous loading of two identical samples. One sample is ¯ooded at a critical point in the test, and thus undergoes hydrocollapse. The other sample is loaded in its dry state. Figure 5 presents an example of the collapse curves for dry and saturated samples.
Figure 5. Double-oedometer test results for loess samples from Pegwell Bay, Kent.
350
S. C. Dibben, I. F. Jeerson and I. J. Smalley
Figure 6. Voids ratios of loess structures. RESULTS
Computer model A simple plot of the fractional density and voids ratio provides a curve upon which a series of states may be plotted (Fig. 6). The initial unstable structure has a void ratio greater than 1 (Fig. 6, point a). This then adjusts to form the metastable structure, with a void ratio less than 1 (Fig. 6, point b), due to the disappearance of some of the largest pores. Hydrocollapse causes a signi®cant densi®cation of the soil structure (Fig. 6, point c).
One important function of the program is that it calculates the void ratio of each structure generated. By varying the particle width and shape, changes in the void ratio are generated. Table 1 shows the average void ratio for each particle width and the deviation from this average for ten runs of the program. Typically, loess grains have an aspect ratio of 8:5:2 and so the larger particle widths are clearly not applicable. However, use of this method makes it possible to demonstrate for the ®rst time the eect of particle shape on the void ratio for a model soil. It can be seen that the void ratio for a rectangular particle of 3 units width is 1.19, which
Figure 7. Average void ratios and deviations from this average.
Monte Carlo model of soil structure
is a realistic value for loess (Rogers and Smalley, 1993). The void ratio increases with increasing particle width, which is not surprising because of the increased likelihood of giant pores developing. Figure 7 shows how this increase levels o for particle widths greater than 6. Ten runs of the program show that deviation from the average particle width increases with increasing particle width. This is not surprising since high void ratios are dependent on the presence of one or two giant pores. It is also true that as the particle width increases, so the array size decreases in relative terms, with the results becoming less representative of an in®nite structure. As would be expected, the void ratio for elliptical particles is lower, because of the closer packing obtained. It is anticipated that the void ratio will be similar if the third dimension is included, and the development of a three dimensional version of the model will clarify this. One major use of the model is its ability to display the structure of the soil. Figures 2 and 4 show the characteristic ``staircase'' structures and the existence of long vertical voids. It has been suggested that these voids may have been created by vegetation roots (Muxart and others, 1995). It is now apparent that, if they can be generated without the input of any vegetation, this may not be true for all cases. Suzuki and Matsukura (1992) have used mercury intrusion porosimetry to look at the pore-size distributions (PSD) of samples from the Loess Plateau, China. However, their samples are in a metastable state, compared to the more open, initial structure of the computer model. From these structures, Suzuski and Matsukura found a peak at around 10 mm, with pores up to 40 mm present. Many researchers (including Zhang and Wang, 1995; Klukanova and Frankovska, 1995) have looked at loess using a scanning electron microscope (SEM) and have identi®ed macropores of 150±200 mm. Experimental The need for the computer simulation to be validated by experimental work cannot be overstressed. To this end, an extensive experimental test program has recently been initiated. Airfall sedimentation samples of loess from Pegwell Bay, Kent, U.K. have been examined. Initial voids ratios range from 1.15 to 1.29 with metastable void ratios in the region of 1.05 to 1.10. Upon saturation and hydrocollapse, the void ratio falls to between 0.35 and 0.67. For the same loess Fookes and Best (1969) found ®eld void ratios of between 0.63 and 0.77, with these ®gures falling to 0.36 to 0.45 upon the collapse of the undisturbed samples. It should be noted that Fookes and Best excavated their samples from a depth of 1 m, so that overburden pressures must be taken into account when considering the void ratios obtained. This accounts for the slightly
351
lower value of void ratio obtained by Fookes and Best. Loess void ratios have been quoted as 0.89 to 0.81 for Chinese loess (Derbyshire and Mellors, 1988) and 0.87 to 0.96 for Libyan loess (Assallay, Rogers and Smalley, 1996b), both in their metastable states. Assallay (1995) also obtained void ratios of 1.85 to 1.95 for the unstable structure and 1.32 to 1.41 for the metastable structure of samples manufactured from Malan loess from China. The void ratio for undisturbed samples of Malan loess was between 1.02 and 1.06. Recently, Assallay and others (1996b) obtained initial void ratios of 1.45 to 1.50 from manufactured loess formed of crushed sand, with void ratios of 1.05 to 1.10 for the corresponding metastable structure.
CONCLUSION
The problem of loess collapse is one of great importance which requires collapse mechanisms to be studied at a microstructural level. Despite signi®cant research into particle packing systems and the structure of loess, the exact mechanisms of collapse are still far from understood. The ``Loughborough loess'' has been developed as an archetypal collapsing soil which may be used as a tool in the understanding of collapse. Using the simple Monte Carlo method, a computer simulation has been developed. This models the formation of the initial, unstable loess structure. The computer program allows the particle shape and size to be varied and calculates the void ratio accordingly. The importance of validating a computer simulation with the true soil is identi®ed. An extensive program of experimental work is currently comparing the model to undisturbed samples of natural loess. Initial results indicate that void ratios characteristic of natural loess have been achieved. At present, the model described in this paper is idealized, but it is planned to develop it into three dimensions. Subsequently, it is intended to model the formation of a metastable structure and hydrocollapse.
REFERENCES Assallay, A. M. (1995) Collapse behaviour of loess deposits. MPhil/Ph.D. Internal Report, Department of Civil and Building Engineering, Loughborough University of Technology. Assallay, A. M., Rogers, C. D. F. and Smalley, I. J. (1996a) Formation and collapse of metastable particle packings and open structures in loess deposits. Engineering Geology. In press. Assallay, A. M., Rogers, C. D. F. and Smalley, I. J. (1996b) Engineering properties of loess in Libya. Journal of Arid Environments 32, 373±386. Derbyshire, E. and Mellors, T. W. (1988) Geological and geotechnical characteristics of some engineering loess and loessic soils from China and Britain: a comparison. Engineering Geology 25, 135±175.
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Fookes, P. G. and Best, R. (1969) Consolidation characteristics of some late Pleistocene periglacial metastable soils of East Kent. Engineering Geology 2, 103±128. Graton, L. C. and Fraser, H. J. (1935) Particle packing themes. Journal of Geology 43, 785±909. Klukanova, A. and Frankovska, J. (1995) The Slovak Carpathians loess sediments, their fabric and properties. In Proceedings of the NATO Advanced Workshop on Genesis and Properties of Collapsible Soils, eds. E. Derbyshire, T. Dijkstra and I. J. Smalley, pp. 129±247. Muxart, T., Billard, A., Andrieu, A., Derbyshire, E. and Meng, X. (1995) Changes in water chemistry and loess porosity with leaching: implications for collapsibility in the loess of N. China. In Proceedings of the NATO Advanced Workshop on Genesis and Properties of Collapsible Soils, eds. E. Derbyshire, T. Dijkstra and I. J. Smalley, pp. 313±332. Rogers, C. D. F. and Smalley, I. J. (1993) The shape of loess particles. Naturwissenschaften 80, 461±462. Smalley, I. J. (1962) Packing of equal spheres. Nature 194, 1271.
Smalley, I. J. (1970) Variations on the particle packing theme of Graton and Fraser. Powder Technology 4, 97±101. Smalley, I. J. (1978) Mineralogy, interparticle forces and soil structure in the Leda/Champlain clays of eastern Canada. In Modi®cation of Soil Structure, eds. E. W. Emerson, R. D. Bond and A. R. Dexter, pp. 59±67. John Wiley & Sons, Chichester. Smalley, I. J. and Unwin, D. J. (1968) The formation and shape of drumlins and their distribution and orientation in drumlin ®elds. Journal of Glaciology 7, 377± 390. Suzuki, T. and Matsukura, Y. (1992) Pore-size distribution of loess from the Loess Plateau, China. Transactions Japanese Geomorphological Union 13(3), 169±183. Ulam, S. M. (1991) Adventures of a Mathematician. Charles Scribner's Sons, New York, 198 pp. Zhang, Z. and Wang, L. (1995) Geological disasters in loess areas during the 1920 Haiyuan earthquake, China. Geojournal 36(2 (Part 3)), 269±274.
Computers & Geosciences Vol. 24, No. 4, pp. 345±352, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0098-3004(97)00118-0 0098-3004/98 $19.00 + 0.00
THE ``LOUGHBOROUGH LOESS'' MONTE CARLO MODEL OF SOIL STRUCTURE S. C. DIBBEN, I. F. JEFFERSON, and I. J. SMALLEY Collapsing Soils Research Group, Department of Civil and Structural Engineering, Nottingham Trent University, Nottingham NG1 4BU, U.K. (e-mail: [email protected]) (Received 20 November 1996; revised 26 July 1997) AbstractÐA major problem encountered with loess deposits is that of structural collapse when loaded and wetted, the process known as hydrocollapse. To elucidate this problem it is bene®cial to model the soil at a microstructural level. The ``Loughborough Loess'' has been developed as a Monte Carlo computer-based simulation to model soil structure. The simulation creates a random packing, representative of the initial aeolian deposition of loess. From this the in¯uence of particle shape and size on the structure and void ratio are analysed. Secondly, an archetypal soil has been manufactured from crushed sand and clay mineral. This serves to validate the computer simulation and to provide a benchmark for the comparison of worldwide loess deposits. This has been achieved by the manufacture of loess from crushed sand and clay mineral using a recently developed airfall method. Both the arti®cial and natural loess samples have shown that the computer simulation closely reproduces characteristic initial loess structures with typical void ratios. Further work is currently being undertaken to improve the computer model to take account of three dimensions. The manufactured samples are tested alongside samples of actual loess using double oedometer tests. # 1998 Elsevier Science Ltd. All rights reserved Code available from http://www.iamg.org/CGEditor/index.htm Key Words: Loess, Collapse, Structure, Particulate, Monte Carlo.
INTRODUCTION
of this process is still incomplete in certain key areas. To understand the collapse mechanisms it is necessary to model the soil on a microstructural scale. This is made dicult by the scales of analysis involved and the complexities of the system. Loess is a relatively simple soil in which the structure is dominated by 20±30 mm quartz particles of an 8:5:2 aspect ratio, tending to be blade shaped (Rogers and Smalley, 1993; Assallay, Rogers and Smalley, 1996a). To study the loess collapse phenomenon an ideal, archetypal collapsing soil has been developed called ``the Loughborough Loess''. It can be manufactured from crushed sand and clay mineral to form a material which exhibits ideal collapse properties, but it can also be modelled theoretically using Monte Carlo simulation. The aims of this paper are to discuss the computer model developed to aid understanding of the collapse mechanisms in loess. The model is evaluated and considered alongside experimental
Loess is a partially saturated, aeolian, collapsible soil. Upon deposition it forms a metastable structure consisting of an open skeleton of quartz particles, bonded by various clay and carbonate bridges and skins. In its dry state this structure has signi®cant strength and can withstand large loads. However, on saturation the bonding disintegrates and a denser structure is achieved, a mechanism known as hydrocollapse. Loess occurs as extensive sedimentary drapes in many regions of the world, including the Midwest of the U.S.A., central Europe, the Ukraine and much of China. Many of the loess regions are densely populated, a fact partly rooted in the favourable agricultural properties of the material. The loess collapse phenomenon has caused many subsidence problems, two of the best documented being the 1976 Teton Dam failure in Idaho, U.S.A. and the 1920 Kansu Province earth¯ow in China. The grave consequences of such events have led to much research into the collapse mechanisms. Unfortunately current understanding 345
346
S. C. Dibben, I. F. Jeerson and I. J. Smalley
evidence of the behaviour of loess. Future developments of the model are then considered.
ORIGIN OF THE MONTE CARLO APPROACH
The Monte Carlo method was invented by Ulam in the mid 1940s, at a time when computers were becoming available to tackle complex problems using mathematical processes (Ulam, 1991). However, the basic method can be used in simple applications such as the collapsing soil problem and such an approach has been promoted by Smalley since the early 1960s. The ®rst application was to the problem of the closest random packing of spheres in one dimension (Smalley, 1962) and these original experiments have recently been repeated with the 0.78 packing density con®rmed (Assallay, Rogers and Smalley, 1996a). The method allowed model drumlin ®elds to be constructed (Smalley and Unwin, 1968) and produced some initial soil structures for post-glacial marine clays (Smalley, 1978). However, the most successful application was in the determination of the modal shape of loess particles (Rogers and Smalley, 1993). Assallay, Rogers and Smalley (1996a) developed a simple structural model from randomly placed particles. Figure 1 illustrates their initial ideal structure, showing the eect that pure randomness can have. The two-dimensional structure is composed of a series of one-dimensional packings, developed sequentially and placed on top of each other to ®ll a 100 100 unit reference frame. Assallay, Rogers and Smalley (1996a) then went on to develop a
more realistic model which allowed rectangular particles to fall freely according to a random number allocation until contacting the existing structure. A major problem with much of the soil structure work to date that uses the Monte Carlo approach is that it has been conducted by hand using random number tables (Assallay, Rogers and Smalley, 1996a). The work is time consuming and thus the structures obtained are small and few. This raises questions about the repeatability of the results and the behaviour of the structure at a larger scale. The application of computing to the model allows the scale of the model to be increased signi®cantly and the time taken to produce a structure becomes negligible.
PARTICLE PACKING
Alongside the use of Monte Carlo simulation, much work has been conducted on the subject of particle packing particularly in relation to powder technology, but, as Smalley (1970) noted, it is equally applicable to silty soils. Smalley studied the work of Graton and Fraser on simple, regular packings (Graton and Fraser, 1935). Starting from a simple cubic packing, certain movements of the unit cell layers can be described so as to give a total of nine basic packings of that type. Smalley (1970) showed that, if classi®ed in terms of packing density, these regular packings in three dimensions have a range from 0.056 to 0.740. These are empirically established limits and are not proven density bounds. By studying these packing struc-
Figure 1. Ideal structure of two-dimensional particle packing (after Assallay, Rogers and Smalley, 1996a).
Monte Carlo model of soil structure
tures it may be possible to understand the mechanisms of collapse from the most open packing to a denser structure. THE ``LOUGHBOROUGH LOESS'' MODEL
The computer model An original model, written in FORTRAN 90, simulates the initial formation of the loess deposit. At this stage in the formation an unstable structure is developed which in reality will undergo some rapid adjustment to form a stronger metastable structure. The program sets up a two dimensional array of up to 1000 units wide by up to 500 units deep. The unit is an arbitrary integer value, all scaled to ®t the computer screen. Default values of 200 by 100 have been set as these appear to produce a suitable scale for the graphics on a standard PC screen and on A4 size printouts. Rectangular blocks are used to represent the quartz particles and have a dimension of one unit high by any value greater than two units wide. A default width of 3 units has been set as this best represents the aspect ratio of the quartz particles in a simpli®ed two dimensional form. A pseudo-random number generator is used to drop the particles into the array to form an open, random packing. The randomness of the generator has been tested using probability theory and found to be satisfactory. For a particle width of 3 units, a random number (r1) is generated and converted to an integer of value between 0 and the array width. A block is
347
placed at the location, x1 coinciding with r1 on the ®rst row of the empty array. Blocks are also placed at locations on either side of r1, i.e. at (x1ÿ1) and (x1+1) so that a particle 3 units wide is created. A second random number (r2) is generated and placed at location x2. If any of the locations x2, (x2ÿ1) or (x2+1) coincide with the locations x1, (x1ÿ1) or (x1+1) then the particles are overlapping and the blocks will be placed at the three locations in row 2 of the array. In this way the random, open structure is built up. Figure 2 is an example of the structure created using a particle width of 3 units. The method of particle placement used by the program means that the array formed is only pseudo-in®nite. For example, if an array of 10 10 is created and a particle of width 3 units is located at each position in turn then a de®ciency of particles occurs in columns 1 and 10 due to the nonexistence of columns 0 and 11 (Fig. 3). The decrease in the probability of these outermost columns receiving particles will increase the porosity of the structure. Thus, side boundaries are introduced to eliminate these columns from the calculations. After a number of particles equal to the width of the array divided by 10 have been located in the top row of the array, the program will stop generating random numbers. This is necessary to prevent the upper part of the array from ®lling up inde®nitely. However, errors may result from columns which are un®lled. To allow for this a boundary is located at 80% of the height of the array. This value has been obtained from studies of numerous runs of the program. Finally, the void
Figure 2. Initial structure of loess using rectangular particles.
348
S. C. Dibben, I. F. Jeerson and I. J. Smalley
Figure 3. De®ciency of particles in columns 1 and 10 for 10 10 array and particle width of 3 units.
ratio is calculated for the area within these boundaries and the two dimensional structure is displayed on screen. The ®rst alteration to the initial model has been to change the particle shape from a rectangle to an ellipse. This involved converting the vertical axis of the array into real numbers. Only ellipses landing directly on top of each other can gain the same height in the array as is obtained using rectangles. By using the geometry of an ellipse it has been possible to calculate the points of intersection of ellipses contacting from adjacent columns. This
procedure gives three times the number of vertical locations for the particles compared to the number available when using rectangles. The use of elliptical particles has two main eects: ®rst, it changes the appearance of the structure (Fig. 4) and, second, it changes the void ratio (Table 1). Experimental work A computer simulation is only of use if it matches the behaviour of the natural material being modelled. Therefore, an extensive pro-
Figure 4. Initial structure of loess using elliptical particles.
Monte Carlo model of soil structure
349
Table 1. Void ratios for dierent particle widths and shapes Rectangular particles
Elliptical particles
Particle width
average void ratio
deviation from mean
average void ratio
deviation from mean
2 3 4 5 6 7 9 10
0.9507 1.1895 1.2762 1.3443 1.3993 1.4096 1.4593 1.4740
20.0379 20.0470 20.0760 20.0788 20.0610 20.1307 20.1182 20.1094
0.8993 1.1361 1.2038 1.2884 1.3163 0.3456 1.3721 1.3892
20.0382 20.0455 20.0693 20.0751 20.0790 20.0873 20.1174 20.1183
gramme of experimental testing is being conducted in order to test the model. This will be used in connection with the ®ndings of other researchers who have undertaken similar experimental work on loess. Oedometer tests are being conducted on both undisturbed and manufactured samples of loess from several sources. Manufactured samples of ``Loughborough Loess'' are being created from crushed sand and clay mineral in order to obtain an ideal collapsible material allowing both repeatable and reproducible results. This material may be used as a benchmark against which loess deposits from around the world can be compared. Collapse is typically measured in an oedometer. Samples are either carefully cut blocks of undisturbed material or are manufactured samples. The latter are prepared by direct airfall sedimentation into an oedometer specimen ring in the laboratory. Using the ``Loughborough Loess'', a whole range of variables may be tested, including the quantity and type of clay mineral present, the particle shape
and size, the void ratio and the moisture content. This ideal soil may be directly compared with natural, undisturbed loess and the computer generated simulation. Using the airfall sedimentation method some collapse occurs almost instantaneously. This relates to the collapse of the largest pores shown in Figure 2, indicating the formation of a metastable structure. Applying a degree of moisture to the sample and then oven drying allows clay and carbonate bondings to form, a characteristic feature of the metastable structure. Once formed, the samples are tested in the oedometer rig. The samples are stressed axially and the subsequent deformation is monitored. Use of a double oedometer test demonstrates the eect of saturation on the sample. This test involves simultaneous loading of two identical samples. One sample is ¯ooded at a critical point in the test, and thus undergoes hydrocollapse. The other sample is loaded in its dry state. Figure 5 presents an example of the collapse curves for dry and saturated samples.
Figure 5. Double-oedometer test results for loess samples from Pegwell Bay, Kent.
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Figure 6. Voids ratios of loess structures. RESULTS
Computer model A simple plot of the fractional density and voids ratio provides a curve upon which a series of states may be plotted (Fig. 6). The initial unstable structure has a void ratio greater than 1 (Fig. 6, point a). This then adjusts to form the metastable structure, with a void ratio less than 1 (Fig. 6, point b), due to the disappearance of some of the largest pores. Hydrocollapse causes a signi®cant densi®cation of the soil structure (Fig. 6, point c).
One important function of the program is that it calculates the void ratio of each structure generated. By varying the particle width and shape, changes in the void ratio are generated. Table 1 shows the average void ratio for each particle width and the deviation from this average for ten runs of the program. Typically, loess grains have an aspect ratio of 8:5:2 and so the larger particle widths are clearly not applicable. However, use of this method makes it possible to demonstrate for the ®rst time the eect of particle shape on the void ratio for a model soil. It can be seen that the void ratio for a rectangular particle of 3 units width is 1.19, which
Figure 7. Average void ratios and deviations from this average.
Monte Carlo model of soil structure
is a realistic value for loess (Rogers and Smalley, 1993). The void ratio increases with increasing particle width, which is not surprising because of the increased likelihood of giant pores developing. Figure 7 shows how this increase levels o for particle widths greater than 6. Ten runs of the program show that deviation from the average particle width increases with increasing particle width. This is not surprising since high void ratios are dependent on the presence of one or two giant pores. It is also true that as the particle width increases, so the array size decreases in relative terms, with the results becoming less representative of an in®nite structure. As would be expected, the void ratio for elliptical particles is lower, because of the closer packing obtained. It is anticipated that the void ratio will be similar if the third dimension is included, and the development of a three dimensional version of the model will clarify this. One major use of the model is its ability to display the structure of the soil. Figures 2 and 4 show the characteristic ``staircase'' structures and the existence of long vertical voids. It has been suggested that these voids may have been created by vegetation roots (Muxart and others, 1995). It is now apparent that, if they can be generated without the input of any vegetation, this may not be true for all cases. Suzuki and Matsukura (1992) have used mercury intrusion porosimetry to look at the pore-size distributions (PSD) of samples from the Loess Plateau, China. However, their samples are in a metastable state, compared to the more open, initial structure of the computer model. From these structures, Suzuski and Matsukura found a peak at around 10 mm, with pores up to 40 mm present. Many researchers (including Zhang and Wang, 1995; Klukanova and Frankovska, 1995) have looked at loess using a scanning electron microscope (SEM) and have identi®ed macropores of 150±200 mm. Experimental The need for the computer simulation to be validated by experimental work cannot be overstressed. To this end, an extensive experimental test program has recently been initiated. Airfall sedimentation samples of loess from Pegwell Bay, Kent, U.K. have been examined. Initial voids ratios range from 1.15 to 1.29 with metastable void ratios in the region of 1.05 to 1.10. Upon saturation and hydrocollapse, the void ratio falls to between 0.35 and 0.67. For the same loess Fookes and Best (1969) found ®eld void ratios of between 0.63 and 0.77, with these ®gures falling to 0.36 to 0.45 upon the collapse of the undisturbed samples. It should be noted that Fookes and Best excavated their samples from a depth of 1 m, so that overburden pressures must be taken into account when considering the void ratios obtained. This accounts for the slightly
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lower value of void ratio obtained by Fookes and Best. Loess void ratios have been quoted as 0.89 to 0.81 for Chinese loess (Derbyshire and Mellors, 1988) and 0.87 to 0.96 for Libyan loess (Assallay, Rogers and Smalley, 1996b), both in their metastable states. Assallay (1995) also obtained void ratios of 1.85 to 1.95 for the unstable structure and 1.32 to 1.41 for the metastable structure of samples manufactured from Malan loess from China. The void ratio for undisturbed samples of Malan loess was between 1.02 and 1.06. Recently, Assallay and others (1996b) obtained initial void ratios of 1.45 to 1.50 from manufactured loess formed of crushed sand, with void ratios of 1.05 to 1.10 for the corresponding metastable structure.
CONCLUSION
The problem of loess collapse is one of great importance which requires collapse mechanisms to be studied at a microstructural level. Despite signi®cant research into particle packing systems and the structure of loess, the exact mechanisms of collapse are still far from understood. The ``Loughborough loess'' has been developed as an archetypal collapsing soil which may be used as a tool in the understanding of collapse. Using the simple Monte Carlo method, a computer simulation has been developed. This models the formation of the initial, unstable loess structure. The computer program allows the particle shape and size to be varied and calculates the void ratio accordingly. The importance of validating a computer simulation with the true soil is identi®ed. An extensive program of experimental work is currently comparing the model to undisturbed samples of natural loess. Initial results indicate that void ratios characteristic of natural loess have been achieved. At present, the model described in this paper is idealized, but it is planned to develop it into three dimensions. Subsequently, it is intended to model the formation of a metastable structure and hydrocollapse.
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Fookes, P. G. and Best, R. (1969) Consolidation characteristics of some late Pleistocene periglacial metastable soils of East Kent. Engineering Geology 2, 103±128. Graton, L. C. and Fraser, H. J. (1935) Particle packing themes. Journal of Geology 43, 785±909. Klukanova, A. and Frankovska, J. (1995) The Slovak Carpathians loess sediments, their fabric and properties. In Proceedings of the NATO Advanced Workshop on Genesis and Properties of Collapsible Soils, eds. E. Derbyshire, T. Dijkstra and I. J. Smalley, pp. 129±247. Muxart, T., Billard, A., Andrieu, A., Derbyshire, E. and Meng, X. (1995) Changes in water chemistry and loess porosity with leaching: implications for collapsibility in the loess of N. China. In Proceedings of the NATO Advanced Workshop on Genesis and Properties of Collapsible Soils, eds. E. Derbyshire, T. Dijkstra and I. J. Smalley, pp. 313±332. Rogers, C. D. F. and Smalley, I. J. (1993) The shape of loess particles. Naturwissenschaften 80, 461±462. Smalley, I. J. (1962) Packing of equal spheres. Nature 194, 1271.
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